The Generating Functions for the Number of Perfect Matchings in Cartesian products of Connected Graphs with Path Graphs By Shalosh B. Ekhad This is the story of the generating functions, in , z of the sequences enumerating the number of PERFECT matchings (alias domino tilings) in Cartesian products P_n x G, for general n, and all connected graphs G with at most, 7, vertices ------------------------------------------------------------- There are, 1, different connected graphs with , 2, vertices (up to isomorphism) -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 1 f(z) = - ----------- 2 -1 + z + z And in Maple-input format, it is: -1/(-1+z+z^2) The first , 40, terms are: [1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141] ------------------------------------------------------------- There are, 2, different connected graphs with , 3, vertices (up to isomorphism) -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 -1 + z f(z) = - -------------- 2 4 -4 z + 1 + z And in Maple-input format, it is: -(-1+z^2)/(-4*z^2+1+z^4) The first , 40, terms are: [0, 3, 0, 11, 0, 41, 0, 153, 0, 571, 0, 2131, 0, 7953, 0, 29681, 0, 110771, 0, 413403, 0, 1542841, 0, 5757961, 0, 21489003, 0, 80198051, 0, 299303201, 0, 1117014753, 0, 4168755811, 0, 15558008491, 0, 58063278153, 0, 216695104121] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {2, 3}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 -1 + z f(z) = - -------------- 2 4 -5 z + 1 + z And in Maple-input format, it is: -(-1+z^2)/(-5*z^2+1+z^4) The first , 40, terms are: [0, 4, 0, 19, 0, 91, 0, 436, 0, 2089, 0, 10009, 0, 47956, 0, 229771, 0, 1100899, 0, 5274724, 0, 25272721, 0, 121088881, 0, 580171684, 0, 2779769539, 0, 13318676011, 0, 63813610516, 0, 305749376569, 0, 1464933272329, 0, 7018916985076, 0, 33629651653051] ------------------------------------------------------------- There are, 6, different connected graphs with , 4, vertices (up to isomorphism) -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 -1 + z f(z) = - -------------- 2 4 -5 z + 1 + z And in Maple-input format, it is: -(-1+z^2)/(-5*z^2+1+z^4) The first , 40, terms are: [0, 4, 0, 19, 0, 91, 0, 436, 0, 2089, 0, 10009, 0, 47956, 0, 229771, 0, 1100899, 0, 5274724, 0, 25272721, 0, 121088881, 0, 580171684, 0, 2779769539, 0, 13318676011, 0, 63813610516, 0, 305749376569, 0, 1464933272329, 0, 7018916985076, 0, 33629651653051] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {2, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 -1 + z f(z) = - ---------------------- 4 3 2 z - z - 5 z - z + 1 And in Maple-input format, it is: -(-1+z^2)/(z^4-z^3-5*z^2-z+1) The first , 40, terms are: [1, 5, 11, 36, 95, 281, 781, 2245, 6336, 18061, 51205, 145601, 413351, 1174500, 3335651, 9475901, 26915305, 76455961, 217172736, 616891945, 1752296281, 4977472781, 14138673395, 40161441636, 114079985111, 324048393905, 920471087701, 2614631600701, 7426955448000, 21096536145301, 59925473898301, 170220478472105, 483517428660911, 1373448758774436, 3901330906652795, 11081871650713781, 31478457514091281, 89415697915538545, 253988526230055936, 721463601671126161] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 -1 + z f(z) = - ---------------------- 4 3 2 z - z - 6 z - z + 1 And in Maple-input format, it is: -(-1+z^2)/(z^4-z^3-6*z^2-z+1) The first , 40, terms are: [1, 6, 13, 49, 132, 433, 1261, 3942, 11809, 36289, 109824, 335425, 1018849, 3104934, 9443629, 28756657, 87504516, 266383153, 810723277, 2467770054, 7510988353, 22861948801, 69584925696, 211799836801, 644660351425, 1962182349126, 5972359368781, 18178313978161, 55329992188548, 168410053077169, 512595960817837, 1560207957491238, 4748863783286881, 14454297435974977, 43995092132369664, 133909532574015169, 407585519020921249, 1240583509161406950, 3776011063728579949, 11493188105143927729] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {2, 4}, {3, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : -1 + z f(z) = - ------------------- 2 3 1 - 3 z - 3 z + z And in Maple-input format, it is: -(-1+z)/(1-3*z-3*z^2+z^3) The first , 40, terms are: [2, 9, 32, 121, 450, 1681, 6272, 23409, 87362, 326041, 1216800, 4541161, 16947842, 63250209, 236052992, 880961761, 3287794050, 12270214441, 45793063712, 170902040409, 637815097922, 2380358351281, 8883618307200, 33154114877521, 123732841202882, 461777249934009, 1723376158533152, 6431727384198601, 24003533378261250, 89582406128846401, 334326091137124352, 1247721958419651009, 4656561742541479682, 17378525011746267721, 64857538304443591200, 242051628206028097081, 903348974519668797122, 3371344269872647091409, 12582028104970919568512, 46956768150011031182641] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 -1 + z f(z) = - --------------------------- 3 2 4 -2 z - 7 z - 2 z + z + 1 And in Maple-input format, it is: -(-1+z^2)/(-2*z^3-7*z^2-2*z+z^4+1) The first , 40, terms are: [2, 10, 36, 145, 560, 2197, 8568, 33490, 130790, 510949, 1995840, 7796413, 30454814, 118965250, 464711184, 1815292333, 7091038640, 27699580729, 108202305420, 422668460890, 1651061182538, 6449506621417, 25193576136960, 98413152528025, 384429290075066, 1501688293498810, 5866014346442172, 22914292174998121, 89509632072014000, 349649649768400381, 1365829294044452832, 5335311108436738210, 20841216942649433006, 81411620582676538765, 318016552686728132160, 1242261572229054163477, 4852621037487908896598, 18955694565369976663090, 74046242984926695797160, 289245328430189991865669] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : -1 + z f(z) = - ------------------- 2 3 1 - 4 z - 4 z + z And in Maple-input format, it is: -(-1+z)/(1-4*z-4*z^2+z^3) The first , 40, terms are: [3, 16, 75, 361, 1728, 8281, 39675, 190096, 910803, 4363921, 20908800, 100180081, 479991603, 2299777936, 11018898075, 52794712441, 252954664128, 1211978608201, 5806938376875, 27822713276176, 133306628004003, 638710426743841, 3060245505715200, 14662517101832161, 70252340003445603, 336599182915395856, 1612743574573533675, 7727118689952272521, 37022849875187828928, 177387130685986872121, 849912803554746531675, 4072176887087745786256, 19510971631883982399603, 93482681272332166211761, 447902434729776848659200, 2146029492376552077084241, 10282245027152983536762003, 49265195643388365606725776, 236043733189788844496866875, 1130953470305555856877608601] ------------------------------------------------------------- There are, 21, different connected graphs with , 5, vertices (up to isomorphism) -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 -1 + z f(z) = - -------------- 2 4 -6 z + 1 + z And in Maple-input format, it is: -(-1+z^2)/(-6*z^2+1+z^4) The first , 40, terms are: [0, 5, 0, 29, 0, 169, 0, 985, 0, 5741, 0, 33461, 0, 195025, 0, 1136689, 0, 6625109, 0, 38613965, 0, 225058681, 0, 1311738121, 0, 7645370045, 0, 44560482149, 0, 259717522849, 0, 1513744654945, 0, 8822750406821, 0, 51422757785981, 0, 299713796309065, 0, 1746860020068409] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 4 8 6 2 10 -34 z - 11 z + 34 z - 1 + 11 z + z f(z) = - -------------------------------------------------- 10 8 6 4 12 2 -18 z + 89 z - 152 z + 89 z + z - 18 z + 1 And in Maple-input format, it is: -(-34*z^4-11*z^8+34*z^6-1+11*z^2+z^10)/(-18*z^10+89*z^8-152*z^6+89*z^4+z^12-18* z^2+1) The first , 40, terms are: [0, 7, 0, 71, 0, 773, 0, 8581, 0, 95847, 0, 1072839, 0, 12017505, 0, 134651297, 0, 1508860231, 0, 16908413479, 0, 189479536517, 0, 2123360889605, 0, 23795019256967, 0, 266654293034375, 0, 2988210579801025, 0, 33486815887487041, 0, 375263671216306823, 0, 4205321430942608391, 0, 47126140274266744581, 0, 528110190129944930181] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {2, 4}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 6 4 2 z - 7 z + 7 z - 1 f(z) = - ------------------------------ 8 6 4 2 z - 15 z + 32 z - 15 z + 1 And in Maple-input format, it is: -(z^6-7*z^4+7*z^2-1)/(z^8-15*z^6+32*z^4-15*z^2+1) The first , 40, terms are: [0, 8, 0, 95, 0, 1183, 0, 14824, 0, 185921, 0, 2332097, 0, 29253160, 0, 366944287, 0, 4602858719, 0, 57737128904, 0, 724240365697, 0, 9084693297025, 0, 113956161827912, 0, 1429438110270431, 0, 17930520634652959, 0, 224916047725262248, 0, 2821291671062267585, 0, 35389589910135145793, 0, 443918325373278904936, 0, 5568402462493067660191] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : f(z) = 10 8 2 12 6 4 14 94 z - 210 z + 17 z - 17 z + 210 z - 94 z - 1 + z - ------------------------------------------------------------------------- 2 4 6 8 10 12 14 16 -25 z + 205 z - 696 z + 1044 z - 696 z + 205 z - 25 z + z + 1 And in Maple-input format, it is: -(94*z^10-210*z^8+17*z^2-17*z^12+210*z^6-94*z^4-1+z^14)/(-25*z^2+205*z^4-696*z^ 6+1044*z^8-696*z^10+205*z^12-25*z^14+z^16+1) The first , 40, terms are: [0, 8, 0, 89, 0, 1071, 0, 13264, 0, 166239, 0, 2094735, 0, 26463328, 0, 334744623, 0, 4237029785, 0, 53647679768, 0, 679379618545, 0, 8604208653457, 0, 108975330035192, 0, 1380241523321753, 0, 17481830863039407, 0, 221422251196465216, 0, 2804509402019720367, 0, 35521655583318475839, 0, 449914340164057313776, 0, 5698579943625197681583] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : f(z) = 12 10 8 6 4 2 14 -17 z + 99 z - 242 z + 242 z - 99 z + 17 z - 1 + z - ------------------------------------------------------------------------ 16 14 12 10 8 6 4 2 z - 26 z + 217 z - 770 z + 1203 z - 770 z + 217 z - 26 z + 1 And in Maple-input format, it is: -(-17*z^12+99*z^10-242*z^8+242*z^6-99*z^4+17*z^2-1+z^14)/(z^16-26*z^14+217*z^12 -770*z^10+1203*z^8-770*z^6+217*z^4-26*z^2+1) The first , 40, terms are: [0, 9, 0, 116, 0, 1591, 0, 22163, 0, 310155, 0, 4346911, 0, 60954980, 0, 854907185, 0, 11991084425, 0, 168193387801, 0, 2359192846593, 0, 33091730619076, 0, 464168946440111, 0, 6510778110086075, 0, 91325022076889379, 0, 1280992852374594279, 0, 17968161322019256468, 0, 252034836694020152825, 0, 3535228674445477014833, 0, 49587755252978412752721] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 4 6 8 10 12 14 f(z) = - (-1 + 22 z - 184 z + 737 z - 1482 z + 1482 z - 737 z + 184 z 16 18 / 2 4 6 8 10 - 22 z + z ) / (1 - 32 z + 359 z - 1888 z + 5081 z - 7112 z / 12 14 16 18 20 + 5081 z - 1888 z + 359 z - 32 z + z ) And in Maple-input format, it is: -(-1+22*z^2-184*z^4+737*z^6-1482*z^8+1482*z^10-737*z^12+184*z^14-22*z^16+z^18)/ (1-32*z^2+359*z^4-1888*z^6+5081*z^8-7112*z^10+5081*z^12-1888*z^14+359*z^16-32*z ^18+z^20) The first , 40, terms are: [0, 10, 0, 145, 0, 2201, 0, 33658, 0, 515477, 0, 7897561, 0, 121010866, 0, 1854262901, 0, 28413431085, 0, 435389580706, 0, 6671648045989, 0, 102232389065885, 0, 1566549074783858, 0, 24004880231519541, 0, 367836736108241917, 0, 5636514965881852098, 0, 86370658655129449233, 0, 1323493460453613345757, 0, 20280439782633665595754, 0, 310765598967697148467345] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 10 8 6 4 2 z - 14 z + 55 z - 55 z + 14 z - 1 f(z) = - ---------------------------------------------------- 10 8 6 4 12 2 -26 z + 154 z - 268 z + 154 z + z - 26 z + 1 And in Maple-input format, it is: -(z^10-14*z^8+55*z^6-55*z^4+14*z^2-1)/(-26*z^10+154*z^8-268*z^6+154*z^4+z^12-26 *z^2+1) The first , 40, terms are: [0, 12, 0, 213, 0, 3903, 0, 71752, 0, 1319751, 0, 24277231, 0, 446600352, 0, 8215660303, 0, 151135631573, 0, 2780299176772, 0, 51146541729081, 0, 940894736640041, 0, 17308754179042772, 0, 318412847633477173, 0, 5857541253534036623, 0, 107755669416656143472, 0, 1982279558927628297791, 0, 36466130006078774450231, 0, 670833047559097274020472, 0, 12340683741962581448385023] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 4 2 6 -1 - 8 z + 8 z + z f(z) = - ------------------------------ 4 2 6 8 1 + 41 z - 19 z - 19 z + z And in Maple-input format, it is: -(-1-8*z^4+8*z^2+z^6)/(1+41*z^4-19*z^2-19*z^6+z^8) The first , 40, terms are: [0, 11, 0, 176, 0, 2911, 0, 48301, 0, 801701, 0, 13307111, 0, 220880176, 0, 3666315811, 0, 60855946601, 0, 1010127453401, 0, 16766766924211, 0, 278305942640176, 0, 4619507031938711, 0, 76677648402694901, 0, 1272746577484955101, 0, 21125893715367851311, 0, 350661626727725280176, 0, 5820514772820509871611, 0, 96612773221767455293201, 0, 1603643030542052158306801] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 16 14 12 10 8 6 4 f(z) = - (-25 z + 228 z - 956 z + 1930 z - 1930 z + 956 z - 228 z 2 18 / 16 12 20 8 10 + 25 z - 1 + z ) / (477 z + 7358 z + z + 7358 z - 10300 z / 14 18 4 2 6 - 2676 z - 38 z + 477 z - 38 z - 2676 z + 1) And in Maple-input format, it is: -(-25*z^16+228*z^14-956*z^12+1930*z^10-1930*z^8+956*z^6-228*z^4+25*z^2-1+z^18)/ (477*z^16+7358*z^12+z^20+7358*z^8-10300*z^10-2676*z^14-38*z^18+477*z^4-38*z^2-\ 2676*z^6+1) The first , 40, terms are: [0, 13, 0, 245, 0, 4829, 0, 95997, 0, 1912789, 0, 38142605, 0, 760805121, 0, 15176819777, 0, 302764305101, 0, 6039969963221, 0, 120494499836029, 0, 2403812274318557, 0, 47955034267442037, 0, 956682848980371533, 0, 19085424234544653633, 0, 380746277170430152641, 0, 7595730021342875759821, 0, 151531658625523988015477, 0, 3022993649336995071440861, 0, 60307467722963389108943997] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 10 8 6 4 2 z - 14 z + 47 z - 47 z + 14 z - 1 f(z) = - ------------------------------------------------- 2 4 8 6 4 2 (-3 z + 1 + z ) (z - 23 z + 76 z - 23 z + 1) And in Maple-input format, it is: -(z^10-14*z^8+47*z^6-47*z^4+14*z^2-1)/(-3*z^2+1+z^4)/(z^8-23*z^6+76*z^4-23*z^2+ 1) The first , 40, terms are: [0, 12, 0, 213, 0, 4013, 0, 76396, 0, 1457033, 0, 27797817, 0, 530368556, 0, 10119278525, 0, 193073289797, 0, 3683791179596, 0, 70285840152017, 0, 1341036760255793, 0, 25586655758228172, 0, 488187179192274533, 0, 9314492843443097245, 0, 177718261826258392748, 0, 3390821284367934421209, 0, 64696046789886859850217, 0, 1234384864084595278933996, 0, 23551763489192926348476109] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 14 12 10 8 6 18 4 f(z) = - (252 z - 1141 z + 2490 z - 2490 z + 1141 z + z - 252 z 16 2 / 14 12 10 8 - 26 z + 26 z - 1) / (-3150 z + 9213 z - 13234 z + 9213 z / 20 6 18 4 16 2 + z - 3150 z - 40 z + 529 z + 529 z - 40 z + 1) And in Maple-input format, it is: -(252*z^14-1141*z^12+2490*z^10-2490*z^8+1141*z^6+z^18-252*z^4-26*z^16+26*z^2-1) /(-3150*z^14+9213*z^12-13234*z^10+9213*z^8+z^20-3150*z^6-40*z^18+529*z^4+529*z^ 16-40*z^2+1) The first , 40, terms are: [0, 14, 0, 283, 0, 5923, 0, 124590, 0, 2623545, 0, 55261065, 0, 1164096334, 0, 24522890355, 0, 516605427723, 0, 10882982026542, 0, 229264838439985, 0, 4829778483233105, 0, 101745931173477486, 0, 2143418221601019691, 0, 45154059070747017939, 0, 951232497972869937550, 0, 20039023885211012149865, 0, 422149663329802263709273, 0, 8893164624747905119777646, 0, 187346772782598190046659779] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 8 6 4 2 10 -15 z + 60 z - 60 z + 15 z + z - 1 f(z) = - --------------------------------------------------- 12 8 6 4 2 10 z + 183 z - 332 z + 183 z - 30 z - 30 z + 1 And in Maple-input format, it is: -(-15*z^8+60*z^6-60*z^4+15*z^2+z^10-1)/(z^12+183*z^8-332*z^6+183*z^4-30*z^2-30* z^10+1) The first , 40, terms are: [0, 15, 0, 327, 0, 7337, 0, 165081, 0, 3715607, 0, 83634879, 0, 1882564305, 0, 42375337777, 0, 953842634335, 0, 21470409686391, 0, 483285696713337, 0, 10878463389870857, 0, 244867511388952551, 0, 5511816880562120303, 0, 124067603554071833505, 0, 2792685349560564583521, 0, 62861627356800237010351, 0, 1414976518772111050940071, 0, 31850250031111505926161993, 0, 716929513377826692424363833] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 4 2 z - 10 z + 1 f(z) = - ----------------------- 4 2 6 -29 z + 29 z + z - 1 And in Maple-input format, it is: -(z^4-10*z^2+1)/(-29*z^4+29*z^2+z^6-1) The first , 40, terms are: [0, 19, 0, 523, 0, 14617, 0, 408745, 0, 11430235, 0, 319637827, 0, 8938428913, 0, 249956371729, 0, 6989839979491, 0, 195465563054011, 0, 5466045925532809, 0, 152853820351864633, 0, 4274440923926676907, 0, 119531492049595088755, 0, 3342607336464735808225, 0, 93473473928963007541537, 0, 2613914662674499475354803, 0, 73096137080957022302392939, 0, 2044077923604122124991647481, 0, 57161085723834462477463736521] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 8 6 4 2 10 -20 z + 103 z - 103 z + 20 z - 1 + z f(z) = - ---------------------------------------------------- 10 6 4 8 12 2 -40 z - 626 z + 328 z + 328 z + z - 40 z + 1 And in Maple-input format, it is: -(-20*z^8+103*z^6-103*z^4+20*z^2-1+z^10)/(-40*z^10-626*z^6+328*z^4+328*z^8+z^12 -40*z^2+1) The first , 40, terms are: [0, 20, 0, 575, 0, 16963, 0, 502132, 0, 14874845, 0, 440725541, 0, 13058866228, 0, 386944303291, 0, 11465499316679, 0, 339733127960660, 0, 10066602782439865, 0, 298282652081183305, 0, 8838388089647824340, 0, 261889532812149472439, 0, 7760026691431292096971, 0, 229936697453481736465588, 0, 6813234920836287199072661, 0, 201882390248024016059727725, 0, 5981960106480672940644001012, 0, 177250956220731938574762526003] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 14 12 10 8 6 4 2 / f(z) = - (z - 24 z + 181 z - 493 z + 493 z - 181 z + 24 z - 1) / ( / 14 12 10 8 6 4 16 2 -44 z + 498 z - 1876 z + 2851 z - 1876 z + 498 z + z - 44 z + 1 ) And in Maple-input format, it is: -(z^14-24*z^12+181*z^10-493*z^8+493*z^6-181*z^4+24*z^2-1)/(-44*z^14+498*z^12-\ 1876*z^10+2851*z^8-1876*z^6+498*z^4+z^16-44*z^2+1) The first , 40, terms are: [0, 20, 0, 563, 0, 16195, 0, 467368, 0, 13499945, 0, 390062069, 0, 11271415120, 0, 325714887895, 0, 9412427177879, 0, 271998983320940, 0, 7860197910738349, 0, 227143266136356565, 0, 6563965858392133820, 0, 189684909415369193999, 0, 5481497956849410928495, 0, 158403850400717044975168, 0, 4577540676868755067099805, 0, 132281372001154928711166593, 0, 3822655573668453611106153400, 0, 110466768032819659745325593131] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 12 14 8 10 4 6 2 / f(z) = - (-21 z + z - 336 z + 135 z - 135 z + 336 z + 21 z - 1) / ( / 12 14 8 16 10 4 6 2 371 z - 40 z + 1987 z + z - 1306 z + 371 z - 1306 z - 40 z + 1) And in Maple-input format, it is: -(-21*z^12+z^14-336*z^8+135*z^10-135*z^4+336*z^6+21*z^2-1)/(371*z^12-40*z^14+ 1987*z^8+z^16-1306*z^10+371*z^4-1306*z^6-40*z^2+1) The first , 40, terms are: [0, 19, 0, 524, 0, 14881, 0, 423999, 0, 12086871, 0, 344589073, 0, 9824185260, 0, 280087072771, 0, 7985275138825, 0, 227660017761081, 0, 6490582260911747, 0, 185046363191272620, 0, 5275667913063556689, 0, 150409181005689727911, 0, 4288162580496158178543, 0, 122255424795232464287393, 0, 3485499584338121823854988, 0, 99371519691452653256997331, 0, 2833079932116752295283376209, 0, 80771049156584262679447720113] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 8 2 4 6 10 12 14 / f(z) = - (-634 z - 1 + 25 z - 206 z + 634 z + 206 z - 25 z + z ) / ( / 8 2 4 6 10 12 14 16 3724 z + 1 - 47 z + 549 z - 2304 z - 2304 z + 549 z - 47 z + z ) And in Maple-input format, it is: -(-634*z^8-1+25*z^2-206*z^4+634*z^6+206*z^10-25*z^12+z^14)/(3724*z^8+1-47*z^2+ 549*z^4-2304*z^6-2304*z^10+549*z^12-47*z^14+z^16) The first , 40, terms are: [0, 22, 0, 691, 0, 22069, 0, 705482, 0, 22554007, 0, 721052567, 0, 23052146410, 0, 736980707797, 0, 23561392226771, 0, 753261549805878, 0, 24081894822012545, 0, 769902114672609153, 0, 24613896500217292534, 0, 786910296141401866643, 0, 25157650849936572267669, 0, 804294211678576644733098, 0, 25713417472912584686104983, 0, 822062161503147524706151319, 0, 26281461734415650128255496714, 0, 840222629435496046284551157621] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 4 6 12 14 8 10 / f(z) = - (-1 + 27 z - 228 z + 716 z - 27 z + z - 716 z + 228 z ) / ( / 2 4 6 12 14 8 10 16 1 - 55 z + 665 z - 2838 z + 665 z - 55 z + 4546 z - 2838 z + z ) And in Maple-input format, it is: -(-1+27*z^2-228*z^4+716*z^6-27*z^12+z^14-716*z^8+228*z^10)/(1-55*z^2+665*z^4-\ 2838*z^6+665*z^12-55*z^14+4546*z^8-2838*z^10+z^16) The first , 40, terms are: [0, 28, 0, 1103, 0, 44167, 0, 1771324, 0, 71057401, 0, 2850637129, 0, 114361216508, 0, 4587926296279, 0, 184057816251775, 0, 7384007528285020, 0, 296230657939219249, 0, 11884143215292318161, 0, 476766520485422862556, 0, 19126857608935338173983, 0, 767328799909784354739511, 0, 30783597557537867992790140, 0, 1234972385627601727363831721, 0, 49544462449924362381038237593, 0, 1987618337072877053813720017084, 0, 79739015391707212040209333543975] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 6 4 2 z - 15 z + 15 z - 1 f(z) = - -------------------------------- 6 4 8 2 -44 z + 102 z + z - 44 z + 1 And in Maple-input format, it is: -(z^6-15*z^4+15*z^2-1)/(-44*z^6+102*z^4+z^8-44*z^2+1) The first , 40, terms are: [0, 29, 0, 1189, 0, 49401, 0, 2053641, 0, 85373589, 0, 3549138989, 0, 147544320241, 0, 6133692298001, 0, 254989017189389, 0, 10600368542888629, 0, 440677071050573801, 0, 18319766917914642201, 0, 761586844367955639429, 0, 31660584117320436988989, 0, 1316189472103884945976801, 0, 54716448693989525183595041, 0, 2274664720495260574860424189, 0, 94562050611192987342317960389, 0, 3931120632955038299649359194841, 0, 163424009218933097951424047682601] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 6 4 8 2 10 145 z - 145 z - 25 z + 25 z + z - 1 f(z) = - --------------------------------------------------- 12 6 4 8 2 10 z - 995 z + 548 z + 548 z - 65 z - 65 z + 1 And in Maple-input format, it is: -(145*z^6-145*z^4-25*z^8+25*z^2+z^10-1)/(z^12-995*z^6+548*z^4+548*z^8-65*z^2-65 *z^10+1) The first , 40, terms are: [0, 40, 0, 2197, 0, 121735, 0, 6748096, 0, 374079619, 0, 20737143595, 0, 1149566489968, 0, 63726386332735, 0, 3532681575875629, 0, 195834721732832344, 0, 10856126548559080585, 0, 601810968956118729913, 0, 33361479413223474759160, 0, 1849398508920455533993789, 0, 102521677843870104359906191, 0, 5683304262020707489694083600, 0, 315054806105419432032201005467, 0, 17465109428231028612515926221715, 0, 968180905128040774611331672842016, 0, 53671250610051012999881846667766711] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 4 2 z - 20 z + 1 f(z) = - ---------------------- 6 4 2 z - 76 z + 76 z - 1 And in Maple-input format, it is: -(z^4-20*z^2+1)/(z^6-76*z^4+76*z^2-1) The first , 40, terms are: [0, 56, 0, 4181, 0, 313501, 0, 23508376, 0, 1762814681, 0, 132187592681, 0, 9912306636376, 0, 743290810135501, 0, 55736898453526181, 0, 4179524093204328056, 0, 313408570091871078001, 0, 23501463232797126522001, 0, 1762296333889692618072056, 0, 132148723578494149228882181, 0, 9909391972053171499548091501, 0, 743072249180409368316877980376, 0, 55720509296558649452266300436681, 0, 4178295124992718299551655654770681, 0, 313316413865157313816921907807364376, 0, 23494552744761805817969591429897557501] ------------------------------------------------------------- There are, 112, different connected graphs with , 6, vertices (up to isomorphism) -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 -1 + z f(z) = - -------------- 2 4 -7 z + 1 + z And in Maple-input format, it is: -(-1+z^2)/(-7*z^2+1+z^4) The first , 40, terms are: [0, 6, 0, 41, 0, 281, 0, 1926, 0, 13201, 0, 90481, 0, 620166, 0, 4250681, 0, 29134601, 0, 199691526, 0, 1368706081, 0, 9381251041, 0, 64300051206, 0, 440719107401, 0, 3020733700601, 0, 20704416796806, 0, 141910183877041, 0, 972666870342481, 0, 6666757908520326, 0, 45694638489299801] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 8 10 6 4 2 -14 z + z + 53 z - 53 z + 14 z - 1 f(z) = - --------------------------------------------------- 12 10 8 6 4 2 z - 23 z + 144 z - 276 z + 144 z - 23 z + 1 And in Maple-input format, it is: -(-14*z^8+z^10+53*z^6-53*z^4+14*z^2-1)/(z^12-23*z^10+144*z^8-276*z^6+144*z^4-23 *z^2+1) The first , 40, terms are: [0, 9, 0, 116, 0, 1595, 0, 22335, 0, 314767, 0, 4447123, 0, 62894820, 0, 889891169, 0, 12593234417, 0, 178225865169, 0, 2522424220465, 0, 35700262990852, 0, 505274278675299, 0, 7151283853613615, 0, 101214162603262751, 0, 1432513565529588075, 0, 20274786023403373396, 0, 286955034713743605049, 0, 4061359489971668057505, 0, 57481623034048377999713] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 4 3 2 6 5 -7 z - 2 z + 7 z + z - 1 + z + z f(z) = - ----------------------------------------------------- 8 7 6 5 4 3 2 z + z - 17 z - 8 z + 36 z + 8 z - 17 z - z + 1 And in Maple-input format, it is: -(-7*z^4-2*z^3+7*z^2+z-1+z^6+z^5)/(z^8+z^7-17*z^6-8*z^5+36*z^4+8*z^3-17*z^2-z+1 ) The first , 40, terms are: [0, 10, 4, 145, 140, 2229, 3384, 35138, 71012, 564613, 1388952, 9218989, 26094100, 152539058, 478070160, 2551201629, 8615422756, 43033405129, 153528757412, 730712416090, 2714637004440, 12470978634553, 47734773270576, 213667298365321, 836067722105688, 3671521022254138, 14601914490365932, 63227646824220697, 254495230903487036, 1090635834288313293, 4428888580782847680, 18835722069065109074, 76989459087727106588, 325595293988918928925, 1337264009057927551176, 5632036534632824963221, 23213773560908703547468, 97469455417063078617506, 402796127895115911663000, 1687450773140255171090373] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : f(z) = - ( 2 3 4 6 8 10 5 7 9 2 z - 1 + 9 z - 14 z - 25 z + 25 z - 9 z + z + 26 z - 14 z + 2 z ) / 12 11 10 9 8 7 6 5 4 / (z + 3 z - 18 z - 29 z + 84 z + 81 z - 138 z - 81 z + 84 z / 3 2 + 29 z - 18 z - 3 z + 1) And in Maple-input format, it is: -(2*z-1+9*z^2-14*z^3-25*z^4+25*z^6-9*z^8+z^10+26*z^5-14*z^7+2*z^9)/(z^12+3*z^11 -18*z^10-29*z^9+84*z^8+81*z^7-138*z^6-81*z^5+84*z^4+29*z^3-18*z^2-3*z+1) The first , 40, terms are: [1, 12, 39, 245, 1060, 5645, 26677, 134980, 656579, 3267113, 16045240, 79399393, 391189877, 1932219396, 9529945235, 47042606021, 232102701916, 1145490952141, 5652391472753, 27894192222380, 137648553351847, 679271216154169, 3352021945672720, 16541506992229129, 81628280034679401, 402816961004827500, 1987805862273360319, 9809360531853749213, 48406884378357418404, 238876669866928828565, 1178800312337129404157, 5817103695839142167556, 28706043323007964855611, 141657601887913712503729, 699047076494520796870952, 3449633594118868205123545, 17023133681176084993705325, 84005177316265342890571076, 414545871992332947684973147, 2045686775005759479359679421] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 18 16 14 12 10 8 6 f(z) = - (z - 21 z + 168 z - 649 z + 1281 z - 1281 z + 649 z 4 2 / 20 18 16 14 12 - 168 z + 21 z - 1) / (z - 32 z + 344 z - 1716 z + 4420 z / 10 8 6 4 2 - 6054 z + 4420 z - 1716 z + 344 z - 32 z + 1) And in Maple-input format, it is: -(z^18-21*z^16+168*z^14-649*z^12+1281*z^10-1281*z^8+649*z^6-168*z^4+21*z^2-1)/( z^20-32*z^18+344*z^16-1716*z^14+4420*z^12-6054*z^10+4420*z^8-1716*z^6+344*z^4-\ 32*z^2+1) The first , 40, terms are: [0, 11, 0, 176, 0, 2915, 0, 48473, 0, 806545, 0, 13421771, 0, 223358992, 0, 3717067123, 0, 61858354329, 0, 1029429565385, 0, 17131485296835, 0, 285097516910544, 0, 4744515440566427, 0, 78956938104391809, 0, 1313979936535256361, 0, 21866897543191847123, 0, 363902975296791179760, 0, 6055974570245604154107, 0, 100781885522419192166033, 0, 1677184791952803345393393] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 6 4 3 2 z - 8 z + 2 z + 8 z - 1 f(z) = - ------------------------------------------------------- 8 7 6 5 4 3 2 z + z - 20 z + 10 z + 38 z - 10 z - 20 z - z + 1 And in Maple-input format, it is: -(z^6-8*z^4+2*z^3+8*z^2-1)/(z^8+z^7-20*z^6+10*z^5+38*z^4-10*z^3-20*z^2-z+1) The first , 40, terms are: [1, 13, 41, 281, 1183, 6728, 31529, 167089, 817991, 4213133, 21001799, 106912793, 536948224, 2720246633, 13704300553, 69289288909, 349519610713, 1765722581057, 8911652846951, 45005025662792, 227191499132401, 1147185247901449, 5791672851807479, 29242880940226381, 147640981046478543, 745439797095329713, 3763622719883603968, 19002353776441540177, 95940879136187583953, 484398978524471931341, 2445685822753246301257, 12348080425980866090537, 62344389094970498108207, 314771823879840325570888, 1589256410595418296414137, 8024025901064701223963681, 40512638138500187085447911, 204544978591083096779665229, 1032730632545960228058586519, 5214171781414287060178827977] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 5 4 3 2 6 7 8 / f(z) = - (-14 z + 18 z + 14 z - 10 z - 2 z - 10 z + 1 + 2 z + z ) / ( / 10 9 8 7 6 5 4 3 2 z + 2 z - 20 z - 34 z + 79 z + 88 z - 79 z - 34 z + 20 z + 2 z - 1) And in Maple-input format, it is: -(-14*z^5+18*z^4+14*z^3-10*z^2-2*z-10*z^6+1+2*z^7+z^8)/(z^10+2*z^9-20*z^8-34*z^ 7+79*z^6+88*z^5-79*z^4-34*z^3+20*z^2+2*z-1) The first , 40, terms are: [0, 10, 0, 139, 12, 2083, 528, 32098, 14880, 501049, 346200, 7879561, 7267488, 124558258, 143381520, 1977252019, 2716504740, 31503270139, 50044372512, 503644260922, 903422005440, 8077373827249, 16062831295152, 129930062579281, 282268559500992, 2095854443484634, 4914614442927648, 33895695833688859, 84936580608827580, 549516420724148371, 1459068840931430448, 8928765427558988050, 24939806776072021728, 145378643826138043369, 424530123125192853768, 2371563246274666805977, 7201298541802894159200, 38754605525500059693058, 121796141746905356173872, 634307249739261662191171 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 15 8 10 f(z) = - (-1 + 23 z + 2 z - 195 z + 777 z + 2 z - 1544 z + 1544 z 12 14 18 16 5 7 9 11 - 777 z + 195 z + z - 23 z - 24 z + 94 z - 144 z + 94 z 13 / 2 4 16 14 13 12 11 - 24 z ) / ((-3 z + 1 + z ) (z - 32 z + 6 z + 313 z - 50 z / 10 9 8 7 6 5 4 3 - 1224 z + 136 z + 1940 z - 136 z - 1224 z + 50 z + 313 z - 6 z 2 - 32 z + 1)) And in Maple-input format, it is: -(-1+23*z^2+2*z^3-195*z^4+777*z^6+2*z^15-1544*z^8+1544*z^10-777*z^12+195*z^14+z ^18-23*z^16-24*z^5+94*z^7-144*z^9+94*z^11-24*z^13)/(-3*z^2+1+z^4)/(z^16-32*z^14 +6*z^13+313*z^12-50*z^11-1224*z^10+136*z^9+1940*z^8-136*z^7-1224*z^6+50*z^5+313 *z^4-6*z^3-32*z^2+1) The first , 40, terms are: [0, 12, 4, 205, 168, 3697, 4852, 68040, 121016, 1270085, 2801376, 23993773, 62077816, 458014656, 1337793596, 8821670265, 28291098936, 171209045189, 590404047340, 3344026607060, 12203495799408, 65660075288537, 250465851176768, 1294810561705577, 5113467554100848, 25623066756233092, 103978858008531188, 508488064014203541, 2107898736611171720, 10113738318325703593, 42632216422694859300, 201523314474793309072, 860681317978022686184, 4021247518644333470109, 17351635339592211617440, 80332095840039614821333, 349436172108931391897768, 1606227110469977632336504, 7031197937720166098767212, 32138988861976554623871425] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-34 z - 1 + 26 z + z + 2 z - 264 z + 1364 z + 212 z 8 10 12 14 18 16 20 - 3932 z + 6607 z - 6607 z + 3932 z + 264 z - 1364 z - 26 z 5 19 7 9 11 13 / 17 - 34 z + 2 z + 212 z - 608 z + 858 z - 608 z ) / (1008 z - z / 2 24 23 22 21 3 4 6 + 1 - 39 z + z + z - 39 z - 6 z + 6 z + 529 z - 3479 z 15 8 10 12 14 18 - 3940 z + 12582 z - 26647 z + 34101 z - 26647 z - 3479 z 16 20 5 19 7 9 11 + 12582 z + 529 z + 88 z - 88 z - 1008 z + 3940 z - 7493 z 13 + 7493 z ) And in Maple-input format, it is: -(-34*z^17-1+26*z^2+z^22+2*z^3-264*z^4+1364*z^6+212*z^15-3932*z^8+6607*z^10-\ 6607*z^12+3932*z^14+264*z^18-1364*z^16-26*z^20-34*z^5+2*z^19+212*z^7-608*z^9+ 858*z^11-608*z^13)/(1008*z^17-z+1-39*z^2+z^24+z^23-39*z^22-6*z^21+6*z^3+529*z^4 -3479*z^6-3940*z^15+12582*z^8-26647*z^10+34101*z^12-26647*z^14-3479*z^18+12582* z^16+529*z^20+88*z^5-88*z^19-1008*z^7+3940*z^9-7493*z^11+7493*z^13) The first , 40, terms are: [1, 14, 45, 320, 1408, 8239, 40469, 221166, 1131784, 6030701, 31343072, 165423883, 864826479, 4547882344, 23829461861, 125139189591, 656251333800, 3444448129881, 18069187420384, 94820061711478, 497477652099839, 2610366941749499, 13696062222048848, 71863906881063816, 377062167181856223, 1978441101583017358, 10380740117934227587, 54467393387473867453, 285787320974259816272, 1499513930020198926789, 7867872611384236350261, 41282366803096268894030, 216606545411479170361385, 1136524331288652281168872, 5963288100620572395403024, 31289088552600856084751875, 164172341261420747180753113, 861404421624367140580374182, 4519747630602359259455921168, 23714899291170961865030874817] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 14 12 10 8 6 4 2 / f(z) = - (z - 19 z + 117 z - 287 z + 287 z - 117 z + 19 z - 1) / ( / 16 14 12 10 8 6 4 2 z - 34 z + 300 z - 1022 z + 1526 z - 1022 z + 300 z - 34 z + 1) And in Maple-input format, it is: -(z^14-19*z^12+117*z^10-287*z^8+287*z^6-117*z^4+19*z^2-1)/(z^16-34*z^14+300*z^ 12-1022*z^10+1526*z^8-1022*z^6+300*z^4-34*z^2+1) The first , 40, terms are: [0, 15, 0, 327, 0, 7353, 0, 165993, 0, 3750071, 0, 84735327, 0, 1914733713, 0, 43267032561, 0, 977703339071, 0, 22093138530135, 0, 499238212053897, 0, 11281276536227481, 0, 254922798262355367, 0, 5760485801092908079, 0, 130169592212622953377, 0, 2941439893440958578273, 0, 66467663455119718416879, 0, 1501968575019485063557863, 0, 33939956410220175941172441, 0, 766940574048976749533892425] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 8 7 6 5 4 3 2 / 10 f(z) = - (z + 3 z - 6 z - 17 z + 8 z + 17 z - 6 z - 3 z + 1) / (z / 9 8 7 6 5 4 3 2 + 4 z - 16 z - 44 z + 48 z + 93 z - 48 z - 44 z + 16 z + 4 z - 1) And in Maple-input format, it is: -(z^8+3*z^7-6*z^6-17*z^5+8*z^4+17*z^3-6*z^2-3*z+1)/(z^10+4*z^9-16*z^8-44*z^7+48 *z^6+93*z^5-48*z^4-44*z^3+16*z^2+4*z-1) The first , 40, terms are: [1, 14, 45, 320, 1412, 8251, 40665, 221986, 1139056, 6067169, 31594644, 166790327, 873257795, 4594960192, 24105185985, 126686855011, 665079564796, 3493858116605, 18346863746448, 96366095803274, 506084101741523, 2658029626219375, 13959618430025100, 73316448707615872, 385053205465592143, 2022298385535511718, 10621029176413956963, 55781457555126886493, 292962458793734842496, 1538632056452878671157, 8080852197899561356341, 42440432673469621513942, 222896007552502452663289, 1170644067670008097576064, 6148191581307945731378388, 32290141555336944947317751, 169586971863441061718533589, 890666329005836538405112282, 4677756122974512588052466928, 24567452322439416664569464149] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 28 25 26 2 24 f(z) = - (-16068 z - 1 - 33 z - 48 z + 461 z + 33 z - 3595 z 23 22 21 3 4 6 15 + 476 z + 17402 z - 2548 z + 2 z - 461 z + 3595 z + 20142 z 8 10 12 14 18 16 - 17402 z + 55094 z - 117337 z + 170663 z + 117337 z - 170663 z 20 5 19 7 9 11 13 - 55094 z - 48 z + 8116 z + 476 z - 2548 z + 8116 z - 16068 z 30 27 / 17 28 25 26 + z + 2 z ) / (185141 z - z + 1 + 839 z + 1573 z - 7941 z / 29 2 24 23 22 21 3 - 13 z - 47 z + 45503 z - 12330 z - 168902 z + 49650 z + 13 z 4 6 15 8 10 12 + 839 z - 7941 z - 185141 z + 45503 z - 168902 z + 421949 z 14 18 16 20 5 19 - 725243 z - 725243 z + 867681 z + 421949 z + 35 z - 120223 z 7 9 11 13 30 27 31 - 1573 z + 12330 z - 49650 z + 120223 z - 47 z - 35 z + z 32 + z ) And in Maple-input format, it is: -(-16068*z^17-1-33*z^28-48*z^25+461*z^26+33*z^2-3595*z^24+476*z^23+17402*z^22-\ 2548*z^21+2*z^3-461*z^4+3595*z^6+20142*z^15-17402*z^8+55094*z^10-117337*z^12+ 170663*z^14+117337*z^18-170663*z^16-55094*z^20-48*z^5+8116*z^19+476*z^7-2548*z^ 9+8116*z^11-16068*z^13+z^30+2*z^27)/(185141*z^17-z+1+839*z^28+1573*z^25-7941*z^ 26-13*z^29-47*z^2+45503*z^24-12330*z^23-168902*z^22+49650*z^21+13*z^3+839*z^4-\ 7941*z^6-185141*z^15+45503*z^8-168902*z^10+421949*z^12-725243*z^14-725243*z^18+ 867681*z^16+421949*z^20+35*z^5-120223*z^19-1573*z^7+12330*z^9-49650*z^11+120223 *z^13-47*z^30-35*z^27+z^31+z^32) The first , 40, terms are: [1, 15, 47, 361, 1549, 9631, 46821, 267404, 1372079, 7556335, 39703199, 215212049, 1142673513, 6150888727, 32809199295, 176067837789, 941066735208, 5043342311821, 26980369377105, 144506555403759, 773372053574455, 4141086221574825, 22166176115928465, 118676955881261935, 635296140140892585, 3401180801093942044, 18207663566691855971, 97476061817003284207, 521829884729029889827, 2793626910483059670161, 14955544526057857744737, 80064462973366218345095, 428622412854305629804527, 2294624398048763963505185, 12284210301890663362354192, 65763301099988719499817441, 352062270325760005138284881, 1884758391231764425945827191, 10090011778423015979284619903, 54016669324867632841214549921] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {4, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (z - 1 + 18 z - 13 z - 112 z + 285 z - 285 z + 112 z - 18 z 14 5 7 9 11 13 / 16 15 14 + z + 53 z - 82 z + 53 z - 13 z + z ) / (z + z - 32 z / 13 12 11 10 9 8 7 - 23 z + 285 z + 148 z - 1002 z - 368 z + 1520 z + 368 z 6 5 4 3 2 - 1002 z - 148 z + 285 z + 23 z - 32 z - z + 1) And in Maple-input format, it is: -(z-1+18*z^2-13*z^3-112*z^4+285*z^6-285*z^8+112*z^10-18*z^12+z^14+53*z^5-82*z^7 +53*z^9-13*z^11+z^13)/(z^16+z^15-32*z^14-23*z^13+285*z^12+148*z^11-1002*z^10-\ 368*z^9+1520*z^8+368*z^7-1002*z^6-148*z^5+285*z^4+23*z^3-32*z^2-z+1) The first , 40, terms are: [0, 14, 4, 279, 180, 5743, 5732, 119238, 159736, 2491097, 4153144, 52339225, 103528136, 1105518294, 2509367708, 23466605007, 59630793628, 500401013207, 1396520619356, 10715291264446, 32346213079584, 230325556190817, 742805453104880, 4967876948008161, 16942841624763744, 107481802540481118, 384364897973155620, 2331782957943583255, 8681531280895347204, 50710078515631448015, 195386066857781884580, 1105168194816386575286, 4384414649799085200504, 24131077785740381810393, 98145693640149806189992, 527760477350954337227417, 2192556836462515258366408, 11558918775863353705488614, 48898526714126418140898652, 253475743015304318723231215] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {4, 5}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-34 z - 1 + 26 z + z + 2 z - 268 z + 1416 z + 212 z 8 10 12 14 18 16 20 - 4176 z + 7107 z - 7107 z + 4176 z + 268 z - 1416 z - 26 z 5 19 7 9 11 13 / 17 - 34 z + 2 z + 212 z - 616 z + 874 z - 616 z ) / (1044 z - z / 2 24 23 22 21 3 4 6 + 1 - 41 z + z + z - 41 z - 6 z + 6 z + 563 z - 3713 z 15 8 10 12 14 18 - 4096 z + 13436 z - 28405 z + 36321 z - 28405 z - 3713 z 16 20 5 19 7 9 11 + 13436 z + 563 z + 92 z - 92 z - 1044 z + 4096 z - 7777 z 13 + 7777 z ) And in Maple-input format, it is: -(-34*z^17-1+26*z^2+z^22+2*z^3-268*z^4+1416*z^6+212*z^15-4176*z^8+7107*z^10-\ 7107*z^12+4176*z^14+268*z^18-1416*z^16-26*z^20-34*z^5+2*z^19+212*z^7-616*z^9+ 874*z^11-616*z^13)/(1044*z^17-z+1-41*z^2+z^24+z^23-41*z^22-6*z^21+6*z^3+563*z^4 -3713*z^6-4096*z^15+13436*z^8-28405*z^10+36321*z^12-28405*z^14-3713*z^18+13436* z^16+563*z^20+92*z^5-92*z^19-1044*z^7+4096*z^9-7777*z^11+7777*z^13) The first , 40, terms are: [1, 16, 49, 404, 1696, 11163, 53761, 320500, 1647432, 9377999, 49735704, 276984597, 1491189571, 8218268752, 44564061809, 244377583181, 1329734425464, 7274488149155, 39648281688912, 216652821305532, 1181763200798391, 6454047349933651, 35217874855735528, 192287340883070716, 1049447406407725395, 5729194349774253472, 31270971686895860759, 170705716485111523317, 931781343522281896720, 5086372407939249677389, 27764047736952489942857, 151555226385023762888960, 827274626487622377889525, 4515802721973839483627980, 24649932803778219613871144, 134554930968240572799064723, 734482327685877296392135337, 4009263205432876172677383468, 21885010453649052071668255904, 119461966527710769804994815739] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (-2 z + 1 - 13 z + 21 z + 53 z - 84 z + 53 z - 13 z + z 5 7 9 11 / 14 13 12 11 - 65 z + 65 z - 21 z + 2 z ) / (z + 3 z - 27 z - 44 z / 10 9 8 7 6 5 4 3 + 168 z + 190 z - 393 z - 297 z + 393 z + 190 z - 168 z - 44 z 2 + 27 z + 3 z - 1) And in Maple-input format, it is: -(-2*z+1-13*z^2+21*z^3+53*z^4-84*z^6+53*z^8-13*z^10+z^12-65*z^5+65*z^7-21*z^9+2 *z^11)/(z^14+3*z^13-27*z^12-44*z^11+168*z^10+190*z^9-393*z^8-297*z^7+393*z^6+ 190*z^5-168*z^4-44*z^3+27*z^2+3*z-1) The first , 40, terms are: [1, 17, 55, 465, 2089, 14045, 72229, 442360, 2413023, 14202137, 79492883, 459900109, 2603265149, 14949274829, 85035529131, 486740673989, 2774615551936, 15859502935821, 90489266969981, 516914139433157, 2950535804944923, 16850268372352229, 96197856765016853, 549314602500902577, 3136265428771463929, 17907985544879840824, 102247522321538610771, 583817658857288137637, 3333416209846830842847, 19033114058774274229529, 108673808578757784953537, 620502306435937958790713, 3542905928108219607134663, 20229136883099005023421849, 115503219200652562780383872, 659494960677521767445661865, 3765550191272545656025694729, 21500357726716353025723106473, 122761657208534170392831604975, 700938513891285176297741510089] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 5 f(z) = - (1 - 16 z - 4 z + 77 z - 124 z + 77 z - 16 z + z + 28 z 7 9 / 9 8 10 2 7 - 28 z + 4 z ) / (-158 z - 489 z + 225 z - 1 + 2 z + 33 z + 304 z / 11 13 12 14 5 6 4 3 + 8 z + 2 z - 33 z + z - 158 z + 489 z - 225 z + 8 z ) And in Maple-input format, it is: -(1-16*z^2-4*z^3+77*z^4-124*z^6+77*z^8-16*z^10+z^12+28*z^5-28*z^7+4*z^9)/(-158* z^9-489*z^8+225*z^10-1+2*z+33*z^2+304*z^7+8*z^11+2*z^13-33*z^12+z^14-158*z^5+ 489*z^6-225*z^4+8*z^3) The first , 40, terms are: [2, 21, 112, 785, 4854, 31833, 202864, 1311193, 8415402, 54201265, 348480944, 2242501013, 14424254750, 92800659937, 596980321376, 3840550392097, 24706691996962, 158943191519381, 1022506652723856, 6577970505508465, 42317198990771862, 272233960642282969, 1751327996739737936, 11266597497009909337, 72479972154641484874, 466276235689455478865, 2999635832202734897104, 19297177445121697490197, 124142087712421658967614, 798627573535318060048065, 5137709633916909117307072, 33051776800655571710242113, 212627810250418004426886722, 1367871566341149995439391509, 8799754931288435019985492144, 56610349071392962740459589073, 364184190004315824625453175350, 2342860032250618662015211439705, 15072024764749458245660853717552, 96960948320875578916990678800857] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 10 6 4 8 12 2 -17 z - 162 z + 89 z + 89 z + z - 17 z + 1 f(z) = - ---------------------------------------------------------------- 2 10 6 4 8 12 2 (-1 + z ) (-32 z - 442 z + 216 z + 216 z + z - 32 z + 1) And in Maple-input format, it is: -(-17*z^10-162*z^6+89*z^4+89*z^8+z^12-17*z^2+1)/(-1+z^2)/(-32*z^10-442*z^6+216* z^4+216*z^8+z^12-32*z^2+1) The first , 40, terms are: [0, 16, 0, 369, 0, 8705, 0, 205712, 0, 4862161, 0, 114923761, 0, 2716389776, 0, 64205860001, 0, 1517599978065, 0, 35870709669904, 0, 847857036801889, 0, 20040349423054753, 0, 473683165538509840, 0, 11196199057823661969, 0, 264638649764189509217, 0, 6255124135211262822800, 0, 147849068840821717753777, 0, 3494630431720891859744785, 0, 82600735669552046529377168, 0, 1952390007028983809953273409] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 3 4 6 15 8 f(z) = - (2 z + 2 z - 1 + 19 z - 34 z - 136 z + 471 z - 34 z - 863 z 10 12 14 18 16 5 7 9 + 863 z - 471 z + 136 z + z - 19 z + 210 z - 594 z + 834 z 11 13 / 20 19 18 17 16 - 594 z + 210 z ) / (z + 3 z - 32 z - 65 z + 326 z / 15 14 13 12 11 10 9 + 496 z - 1523 z - 1755 z + 3711 z + 3221 z - 4971 z - 3221 z 8 7 6 5 4 3 2 + 3711 z + 1755 z - 1523 z - 496 z + 326 z + 65 z - 32 z - 3 z + 1) And in Maple-input format, it is: -(2*z^17+2*z-1+19*z^2-34*z^3-136*z^4+471*z^6-34*z^15-863*z^8+863*z^10-471*z^12+ 136*z^14+z^18-19*z^16+210*z^5-594*z^7+834*z^9-594*z^11+210*z^13)/(z^20+3*z^19-\ 32*z^18-65*z^17+326*z^16+496*z^15-1523*z^14-1755*z^13+3711*z^12+3221*z^11-4971* z^10-3221*z^9+3711*z^8+1755*z^7-1523*z^6-496*z^5+326*z^4+65*z^3-32*z^2-3*z+1) The first , 40, terms are: [1, 16, 49, 404, 1700, 11175, 53989, 321432, 1656976, 9424867, 50099260, 278949481, 1504460087, 8293254944, 45035246125, 247087522601, 1346107503980, 7369106663823, 40207234009312, 219879997100160, 1200566550835787, 6562270615197615, 35842743325674836, 195870266462253068, 1070002213222041955, 5846615710929133640, 31941348043604130487, 174522243434733043661, 953487057805695814912, 5209569439390658646453, 28462501401429861885385, 155508860627787905685592, 849629872429963634510437, 4642046795530291338881372, 25362131501094535306697564, 138568460298207968860383599, 757079317046034937243485157, 4136371195423333453851942992, 22599396975085926131024173664, 123473776222088923422113675191] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 4 3 2 z + 2 z - 4 z - 2 z + 1 f(z) = - ------------------------------------------- 2 4 3 2 (z - 2 z - 1) (z + 6 z - 3 z - 6 z + 1) And in Maple-input format, it is: -(z^4+2*z^3-4*z^2-2*z+1)/(z^2-2*z-1)/(z^4+6*z^3-3*z^2-6*z+1) The first , 40, terms are: [2, 20, 108, 725, 4480, 28561, 179928, 1138660, 7193450, 45474461, 287400960, 1816564229, 11481464878, 72568802500, 458669938608, 2899021855801, 18323243845760, 115811947027949, 731988596166300, 4626528972901940, 29241944603930518, 184823511249946969, 1168175738318561280, 7383446768415075625, 46667024810082486938, 294958611232449262580, 1864283885457133125492, 11783193550857108763709, 74475594269198528560000, 470722484344054612528489, 2975198243680678004021088, 18804720155951741886725380, 118855105166415197047288322, 751222879519927367775180725, 4748099073439396741451320320, 30010327728041772564365449709, 189680071206282718406826149062, 1198871593101665628854116856740, 7577459706787151151344419999800, 47893282265062167509906326079329] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (z - 1 + 20 z - 13 z - 128 z + 315 z - 315 z + 128 z - 20 z 14 5 7 9 11 13 / 13 6 + z + 59 z - 94 z + 59 z - 13 z + z ) / (-25 z - 1230 z / 8 7 14 15 9 10 2 + 1888 z + 450 z - 36 z + z - 450 z - 1230 z + 1 - 36 z - z 16 4 3 5 11 12 + z + 341 z + 25 z - 184 z + 184 z + 341 z ) And in Maple-input format, it is: -(z-1+20*z^2-13*z^3-128*z^4+315*z^6-315*z^8+128*z^10-20*z^12+z^14+59*z^5-94*z^7 +59*z^9-13*z^11+z^13)/(-25*z^13-1230*z^6+1888*z^8+450*z^7-36*z^14+z^15-450*z^9-\ 1230*z^10+1-36*z^2-z+z^16+341*z^4+25*z^3-184*z^5+184*z^11+341*z^12) The first , 40, terms are: [0, 16, 4, 367, 236, 8807, 9352, 214200, 315860, 5254001, 9826024, 129806801, 291159572, 3228239144, 8359521872, 80771758775, 234839171588, 2032100252511, 6494287618964, 51379290323104, 177502209720104, 1304843355450081, 4808300869261904, 33268661062753569, 129348036491835176, 851161027637470336, 3460565893787130780, 21842204630630436735, 92179867679679404252, 561973582182284738839, 2446797375608412960864, 14491530950247786948424, 64762378189009769787580, 374415130829181041500177, 1710170801107541232406200, 9689732407613053777844529, 45074496549141983656216828, 251121253583547506155009176, 1186162039085036412858603048, 6515932955103707286577915911] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 5 7 f(z) = - (2 z - 1 + 13 z - 18 z - 43 z + 43 z - 13 z + z + 34 z - 18 z 9 / 8 3 2 9 10 4 + 2 z ) / (158 z + 41 z - 26 z - 41 z - 26 z + 1 - 3 z + 158 z / 5 11 12 6 7 - 131 z + 3 z + z - 294 z + 131 z ) And in Maple-input format, it is: -(2*z-1+13*z^2-18*z^3-43*z^4+43*z^6-13*z^8+z^10+34*z^5-18*z^7+2*z^9)/(158*z^8+ 41*z^3-26*z^2-41*z^9-26*z^10+1-3*z+158*z^4-131*z^5+3*z^11+z^12-294*z^6+131*z^7) The first , 40, terms are: [1, 16, 51, 413, 1848, 12045, 61469, 367664, 1982239, 11437009, 63122640, 358656273, 1999430529, 11286176336, 63189129411, 355680239853, 1995062051304, 11216287088061, 62963585934957, 353799397875504, 1986757479365407, 11161329627945985, 62685478893803424, 352124790882388993, 1977765279323701793, 11109298379083461744, 62398825829916777043, 350494071834895501309, 1968681483128950287704, 11057996422200836519341, 62111692973955368352893, 348877492664639353272720, 1959615172342854012627903, 11007020540154269335306897, 61825553345769526754689648, 347269576472521197765603409, 1950586174419599653998605345, 10956295984710347037483926512, 61540671827126563692345630243, 345669291976450403456789655885] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (z - 1 + 19 z - 13 z - 116 z + 276 z - 276 z + 116 z - 19 z 14 5 7 9 11 13 / 16 15 14 + z + 59 z - 94 z + 59 z - 13 z + z ) / (z + z - 36 z / 13 12 11 10 9 8 7 - 22 z + 324 z + 173 z - 1093 z - 464 z + 1624 z + 464 z 6 5 4 3 2 - 1093 z - 173 z + 324 z + 22 z - 36 z - z + 1) And in Maple-input format, it is: -(z-1+19*z^2-13*z^3-116*z^4+276*z^6-276*z^8+116*z^10-19*z^12+z^14+59*z^5-94*z^7 +59*z^9-13*z^11+z^13)/(z^16+z^15-36*z^14-22*z^13+324*z^12+173*z^11-1093*z^10-\ 464*z^9+1624*z^8+464*z^7-1093*z^6-173*z^5+324*z^4+22*z^3-36*z^2-z+1) The first , 40, terms are: [0, 17, 8, 412, 440, 10405, 17160, 267189, 586016, 6952173, 18695304, 183002029, 572850072, 4866328380, 17102241064, 130541016169, 501615665104, 3527869512393, 14529545603520, 95932687220921, 417051745374928, 2622031602971257, 11890891377307768, 71964621115808892, 337331083866321352, 1981823918992461917, 9533369943596654744, 54725377143584268541, 268643630950164214112, 1514446900581598057125, 7553354728577900387352, 41982555918205108113781, 212010831743734565938728, 1165408749558013925987676, 5942919603951743052729112, 32386052643179536092901249, 166415837089636667542846112, 900759839345550810028346129, 4656311693694917457934339968, 25069896717139424144583521009] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 28 25 26 2 24 f(z) = - (-12180 z - 1 - 36 z - 44 z + 537 z + 36 z - 4395 z 23 22 21 3 4 6 15 + 406 z + 22009 z - 2056 z + 2 z - 537 z + 4395 z + 15150 z 8 10 12 14 18 16 - 22009 z + 71247 z - 153753 z + 225001 z + 153753 z - 225001 z 20 5 19 7 9 11 13 - 71247 z - 44 z + 6298 z + 406 z - 2056 z + 6298 z - 12180 z 30 27 / 17 28 25 26 + z + 2 z ) / (196041 z - z + 1 + 1049 z + 2023 z - 10524 z / 29 2 24 23 22 21 3 - 10 z - 54 z + 62788 z - 14213 z - 239400 z + 54507 z + 10 z 4 6 15 8 10 12 + 1049 z - 10524 z - 196041 z + 62788 z - 239400 z + 608092 z 14 18 16 20 5 - 1054574 z - 1054574 z + 1265244 z + 608092 z + 93 z 19 7 9 11 13 30 - 128709 z - 2023 z + 14213 z - 54507 z + 128709 z - 54 z 27 31 32 - 93 z + z + z ) And in Maple-input format, it is: -(-12180*z^17-1-36*z^28-44*z^25+537*z^26+36*z^2-4395*z^24+406*z^23+22009*z^22-\ 2056*z^21+2*z^3-537*z^4+4395*z^6+15150*z^15-22009*z^8+71247*z^10-153753*z^12+ 225001*z^14+153753*z^18-225001*z^16-71247*z^20-44*z^5+6298*z^19+406*z^7-2056*z^ 9+6298*z^11-12180*z^13+z^30+2*z^27)/(196041*z^17-z+1+1049*z^28+2023*z^25-10524* z^26-10*z^29-54*z^2+62788*z^24-14213*z^23-239400*z^22+54507*z^21+10*z^3+1049*z^ 4-10524*z^6-196041*z^15+62788*z^8-239400*z^10+608092*z^12-1054574*z^14-1054574* z^18+1265244*z^16+608092*z^20+93*z^5-128709*z^19-2023*z^7+14213*z^9-54507*z^11+ 128709*z^13-54*z^30-93*z^27+z^31+z^32) The first , 40, terms are: [1, 19, 61, 565, 2571, 18576, 98145, 638381, 3608383, 22404611, 130579503, 794057025, 4692654224, 28269632497, 168118571617, 1008505756443, 6014616385761, 36011407618989, 215044098662527, 1286424959570144, 7686418987636037, 45963323271307925, 274704003874200147, 1642387874303705819, 9817045580295006623, 58689028161908055425, 350820752052863275552, 2097228424895035852385, 12536739842480214270657, 74944195274396664340011, 448003734212014705058765, 2678131047841739101577365, 16009494033144725345381211, 95703176403365891210407360, 572101525498494822022896961, 3419961354714669973122267629, 20444118567105198591333453471, 122212657734495834266108563723, 730572935548503019454096945247, 4367282082244426845870946504113] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 3 4 6 15 8 f(z) = - (4 z + 1 - 28 z - 4 z + 295 z - 1495 z - 72 z + 3896 z 10 12 14 18 16 20 5 7 - 5338 z + 3896 z - 1495 z - 28 z + 295 z + z + 72 z - 436 z 9 11 13 / 3 21 22 20 + 1072 z - 1072 z + 436 z ) / (-12 z - 1 + 2 z + z - 47 z / 19 14 4 6 17 16 - 12 z + 14248 z + 2 z - 680 z + 4376 z - 230 z - 4376 z 9 2 10 8 5 13 12 - 8300 z + 47 z + 25176 z - 14248 z - 230 z - 8300 z - 25176 z 15 11 7 18 + 2438 z + 12284 z + 2438 z + 680 z ) And in Maple-input format, it is: -(4*z^17+1-28*z^2-4*z^3+295*z^4-1495*z^6-72*z^15+3896*z^8-5338*z^10+3896*z^12-\ 1495*z^14-28*z^18+295*z^16+z^20+72*z^5-436*z^7+1072*z^9-1072*z^11+436*z^13)/(-\ 12*z^3-1+2*z^21+z^22-47*z^20-12*z^19+14248*z^14+2*z-680*z^4+4376*z^6-230*z^17-\ 4376*z^16-8300*z^9+47*z^2+25176*z^10-14248*z^8-230*z^5-8300*z^13-25176*z^12+ 2438*z^15+12284*z^11+2438*z^7+680*z^18) The first , 40, terms are: [2, 23, 124, 920, 5874, 40281, 266744, 1797259, 12005168, 80542247, 539161228, 3613294253, 24201269658, 162144069480, 1086171265040, 7276613944339, 48746448767490, 326561955283269, 2187678891833504, 14655611637954621, 98180014042034054, 657722754540074043, 4406180822586799048, 29517660575648830344, 197743157882525168062, 1324710692028464536485, 8874432580801619093372, 59451136237655512811503, 398271947733726118265744, 2668082656995987816216259, 17873880090272721679252496, 119739764913550370135356833, 802154384741887890937566694, 5373750798395958938068166904, 35999550940040556602312600028, 241166313193458202010820145199, 1615608781117914525981566597766, 10823202043276897847457478340105, 72506230367368716177255357362496, 485729955064940257299094677672953] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 14 f(z) = - (-1 + 21 z + 4 z - 142 z + 366 z - 366 z + 142 z - 21 z + z 5 7 9 11 / 16 14 13 12 - 36 z + 64 z - 36 z + 4 z ) / (z - 39 z + 12 z + 385 z / 11 10 9 8 7 6 5 - 116 z - 1412 z + 268 z + 2156 z - 268 z - 1412 z + 116 z 4 3 2 + 385 z - 12 z - 39 z + 1) And in Maple-input format, it is: -(-1+21*z^2+4*z^3-142*z^4+366*z^6-366*z^8+142*z^10-21*z^12+z^14-36*z^5+64*z^7-\ 36*z^9+4*z^11)/(z^16-39*z^14+12*z^13+385*z^12-116*z^11-1412*z^10+268*z^9+2156*z ^8-268*z^7-1412*z^6+116*z^5+385*z^4-12*z^3-39*z^2+1) The first , 40, terms are: [0, 18, 8, 459, 448, 12113, 18016, 323766, 638144, 8740307, 21165168, 238046859, 674692288, 6535158950, 20953518608, 180690754009, 639030844352, 5027315030387, 19234727992072, 140638959486914, 573345177481216, 3952907775932185, 16964372131062304, 111550176169418281, 499104057180740864, 3158633819379348706, 14619192161826664216, 89695175566611260387, 426723962208588880192, 2553150114050602506249, 12421600374650217383120, 72819369152183685551686, 360793890120235501704000, 2080333538348981064553179, 10461230754943310855184720, 59512756041549382628887299, 302899612043259605989988672, 1704409863334390888778429974, 8760444052258817044911097984, 48858282303841818995632132513] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-70 z - 1 + 28 z + z + 4 z - 304 z + 1665 z + 450 z 8 10 12 14 18 16 20 - 5013 z + 8596 z - 8596 z + 5013 z + 304 z - 1665 z - 28 z 5 19 7 9 11 13 / 24 23 - 70 z + 4 z + 450 z - 1328 z + 1890 z - 1328 z ) / (z + z / 22 20 19 18 17 16 15 - 45 z + 658 z - 206 z - 4575 z + 1907 z + 17276 z - 7269 z 14 13 12 11 10 9 - 37601 z + 13806 z + 48568 z - 13806 z - 37601 z + 7269 z 8 7 6 5 4 2 + 17276 z - 1907 z - 4575 z + 206 z + 658 z - 45 z - z + 1) And in Maple-input format, it is: -(-70*z^17-1+28*z^2+z^22+4*z^3-304*z^4+1665*z^6+450*z^15-5013*z^8+8596*z^10-\ 8596*z^12+5013*z^14+304*z^18-1665*z^16-28*z^20-70*z^5+4*z^19+450*z^7-1328*z^9+ 1890*z^11-1328*z^13)/(z^24+z^23-45*z^22+658*z^20-206*z^19-4575*z^18+1907*z^17+ 17276*z^16-7269*z^15-37601*z^14+13806*z^13+48568*z^12-13806*z^11-37601*z^10+ 7269*z^9+17276*z^8-1907*z^7-4575*z^6+206*z^5+658*z^4-45*z^2-z+1) The first , 40, terms are: [1, 18, 59, 515, 2376, 16411, 86833, 546298, 3065319, 18540241, 106714512, 634609673, 3693416785, 21803357706, 127509261199, 750326850715, 4397264808712, 25839710483435, 151571704244917, 890143528629218, 5223531534182631, 30668420886088465, 179999515790402800, 1056692224938515089, 6202424164557591353, 36409676984980727234, 213719515461279640475, 1254555942311896629435, 7364167672491180333464, 43228022711438124423643, 253747485202113294885377, 1489503885457568114615498, 8743377123007743994577935, 51323742293592500162711609, 301270360531906346866399280, 1768459701215495147571532833, 10380863645382502683031781209, 60935741057787068160186143546, 357693060774249606965742781631, 2099660412640995168927751249755] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 28 25 26 2 24 f(z) = - (-12612 z - 1 - 36 z - 44 z + 539 z + 36 z - 4437 z 23 22 21 3 4 6 15 + 406 z + 22362 z - 2072 z + 2 z - 539 z + 4437 z + 15762 z 8 10 12 14 18 16 - 22362 z + 72795 z - 157699 z + 231214 z + 157699 z - 231214 z 20 5 19 7 9 11 13 - 72795 z - 44 z + 6440 z + 406 z - 2072 z + 6440 z - 12612 z 30 27 / 17 28 25 26 + z + 2 z ) / (201696 z - z + 1 + 1071 z + 2061 z - 10748 z / 29 2 24 23 22 21 3 - 10 z - 55 z + 64183 z - 14524 z - 245059 z + 55897 z + 10 z 4 6 15 8 10 12 + 1071 z - 10748 z - 201696 z + 64183 z - 245059 z + 623246 z 14 18 16 20 5 - 1081751 z - 1081751 z + 1298228 z + 623246 z + 95 z 19 7 9 11 13 30 - 132293 z - 2061 z + 14524 z - 55897 z + 132293 z - 55 z 27 31 32 - 95 z + z + z ) And in Maple-input format, it is: -(-12612*z^17-1-36*z^28-44*z^25+539*z^26+36*z^2-4437*z^24+406*z^23+22362*z^22-\ 2072*z^21+2*z^3-539*z^4+4437*z^6+15762*z^15-22362*z^8+72795*z^10-157699*z^12+ 231214*z^14+157699*z^18-231214*z^16-72795*z^20-44*z^5+6440*z^19+406*z^7-2072*z^ 9+6440*z^11-12612*z^13+z^30+2*z^27)/(201696*z^17-z+1+1071*z^28+2061*z^25-10748* z^26-10*z^29-55*z^2+64183*z^24-14524*z^23-245059*z^22+55897*z^21+10*z^3+1071*z^ 4-10748*z^6-201696*z^15+64183*z^8-245059*z^10+623246*z^12-1081751*z^14-1081751* z^18+1298228*z^16+623246*z^20+95*z^5-132293*z^19-2061*z^7+14524*z^9-55897*z^11+ 132293*z^13-55*z^30-95*z^27+z^31+z^32) The first , 40, terms are: [1, 20, 63, 621, 2764, 21085, 109925, 746084, 4203579, 26972765, 157923880, 985928385, 5883425645, 36239056540, 218296898851, 1335710045433, 8083541740244, 49299795410137, 299041637640905, 1820848814339916, 11057392824782183, 67274256433332309, 408762275176242640, 2485971635962124781, 15109081249351472185, 91871140311892624172, 558444766793559842823, 3395311762714437188577, 20640015336262340982828, 125484159039667796452465, 762840418309437961010349, 4637697774184542506719708, 28193848355856312790903571, 171402874679781267218979017, 1042013953412538492574524024, 6334827675077850801121907477, 38511636580464344420237452005, 234127283221321268759146835844, 1423344580281555875698733831851, 8653055731202488740731314221093] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (1 - 22 z - 4 z + 165 z - 530 z + 772 z - 530 z + 165 z 14 16 5 7 9 11 13 / 4 - 22 z + z + 48 z - 156 z + 156 z - 48 z + 4 z ) / (-442 z / 6 12 11 14 13 2 10 + 1831 z - 1831 z + 1120 z + 442 z - 226 z + 42 z + 3548 z 16 9 17 18 7 5 8 - 42 z - 1808 z + 2 z + z + 1120 z - 226 z - 3548 z - 1 + 2 z) And in Maple-input format, it is: -(1-22*z^2-4*z^3+165*z^4-530*z^6+772*z^8-530*z^10+165*z^12-22*z^14+z^16+48*z^5-\ 156*z^7+156*z^9-48*z^11+4*z^13)/(-442*z^4+1831*z^6-1831*z^12+1120*z^11+442*z^14 -226*z^13+42*z^2+3548*z^10-42*z^16-1808*z^9+2*z^17+z^18+1120*z^7-226*z^5-3548*z ^8-1+2*z) The first , 40, terms are: [2, 24, 128, 987, 6288, 44271, 295264, 2028136, 13707502, 93472653, 634277856, 4315696549, 29320583154, 199366985032, 1354988316800, 9211438791383, 62612178253232, 425621578487251, 2893143378337312, 19666474117412664, 133683374273771230, 908722786297703049, 6177086884469660608, 41989137765146609465, 285423465910356386530, 1940182960349365815032, 13188504000793961434528, 89649624302246086412579, 609398475021400985245392, 4142421427109530911045959, 28158348582188151419474112, 191408002303947047260912008, 1301106947062010052279389646, 8844349646950240089557747765, 60119977532287922657633525664, 408669020247498474385770730621, 2777951270113123131640147006674, 18883284225129637391868834921000, 128360215262105679109031131630816, 872535977770091067616097647708415] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {3, 6}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 8 10 4 6 2 -20 z + z - 107 z + 107 z + 20 z - 1 f(z) = - --------------------------------------------------- 12 8 10 4 6 2 z + 344 z - 43 z + 344 z - 632 z - 43 z + 1 And in Maple-input format, it is: -(-20*z^8+z^10-107*z^4+107*z^6+20*z^2-1)/(z^12+344*z^8-43*z^10+344*z^4-632*z^6-\ 43*z^2+1) The first , 40, terms are: [0, 23, 0, 752, 0, 24949, 0, 828331, 0, 27503171, 0, 913200557, 0, 30321488176, 0, 1006781130223, 0, 33428713921529, 0, 1109952211172681, 0, 36854361813908479, 0, 1223695914755062960, 0, 40631057451622430333, 0, 1349095645277687098067, 0, 44794774595760010960027, 0, 1487345866184511530750245, 0, 49385173731114664835753840, 0, 1639763447024777928933774215, 0, 54445979614011365573239662385, 0, 1807800205272274080708489678289] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (2 z - 1 + 18 z - 30 z - 107 z + 259 z - 259 z + 107 z - 18 z 14 5 7 9 11 13 / 4 + z + 136 z - 218 z + 136 z - 30 z + 2 z ) / (1 - 3 z + 309 z / 5 7 6 9 10 8 12 - 393 z + 869 z - 1090 z - 869 z - 1090 z + 1642 z + 309 z 14 16 15 13 11 3 2 - 35 z + z + 3 z - 66 z + 393 z + 66 z - 35 z ) And in Maple-input format, it is: -(2*z-1+18*z^2-30*z^3-107*z^4+259*z^6-259*z^8+107*z^10-18*z^12+z^14+136*z^5-218 *z^7+136*z^9-30*z^11+2*z^13)/(1-3*z+309*z^4-393*z^5+869*z^7-1090*z^6-869*z^9-\ 1090*z^10+1642*z^8+309*z^12-35*z^14+z^16+3*z^15-66*z^13+393*z^11+66*z^3-35*z^2) The first , 40, terms are: [1, 20, 59, 609, 2520, 20025, 98149, 683556, 3670911, 23843545, 134578416, 842042729, 4883053057, 29943219780, 176213493339, 1068852773001, 6340472661512, 38233376773649, 227784345105157, 1369179816442228, 8176343794227199, 49062104952714993, 293356746169012960, 1758640957575122961, 10522664262273913857, 63050272702134910900, 377395975901789046651, 2260681844264326451377, 13534348851799840584632, 81061577910968999992809, 485356037172124100277029, 2906719941061286326616836, 17405010171875376635153151, 104231290334425828384037705, 624141601192889405211793104, 3737633510068978498805852921, 22381504362170278346959094145, 134028546221257049253199856484, 802590383265703820524013259291, 4806168333825307030533620604121] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-80 z - 1 + 30 z + z + 4 z - 351 z + 2065 z + 572 z 8 10 12 14 18 16 20 - 6622 z + 11851 z - 11851 z + 6622 z + 351 z - 2065 z - 30 z 5 19 7 9 11 13 / 17 - 80 z + 4 z + 572 z - 1840 z + 2752 z - 1840 z ) / (2946 z / 2 24 23 22 21 3 4 - 2 z + 1 - 50 z + z + 2 z - 50 z - 16 z + 16 z + 792 z 6 15 8 10 12 14 - 5806 z - 12912 z + 22596 z - 49766 z + 64354 z - 49766 z 18 16 20 5 19 7 9 - 5806 z + 22596 z + 792 z + 214 z - 214 z - 2946 z + 12912 z 11 13 - 26162 z + 26162 z ) And in Maple-input format, it is: -(-80*z^17-1+30*z^2+z^22+4*z^3-351*z^4+2065*z^6+572*z^15-6622*z^8+11851*z^10-\ 11851*z^12+6622*z^14+351*z^18-2065*z^16-30*z^20-80*z^5+4*z^19+572*z^7-1840*z^9+ 2752*z^11-1840*z^13)/(2946*z^17-2*z+1-50*z^2+z^24+2*z^23-50*z^22-16*z^21+16*z^3 +792*z^4-5806*z^6-12912*z^15+22596*z^8-49766*z^10+64354*z^12-49766*z^14-5806*z^ 18+22596*z^16+792*z^20+214*z^5-214*z^19-2946*z^7+12912*z^9-26162*z^11+26162*z^ 13) The first , 40, terms are: [2, 24, 128, 983, 6264, 43935, 292816, 2005492, 13534982, 92076665, 623687664, 4234506533, 28713421866, 194835564428, 1321562305440, 8965948923667, 60821386148552, 412612907791211, 2799077404544400, 18988683114838208, 128816209770232686, 873873381434474269, 5928232652096950112, 40216350453951921317, 272822195216735237458, 1850789154064174890672, 12555501989590865862736, 85174830350958853448963, 577814505540369638372024, 3919815457869053275172331, 26591497323326832910824192, 180393119567742356456786780, 1223762504766882845480027094, 8301839211563016538448664141, 56318553550702808545308125136, 382057444939269690296414670289, 2591825995962889877074412490042, 17582596765171312624301411469220, 119277956712478235139942328279888, 809165514563591576739152362679623] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 3 4 6 15 8 f(z) = - (2 z + 2 z - 1 + 21 z - 36 z - 163 z + 607 z - 36 z - 1160 z 10 12 14 18 16 5 7 9 + 1160 z - 607 z + 163 z + z - 21 z + 230 z - 668 z + 944 z 11 13 / 12 14 4 3 - 668 z + 230 z ) / (5647 z - 4 z - 2163 z + 1 + 420 z + 78 z / 5 6 7 8 11 13 17 - 550 z - 2163 z + 1848 z + 5647 z + 3312 z - 1848 z - 78 z 16 18 19 20 15 9 10 2 + 420 z - 37 z + 4 z + z + 550 z - 3312 z - 7752 z - 37 z ) And in Maple-input format, it is: -(2*z^17+2*z-1+21*z^2-36*z^3-163*z^4+607*z^6-36*z^15-1160*z^8+1160*z^10-607*z^ 12+163*z^14+z^18-21*z^16+230*z^5-668*z^7+944*z^9-668*z^11+230*z^13)/(5647*z^12-\ 4*z-2163*z^14+1+420*z^4+78*z^3-550*z^5-2163*z^6+1848*z^7+5647*z^8+3312*z^11-\ 1848*z^13-78*z^17+420*z^16-37*z^18+4*z^19+z^20+550*z^15-3312*z^9-7752*z^10-37*z ^2) The first , 40, terms are: [2, 24, 128, 987, 6292, 44279, 295520, 2029216, 13721138, 93554429, 635011176, 4320747685, 29360372910, 199653016272, 1357129298280, 9226950298767, 62725693427708, 426441722262419, 2899074036822936, 19709102140682216, 133989288835399918, 910910379777307049, 6192698809916282640, 42100276607977890585, 286213140868928295186, 1945782938633928344264, 13228149770906504259512, 89929851220580663673763, 611376321238164715898660, 4156362053904773931678367, 28256483851681479350418568, 192098012323433561416085744, 1305953221107662695284298738, 8878352258778613386464299093, 60358317135071375630089284664, 410338128618257739873700265677, 2789630123854201924269640577966, 18964935713979944131615225955200, 128930636180612635016935516035024, 876518075151921442685482199900647] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 28 25 26 2 24 f(z) = - (-10684 z - 1 - 36 z - 44 z + 541 z + 36 z - 4479 z 23 22 21 3 4 6 15 + 394 z + 22741 z - 1916 z + 2 z - 541 z + 4479 z + 13182 z 8 10 12 14 18 16 - 22741 z + 74659 z - 163005 z + 240125 z + 163005 z - 240125 z 20 5 19 7 9 11 13 - 74659 z - 44 z + 5658 z + 394 z - 1916 z + 5658 z - 10684 z 30 27 / 17 28 25 26 + z + 2 z ) / (192901 z - z + 1 + 1097 z + 2111 z - 11028 z / 29 2 24 23 22 21 3 - 10 z - 56 z + 65944 z - 14525 z - 252400 z + 54723 z + 10 z 4 6 15 8 10 12 + 1097 z - 11028 z - 192901 z + 65944 z - 252400 z + 643940 z 14 18 16 20 5 - 1120558 z - 1120558 z + 1346120 z + 643940 z + 101 z 19 7 9 11 13 30 - 127533 z - 2111 z + 14525 z - 54723 z + 127533 z - 56 z 27 31 32 - 101 z + z + z ) And in Maple-input format, it is: -(-10684*z^17-1-36*z^28-44*z^25+541*z^26+36*z^2-4479*z^24+394*z^23+22741*z^22-\ 1916*z^21+2*z^3-541*z^4+4479*z^6+13182*z^15-22741*z^8+74659*z^10-163005*z^12+ 240125*z^14+163005*z^18-240125*z^16-74659*z^20-44*z^5+5658*z^19+394*z^7-1916*z^ 9+5658*z^11-10684*z^13+z^30+2*z^27)/(192901*z^17-z+1+1097*z^28+2111*z^25-11028* z^26-10*z^29-56*z^2+65944*z^24-14525*z^23-252400*z^22+54723*z^21+10*z^3+1097*z^ 4-11028*z^6-192901*z^15+65944*z^8-252400*z^10+643940*z^12-1120558*z^14-1120558* z^18+1346120*z^16+643940*z^20+101*z^5-127533*z^19-2111*z^7+14525*z^9-54723*z^11 +127533*z^13-56*z^30-101*z^27+z^31+z^32) The first , 40, terms are: [1, 21, 65, 675, 2951, 23512, 121337, 851955, 4788883, 31553785, 185408915, 1182620133, 7110656960, 44604436629, 271400369229, 1687897982937, 10333566895861, 63982496746579, 392957471746975, 2427522826204056, 14933464208182177, 92143723016335251, 567323930730967015, 3498418063259378309, 21548999564414015183, 132840692276324744913, 818435880699622191808, 5044500477615483818993, 31082959155206544761553, 191566455160934346646501, 1180456008690941850542377, 7274918290572271426715891, 44830333168441161737692783, 276274357354379479533282072, 1702516755553574690534244625, 10491920641941164007955198611, 64656086365244282864544863067, 398446879305335146036272972761, 2455425172297528906555122536371, 15131653414386714275464862479157] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 15 8 10 f(z) = - (-1 + 23 z + 4 z - 194 z + 767 z + 4 z - 1511 z + 1511 z 12 14 18 16 5 7 9 11 - 767 z + 194 z + z - 23 z - 52 z + 224 z - 368 z + 224 z 13 / 20 19 18 17 16 15 14 - 52 z ) / (z + 2 z - 42 z - 2 z + 488 z - 220 z - 2456 z / 13 12 11 10 9 8 7 + 1446 z + 6150 z - 3386 z - 8254 z + 3386 z + 6150 z - 1446 z 6 5 4 3 2 - 2456 z + 220 z + 488 z + 2 z - 42 z - 2 z + 1) And in Maple-input format, it is: -(-1+23*z^2+4*z^3-194*z^4+767*z^6+4*z^15-1511*z^8+1511*z^10-767*z^12+194*z^14+z ^18-23*z^16-52*z^5+224*z^7-368*z^9+224*z^11-52*z^13)/(z^20+2*z^19-42*z^18-2*z^ 17+488*z^16-220*z^15-2456*z^14+1446*z^13+6150*z^12-3386*z^11-8254*z^10+3386*z^9 +6150*z^8-1446*z^7-2456*z^6+220*z^5+488*z^4+2*z^3-42*z^2-2*z+1) The first , 40, terms are: [2, 23, 124, 916, 5850, 39949, 264328, 1775267, 11838576, 79215119, 529190380, 3537967961, 23644682242, 158048791332, 1056358360256, 7060728140811, 47193150320306, 315437021961189, 2108357843009312, 14092140562078525, 94190940709730102, 629566385428587587, 4207981463315211576, 28125882495479459652, 187991612101338930870, 1256524030666620587745, 8398526934189259876604, 56135221785077742597143, 375204263709618730280016, 2507841516451230597876699, 16762253724569608111299680, 112037841381645417133615941, 748853829889270243769045726, 5005291530724579059877389044, 33455051315157284840357852540, 223611442340977613309308307359, 1494604706305439250107472520422, 9989843116858966767925950404713, 66771478156199377687454878790720, 446296327481323924862278106801689] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {4, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 5 7 f(z) = - (2 z - 1 + 13 z - 18 z - 51 z + 51 z - 13 z + z + 34 z - 18 z 9 / 12 11 10 9 8 7 6 + 2 z ) / (z + 3 z - 30 z - 45 z + 186 z + 123 z - 318 z / 5 4 3 2 - 123 z + 186 z + 45 z - 30 z - 3 z + 1) And in Maple-input format, it is: -(2*z-1+13*z^2-18*z^3-51*z^4+51*z^6-13*z^8+z^10+34*z^5-18*z^7+2*z^9)/(z^12+3*z^ 11-30*z^10-45*z^9+186*z^8+123*z^7-318*z^6-123*z^5+186*z^4+45*z^3-30*z^2-3*z+1) The first , 40, terms are: [1, 20, 63, 609, 2720, 20265, 105945, 703924, 3964615, 24991577, 145802688, 896848937, 5319891977, 32350425172, 193399947447, 1169771397561, 7018944658720, 42346960884369, 254531756866641, 1533835873937972, 9226782497822351, 55570737891151153, 334412626351387776, 2013563819470952017, 12119352549880539409, 72964067247630250868, 439197083216268919503, 2644016293993595243313, 15915915485224379453792, 95813035184974150560153, 576766617318380935273833, 3472062977848891304981140, 20900990953112450612887063, 125820606818471370360533513, 757413124603280956747605312, 4559492580496787840177482361, 27447221948087491862654301785, 165227188325627343674407855732, 994635104238408096875685592647, 5987515275038141783713201182537] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 8 7 6 5 4 3 2 / 10 f(z) = - (z + z - 14 z - 7 z + 42 z + 7 z - 14 z - z + 1) / (z / 9 8 7 6 5 4 3 2 + 3 z - 36 z - 11 z + 169 z - 12 z - 169 z - 11 z + 36 z + 3 z - 1 ) And in Maple-input format, it is: -(z^8+z^7-14*z^6-7*z^5+42*z^4+7*z^3-14*z^2-z+1)/(z^10+3*z^9-36*z^8-11*z^7+169*z ^6-12*z^5-169*z^4-11*z^3+36*z^2+3*z-1) The first , 40, terms are: [2, 28, 152, 1315, 8752, 67323, 476880, 3538612, 25573406, 187601521, 1364581088, 9973577025, 72697205746, 530708843940, 3870917193200, 28247988665467, 206081318010128, 1503693146496515, 10970853082638536, 80046792290830956, 584029506022594030, 4261208991596791185, 31090435314353921216, 226841799403283884337, 1655076425827509027122, 12075740335942635174476, 88106724706409535355320, 642842508932877183214851, 4690293091294548793049968, 34221217287734794590614075, 249684096960288545183794640, 1821739750032771664886274116, 13291738040163838184923379438, 96978892491842274312342973025, 707575296982361447618524740768, 5162595601855757935999513950929, 37667218418813027574279073143106, 274826745211425481679297965362452, 2005184958509202620752911329307760, 14630187164609757001800296823198395] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 8 7 6 5 4 3 2 / 10 f(z) = - (z + 3 z - 7 z - 16 z + 14 z + 16 z - 7 z - 3 z + 1) / (z / 9 8 7 6 5 4 3 2 + 6 z - 21 z - 42 z + 89 z + 68 z - 89 z - 42 z + 21 z + 6 z - 1) And in Maple-input format, it is: -(z^8+3*z^7-7*z^6-16*z^5+14*z^4+16*z^3-7*z^2-3*z+1)/(z^10+6*z^9-21*z^8-42*z^7+ 89*z^6+68*z^5-89*z^4-42*z^3+21*z^2+6*z-1) The first , 40, terms are: [3, 32, 229, 1845, 14320, 112485, 880163, 6895792, 54003765, 422983905, 3312866080, 25947198337, 203223953179, 1591695681488, 12466511517581, 97640484615909, 764741896529104, 5989627994067061, 46912093390144139, 367425909133064576, 2877761124002870925, 22539262700246222113, 176532499098916036672, 1382641644200638958689, 10829155685321086186003, 84816346555990188238784, 664300417515471653618389, 5202948047521218011848053, 40750641835283028983859376, 319168055267960300180798309, 2499794921397538006954142035, 19578947660656599519048845968, 153346655846641313847184603141, 1201044982953882474872676371393, 9406850401232837172800814506336, 73676536455397835337567501968289, 577050956752980388793647989462507, 4519590940476871955485103016368112, 35398437573312479907810743706259389, 277248405692991486235105886058721765] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (z - 1 + 19 z - 13 z - 120 z + 304 z - 304 z + 120 z - 19 z 14 5 7 9 11 13 / 15 3 13 + z + 51 z - 78 z + 51 z - 13 z + z ) / (3 z + 32 z - 32 z / 2 9 6 8 10 5 7 4 - 38 z - 16 z - 1231 z + 1852 z - 1231 z - 77 z + 16 z + 350 z 14 16 12 11 - 38 z + z + 1 - 3 z + 350 z + 77 z ) And in Maple-input format, it is: -(z-1+19*z^2-13*z^3-120*z^4+304*z^6-304*z^8+120*z^10-19*z^12+z^14+51*z^5-78*z^7 +51*z^9-13*z^11+z^13)/(3*z^15+32*z^3-32*z^13-38*z^2-16*z^9-1231*z^6+1852*z^8-\ 1231*z^10-77*z^5+16*z^7+350*z^4-38*z^14+z^16+1-3*z+350*z^12+77*z^11) The first , 40, terms are: [2, 25, 132, 1052, 6698, 48177, 323640, 2258469, 15427120, 106648241, 732326644, 5047684197, 34718533506, 239081307948, 1645284049600, 11326566532781, 77958721806562, 536638928335117, 3693779822585120, 25425874549465365, 175013593560769126, 1204682832234039077, 8292220144246498184, 57078232618784332172, 392888500684234562006, 2704385629448962379037, 18615197151959923818276, 128134718314397872232633, 881994563818868612860560, 6071067309156984578834653, 41789209843865036551947488, 287649277018383145666507065, 1979987303135013604557622190, 13628922713149302934347316412, 93812487002618904378659642308, 645743096077633016742362183265, 4444868253905796400081539162054, 30595532397879786628929179863737, 210599403322650586311503172880960, 1449626962484126320960842931560905] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 5 8 7 10 / f(z) = - (-1 + 21 z + 4 z - 110 z + 110 z - 8 z - 21 z + 4 z + z ) / / 7 6 9 8 2 10 12 5 (-32 z - 742 z + 1 + 16 z + 375 z - 45 z - 45 z + z + 32 z 4 3 + 375 z - 16 z ) And in Maple-input format, it is: -(-1+21*z^2+4*z^3-110*z^4+110*z^6-8*z^5-21*z^8+4*z^7+z^10)/(-32*z^7-742*z^6+1+ 16*z^9+375*z^8-45*z^2-45*z^10+z^12+32*z^5+375*z^4-16*z^3) The first , 40, terms are: [0, 24, 12, 815, 900, 28499, 48300, 1008300, 2275560, 36026533, 100197528, 1298815165, 4236113256, 47206114404, 174381420228, 1728217854251, 7046989304076, 63675483798743, 281009086080612, 2359162152867552, 11095589446612560, 87824838367314409, 434850308011100208, 3282793023409700761, 16945005970811542608, 123129495004105868496, 657373037012552525436, 4631619547729445132807, 25413715338924067786260, 174641789310449061041915, 979782339060104834591772, 6598276967064019310529396, 37691442401592322400075064, 249706471659317370620621581, 1447437890875832740205629896, 9462803014400518432225145173, 55507585220935713275948766648, 359000674426666422314955674268, 2126275492439948218807667629620, 13632288721823629253693979388547] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 8 7 6 4 10 5 3 9 2 f(z) = - (-16 z - 20 z + 67 z - 67 z + z + 36 z - 20 z + 2 z + 16 z / 11 8 7 6 12 4 10 + 2 z - 1) / (2 z + 248 z + 182 z - 430 z + z + 248 z - 40 z / 5 3 9 2 - 182 z + 56 z - 56 z - 40 z - 2 z + 1) And in Maple-input format, it is: -(-16*z^8-20*z^7+67*z^6-67*z^4+z^10+36*z^5-20*z^3+2*z^9+16*z^2+2*z-1)/(2*z^11+ 248*z^8+182*z^7-430*z^6+z^12+248*z^4-40*z^10-182*z^5+56*z^3-56*z^9-40*z^2-2*z+1 ) The first , 40, terms are: [0, 24, 12, 803, 888, 27635, 47052, 962904, 2187096, 33900769, 94957488, 1204755553, 3957244248, 43176465624, 160551988452, 1559004026387, 6394249581672, 56662719949955, 251293205309604, 2071158246099096, 9779166260271792, 76074354821502721, 377750943747994464, 2805756180121456129, 14509355133751993008, 103839503287877193624, 554863359946993908924, 3854140944660614101187, 21146419711563353637720, 143394122837554578215123, 803745157353780538966140, 5345552533241043187721688, 30484248250752375673679880, 199599242143562533919267425, 1154246693896322358864912144, 7462819953469812589405852897, 43645253277072458478479729160, 279331092344543750165123797080, 1648573155413788801065674300820, 10464553914180483205690307751731] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 15 8 10 f(z) = - (-z + 1 - 23 z + 16 z + 171 z - 527 z + z + 758 z - 527 z 12 14 16 5 7 9 11 13 / + 171 z - 23 z + z - 97 z + 228 z - 228 z + 97 z - 16 z ) / / 18 17 16 15 14 13 12 11 (z + 2 z - 45 z - 32 z + 489 z + 237 z - 2083 z - 719 z 10 9 8 7 6 5 4 + 4159 z + 1023 z - 4159 z - 719 z + 2083 z + 237 z - 489 z 3 2 - 32 z + 45 z + 2 z - 1) And in Maple-input format, it is: -(-z+1-23*z^2+16*z^3+171*z^4-527*z^6+z^15+758*z^8-527*z^10+171*z^12-23*z^14+z^ 16-97*z^5+228*z^7-228*z^9+97*z^11-16*z^13)/(z^18+2*z^17-45*z^16-32*z^15+489*z^ 14+237*z^13-2083*z^12-719*z^11+4159*z^10+1023*z^9-4159*z^8-719*z^7+2083*z^6+237 *z^5-489*z^4-32*z^3+45*z^2+2*z-1) The first , 40, terms are: [1, 24, 77, 884, 4116, 35605, 197769, 1497896, 9092592, 64517391, 409643972, 2813712915, 18276011767, 123504060544, 811457669841, 5438913723903, 35942578101060, 239920976196695, 1590124892078576, 10592311944209744, 70305848888722539, 467840910752411801, 3107563695102010212, 20668017407357104748, 137335364417943533987, 913158964088653237728, 6068919662008719402131, 40347590025676552155169, 268178284026645976805696, 1782792173967219354986737, 11850246259639048717165813, 78775258074178062182105744, 523632854985571345580651661, 3480823237191712532149724028, 23137916977040967923986025932, 153806831218556253222199647793, 1022399314782354135799302332125, 6796260946518479506625311414048, 45176886598052077473074781033968, 300306582505999792445853183402423] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 15 8 10 f(z) = - (-z + 1 - 23 z + 12 z + 182 z - 613 z + z + 906 z - 613 z 12 14 16 5 7 9 11 13 / + 182 z - 23 z + z - 42 z + 41 z - 41 z + 42 z - 12 z ) / ( / 18 17 16 15 14 13 12 11 z + 3 z - 43 z - 32 z + 499 z + 37 z - 2323 z + 364 z 10 9 8 7 6 5 4 3 + 4862 z - 808 z - 4862 z + 364 z + 2323 z + 37 z - 499 z - 32 z 2 + 43 z + 3 z - 1) And in Maple-input format, it is: -(-z+1-23*z^2+12*z^3+182*z^4-613*z^6+z^15+906*z^8-613*z^10+182*z^12-23*z^14+z^ 16-42*z^5+41*z^7-41*z^9+42*z^11-12*z^13)/(z^18+3*z^17-43*z^16-32*z^15+499*z^14+ 37*z^13-2323*z^12+364*z^11+4862*z^10-808*z^9-4862*z^8+364*z^7+2323*z^6+37*z^5-\ 499*z^4-32*z^3+43*z^2+3*z-1) The first , 40, terms are: [2, 26, 144, 1169, 7864, 58061, 409084, 2951394, 21039362, 150900093, 1078946160, 7726736565, 55289613526, 395793570130, 2832718884876, 20276093483365, 145124803764904, 1038749784932121, 7434883527998936, 53215786789500266, 380894963060378806, 2726281987606410345, 19513533135684650592, 139669395753334969561, 999692634641804264954, 7155364870723096774858, 51214984907364899381688, 366574564807990010138825, 2623781098518132665631000, 18779882652312824513565685, 134418222287883583399571780, 962107104536117913703368050, 6886343710127363885828126874, 49289449685009775369058931589, 352792418128076329277436184656, 2525134508544060225273010092845, 18073813262769277753374984703726, 129364485244045055614540001619138, 925934654675697538407615988386004, 6627437067603174930928126115093437] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 15 8 10 f(z) = - (-z + 1 - 27 z + 20 z + 250 z - 977 z + z + 1570 z - 977 z 12 14 16 5 7 9 11 13 + 250 z - 27 z + z - 130 z + 329 z - 329 z + 130 z - 20 z ) / 18 17 16 15 14 13 12 / (z + 3 z - 51 z - 48 z + 691 z + 245 z - 3667 z / 11 10 9 8 7 6 5 - 404 z + 8310 z + 280 z - 8310 z - 404 z + 3667 z + 245 z 4 3 2 - 691 z - 48 z + 51 z + 3 z - 1) And in Maple-input format, it is: -(-z+1-27*z^2+20*z^3+250*z^4-977*z^6+z^15+1570*z^8-977*z^10+250*z^12-27*z^14+z^ 16-130*z^5+329*z^7-329*z^9+130*z^11-20*z^13)/(z^18+3*z^17-51*z^16-48*z^15+691*z ^14+245*z^13-3667*z^12-404*z^11+8310*z^10+280*z^9-8310*z^8-404*z^7+3667*z^6+245 *z^5-691*z^4-48*z^3+51*z^2+3*z-1) The first , 40, terms are: [2, 30, 164, 1485, 10112, 80649, 587664, 4507222, 33570410, 254258493, 1907345584, 14386932773, 108178241902, 814883211110, 6131988437208, 46170556291377, 347520940454528, 2616264869480293, 19693991709928644, 148256495468863502, 1116034647972771286, 8401383797154295897, 63243907244199231584, 476090582194888281641, 3583923654500083603162, 26979189981832500709934, 203094653450928277737740, 1528862411729792363597237, 11509014243386672355037568, 86637908575342560087545953, 652195398363812820561978616, 4909616119983128446357177158, 36958754034826338525952603458, 278219213749037985622802229973, 2094386883985781402492312952336, 15766188106608826381107491829645, 118685180872940174541285578507846, 893441845604557592749414901010038, 6725678170298165505413254348924992, 50629760757922499811689786419901145] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (3 z - 1 + 17 z - 45 z - 92 z + 200 z - 200 z + 92 z - 17 z 14 5 7 9 11 13 / 16 15 + z + 205 z - 328 z + 205 z - 45 z + 3 z ) / (z + 4 z / 14 13 12 11 10 9 8 - 39 z - 107 z + 319 z + 683 z - 995 z - 1553 z + 1429 z 7 6 5 4 3 2 + 1553 z - 995 z - 683 z + 319 z + 107 z - 39 z - 4 z + 1) And in Maple-input format, it is: -(3*z-1+17*z^2-45*z^3-92*z^4+200*z^6-200*z^8+92*z^10-17*z^12+z^14+205*z^5-328*z ^7+205*z^9-45*z^11+3*z^13)/(z^16+4*z^15-39*z^14-107*z^13+319*z^12+683*z^11-995* z^10-1553*z^9+1429*z^8+1553*z^7-995*z^6-683*z^5+319*z^4+107*z^3-39*z^2-4*z+1) The first , 40, terms are: [1, 26, 81, 1004, 4552, 41881, 229313, 1821682, 11022448, 81186365, 518049848, 3667888213, 24074846031, 166950179560, 1112327497457, 7629583936509, 51237384087504, 349418043228293, 2356412857298976, 16020800431822058, 108281095044613999, 734996277744799945, 4973502497664945280, 33730654922203955948, 228386753319972654183, 1548238185022699797154, 10486391356731580156167, 71070532231811272986929, 481451991921831019147616, 3262585249758840925092177, 22103701084506992563684105, 149776959087597436571515090, 1014773523443972088690971337, 6875968506851379464203165548, 46587466158069920671696559952, 315664524547169487701572845993, 2138783629982261567469177333681, 14491698033129829098930441391802, 98189138338270676371677926642176, 665294021398706570609800316596149] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 28 25 26 2 24 23 f(z) = - (1644 z - 1 - 39 z - 32 z + 623 z + 39 z - 5415 z + 180 z 22 21 3 4 6 15 8 + 28542 z - 380 z + 2 z - 623 z + 5415 z - 2642 z - 28542 z 10 12 14 18 16 + 96194 z - 213373 z + 316601 z + 213373 z - 316601 z 20 5 19 7 9 11 13 30 - 96194 z - 32 z - 92 z + 180 z - 380 z - 92 z + 1644 z + z 27 / 17 28 25 26 29 + 2 z ) / (145155 z - z + 1 + 1323 z + 2761 z - 14081 z - 3 z / 2 24 23 22 21 3 4 - 63 z + 88267 z - 15406 z - 350346 z + 49558 z + 3 z + 1323 z 6 15 8 10 12 14 - 14081 z - 145155 z + 88267 z - 350346 z + 916605 z - 1618487 z 18 16 20 5 19 7 - 1618487 z + 1953573 z + 916605 z + 217 z - 102523 z - 2761 z 9 11 13 30 27 31 32 + 15406 z - 49558 z + 102523 z - 63 z - 217 z + z + z ) And in Maple-input format, it is: -(1644*z^17-1-39*z^28-32*z^25+623*z^26+39*z^2-5415*z^24+180*z^23+28542*z^22-380 *z^21+2*z^3-623*z^4+5415*z^6-2642*z^15-28542*z^8+96194*z^10-213373*z^12+316601* z^14+213373*z^18-316601*z^16-96194*z^20-32*z^5-92*z^19+180*z^7-380*z^9-92*z^11+ 1644*z^13+z^30+2*z^27)/(145155*z^17-z+1+1323*z^28+2761*z^25-14081*z^26-3*z^29-\ 63*z^2+88267*z^24-15406*z^23-350346*z^22+49558*z^21+3*z^3+1323*z^4-14081*z^6-\ 145155*z^15+88267*z^8-350346*z^10+916605*z^12-1618487*z^14-1618487*z^18+1953573 *z^16+916605*z^20+217*z^5-102523*z^19-2761*z^7+15406*z^9-49558*z^11+102523*z^13 -63*z^30-217*z^27+z^31+z^32) The first , 40, terms are: [1, 25, 83, 955, 4601, 39891, 228317, 1741232, 10845627, 77792671, 505235859, 3517382351, 23320790017, 160000672869, 1071690038607, 7300128399197, 49142584486496, 333569528276485, 2251063377074217, 15253230820756949, 103060441446405399, 697742047456886135, 4717212226853006533, 31923138665918095719, 215885905295616512253, 1460678065648008645056, 9879530787542788986843, 66837817016923662454859, 452100654345219115979407, 3058435141097325120070323, 20688425889957069973393141, 139952562980264048320184313, 946709114729168126852878535, 6404197014522605290149481177, 43321560556499498402177895808, 293055325088081548166672315785, 1982398000195998264308480382185, 13410195068905960901180769637593, 90714607784573187318060301441403, 613650145122305073701385475651683] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (-2 z + 1 - 18 z + 26 z + 103 z - 204 z + 103 z - 18 z + z 5 7 9 11 / 14 13 12 11 - 100 z + 100 z - 26 z + 2 z ) / (z + 4 z - 39 z - 60 z / 10 9 8 7 6 5 4 3 + 353 z + 276 z - 1079 z - 408 z + 1079 z + 276 z - 353 z - 60 z 2 + 39 z + 4 z - 1) And in Maple-input format, it is: -(-2*z+1-18*z^2+26*z^3+103*z^4-204*z^6+103*z^8-18*z^10+z^12-100*z^5+100*z^7-26* z^9+2*z^11)/(z^14+4*z^13-39*z^12-60*z^11+353*z^10+276*z^9-1079*z^8-408*z^7+1079 *z^6+276*z^5-353*z^4-60*z^3+39*z^2+4*z-1) The first , 40, terms are: [2, 29, 160, 1401, 9574, 74525, 540800, 4074341, 30083018, 224534465, 1666176352, 12402635733, 92167793518, 685543914681, 5096610302464, 37900098072905, 281798500784370, 2095412596118629, 15580556009489568, 115852587207193393, 861436921932198166, 6405365762798295253, 47628064416253224320, 354146302124329886125, 2633310501457696292474, 19580404347412502876681, 145593209532693345629536, 1082581618722874281541517, 8049708298720826613281118, 59854892574096154336925553, 445060609534320995115916288, 3309319257071748916695752849, 24606971808726896121788016482, 182969069084670845333060362413, 1360495734190774846187270712992, 10116183319556910897505166121193, 75220496710192377017568293684038, 559314016787426224513392890944653, 4158868699444164627233912292914816, 30923932427177904123621430312482293] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 14 f(z) = - (-1 + 25 z + 6 z - 196 z + 542 z - 542 z + 196 z - 25 z + z 5 7 9 11 / 7 9 8 16 - 84 z + 210 z - 84 z + 6 z ) / (-1854 z + 1854 z + 2997 z + z / 3 12 15 14 13 11 4 6 + 3 z + 608 z + 3 z - 52 z - 3 z - 468 z + 608 z + 1 - 2132 z 5 2 10 + 468 z - 3 z - 52 z - 2132 z ) And in Maple-input format, it is: -(-1+25*z^2+6*z^3-196*z^4+542*z^6-542*z^8+196*z^10-25*z^12+z^14-84*z^5+210*z^7-\ 84*z^9+6*z^11)/(-1854*z^7+1854*z^9+2997*z^8+z^16+3*z^3+608*z^12+3*z^15-52*z^14-\ 3*z^13-468*z^11+608*z^4+1-2132*z^6+468*z^5-3*z-52*z^2-2132*z^10) The first , 40, terms are: [3, 36, 255, 2216, 17592, 145541, 1180911, 9671280, 78851208, 644281231, 5258776080, 42945466471, 350623480641, 2862977614416, 23375928658167, 190867741333139, 1558439562363336, 12724784759826959, 103898537177197632, 848338450526978976, 6926734039967587773, 56557219312878730729, 461793163551145350840, 3770569859846212608136, 30786935619732937914537, 251377235277726913317180, 2052510664929226927096569, 16758876548063993863929281, 136837263352493493908757168, 1117284718518302893871434865, 9122698823686763544353622591, 74487400100282535959413696716, 608194228491712972750720459575, 4965943489783082045082161819656, 40547235714410389249629224626344, 331070687270718059820584176766593, 2703212636761072277824310711985243, 22071898360489773084619584220269312, 180218415158956944764478134258509008, 1471494505455831885690478652936490335] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 28 25 26 2 24 23 f(z) = - (16408 z + 1 + z + 4 z - 38 z - 38 z + 587 z - 100 z 22 21 3 4 6 15 8 - 4868 z + 1020 z - 4 z + 587 z - 4868 z - 28528 z + 24017 z 10 12 14 18 16 20 - 73782 z + 143987 z - 179808 z - 73782 z + 143987 z + 24017 z 5 19 7 9 11 13 / + 100 z - 5444 z - 1020 z + 5444 z - 16408 z + 28528 z ) / ( / 17 28 25 26 29 2 -287810 z + 2 z - 1 - 63 z - 178 z + 1289 z + 2 z + 63 z 24 23 22 21 3 4 - 13075 z + 4744 z + 76413 z - 35086 z - 24 z - 1289 z 6 15 8 10 12 14 + 13075 z + 371584 z - 76413 z + 275899 z - 638485 z + 966375 z 18 16 20 5 19 7 + 638485 z - 966375 z - 275899 z - 178 z + 132624 z + 4744 z 9 11 13 30 27 - 35086 z + 132624 z - 287810 z + z - 24 z ) And in Maple-input format, it is: -(16408*z^17+1+z^28+4*z^25-38*z^26-38*z^2+587*z^24-100*z^23-4868*z^22+1020*z^21 -4*z^3+587*z^4-4868*z^6-28528*z^15+24017*z^8-73782*z^10+143987*z^12-179808*z^14 -73782*z^18+143987*z^16+24017*z^20+100*z^5-5444*z^19-1020*z^7+5444*z^9-16408*z^ 11+28528*z^13)/(-287810*z^17+2*z-1-63*z^28-178*z^25+1289*z^26+2*z^29+63*z^2-\ 13075*z^24+4744*z^23+76413*z^22-35086*z^21-24*z^3-1289*z^4+13075*z^6+371584*z^ 15-76413*z^8+275899*z^10-638485*z^12+966375*z^14+638485*z^18-966375*z^16-275899 *z^20-178*z^5+132624*z^19+4744*z^7-35086*z^9+132624*z^11-287810*z^13+z^30-24*z^ 27) The first , 40, terms are: [2, 29, 156, 1389, 9254, 72741, 518776, 3916217, 28588698, 212855289, 1566474740, 11607508393, 85664049358, 633714652629, 4681424080400, 34611671955453, 255772931343186, 1890653731371825, 13973185465363628, 103281491198932193, 763350202248336678, 5642094321736957697, 41701137039447489832, 308219945230977570173, 2278087817736554356378, 16837671206491852207045, 124449312568809152630532, 919821633986114073742165, 6798519691963643274250910, 50248756955812903690234745, 371395086771504705354285216, 2745029797342097944984137929, 20288873122998797485933908898, 149957715795506320178695656805, 1108357056858503716367632084028, 8192011898559717638804700444949, 60548230079757465670635840887014, 447519881416972037022475625334413, 3307677914111301639836596911514904, 24447479690836943056572391558843697] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-72 z - 1 + 33 z + z + 4 z - 413 z + 2537 z + 472 z 8 10 12 14 18 16 20 - 8330 z + 15038 z - 15038 z + 8330 z + 413 z - 2537 z - 33 z 5 19 7 9 11 13 / 17 - 72 z + 4 z + 472 z - 1416 z + 2024 z - 1416 z ) / (3326 z / 2 24 23 22 21 3 4 - 2 z + 1 - 60 z + z + 2 z - 60 z - 10 z + 10 z + 1018 z 6 15 8 10 12 14 - 7756 z - 13692 z + 31019 z - 69736 z + 91092 z - 69736 z 18 16 20 5 19 7 9 - 7756 z + 31019 z + 1018 z + 294 z - 294 z - 3326 z + 13692 z 11 13 - 26908 z + 26908 z ) And in Maple-input format, it is: -(-72*z^17-1+33*z^2+z^22+4*z^3-413*z^4+2537*z^6+472*z^15-8330*z^8+15038*z^10-\ 15038*z^12+8330*z^14+413*z^18-2537*z^16-33*z^20-72*z^5+4*z^19+472*z^7-1416*z^9+ 2024*z^11-1416*z^13)/(3326*z^17-2*z+1-60*z^2+z^24+2*z^23-60*z^22-10*z^21+10*z^3 +1018*z^4-7756*z^6-13692*z^15+31019*z^8-69736*z^10+91092*z^12-69736*z^14-7756*z ^18+31019*z^16+1018*z^20+294*z^5-294*z^19-3326*z^7+13692*z^9-26908*z^11+26908*z ^13) The first , 40, terms are: [2, 31, 168, 1571, 10654, 86961, 635680, 4958209, 37211762, 285709859, 2164240856, 16525622143, 125589231822, 957118571569, 7282104797248, 55459496513233, 422125356903122, 3214083342787007, 24467176294539048, 186278736653384995, 1418115107683166638, 10796380651266894945, 82192829075890464096, 625743238716631290193, 4763811337775521031810, 36267299759624813463843, 276105170229526185515096, 2102009414315615267186655, 16002738194149084928751518, 121829994500034323993852769, 927500137205059216596490368, 7061124014714479991092924577, 53756828320107230573219495842, 409254506135713131179584810079, 3115683125581301404251072055720, 23719913807031697172084741504803, 180581361777570553545900790716862, 1374778542917667014481245928081937, 10466285182826383289011409895816096, 79680561258607614790960155597813665] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 28 25 26 2 24 f(z) = - (-68032 z - 1 - 39 z - 160 z + 627 z + 39 z - 5491 z 23 22 21 3 4 6 15 + 1736 z + 29130 z - 9980 z + 6 z - 627 z + 5491 z + 86050 z 8 10 12 14 18 16 - 29130 z + 98622 z - 219357 z + 325873 z + 219357 z - 325873 z 20 5 19 7 9 11 - 98622 z - 160 z + 33408 z + 1736 z - 9980 z + 33408 z 13 30 27 / 17 28 25 - 68032 z + z + 6 z ) / (994729 z - 3 z + 1 + 1383 z + 7511 z / 26 29 2 24 23 22 - 14671 z - 41 z - 65 z + 90531 z - 61106 z - 351058 z 21 3 4 6 15 8 + 256038 z + 41 z + 1383 z - 14671 z - 994729 z + 90531 z 10 12 14 18 16 - 351058 z + 896485 z - 1554373 z - 1554373 z + 1863505 z 20 5 19 7 9 11 + 896485 z + 219 z - 637233 z - 7511 z + 61106 z - 256038 z 13 30 27 31 32 + 637233 z - 65 z - 219 z + 3 z + z ) And in Maple-input format, it is: -(-68032*z^17-1-39*z^28-160*z^25+627*z^26+39*z^2-5491*z^24+1736*z^23+29130*z^22 -9980*z^21+6*z^3-627*z^4+5491*z^6+86050*z^15-29130*z^8+98622*z^10-219357*z^12+ 325873*z^14+219357*z^18-325873*z^16-98622*z^20-160*z^5+33408*z^19+1736*z^7-9980 *z^9+33408*z^11-68032*z^13+z^30+6*z^27)/(994729*z^17-3*z+1+1383*z^28+7511*z^25-\ 14671*z^26-41*z^29-65*z^2+90531*z^24-61106*z^23-351058*z^22+256038*z^21+41*z^3+ 1383*z^4-14671*z^6-994729*z^15+90531*z^8-351058*z^10+896485*z^12-1554373*z^14-\ 1554373*z^18+1863505*z^16+896485*z^20+219*z^5-637233*z^19-7511*z^7+61106*z^9-\ 256038*z^11+637233*z^13-65*z^30-219*z^27+3*z^31+z^32) The first , 40, terms are: [3, 35, 253, 2155, 17267, 141621, 1151087, 9389524, 76482489, 623345465, 5079202489, 41390718919, 337282517283, 2748473096787, 22396820038129, 182508199456949, 1487229277442552, 12119192071009141, 98757327589222823, 804757483292899747, 6557838307509396853, 53438762282613837935, 435463816647147557735, 3548524104046594427753, 28916348094632453703631, 235634636528094606462596, 1920148482811493208885113, 15646978944300414966041797, 127504696781419039584788669, 1039015119748929106150151779, 8466765902059415648327220891, 68994303815089907470740711523, 562223405487898839682964428213, 4581467457453998117090948716737, 37333636164598448545194048658704, 304225753476215601332215948674913, 2479086383927748306580208248464283, 20201673358519642872167199336115299, 164620163754735237131775128147460949, 1341463047822576200964649053349405139] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (1 - 24 z - 8 z + 195 z - 652 z + 944 z - 652 z + 195 z 14 16 5 7 9 11 13 / - 24 z + z + 100 z - 336 z + 336 z - 100 z + 8 z ) / ((1 + z) / 16 15 14 13 12 11 10 9 (z + 2 z - 47 z + 14 z + 509 z - 412 z - 1886 z + 1580 z 8 7 6 5 4 3 2 + 2734 z - 1580 z - 1886 z + 412 z + 509 z - 14 z - 47 z - 2 z + 1) (-1 + z)) And in Maple-input format, it is: -(1-24*z^2-8*z^3+195*z^4-652*z^6+944*z^8-652*z^10+195*z^12-24*z^14+z^16+100*z^5 -336*z^7+336*z^9-100*z^11+8*z^13)/(1+z)/(z^16+2*z^15-47*z^14+14*z^13+509*z^12-\ 412*z^11-1886*z^10+1580*z^9+2734*z^8-1580*z^7-1886*z^6+412*z^5+509*z^4-14*z^3-\ 47*z^2-2*z+1)/(-1+z) The first , 40, terms are: [2, 28, 156, 1319, 9024, 68555, 493872, 3654220, 26675050, 196056365, 1436010368, 10536400501, 77240246894, 566485679148, 4153709801576, 30460216121507, 223359646188992, 1637906638324623, 12010667644216316, 88074138946569404, 645844578351701718, 4735964762021033209, 34728696713816533760, 254664683455818897801, 1867449430646762774234, 13693959313867383770236, 100417449971048298701108, 736358590802175634429759, 5399698613978490320489536, 39595851431603655355612019, 290355361687262612001231560, 2129168415126342139437146476, 15613137325221547488556027202, 114490735225982058846483574373, 839557622274434563651978520192, 6156454492367666644887175264829, 45145122750794660318119291418694, 331048026227282762064236499984140, 2427566678065776613800096788333056, 17801284132903278310217951547730427] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-26 z - 1 + 32 z + z + 2 z - 384 z + 2246 z + 124 z 8 10 12 14 18 16 20 - 7044 z + 12353 z - 12353 z + 7044 z + 384 z - 2246 z - 32 z 5 19 7 9 11 13 / 15 - 26 z + 2 z + 124 z - 296 z + 394 z - 296 z ) / (-4764 z / 17 23 24 18 20 19 10 + 1378 z + z + z - 7007 z + 961 z - 168 z - 58951 z 11 16 14 4 12 13 2 - 8591 z + 26958 z - 58951 z + 961 z + 76201 z + 8591 z - 59 z 6 8 22 9 5 7 - 7007 z + 26958 z - 59 z + 4764 z + 168 z - 1378 z - z + 1) And in Maple-input format, it is: -(-26*z^17-1+32*z^2+z^22+2*z^3-384*z^4+2246*z^6+124*z^15-7044*z^8+12353*z^10-\ 12353*z^12+7044*z^14+384*z^18-2246*z^16-32*z^20-26*z^5+2*z^19+124*z^7-296*z^9+ 394*z^11-296*z^13)/(-4764*z^15+1378*z^17+z^23+z^24-7007*z^18+961*z^20-168*z^19-\ 58951*z^10-8591*z^11+26958*z^16-58951*z^14+961*z^4+76201*z^12+8591*z^13-59*z^2-\ 7007*z^6+26958*z^8-59*z^22+4764*z^9+168*z^5-1378*z^7-z+1) The first , 40, terms are: [1, 28, 85, 1160, 5072, 51197, 272317, 2341560, 13941944, 109573831, 696631104, 5201144463, 34352373691, 249022453920, 1681576765221, 11984116627251, 81972109006888, 578473779155007, 3986390754840224, 27972210956738584, 193596791038645039, 1353990003343580793, 9394486545589171088, 65578712245147340760, 455669100074136273339, 3177315943977171424212, 22095885455144695267371, 153973121130923339876441, 1071289579165787367757680, 7462422336605586795646297, 51935455520484833487340109, 361696184463347511434657444, 2517669845882874284515164213, 17531736566028372138817287608, 122045210666993027779313514032, 849798059140692389093598766241, 5916096550069794339208280860553, 41191935822670326518712879998872, 286777748316051439080305093499056, 1996696018224086352010729186692511] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 15 8 10 f(z) = - (-1 + 25 z + 4 z - 219 z + 883 z + 4 z - 1768 z + 1768 z 12 14 18 16 5 7 9 11 - 883 z + 219 z + z - 25 z - 40 z + 152 z - 248 z + 152 z 13 / 2 18 17 16 15 14 13 - 40 z ) / ((-1 + z ) (z + 2 z - 48 z + 8 z + 540 z - 250 z / 12 11 10 9 8 7 6 - 2511 z + 1168 z + 5428 z - 1968 z - 5428 z + 1168 z + 2511 z 5 4 3 2 - 250 z - 540 z + 8 z + 48 z + 2 z - 1)) And in Maple-input format, it is: -(-1+25*z^2+4*z^3-219*z^4+883*z^6+4*z^15-1768*z^8+1768*z^10-883*z^12+219*z^14+z ^18-25*z^16-40*z^5+152*z^7-248*z^9+152*z^11-40*z^13)/(-1+z^2)/(z^18+2*z^17-48*z ^16+8*z^15+540*z^14-250*z^13-2511*z^12+1168*z^11+5428*z^10-1968*z^9-5428*z^8+ 1168*z^7+2511*z^6-250*z^5-540*z^4+8*z^3+48*z^2+2*z-1) The first , 40, terms are: [2, 28, 156, 1327, 9072, 69291, 499488, 3710236, 27131322, 200027861, 1468434656, 10804085613, 79398950846, 583856767132, 4291997295064, 31556216665523, 231991773299280, 1705611124944423, 12539410325624412, 92189108409338524, 677765235177840758, 4982880139694436073, 36633702964598717888, 269328060049110062617, 1980077436187352369274, 14557367879818680311708, 107024567592157466377620, 786835845551255449658423, 5784752410267752340932912, 42529024666820041429421987, 312669893688815875865828888, 2298723370528598646764412828, 16900025319086024194618492722, 124247597537509049515307948861, 913458127482301019484374026912, 6715669095618873352033947275525, 49373047363427992959478629740854, 362986587263571850315158451200540, 2668647563033301761326329415068880, 19619677601803804685653690186810555] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 3 4 6 15 8 f(z) = - (6 z + 1 - 30 z - 6 z + 330 z - 1702 z - 106 z + 4441 z 10 12 14 18 16 20 5 - 6080 z + 4441 z - 1702 z - 30 z + 330 z + z + 106 z 7 9 11 13 / 5 17 22 - 622 z + 1474 z - 1474 z + 622 z ) / (-406 z - 406 z + z / 10 11 7 9 16 15 + 30865 z + 15868 z + 3476 z - 10980 z + 2 z - 5555 z + 3476 z 3 12 2 14 13 4 8 - 2 z - 30865 z + 58 z + 17706 z - 10980 z - 871 z - 1 - 17706 z 6 21 18 20 19 + 5555 z + 2 z + 871 z - 58 z - 2 z ) And in Maple-input format, it is: -(6*z^17+1-30*z^2-6*z^3+330*z^4-1702*z^6-106*z^15+4441*z^8-6080*z^10+4441*z^12-\ 1702*z^14-30*z^18+330*z^16+z^20+106*z^5-622*z^7+1474*z^9-1474*z^11+622*z^13)/(-\ 406*z^5-406*z^17+z^22+30865*z^10+15868*z^11+3476*z^7-10980*z^9+2*z-5555*z^16+ 3476*z^15-2*z^3-30865*z^12+58*z^2+17706*z^14-10980*z^13-871*z^4-1-17706*z^8+ 5555*z^6+2*z^21+871*z^18-58*z^20-2*z^19) The first , 40, terms are: [2, 32, 172, 1655, 11180, 93175, 682640, 5407180, 40833194, 317450673, 2424540760, 18718998353, 143570993374, 1105607042580, 8493113304840, 65340822581927, 502232361378180, 3862490013810855, 29694911001629772, 228342630810555544, 1755645654255697606, 13499583061883773345, 103796647988882828848, 798102773777039840161, 6136583859964606760906, 47184481536104023064104, 362801319184773533707492, 2789589263308142954651591, 21449175832333757806797276, 164923142726986544922250695, 1268096197874124042343153000, 9750413622981847248407893764, 74971073846067481395607485618, 576453809018856758134983706193, 4432362177591050537917181066504, 34080503370745490592274823839793, 262045520890958999852283999608806, 2014872146528490211914369384584668, 15492383469343524882451822583360352, 119121180233870394549519870607561943] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {3, 6}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 4 2 z - 14 z + 1 f(z) = - ----------------------- 4 2 6 -47 z + 47 z + z - 1 And in Maple-input format, it is: -(z^4-14*z^2+1)/(-47*z^4+47*z^2+z^6-1) The first , 40, terms are: [0, 33, 0, 1505, 0, 69185, 0, 3180993, 0, 146256481, 0, 6724617121, 0, 309186131073, 0, 14215837412225, 0, 653619334831265, 0, 30052273564825953, 0, 1381750964647162561, 0, 63530492100204651841, 0, 2921020885644766822113, 0, 134303430247559069165345, 0, 6175036770502072414783745, 0, 283917388012847772010886913, 0, 13054024811820495440086014241, 0, 600201223955729942471945768161, 0, 27596202277151756858269419321153, 0, 1268825103525025085537921343004865] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {4, 5}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 15 8 10 f(z) = - (-z + 1 - 27 z + 20 z + 252 z - 977 z + z + 1566 z - 977 z 12 14 16 5 7 9 11 13 + 252 z - 27 z + z - 130 z + 321 z - 321 z + 130 z - 20 z ) / 18 17 16 15 14 13 12 / (z + 3 z - 53 z - 50 z + 713 z + 239 z - 3699 z / 11 10 9 8 7 6 5 - 362 z + 8246 z + 276 z - 8246 z - 362 z + 3699 z + 239 z 4 3 2 - 713 z - 50 z + 53 z + 3 z - 1) And in Maple-input format, it is: -(-z+1-27*z^2+20*z^3+252*z^4-977*z^6+z^15+1566*z^8-977*z^10+252*z^12-27*z^14+z^ 16-130*z^5+321*z^7-321*z^9+130*z^11-20*z^13)/(z^18+3*z^17-53*z^16-50*z^15+713*z ^14+239*z^13-3699*z^12-362*z^11+8246*z^10+276*z^9-8246*z^8-362*z^7+3699*z^6+239 *z^5-713*z^4-50*z^3+53*z^2+3*z-1) The first , 40, terms are: [2, 32, 172, 1651, 11152, 92743, 679104, 5370000, 40516682, 314533149, 2399708384, 18502756165, 141749519694, 1090200116592, 8364812136856, 64274083933823, 493437607862448, 3790198356151035, 29103792828829356, 223523764060050112, 1716505439853677462, 13182527995877589913, 101235473760268718528, 777460990508921333225, 5970590276395710143386, 45852216010917835293824, 352128123833161862748676, 2704224390399930901766507, 20767477388341346392825808, 159487030101600266135042447, 1224804089902733319773950232, 9406067882490920004878346928, 72235295418578544696475758050, 554741786416820213879635513461, 4260222264675160642841083473568, 32717014424717115109341999378157, 251255197564641763031691666835174, 1929551857582493400988039281643856, 14818281762091596914693141374839280, 113799209723342215412961144234925623] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (3 z - 1 + 17 z - 45 z - 100 z + 252 z - 252 z + 100 z - 17 z 14 5 7 9 11 13 / 16 15 + z + 213 z - 372 z + 213 z - 45 z + 3 z ) / (z + 6 z / 14 13 12 11 10 9 8 - 33 z - 105 z + 337 z + 607 z - 1385 z - 1409 z + 2269 z 7 6 5 4 3 2 + 1409 z - 1385 z - 607 z + 337 z + 105 z - 33 z - 6 z + 1) And in Maple-input format, it is: -(3*z-1+17*z^2-45*z^3-100*z^4+252*z^6-252*z^8+100*z^10-17*z^12+z^14+213*z^5-372 *z^7+213*z^9-45*z^11+3*z^13)/(z^16+6*z^15-33*z^14-105*z^13+337*z^12+607*z^11-\ 1385*z^10-1409*z^9+2269*z^8+1409*z^7-1385*z^6-607*z^5+337*z^4+105*z^3-33*z^2-6* z+1) The first , 40, terms are: [3, 34, 243, 2028, 16000, 128905, 1030355, 8260906, 66154160, 530013357, 4245590672, 34011073405, 272452231133, 2182555155752, 17483894403083, 140059277316861, 1121980441227272, 8987911987852893, 71999965492288864, 576774145520997458, 4620396786340138229, 37012870247738229953, 296501063343487140760, 2375197602113809089740, 19027127872730952841741, 152421674232763084951514, 1221012804965511083080221, 9781235361933160373718713, 78355087526860001259768032, 627683468824647621747599113, 5028218964084553246578388835, 40279834035043017045785725098, 322671912555732396583527372403, 2584845882478183610135517241740, 20706569044834677845538800284120, 165875267270265151852459862339793, 1328786252923215101408368994475163, 10644582130986796172459288818308546, 85271147631199053351422918734278080, 683086337149402039480559075408676989] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 3 4 6 15 8 f(z) = - (2 z + 2 z - 1 + 23 z - 36 z - 194 z + 773 z - 36 z - 1533 z 10 12 14 18 16 5 7 9 + 1533 z - 773 z + 194 z + z - 23 z + 220 z - 600 z + 826 z 11 13 / 20 19 18 17 16 - 600 z + 220 z ) / (z + 5 z - 44 z - 89 z + 554 z / 15 14 13 12 11 10 + 526 z - 3011 z - 1345 z + 7971 z + 1875 z - 10939 z 9 8 7 6 5 4 3 2 - 1875 z + 7971 z + 1345 z - 3011 z - 526 z + 554 z + 89 z - 44 z - 5 z + 1) And in Maple-input format, it is: -(2*z^17+2*z-1+23*z^2-36*z^3-194*z^4+773*z^6-36*z^15-1533*z^8+1533*z^10-773*z^ 12+194*z^14+z^18-23*z^16+220*z^5-600*z^7+826*z^9-600*z^11+220*z^13)/(z^20+5*z^ 19-44*z^18-89*z^17+554*z^16+526*z^15-3011*z^14-1345*z^13+7971*z^12+1875*z^11-\ 10939*z^10-1875*z^9+7971*z^8+1345*z^7-3011*z^6-526*z^5+554*z^4+89*z^3-44*z^2-5* z+1) The first , 40, terms are: [3, 36, 259, 2252, 18096, 150389, 1231479, 10150516, 83420968, 686489395, 5645931560, 46446495191, 382048295845, 3142729575728, 25851467834363, 212651355144083, 1749238232851856, 14389003958105843, 118361938283759480, 973629223367638204, 8008940085440955429, 65880445677141193081, 541923512656354884968, 4457788589673006539956, 36669158063667553847029, 301635470394382459659660, 2481212049854650467125833, 20410110366007323729914777, 167890771380659672004888416, 1381046482047113610721822425, 11360299138282841627046279599, 93448264191684474288891696508, 768692617511030073690200820523, 6323160149959390415510764908772, 52013449031654835561418521391688, 427855505160570798627478630000785, 3519480763228980192107144206308787, 28950766539964138284888745890680172, 238145038895579039331708785912882664, 1958948460735649170137533715019234499] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (-2 z + 1 - 16 z + 24 z + 79 z - 136 z + 79 z - 16 z + z 5 7 9 11 / 12 9 14 7 - 86 z + 86 z - 24 z + 2 z ) / (-37 z + 252 z + z - 396 z / 6 5 13 4 11 3 10 2 + 767 z + 252 z + 4 z - 291 z - 54 z - 54 z + 291 z + 37 z 8 + 4 z - 767 z - 1) And in Maple-input format, it is: -(-2*z+1-16*z^2+24*z^3+79*z^4-136*z^6+79*z^8-16*z^10+z^12-86*z^5+86*z^7-24*z^9+ 2*z^11)/(-37*z^12+252*z^9+z^14-396*z^7+767*z^6+252*z^5+4*z^13-291*z^4-54*z^11-\ 54*z^3+291*z^10+37*z^2+4*z-767*z^8-1) The first , 40, terms are: [2, 29, 160, 1393, 9510, 73637, 533168, 3999421, 29442058, 218907497, 1619049696, 12008125669, 88929433198, 659109749929, 4883021060448, 36183922211801, 268096824568562, 1986528301195477, 14719175786090528, 109063602562713849, 808113103634000342, 5987790545044479757, 44366984875176640528, 328740970902062232661, 2435832562076768444858, 18048503566497637777121, 133731857698268582123616, 990897204349421572399917, 7342133948604174536973406, 54402144185536581758986705, 403097146445730347846468288, 2986781372628310395583178801, 22130801484033019855446195810, 163979988629623478702112195341, 1215023172816979762338361115040, 9002813842314372979445817393665, 66707087455703043735856710097414, 494271635026904692501318319396085, 3662346213651363769426803048179056, 27136454610927207383837110832740461] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 6}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 6 5 4 3 2 z + 2 z - 11 z - 12 z + 11 z + 2 z - 1 f(z) = - ------------------------------------------------------------ 8 7 6 5 4 3 2 z + 6 z - 36 z - 18 z + 102 z + 18 z - 36 z - 6 z + 1 And in Maple-input format, it is: -(z^6+2*z^5-11*z^4-12*z^3+11*z^2+2*z-1)/(z^8+6*z^7-36*z^6-18*z^5+102*z^4+18*z^3 -36*z^2-6*z+1) The first , 40, terms are: [4, 49, 432, 4193, 39436, 374897, 3550560, 33671633, 319170068, 3025907777, 28685464848, 271943066833, 2578045069340, 24440175423265, 231695533300416, 2196499833330145, 20823063621425188, 197404977858813073, 1871421288563849328, 17741283431816200577, 168189353705452923436, 1594453909368162852881, 15115601625596374779936, 143297596241604210979889, 1358477260539741844297268, 12878516568557449597960481, 122089779360595841732815824, 1157424781417876577281174129, 10972516550157682436046479420, 104020685729596816546599585601, 986127750201389302306979755392, 9348601510330158828069387721921, 88625789286527972889406074527812, 840182407815850822125786575870449, 7965023319804097940443008270118704, 75509313090649845655579989123539873, 715836744538497696891651239104203340, 6786212506215032529463172082293104817, 64334054560442585192632862507476706016, 609894042132558054310080826772693866385] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 5 z - 1 + 15 z - 2 z - 15 z + z + z f(z) = - -------------------------------------------------------- 7 6 8 4 3 2 5 z - 49 z + z + 108 z + 12 z - 49 z - 12 z - z + 1 And in Maple-input format, it is: -(z-1+15*z^2-2*z^3-15*z^4+z^6+z^5)/(z^7-49*z^6+z^8+108*z^4+12*z^3-49*z^2-12*z^5 -z+1) The first , 40, terms are: [0, 34, 24, 1597, 2376, 76717, 171792, 3731890, 10992792, 183615169, 659961168, 9127963489, 38128578552, 457963344850, 2148190451040, 23161933301869, 118951497832152, 1179551872544701, 6505581570547992, 60422887787418754, 352577708947661520, 3110400376001758369, 18978988587109523616, 160767725529952652257, 1016377518866986264656, 8337508935056674402498, 54215598069552839541192, 433573835784563829007165, 2883174982103381206177128, 22597306853114662321418605, 152964383214184914163185408, 1179866292936237914087097490, 8100420586575126521383966632, 61693567866375861403561842337, 428347583173116822762998167536, 3229641325835073768038186865793, 22625132833382068779992400229896, 169229324405357398385593290443506, 1193979508940823807643921163415120, 8874067423149213843947454448103341] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-34 z - 1 + 34 z + z + 4 z - 418 z + 2481 z + 42 z 8 10 12 14 18 16 20 - 7829 z + 13758 z - 13758 z + 7829 z + 418 z - 2481 z - 34 z 5 19 7 9 11 13 / 17 - 34 z + 4 z + 42 z + 152 z - 326 z + 152 z ) / (1495 z - z / 2 24 23 22 21 3 4 6 + 1 - 67 z + z + z - 67 z + 14 z - 14 z + 1126 z - 8369 z 15 8 10 12 14 18 - 3621 z + 32488 z - 71315 z + 92276 z - 71315 z - 8369 z 16 20 5 19 7 9 11 + 32488 z + 1126 z + 292 z - 292 z - 1495 z + 3621 z - 5312 z 13 + 5312 z ) And in Maple-input format, it is: -(-34*z^17-1+34*z^2+z^22+4*z^3-418*z^4+2481*z^6+42*z^15-7829*z^8+13758*z^10-\ 13758*z^12+7829*z^14+418*z^18-2481*z^16-34*z^20-34*z^5+4*z^19+42*z^7+152*z^9-\ 326*z^11+152*z^13)/(1495*z^17-z+1-67*z^2+z^24+z^23-67*z^22+14*z^21-14*z^3+1126* z^4-8369*z^6-3621*z^15+32488*z^8-71315*z^10+92276*z^12-71315*z^14-8369*z^18+ 32488*z^16+1126*z^20+292*z^5-292*z^19-1495*z^7+3621*z^9-5312*z^11+5312*z^13) The first , 40, terms are: [1, 34, 111, 1695, 8224, 90655, 540301, 5049722, 33707423, 288424817, 2050376432, 16720857217, 123096936697, 977703012842, 7338696210467, 57446322280839, 435852846441408, 3384524757229471, 25831972616575169, 199705673503279122, 1529250705390837815, 11793641716379147033, 90474490060915486416, 696799114209338044617, 5350847475452364950153, 41179316212562185547378, 316399354515779329457167, 2433955105911737032144031, 18706929941964585685734528, 143873306079105903647487031, 1105971616095613889550384845, 8504852941170882978787572266, 65383981834798484167502444327, 502763634264293138666227217249, 3865369075061301486788015984720, 29721223924398140492997790592913, 228510556938125194992202312138721, 1757003879687346439674492885291706, 13508875278018188511497357773754563, 103867700980924764418847828885086511] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 3 4 6 15 8 f(z) = - (8 z + 1 - 32 z - 8 z + 366 z - 1962 z - 112 z + 5301 z 10 12 14 18 16 20 5 - 7348 z + 5301 z - 1962 z - 32 z + 366 z + z + 112 z 7 9 11 13 / 22 21 20 - 560 z + 1272 z - 1272 z + 560 z ) / (z + 2 z - 64 z / 19 18 17 16 15 14 + 12 z + 1001 z - 524 z - 6741 z + 3760 z + 22780 z 13 12 11 10 9 8 - 11470 z - 41145 z + 16520 z + 41145 z - 11470 z - 22780 z 7 6 5 4 3 2 + 3760 z + 6741 z - 524 z - 1001 z + 12 z + 64 z + 2 z - 1) And in Maple-input format, it is: -(8*z^17+1-32*z^2-8*z^3+366*z^4-1962*z^6-112*z^15+5301*z^8-7348*z^10+5301*z^12-\ 1962*z^14-32*z^18+366*z^16+z^20+112*z^5-560*z^7+1272*z^9-1272*z^11+560*z^13)/(z ^22+2*z^21-64*z^20+12*z^19+1001*z^18-524*z^17-6741*z^16+3760*z^15+22780*z^14-\ 11470*z^13-41145*z^12+16520*z^11+41145*z^10-11470*z^9-22780*z^8+3760*z^7+6741*z ^6-524*z^5-1001*z^4+12*z^3+64*z^2+2*z-1) The first , 40, terms are: [2, 36, 204, 2101, 15276, 135159, 1066808, 8989824, 72939834, 604851975, 4952945864, 40853697255, 335567071814, 2762974614632, 22717936429168, 186943420444671, 1537623339060500, 12650450612639837, 104062669212433740, 856096294303368172, 7042517796653867902, 57935687521175703713, 476603194607725436592, 3920775882766296447265, 32254076757726840829762, 265337503266091597996700, 2182789412978079786067876, 17956658103441962542022157, 147719869379487306828302188, 1215213188324408039696377455, 9996913681630868977764426528, 82239310940771611998055504216, 676539180707844465739201190650, 5565529139694529084215788826855, 45784656333737740755111572118872, 376646089003130564663846173044615, 3098467622651060852391821567972486, 25489449994971918878238234556291856, 209688187355708963045357129891705560, 1724993516552029935697792868849231751] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 10 9 8 7 6 5 4 3 2 f(z) = - (z + 2 z - 17 z - 16 z + 85 z + 10 z - 85 z - 16 z + 17 z / 12 11 10 9 8 7 6 + 2 z - 1) / (z + 5 z - 42 z - 45 z + 354 z - 55 z - 614 z / 5 4 3 2 + 55 z + 354 z + 45 z - 42 z - 5 z + 1) And in Maple-input format, it is: -(z^10+2*z^9-17*z^8-16*z^7+85*z^6+10*z^5-85*z^4-16*z^3+17*z^2+2*z-1)/(z^12+5*z^ 11-42*z^10-45*z^9+354*z^8-55*z^7-614*z^6+55*z^5+354*z^4+45*z^3-42*z^2-5*z+1) The first , 40, terms are: [3, 40, 297, 2761, 23352, 205561, 1778919, 15507976, 134769309, 1172774833, 10199669808, 88729145521, 771791150019, 6713568252232, 58398054059673, 507980530588729, 4418696345053992, 38436332124522313, 334340833389252375, 2908285456702014184, 25297906757502115485, 220055469336068158273, 1914166611303057596256, 16650501184083854173249, 144835453111714478285091, 1259860486197319268786536, 10958977303596161359446249, 95327367512508037461257929, 829211225077749662350684056, 7212947066315420263006207033, 62742283035493583607490076391, 545767776244547734293080614600, 4747396033021287002385379889277, 41295529115843108691042620353585, 359211810662982563033968901161104, 3124626991892132251847798005820209, 27179768450373213106209366184615587, 236424960461796219327440975458071688, 2056557694059110475682330008192913305, 17889099106679203301444909396624559097] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 5 7 f(z) = - (2 z - 1 + 18 z - 26 z - 69 z + 69 z - 18 z + z + 64 z - 26 z 9 / 12 11 10 9 8 7 6 + 2 z ) / (z + 4 z - 48 z - 66 z + 304 z + 232 z - 558 z / 5 4 3 2 - 232 z + 304 z + 66 z - 48 z - 4 z + 1) And in Maple-input format, it is: -(2*z-1+18*z^2-26*z^3-69*z^4+69*z^6-18*z^8+z^10+64*z^5-26*z^7+2*z^9)/(z^12+4*z^ 11-48*z^10-66*z^9+304*z^8+232*z^7-558*z^6-232*z^5+304*z^4+66*z^3-48*z^2-4*z+1) The first , 40, terms are: [2, 38, 208, 2289, 16192, 150313, 1173888, 10214758, 83000382, 704112329, 5812747744, 48816850937, 405560984898, 3392400349542, 28254493473440, 235965018389369, 1967267964189120, 16419079771120673, 136942421875270736, 1142651953794253094, 9531738401411165054, 79525164695620045393, 663422448794618555328, 5534843146936437702833, 46174459550597916635010, 385221119674242602776870, 3213741464598155020002288, 26811213094476188007174337, 223675820847189265789741120, 1866050865690547041826719193, 15567781979589903436005576608, 129876550416323131273817873254, 1083513387668475531293906545086, 9039368959391236812219945306265, 75412225744574059543811762108832, 629137426926814628857389309033513, 5248669305279191160427251921500866, 43787781903298220223952066707148646, 365305871351891113378893456127007936, 3047616927629054313501295213908575945] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 3 4 6 15 8 f(z) = - (2 z + 2 z - 1 + 25 z - 38 z - 220 z + 880 z - 38 z - 1749 z 10 12 14 18 16 5 7 9 + 1749 z - 880 z + 220 z + z - 25 z + 260 z - 784 z + 1118 z 11 13 / 11 20 10 5 - 784 z + 260 z ) / (3565 z + z + 1 - 13770 z - 630 z / 12 4 6 19 8 9 7 + 9917 z + 655 z - 3644 z + 5 z + 9917 z - 3565 z + 2076 z 2 3 18 13 15 14 17 - 50 z + 91 z - 5 z - 50 z - 2076 z + 630 z - 3644 z - 91 z 16 + 655 z ) And in Maple-input format, it is: -(2*z^17+2*z-1+25*z^2-38*z^3-220*z^4+880*z^6-38*z^15-1749*z^8+1749*z^10-880*z^ 12+220*z^14+z^18-25*z^16+260*z^5-784*z^7+1118*z^9-784*z^11+260*z^13)/(3565*z^11 +z^20+1-13770*z^10-630*z^5+9917*z^12+655*z^4-3644*z^6+5*z^19+9917*z^8-3565*z^9+ 2076*z^7-50*z^2+91*z^3-5*z-50*z^18-2076*z^13+630*z^15-3644*z^14-91*z^17+655*z^ 16) The first , 40, terms are: [3, 40, 297, 2777, 23500, 207777, 1801483, 15757304, 137281897, 1198179649, 10449111304, 91158190265, 795133656111, 6936128373672, 60503343965229, 527774378359593, 4603776253492820, 40158869709801985, 350306450531306399, 3055730635322903896, 26655195683761175973, 232513802981453893465, 2028222414158164587952, 17692223945753118727657, 154329614996408869269739, 1346220253053176943974200, 11743105593905205559892465, 102435339859129206677056177, 893545473361622012894009964, 7794414644882988195011459225, 67990831425043126116184594883, 593085352615266379763133655944, 5173495133192133351122831088705, 45128499254622338285368390728585, 393656781833829690131583638464024, 3433875808937004915390473271629041, 29953765857308283373677672012306567, 261287285550540857172221614486673304, 2279214103347875997913057027819180261, 19881629211136901090424812811312039217] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-64 z - 1 + 36 z + z + 4 z - 481 z + 3139 z + 404 z 8 10 12 14 18 16 - 10856 z + 20199 z - 20199 z + 10856 z + 481 z - 3139 z 20 5 19 7 9 11 13 / - 36 z - 64 z + 4 z + 404 z - 1256 z + 1888 z - 1256 z ) / ( / 17 2 24 23 22 21 3 4 3838 z - 2 z + 1 - 70 z + z + 2 z - 70 z - 4 z + 4 z + 1276 z 6 15 8 10 12 14 - 10350 z - 15948 z + 43588 z - 101070 z + 133202 z - 101070 z 18 16 20 5 19 7 9 - 10350 z + 43588 z + 1276 z + 370 z - 370 z - 3838 z + 15948 z 11 13 - 31930 z + 31930 z ) And in Maple-input format, it is: -(-64*z^17-1+36*z^2+z^22+4*z^3-481*z^4+3139*z^6+404*z^15-10856*z^8+20199*z^10-\ 20199*z^12+10856*z^14+481*z^18-3139*z^16-36*z^20-64*z^5+4*z^19+404*z^7-1256*z^9 +1888*z^11-1256*z^13)/(3838*z^17-2*z+1-70*z^2+z^24+2*z^23-70*z^22-4*z^21+4*z^3+ 1276*z^4-10350*z^6-15948*z^15+43588*z^8-101070*z^10+133202*z^12-101070*z^14-\ 10350*z^18+43588*z^16+1276*z^20+370*z^5-370*z^19-3838*z^7+15948*z^9-31930*z^11+ 31930*z^13) The first , 40, terms are: [2, 38, 208, 2273, 16096, 148453, 1159200, 10036662, 81440782, 688146041, 5668810592, 47446060021, 393175853730, 3278525560062, 27231769052928, 226743120202761, 1885071634333792, 15686922177070629, 130462890099729872, 1085424126115691182, 9028377752002656814, 75107618794114357469, 624767126493938643776, 5197296065573856158885, 43233561982299687823442, 359645332323338515640734, 2991724995160537204267376, 24887012998863891015196381, 207024364442569423530101536, 1722152714884785192315551297, 14325866914847058976927471296, 119171047914869326072100175086, 991334526176708023138253300254, 8246505233038359146077138194365, 68599270885392961518149349124640, 570649120287657691734460903647665, 4746994703419278537783509583824754, 39488294599643268353154956591880134, 328486848453431303607109530250597024, 2732546801412759379424333422381934125] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 15 8 10 f(z) = - (-z + 1 - 29 z + 20 z + 279 z - 1110 z + z + 1772 z - 1110 z 12 14 16 5 7 9 11 13 + 279 z - 29 z + z - 113 z + 213 z - 213 z + 113 z - 20 z ) / 9 14 6 8 16 4 / (-1640 z + 856 z + 4548 z - 10037 z - 61 z + 4 z - 1 - 856 z / 5 15 2 10 12 7 11 + 164 z - 62 z + 61 z + 10037 z - 4548 z + 612 z + 612 z 13 18 3 17 + 164 z + z - 62 z + 4 z ) And in Maple-input format, it is: -(-z+1-29*z^2+20*z^3+279*z^4-1110*z^6+z^15+1772*z^8-1110*z^10+279*z^12-29*z^14+ z^16-113*z^5+213*z^7-213*z^9+113*z^11-20*z^13)/(-1640*z^9+856*z^14+4548*z^6-\ 10037*z^8-61*z^16+4*z-1-856*z^4+164*z^5-62*z^15+61*z^2+10037*z^10-4548*z^12+612 *z^7+612*z^11+164*z^13+z^18-62*z^3+4*z^17) The first , 40, terms are: [3, 44, 317, 3189, 26848, 248533, 2184475, 19749724, 175820613, 1578496945, 14106244416, 126379573841, 1130684490475, 10123598098236, 90604196815029, 811073634064149, 7259693881204832, 64983913329978613, 581670783554411539, 5206639242989306572, 46605039598286733357, 417167941372065168033, 3734113549070757609728, 33424499146332706046561, 299186409303918783504819, 2678052292586159387473292, 23971549867704935540502285, 214572097419101694282523637, 1920659327684995926379965216, 17192041644249033211417682325, 153887929877524018548805469035, 1377468470653031088351272993660, 12329877813300709515979088578293, 110366146885659314765036923277073, 987900004637125419336927403459008, 8842805950857611194029388829575409, 79152967568406994456238264540401051, 708507267058401386347824638363120412, 6341929592617775641000996468669295621, 56767337234926268498211655637308537941] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 16 14 13 12 11 10 9 f(z) = - (z - 30 z + 8 z + 299 z - 112 z - 1196 z + 408 z 8 7 6 5 4 3 2 / + 1916 z - 408 z - 1196 z + 112 z + 299 z - 8 z - 30 z + 1) / ( / 13 18 14 9 7 6 12 -700 z + z + 918 z - 1 + 4 z - 7136 z + 4136 z + 4573 z - 4573 z 11 2 8 17 10 16 4 5 + 4136 z + 66 z - 9532 z + 4 z + 9532 z - 66 z - 918 z - 700 z ) And in Maple-input format, it is: -(z^16-30*z^14+8*z^13+299*z^12-112*z^11-1196*z^10+408*z^9+1916*z^8-408*z^7-1196 *z^6+112*z^5+299*z^4-8*z^3-30*z^2+1)/(-700*z^13+z^18+918*z^14-1+4*z-7136*z^9+ 4136*z^7+4573*z^6-4573*z^12+4136*z^11+66*z^2-9532*z^8+4*z^17+9532*z^10-66*z^16-\ 918*z^4-700*z^5) The first , 40, terms are: [4, 52, 464, 4669, 45040, 441155, 4296928, 41939724, 409033212, 3990400911, 38924934368, 379714060543, 3704068063540, 36132971106764, 352474317888000, 3438362205112243, 33540971887056912, 327189764026969325, 3191712442141323600, 31134924101025233140, 303718931698260681420, 2962756208127481156273, 28901472465582786485824, 281931773069340116356305, 2750224050063677889904852, 26828236645771652074661236, 261707507609628267241191664, 2552937803701706022512991501, 24903723584715360988033605616, 242934021927658130505870439763, 2369803809023616992392811784128, 23117264715338369924330242328140, 225507244896862107070013893302060, 2199806859815586527198307772586655, 21458956774115678843345179499210400, 209330570899287765052856044124845807, 2042004575258644776777553130042655332, 19919606904351241186814710624899908236, 194314324282852895297121400655323261152, 1895522175854476418477969508699370930851] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (4 z - 1 + 15 z - 58 z - 80 z + 184 z - 184 z + 80 z - 15 z 14 5 7 9 11 13 / 2 14 + z + 274 z - 472 z + 274 z - 58 z + 4 z ) / ((-z - 1 + z ) (z / 13 12 11 10 9 8 7 6 + 9 z - 23 z - 164 z + 154 z + 888 z - 376 z - 1590 z + 376 z 5 4 3 2 + 888 z - 154 z - 164 z + 23 z + 9 z - 1)) And in Maple-input format, it is: -(4*z-1+15*z^2-58*z^3-80*z^4+184*z^6-184*z^8+80*z^10-15*z^12+z^14+274*z^5-472*z ^7+274*z^9-58*z^11+4*z^13)/(-z-1+z^2)/(z^14+9*z^13-23*z^12-164*z^11+154*z^10+ 888*z^9-376*z^8-1590*z^7+376*z^6+888*z^5-154*z^4-164*z^3+23*z^2+9*z-1) The first , 40, terms are: [4, 50, 440, 4309, 40752, 389989, 3717280, 35481162, 338505516, 3230017449, 30819073232, 294064619353, 2805840996948, 26772218002938, 255449642797648, 2437397537495461, 23256662718371280, 221905688564404117, 2117334488031721080, 20202750912433582914, 192766493047922552572, 1839300054770369819409, 17549858574639615715872, 167453665435740621274353, 1597775272550155060037508, 15245326609884414390266018, 145464751792373892828741352, 1387965935758635847749148565, 13243410620708162844163808560, 126363277620959204932380754853, 1205707380706227187942935152016, 11504373068338056249409832486426, 109770083366313002222215657764652, 1047381819997603137338579016930041, 9993694485961815093331147683375216, 95355798212128752724057802719200265, 909846530274145396016470110662267732, 8681388276047244912398586181013275946, 82834302150695581532732284136177275200, 790371469932326942245000674903770670181] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 8 7 6 5 4 3 2 / 10 f(z) = - (z + 2 z - 13 z - 12 z + 32 z + 12 z - 13 z - 2 z + 1) / (z / 9 8 7 6 5 4 3 2 + 4 z - 40 z - 28 z + 168 z + 58 z - 168 z - 28 z + 40 z + 4 z - 1 ) And in Maple-input format, it is: -(z^8+2*z^7-13*z^6-12*z^5+32*z^4+12*z^3-13*z^2-2*z+1)/(z^10+4*z^9-40*z^8-28*z^7 +168*z^6+58*z^5-168*z^4-28*z^3+40*z^2+4*z-1) The first , 40, terms are: [2, 35, 204, 2024, 14986, 129583, 1029168, 8557969, 69403232, 570547157, 4655763324, 38144872259, 311840001586, 2552370789448, 20877329781096, 170827896950551, 1397525131315874, 11434195038330629, 93546387578808000, 765353006782516637, 6261657473827502566, 51229576717707949327, 419131288302083233536, 3429103264998163497352, 28055008901606357718278, 229530605778214845049979, 1877891980622257431115740, 15363873184553256688652525, 125698693812159449493440800, 1028397106220259359139457081, 8413775323123003622447347368, 68836849319578693192315693111, 563184966244299990257490197678, 4607667423253460529774668320424, 37697382362323231174768485006876, 308419099944232269161370492873323, 2523314223871197324448246313440390, 20644359175295659020958647136984873, 168900710662092711366617754237708672, 1381852050935764763943113425834997593] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-60 z - 1 + 35 z + z + 4 z - 451 z + 2806 z + 332 z 8 10 12 14 18 16 20 - 9259 z + 16728 z - 16728 z + 9259 z + 451 z - 2806 z - 35 z 5 19 7 9 11 13 / 17 - 60 z + 4 z + 332 z - 884 z + 1216 z - 884 z ) / (3484 z / 2 24 23 22 21 3 4 - 2 z + 1 - 70 z + z + 2 z - 70 z - 2 z + 2 z + 1217 z 6 15 8 10 12 14 - 9336 z - 13082 z + 37544 z - 84794 z + 110940 z - 84794 z 18 16 20 5 19 7 9 - 9336 z + 37544 z + 1217 z + 378 z - 378 z - 3484 z + 13082 z 11 13 - 24612 z + 24612 z ) And in Maple-input format, it is: -(-60*z^17-1+35*z^2+z^22+4*z^3-451*z^4+2806*z^6+332*z^15-9259*z^8+16728*z^10-\ 16728*z^12+9259*z^14+451*z^18-2806*z^16-35*z^20-60*z^5+4*z^19+332*z^7-884*z^9+ 1216*z^11-884*z^13)/(3484*z^17-2*z+1-70*z^2+z^24+2*z^23-70*z^22-2*z^21+2*z^3+ 1217*z^4-9336*z^6-13082*z^15+37544*z^8-84794*z^10+110940*z^12-84794*z^14-9336*z ^18+37544*z^16+1217*z^20+378*z^5-378*z^19-3484*z^7+13082*z^9-24612*z^11+24612*z ^13) The first , 40, terms are: [2, 39, 212, 2384, 16778, 158323, 1235416, 10881209, 88632736, 759031413, 6293913284, 53280883183, 445081527682, 3749923302128, 31420774202480, 264208081724963, 2216607333324562, 18623678406754685, 156327228004946240, 1313002993973158885, 11023727104111802198, 92576209438094376139, 777321694508877001112, 6527500030956820561584, 54810560899416800001494, 460256654009013994971159, 3864773413573120284031348, 32453058015654049164026717, 272509917645520156606274272, 2288295729386342166874603537, 19214981127848110645676618720, 161350039563676181788532732427, 1354869148060221142185472136910, 11376958359654044360908798460816, 95533268793761762414513484209332, 802201241010961269362569179483711, 6736151998400761159221850300904550, 56564052777018616615941894888000473, 474973194304463433130879686336433792, 3988390928975361352558262238067168681] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 3 4 6 15 8 f(z) = - (2 z + 2 z - 1 + 25 z - 38 z - 225 z + 934 z - 38 z - 1894 z 10 12 14 18 16 5 7 9 + 1894 z - 934 z + 225 z + z - 25 z + 260 z - 798 z + 1146 z 11 13 / 2 18 17 16 15 - 798 z + 260 z ) / ((-1 + z ) (z + 5 z - 50 z - 84 z / 14 13 12 11 10 9 8 + 637 z + 525 z - 3252 z - 1499 z + 7320 z + 2017 z - 7320 z 7 6 5 4 3 2 - 1499 z + 3252 z + 525 z - 637 z - 84 z + 50 z + 5 z - 1)) And in Maple-input format, it is: -(2*z^17+2*z-1+25*z^2-38*z^3-225*z^4+934*z^6-38*z^15-1894*z^8+1894*z^10-934*z^ 12+225*z^14+z^18-25*z^16+260*z^5-798*z^7+1146*z^9-798*z^11+260*z^13)/(-1+z^2)/( z^18+5*z^17-50*z^16-84*z^15+637*z^14+525*z^13-3252*z^12-1499*z^11+7320*z^10+ 2017*z^9-7320*z^8-1499*z^7+3252*z^6+525*z^5-637*z^4-84*z^3+50*z^2+5*z-1) The first , 40, terms are: [3, 41, 307, 2897, 24781, 220944, 1935219, 17080153, 150254237, 1323662505, 11653653973, 102627040641, 903674142768, 7957623323073, 70072177123899, 617037934139209, 5433458793158243, 47845560246762665, 421314673395328397, 3709980838210333456, 32669063653063091379, 287674743222327629505, 2533184194801835366413, 22306519391355636108745, 196425039251767449263693, 1729664563564371580342209, 15230947699296336333345120, 134119512417947657121980225, 1181019327440223522742777811, 10399729515462580864395473673, 91577141439060326082823993843, 806402976327316152659831247201, 7100961550086259231858856204877, 62529103210390622969770834985744, 550613986671101638446836287938163, 4848554461078774728812202185847689, 42695029423733399693935177907918877, 375960619216815729809854917607794313, 3310605218211043064341948039756093541, 29152273803777591493073692893284859457] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 28 25 26 2 24 f(z) = - (-55212 z - 1 - 42 z - 148 z + 715 z + 42 z - 6557 z 23 22 21 3 4 6 15 + 1514 z + 36107 z - 8368 z + 6 z - 715 z + 6557 z + 69626 z 8 10 12 14 18 - 36107 z + 125833 z - 285487 z + 428407 z + 285487 z 16 20 5 19 7 9 - 428407 z - 125833 z - 148 z + 27398 z + 1514 z - 8368 z 11 13 30 27 / 17 + 27398 z - 55212 z + z + 6 z ) / (1087969 z - 3 z + 1 / 28 25 26 29 2 24 + 1717 z + 9223 z - 18992 z - 32 z - 76 z + 121624 z 23 22 21 3 4 6 - 69921 z - 488068 z + 285075 z + 32 z + 1717 z - 18992 z 15 8 10 12 14 - 1087969 z + 121624 z - 488068 z + 1282692 z - 2267506 z 18 16 20 5 19 7 - 2267506 z + 2737208 z + 1282692 z + 409 z - 700645 z - 9223 z 9 11 13 30 27 31 32 + 69921 z - 285075 z + 700645 z - 76 z - 409 z + 3 z + z ) And in Maple-input format, it is: -(-55212*z^17-1-42*z^28-148*z^25+715*z^26+42*z^2-6557*z^24+1514*z^23+36107*z^22 -8368*z^21+6*z^3-715*z^4+6557*z^6+69626*z^15-36107*z^8+125833*z^10-285487*z^12+ 428407*z^14+285487*z^18-428407*z^16-125833*z^20-148*z^5+27398*z^19+1514*z^7-\ 8368*z^9+27398*z^11-55212*z^13+z^30+6*z^27)/(1087969*z^17-3*z+1+1717*z^28+9223* z^25-18992*z^26-32*z^29-76*z^2+121624*z^24-69921*z^23-488068*z^22+285075*z^21+ 32*z^3+1717*z^4-18992*z^6-1087969*z^15+121624*z^8-488068*z^10+1282692*z^12-\ 2267506*z^14-2267506*z^18+2737208*z^16+1282692*z^20+409*z^5-700645*z^19-9223*z^ 7+69921*z^9-285075*z^11+700645*z^13-76*z^30-409*z^27+3*z^31+z^32) The first , 40, terms are: [3, 43, 319, 3127, 26837, 245332, 2174919, 19570483, 174825209, 1567276311, 14026058497, 125629792149, 1124787319704, 10072464225061, 90189978124999, 807610075099903, 7231609450844063, 64754982225958963, 579841073059751265, 5192134731705608900, 46492441102240436499, 416312096488317282471, 3727825472334164445497, 33380449530035651459283, 298901956097736413287669, 2676488302386581233170433, 23966352029299025143521808, 214604350858619802677518305, 1921653622981818565805529627, 17207259031831970463821030627, 154080714411491491107074661847, 1379700655502919991631651028519, 12354394289985449664262343758077, 110626212781873947119717464439396, 990591579520387716619587021827567, 8870155207970088863385444761157715, 79426935418185487896700144514925233, 711220708325350559875927841109925007, 6368556123761597496241032373776638313, 57026611608466768128884238524002637669] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 15 8 10 f(z) = - (-1 + 27 z + 8 z - 246 z + 1009 z + 8 z - 2029 z + 2029 z 12 14 18 16 5 7 9 11 - 1009 z + 246 z + z - 27 z - 100 z + 412 z - 672 z + 412 z 13 / 20 19 18 17 16 15 - 100 z ) / (z + 4 z - 60 z + 4 z + 742 z - 564 z / 14 13 12 11 10 9 - 3730 z + 3492 z + 9196 z - 8132 z - 12174 z + 8132 z 8 7 6 5 4 3 2 + 9196 z - 3492 z - 3730 z + 564 z + 742 z - 4 z - 60 z - 4 z + 1) And in Maple-input format, it is: -(-1+27*z^2+8*z^3-246*z^4+1009*z^6+8*z^15-2029*z^8+2029*z^10-1009*z^12+246*z^14 +z^18-27*z^16-100*z^5+412*z^7-672*z^9+412*z^11-100*z^13)/(z^20+4*z^19-60*z^18+4 *z^17+742*z^16-564*z^15-3730*z^14+3492*z^13+9196*z^12-8132*z^11-12174*z^10+8132 *z^9+9196*z^8-3492*z^7-3730*z^6+564*z^5+742*z^4-4*z^3-60*z^2-4*z+1) The first , 40, terms are: [4, 49, 432, 4188, 39436, 374859, 3552168, 33696383, 319529728, 3030380079, 28738366448, 272542780083, 2584666310796, 24511806547724, 232458683276952, 2204531811442713, 20906768634527780, 198270214391384497, 1880303834805510080, 17831939834567319825, 169109944684790634220, 1603761210643815521545, 15209336297082291674168, 144238374822859309321516, 1367890640601941383043172, 12972447914626320352919459, 123024750591545275217304576, 1166710350872586374566210575, 11064546250138840502814381824, 104931085620302751821044140063, 995118324831050534201596887496, 9437246117874944179304948333787, 89498516977330075773486483904612, 848762916754910752322932354302268, 8049278504143015025680515492796832, 76335668251123568937405776402170049, 723932492129118092775946999624703628, 6865443968292789555715247153490847521, 65108724078876029888869158154062342016, 617461299044488129152168966471585384417] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-152 z - 1 + 35 z + z + 8 z - 455 z + 2842 z + 984 z 8 10 12 14 18 16 20 - 9347 z + 16840 z - 16840 z + 9347 z + 455 z - 2842 z - 35 z 5 19 7 9 11 13 / 17 - 152 z + 8 z + 984 z - 2824 z + 3968 z - 2824 z ) / (7624 z / 2 24 23 22 21 3 4 - 4 z + 1 - 72 z + z + 4 z - 72 z - 20 z + 20 z + 1267 z 6 15 8 10 12 14 - 9512 z - 30516 z + 36776 z - 80388 z + 103920 z - 80388 z 18 16 20 5 19 7 9 - 9512 z + 36776 z + 1267 z + 692 z - 692 z - 7624 z + 30516 z 11 13 - 58920 z + 58920 z ) And in Maple-input format, it is: -(-152*z^17-1+35*z^2+z^22+8*z^3-455*z^4+2842*z^6+984*z^15-9347*z^8+16840*z^10-\ 16840*z^12+9347*z^14+455*z^18-2842*z^16-35*z^20-152*z^5+8*z^19+984*z^7-2824*z^9 +3968*z^11-2824*z^13)/(7624*z^17-4*z+1-72*z^2+z^24+4*z^23-72*z^22-20*z^21+20*z^ 3+1267*z^4-9512*z^6-30516*z^15+36776*z^8-80388*z^10+103920*z^12-80388*z^14-9512 *z^18+36776*z^16+1267*z^20+692*z^5-692*z^19-7624*z^7+30516*z^9-58920*z^11+58920 *z^13) The first , 40, terms are: [4, 53, 472, 4812, 46564, 460031, 4506480, 44300647, 434860976, 4271241951, 41941897864, 411896246807, 4044906117668, 39722544011788, 390087759569248, 3830795660925517, 37619677442788676, 369437853198955321, 3628002807401530528, 35628199834197503433, 349880812900140147148, 3435946410417450558237, 33742140796477184070032, 331359087987696891187852, 3254056869635039214065772, 31955924856770595439221479, 313817850805873462224803816, 3081796065511875553635508975, 30264266244427341588456012560, 297205198480913258257639899991, 2918654273329905115331078200832, 28662159379414917567008547204655, 281471974189239589408423701132204, 2764148758133227740866396789708876, 27144863637297805981338567535148056, 266571623440839559977776899445136741, 2617822339186232566104497807693148684, 25707889351033906593774316235006329713, 252460056204746966078351524419920193856, 2479242037672021372574207907793008820369] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-154 z - 1 + 34 z + z + 8 z - 434 z + 2705 z + 1074 z 8 10 12 14 18 16 20 - 9021 z + 16510 z - 16510 z + 9021 z + 434 z - 2705 z - 34 z 5 19 7 9 11 13 / 17 - 154 z + 8 z + 1074 z - 3432 z + 5158 z - 3432 z ) / (6773 z / 2 24 23 22 21 3 4 - 3 z + 1 - 67 z + z + 3 z - 67 z - 6 z + 6 z + 1162 z 6 15 8 10 12 14 - 8869 z - 27699 z + 35348 z - 78979 z + 102572 z - 78979 z 18 16 20 5 19 7 9 - 8869 z + 35348 z + 1162 z + 656 z - 656 z - 6773 z + 27699 z 11 13 - 55044 z + 55044 z ) And in Maple-input format, it is: -(-154*z^17-1+34*z^2+z^22+8*z^3-434*z^4+2705*z^6+1074*z^15-9021*z^8+16510*z^10-\ 16510*z^12+9021*z^14+434*z^18-2705*z^16-34*z^20-154*z^5+8*z^19+1074*z^7-3432*z^ 9+5158*z^11-3432*z^13)/(6773*z^17-3*z+1-67*z^2+z^24+3*z^23-67*z^22-6*z^21+6*z^3 +1162*z^4-8869*z^6-27699*z^15+35348*z^8-78979*z^10+102572*z^12-78979*z^14-8869* z^18+35348*z^16+1162*z^20+656*z^5-656*z^19-6773*z^7+27699*z^9-55044*z^11+55044* z^13) The first , 40, terms are: [3, 42, 313, 3007, 25752, 232239, 2045107, 18207850, 161286021, 1432042129, 12701100016, 112705875625, 999884478259, 8871568675850, 78709842451133, 698341523242655, 6195865106314920, 54971582591681367, 487723302487410327, 4327222686751367754, 38392356076197499573, 340628032905352428497, 3022149614928165309776, 26813379627420725531057, 237896001763311061232843, 2110681646992266345291050, 18726573625979256670402169, 166147538684974146197060519, 1474108672105784875076601944, 13078715432559222686147862975, 116038118873002920308624345203, 1029523511119335795069935128010, 9134228219035847486282796514157, 81041495661120742499002502330009, 719023420635832447678305363400528, 6379382256064512032733462161141601, 56599711220681388417976595628784635, 502168890604200701992328981742774634, 4455386595654250333358553354374194781, 39529469244613159510114933898828909135] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (-z + 1 - 23 z + 14 z + 143 z - 258 z + 143 z - 23 z + z 5 7 9 11 / 13 9 10 7 - 41 z + 41 z - 14 z + z ) / (3 z + 34 z + 528 z + 6 z / 6 8 5 12 4 14 3 2 + 1392 z - 1392 z + 34 z - 59 z - 528 z + z - 34 z + 59 z + 3 z 11 - 34 z - 1) And in Maple-input format, it is: -(-z+1-23*z^2+14*z^3+143*z^4-258*z^6+143*z^8-23*z^10+z^12-41*z^5+41*z^7-14*z^9+ z^11)/(3*z^13+34*z^9+528*z^10+6*z^7+1392*z^6-1392*z^8+34*z^5-59*z^12-528*z^4+z^ 14-34*z^3+59*z^2+3*z-34*z^11-1) The first , 40, terms are: [2, 42, 224, 2697, 18816, 186981, 1465376, 13429090, 110853246, 980664717, 8284284128, 72158989477, 615848148610, 5327681066066, 45676283581760, 393952120669789, 3384286042887552, 29150103573115313, 250639131755959232, 2157574231271026554, 18558582167262712830, 159716004683163862537, 1374050691876647023040, 11823780303995199558585, 101728804653081136888322, 875337261119714677440602, 7531434670488143481868032, 64803641243837839057828641, 557581309903932807012347008, 4797617071557371087369191725, 41279768558990959011795794944, 355183445992564064291069379570, 3056086618275749748532241687934, 26295430021541469006519052984117, 226252705800300526591271311093408, 1946740332175012627993615506986621, 16750268735272356149536724125166466, 144123851319739917186529575599595714, 1240080051116955105521765430434084896, 10669983014474172661124816149220254485] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 6 5 2 4 3 z + 4 z + 4 z - 4 z + 4 z - 16 z - 1 f(z) = - ----------------------------------------------------------- 8 7 6 5 2 4 3 z + 8 z - 22 z - 60 z - 22 z + 67 z - 8 z + 60 z + 1 And in Maple-input format, it is: -(z^6+4*z^5+4*z^2-4*z^4+4*z-16*z^3-1)/(z^8+8*z^7-22*z^6-60*z^5-22*z^2+67*z^4-8* z+60*z^3+1) The first , 40, terms are: [4, 50, 444, 4349, 41348, 396733, 3795912, 36350866, 348013000, 3332060177, 31902067752, 305441725601, 2924400160544, 27999196885618, 268073721835248, 2566628109851821, 24573761479828684, 235277465932911917, 2252625667246251996, 21567396521342873618, 206493515265533535236, 1977038434259766176785, 18928831568543726184912, 181231006106768031422641, 1735166666546200556818252, 16613069834867070893310674, 159059123633048057575880724, 1522885599253682221561297037, 14580619429065820048452765908, 139599759193624730378090740621, 1336575092829686348917296020160, 12796819916393300430546311308594, 122521062116984411862411867053680, 1173057897223642497461193958520129, 11231251235194749298810296570505848, 107531780491491722638798071344671985, 1029545468579294633303603748716968520, 9857214927786183845429564784280440274, 94376294294851969365586478005912518504, 903590414744957928933403987818428755549] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 z + 4 z - 1 f(z) = - ----------------------------- 3 2 4 10 z - 23 z + z + 1 - 10 z And in Maple-input format, it is: -(z^2+4*z-1)/(10*z^3-23*z^2+z^4+1-10*z) The first , 40, terms are: [6, 82, 948, 11305, 134028, 1590733, 18875976, 223995034, 2658056430, 31542099589, 374297467464, 4441638405853, 52707146757882, 625454634139210, 7422020035297392, 88074143832191077, 1045140645645600804, 12402266109609289681, 147172732487584507140, 1746439964796570534946, 20724304693438410516066, 245927036533719773103337, 2918317800905827883046288, 34630510362434652936144025, 410946418375149934529955438, 4876536817041904364227300738, 57867912371622239779217495916, 686695375814072264171167177729, 8148739428099239963056871220540, 96697832264182797404349155033269, 1147474507817598148312308699131904, 13616621130595379351528377677701098, 161582997923688623712460328435491542, 1917440822330139797929228534795688365, 22753503489732464933015276816497046232, 270006727210550383066048892803238678197, 3204061861148125437516835549135426470114, 38021320861604258396559398029128988148986, 451183810646854232732887693716555806474280, 5354017860946748465691508687451367288790141] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 5 7 f(z) = - (-1 + 22 z + 14 z - 95 z + 95 z - 22 z + z - 36 z + 14 z ) / 12 11 10 9 8 7 6 5 / (z + 2 z - 66 z + 62 z + 398 z - 282 z - 658 z + 282 z / 4 3 2 + 398 z - 62 z - 66 z - 2 z + 1) And in Maple-input format, it is: -(-1+22*z^2+14*z^3-95*z^4+95*z^6-22*z^8+z^10-36*z^5+14*z^7)/(z^12+2*z^11-66*z^ 10+62*z^9+398*z^8-282*z^7-658*z^6+282*z^5+398*z^4-62*z^3-66*z^2-2*z+1) The first , 40, terms are: [2, 48, 276, 3541, 27232, 286177, 2467408, 24055504, 217060298, 2055573933, 18884643008, 176819565045, 1635990718350, 15250000143760, 141491052311928, 1316617952845289, 12229075991702816, 113717375424149389, 1056688856268322100, 9823437219683807600, 91297098489375126870, 848647723506554717305, 7887687457197138957696, 73316573350385808080393, 681452653309476899126362, 6334044071189469281894128, 58873389434339201343818588, 547219616236678600292395165, 5086292779059869842060425696, 47276235043136059443026311257, 439423485496655728940763893784, 4084363290491913834366666483728, 37963392384284055260473336536610, 352862856356933241894680424323557, 3279795005659306581069611418085952, 30485095916451338033529454689730237, 283353356835573850926998132504281062, 2633717574194153601271625669742603344, 24479921124116093209014541189261543232, 227536379627309567141236293305197042801] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 15 8 10 f(z) = - (-z + 1 - 31 z + 22 z + 298 z - 1133 z + z + 1794 z - 1133 z 12 14 16 5 7 9 11 13 + 298 z - 31 z + z - 178 z + 519 z - 519 z + 178 z - 22 z ) / 16 15 12 14 13 7 8 / (-75 z - 46 z - 5075 z + 1017 z + 379 z - 1314 z - 11072 z / 9 4 5 6 2 11 18 3 + 1876 z - 1017 z + 379 z + 5075 z + 75 z - 1314 z + z - 46 z 10 17 + 3 z + 11072 z - 1 + 3 z ) And in Maple-input format, it is: -(-z+1-31*z^2+22*z^3+298*z^4-1133*z^6+z^15+1794*z^8-1133*z^10+298*z^12-31*z^14+ z^16-178*z^5+519*z^7-519*z^9+178*z^11-22*z^13)/(-75*z^16-46*z^15-5075*z^12+1017 *z^14+379*z^13-1314*z^7-11072*z^8+1876*z^9-1017*z^4+379*z^5+5075*z^6+75*z^2-\ 1314*z^11+z^18-46*z^3+3*z+11072*z^10-1+3*z^17) The first , 40, terms are: [2, 50, 276, 3767, 27868, 307283, 2586280, 26038546, 232351770, 2247247493, 20590104808, 195606892237, 1813695105390, 17092506418802, 159335519306816, 1496205375337611, 13981178108743124, 131075302308845663, 1226148897417137972, 11486966055103836370, 107507483053476535654, 1006837495149297245977, 9425133475333219300560, 88256088996925699052457, 826257415311589970693290, 7736490811130496856756242, 72432557767199220645955100, 678187303510559555134627407, 6349624560193556844376599436, 59450858111762829010408223835, 556622009857651723630733865168, 5211561557406406766161514906418, 48794609968865377003419141333346, 456854794813788587762657099250941, 4277430323579388880510253466864824, 40048730982371883798119923008445397, 374967705312030497509164221953270070, 3510746296012605882354671229895041362, 32870378802104872459161619044015364072, 307758591115947944238887058451131229443] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 15 8 10 f(z) = - (-z + 1 - 27 z + 10 z + 231 z - 809 z + z + 1206 z - 809 z 12 14 16 5 7 9 11 13 / 18 + 231 z - 27 z + z - 29 z + 8 z - 8 z + 29 z - 10 z ) / (z / 17 16 15 14 13 12 11 + 4 z - 67 z - 20 z + 799 z - 147 z - 3617 z + 1105 z 10 9 8 7 6 5 4 + 7285 z - 1885 z - 7285 z + 1105 z + 3617 z - 147 z - 799 z 3 2 - 20 z + 67 z + 4 z - 1) And in Maple-input format, it is: -(-z+1-27*z^2+10*z^3+231*z^4-809*z^6+z^15+1206*z^8-809*z^10+231*z^12-27*z^14+z^ 16-29*z^5+8*z^7-8*z^9+29*z^11-10*z^13)/(z^18+4*z^17-67*z^16-20*z^15+799*z^14-\ 147*z^13-3617*z^12+1105*z^11+7285*z^10-1885*z^9-7285*z^8+1105*z^7+3617*z^6-147* z^5-799*z^4-20*z^3+67*z^2+4*z-1) The first , 40, terms are: [3, 52, 399, 4452, 40928, 414835, 3997995, 39536884, 385837480, 3791057447, 37119198808, 364096096449, 3568069029781, 34982913752944, 342904489481743, 3361588589450057, 32952480939957232, 323032313306811003, 3166623841383628008, 31042075901878855308, 304300804078815083005, 2983021870654549024035, 29242147576691346620344, 286656869503461412722140, 2810058158130529852128341, 27546620888234997240482316, 270035784891275389315676877, 2647124250894451612814839749, 25949400193041589141935933440, 254378455264114133030168609565, 2493637527389017833569171797059, 24444790851901421517095985337500, 239628972669004679500915993836307, 2349050353284483378646353563391212, 23027422346402206561586554255302776, 225734701380861247720413455334508371, 2212846693635827245976775043589474115, 21692236328058211044506594081696935964, 212646053724689274418655431160822757272, 2084540454080970027226967262367581734067] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-216 z - 1 + 36 z + z + 12 z - 469 z + 2947 z + 1428 z 8 10 12 14 18 16 20 - 9816 z + 17907 z - 17907 z + 9816 z + 469 z - 2947 z - 36 z 5 19 7 9 11 13 / 24 - 216 z + 12 z + 1428 z - 4352 z + 6384 z - 4352 z ) / (z / 23 22 20 19 18 17 16 + 4 z - 78 z + 1374 z - 996 z - 10432 z + 9820 z + 40822 z 15 14 13 12 11 10 - 39184 z - 89502 z + 76756 z + 115246 z - 76756 z - 89502 z 9 8 7 6 5 4 2 + 39184 z + 40822 z - 9820 z - 10432 z + 996 z + 1374 z - 78 z - 4 z + 1) And in Maple-input format, it is: -(-216*z^17-1+36*z^2+z^22+12*z^3-469*z^4+2947*z^6+1428*z^15-9816*z^8+17907*z^10 -17907*z^12+9816*z^14+469*z^18-2947*z^16-36*z^20-216*z^5+12*z^19+1428*z^7-4352* z^9+6384*z^11-4352*z^13)/(z^24+4*z^23-78*z^22+1374*z^20-996*z^19-10432*z^18+ 9820*z^17+40822*z^16-39184*z^15-89502*z^14+76756*z^13+115246*z^12-76756*z^11-\ 89502*z^10+39184*z^9+40822*z^8-9820*z^7-10432*z^6+996*z^5+1374*z^4-78*z^2-4*z+1 ) The first , 40, terms are: [4, 58, 532, 5747, 58208, 604907, 6221236, 64254770, 662494948, 6835458593, 70506074480, 727339606401, 7502856722364, 77397123366658, 798397903366124, 8235982900342523, 84959290267773024, 876408469428695683, 9040701969755296236, 93260509165412620458, 962040585886415437372, 9924051589539142333057, 102372811215072541371488, 1056039704242226744754817, 10893711351776387326252292, 112375459554766354372891210, 1159223289463354037768450004, 11958114701081580682633960803, 123355447133633604166370202912, 1272488742424462630550345185627, 13126518830041843286948863954580, 135408267948642342543325078206690, 1396821142470061174754070554572484, 14409085454014895732310003251053889, 148638746442456661233130032603033680, 1533301819500997403698990563952865377, 15816969168232093090063603803218479900, 163161949256817351841473905913874419794, 1683117758031244369600105476408418143820, 17362414461849549197716564389763518566539] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-164 z - 1 + 37 z + z + 10 z - 494 z + 3146 z + 960 z 8 10 12 14 18 16 - 10481 z + 18977 z - 18977 z + 10481 z + 494 z - 3146 z 20 5 19 7 9 11 13 / - 37 z - 164 z + 10 z + 960 z - 2588 z + 3564 z - 2588 z ) / ( / 17 2 24 23 22 21 3 4 8574 z - 4 z + 1 - 81 z + z + 4 z - 81 z - 2 z + 2 z + 1449 z 6 15 8 10 12 14 - 11162 z - 32862 z + 44437 z - 99213 z + 129202 z - 99213 z 18 16 20 5 19 7 9 - 11162 z + 44437 z + 1449 z + 900 z - 900 z - 8574 z + 32862 z 11 13 - 62470 z + 62470 z ) And in Maple-input format, it is: -(-164*z^17-1+37*z^2+z^22+10*z^3-494*z^4+3146*z^6+960*z^15-10481*z^8+18977*z^10 -18977*z^12+10481*z^14+494*z^18-3146*z^16-37*z^20-164*z^5+10*z^19+960*z^7-2588* z^9+3564*z^11-2588*z^13)/(8574*z^17-4*z+1-81*z^2+z^24+4*z^23-81*z^22-2*z^21+2*z ^3+1449*z^4-11162*z^6-32862*z^15+44437*z^8-99213*z^10+129202*z^12-99213*z^14-\ 11162*z^18+44437*z^16+1449*z^20+900*z^5-900*z^19-8574*z^7+32862*z^9-62470*z^11+ 62470*z^13) The first , 40, terms are: [4, 60, 552, 6105, 62480, 660797, 6890272, 72287800, 756381460, 7923489669, 82961260752, 868816860637, 9097882608940, 95273062608688, 997682233400352, 10447627644743445, 109406137519747120, 1145687867765048865, 11997497546058044728, 125636301674912573508, 1315647564983936500188, 13777296696794362018057, 144274124086514820491424, 1510820557921245163726201, 15821123594517726913121060, 165676824382804496917864308, 1734946949368189071908258120, 18168147110299352222370610449, 190254560517004137454647755600, 1992321923644945690010341814885, 20863345596229654020574162265504, 218478341430218391354860358754304, 2287877821593157593929742783739924, 23958369933990451316630251843373101, 250889048565391085270764548174760048, 2627278686467393648935278943330159157, 27512533272513352383599194957613301932, 288107801798897569609676794505327911400, 3017028807751335833670383394241850915360, 31593947716678654828744145901199911635181] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 6 5 4 3 2 z + 4 z - 5 z - 14 z + 5 z + 4 z - 1 f(z) = - ----------------------------------------------------------- 8 7 6 5 4 3 2 z + 9 z - 26 z - 53 z + 85 z + 53 z - 26 z - 9 z + 1 And in Maple-input format, it is: -(z^6+4*z^5-5*z^4-14*z^3+5*z^2+4*z-1)/(z^8+9*z^7-26*z^6-53*z^5+85*z^4+53*z^3-26 *z^2-9*z+1) The first , 40, terms are: [5, 66, 685, 7536, 81760, 890151, 9683105, 105356006, 1146253800, 12471195941, 135685665280, 1476250975191, 16061508652055, 174748113978096, 1901247485618885, 20685442212264411, 225056192002172080, 2448595927954214761, 26640555697438138440, 289847418177467622286, 3153520023306183840755, 34310081490227632601091, 373291332595267681378720, 4061384087077935887690496, 44187580215407288136214855, 480757841028984027426000426, 5230612325547986114071781015, 56908703229085916090487472181, 619162786620184283041983463200, 6736448637601991224321045346141, 73292098988641600451743489762225, 797413008417598877134351794955386, 8675798820990906631935139622812145, 94392095924887611454860233188126176, 1026979527410888442002554065638184800, 11173466797054252765651681634852448891, 121566552139187416296104025237830000005, 1322635746580096624417984229881043829406, 14390186176610131589258849747966068407960, 156564238289291578818988728612967621890481] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 14 f(z) = - (-1 + 31 z + 6 z - 286 z + 944 z - 944 z + 286 z - 31 z + z 5 7 9 11 / 4 6 5 - 60 z + 162 z - 60 z + 6 z ) / (-3 z + 992 z - 4240 z + 546 z / 16 8 7 2 13 14 15 10 + z + 6605 z - 2364 z - 78 z + 15 z - 78 z + 3 z - 4240 z 12 11 9 3 + 992 z - 546 z + 1 + 2364 z - 15 z ) And in Maple-input format, it is: -(-1+31*z^2+6*z^3-286*z^4+944*z^6-944*z^8+286*z^10-31*z^12+z^14-60*z^5+162*z^7-\ 60*z^9+6*z^11)/(-3*z+992*z^4-4240*z^6+546*z^5+z^16+6605*z^8-2364*z^7-78*z^2+15* z^13-78*z^14+3*z^15-4240*z^10+992*z^12-546*z^11+1+2364*z^9-15*z^3) The first , 40, terms are: [3, 56, 411, 4940, 44256, 470359, 4513779, 46007164, 452405352, 4544265815, 45072527400, 450418347185, 4481118278073, 44699699536000, 445185707313243, 4437950498550505, 44216407023284856, 440683790999853299, 4391228343313870992, 43761806536558500260, 436088478566919635349, 4345819715914628793439, 43307017667754460132056, 431569848567658797450148, 4300710453938316511165977, 42857962042723831910807480, 427092081851400070257005877, 4256104669668635123106091469, 42413354187366197227593546960, 422662008175290870354311780533, 4211954224340579746734899087307, 41973401877374033390081545990072, 418277630537116427829330233792655, 4168263282865123413278137006593428, 41538004233069722439405086696129304, 413938786056355747108299238060464383, 4125025237937761000576184491598631243, 41107124899372714683408337456903899604, 409644937973806513053703106838603738912, 4082235772840621623113178477357803048443] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 5 f(z) = - (4 z - 1 + 14 z - 52 z - 43 z + 43 z - 14 z + z + 144 z 7 9 / 12 11 10 9 8 7 - 52 z + 4 z ) / (z + 8 z - 42 z - 148 z + 298 z + 624 z / 6 5 4 3 2 - 642 z - 624 z + 298 z + 148 z - 42 z - 8 z + 1) And in Maple-input format, it is: -(4*z-1+14*z^2-52*z^3-43*z^4+43*z^6-14*z^8+z^10+144*z^5-52*z^7+4*z^9)/(z^12+8*z ^11-42*z^10-148*z^9+298*z^8+624*z^7-642*z^6-624*z^5+298*z^4+148*z^3-42*z^2-8*z+ 1) The first , 40, terms are: [4, 60, 552, 6089, 62304, 657689, 6852048, 71783996, 750268852, 7849409361, 82087067072, 858597615089, 8979909222076, 93922154586172, 982331999821856, 10274270969520057, 107458974025556512, 1123918470452844457, 11755111125227541960, 122947230194316935740, 1285910450175469931628, 13449394001043989273569, 140667801909195190209408, 1471250720218333517379873, 15387875881799583670354036, 160942469661837546263565884, 1683304357809970922774549016, 17605754202181119423516380393, 184139356358616570332316936416, 1925921614713621241168600184697, 20143298745674623764142564449216, 210679646180874309574882607260732, 2203507671468911460029989718681124, 23046583503653222550770095715595697, 241045228962847992555294014613748032, 2521102635301801354487380780344513937, 26368323177658903798505295273979657004, 275787450088582000726364267337606283836, 2884473051771576845227070640856132394512, 30168830321046409814189922560002533546841] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-202 z - 1 + 38 z + z + 10 z - 528 z + 3520 z + 1484 z 8 10 12 14 18 16 - 12172 z + 22307 z - 22307 z + 12172 z + 528 z - 3520 z 20 5 19 7 9 11 13 / - 38 z - 202 z + 10 z + 1484 z - 4856 z + 7138 z - 4856 z ) / / 17 2 24 23 22 21 3 (11740 z - 5 z + 1 - 83 z + z + 5 z - 83 z - 26 z + 26 z 4 6 15 8 10 12 + 1565 z - 12563 z - 51524 z + 50482 z - 109691 z + 140549 z 14 18 16 20 5 19 - 109691 z - 12563 z + 50482 z + 1565 z + 952 z - 952 z 7 9 11 13 - 11740 z + 51524 z - 103077 z + 103077 z ) And in Maple-input format, it is: -(-202*z^17-1+38*z^2+z^22+10*z^3-528*z^4+3520*z^6+1484*z^15-12172*z^8+22307*z^ 10-22307*z^12+12172*z^14+528*z^18-3520*z^16-38*z^20-202*z^5+10*z^19+1484*z^7-\ 4856*z^9+7138*z^11-4856*z^13)/(11740*z^17-5*z+1-83*z^2+z^24+5*z^23-83*z^22-26*z ^21+26*z^3+1565*z^4-12563*z^6-51524*z^15+50482*z^8-109691*z^10+140549*z^12-\ 109691*z^14-12563*z^18+50482*z^16+1565*z^20+952*z^5-952*z^19-11740*z^7+51524*z^ 9-103077*z^11+103077*z^13) The first , 40, terms are: [5, 70, 729, 8288, 91552, 1021443, 11356089, 126414934, 1406584632, 15653365197, 174189574368, 1938414942911, 21570865832843, 240043419085064, 2671231398280577, 29725790391962395, 330792183898085416, 3681095582084949393, 40963677596507546016, 455848770889815468222, 5072740375127079105947, 56450069820043641869311, 628183219645279431883232, 6990499015630447473378888, 77791120418825618970983563, 865669017708869030920276486, 9633269763694633918209809471, 107200193656023654034196273701, 1192936749595163622890025659312, 13275144754883968260273882061885, 147727420018621684047344917022777, 1643928637187230925882807393489990, 18293837148334018412328219605855469, 203576037328713365708091181381743560, 2265418820470680236157311664832007392, 25209855243699227301582260441565169031, 280538325039705151774364683632630997373, 3121864487331930642176548445324792012462, 34740486441148036527520893386443483274864, 386596344288811990186950577432752315918633] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 5 f(z) = - (2 z - 1 + 23 z - 28 z - 136 z + 136 z - 23 z + z + 52 z 7 9 / 2 10 9 8 7 6 - 28 z + 2 z ) / ((z - 2 z - 1) (z + 10 z - 40 z - 144 z + 303 z / 5 4 3 2 + 292 z - 303 z - 144 z + 40 z + 10 z - 1)) And in Maple-input format, it is: -(2*z-1+23*z^2-28*z^3-136*z^4+136*z^6-23*z^8+z^10+52*z^5-28*z^7+2*z^9)/(z^2-2*z -1)/(z^10+10*z^9-40*z^8-144*z^7+303*z^6+292*z^5-303*z^4-144*z^3+40*z^2+10*z-1) The first , 40, terms are: [6, 86, 1008, 12371, 150084, 1826479, 22208172, 270099670, 3284740758, 39947417765, 485817368520, 5908243632701, 71852755763874, 873833249237686, 10627073135425740, 129240545829279703, 1571751539532508044, 19114766902498140011, 232463149723589492616, 2827087364858469216950, 34381462083102524548530, 418128194301214241452201, 5085042237171044540615952, 61841451751631918920028377, 752081295764047966758775230, 9146393873642195791571345270, 111233348526246080394183891048, 1352758037243220172907076329147, 16451489877555275663121909653268, 200073857807464172640469421597959, 2433186834499018196633841789706020, 29591063202653361382704137718925942, 359870030960328649874468107533438414, 4376538899480143209902235013293392525, 53225028734816686286886463201802300760, 647293157649886693589398999236597318485, 7872018895995136512175453585429157576154, 95735109766790743237289527808456183095318, 1164277088654124440343746811181453344029508, 14159289548703723572144176142611340942853247] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (-1 + 25 z + 12 z - 171 z + 431 z - 431 z + 171 z - 25 z 14 5 7 9 11 / 16 15 14 + z - 88 z + 168 z - 88 z + 12 z ) / (z + 4 z - 68 z / 13 12 11 10 9 8 7 + 48 z + 612 z - 648 z - 1952 z + 1852 z + 2790 z - 1852 z 6 5 4 3 2 - 1952 z + 648 z + 612 z - 48 z - 68 z - 4 z + 1) And in Maple-input format, it is: -(-1+25*z^2+12*z^3-171*z^4+431*z^6-431*z^8+171*z^10-25*z^12+z^14-88*z^5+168*z^7 -88*z^9+12*z^11)/(z^16+4*z^15-68*z^14+48*z^13+612*z^12-648*z^11-1952*z^10+1852* z^9+2790*z^8-1852*z^7-1952*z^6+648*z^5+612*z^4-48*z^3-68*z^2-4*z+1) The first , 40, terms are: [4, 59, 544, 5939, 60572, 635073, 6582592, 68555825, 712562756, 7412526387, 77082595936, 801697887547, 8337543236636, 86711526896913, 901801148519808, 9378788600358321, 97539800533967524, 1014418907935576699, 10550004820281835168, 109720567103982944499, 1141099204765400025852, 11867487226185990339729, 123422442986508045025216, 1283599401596986209710049, 13349496094563387504593828, 138835407604886635952203251, 1443894980136558209364778720, 15016577902760803398194280059, 156173139319906367301336502716, 1624208231967765135717023096801, 16891844476277616432542815896320, 175676002741553446544412593838881, 1827038958506522331726552981228484, 19001293880838412732991903948513019, 197614378973615386489886374929171232, 2055199136544802025571353274794598259, 21374170810810284739392564841225472732, 222292414261018995379251558592511377057, 2311851901782322716715991054187475702336, 24043372031124682841774521428347235826897] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 5}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (1 - 32 z - 8 z + 328 z - 1336 z + 2142 z - 1336 z + 328 z 14 16 5 7 9 11 13 / 5 - 32 z + z + 88 z - 256 z + 256 z - 88 z + 8 z ) / (-768 z / 6 18 13 15 16 17 14 12 + 5510 z + z - 768 z + 16 z - 79 z + 4 z + 1090 z - 5510 z 8 4 2 9 11 10 - 11992 z - 1090 z + 79 z - 6536 z + 3952 z + 11992 z - 1 + 4 z 7 3 + 3952 z + 16 z ) And in Maple-input format, it is: -(1-32*z^2-8*z^3+328*z^4-1336*z^6+2142*z^8-1336*z^10+328*z^12-32*z^14+z^16+88*z ^5-256*z^7+256*z^9-88*z^11+8*z^13)/(-768*z^5+5510*z^6+z^18-768*z^13+16*z^15-79* z^16+4*z^17+1090*z^14-5510*z^12-11992*z^8-1090*z^4+79*z^2-6536*z^9+3952*z^11+ 11992*z^10-1+4*z+3952*z^7+16*z^3) The first , 40, terms are: [4, 63, 576, 6583, 67804, 732921, 7743040, 82693033, 878605412, 9357880231, 99553988800, 1059688533519, 11276768840252, 120017562542481, 1277260414959744, 13593340363008753, 144666239881125956, 1539610690291715055, 16385260507986480448, 174379888730184294727, 1855834102322125856796, 19750679594518892411593, 210196205102661880652480, 2237008989511838549590233, 23807323480195878367893412, 253368968602010783074897687, 2696474209410640263363546048, 28697173251030695048980139359, 305409096184630392580943920764, 3250310241103564583595244097441, 34591362180446089661755463983360, 368137884989329678808380374499937, 3917900129101896831317240447440516, 41696174308976010795770109064821023, 443750706930054194148781585968836672, 4722608084942951140619431969919937111, 50260262745643905615649553785894244700, 534893848024440684682871213327606210329, 5692597153780410495247851304174155935552, 60583352145346545134395438498387216834185] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 3 4 6 15 8 f(z) = - (4 z + 4 z - 1 + 21 z - 80 z - 157 z + 550 z - 80 z - 1005 z 10 12 14 18 16 5 7 9 + 1005 z - 550 z + 157 z + z - 21 z + 562 z - 1770 z + 2626 z 11 13 / 20 19 18 17 16 - 1770 z + 562 z ) / (z + 9 z - 45 z - 203 z + 605 z / 15 14 13 12 11 10 + 1691 z - 3536 z - 6664 z + 10072 z + 13201 z - 14403 z 9 8 7 6 5 4 3 - 13201 z + 10072 z + 6664 z - 3536 z - 1691 z + 605 z + 203 z 2 - 45 z - 9 z + 1) And in Maple-input format, it is: -(4*z^17+4*z-1+21*z^2-80*z^3-157*z^4+550*z^6-80*z^15-1005*z^8+1005*z^10-550*z^ 12+157*z^14+z^18-21*z^16+562*z^5-1770*z^7+2626*z^9-1770*z^11+562*z^13)/(z^20+9* z^19-45*z^18-203*z^17+605*z^16+1691*z^15-3536*z^14-6664*z^13+10072*z^12+13201*z ^11-14403*z^10-13201*z^9+10072*z^8+6664*z^7-3536*z^6-1691*z^5+605*z^4+203*z^3-\ 45*z^2-9*z+1) The first , 40, terms are: [5, 69, 723, 8149, 89973, 999389, 11081089, 122931832, 1363562747, 15125431321, 167777480551, 1861065505209, 20643769507033, 228990015037697, 2540060300079511, 28175493240936077, 312535260711459872, 3466781886878298949, 38455106176302189457, 426561358610663020953, 4731610720324477341455, 52485157311397269584161, 582189005137859676847393, 6457902673204448683184721, 71633965204566327729671501, 794596207252049333045797496, 8814019031003087003122012135, 97769069081706675776323890629, 1084498551169518545111179671347, 12029746406871061596621635655693, 133439365555230318307688637451701, 1480169546185284363234914373474445, 16418707300789285923346464571837443, 182123696656063167110586659332712873, 2020198075038173056338638873491780672, 22408946981211358581383897505962065561, 248570133301039294653538387989390975693, 2757251878952699606468313034851616003901, 30584679756280301065805838168183701032603, 339259043736522245566519944736997006991197] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 8 7 6 5 4 3 2 / 10 f(z) = - (z + z - 18 z - 5 z + 50 z + 5 z - 18 z - z + 1) / (z / 9 8 7 6 5 4 3 2 + 5 z - 58 z + 15 z + 261 z - 72 z - 261 z + 15 z + 58 z + 5 z - 1 ) And in Maple-input format, it is: -(z^8+z^7-18*z^6-5*z^5+50*z^4+5*z^3-18*z^2-z+1)/(z^10+5*z^9-58*z^8+15*z^7+261*z ^6-72*z^5-261*z^4+15*z^3+58*z^2+5*z-1) The first , 40, terms are: [4, 60, 552, 6089, 62240, 656937, 6838608, 71615676, 748033396, 7822107521, 81754837824, 854664898881, 8933834807804, 93389388759484, 976224049378848, 10204807101834793, 106674016770248032, 1115098233916675401, 11656477584843019848, 121848912441822338876, 1273725752660769886316, 13314664407396090697665, 139182460500815890058880, 1454918945734541466413889, 15208734792721137829700212, 158981787340387789509130556, 1661887660599333951562759320, 17372245234094389373627112777, 181597656437310238300246148000, 1898298601093784923792919394601, 19843524688068825176195890589376, 207430733934127363702643092512700, 2168340053321428427683935127575268, 22666354679834532201032572880974017, 236938682050604710502198393938319808, 2476796107926828199628222605038068225, 25890744842288225037694314522525926252, 270644267545149067928602748433830368956, 2829131413608793557708614882103768764304, 29573818902825981901090024965262253734185] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (1 - 34 z - 12 z + 351 z - 1456 z + 2340 z - 1456 z + 351 z 14 16 5 7 9 11 13 / 18 17 - 34 z + z + 88 z - 92 z + 92 z - 88 z + 12 z ) / (z + 4 z / 16 15 14 13 12 11 10 - 92 z + 52 z + 1248 z - 932 z - 6453 z + 3632 z + 14412 z 9 8 7 6 5 4 3 - 5704 z - 14412 z + 3632 z + 6453 z - 932 z - 1248 z + 52 z 2 + 92 z + 4 z - 1) And in Maple-input format, it is: -(1-34*z^2-12*z^3+351*z^4-1456*z^6+2340*z^8-1456*z^10+351*z^12-34*z^14+z^16+88* z^5-92*z^7+92*z^9-88*z^11+12*z^13)/(z^18+4*z^17-92*z^16+52*z^15+1248*z^14-932*z ^13-6453*z^12+3632*z^11+14412*z^10-5704*z^9-14412*z^8+3632*z^7+6453*z^6-932*z^5 -1248*z^4+52*z^3+92*z^2+4*z-1) The first , 40, terms are: [4, 74, 704, 8935, 98520, 1161625, 13256752, 153692518, 1768263796, 20419727439, 235381823456, 2715650459119, 31317825172332, 361241909938694, 4166406260834736, 48055817277349593, 554268509489043240, 6392921115867068711, 73735411688561388384, 850460260832505201898, 9809150157759439559708, 113138133812773875944097, 1304927873390142511036480, 15050955383140633692186465, 173596753109666213925909540, 2002253874266165363303076778, 23093868049506122184272869600, 266363198321201922470987798183, 3072216096052429815228246993112, 35434744057229134513122158127705, 408702072376449321517061481986576, 4713943573843845738380799421698246, 54370323799354760507015522777664596, 627103838654201386246254781309053103, 7232975582159488568121343184280160160, 83424677937991425468100875567932737935, 962214901726040065603377933215366709068, 11098125159046991391709384123211981058726, 128005066045624686878864136795235522601872, 1476402247996819312269369875465751346780697] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 3 4 7 6 5 8 2 / 10 f(z) = - (24 z + 42 z + 3 z - 15 z - 24 z + z + 1 - 3 z - 15 z ) / (z / 9 3 4 7 6 5 8 2 + 8 z - 68 z - 289 z - 68 z + 289 z + 68 z - 55 z - 1 + 8 z + 55 z ) And in Maple-input format, it is: -(24*z^3+42*z^4+3*z^7-15*z^6-24*z^5+z^8+1-3*z-15*z^2)/(z^10+8*z^9-68*z^3-289*z^ 4-68*z^7+289*z^6+68*z^5-55*z^8-1+8*z+55*z^2) The first , 40, terms are: [5, 80, 871, 10781, 127312, 1529717, 18261889, 218538576, 2612877699, 31250455945, 373713645472, 4469325562297, 53448729713357, 639198342391376, 7644213791585295, 91417724818175749, 1093271037321893872, 13074507343322512653, 156358970680277724841, 1869908173867830512720, 22362366163260266175435, 267433143862956561675345, 3198252181692475649599296, 38248127638668899848902833, 457412106539503322839871701, 5470224247202358666799649872, 65418804807146924057401441399, 782348187026072672271011188781, 9356158180293626663596617517712, 111890967917385369260074497911013, 1338112124681071789156597574698001, 16002579042311781264023911112293968, 191375992551041777561078260835262675, 2288679245267839936080674830396403289, 27370479535580697362836119894104485600, 327325531332798383284816304591767587561, 3914509547522457265400051527061934312477, 46813900936022274847622708050618763823696, 559850804868967867825428212042938808245919, 6695296000664011292945475402315100897253269] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 8 7 6 5 4 3 2 / 10 f(z) = - (z + z - 25 z - 12 z + 102 z + 12 z - 25 z - z + 1) / (z / 9 8 7 6 5 4 3 2 + 4 z - 85 z - 18 z + 511 z - 80 z - 511 z - 18 z + 85 z + 4 z - 1 ) And in Maple-input format, it is: -(z^8+z^7-25*z^6-12*z^5+102*z^4+12*z^3-25*z^2-z+1)/(z^10+4*z^9-85*z^8-18*z^7+ 511*z^6-80*z^5-511*z^4-18*z^3+85*z^2+4*z-1) The first , 40, terms are: [3, 72, 537, 7805, 73944, 912989, 9518055, 110350632, 1195866477, 13523464321, 148857030768, 1666710921601, 18461194162419, 205885759393224, 2286204221071161, 25456086718238525, 282954476226835272, 3148596946649137373, 35011978559092948743, 389499251738838122856, 4331877197176691952669, 48186109658467458909313, 535943943105597562822368, 5961387913362167306320129, 66306498154954379792533923, 737525459497007986274276520, 8203330570139388195058096185, 91244836795843578551552903453, 1014900089172425839010693692920, 11288604048548970351493165243517, 125561342710841798325693202098951, 1396601312783464216580837748941832, 15534183990962973874317713663329101, 172784490738884620467198908391604993, 1921856121379979487429107817452679504, 21376525194445483114686650790742858753, 237767926895302973309503857435127778259, 2644657777223555870129342750245682867688, 29416138406861615067156516619009230013017, 327191384205139336484436678840021332424989] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 22 3 4 6 15 f(z) = - (-190 z - 1 + 40 z + z + 12 z - 564 z + 3801 z + 1122 z 8 10 12 14 18 16 - 13421 z + 25312 z - 25312 z + 13421 z + 564 z - 3801 z 20 5 19 7 9 11 13 / - 40 z - 190 z + 12 z + 1122 z - 3136 z + 4394 z - 3136 z ) / / 17 2 24 23 22 21 3 (12271 z - 5 z + 1 - 97 z + z + 5 z - 97 z + 8 z - 8 z 4 6 15 8 10 12 + 1802 z - 14443 z - 48753 z + 60116 z - 138901 z + 183080 z 14 18 16 20 5 19 - 138901 z - 14443 z + 60116 z + 1802 z + 1294 z - 1294 z 7 9 11 13 - 12271 z + 48753 z - 95454 z + 95454 z ) And in Maple-input format, it is: -(-190*z^17-1+40*z^2+z^22+12*z^3-564*z^4+3801*z^6+1122*z^15-13421*z^8+25312*z^ 10-25312*z^12+13421*z^14+564*z^18-3801*z^16-40*z^20-190*z^5+12*z^19+1122*z^7-\ 3136*z^9+4394*z^11-3136*z^13)/(12271*z^17-5*z+1-97*z^2+z^24+5*z^23-97*z^22+8*z^ 21-8*z^3+1802*z^4-14443*z^6-48753*z^15+60116*z^8-138901*z^10+183080*z^12-138901 *z^14-14443*z^18+60116*z^16+1802*z^20+1294*z^5-1294*z^19-12271*z^7+48753*z^9-\ 95454*z^11+95454*z^13) The first , 40, terms are: [5, 82, 891, 11211, 133024, 1616123, 19445305, 234898458, 2832928315, 34188761985, 412487401648, 4977229413297, 60054306819021, 724617966563002, 8743202908711847, 105495382687067475, 1272904303022412784, 15358835938990268155, 185319341120788497589, 2236058879149061157842, 26980233599897951304019, 325542866388986713998841, 3927992571089076816119728, 47395066131563444674063561, 571867754432337302419236461, 6900142892624206192222994162, 83256962051773068868146949275, 1004576548415233308472091592763, 12121197035330857434756588245040, 146254078699218022158396448609827, 1764698278040433848475501333649641, 21292807969687893945233916529976570, 256918520789299458979656541076131955, 3099972836769783873194668772859359825, 37404199429398196157440714678161304080, 451318191682008789517600485748623147937, 5445594699268503839842796289207471615077, 65706417723120780354082909450295287736090, 792812092788535907001434602106613718953655, 9566052088250805593041171484767827643445035] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 17 2 3 4 6 15 8 f(z) = - (2 z + 2 z - 1 + 29 z - 36 z - 287 z + 1267 z - 36 z - 2668 z 10 12 14 18 16 5 7 9 + 2668 z - 1267 z + 287 z + z - 29 z + 246 z - 772 z + 1136 z 11 13 / 20 19 18 17 16 - 772 z + 246 z ) / (z + 8 z - 73 z - 102 z + 1048 z / 15 14 13 12 11 10 + 378 z - 6147 z - 156 z + 17311 z - 1220 z - 24364 z 9 8 7 6 5 4 3 + 1220 z + 17311 z + 156 z - 6147 z - 378 z + 1048 z + 102 z 2 - 73 z - 8 z + 1) And in Maple-input format, it is: -(2*z^17+2*z-1+29*z^2-36*z^3-287*z^4+1267*z^6-36*z^15-2668*z^8+2668*z^10-1267*z ^12+287*z^14+z^18-29*z^16+246*z^5-772*z^7+1136*z^9-772*z^11+246*z^13)/(z^20+8*z ^19-73*z^18-102*z^17+1048*z^16+378*z^15-6147*z^14-156*z^13+17311*z^12-1220*z^11 -24364*z^10+1220*z^9+17311*z^8+156*z^7-6147*z^6-378*z^5+1048*z^4+102*z^3-73*z^2 -8*z+1) The first , 40, terms are: [6, 92, 1108, 14207, 179000, 2266827, 28663592, 362608940, 4586565930, 58016900325, 733864771856, 9282804047565, 117419934454606, 1485267346877644, 18787430536939384, 237645807036347571, 3006027298453417128, 38023814760060468615, 480970511604987269852, 6083888072647097417724, 76956264852889008494338, 973434525674393876214793, 12313159657362276548209312, 155751513583307825902290809, 1970130710436298483625302638, 24920560493487356069329592508, 315224940162468453640602783972, 3987340610834248508433376263607, 50436634673048343910789140037144, 637982646937782128825056326552803, 8069964628532947282616012882266856, 102078527399388700024010588330196108, 1291210833711178658342803664304721218, 16332773008862011794299410145597167549, 206596372330841900472992955918856170288, 2613277061837874359162379911009656207093, 33055841808256612942294308212574709293606, 418129670829474170577201433379153441025388, 5289002247835515073888804959006039034238712, 66901601893297790792252192002336018178620187] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 5 f(z) = - (2 z - 1 + 24 z - 26 z - 139 z + 139 z - 24 z + z + 48 z 7 9 / 12 11 10 9 8 7 - 26 z + 2 z ) / (z + 8 z - 72 z - 62 z + 680 z - 172 z / 6 5 4 3 2 - 1266 z + 172 z + 680 z + 62 z - 72 z - 8 z + 1) And in Maple-input format, it is: -(2*z-1+24*z^2-26*z^3-139*z^4+139*z^6-24*z^8+z^10+48*z^5-26*z^7+2*z^9)/(z^12+8* z^11-72*z^10-62*z^9+680*z^8-172*z^7-1266*z^6+172*z^5+680*z^4+62*z^3-72*z^2-8*z+ 1) The first , 40, terms are: [6, 96, 1164, 15311, 196044, 2533391, 32632776, 420821856, 5424607182, 69935892625, 901591912152, 11623252881505, 149845117701306, 1931783674954080, 24904282636504224, 321062590540622111, 4139094390795105060, 53360632784520085343, 687917893574177763228, 8868542327239512184416, 114332020750224246833922, 1473952594916077185829777, 19001992944404210643392304, 244971064287182127791043889, 3158133071200628673849390222, 40714214654389989685837306464, 524882022874402523989580182452, 6766706426138530514176688542079, 87235443132107743268929972482492, 1124627264611364583025833919645247, 14498539113188261811817073044566576, 186913160503266680035873470275548512, 2409658607434489251187319349255684054, 31064985412205130230086991847897724801, 400485494369663669851536074295188146824, 5163003589806896451937258566170250979057, 66560728024156135088905896035235863041986, 858091697602593491757316435350845420480352, 11062399456136896937768342770373256326968584, 142614923403925080818517111714788431988169071] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 5 7 f(z) = - (-1 + 27 z + 12 z - 157 z + 157 z - 27 z + z - 60 z + 12 z ) / 9 12 11 10 2 6 4 / (66 z + z + 6 z + 1 - 6 z - 87 z - 87 z - 999 z + 674 z / 8 5 3 7 + 674 z + 810 z - 66 z - 810 z ) And in Maple-input format, it is: -(-1+27*z^2+12*z^3-157*z^4+157*z^6-27*z^8+z^10-60*z^5+12*z^7)/(66*z^9+z^12+6*z^ 11+1-6*z-87*z^10-87*z^2-999*z^6+674*z^4+674*z^8+810*z^5-66*z^3-810*z^7) The first , 40, terms are: [6, 96, 1152, 15143, 192624, 2480495, 31793280, 408236544, 5238277386, 67232717017, 862836089856, 11073709005481, 142118722573974, 1823946242506944, 23408403222984576, 300422132477243999, 3855599480922201936, 49482536998310843447, 635055938117383377792, 8150270313049835223456, 104600085906002237365914, 1342431303826208578438705, 17228683780642534390514688, 221111906504049325047174097, 2837737102399192630856591526, 36419349777054688363373834016, 467403776420907994629030074496, 5998632363069272903767042845719, 76986092202061242209574109648944, 988034944271865033680267530438271, 12680381912872885663808225292356736, 162739269889733868861260752695499584, 2088586144030519325908630840669114026, 26804790779705896339067010157334320393, 344011096117517239218954817504569351168, 4415018017658808233710184254688285213305, 56662079555692898697708991667223477519414, 727197770594418271699003480083355975657344, 9332813086002617163648177731726208674524032, 119776769979732202167398613854867362390960463] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : f(z) = - ( 2 3 4 6 8 10 5 7 9 6 z - 1 + 5 z - 62 z + 17 z - 17 z - 5 z + z + 142 z - 62 z + 6 z / 12 11 10 9 8 7 6 5 ) / (z + 13 z - 22 z - 203 z + 166 z + 865 z - 430 z - 865 z / 4 3 2 + 166 z + 203 z - 22 z - 13 z + 1) And in Maple-input format, it is: -(6*z-1+5*z^2-62*z^3+17*z^4-17*z^6-5*z^8+z^10+142*z^5-62*z^7+6*z^9)/(z^12+13*z^ 11-22*z^10-203*z^9+166*z^8+865*z^7-430*z^6-865*z^5+166*z^4+203*z^3-22*z^2-13*z+ 1) The first , 40, terms are: [7, 108, 1417, 19193, 258320, 3481329, 46904543, 631989228, 8515285873, 114733414121, 1545896317440, 20829117051577, 280647609463647, 3781393179138252, 50949781418277937, 686487784871972353, 9249607469007366000, 124627473666982236649, 1679207171181428841543, 22625322015942921530572, 304849339087138218444873, 4107482734450477690389617, 55343450847981441989585920, 745687261463976459736881297, 10047250097161687392881998167, 135374760615770854485462427532, 1824014096847692243880166197977, 24576423333017596753316592831369, 331138111754476567357558282856080, 4461692719501858561955651469939553, 60116009654653643104019769529998703, 809991822386616647427338877066440332, 10913677672589899995768608291188894273, 147048843024412369574879938487083397657, 1981308490457449945128443702348218206720, 26695778447621460107121872126631447910281, 359693904486296577473324173737946363281967, 4846448107083557074709813176296248779834668, 65300131477620379843829009259562734642347457, 879841706086174538935708687109300247253144209] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (-z + 1 - 26 z + 8 z + 174 z - 346 z + 174 z - 26 z + z 5 7 9 11 / 14 13 12 11 10 - 19 z + 19 z - 8 z + z ) / (z + 7 z - 77 z + 3 z + 721 z / 9 8 7 6 5 4 3 2 - 424 z - 2090 z + 984 z + 2090 z - 424 z - 721 z + 3 z + 77 z + 7 z - 1) And in Maple-input format, it is: -(-z+1-26*z^2+8*z^3+174*z^4-346*z^6+174*z^8-26*z^10+z^12-19*z^5+19*z^7-8*z^9+z^ 11)/(z^14+7*z^13-77*z^12+3*z^11+721*z^10-424*z^9-2090*z^8+984*z^7+2090*z^6-424* z^5-721*z^4+3*z^3+77*z^2+7*z-1) The first , 40, terms are: [6, 93, 1124, 14500, 183558, 2336925, 29699648, 377658717, 4801429072, 61047286937, 776165240804, 9868351561025, 125468351497502, 1595232706216820, 20282141547347912, 257871651947924065, 3278637380584330118, 41685323175431498109, 529996447469639794144, 6738492427132598872869, 85674687814001955520658, 1089286989928438896467449, 13849436475016017907641936, 176084808182377572133660244, 2238781319981790938572467626, 28464362431042810349671607321, 361902219468276558620169337380, 4601305115242611014590750220225, 58502014148086918783170795053808, 743807587991812632728554116643317, 9456934705764967088172730405406664, 120237566103031537816692129290505093, 1528727093100030853691319692276105794, 19436575447439533188630799116869370052, 247120932721828563977043399911157315252, 3141950368492067867292016598528786984325, 39947454104099233444989871262563175284882, 507900794806314722436958392899028444487705, 6457563395470930703968709324602787812727744, 82102893779538536757348306011308589521581673] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 6 5 4 3 2 z + 4 z - 15 z - 44 z + 15 z + 4 z - 1 f(z) = - ---------------------------------------------------------------- 8 7 6 5 4 3 2 z + 10 z - 67 z - 150 z + 276 z + 150 z - 67 z - 10 z + 1 And in Maple-input format, it is: -(z^6+4*z^5-15*z^4-44*z^3+15*z^2+4*z-1)/(z^8+10*z^7-67*z^6-150*z^5+276*z^4+150* z^3-67*z^2-10*z+1) The first , 40, terms are: [6, 112, 1416, 20503, 281592, 3947275, 54890340, 765693040, 10667941734, 148701450349, 2072373661296, 28883706429205, 402555021982386, 5610511840932976, 78194786045078964, 1089817709122737955, 15189015701357827560, 211692521939132423935, 2950403111118621011472, 41120389105952710809712, 573103508079390091540626, 7987464130753081444784473, 111322967290259288126021856, 1551531606943560868469314729, 21624022293300209868700886430, 301378546281152307396801964144, 4200376179849351552084921574944, 58541526164854095286254361139503, 815905561535428968711321918928344, 11371447397393036518350535679333107, 158486253811234358000153083270763388, 2208856249283686651117061000198857840, 30785294072316127540087605386801455230, 429061117683145664041654479627245466565, 5979915029398777505955164780467527990096, 83343333350558986871624018344533202515901, 1161573564144895231041783289895742636430794, 16189095044291603518047892238607567140379760, 225630822225232675419941739448743327200152524, 3144664219880873239816212465500777997479325499] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 5 7 f(z) = - (-1 + 28 z + 20 z - 139 z + 139 z - 28 z + z - 76 z + 20 z ) / 12 11 10 9 8 7 6 5 / (z + 6 z - 104 z + 128 z + 620 z - 846 z - 890 z + 846 z / 4 3 2 + 620 z - 128 z - 104 z - 6 z + 1) And in Maple-input format, it is: -(-1+28*z^2+20*z^3-139*z^4+139*z^6-28*z^8+z^10-76*z^5+20*z^7)/(z^12+6*z^11-104* z^10+128*z^9+620*z^8-846*z^7-890*z^6+846*z^5+620*z^4-128*z^3-104*z^2-6*z+1) The first , 40, terms are: [6, 112, 1404, 20359, 278016, 3891379, 53908824, 750036208, 10416578286, 144774945493, 2011526645568, 27952132649389, 388401047018682, 5397041289803632, 74994074876276496, 1042077194237811547, 14480118273899685888, 201207714676326152671, 2795870305888623576684, 38849860834185442665712, 539836060105324895829282, 7501261824086029027374889, 104233363821624953539119744, 1448368878250155150277642969, 20125728723908174970339144750, 279655937800954371793401907696, 3885943440816799208930489756772, 53996909722934954273912162142511, 750311038720970059541174091783168, 10425905069868086077120331531744779, 144872580723734304126131344680276672, 2013068841983241815544206789766463600, 27972485492515732882073563572430526358, 388690107516940474761737189642701003261, 5401021647572785613356052873533292758464, 75049594197046814400468901595300216845957, 1042847438256866230882309181766067948454754, 14490828246499995621623807442105287627503600, 201356493352998835054493715981590023446225528, 2797937890486638478180455716903987479645752835] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 12 f(z) = - (-2 z + 1 - 26 z + 25 z + 171 z - 330 z + 171 z - 26 z + z 5 7 9 11 / 6 11 14 - 109 z + 109 z - 25 z + 2 z ) / (-1 + 2479 z - 47 z + 9 z + z / 12 13 10 4 5 7 2 3 - 85 z + 9 z + 815 z - 815 z - 51 z + 375 z + 85 z - 47 z 9 8 - 51 z - 2479 z ) And in Maple-input format, it is: -(-2*z+1-26*z^2+25*z^3+171*z^4-330*z^6+171*z^8-26*z^10+z^12-109*z^5+109*z^7-25* z^9+2*z^11)/(-1+2479*z^6-47*z^11+9*z+z^14-85*z^12+9*z^13+815*z^10-815*z^4-51*z^ 5+375*z^7+85*z^2-47*z^3-51*z^9-2479*z^8) The first , 40, terms are: [7, 122, 1671, 24436, 350360, 5054125, 72768983, 1048356746, 15100411488, 217517805829, 3133231543328, 45132856077185, 650118168619417, 9364660906386760, 134893717824637239, 1943083301835831329, 27989240007758149720, 403172401488652897021, 5807516922775785193696, 83654666601581653991218, 1205007808997191269026433, 17357594965401737821074821, 250028340669118312945864880, 3601545678578037556982908324, 51878643997496403426377839329, 747288509772090606744865659826, 10764354536025035323459166095081, 155055680720407273613156808879473, 2233507270984638153454256227718784, 32172666659898271378673967641098321, 463432778328325774155541357584683255, 6675540523248325895206978265798840578, 96158155748662479152456147811585706607, 1385114940847475653677085682865739764548, 19951957110882839467943590971784633426080, 287398959331811614288304100356599713873685, 4139852615258227202405133222016830836508911, 59632713061683555260953111204527065575970594, 858982383573399721288666915006230745794944032, 12373254500867885712737693987065752948454056973] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 15 8 10 f(z) = - (-z + 1 - 35 z + 12 z + 374 z - 1565 z + z + 2514 z - 1565 z 12 14 16 5 7 9 11 13 / + 374 z - 35 z + z - 42 z + 41 z - 41 z + 42 z - 12 z ) / ( / 18 17 16 15 14 13 12 11 z + 9 z - 101 z - 34 z + 1471 z - 875 z - 7751 z + 6034 z 10 9 8 7 6 5 4 + 16916 z - 10588 z - 16916 z + 6034 z + 7751 z - 875 z - 1471 z 3 2 - 34 z + 101 z + 9 z - 1) And in Maple-input format, it is: -(-z+1-35*z^2+12*z^3+374*z^4-1565*z^6+z^15+2514*z^8-1565*z^10+374*z^12-35*z^14+ z^16-42*z^5+41*z^7-41*z^9+42*z^11-12*z^13)/(z^18+9*z^17-101*z^16-34*z^15+1471*z ^14-875*z^13-7751*z^12+6034*z^11+16916*z^10-10588*z^9-16916*z^8+6034*z^7+7751*z ^6-875*z^5-1471*z^4-34*z^3+101*z^2+9*z-1) The first , 40, terms are: [8, 138, 2028, 30821, 464840, 7023717, 106078524, 1602288890, 24201382304, 365547052865, 5521351328416, 83396486840913, 1259650471050048, 19026213757543386, 287378771733349764, 4340672284787860693, 65563074654126350680, 990288249667093927509, 14957669732972767661492, 225926020954562484387818, 3412467841267427543910584, 51543141062289475207925201, 778526132448236780170214336, 11759138585916053203076415217, 177614256631223746785221858056, 2682749584773683944884984907306, 40521214181283977919213992397900, 612046986436867517042959957373109, 9244577715034825864670600563537896, 139633425248690333024051348888638773, 2109073453400909659677918671158601532, 31856203655522373468869553261895058970, 481167552370312021299926903783541756224, 7267727691523024751733923713987639275249, 109774371812752296535638797075276182806496, 1658071575925968942374404387814575774907937, 25044109162228485507942747434559205852036960, 378275228184492374089721392242660247867766970, 5713605037061621231045163947546108714077085956, 86300344530131840074007774144257531305344083589] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 4 3 2 z + 4 z - 6 z - 4 z + 1 f(z) = - -------------------------------------------- 6 5 4 3 2 z + 12 z - 47 z - 8 z + 47 z + 12 z - 1 And in Maple-input format, it is: -(z^4+4*z^3-6*z^2-4*z+1)/(z^6+12*z^5-47*z^4-8*z^3+47*z^2+12*z-1) The first , 40, terms are: [8, 137, 2016, 30521, 459544, 6926545, 104379840, 1573019185, 23705440040, 357242140889, 5383654944672, 81131924020457, 1222661758446136, 18425567948435617, 277674141464763264, 4184561857758579553, 63061536262455564872, 950340200711850811433, 14321669762843848449888, 215828210405434397372825, 3252540882325277995296088, 49015938052419552069697393, 738672401080170359224459584, 11131826458855460455362611089, 167757127691339515922181362024, 2528107493883974059417488537209, 38098694157376545630728767637280, 574149042320719420056450442491209, 8652452008882673281293798532326520, 130392842707553087901991822621613377, 1965026030991214545067843478903633664, 29613031070526816875736362013703918273, 446269716204032981789174552695003079816, 6725304786481120375936562865522506628041, 101350646993906692556433984018987113846496, 1527358829406760546226381973584768534095865, 23017366568040179904734451129196414912812952, 346872754147309557253931221813003999074847249, 5227388077348789421127223439719647912259439808, 78776974508651296420981986284426000831098875121] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 3 4 6 8 10 5 f(z) = - (2 z - 1 + 29 z - 28 z - 181 z + 181 z - 29 z + z + 142 z 7 9 / 7 6 10 12 11 4 - 28 z + 2 z ) / (-137 z - 2342 z - 106 z + z + 11 z + 970 z / 5 9 2 8 3 + 137 z - 35 z - 106 z + 970 z + 35 z - 11 z + 1) And in Maple-input format, it is: -(2*z-1+29*z^2-28*z^3-181*z^4+181*z^6-29*z^8+z^10+142*z^5-28*z^7+2*z^9)/(-137*z ^7-2342*z^6-106*z^10+z^12+11*z^11+970*z^4+137*z^5-35*z^9-106*z^2+970*z^8+35*z^3 -11*z+1) The first , 40, terms are: [9, 176, 2883, 49265, 832344, 14107177, 238883757, 4046180512, 68528557095, 1160665941305, 19658039470512, 332946107656361, 5639069626132665, 95508282774372160, 1617612884671465395, 27397325145819643609, 464025373161746280072, 7859144862356997240065, 133109440822322430737565, 2254459428841458324739088, 38183522406564986554256727, 646709967245883446561316433, 10953253010057200647129566304, 185513997895110543203829747505, 3142029439420511177306333123433, 53216194520089470304389741322064, 901316621566175418844423909414179, 15265496896904042194933513612590689, 258549981142533499494386804408729400, 4379031563811181850129893507509103609, 74167158520438252184681137768357499661, 1256160711070180021267912370515361142784, 21275450799446594001903773667588982319367, 360339885438738304974408108513396568475465, 6103035571936294873314593423366242255212816, 103366417922259621336677587400214083417749913, 1750705239735208273125466864029405862319612313, 29651495118476791283131229378657700988304440288, 502203993456963917808785803924421428690958809875, 8505771801266199467163808482761220758955457225225] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 8 7 6 5 4 3 2 / 10 f(z) = - (z + 4 z - 15 z - 34 z + 36 z + 34 z - 15 z - 4 z + 1) / (z / 9 8 7 6 5 4 3 2 + 14 z - 69 z - 88 z + 409 z + 102 z - 409 z - 88 z + 69 z + 14 z - 1) And in Maple-input format, it is: -(z^8+4*z^7-15*z^6-34*z^5+36*z^4+34*z^3-15*z^2-4*z+1)/(z^10+14*z^9-69*z^8-88*z^ 7+409*z^6+102*z^5-409*z^4-88*z^3+69*z^2+14*z-1) The first , 40, terms are: [10, 194, 3352, 59061, 1037048, 18220973, 320105392, 5623726754, 98799225790, 1735734281257, 30493892723376, 535725735261721, 9411788260540818, 165349081713268066, 2904901604197117728, 51034171118835008253, 896583422308371960552, 15751442916315635049445, 276726010957311546849592, 4861604460441656627806978, 85410106003486100739176390, 1500510020279199218088492177, 26361404127825141902404917600, 463124949649594738332592929905, 8136316182093862893141165276890, 142941210714494665315392670316674, 2511233494771510447496780631771240, 44118093261839545865164318463585925, 775079719632948571311945400132116696, 13616829907422916124464953314685815133, 239224497856162300605298911120358934912, 4202766779317425112639581444642014415330, 73835450631625878174766600572971575028238, 1297163049066613497505136117018187102024505, 22788944354909552866752074832490436183169040, 400362919052356079210403737505418053116868873, 7033694253485292863810505554267421651039055330, 123570022340261709866215501862413896170718282274, 2170914724308140699612788905499880759700841690864, 38139272381456373830162505973401094452792633936717] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 6 5 4 3 2 z + 3 z - 21 z - 24 z + 21 z + 3 z - 1 f(z) = - ---------------------------------------------------------------- 8 7 6 5 4 3 2 z + 15 z - 105 z + 12 z + 364 z - 12 z - 105 z - 15 z + 1 And in Maple-input format, it is: -(z^6+3*z^5-21*z^4-24*z^3+21*z^2+3*z-1)/(z^8+15*z^7-105*z^6+12*z^5+364*z^4-12*z ^3-105*z^2-15*z+1) The first , 40, terms are: [12, 264, 5256, 106361, 2146080, 43326041, 874590240, 17655089928, 356396335068, 7194439208257, 145231417363008, 2931731702452993, 59181758683304436, 1194679774009923336, 24116548639048860720, 486831643896790019801, 9827486222924786158944, 198383746564022576955257, 4004697641695755569681352, 80841316283089860495824136, 1631913068876527860833494980, 32942811755368422819163759297, 665004078370885022714283308160, 13424185753598997247371128024641, 270988959328794547320798666829788, 5470351604633844269253010173766920, 110427918364054003052487361580574936, 2229166612231755707261798653151918201, 44999343089187027201001964568519478176, 908384715322409881041639440595486572249, 18337218598856735018046227987306065780848, 370166494735561490249128724595570160467720, 7472410992218602787068494549258070926832044, 150842733825810008930282292709597171263944833, 3045005202703459706811741029577347840104914624, 61468368076637435013030024352120377569356286401, 1240838692377412923397312291367662361783324120004, 25048341263611348113108551250142642063102708769544, 505641389096447802141129335952641528861357427887232, 10207191433422829403249215381658673096514567432584281] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 z + 6 z - 1 f(z) = - ----------------------------- 3 4 2 21 z + 1 - 21 z + z - 62 z And in Maple-input format, it is: -(z^2+6*z-1)/(21*z^3+1-21*z+z^4-62*z^2) The first , 40, terms are: [15, 376, 8805, 207901, 4903920, 115686901, 2729093235, 64380355576, 1518756918825, 35828050696201, 845197277027040, 19938523685081401, 470357320740846855, 11095907233164566776, 261756651587724670845, 6174938489539178362501, 145669136269544478455760, 3436390062421287723391501, 81065742294626889385195275, 1912371545257134542713440376, 45113568612219575952194677665, 1064246159684303850679509042001, 25105969739135300121019392454080, 592259329109866273053998000468401, 13971621752211597741196938457429695, 329595845590573735888091496884486776, 7775290754158121186569876641532572885, 183422051947810563124190892573415718701, 4326995633282671079123949977443717959600, 102075464818017436456648929364900783193701, 2407998852060233592548517836643743697897315, 56805604381631102600606381049723574720982776, 1340065709085175988514073356449466818245244505, 31612657311091741197153160587022781297412029401, 745754551805341163039883637907464511164019241120, 17592632155704682051199656205585678648126078769001, 415016851612487463442536835421878123428752949236535, 9790404619270703763302617519049108890671887001552376, 230959350774837398098554019032232523640570498211874925, 5448418506150351051980052212657862721402985518532884501] ------------------------------------------------------------- There are, 853, different connected graphs with , 7, vertices (up to isomorphism) -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 -1 + z f(z) = - -------------- 2 4 -8 z + 1 + z And in Maple-input format, it is: -(-1+z^2)/(-8*z^2+1+z^4) The first , 40, terms are: [0, 7, 0, 55, 0, 433, 0, 3409, 0, 26839, 0, 211303, 0, 1663585, 0, 13097377, 0, 103115431, 0, 811826071, 0, 6391493137, 0, 50320119025, 0, 396169459063, 0, 3119035553479, 0, 24556114968769, 0, 193329884196673, 0, 1522082958604615, 0, 11983333784640247, 0, 94344587318517361, 0, 742773364763498641] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 4 2 6 8 10 -1 - 76 z + 17 z + 76 z - 17 z + z f(z) = - --------------------------------------------------- 8 2 10 12 6 4 1 + 213 z - 28 z - 28 z + z - 456 z + 213 z And in Maple-input format, it is: -(-1-76*z^4+17*z^2+76*z^6-17*z^8+z^10)/(1+213*z^8-28*z^2-28*z^10+z^12-456*z^6+ 213*z^4) The first , 40, terms are: [0, 11, 0, 171, 0, 2825, 0, 47497, 0, 803851, 0, 13643051, 0, 231846849, 0, 3942260033, 0, 67051240171, 0, 1140571991371, 0, 19402769999753, 0, 330077868030729, 0, 5615319360255851, 0, 95528950974808651, 0, 1625162190336729217, 0, 27647695951168299905, 0, 470350301489302183371, 0, 8001732395201746668011, 0, 136127751850926497466121, 0, 2315844238901871290111881] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 16 14 12 10 8 6 4 f(z) = - (-39 z + 446 z - 2094 z + 4376 z - 4376 z + 2094 z - 446 z 2 18 / 18 20 12 16 14 + 39 z - 1 + z ) / (-52 z + z + 16490 z + 881 z - 5720 z / 8 6 10 4 2 + 16490 z - 5720 z - 23272 z + 881 z - 52 z + 1) And in Maple-input format, it is: -(-39*z^16+446*z^14-2094*z^12+4376*z^10-4376*z^8+2094*z^6-446*z^4+39*z^2-1+z^18 )/(-52*z^18+z^20+16490*z^12+881*z^16-5720*z^14+16490*z^8-5720*z^6-23272*z^10+ 881*z^4-52*z^2+1) The first , 40, terms are: [0, 13, 0, 241, 0, 4705, 0, 94585, 0, 1956361, 0, 41628037, 0, 909944089, 0, 20379064489, 0, 466027553557, 0, 10841906256601, 0, 255709165686793, 0, 6095284916379217, 0, 146465443115749153, 0, 3540644935006450813, 0, 85970610631670642929, 0, 2094209379897799824913, 0, 51133748092066065997213, 0, 1250632580605213480865473, 0, 30625301720143814194696561, 0, 750603067587705068165841385] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 4 6 f(z) = - (1 + z - 50 z - 50 z + 903 z - 8116 z + 903 z - 8116 z 8 10 12 14 18 + 41408 z - 128312 z + 250244 z - 312220 z - 128312 z 16 20 / 2 28 26 24 + 250244 z + 41408 z ) / ((-1 + z ) (z - 65 z + 1449 z / 22 20 18 16 14 - 15476 z + 90928 z - 313620 z + 654234 z - 835410 z 12 10 8 6 4 2 + 654234 z - 313620 z + 90928 z - 15476 z + 1449 z - 65 z + 1)) And in Maple-input format, it is: -(1+z^28-50*z^26-50*z^2+903*z^24-8116*z^22+903*z^4-8116*z^6+41408*z^8-128312*z^ 10+250244*z^12-312220*z^14-128312*z^18+250244*z^16+41408*z^20)/(-1+z^2)/(z^28-\ 65*z^26+1449*z^24-15476*z^22+90928*z^20-313620*z^18+654234*z^16-835410*z^14+ 654234*z^12-313620*z^10+90928*z^8-15476*z^6+1449*z^4-65*z^2+1) The first , 40, terms are: [0, 16, 0, 445, 0, 13955, 0, 453104, 0, 14882391, 0, 490832103, 0, 16212611824, 0, 535822333315, 0, 17712602919805, 0, 585570979310224, 0, 19359325820664529, 0, 640038440054657713, 0, 21160398615534447504, 0, 699588039789144265341, 0, 23129230873004218794755, 0, 764680673879939562033392, 0, 25281281142320185328153671, 0, 835830198192954029770737655, 0, 27633573071304821315018510320, 0, 913599875094182135742957724675] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 4 6 8 10 12 f(z) = - (-1 + 25 z - 235 z + 1041 z - 2246 z + 2246 z - 1041 z 14 16 18 / 2 4 6 8 + 235 z - 25 z + z ) / (1 - 39 z + 502 z - 2911 z + 8337 z / 10 12 14 16 18 20 - 11900 z + 8337 z - 2911 z + 502 z - 39 z + z ) And in Maple-input format, it is: -(-1+25*z^2-235*z^4+1041*z^6-2246*z^8+2246*z^10-1041*z^12+235*z^14-25*z^16+z^18 )/(1-39*z^2+502*z^4-2911*z^6+8337*z^8-11900*z^10+8337*z^12-2911*z^14+502*z^16-\ 39*z^18+z^20) The first , 40, terms are: [0, 14, 0, 279, 0, 5723, 0, 117802, 0, 2426437, 0, 49987373, 0, 1029851202, 0, 21217596419, 0, 437139930799, 0, 9006285024310, 0, 185554384984601, 0, 3822934658426025, 0, 78763058346715430, 0, 1622737540407402175, 0, 33432896274470250355, 0, 688810439706103160786, 0, 14191406535648908777373, 0, 292382356571215505974613, 0, 6023887924004569261856026, 0, 124108807897041291568008043] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 2039027324 z - 2989422384 z - 80 z + 3394736660 z 22 4 6 8 10 - 2989422384 z + 2692 z - 51128 z + 620390 z - 5150072 z 12 14 18 16 48 + 30522848 z - 132874832 z - 1073368256 z + 433325199 z + z 20 36 34 30 + 2039027324 z + 30522848 z - 132874832 z - 1073368256 z 42 44 46 32 38 40 - 51128 z + 2692 z - 80 z + 433325199 z - 5150072 z + 620390 z / 2 28 26 2 ) / ((-1 + z ) (1 + 5195047640 z - 7756818536 z - 96 z / 24 22 4 6 8 + 8862489228 z - 7756818536 z + 3754 z - 81024 z + 1096866 z 10 12 14 18 - 9997608 z + 64139758 z - 298347416 z - 2653245464 z 16 48 20 36 34 + 1026958471 z + z + 5195047640 z + 64139758 z - 298347416 z 30 42 44 46 32 - 2653245464 z - 81024 z + 3754 z - 96 z + 1026958471 z 38 40 - 9997608 z + 1096866 z )) And in Maple-input format, it is: -(1+2039027324*z^28-2989422384*z^26-80*z^2+3394736660*z^24-2989422384*z^22+2692 *z^4-51128*z^6+620390*z^8-5150072*z^10+30522848*z^12-132874832*z^14-1073368256* z^18+433325199*z^16+z^48+2039027324*z^20+30522848*z^36-132874832*z^34-\ 1073368256*z^30-51128*z^42+2692*z^44-80*z^46+433325199*z^32-5150072*z^38+620390 *z^40)/(-1+z^2)/(1+5195047640*z^28-7756818536*z^26-96*z^2+8862489228*z^24-\ 7756818536*z^22+3754*z^4-81024*z^6+1096866*z^8-9997608*z^10+64139758*z^12-\ 298347416*z^14-2653245464*z^18+1026958471*z^16+z^48+5195047640*z^20+64139758*z^ 36-298347416*z^34-2653245464*z^30-81024*z^42+3754*z^44-96*z^46+1026958471*z^32-\ 9997608*z^38+1096866*z^40) The first , 40, terms are: [0, 17, 0, 491, 0, 15827, 0, 528595, 0, 17886035, 0, 608283601, 0, 20729700281, 0, 707065668649, 0, 24126478366913, 0, 823387559735955, 0, 28102838060407571, 0, 959208274506294963, 0, 32740383079517703915, 0, 1117528296326416115937, 0, 38144789243379771937521, 0, 1302005598654324989976465, 0, 44441728942710444856787585, 0, 1516943021788827028822390571, 0, 51778289935056333242252701555, 0, 1767364767317859610772678167699] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 18 16 14 12 10 8 6 f(z) = - (z - 39 z + 431 z - 1919 z + 3970 z - 3970 z + 1919 z 4 2 / 20 18 16 14 12 - 431 z + 39 z - 1) / (z - 59 z + 886 z - 5473 z + 15647 z / 10 8 6 4 2 - 22012 z + 15647 z - 5473 z + 886 z - 59 z + 1) And in Maple-input format, it is: -(z^18-39*z^16+431*z^14-1919*z^12+3970*z^10-3970*z^8+1919*z^6-431*z^4+39*z^2-1) /(z^20-59*z^18+886*z^16-5473*z^14+15647*z^12-22012*z^10+15647*z^8-5473*z^6+886* z^4-59*z^2+1) The first , 40, terms are: [0, 20, 0, 725, 0, 28609, 0, 1143364, 0, 45783929, 0, 1833890801, 0, 73460633116, 0, 2942653836913, 0, 117875682687509, 0, 4721819196303260, 0, 189144841099319065, 0, 7576692293828591785, 0, 303504266030365497932, 0, 12157658770082097363029, 0, 487006883650891248843409, 0, 19508337025240565859483820, 0, 781457565112281152695315841, 0, 31303330739119974005639732585, 0, 1253936949502783020987990718516, 0, 50229730709242678263248772774433] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4748481129903 z - 2926335122612 z - 104 z 24 22 4 6 8 + 1481198168024 z - 613659153190 z + 4780 z - 129918 z + 2355754 z 10 12 14 18 - 30459007 z + 292894366 z - 2155812018 z - 56568441817 z 16 50 48 20 + 12398397522 z - 2155812018 z + 12398397522 z + 207101750162 z 36 34 64 30 + 4748481129903 z - 6343208431768 z + z - 6343208431768 z 42 44 46 58 - 613659153190 z + 207101750162 z - 56568441817 z - 129918 z 56 54 52 60 + 2355754 z - 30459007 z + 292894366 z + 4780 z 32 38 40 62 + 6984965701220 z - 2926335122612 z + 1481198168024 z - 104 z ) / 2 28 26 2 / ((-1 + z ) (1 + 12275420850884 z - 7438425655478 z - 122 z / 24 22 4 6 8 + 3677811513409 z - 1478834188638 z + 6364 z - 192258 z + 3818313 z 10 12 14 18 - 53469320 z + 551784724 z - 4324305780 z - 125986296464 z 16 50 48 20 + 26292690537 z - 4324305780 z + 26292690537 z + 481314967028 z 36 34 64 30 + 12275420850884 z - 16565683464896 z + z - 16565683464896 z 42 44 46 58 - 1478834188638 z + 481314967028 z - 125986296464 z - 192258 z 56 54 52 60 + 3818313 z - 53469320 z + 551784724 z + 6364 z 32 38 40 62 + 18303823900309 z - 7438425655478 z + 3677811513409 z - 122 z )) And in Maple-input format, it is: -(1+4748481129903*z^28-2926335122612*z^26-104*z^2+1481198168024*z^24-\ 613659153190*z^22+4780*z^4-129918*z^6+2355754*z^8-30459007*z^10+292894366*z^12-\ 2155812018*z^14-56568441817*z^18+12398397522*z^16-2155812018*z^50+12398397522*z ^48+207101750162*z^20+4748481129903*z^36-6343208431768*z^34+z^64-6343208431768* z^30-613659153190*z^42+207101750162*z^44-56568441817*z^46-129918*z^58+2355754*z ^56-30459007*z^54+292894366*z^52+4780*z^60+6984965701220*z^32-2926335122612*z^ 38+1481198168024*z^40-104*z^62)/(-1+z^2)/(1+12275420850884*z^28-7438425655478*z ^26-122*z^2+3677811513409*z^24-1478834188638*z^22+6364*z^4-192258*z^6+3818313*z ^8-53469320*z^10+551784724*z^12-4324305780*z^14-125986296464*z^18+26292690537*z ^16-4324305780*z^50+26292690537*z^48+481314967028*z^20+12275420850884*z^36-\ 16565683464896*z^34+z^64-16565683464896*z^30-1478834188638*z^42+481314967028*z^ 44-125986296464*z^46-192258*z^58+3818313*z^56-53469320*z^54+551784724*z^52+6364 *z^60+18303823900309*z^32-7438425655478*z^38+3677811513409*z^40-122*z^62) The first , 40, terms are: [0, 19, 0, 631, 0, 23083, 0, 865544, 0, 32703833, 0, 1238879749, 0, 46974166337, 0, 1781705711495, 0, 67587777582223, 0, 2564020820222641, 0, 97270953831594017, 0, 3690184314906757637, 0, 139995555126886201736, 0, 5311057030210242912503, 0, 201487403549486381232419, 0, 7643898759600132221049251, 0, 289989311144566215377381273, 0, 11001428242642773407884033497, 0, 417365125698116423354519858163, 0, 15833730448046547149345359737851] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 10 8 6 4 12 2 -16 z + 81 z - 148 z + 81 z + z - 16 z + 1 f(z) = - --------------------------------------------------------------- 12 10 8 6 4 2 2 (z - 30 z + 197 z - 404 z + 197 z - 30 z + 1) (-1 + z ) And in Maple-input format, it is: -(-16*z^10+81*z^8-148*z^6+81*z^4+z^12-16*z^2+1)/(z^12-30*z^10+197*z^8-404*z^6+ 197*z^4-30*z^2+1)/(-1+z^2) The first , 40, terms are: [0, 15, 0, 319, 0, 6937, 0, 151129, 0, 3293215, 0, 71764175, 0, 1563860961, 0, 34079180577, 0, 742643298255, 0, 16183461305055, 0, 352665167371225, 0, 7685174282026905, 0, 167473028937718143, 0, 3649522365532890127, 0, 79529304398418821313, 0, 1733078914071423945537, 0, 37766739507192061140495, 0, 823001538720336390916863, 0, 17934604405209559498012313, 0, 390825557472823625144985817] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 2124465537 z - 3117523603 z - 81 z + 3541275382 z 22 4 6 8 10 - 3117523603 z + 2739 z - 52200 z + 635597 z - 5295427 z 12 14 18 16 48 + 31495932 z - 137555361 z - 1116631952 z + 449821159 z + z 20 36 34 30 + 2124465537 z + 31495932 z - 137555361 z - 1116631952 z 42 44 46 32 38 40 - 52200 z + 2739 z - 81 z + 449821159 z - 5295427 z + 635597 z / 2 28 26 2 ) / ((-1 + z ) (1 + 5413882644 z - 8074827246 z - 98 z / 24 22 4 6 8 + 9222269254 z - 8074827246 z + 3884 z - 84472 z + 1148104 z 10 12 14 18 - 10481250 z + 67241860 z - 312452878 z - 2769356440 z 16 48 20 36 34 + 1073791576 z + z + 5413882644 z + 67241860 z - 312452878 z 30 42 44 46 32 - 2769356440 z - 84472 z + 3884 z - 98 z + 1073791576 z 38 40 - 10481250 z + 1148104 z )) And in Maple-input format, it is: -(1+2124465537*z^28-3117523603*z^26-81*z^2+3541275382*z^24-3117523603*z^22+2739 *z^4-52200*z^6+635597*z^8-5295427*z^10+31495932*z^12-137555361*z^14-1116631952* z^18+449821159*z^16+z^48+2124465537*z^20+31495932*z^36-137555361*z^34-\ 1116631952*z^30-52200*z^42+2739*z^44-81*z^46+449821159*z^32-5295427*z^38+635597 *z^40)/(-1+z^2)/(1+5413882644*z^28-8074827246*z^26-98*z^2+9222269254*z^24-\ 8074827246*z^22+3884*z^4-84472*z^6+1148104*z^8-10481250*z^10+67241860*z^12-\ 312452878*z^14-2769356440*z^18+1073791576*z^16+z^48+5413882644*z^20+67241860*z^ 36-312452878*z^34-2769356440*z^30-84472*z^42+3884*z^44-98*z^46+1073791576*z^32-\ 10481250*z^38+1148104*z^40) The first , 40, terms are: [0, 18, 0, 539, 0, 17841, 0, 613390, 0, 21454191, 0, 756548055, 0, 26785312510, 0, 950212921249, 0, 33742704834619, 0, 1198833892338050, 0, 42603966407818841, 0, 1514251634134458569, 0, 53823896313420138850, 0, 1913229401603376127067, 0, 68009022077808346100961, 0, 2417518905734106027971166, 0, 85936013216017625941081959, 0, 3054791211103502969788372895, 0, 108589636095639164537947767086, 0, 3860072714961658935490271831985] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 14 12 10 8 6 4 2 / f(z) = - (z - 35 z + 277 z - 727 z + 727 z - 277 z + 35 z - 1) / ( / 16 14 12 10 8 6 4 2 z - 56 z + 672 z - 2632 z + 4094 z - 2632 z + 672 z - 56 z + 1) And in Maple-input format, it is: -(z^14-35*z^12+277*z^10-727*z^8+727*z^6-277*z^4+35*z^2-1)/(z^16-56*z^14+672*z^ 12-2632*z^10+4094*z^8-2632*z^6+672*z^4-56*z^2+1) The first , 40, terms are: [0, 21, 0, 781, 0, 31529, 0, 1292697, 0, 53175517, 0, 2188978117, 0, 90124167441, 0, 3710708201969, 0, 152783289861989, 0, 6290652543875133, 0, 259009513044645817, 0, 10664383939345916681, 0, 439092316687230373293, 0, 18079062471131097321077, 0, 744382189686310539093281, 0, 30648981125778378496845537, 0, 1261932455010147392171339189, 0, 51958448944552497866018608237, 0, 2139322438385927464134216597321, 0, 88083855241102209799967552571257] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 26 2 24 22 4 6 8 f(z) = - (-1 + z + 57 z - 57 z + 1204 z - 1204 z + 12450 z - 69362 z 10 12 14 18 16 + 216188 z - 381682 z + 381682 z + 69362 z - 216188 z 20 / 28 26 24 22 20 - 12450 z ) / (z - 69 z + 1833 z - 24262 z + 175526 z / 18 16 14 12 10 - 719678 z + 1692386 z - 2262874 z + 1692386 z - 719678 z 8 6 4 2 + 175526 z - 24262 z + 1833 z - 69 z + 1) And in Maple-input format, it is: -(-1+z^26+57*z^2-57*z^24+1204*z^22-1204*z^4+12450*z^6-69362*z^8+216188*z^10-\ 381682*z^12+381682*z^14+69362*z^18-216188*z^16-12450*z^20)/(z^28-69*z^26+1833*z ^24-24262*z^22+175526*z^20-719678*z^18+1692386*z^16-2262874*z^14+1692386*z^12-\ 719678*z^10+175526*z^8-24262*z^6+1833*z^4-69*z^2+1) The first , 40, terms are: [0, 12, 0, 199, 0, 3547, 0, 64956, 0, 1205629, 0, 22577125, 0, 426063900, 0, 8105634259, 0, 155635770703, 0, 3020962625580, 0, 59388655754905, 0, 1184718643355497, 0, 24024128328114924, 0, 495922215784467775, 0, 10430157742413569251, 0, 223546782711165714012, 0, 4880140351110676207381, 0, 108391698202025847853645, 0, 2445486804047863156280700, 0, 55939763328471550377387211] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 475434700668 z + 475434700668 z + 107 z - 364021797942 z 22 4 6 8 10 + 213468792294 z - 4969 z + 134078 z - 2368758 z + 29207137 z 12 14 18 16 - 261111247 z + 1733303425 z + 32990085632 z - 8671648880 z 50 48 20 36 + 4969 z - 134078 z - 95900752382 z - 32990085632 z 34 30 42 44 + 95900752382 z + 364021797942 z + 261111247 z - 29207137 z 46 54 52 32 38 + 2368758 z + z - 107 z - 213468792294 z + 8671648880 z 40 / 28 26 2 - 1733303425 z ) / (1 + 2683353143582 z - 2352623287260 z - 122 z / 24 22 4 6 8 + 1585846527946 z - 822160610096 z + 6483 z - 199768 z + 4011557 z 10 12 14 18 - 55933444 z + 562997995 z - 4195301646 z - 100377374316 z 16 50 48 20 + 23529313649 z - 199768 z + 4011557 z + 327794355286 z 36 34 30 42 + 327794355286 z - 822160610096 z - 2352623287260 z - 4195301646 z 44 46 56 54 52 + 562997995 z - 55933444 z + z - 122 z + 6483 z 32 38 40 + 1585846527946 z - 100377374316 z + 23529313649 z ) And in Maple-input format, it is: -(-1-475434700668*z^28+475434700668*z^26+107*z^2-364021797942*z^24+213468792294 *z^22-4969*z^4+134078*z^6-2368758*z^8+29207137*z^10-261111247*z^12+1733303425*z ^14+32990085632*z^18-8671648880*z^16+4969*z^50-134078*z^48-95900752382*z^20-\ 32990085632*z^36+95900752382*z^34+364021797942*z^30+261111247*z^42-29207137*z^ 44+2368758*z^46+z^54-107*z^52-213468792294*z^32+8671648880*z^38-1733303425*z^40 )/(1+2683353143582*z^28-2352623287260*z^26-122*z^2+1585846527946*z^24-\ 822160610096*z^22+6483*z^4-199768*z^6+4011557*z^8-55933444*z^10+562997995*z^12-\ 4195301646*z^14-100377374316*z^18+23529313649*z^16-199768*z^50+4011557*z^48+ 327794355286*z^20+327794355286*z^36-822160610096*z^34-2352623287260*z^30-\ 4195301646*z^42+562997995*z^44-55933444*z^46+z^56-122*z^54+6483*z^52+ 1585846527946*z^32-100377374316*z^38+23529313649*z^40) The first , 40, terms are: [0, 15, 0, 316, 0, 6997, 0, 158727, 0, 3682783, 0, 87511981, 0, 2132687100, 0, 53338240951, 0, 1368468122857, 0, 35965414989705, 0, 966010748854183, 0, 26441253000509692, 0, 735291993667337757, 0, 20712607182564153439, 0, 589460326812069817639, 0, 16909700265548035068069, 0, 488058713074194388248508, 0, 14152090638121626147297055, 0, 411794926142701243199512449, 0, 12013502616708031766773216321] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 63879000540216 z - 32623909595670 z - 123 z 24 22 4 6 + 13688765075354 z - 4701067502344 z + 6742 z - 219867 z 8 10 12 14 + 4803017 z - 74995204 z + 871866029 z - 7760191567 z 18 16 50 - 297411824795 z + 53952840850 z - 297411824795 z 48 20 36 + 1314701296393 z + 1314701296393 z + 137184841679318 z 34 66 64 30 - 150899015931272 z - 123 z + 6742 z - 103042405817834 z 42 44 46 - 32623909595670 z + 13688765075354 z - 4701067502344 z 58 56 54 52 - 74995204 z + 871866029 z - 7760191567 z + 53952840850 z 60 68 32 38 + 4803017 z + z + 137184841679318 z - 103042405817834 z 40 62 / 2 28 + 63879000540216 z - 219867 z ) / ((-1 + z ) (1 + 175925651070718 z / 26 2 24 22 - 87398048682624 z - 140 z + 35409854420186 z - 11659780272392 z 4 6 8 10 12 + 8595 z - 310354 z + 7440565 z - 126572626 z + 1592805765 z 14 18 16 50 - 15253470706 z - 664821399908 z + 113430465435 z - 664821399908 z 48 20 36 + 3105547835929 z + 3105547835929 z + 390127561829002 z 34 66 64 30 - 430868530942788 z - 140 z + 8595 z - 289511349767764 z 42 44 46 - 87398048682624 z + 35409854420186 z - 11659780272392 z 58 56 54 52 - 126572626 z + 1592805765 z - 15253470706 z + 113430465435 z 60 68 32 38 + 7440565 z + z + 390127561829002 z - 289511349767764 z 40 62 + 175925651070718 z - 310354 z )) And in Maple-input format, it is: -(1+63879000540216*z^28-32623909595670*z^26-123*z^2+13688765075354*z^24-\ 4701067502344*z^22+6742*z^4-219867*z^6+4803017*z^8-74995204*z^10+871866029*z^12 -7760191567*z^14-297411824795*z^18+53952840850*z^16-297411824795*z^50+ 1314701296393*z^48+1314701296393*z^20+137184841679318*z^36-150899015931272*z^34 -123*z^66+6742*z^64-103042405817834*z^30-32623909595670*z^42+13688765075354*z^ 44-4701067502344*z^46-74995204*z^58+871866029*z^56-7760191567*z^54+53952840850* z^52+4803017*z^60+z^68+137184841679318*z^32-103042405817834*z^38+63879000540216 *z^40-219867*z^62)/(-1+z^2)/(1+175925651070718*z^28-87398048682624*z^26-140*z^2 +35409854420186*z^24-11659780272392*z^22+8595*z^4-310354*z^6+7440565*z^8-\ 126572626*z^10+1592805765*z^12-15253470706*z^14-664821399908*z^18+113430465435* z^16-664821399908*z^50+3105547835929*z^48+3105547835929*z^20+390127561829002*z^ 36-430868530942788*z^34-140*z^66+8595*z^64-289511349767764*z^30-87398048682624* z^42+35409854420186*z^44-11659780272392*z^46-126572626*z^58+1592805765*z^56-\ 15253470706*z^54+113430465435*z^52+7440565*z^60+z^68+390127561829002*z^32-\ 289511349767764*z^38+175925651070718*z^40-310354*z^62) The first , 40, terms are: [0, 18, 0, 545, 0, 18697, 0, 668882, 0, 24322717, 0, 890682501, 0, 32722075082, 0, 1204042292421, 0, 44339064387257, 0, 1633462473315458, 0, 60190175499757189, 0, 2218156825059303589, 0, 81749670848188213634, 0, 3012968179719947433273, 0, 111048106394857046212405, 0, 4092910104355505083174090, 0, 150853606223242050278765157, 0, 5560073512148949219398492557, 0, 204930270718848346489044683186, 0, 7553219479264880897667558235593] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 7}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 14 12 6 10 4 8 2 / f(z) = - (z - 22 z + 415 z + 154 z - 154 z - 415 z + 22 z - 1) / ( / 16 14 12 6 10 4 8 2 z - 40 z + 411 z - 1570 z - 1570 z + 411 z + 2468 z - 40 z + 1) And in Maple-input format, it is: -(z^14-22*z^12+415*z^6+154*z^10-154*z^4-415*z^8+22*z^2-1)/(z^16-40*z^14+411*z^ 12-1570*z^6-1570*z^10+411*z^4+2468*z^8-40*z^2+1) The first , 40, terms are: [0, 18, 0, 463, 0, 12277, 0, 326994, 0, 8717815, 0, 232478143, 0, 6199904250, 0, 165346686901, 0, 4409690299615, 0, 117603770349498, 0, 3136422456313753, 0, 83646525435571657, 0, 2230803247095371562, 0, 59494200585030713695, 0, 1586675075301099136789, 0, 42315684069090148892010, 0, 1128534219994566480175567, 0, 30097338936280758051367975, 0, 802678195313600456782763106, 0, 21406951843741847235009166933] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 2297788 z + 9844728 z + 76 z - 28922859 z + 58982912 z 4 6 8 10 12 - 2276 z + 36685 z - 360391 z + 2297788 z - 9844728 z 14 18 16 20 36 + 28922859 z + 84091548 z - 58982912 z - 84091548 z - 76 z 34 30 32 38 / 10 2 + 2276 z + 360391 z - 36685 z + z ) / (-5830848 z - 92 z / 28 40 38 4 14 16 + 29199203 z + z - 92 z + 3389 z - 100128260 z + 239049023 z 32 30 26 6 22 + 776976 z - 5830848 z - 100128260 z - 66312 z - 401456504 z 20 8 24 34 18 + 476910688 z + 776976 z + 239049023 z - 66312 z - 401456504 z 12 36 + 1 + 29199203 z + 3389 z ) And in Maple-input format, it is: -(-1-2297788*z^28+9844728*z^26+76*z^2-28922859*z^24+58982912*z^22-2276*z^4+ 36685*z^6-360391*z^8+2297788*z^10-9844728*z^12+28922859*z^14+84091548*z^18-\ 58982912*z^16-84091548*z^20-76*z^36+2276*z^34+360391*z^30-36685*z^32+z^38)/(-\ 5830848*z^10-92*z^2+29199203*z^28+z^40-92*z^38+3389*z^4-100128260*z^14+ 239049023*z^16+776976*z^32-5830848*z^30-100128260*z^26-66312*z^6-401456504*z^22 +476910688*z^20+776976*z^8+239049023*z^24-66312*z^34-401456504*z^18+1+29199203* z^12+3389*z^36) The first , 40, terms are: [0, 16, 0, 359, 0, 8431, 0, 203408, 0, 5048329, 0, 129177737, 0, 3410564048, 0, 92809551823, 0, 2595982107303, 0, 74353185326288, 0, 2171467144830433, 0, 64400672605301473, 0, 1932630317996588624, 0, 58511157726041964647, 0, 1782970520256334405711, 0, 54587274789777747724368, 0, 1676892098995991398133321, 0, 51637450432051104566618057, 0, 1592820469355856529036358672, 0, 49191934341168093067127516399] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 403230566700 z - 359369524360 z - 104 z + 254228764312 z 22 4 6 8 10 - 142475588480 z + 4650 z - 119695 z + 2003550 z - 23320505 z 12 14 18 16 + 196821094 z - 1238788192 z - 21899214308 z + 5930129967 z 50 48 20 36 - 119695 z + 2003550 z + 63033551152 z + 63033551152 z 34 30 42 44 - 142475588480 z - 359369524360 z - 1238788192 z + 196821094 z 46 56 54 52 32 - 23320505 z + z - 104 z + 4650 z + 254228764312 z 38 40 / 28 - 21899214308 z + 5930129967 z ) / (-1 - 2176848322730 z / 26 2 24 22 + 1734610631838 z + 123 z - 1100014093262 z + 553694537294 z 4 6 8 10 12 - 6391 z + 188851 z - 3594401 z + 47213817 z - 447074043 z 14 18 16 50 + 3143608679 z + 68850758337 z - 16763833027 z + 3594401 z 48 20 36 34 - 47213817 z - 220287013106 z - 553694537294 z + 1100014093262 z 30 42 44 46 + 2176848322730 z + 16763833027 z - 3143608679 z + 447074043 z 58 56 54 52 32 + z - 123 z + 6391 z - 188851 z - 1734610631838 z 38 40 + 220287013106 z - 68850758337 z ) And in Maple-input format, it is: -(1+403230566700*z^28-359369524360*z^26-104*z^2+254228764312*z^24-142475588480* z^22+4650*z^4-119695*z^6+2003550*z^8-23320505*z^10+196821094*z^12-1238788192*z^ 14-21899214308*z^18+5930129967*z^16-119695*z^50+2003550*z^48+63033551152*z^20+ 63033551152*z^36-142475588480*z^34-359369524360*z^30-1238788192*z^42+196821094* z^44-23320505*z^46+z^56-104*z^54+4650*z^52+254228764312*z^32-21899214308*z^38+ 5930129967*z^40)/(-1-2176848322730*z^28+1734610631838*z^26+123*z^2-\ 1100014093262*z^24+553694537294*z^22-6391*z^4+188851*z^6-3594401*z^8+47213817*z ^10-447074043*z^12+3143608679*z^14+68850758337*z^18-16763833027*z^16+3594401*z^ 50-47213817*z^48-220287013106*z^20-553694537294*z^36+1100014093262*z^34+ 2176848322730*z^30+16763833027*z^42-3143608679*z^44+447074043*z^46+z^58-123*z^ 56+6391*z^54-188851*z^52-1734610631838*z^32+220287013106*z^38-68850758337*z^40) The first , 40, terms are: [0, 19, 0, 596, 0, 21035, 0, 775587, 0, 29117405, 0, 1101691661, 0, 41830743052, 0, 1590931726021, 0, 60556515357631, 0, 2305947266533359, 0, 87827614979250357, 0, 3345509434458687564, 0, 127444185578239715901, 0, 4855033579599994912525, 0, 184957670064630537257299, 0, 7046229586090222359809531, 0, 268437769341466750153583124, 0, 10226611803579914404695381475, 0, 389601525106881189450984864881, 0, 14842598118714699456747033493137] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2378163725298537414 z - 410327208125247815 z - 191 z 24 22 4 6 + 61542957887903056 z - 7979760014560513 z + 17034 z - 946665 z 102 8 10 12 - 1077770683 z + 36911752 z - 1077770683 z + 24559820830 z 14 18 16 - 449419577101 z - 84283132066611 z + 6743134180332 z 50 48 - 141693710503531860521939 z + 94702789544393831265436 z 20 36 + 888544672043578 z + 717474517872954175674 z 34 66 - 208448060553591297659 z - 56366207998279747697465 z 80 100 90 + 53410865113972395748 z + 24559820830 z - 7979760014560513 z 88 84 94 + 61542957887903056 z + 2378163725298537414 z - 84283132066611 z 86 96 98 - 410327208125247815 z + 6743134180332 z - 449419577101 z 92 82 + 888544672043578 z - 12036907156839463629 z 64 112 110 106 108 + 94702789544393831265436 z + z - 191 z - 946665 z + 17034 z 30 42 - 12036907156839463629 z - 14061589630434402186063 z 44 46 + 29856042705134155493478 z - 56366207998279747697465 z 58 56 - 224411494325245513227877 z + 237676070308751833571850 z 54 52 - 224411494325245513227877 z + 188884178017676057204698 z 60 70 + 188884178017676057204698 z - 14061589630434402186063 z 68 78 + 29856042705134155493478 z - 208448060553591297659 z 32 38 + 53410865113972395748 z - 2182986631597253024657 z 40 62 + 5882521041938549936296 z - 141693710503531860521939 z 76 74 + 717474517872954175674 z - 2182986631597253024657 z 72 104 / + 5882521041938549936296 z + 36911752 z ) / (-1 / 28 26 2 - 6225664738247187796 z + 1017941483880104490 z + 212 z 24 22 4 6 - 144573571415925980 z + 17733782849867174 z - 20735 z + 1252804 z 102 8 10 12 + 40374594042 z - 52759710 z + 1655424960 z - 40374594042 z 14 18 16 + 788189522183 z + 166989650480661 z - 12583275156835 z 50 48 + 664128699274346224595989 z - 420386294891190390547507 z 20 36 - 1865844874871637 z - 2318883408057230098871 z 34 66 + 639356925688684628823 z + 420386294891190390547507 z 80 100 90 - 639356925688684628823 z - 788189522183 z + 144573571415925980 z 88 84 - 1017941483880104490 z - 33232008758732667969 z 94 86 96 + 1865844874871637 z + 6225664738247187796 z - 166989650480661 z 98 92 82 + 12583275156835 z - 17733782849867174 z + 155447660488653091849 z 64 112 114 110 - 664128699274346224595989 z - 212 z + z + 20735 z 106 108 30 + 52759710 z - 1252804 z + 33232008758732667969 z 42 44 + 53189475259957956316856 z - 119069593721338489972462 z 46 58 + 237104156375556133270179 z + 1316951308354380377244582 z 56 54 - 1316951308354380377244582 z + 1175073135375101986777822 z 52 60 - 935401180685872460353561 z - 1175073135375101986777822 z 70 68 + 119069593721338489972462 z - 237104156375556133270179 z 78 32 + 2318883408057230098871 z - 155447660488653091849 z 38 40 + 7434303756553134047432 z - 21111050962969656519518 z 62 76 + 935401180685872460353561 z - 7434303756553134047432 z 74 72 + 21111050962969656519518 z - 53189475259957956316856 z 104 - 1655424960 z ) And in Maple-input format, it is: -(1+2378163725298537414*z^28-410327208125247815*z^26-191*z^2+61542957887903056* z^24-7979760014560513*z^22+17034*z^4-946665*z^6-1077770683*z^102+36911752*z^8-\ 1077770683*z^10+24559820830*z^12-449419577101*z^14-84283132066611*z^18+ 6743134180332*z^16-141693710503531860521939*z^50+94702789544393831265436*z^48+ 888544672043578*z^20+717474517872954175674*z^36-208448060553591297659*z^34-\ 56366207998279747697465*z^66+53410865113972395748*z^80+24559820830*z^100-\ 7979760014560513*z^90+61542957887903056*z^88+2378163725298537414*z^84-\ 84283132066611*z^94-410327208125247815*z^86+6743134180332*z^96-449419577101*z^ 98+888544672043578*z^92-12036907156839463629*z^82+94702789544393831265436*z^64+ z^112-191*z^110-946665*z^106+17034*z^108-12036907156839463629*z^30-\ 14061589630434402186063*z^42+29856042705134155493478*z^44-\ 56366207998279747697465*z^46-224411494325245513227877*z^58+ 237676070308751833571850*z^56-224411494325245513227877*z^54+ 188884178017676057204698*z^52+188884178017676057204698*z^60-\ 14061589630434402186063*z^70+29856042705134155493478*z^68-208448060553591297659 *z^78+53410865113972395748*z^32-2182986631597253024657*z^38+ 5882521041938549936296*z^40-141693710503531860521939*z^62+717474517872954175674 *z^76-2182986631597253024657*z^74+5882521041938549936296*z^72+36911752*z^104)/( -1-6225664738247187796*z^28+1017941483880104490*z^26+212*z^2-144573571415925980 *z^24+17733782849867174*z^22-20735*z^4+1252804*z^6+40374594042*z^102-52759710*z ^8+1655424960*z^10-40374594042*z^12+788189522183*z^14+166989650480661*z^18-\ 12583275156835*z^16+664128699274346224595989*z^50-420386294891190390547507*z^48 -1865844874871637*z^20-2318883408057230098871*z^36+639356925688684628823*z^34+ 420386294891190390547507*z^66-639356925688684628823*z^80-788189522183*z^100+ 144573571415925980*z^90-1017941483880104490*z^88-33232008758732667969*z^84+ 1865844874871637*z^94+6225664738247187796*z^86-166989650480661*z^96+ 12583275156835*z^98-17733782849867174*z^92+155447660488653091849*z^82-\ 664128699274346224595989*z^64-212*z^112+z^114+20735*z^110+52759710*z^106-\ 1252804*z^108+33232008758732667969*z^30+53189475259957956316856*z^42-\ 119069593721338489972462*z^44+237104156375556133270179*z^46+ 1316951308354380377244582*z^58-1316951308354380377244582*z^56+ 1175073135375101986777822*z^54-935401180685872460353561*z^52-\ 1175073135375101986777822*z^60+119069593721338489972462*z^70-\ 237104156375556133270179*z^68+2318883408057230098871*z^78-155447660488653091849 *z^32+7434303756553134047432*z^38-21111050962969656519518*z^40+ 935401180685872460353561*z^62-7434303756553134047432*z^76+ 21111050962969656519518*z^74-53189475259957956316856*z^72-1655424960*z^104) The first , 40, terms are: [0, 21, 0, 751, 0, 29916, 0, 1231133, 0, 51248107, 0, 2142549131, 0, 89727394859, 0, 3760276836579, 0, 157630028296893, 0, 6608615123048876, 0, 277079472841029423, 0, 11617373269438059181, 0, 487097372500122106321, 0, 20423281124829438166241, 0, 856320001213907582235005, 0, 35904347786301605181649663, 0, 1505421692700130454733444940, 0, 63120346943979935868724637581, 0, 2646553159077112678438003681459, 0, 110966498753082792237612120834139] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 18084356988467786 z - 5059523931959079 z - 155 z 24 22 4 6 + 1204023122540174 z - 242496853703317 z + 10943 z - 470460 z 8 10 12 14 + 13884859 z - 300534691 z + 4974926028 z - 64826768761 z 18 16 50 - 5808947633865 z + 679054021572 z - 1046943106637643557 z 48 20 36 + 1507591257689502849 z + 41080770029610 z + 627656730507620935 z 34 66 80 88 - 324461441682031302 z - 242496853703317 z + 13884859 z + z 84 86 82 64 + 10943 z - 155 z - 470460 z + 1204023122540174 z 30 42 - 55204590870617711 z - 1875688533493095150 z 44 46 + 2017267449460046132 z - 1875688533493095150 z 58 56 54 - 55204590870617711 z + 144393709693497809 z - 324461441682031302 z 52 60 70 + 627656730507620935 z + 18084356988467786 z - 5808947633865 z 68 78 32 + 41080770029610 z - 300534691 z + 144393709693497809 z 38 40 62 - 1046943106637643557 z + 1507591257689502849 z - 5059523931959079 z 76 74 72 / + 4974926028 z - 64826768761 z + 679054021572 z ) / (-1 / 28 26 2 - 57635715696000660 z + 15104511835485821 z + 175 z 24 22 4 6 - 3365202373103537 z + 633971172237670 z - 13794 z + 656208 z 8 10 12 14 - 21273698 z + 502732984 z - 9039990352 z + 127420732304 z 18 16 50 + 13228762589819 z - 1438724065399 z + 7155163923532357349 z 48 20 36 - 9544447841768892491 z - 100326756941914 z - 2601070653116023724 z 34 66 80 90 + 1258248708829000270 z + 3365202373103537 z - 502732984 z + z 88 84 86 82 64 - 175 z - 656208 z + 13794 z + 21273698 z - 15104511835485821 z 30 42 + 187790853395570444 z + 9544447841768892491 z 44 46 - 11021638199869503664 z + 11021638199869503664 z 58 56 + 524338467713185784 z - 1258248708829000270 z 54 52 + 2601070653116023724 z - 4640521436298821650 z 60 70 68 - 187790853395570444 z + 100326756941914 z - 633971172237670 z 78 32 38 + 9039990352 z - 524338467713185784 z + 4640521436298821650 z 40 62 76 - 7155163923532357349 z + 57635715696000660 z - 127420732304 z 74 72 + 1438724065399 z - 13228762589819 z ) And in Maple-input format, it is: -(1+18084356988467786*z^28-5059523931959079*z^26-155*z^2+1204023122540174*z^24-\ 242496853703317*z^22+10943*z^4-470460*z^6+13884859*z^8-300534691*z^10+ 4974926028*z^12-64826768761*z^14-5808947633865*z^18+679054021572*z^16-\ 1046943106637643557*z^50+1507591257689502849*z^48+41080770029610*z^20+ 627656730507620935*z^36-324461441682031302*z^34-242496853703317*z^66+13884859*z ^80+z^88+10943*z^84-155*z^86-470460*z^82+1204023122540174*z^64-\ 55204590870617711*z^30-1875688533493095150*z^42+2017267449460046132*z^44-\ 1875688533493095150*z^46-55204590870617711*z^58+144393709693497809*z^56-\ 324461441682031302*z^54+627656730507620935*z^52+18084356988467786*z^60-\ 5808947633865*z^70+41080770029610*z^68-300534691*z^78+144393709693497809*z^32-\ 1046943106637643557*z^38+1507591257689502849*z^40-5059523931959079*z^62+ 4974926028*z^76-64826768761*z^74+679054021572*z^72)/(-1-57635715696000660*z^28+ 15104511835485821*z^26+175*z^2-3365202373103537*z^24+633971172237670*z^22-13794 *z^4+656208*z^6-21273698*z^8+502732984*z^10-9039990352*z^12+127420732304*z^14+ 13228762589819*z^18-1438724065399*z^16+7155163923532357349*z^50-\ 9544447841768892491*z^48-100326756941914*z^20-2601070653116023724*z^36+ 1258248708829000270*z^34+3365202373103537*z^66-502732984*z^80+z^90-175*z^88-\ 656208*z^84+13794*z^86+21273698*z^82-15104511835485821*z^64+187790853395570444* z^30+9544447841768892491*z^42-11021638199869503664*z^44+11021638199869503664*z^ 46+524338467713185784*z^58-1258248708829000270*z^56+2601070653116023724*z^54-\ 4640521436298821650*z^52-187790853395570444*z^60+100326756941914*z^70-\ 633971172237670*z^68+9039990352*z^78-524338467713185784*z^32+ 4640521436298821650*z^38-7155163923532357349*z^40+57635715696000660*z^62-\ 127420732304*z^76+1438724065399*z^74-13228762589819*z^72) The first , 40, terms are: [0, 20, 0, 649, 0, 23443, 0, 885540, 0, 34200083, 0, 1336325263, 0, 52547969348, 0, 2073641704139, 0, 81992761646477, 0, 3245695713490308, 0, 128564160501135753, 0, 5094390097249268713, 0, 201909313765476807316, 0, 8003377414233662829373, 0, 317263864915885208494459, 0, 12577241813247499050201828, 0, 498609201223992408778110127, 0, 19767009791506259787967776675, 0, 783655198215596007848257788196, 0, 31067835368991750756968780192755] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 83782370546544 z - 42774030141430 z - 131 z 24 22 4 6 + 17931012397730 z - 6146711074840 z + 7566 z - 257167 z 8 10 12 14 + 5799365 z - 92752796 z + 1097818697 z - 9901775243 z 18 16 50 - 385834309139 z + 69511680146 z - 385834309139 z 48 20 36 + 1713693956865 z + 1713693956865 z + 179922140722942 z 34 66 64 30 - 197901029333752 z - 131 z + 7566 z - 135152747954986 z 42 44 46 - 42774030141430 z + 17931012397730 z - 6146711074840 z 58 56 54 52 - 92752796 z + 1097818697 z - 9901775243 z + 69511680146 z 60 68 32 38 + 5799365 z + z + 179922140722942 z - 135152747954986 z 40 62 / 2 28 + 83782370546544 z - 257167 z ) / ((-1 + z ) (1 + 231283882067822 z / 26 2 24 22 - 115368088652404 z - 150 z + 46945054354378 z - 15519275513272 z 4 6 8 10 12 + 9787 z - 371560 z + 9262349 z - 162158950 z + 2081603877 z 14 18 16 50 - 20186728504 z - 887993148158 z + 151140556563 z - 887993148158 z 48 20 36 + 4145065650825 z + 4145065650825 z + 510114399655954 z 34 66 64 30 - 562970484381196 z - 150 z + 9787 z - 379360684490752 z 42 44 46 - 115368088652404 z + 46945054354378 z - 15519275513272 z 58 56 54 52 - 162158950 z + 2081603877 z - 20186728504 z + 151140556563 z 60 68 32 38 + 9262349 z + z + 510114399655954 z - 379360684490752 z 40 62 + 231283882067822 z - 371560 z )) And in Maple-input format, it is: -(1+83782370546544*z^28-42774030141430*z^26-131*z^2+17931012397730*z^24-\ 6146711074840*z^22+7566*z^4-257167*z^6+5799365*z^8-92752796*z^10+1097818697*z^ 12-9901775243*z^14-385834309139*z^18+69511680146*z^16-385834309139*z^50+ 1713693956865*z^48+1713693956865*z^20+179922140722942*z^36-197901029333752*z^34 -131*z^66+7566*z^64-135152747954986*z^30-42774030141430*z^42+17931012397730*z^ 44-6146711074840*z^46-92752796*z^58+1097818697*z^56-9901775243*z^54+69511680146 *z^52+5799365*z^60+z^68+179922140722942*z^32-135152747954986*z^38+ 83782370546544*z^40-257167*z^62)/(-1+z^2)/(1+231283882067822*z^28-\ 115368088652404*z^26-150*z^2+46945054354378*z^24-15519275513272*z^22+9787*z^4-\ 371560*z^6+9262349*z^8-162158950*z^10+2081603877*z^12-20186728504*z^14-\ 887993148158*z^18+151140556563*z^16-887993148158*z^50+4145065650825*z^48+ 4145065650825*z^20+510114399655954*z^36-562970484381196*z^34-150*z^66+9787*z^64 -379360684490752*z^30-115368088652404*z^42+46945054354378*z^44-15519275513272*z ^46-162158950*z^58+2081603877*z^56-20186728504*z^54+151140556563*z^52+9262349*z ^60+z^68+510114399655954*z^32-379360684490752*z^38+231283882067822*z^40-371560* z^62) The first , 40, terms are: [0, 20, 0, 649, 0, 23439, 0, 882572, 0, 33839555, 0, 1308122083, 0, 50763491580, 0, 1973765331683, 0, 76821970977465, 0, 2991745205562036, 0, 116549201741304389, 0, 4541318241206676997, 0, 176973830839056930036, 0, 6897163685168413707689, 0, 268815288292695687795235, 0, 10477352087880090959837020, 0, 408374181824912549739090643, 0, 15917357839909318268531840147, 0, 620422548322177524330598131116, 0, 24182807010704156034818269940463] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 122525254468 z - 139839379034 z - 107 z + 122525254468 z 22 4 6 8 10 - 82396185508 z + 4883 z - 126660 z + 2102638 z - 23855504 z 12 14 18 16 + 192762758 z - 1140203496 z - 16754601736 z + 5029744802 z 50 48 20 36 34 - 107 z + 4883 z + 42484699346 z + 5029744802 z - 16754601736 z 30 42 44 46 52 - 82396185508 z - 23855504 z + 2102638 z - 126660 z + z 32 38 40 / + 42484699346 z - 1140203496 z + 192762758 z ) / (-1 / 28 26 2 24 - 769656768983 z + 769656768983 z + 131 z - 593318258609 z 22 4 6 8 10 + 352344394069 z - 7002 z + 208122 z - 3916679 z + 50118249 z 12 14 18 16 - 455712147 z + 3030677421 z + 56329637121 z - 15030895583 z 50 48 20 36 + 7002 z - 208122 z - 160913446395 z - 56329637121 z 34 30 42 44 + 160913446395 z + 593318258609 z + 455712147 z - 50118249 z 46 54 52 32 38 + 3916679 z + z - 131 z - 352344394069 z + 15030895583 z 40 - 3030677421 z ) And in Maple-input format, it is: -(1+122525254468*z^28-139839379034*z^26-107*z^2+122525254468*z^24-82396185508*z ^22+4883*z^4-126660*z^6+2102638*z^8-23855504*z^10+192762758*z^12-1140203496*z^ 14-16754601736*z^18+5029744802*z^16-107*z^50+4883*z^48+42484699346*z^20+ 5029744802*z^36-16754601736*z^34-82396185508*z^30-23855504*z^42+2102638*z^44-\ 126660*z^46+z^52+42484699346*z^32-1140203496*z^38+192762758*z^40)/(-1-\ 769656768983*z^28+769656768983*z^26+131*z^2-593318258609*z^24+352344394069*z^22 -7002*z^4+208122*z^6-3916679*z^8+50118249*z^10-455712147*z^12+3030677421*z^14+ 56329637121*z^18-15030895583*z^16+7002*z^50-208122*z^48-160913446395*z^20-\ 56329637121*z^36+160913446395*z^34+593318258609*z^30+455712147*z^42-50118249*z^ 44+3916679*z^46+z^54-131*z^52-352344394069*z^32+15030895583*z^38-3030677421*z^ 40) The first , 40, terms are: [0, 24, 0, 1025, 0, 47689, 0, 2251096, 0, 106562697, 0, 5047961785, 0, 239175673944, 0, 11333125396153, 0, 537025194205201, 0, 25447470894674968, 0, 1205859435215369937, 0, 57141237036096961585, 0, 2707715152142083406360, 0, 128308810766658649117169, 0, 6080090691963709659580889, 0, 288113539003106816552130648, 0, 13652660505213292964858703193, 0, 646950303660345274177324662249, 0, 30656639993468359336918935610712, 0, 1452707531025209118530279298298665] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 31675561614766505 z - 8643206436214439 z - 163 z 24 22 4 6 + 2000795157795425 z - 390988841341028 z + 12105 z - 547002 z 8 10 12 14 + 16944322 z - 384233194 z + 6649644955 z - 90387633437 z 18 16 50 - 8751811277748 z + 985363064835 z - 1992398272008081049 z 48 20 36 + 2891270851464915163 z + 64106284510620 z + 1181693864233529779 z 34 66 80 88 - 602530707689741062 z - 390988841341028 z + 16944322 z + z 84 86 82 64 + 12105 z - 163 z - 547002 z + 2000795157795425 z 30 42 - 98873292375263990 z - 3613963105541668232 z 44 46 + 3892782899868311912 z - 3613963105541668232 z 58 56 54 - 98873292375263990 z + 263710794757033134 z - 602530707689741062 z 52 60 70 + 1181693864233529779 z + 31675561614766505 z - 8751811277748 z 68 78 32 + 64106284510620 z - 384233194 z + 263710794757033134 z 38 40 62 - 1992398272008081049 z + 2891270851464915163 z - 8643206436214439 z 76 74 72 / 2 + 6649644955 z - 90387633437 z + 985363064835 z ) / ((-1 + z ) (1 / 28 26 2 + 79846738953187548 z - 21085532729425172 z - 184 z 24 22 4 6 + 4705977204598231 z - 883232927308328 z + 15182 z - 753092 z 8 10 12 14 + 25363843 z - 620410620 z + 11503887032 z - 166552634452 z 18 16 50 - 18016002213432 z + 1923719303209 z - 5591405342986507440 z 48 20 36 + 8194938977560114079 z + 138530070257338 z + 3270690566921647134 z 34 66 80 88 - 1638427205447887868 z - 883232927308328 z + 25363843 z + z 84 86 82 64 + 15182 z - 184 z - 753092 z + 4705977204598231 z 30 42 - 256573092976876052 z - 10304718119887439920 z 44 46 + 11121904453620614540 z - 10304718119887439920 z 58 56 - 256573092976876052 z + 701835757367372717 z 54 52 - 1638427205447887868 z + 3270690566921647134 z 60 70 68 + 79846738953187548 z - 18016002213432 z + 138530070257338 z 78 32 38 - 620410620 z + 701835757367372717 z - 5591405342986507440 z 40 62 76 + 8194938977560114079 z - 21085532729425172 z + 11503887032 z 74 72 - 166552634452 z + 1923719303209 z )) And in Maple-input format, it is: -(1+31675561614766505*z^28-8643206436214439*z^26-163*z^2+2000795157795425*z^24-\ 390988841341028*z^22+12105*z^4-547002*z^6+16944322*z^8-384233194*z^10+ 6649644955*z^12-90387633437*z^14-8751811277748*z^18+985363064835*z^16-\ 1992398272008081049*z^50+2891270851464915163*z^48+64106284510620*z^20+ 1181693864233529779*z^36-602530707689741062*z^34-390988841341028*z^66+16944322* z^80+z^88+12105*z^84-163*z^86-547002*z^82+2000795157795425*z^64-\ 98873292375263990*z^30-3613963105541668232*z^42+3892782899868311912*z^44-\ 3613963105541668232*z^46-98873292375263990*z^58+263710794757033134*z^56-\ 602530707689741062*z^54+1181693864233529779*z^52+31675561614766505*z^60-\ 8751811277748*z^70+64106284510620*z^68-384233194*z^78+263710794757033134*z^32-\ 1992398272008081049*z^38+2891270851464915163*z^40-8643206436214439*z^62+ 6649644955*z^76-90387633437*z^74+985363064835*z^72)/(-1+z^2)/(1+ 79846738953187548*z^28-21085532729425172*z^26-184*z^2+4705977204598231*z^24-\ 883232927308328*z^22+15182*z^4-753092*z^6+25363843*z^8-620410620*z^10+ 11503887032*z^12-166552634452*z^14-18016002213432*z^18+1923719303209*z^16-\ 5591405342986507440*z^50+8194938977560114079*z^48+138530070257338*z^20+ 3270690566921647134*z^36-1638427205447887868*z^34-883232927308328*z^66+25363843 *z^80+z^88+15182*z^84-184*z^86-753092*z^82+4705977204598231*z^64-\ 256573092976876052*z^30-10304718119887439920*z^42+11121904453620614540*z^44-\ 10304718119887439920*z^46-256573092976876052*z^58+701835757367372717*z^56-\ 1638427205447887868*z^54+3270690566921647134*z^52+79846738953187548*z^60-\ 18016002213432*z^70+138530070257338*z^68-620410620*z^78+701835757367372717*z^32 -5591405342986507440*z^38+8194938977560114079*z^40-21085532729425172*z^62+ 11503887032*z^76-166552634452*z^74+1923719303209*z^72) The first , 40, terms are: [0, 22, 0, 809, 0, 32885, 0, 1382046, 0, 58869965, 0, 2522900253, 0, 108441146014, 0, 4668255681501, 0, 201128409738793, 0, 8669405852194966, 0, 373779325836669161, 0, 16117703399938829129, 0, 695066083945825688086, 0, 29975672769752933073625, 0, 1292775478894560792056253, 0, 55754987852044271548901246, 0, 2404628695225320803142557645, 0, 103708517365316486617627729085, 0, 4472826218438348418048937659198, 0, 192908031495674208548519375952181] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 3492466615 z - 5187737606 z - 90 z + 5917211890 z 22 4 6 8 10 - 5187737606 z + 3279 z - 66386 z + 851717 z - 7434796 z 12 14 18 16 48 + 46115558 z - 209074436 z - 1799868966 z + 706139453 z + z 20 36 34 30 + 3492466615 z + 46115558 z - 209074436 z - 1799868966 z 42 44 46 32 38 40 - 66386 z + 3279 z - 90 z + 706139453 z - 7434796 z + 851717 z / 2 28 26 2 ) / ((-1 + z ) (1 + 9753197091 z - 14726036139 z - 111 z / 24 22 4 6 8 + 16888014690 z - 14726036139 z + 4843 z - 113800 z + 1649907 z 10 12 14 18 - 15918707 z + 107127314 z - 518652143 z - 4890034760 z 16 48 20 36 34 + 1844771427 z + z + 9753197091 z + 107127314 z - 518652143 z 30 42 44 46 32 - 4890034760 z - 113800 z + 4843 z - 111 z + 1844771427 z 38 40 - 15918707 z + 1649907 z )) And in Maple-input format, it is: -(1+3492466615*z^28-5187737606*z^26-90*z^2+5917211890*z^24-5187737606*z^22+3279 *z^4-66386*z^6+851717*z^8-7434796*z^10+46115558*z^12-209074436*z^14-1799868966* z^18+706139453*z^16+z^48+3492466615*z^20+46115558*z^36-209074436*z^34-\ 1799868966*z^30-66386*z^42+3279*z^44-90*z^46+706139453*z^32-7434796*z^38+851717 *z^40)/(-1+z^2)/(1+9753197091*z^28-14726036139*z^26-111*z^2+16888014690*z^24-\ 14726036139*z^22+4843*z^4-113800*z^6+1649907*z^8-15918707*z^10+107127314*z^12-\ 518652143*z^14-4890034760*z^18+1844771427*z^16+z^48+9753197091*z^20+107127314*z ^36-518652143*z^34-4890034760*z^30-113800*z^42+4843*z^44-111*z^46+1844771427*z^ 32-15918707*z^38+1649907*z^40) The first , 40, terms are: [0, 22, 0, 789, 0, 31637, 0, 1332794, 0, 57484821, 0, 2507161289, 0, 109921870322, 0, 4831130755389, 0, 212573670619153, 0, 9358395583173430, 0, 412098559946595089, 0, 18148929447996640161, 0, 799326711684456623478, 0, 35205339968640733322241, 0, 1550593069328371728672445, 0, 68295093078231041908305730, 0, 3008030717245760866312247289, 0, 132487695582517264101274817605, 0, 5835378959702561827110238909578, 0, 257017500383443583906330790982293] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 185095306 z + 514341096 z + 96 z - 1012702543 z 22 4 6 8 10 + 1419282152 z - 3710 z + 77351 z - 987138 z + 8247382 z 12 14 18 16 - 46864649 z + 185095306 z + 1012702543 z - 514341096 z 20 36 34 30 42 - 1419282152 z - 77351 z + 987138 z + 46864649 z + z 32 38 40 / 2 30 - 8247382 z + 3710 z - 96 z ) / (-114 z - 577687408 z / 36 14 22 20 + 2009633 z - 577687408 z - 7993764348 z + 6797029099 z + 1 40 38 12 16 24 + 5311 z - 133140 z + 127758299 z + 1839100325 z + 6797029099 z 10 44 4 28 8 - 19538358 z + z + 5311 z + 1839100325 z + 2009633 z 26 6 18 32 - 4171663970 z - 133140 z - 4171663970 z + 127758299 z 34 42 - 19538358 z - 114 z ) And in Maple-input format, it is: -(-1-185095306*z^28+514341096*z^26+96*z^2-1012702543*z^24+1419282152*z^22-3710* z^4+77351*z^6-987138*z^8+8247382*z^10-46864649*z^12+185095306*z^14+1012702543*z ^18-514341096*z^16-1419282152*z^20-77351*z^36+987138*z^34+46864649*z^30+z^42-\ 8247382*z^32+3710*z^38-96*z^40)/(-114*z^2-577687408*z^30+2009633*z^36-577687408 *z^14-7993764348*z^22+6797029099*z^20+1+5311*z^40-133140*z^38+127758299*z^12+ 1839100325*z^16+6797029099*z^24-19538358*z^10+z^44+5311*z^4+1839100325*z^28+ 2009633*z^8-4171663970*z^26-133140*z^6-4171663970*z^18+127758299*z^32-19538358* z^34-114*z^42) The first , 40, terms are: [0, 18, 0, 451, 0, 11605, 0, 301734, 0, 7927243, 0, 210738439, 0, 5678410446, 0, 155362736569, 0, 4323524756263, 0, 122547603117882, 0, 3541103307502333, 0, 104340726397486213, 0, 3133746990366660330, 0, 95829927381993188383, 0, 2979035991623064609073, 0, 93962972361143708873214, 0, 3000928963167849553759183, 0, 96849597653435564162982595, 0, 3152625679670396674016409078, 0, 103338515624113898070759292957] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 8237365434 z - 12369945592 z - 103 z + 14158939080 z 22 4 6 8 10 - 12369945592 z + 4352 z - 101125 z + 1459545 z - 14022010 z 12 14 18 16 48 + 93773004 z - 450195276 z - 4163696578 z + 1585888840 z + z 20 36 34 30 + 8237365434 z + 93773004 z - 450195276 z - 4163696578 z 42 44 46 32 38 - 101125 z + 4352 z - 103 z + 1585888840 z - 14022010 z 40 / 28 26 2 + 1459545 z ) / (-1 - 59994959392 z + 78248142564 z + 130 z / 24 22 4 6 8 - 78248142564 z + 59994959392 z - 6562 z + 177426 z - 2939078 z 10 12 14 18 + 32163305 z - 243995140 z + 1325973180 z + 15695102796 z 16 50 48 20 36 - 5283349084 z + z - 130 z - 35180363562 z - 1325973180 z 34 30 42 44 46 + 5283349084 z + 35180363562 z + 2939078 z - 177426 z + 6562 z 32 38 40 - 15695102796 z + 243995140 z - 32163305 z ) And in Maple-input format, it is: -(1+8237365434*z^28-12369945592*z^26-103*z^2+14158939080*z^24-12369945592*z^22+ 4352*z^4-101125*z^6+1459545*z^8-14022010*z^10+93773004*z^12-450195276*z^14-\ 4163696578*z^18+1585888840*z^16+z^48+8237365434*z^20+93773004*z^36-450195276*z^ 34-4163696578*z^30-101125*z^42+4352*z^44-103*z^46+1585888840*z^32-14022010*z^38 +1459545*z^40)/(-1-59994959392*z^28+78248142564*z^26+130*z^2-78248142564*z^24+ 59994959392*z^22-6562*z^4+177426*z^6-2939078*z^8+32163305*z^10-243995140*z^12+ 1325973180*z^14+15695102796*z^18-5283349084*z^16+z^50-130*z^48-35180363562*z^20 -1325973180*z^36+5283349084*z^34+35180363562*z^30+2939078*z^42-177426*z^44+6562 *z^46-15695102796*z^32+243995140*z^38-32163305*z^40) The first , 40, terms are: [0, 27, 0, 1300, 0, 68127, 0, 3636879, 0, 195184885, 0, 10493721853, 0, 564527157692, 0, 30376629662373, 0, 1634674177125199, 0, 87970400858852391, 0, 4734205286136796157, 0, 254776595481657222044, 0, 13711112602695680678269, 0, 737880675572244999418325, 0, 39709980879077752399715135, 0, 2137042983751110114995033551, 0, 115007682319359036594647057460, 0, 6189284566522525687810458710787, 0, 333084215352658419401445211014057, 0, 17925350421374090200215125255518649] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {3, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 22 4 6 8 10 12 f(z) = - (-1 + 31 z + z - 374 z + 2258 z - 7365 z + 13267 z - 13267 z 14 18 16 20 / 22 12 + 7365 z + 374 z - 2258 z - 31 z ) / (-51 z + 75502 z / 10 24 8 20 6 18 4 - 57693 z + z + 25485 z + 831 z - 6316 z - 6316 z + 831 z 16 2 14 + 25485 z - 51 z - 57693 z + 1) And in Maple-input format, it is: -(-1+31*z^2+z^22-374*z^4+2258*z^6-7365*z^8+13267*z^10-13267*z^12+7365*z^14+374* z^18-2258*z^16-31*z^20)/(-51*z^22+75502*z^12-57693*z^10+z^24+25485*z^8+831*z^20 -6316*z^6-6316*z^18+831*z^4+25485*z^16-51*z^2-57693*z^14+1) The first , 40, terms are: [0, 20, 0, 563, 0, 16151, 0, 464048, 0, 13335601, 0, 383245049, 0, 11013953448, 0, 316526805415, 0, 9096574564739, 0, 261423909259132, 0, 7512988664614105, 0, 215913682510564873, 0, 6205083014930723116, 0, 178326147687781278275, 0, 5124865352412450413735, 0, 147282074006277189321016, 0, 4232698389534278106170761, 0, 121642336840227448181499745, 0, 3495845144209967299416154912, 0, 100466117223231296182120628183] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {3, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 436850817716 z - 389345354788 z - 108 z + 275459247612 z 22 4 6 8 10 - 154393336580 z + 4932 z - 128401 z + 2161850 z - 25234243 z 12 14 18 16 + 213237320 z - 1342722904 z - 23737300468 z + 6428268171 z 50 48 20 36 - 128401 z + 2161850 z + 68315750412 z + 68315750412 z 34 30 42 44 - 154393336580 z - 389345354788 z - 1342722904 z + 213237320 z 46 56 54 52 32 - 25234243 z + z - 108 z + 4932 z + 275459247612 z 38 40 / 2 28 - 23737300468 z + 6428268171 z ) / ((-1 + z ) (1 + 1245388476250 z / 26 2 24 22 - 1104957019760 z - 128 z + 771240660906 z - 422645967456 z 4 6 8 10 12 + 6809 z - 202532 z + 3824725 z - 49256876 z + 452699201 z 14 18 16 50 - 3061689952 z - 60451098944 z + 15569225625 z - 202532 z 48 20 36 34 + 3824725 z + 181204594090 z + 181204594090 z - 422645967456 z 30 42 44 46 56 - 1104957019760 z - 3061689952 z + 452699201 z - 49256876 z + z 54 52 32 38 - 128 z + 6809 z + 771240660906 z - 60451098944 z 40 + 15569225625 z )) And in Maple-input format, it is: -(1+436850817716*z^28-389345354788*z^26-108*z^2+275459247612*z^24-154393336580* z^22+4932*z^4-128401*z^6+2161850*z^8-25234243*z^10+213237320*z^12-1342722904*z^ 14-23737300468*z^18+6428268171*z^16-128401*z^50+2161850*z^48+68315750412*z^20+ 68315750412*z^36-154393336580*z^34-389345354788*z^30-1342722904*z^42+213237320* z^44-25234243*z^46+z^56-108*z^54+4932*z^52+275459247612*z^32-23737300468*z^38+ 6428268171*z^40)/(-1+z^2)/(1+1245388476250*z^28-1104957019760*z^26-128*z^2+ 771240660906*z^24-422645967456*z^22+6809*z^4-202532*z^6+3824725*z^8-49256876*z^ 10+452699201*z^12-3061689952*z^14-60451098944*z^18+15569225625*z^16-202532*z^50 +3824725*z^48+181204594090*z^20+181204594090*z^36-422645967456*z^34-\ 1104957019760*z^30-3061689952*z^42+452699201*z^44-49256876*z^46+z^56-128*z^54+ 6809*z^52+771240660906*z^32-60451098944*z^38+15569225625*z^40) The first , 40, terms are: [0, 21, 0, 704, 0, 26079, 0, 1011297, 0, 40198315, 0, 1620425221, 0, 65858268064, 0, 2689766879343, 0, 110179527667795, 0, 4521336027608843, 0, 185741366438114599, 0, 7635573580260687328, 0, 314016890488362705437, 0, 12917355092456142010419, 0, 531448439450001782687945, 0, 21867025154456978775585943, 0, 899794678172440178961642496, 0, 37026490367945316741276069421, 0, 1523670724169954622048773986073, 0, 62701147243800463512082587598377] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 324317744392389 z - 141685614334891 z - 134 z 24 22 4 6 + 51213230704289 z - 15256197369982 z + 8023 z - 286713 z 8 10 12 14 + 6889195 z - 118834518 z + 1533774195 z - 15239414262 z 18 16 50 - 741124589326 z + 118978422210 z - 15256197369982 z 48 20 36 + 51213230704289 z + 3726206548057 z + 1400846986171236 z 34 66 64 30 - 1278838130744068 z - 286713 z + 6889195 z - 615974211479422 z 42 44 46 - 615974211479422 z + 324317744392389 z - 141685614334891 z 58 56 54 - 15239414262 z + 118978422210 z - 741124589326 z 52 60 70 68 + 3726206548057 z + 1533774195 z - 134 z + 8023 z 32 38 40 + 972709443302971 z - 1278838130744068 z + 972709443302971 z 62 72 / 2 28 - 118834518 z + z ) / ((-1 + z ) (1 + 891004337202872 z / 26 2 24 22 - 378384676050078 z - 156 z + 132178840346563 z - 37840010015744 z 4 6 8 10 12 + 10562 z - 418990 z + 11032759 z - 206554996 z + 2871306820 z 14 18 16 - 30524667276 z - 1670045029256 z + 253477529401 z 50 48 20 - 37840010015744 z + 132178840346563 z + 8833093266862 z 36 34 66 64 + 4051519954930610 z - 3686625697419348 z - 418990 z + 11032759 z 30 42 44 - 1730451609830648 z - 1730451609830648 z + 891004337202872 z 46 58 56 - 378384676050078 z - 30524667276 z + 253477529401 z 54 52 60 70 - 1670045029256 z + 8833093266862 z + 2871306820 z - 156 z 68 32 38 + 10562 z + 2776970162658673 z - 3686625697419348 z 40 62 72 + 2776970162658673 z - 206554996 z + z )) And in Maple-input format, it is: -(1+324317744392389*z^28-141685614334891*z^26-134*z^2+51213230704289*z^24-\ 15256197369982*z^22+8023*z^4-286713*z^6+6889195*z^8-118834518*z^10+1533774195*z ^12-15239414262*z^14-741124589326*z^18+118978422210*z^16-15256197369982*z^50+ 51213230704289*z^48+3726206548057*z^20+1400846986171236*z^36-1278838130744068*z ^34-286713*z^66+6889195*z^64-615974211479422*z^30-615974211479422*z^42+ 324317744392389*z^44-141685614334891*z^46-15239414262*z^58+118978422210*z^56-\ 741124589326*z^54+3726206548057*z^52+1533774195*z^60-134*z^70+8023*z^68+ 972709443302971*z^32-1278838130744068*z^38+972709443302971*z^40-118834518*z^62+ z^72)/(-1+z^2)/(1+891004337202872*z^28-378384676050078*z^26-156*z^2+ 132178840346563*z^24-37840010015744*z^22+10562*z^4-418990*z^6+11032759*z^8-\ 206554996*z^10+2871306820*z^12-30524667276*z^14-1670045029256*z^18+253477529401 *z^16-37840010015744*z^50+132178840346563*z^48+8833093266862*z^20+ 4051519954930610*z^36-3686625697419348*z^34-418990*z^66+11032759*z^64-\ 1730451609830648*z^30-1730451609830648*z^42+891004337202872*z^44-\ 378384676050078*z^46-30524667276*z^58+253477529401*z^56-1670045029256*z^54+ 8833093266862*z^52+2871306820*z^60-156*z^70+10562*z^68+2776970162658673*z^32-\ 3686625697419348*z^38+2776970162658673*z^40-206554996*z^62+z^72) The first , 40, terms are: [0, 23, 0, 916, 0, 40137, 0, 1800963, 0, 81395467, 0, 3687924169, 0, 167252009836, 0, 7587885636127, 0, 344298164287165, 0, 15623380928941457, 0, 708967344981020163, 0, 32172289730601957420, 0, 1459955444538708403669, 0, 66251852511177109399119, 0, 3006469377203010925793559, 0, 136431823524389443173637941, 0, 6191197279081626762904754100, 0, 280952969721413308804649068747, 0, 12749484404238069229127205714773, 0, 578564285879616385526439882553773] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 20743247098180562 z - 5791382578341728 z - 159 z 24 22 4 6 + 1374719051652632 z - 276034490911664 z + 11470 z - 502015 z 8 10 12 14 + 15035443 z - 329372222 z + 5506041104 z - 72325063556 z 18 16 50 - 6558831917090 z + 762588766048 z - 1207626437466378354 z 48 20 36 + 1739710783055810588 z + 46590684668414 z + 723540992096172208 z 34 66 80 88 - 373714839798323084 z - 276034490911664 z + 15035443 z + z 84 86 82 64 + 11470 z - 159 z - 502015 z + 1374719051652632 z 30 42 - 63428682535890766 z - 2165016707152516274 z 44 46 + 2328622820198800468 z - 2165016707152516274 z 58 56 54 - 63428682535890766 z + 166132205313512640 z - 373714839798323084 z 52 60 70 + 723540992096172208 z + 20743247098180562 z - 6558831917090 z 68 78 32 + 46590684668414 z - 329372222 z + 166132205313512640 z 38 40 62 - 1207626437466378354 z + 1739710783055810588 z - 5791382578341728 z 76 74 72 / 2 + 5506041104 z - 72325063556 z + 762588766048 z ) / ((-1 + z ) (1 / 28 26 2 + 52466377474131048 z - 14211302997752370 z - 179 z 24 22 4 6 + 3259865327627756 z - 629908733944798 z + 14367 z - 692429 z 8 10 12 14 + 22627077 z - 536353620 z + 9629368646 z - 134923202844 z 18 16 50 - 13673628360204 z + 1508141793318 z - 3363907542743817756 z 48 20 36 + 4888457535077664402 z + 101863583984304 z + 1991016027026444370 z 34 66 80 88 - 1012346483437594228 z - 629908733944798 z + 22627077 z + z 84 86 82 64 + 14367 z - 179 z - 692429 z + 3259865327627756 z 30 42 - 164740109024529956 z - 6115409937883629392 z 44 46 + 6589006625349771034 z - 6115409937883629392 z 58 56 - 164740109024529956 z + 441452594951913138 z 54 52 - 1012346483437594228 z + 1991016027026444370 z 60 70 68 + 52466377474131048 z - 13673628360204 z + 101863583984304 z 78 32 38 - 536353620 z + 441452594951913138 z - 3363907542743817756 z 40 62 76 + 4888457535077664402 z - 14211302997752370 z + 9629368646 z 74 72 - 134923202844 z + 1508141793318 z )) And in Maple-input format, it is: -(1+20743247098180562*z^28-5791382578341728*z^26-159*z^2+1374719051652632*z^24-\ 276034490911664*z^22+11470*z^4-502015*z^6+15035443*z^8-329372222*z^10+ 5506041104*z^12-72325063556*z^14-6558831917090*z^18+762588766048*z^16-\ 1207626437466378354*z^50+1739710783055810588*z^48+46590684668414*z^20+ 723540992096172208*z^36-373714839798323084*z^34-276034490911664*z^66+15035443*z ^80+z^88+11470*z^84-159*z^86-502015*z^82+1374719051652632*z^64-\ 63428682535890766*z^30-2165016707152516274*z^42+2328622820198800468*z^44-\ 2165016707152516274*z^46-63428682535890766*z^58+166132205313512640*z^56-\ 373714839798323084*z^54+723540992096172208*z^52+20743247098180562*z^60-\ 6558831917090*z^70+46590684668414*z^68-329372222*z^78+166132205313512640*z^32-\ 1207626437466378354*z^38+1739710783055810588*z^40-5791382578341728*z^62+ 5506041104*z^76-72325063556*z^74+762588766048*z^72)/(-1+z^2)/(1+ 52466377474131048*z^28-14211302997752370*z^26-179*z^2+3259865327627756*z^24-\ 629908733944798*z^22+14367*z^4-692429*z^6+22627077*z^8-536353620*z^10+ 9629368646*z^12-134923202844*z^14-13673628360204*z^18+1508141793318*z^16-\ 3363907542743817756*z^50+4888457535077664402*z^48+101863583984304*z^20+ 1991016027026444370*z^36-1012346483437594228*z^34-629908733944798*z^66+22627077 *z^80+z^88+14367*z^84-179*z^86-692429*z^82+3259865327627756*z^64-\ 164740109024529956*z^30-6115409937883629392*z^42+6589006625349771034*z^44-\ 6115409937883629392*z^46-164740109024529956*z^58+441452594951913138*z^56-\ 1012346483437594228*z^54+1991016027026444370*z^52+52466377474131048*z^60-\ 13673628360204*z^70+101863583984304*z^68-536353620*z^78+441452594951913138*z^32 -3363907542743817756*z^38+4888457535077664402*z^40-14211302997752370*z^62+ 9629368646*z^76-134923202844*z^74+1508141793318*z^72) The first , 40, terms are: [0, 21, 0, 704, 0, 26035, 0, 1004569, 0, 39600543, 0, 1579052177, 0, 63371656632, 0, 2552979536739, 0, 103088545902355, 0, 4168763389702003, 0, 168736241387778491, 0, 6833927882272583560, 0, 276886668935052342017, 0, 11221335399864545519223, 0, 454841098289096157733921, 0, 18438372579659001285938771, 0, 747509767513385690894473616, 0, 30306224157653353930141239325, 0, 1228741173339266490899890186289, 0, 49819339855742492857549811416817] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 13582989707096 z - 8197411049232 z - 119 z 24 22 4 6 8 + 4030039613406 z - 1608595659104 z + 6192 z - 188589 z + 3794643 z 10 12 14 18 - 53939964 z + 565216910 z - 4495107612 z - 134378167790 z 16 50 48 20 + 27704695416 z - 4495107612 z + 27704695416 z + 518857663476 z 36 34 64 30 + 13582989707096 z - 18374855476440 z + z - 18374855476440 z 42 44 46 58 - 1608595659104 z + 518857663476 z - 134378167790 z - 188589 z 56 54 52 60 + 3794643 z - 53939964 z + 565216910 z + 6192 z 32 38 40 62 + 20319257799932 z - 8197411049232 z + 4030039613406 z - 119 z ) / 2 28 26 2 / ((-1 + z ) (1 + 39043084742918 z - 23106635925707 z - 143 z / 24 22 4 6 + 11056461651022 z - 4264219363489 z + 8558 z - 292268 z 8 10 12 14 + 6486431 z - 100505179 z + 1137390152 z - 9692245205 z 18 16 50 48 - 325678899491 z + 63549697862 z - 9692245205 z + 63549697862 z 20 36 34 64 + 1319671512624 z + 39043084742918 z - 53445006250507 z + z 30 42 44 - 53445006250507 z - 4264219363489 z + 1319671512624 z 46 58 56 54 - 325678899491 z - 292268 z + 6486431 z - 100505179 z 52 60 32 38 + 1137390152 z + 8558 z + 59334844262298 z - 23106635925707 z 40 62 + 11056461651022 z - 143 z )) And in Maple-input format, it is: -(1+13582989707096*z^28-8197411049232*z^26-119*z^2+4030039613406*z^24-\ 1608595659104*z^22+6192*z^4-188589*z^6+3794643*z^8-53939964*z^10+565216910*z^12 -4495107612*z^14-134378167790*z^18+27704695416*z^16-4495107612*z^50+27704695416 *z^48+518857663476*z^20+13582989707096*z^36-18374855476440*z^34+z^64-\ 18374855476440*z^30-1608595659104*z^42+518857663476*z^44-134378167790*z^46-\ 188589*z^58+3794643*z^56-53939964*z^54+565216910*z^52+6192*z^60+20319257799932* z^32-8197411049232*z^38+4030039613406*z^40-119*z^62)/(-1+z^2)/(1+39043084742918 *z^28-23106635925707*z^26-143*z^2+11056461651022*z^24-4264219363489*z^22+8558*z ^4-292268*z^6+6486431*z^8-100505179*z^10+1137390152*z^12-9692245205*z^14-\ 325678899491*z^18+63549697862*z^16-9692245205*z^50+63549697862*z^48+ 1319671512624*z^20+39043084742918*z^36-53445006250507*z^34+z^64-53445006250507* z^30-4264219363489*z^42+1319671512624*z^44-325678899491*z^46-292268*z^58+ 6486431*z^56-100505179*z^54+1137390152*z^52+8558*z^60+59334844262298*z^32-\ 23106635925707*z^38+11056461651022*z^40-143*z^62) The first , 40, terms are: [0, 25, 0, 1091, 0, 51816, 0, 2505307, 0, 121698529, 0, 5920063205, 0, 288121607869, 0, 14024938309705, 0, 682739222433459, 0, 33236863441081864, 0, 1618041797383643691, 0, 78770078794324000001, 0, 3834719306963156186265, 0, 186683611233827405549257, 0, 9088222206321959351650577, 0, 442437301680341114714787099, 0, 21538951375312493087973976904, 0, 1048569898601136962992318548355, 0, 51046999608296839631158207006233, 0, 2485095349976945846599748738433389] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2930167680362506077 z - 502726976559195076 z - 197 z 24 22 4 6 + 74920554698967507 z - 9644126674806825 z + 18051 z - 1027128 z 102 8 10 12 - 1215210617 z + 40880295 z - 1215210617 z + 28128111942 z 14 18 16 - 521808350897 z - 100092874922751 z + 7923856270378 z 50 48 - 179604815558444149029686 z + 119946441618899179235637 z 20 36 + 1065086257317548 z + 898573953777368921847 z 34 66 - 260210893443775458109 z - 71316021113369082157161 z 80 100 90 + 66422678665214395024 z + 28128111942 z - 9644126674806825 z 88 84 94 + 74920554698967507 z + 2930167680362506077 z - 100092874922751 z 86 96 98 - 502726976559195076 z + 7923856270378 z - 521808350897 z 92 82 + 1065086257317548 z - 14904509745455581183 z 64 112 110 106 + 119946441618899179235637 z + z - 197 z - 1027128 z 108 30 42 + 18051 z - 14904509745455581183 z - 17737984950603328316567 z 44 46 + 37724180987905761287708 z - 71316021113369082157161 z 58 56 - 284702248826297708629515 z + 301562636579442125563074 z 54 52 - 284702248826297708629515 z + 239552761869342691975419 z 60 70 + 239552761869342691975419 z - 17737984950603328316567 z 68 78 + 37724180987905761287708 z - 260210893443775458109 z 32 38 + 66422678665214395024 z - 2741680655319074393648 z 40 62 + 7405673419357116684309 z - 179604815558444149029686 z 76 74 + 898573953777368921847 z - 2741680655319074393648 z 72 104 / + 7405673419357116684309 z + 40880295 z ) / (-1 / 28 26 2 - 7856800037371048744 z + 1279691596789419100 z + 221 z 24 22 4 6 - 180887965272160696 z + 22060268598835924 z - 22376 z + 1391560 z 102 8 10 12 + 47689935944 z - 60036974 z + 1922347150 z - 47689935944 z 14 18 16 + 944416020408 z + 204555195983081 z - 15259604816957 z 50 48 + 848358962152995540183158 z - 536977731231878720996602 z 20 36 - 2304878381427692 z - 2952813506316123041776 z 34 66 + 812944684431986895876 z + 536977731231878720996602 z 80 100 90 - 812944684431986895876 z - 944416020408 z + 180887965272160696 z 88 84 - 1279691596789419100 z - 42069403614738418268 z 94 86 96 + 2304878381427692 z + 7856800037371048744 z - 204555195983081 z 98 92 82 + 15259604816957 z - 22060268598835924 z + 197271984028379580424 z 64 112 114 110 - 848358962152995540183158 z - 221 z + z + 22376 z 106 108 30 + 60036974 z - 1391560 z + 42069403614738418268 z 42 44 + 67895212850216812313668 z - 152043661851182530097560 z 46 58 + 302829849401331962234968 z + 1682266060434765076832232 z 56 54 - 1682266060434765076832232 z + 1501042393927331776403268 z 52 60 - 1194892754502269220579136 z - 1501042393927331776403268 z 70 68 + 152043661851182530097560 z - 302829849401331962234968 z 78 32 + 2952813506316123041776 z - 197271984028379580424 z 38 40 + 9477062720641052943944 z - 26933111363286204671332 z 62 76 + 1194892754502269220579136 z - 9477062720641052943944 z 74 72 + 26933111363286204671332 z - 67895212850216812313668 z 104 - 1922347150 z ) And in Maple-input format, it is: -(1+2930167680362506077*z^28-502726976559195076*z^26-197*z^2+74920554698967507* z^24-9644126674806825*z^22+18051*z^4-1027128*z^6-1215210617*z^102+40880295*z^8-\ 1215210617*z^10+28128111942*z^12-521808350897*z^14-100092874922751*z^18+ 7923856270378*z^16-179604815558444149029686*z^50+119946441618899179235637*z^48+ 1065086257317548*z^20+898573953777368921847*z^36-260210893443775458109*z^34-\ 71316021113369082157161*z^66+66422678665214395024*z^80+28128111942*z^100-\ 9644126674806825*z^90+74920554698967507*z^88+2930167680362506077*z^84-\ 100092874922751*z^94-502726976559195076*z^86+7923856270378*z^96-521808350897*z^ 98+1065086257317548*z^92-14904509745455581183*z^82+119946441618899179235637*z^ 64+z^112-197*z^110-1027128*z^106+18051*z^108-14904509745455581183*z^30-\ 17737984950603328316567*z^42+37724180987905761287708*z^44-\ 71316021113369082157161*z^46-284702248826297708629515*z^58+ 301562636579442125563074*z^56-284702248826297708629515*z^54+ 239552761869342691975419*z^52+239552761869342691975419*z^60-\ 17737984950603328316567*z^70+37724180987905761287708*z^68-260210893443775458109 *z^78+66422678665214395024*z^32-2741680655319074393648*z^38+ 7405673419357116684309*z^40-179604815558444149029686*z^62+898573953777368921847 *z^76-2741680655319074393648*z^74+7405673419357116684309*z^72+40880295*z^104)/( -1-7856800037371048744*z^28+1279691596789419100*z^26+221*z^2-180887965272160696 *z^24+22060268598835924*z^22-22376*z^4+1391560*z^6+47689935944*z^102-60036974*z ^8+1922347150*z^10-47689935944*z^12+944416020408*z^14+204555195983081*z^18-\ 15259604816957*z^16+848358962152995540183158*z^50-536977731231878720996602*z^48 -2304878381427692*z^20-2952813506316123041776*z^36+812944684431986895876*z^34+ 536977731231878720996602*z^66-812944684431986895876*z^80-944416020408*z^100+ 180887965272160696*z^90-1279691596789419100*z^88-42069403614738418268*z^84+ 2304878381427692*z^94+7856800037371048744*z^86-204555195983081*z^96+ 15259604816957*z^98-22060268598835924*z^92+197271984028379580424*z^82-\ 848358962152995540183158*z^64-221*z^112+z^114+22376*z^110+60036974*z^106-\ 1391560*z^108+42069403614738418268*z^30+67895212850216812313668*z^42-\ 152043661851182530097560*z^44+302829849401331962234968*z^46+ 1682266060434765076832232*z^58-1682266060434765076832232*z^56+ 1501042393927331776403268*z^54-1194892754502269220579136*z^52-\ 1501042393927331776403268*z^60+152043661851182530097560*z^70-\ 302829849401331962234968*z^68+2952813506316123041776*z^78-197271984028379580424 *z^32+9477062720641052943944*z^38-26933111363286204671332*z^40+ 1194892754502269220579136*z^62-9477062720641052943944*z^76+ 26933111363286204671332*z^74-67895212850216812313668*z^72-1922347150*z^104) The first , 40, terms are: [0, 24, 0, 979, 0, 43767, 0, 2007164, 0, 92839249, 0, 4307888937, 0, 200150380940, 0, 9304400158219, 0, 432641519786499, 0, 20119544490885064, 0, 935690046788313777, 0, 43516857612108159549, 0, 2023898927430449864920, 0, 94128907804476636008343, 0, 4377828051806527528426263, 0, 203608143040020993430198796, 0, 9469607600712874359283326917, 0, 440422026975778614740568461789, 0, 20483595419719172336731372827004, 0, 952672009785677665623200639019251] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {4, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 37050191842539071 z - 10095249823660311 z - 167 z 24 22 4 6 + 2331836846542445 z - 454270014248432 z + 12687 z - 585154 z 8 10 12 14 + 18454060 z - 424955326 z + 7450409409 z - 102368201673 z 18 16 50 - 10069905248704 z + 1125815777835 z - 2333261317831640217 z 48 20 36 + 3385372120928929527 z + 74169164141432 z + 1384064373248666705 z 34 66 80 88 - 705758569358954862 z - 454270014248432 z + 18454060 z + z 84 86 82 64 + 12687 z - 167 z - 585154 z + 2331836846542445 z 30 42 - 115748961833102722 z - 4231068164780081536 z 44 46 + 4557301711204585264 z - 4231068164780081536 z 58 56 54 - 115748961833102722 z + 308852025423190932 z - 705758569358954862 z 52 60 70 + 1384064373248666705 z + 37050191842539071 z - 10069905248704 z 68 78 32 + 74169164141432 z - 424955326 z + 308852025423190932 z 38 40 - 2333261317831640217 z + 3385372120928929527 z 62 76 74 - 10095249823660311 z + 7450409409 z - 102368201673 z 72 / 28 + 1125815777835 z ) / (-1 - 119250852137487020 z / 26 2 24 + 30443082725722875 z + 191 z - 6586235924796317 z 22 4 6 8 + 1200958378232360 z - 16292 z + 832996 z - 28852653 z 10 12 14 18 + 724603147 z - 13779170156 z + 204465645260 z + 23236818045747 z 16 50 48 - 2420209391029 z + 16274596094104129937 z - 21832647476729897567 z 20 36 34 - 183333289613600 z - 5799540049309817524 z + 2765716586964183547 z 66 80 90 88 84 + 6586235924796317 z - 724603147 z + z - 191 z - 832996 z 86 82 64 + 16292 z + 28852653 z - 30443082725722875 z 30 42 + 397648353719550268 z + 21832647476729897567 z 44 46 - 25283269585148253112 z + 25283269585148253112 z 58 56 + 1132886289834841773 z - 2765716586964183547 z 54 52 + 5799540049309817524 z - 10465376731021574228 z 60 70 68 - 397648353719550268 z + 183333289613600 z - 1200958378232360 z 78 32 38 + 13779170156 z - 1132886289834841773 z + 10465376731021574228 z 40 62 76 - 16274596094104129937 z + 119250852137487020 z - 204465645260 z 74 72 + 2420209391029 z - 23236818045747 z ) And in Maple-input format, it is: -(1+37050191842539071*z^28-10095249823660311*z^26-167*z^2+2331836846542445*z^24 -454270014248432*z^22+12687*z^4-585154*z^6+18454060*z^8-424955326*z^10+ 7450409409*z^12-102368201673*z^14-10069905248704*z^18+1125815777835*z^16-\ 2333261317831640217*z^50+3385372120928929527*z^48+74169164141432*z^20+ 1384064373248666705*z^36-705758569358954862*z^34-454270014248432*z^66+18454060* z^80+z^88+12687*z^84-167*z^86-585154*z^82+2331836846542445*z^64-\ 115748961833102722*z^30-4231068164780081536*z^42+4557301711204585264*z^44-\ 4231068164780081536*z^46-115748961833102722*z^58+308852025423190932*z^56-\ 705758569358954862*z^54+1384064373248666705*z^52+37050191842539071*z^60-\ 10069905248704*z^70+74169164141432*z^68-424955326*z^78+308852025423190932*z^32-\ 2333261317831640217*z^38+3385372120928929527*z^40-10095249823660311*z^62+ 7450409409*z^76-102368201673*z^74+1125815777835*z^72)/(-1-119250852137487020*z^ 28+30443082725722875*z^26+191*z^2-6586235924796317*z^24+1200958378232360*z^22-\ 16292*z^4+832996*z^6-28852653*z^8+724603147*z^10-13779170156*z^12+204465645260* z^14+23236818045747*z^18-2420209391029*z^16+16274596094104129937*z^50-\ 21832647476729897567*z^48-183333289613600*z^20-5799540049309817524*z^36+ 2765716586964183547*z^34+6586235924796317*z^66-724603147*z^80+z^90-191*z^88-\ 832996*z^84+16292*z^86+28852653*z^82-30443082725722875*z^64+397648353719550268* z^30+21832647476729897567*z^42-25283269585148253112*z^44+25283269585148253112*z ^46+1132886289834841773*z^58-2765716586964183547*z^56+5799540049309817524*z^54-\ 10465376731021574228*z^52-397648353719550268*z^60+183333289613600*z^70-\ 1200958378232360*z^68+13779170156*z^78-1132886289834841773*z^32+ 10465376731021574228*z^38-16274596094104129937*z^40+119250852137487020*z^62-\ 204465645260*z^76+2420209391029*z^74-23236818045747*z^72) The first , 40, terms are: [0, 24, 0, 979, 0, 43823, 0, 2013636, 0, 93327393, 0, 4338725553, 0, 201931455660, 0, 9402253606007, 0, 437857471145767, 0, 20392135314686608, 0, 949739593550824473, 0, 44233502999859578805, 0, 2060156531262332922496, 0, 95951110442376298471939, 0, 4468895334472910796379883, 0, 208137596088401421992164940, 0, 9693954180687473001752886717, 0, 451493420513757571827069575757, 0, 21028190603618556328704437016612, 0, 979382616436950137084988168257907] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {4, 5}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 97440884453256 z - 49559495008366 z - 135 z 24 22 4 6 + 20674563602274 z - 7045492861624 z + 7962 z - 274715 z 8 10 12 14 + 6267621 z - 101243100 z + 1209321017 z - 11003516103 z 18 16 50 - 435938745127 z + 77904071070 z - 435938745127 z 48 20 36 + 1950930146929 z + 1950930146929 z + 210149728377310 z 34 66 64 30 - 231271265781432 z - 135 z + 7962 z - 157607561708386 z 42 44 46 - 49559495008366 z + 20674563602274 z - 7045492861624 z 58 56 54 52 - 101243100 z + 1209321017 z - 11003516103 z + 77904071070 z 60 68 32 38 + 6267621 z + z + 210149728377310 z - 157607561708386 z 40 62 / 2 28 + 97440884453256 z - 274715 z ) / ((-1 + z ) (1 + 269581988974686 z / 26 2 24 22 - 134465773729816 z - 156 z + 54691162905098 z - 18060816292840 z 4 6 8 10 12 + 10499 z - 407986 z + 10344309 z - 183332834 z + 2374551317 z 14 18 16 - 23181420170 z - 1028723597644 z + 174435626539 z 50 48 20 - 1028723597644 z + 4815084358857 z + 4815084358857 z 36 34 66 64 + 594377165245882 z - 655913621777188 z - 156 z + 10499 z 30 42 44 - 442106732355180 z - 134465773729816 z + 54691162905098 z 46 58 56 54 - 18060816292840 z - 183332834 z + 2374551317 z - 23181420170 z 52 60 68 32 + 174435626539 z + 10344309 z + z + 594377165245882 z 38 40 62 - 442106732355180 z + 269581988974686 z - 407986 z )) And in Maple-input format, it is: -(1+97440884453256*z^28-49559495008366*z^26-135*z^2+20674563602274*z^24-\ 7045492861624*z^22+7962*z^4-274715*z^6+6267621*z^8-101243100*z^10+1209321017*z^ 12-11003516103*z^14-435938745127*z^18+77904071070*z^16-435938745127*z^50+ 1950930146929*z^48+1950930146929*z^20+210149728377310*z^36-231271265781432*z^34 -135*z^66+7962*z^64-157607561708386*z^30-49559495008366*z^42+20674563602274*z^ 44-7045492861624*z^46-101243100*z^58+1209321017*z^56-11003516103*z^54+ 77904071070*z^52+6267621*z^60+z^68+210149728377310*z^32-157607561708386*z^38+ 97440884453256*z^40-274715*z^62)/(-1+z^2)/(1+269581988974686*z^28-\ 134465773729816*z^26-156*z^2+54691162905098*z^24-18060816292840*z^22+10499*z^4-\ 407986*z^6+10344309*z^8-183332834*z^10+2374551317*z^12-23181420170*z^14-\ 1028723597644*z^18+174435626539*z^16-1028723597644*z^50+4815084358857*z^48+ 4815084358857*z^20+594377165245882*z^36-655913621777188*z^34-156*z^66+10499*z^ 64-442106732355180*z^30-134465773729816*z^42+54691162905098*z^44-18060816292840 *z^46-183332834*z^58+2374551317*z^56-23181420170*z^54+174435626539*z^52+ 10344309*z^60+z^68+594377165245882*z^32-442106732355180*z^38+269581988974686*z^ 40-407986*z^62) The first , 40, terms are: [0, 22, 0, 761, 0, 28837, 0, 1140950, 0, 46221553, 0, 1897651033, 0, 78514404686, 0, 3263521736897, 0, 136034620890897, 0, 5680396314710422, 0, 237462715272743141, 0, 9934062594771878613, 0, 415780011761245245462, 0, 17407454856185117567105, 0, 728947422607406301839969, 0, 30529269772006220853898542, 0, 1278721967997482315940078121, 0, 53562668903338844261506573873, 0, 2243705888976643148623382685974, 0, 93989957708513918019054205715701] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {4, 5}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 40103090708264591 z - 10902015239466873 z - 169 z 24 22 4 6 + 2511592165721733 z - 487841874562328 z + 12971 z - 603498 z 8 10 12 14 + 19175572 z - 444447166 z + 7836766853 z - 108225005147 z 18 16 50 - 10738567815560 z + 1195659842215 z - 2543430473825159811 z 48 20 36 + 3692565695295961811 z + 79386773066072 z + 1507430083333138881 z 34 66 80 88 - 767798520248231534 z - 487841874562328 z + 19175572 z + z 84 86 82 64 + 12971 z - 169 z - 603498 z + 2511592165721733 z 30 42 - 125535584588035914 z - 4616680547035154112 z 44 46 + 4973247130545199984 z - 4616680547035154112 z 58 56 54 - 125535584588035914 z + 335531212984091292 z - 767798520248231534 z 52 60 70 + 1507430083333138881 z + 40103090708264591 z - 10738567815560 z 68 78 32 + 79386773066072 z - 444447166 z + 335531212984091292 z 38 40 - 2543430473825159811 z + 3692565695295961811 z 62 76 74 - 10902015239466873 z + 7836766853 z - 108225005147 z 72 / 2 28 + 1195659842215 z ) / ((-1 + z ) (1 + 103026418363751226 z / 26 2 24 - 27134880821078222 z - 194 z + 6034575010547019 z 22 4 6 8 - 1127296691968840 z + 16688 z - 855812 z + 29608257 z 10 12 14 18 - 740219164 z + 13972134458 z - 205250437378 z - 22681898013368 z 16 50 48 + 2398895464349 z - 7251416479032614910 z + 10630107896927717943 z 20 36 34 + 175746226823750 z + 4240187342384918108 z - 2122849565354295996 z 66 80 88 84 86 - 1127296691968840 z + 29608257 z + z + 16688 z - 194 z 82 64 30 - 855812 z + 6034575010547019 z - 331689413716841140 z 42 44 - 13368220761224922448 z + 14428807186321477236 z 46 58 - 13368220761224922448 z - 331689413716841140 z 56 54 + 908523299766536367 z - 2122849565354295996 z 52 60 70 + 4240187342384918108 z + 103026418363751226 z - 22681898013368 z 68 78 32 + 175746226823750 z - 740219164 z + 908523299766536367 z 38 40 - 7251416479032614910 z + 10630107896927717943 z 62 76 74 - 27134880821078222 z + 13972134458 z - 205250437378 z 72 + 2398895464349 z )) And in Maple-input format, it is: -(1+40103090708264591*z^28-10902015239466873*z^26-169*z^2+2511592165721733*z^24 -487841874562328*z^22+12971*z^4-603498*z^6+19175572*z^8-444447166*z^10+ 7836766853*z^12-108225005147*z^14-10738567815560*z^18+1195659842215*z^16-\ 2543430473825159811*z^50+3692565695295961811*z^48+79386773066072*z^20+ 1507430083333138881*z^36-767798520248231534*z^34-487841874562328*z^66+19175572* z^80+z^88+12971*z^84-169*z^86-603498*z^82+2511592165721733*z^64-\ 125535584588035914*z^30-4616680547035154112*z^42+4973247130545199984*z^44-\ 4616680547035154112*z^46-125535584588035914*z^58+335531212984091292*z^56-\ 767798520248231534*z^54+1507430083333138881*z^52+40103090708264591*z^60-\ 10738567815560*z^70+79386773066072*z^68-444447166*z^78+335531212984091292*z^32-\ 2543430473825159811*z^38+3692565695295961811*z^40-10902015239466873*z^62+ 7836766853*z^76-108225005147*z^74+1195659842215*z^72)/(-1+z^2)/(1+ 103026418363751226*z^28-27134880821078222*z^26-194*z^2+6034575010547019*z^24-\ 1127296691968840*z^22+16688*z^4-855812*z^6+29608257*z^8-740219164*z^10+ 13972134458*z^12-205250437378*z^14-22681898013368*z^18+2398895464349*z^16-\ 7251416479032614910*z^50+10630107896927717943*z^48+175746226823750*z^20+ 4240187342384918108*z^36-2122849565354295996*z^34-1127296691968840*z^66+ 29608257*z^80+z^88+16688*z^84-194*z^86-855812*z^82+6034575010547019*z^64-\ 331689413716841140*z^30-13368220761224922448*z^42+14428807186321477236*z^44-\ 13368220761224922448*z^46-331689413716841140*z^58+908523299766536367*z^56-\ 2122849565354295996*z^54+4240187342384918108*z^52+103026418363751226*z^60-\ 22681898013368*z^70+175746226823750*z^68-740219164*z^78+908523299766536367*z^32 -7251416479032614910*z^38+10630107896927717943*z^40-27134880821078222*z^62+ 13972134458*z^76-205250437378*z^74+2398895464349*z^72) The first , 40, terms are: [0, 26, 0, 1159, 0, 56075, 0, 2764890, 0, 137037361, 0, 6802920121, 0, 337904904474, 0, 16787612217699, 0, 834112271282047, 0, 41445663314333402, 0, 2059410179323730841, 0, 102331933203752398985, 0, 5084892982517512351354, 0, 252669980280028451482223, 0, 12555271105135045910421459, 0, 623876845477766406030055194, 0, 31000722188476224690030114825, 0, 1540440206274843756950431602337, 0, 76545193934077474133977204732442, 0, 3803566692524185275305134180701339] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 2687498 z + 11645019 z + 81 z - 34519579 z + 70851084 z 4 6 8 10 12 - 2514 z + 41532 z - 415572 z + 2687498 z - 11645019 z 14 18 16 20 36 + 34519579 z + 101354768 z - 70851084 z - 101354768 z - 81 z 34 30 32 38 / 30 2 + 2514 z + 415572 z - 41532 z + z ) / (-7203412 z - 105 z / 8 34 38 40 6 4 + 958146 z - 81204 z - 105 z + z + 1 - 81204 z + 4071 z 12 36 16 32 28 + 36055409 z + 4071 z + 294583207 z + 958146 z + 36055409 z 10 18 14 26 - 7203412 z - 494475288 z - 123501587 z - 123501587 z 24 20 22 + 294583207 z + 587324548 z - 494475288 z ) And in Maple-input format, it is: -(-1-2687498*z^28+11645019*z^26+81*z^2-34519579*z^24+70851084*z^22-2514*z^4+ 41532*z^6-415572*z^8+2687498*z^10-11645019*z^12+34519579*z^14+101354768*z^18-\ 70851084*z^16-101354768*z^20-81*z^36+2514*z^34+415572*z^30-41532*z^32+z^38)/(-\ 7203412*z^30-105*z^2+958146*z^8-81204*z^34-105*z^38+z^40+1-81204*z^6+4071*z^4+ 36055409*z^12+4071*z^36+294583207*z^16+958146*z^32+36055409*z^28-7203412*z^10-\ 494475288*z^18-123501587*z^14-123501587*z^26+294583207*z^24+587324548*z^20-\ 494475288*z^22) The first , 40, terms are: [0, 24, 0, 963, 0, 43083, 0, 2009664, 0, 95343689, 0, 4554034033, 0, 218102904832, 0, 10456433403923, 0, 501516504955435, 0, 24057880557020856, 0, 1154136454464439593, 0, 55369143754670600217, 0, 2656334208807542679224, 0, 127438101849326985811675, 0, 6113874628578745316945283, 0, 293314829527715092336122944, 0, 14071863498514244756667914849, 0, 675101769321778714587347025369, 0, 32388206103696738524434230979456, 0, 1553833743319083476157411022985947] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 12268977935598 z - 7468701154028 z - 123 z 24 22 4 6 8 + 3714810446356 z - 1504025774180 z + 6512 z - 199175 z + 3986593 z 10 12 14 18 - 56013708 z + 577868772 z - 4514627692 z - 130006716300 z 16 50 48 20 + 27308659432 z - 4514627692 z + 27308659432 z + 493109158232 z 36 34 64 30 + 12268977935598 z - 16509145116574 z + z - 16509145116574 z 42 44 46 58 - 1504025774180 z + 493109158232 z - 130006716300 z - 199175 z 56 54 52 60 + 3986593 z - 56013708 z + 577868772 z + 6512 z 32 38 40 62 + 18223323083864 z - 7468701154028 z + 3714810446356 z - 123 z ) / 2 28 26 2 / ((-1 + z ) (1 + 35725139813378 z - 21349714821513 z - 151 z / 24 22 4 6 + 10350899774282 z - 4056868843883 z + 9262 z - 318366 z 8 10 12 14 + 7029955 z - 107575441 z + 1196704652 z - 9997327327 z 18 16 50 48 - 322060910005 z + 64184099538 z - 9997327327 z + 64184099538 z 20 36 34 64 + 1278921977892 z + 35725139813378 z - 48611623056389 z + z 30 42 44 - 48611623056389 z - 4056868843883 z + 1278921977892 z 46 58 56 54 - 322060910005 z - 318366 z + 7029955 z - 107575441 z 52 60 32 38 + 1196704652 z + 9262 z + 53860046957066 z - 21349714821513 z 40 62 + 10350899774282 z - 151 z )) And in Maple-input format, it is: -(1+12268977935598*z^28-7468701154028*z^26-123*z^2+3714810446356*z^24-\ 1504025774180*z^22+6512*z^4-199175*z^6+3986593*z^8-56013708*z^10+577868772*z^12 -4514627692*z^14-130006716300*z^18+27308659432*z^16-4514627692*z^50+27308659432 *z^48+493109158232*z^20+12268977935598*z^36-16509145116574*z^34+z^64-\ 16509145116574*z^30-1504025774180*z^42+493109158232*z^44-130006716300*z^46-\ 199175*z^58+3986593*z^56-56013708*z^54+577868772*z^52+6512*z^60+18223323083864* z^32-7468701154028*z^38+3714810446356*z^40-123*z^62)/(-1+z^2)/(1+35725139813378 *z^28-21349714821513*z^26-151*z^2+10350899774282*z^24-4056868843883*z^22+9262*z ^4-318366*z^6+7029955*z^8-107575441*z^10+1196704652*z^12-9997327327*z^14-\ 322060910005*z^18+64184099538*z^16-9997327327*z^50+64184099538*z^48+ 1278921977892*z^20+35725139813378*z^36-48611623056389*z^34+z^64-48611623056389* z^30-4056868843883*z^42+1278921977892*z^44-322060910005*z^46-318366*z^58+ 7029955*z^56-107575441*z^54+1196704652*z^52+9262*z^60+53860046957066*z^32-\ 21349714821513*z^38+10350899774282*z^40-151*z^62) The first , 40, terms are: [0, 29, 0, 1507, 0, 84540, 0, 4804173, 0, 273685051, 0, 15599343839, 0, 889223026131, 0, 50690501198547, 0, 2889650746004361, 0, 164727043024316204, 0, 9390412614728520031, 0, 535308968335839320821, 0, 30515773803513909091381, 0, 1739579427976686947050541, 0, 99166307480824591209144149, 0, 5653065567577101680290263255, 0, 322258145505736409427816555628, 0, 18370618763751072758888266759841, 0, 1047233835749294205017253386733651, 0, 59698517557573603455072501738316267] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 4 f(z) = - (-1 - 75 z + 1977 z + 75 z - 24291 z + 161772 z - 1977 z 6 8 10 12 14 + 24291 z - 161772 z + 634816 z - 1541060 z + 2383928 z 18 16 20 30 / 4 2 + 1541060 z - 2383928 z - 634816 z + z ) / (3528 z + 1 - 105 z / 22 20 18 16 14 - 2073112 z + 6189134 z - 11814854 z + 14633472 z - 11814854 z 12 32 30 28 26 24 + 6189134 z + z - 105 z + 3528 z - 53221 z + 431957 z 6 10 8 - 53221 z - 2073112 z + 431957 z ) And in Maple-input format, it is: -(-1-75*z^28+1977*z^26+75*z^2-24291*z^24+161772*z^22-1977*z^4+24291*z^6-161772* z^8+634816*z^10-1541060*z^12+2383928*z^14+1541060*z^18-2383928*z^16-634816*z^20 +z^30)/(3528*z^4+1-105*z^2-2073112*z^22+6189134*z^20-11814854*z^18+14633472*z^ 16-11814854*z^14+6189134*z^12+z^32-105*z^30+3528*z^28-53221*z^26+431957*z^24-\ 53221*z^6-2073112*z^10+431957*z^8) The first , 40, terms are: [0, 30, 0, 1599, 0, 90985, 0, 5238598, 0, 302637675, 0, 17504340859, 0, 1012890552302, 0, 58621091932177, 0, 3392924392275159, 0, 196383805706966150, 0, 11366886287221495297, 0, 657929030571322887265, 0, 38081780088195206844662, 0, 2204223797962941655305799, 0, 127583417814010295031866913, 0, 7384699302684549748661079774, 0, 427436312672005168473522429051, 0, 24740588119472898110281939283371, 0, 1432018491481420392962830427483094, 0, 82887155074070910835780275757211417] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 4150964631 z - 6170413748 z - 92 z + 7039443742 z 22 4 6 8 10 - 6170413748 z + 3473 z - 72660 z + 957463 z - 8529416 z 12 14 18 16 48 + 53679224 z - 245742464 z - 2135564308 z + 834996281 z + z 20 36 34 30 + 4150964631 z + 53679224 z - 245742464 z - 2135564308 z 42 44 46 32 38 40 - 72660 z + 3473 z - 92 z + 834996281 z - 8529416 z + 957463 z / 2 28 26 2 ) / ((-1 + z ) (1 + 11773128661 z - 17843131813 z - 115 z / 24 22 4 6 8 + 20488760014 z - 17843131813 z + 5163 z - 124468 z + 1846109 z 10 12 14 18 - 18165725 z + 124294520 z - 610069531 z - 5866232044 z 16 48 20 36 34 + 2194074451 z + z + 11773128661 z + 124294520 z - 610069531 z 30 42 44 46 32 - 5866232044 z - 124468 z + 5163 z - 115 z + 2194074451 z 38 40 - 18165725 z + 1846109 z )) And in Maple-input format, it is: -(1+4150964631*z^28-6170413748*z^26-92*z^2+7039443742*z^24-6170413748*z^22+3473 *z^4-72660*z^6+957463*z^8-8529416*z^10+53679224*z^12-245742464*z^14-2135564308* z^18+834996281*z^16+z^48+4150964631*z^20+53679224*z^36-245742464*z^34-\ 2135564308*z^30-72660*z^42+3473*z^44-92*z^46+834996281*z^32-8529416*z^38+957463 *z^40)/(-1+z^2)/(1+11773128661*z^28-17843131813*z^26-115*z^2+20488760014*z^24-\ 17843131813*z^22+5163*z^4-124468*z^6+1846109*z^8-18165725*z^10+124294520*z^12-\ 610069531*z^14-5866232044*z^18+2194074451*z^16+z^48+11773128661*z^20+124294520* z^36-610069531*z^34-5866232044*z^30-124468*z^42+5163*z^44-115*z^46+2194074451*z ^32-18165725*z^38+1846109*z^40) The first , 40, terms are: [0, 24, 0, 979, 0, 43863, 0, 2018976, 0, 93789621, 0, 4371753373, 0, 204049391728, 0, 9529075912863, 0, 445107454393867, 0, 20793176717320040, 0, 971392623146747769, 0, 45381256851268239497, 0, 2120125654540603385416, 0, 99048554735161263763483, 0, 4627381642501865711129743, 0, 216183613553472319134338896, 0, 10099743507763557124226837965, 0, 471843492363847402211415437957, 0, 22043757130334985486928365072064, 0, 1029848344596708449263322510832487] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 2980010678 z + 5358436651 z + 103 z - 7176712021 z 22 4 6 8 10 + 7176712021 z - 4294 z + 97618 z - 1370115 z + 12719949 z 12 14 18 16 - 81561185 z + 371725795 z + 2980010678 z - 1228025046 z 20 36 34 30 42 - 5358436651 z - 12719949 z + 81561185 z + 1228025046 z + 4294 z 44 46 32 38 40 / - 103 z + z - 371725795 z + 1370115 z - 97618 z ) / (1 / 28 26 2 24 + 23933214409 z - 36737494319 z - 131 z + 42363210706 z 22 4 6 8 10 - 36737494319 z + 6585 z - 175442 z + 2838851 z - 30085909 z 12 14 18 16 48 + 219026530 z - 1130947977 z - 11672338362 z + 4234365283 z + z 20 36 34 30 + 23933214409 z + 219026530 z - 1130947977 z - 11672338362 z 42 44 46 32 38 - 175442 z + 6585 z - 131 z + 4234365283 z - 30085909 z 40 + 2838851 z ) And in Maple-input format, it is: -(-1-2980010678*z^28+5358436651*z^26+103*z^2-7176712021*z^24+7176712021*z^22-\ 4294*z^4+97618*z^6-1370115*z^8+12719949*z^10-81561185*z^12+371725795*z^14+ 2980010678*z^18-1228025046*z^16-5358436651*z^20-12719949*z^36+81561185*z^34+ 1228025046*z^30+4294*z^42-103*z^44+z^46-371725795*z^32+1370115*z^38-97618*z^40) /(1+23933214409*z^28-36737494319*z^26-131*z^2+42363210706*z^24-36737494319*z^22 +6585*z^4-175442*z^6+2838851*z^8-30085909*z^10+219026530*z^12-1130947977*z^14-\ 11672338362*z^18+4234365283*z^16+z^48+23933214409*z^20+219026530*z^36-\ 1130947977*z^34-11672338362*z^30-175442*z^42+6585*z^44-131*z^46+4234365283*z^32 -30085909*z^38+2838851*z^40) The first , 40, terms are: [0, 28, 0, 1377, 0, 73831, 0, 4047956, 0, 223566867, 0, 12380369899, 0, 686281701980, 0, 38057974934811, 0, 2110856474372145, 0, 117084621297145668, 0, 6494599688283345005, 0, 360254600463034744541, 0, 19983362122666972063924, 0, 1108481229105869604953297, 0, 61487726847770142607360843, 0, 3410740344781927780077451612, 0, 189194684342616060745030222555, 0, 10494680533472960934106848625763, 0, 582142791337587945486597279874484, 0, 32291619655902002412139674041676887] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 9182383318 z - 13781460580 z - 107 z + 15771233648 z 22 4 6 8 10 - 13781460580 z + 4664 z - 110525 z + 1612753 z - 15582602 z 12 14 18 16 48 + 104502816 z - 502295448 z - 4644395054 z + 1769723972 z + z 20 36 34 30 + 9182383318 z + 104502816 z - 502295448 z - 4644395054 z 42 44 46 32 38 - 110525 z + 4664 z - 107 z + 1769723972 z - 15582602 z 40 / 28 26 2 + 1612753 z ) / ((1 + 27150489194 z - 41791957294 z - 139 z / 24 22 4 6 8 + 48237397538 z - 41791957294 z + 7185 z - 193571 z + 3143465 z 10 12 14 18 - 33378156 z + 243695272 z - 1263876068 z - 13184079708 z 16 48 20 36 34 + 4757460408 z + z + 27150489194 z + 243695272 z - 1263876068 z 30 42 44 46 32 - 13184079708 z - 193571 z + 7185 z - 139 z + 4757460408 z 38 40 2 - 33378156 z + 3143465 z ) (-1 + z )) And in Maple-input format, it is: -(1+9182383318*z^28-13781460580*z^26-107*z^2+15771233648*z^24-13781460580*z^22+ 4664*z^4-110525*z^6+1612753*z^8-15582602*z^10+104502816*z^12-502295448*z^14-\ 4644395054*z^18+1769723972*z^16+z^48+9182383318*z^20+104502816*z^36-502295448*z ^34-4644395054*z^30-110525*z^42+4664*z^44-107*z^46+1769723972*z^32-15582602*z^ 38+1612753*z^40)/(1+27150489194*z^28-41791957294*z^26-139*z^2+48237397538*z^24-\ 41791957294*z^22+7185*z^4-193571*z^6+3143465*z^8-33378156*z^10+243695272*z^12-\ 1263876068*z^14-13184079708*z^18+4757460408*z^16+z^48+27150489194*z^20+ 243695272*z^36-1263876068*z^34-13184079708*z^30-193571*z^42+7185*z^44-139*z^46+ 4757460408*z^32-33378156*z^38+3143465*z^40)/(-1+z^2) The first , 40, terms are: [0, 33, 0, 1960, 0, 122939, 0, 7757085, 0, 489885255, 0, 30943839365, 0, 1954691620864, 0, 123478245651375, 0, 7800197264877971, 0, 492744530927671003, 0, 31127084916299268175, 0, 1966324743632280592592, 0, 124214443362822259488453, 0, 7846734775324096184203023, 0, 495685092614089638363835237, 0, 31312860692825949526338661019, 0, 1978060793510401061233165016248, 0, 124955830307897606883935182440737, 0, 7893569088384834934707904248802249, 0, 498643663202811299514488657918203577 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 366985630439125 z - 161321675736727 z - 140 z 24 22 4 6 + 58697471159509 z - 17599857947458 z + 8691 z - 319445 z 8 10 12 14 + 7835803 z - 137080186 z + 1784435693 z - 17800594326 z 18 16 50 - 863839971610 z + 139020882802 z - 17599857947458 z 48 20 36 + 58697471159509 z + 4323720964959 z + 1564922171667700 z 34 66 64 30 - 1429872104662488 z - 319445 z + 7835803 z - 693313874448528 z 42 44 46 - 693313874448528 z + 366985630439125 z - 161321675736727 z 58 56 54 - 17800594326 z + 139020882802 z - 863839971610 z 52 60 70 68 + 4323720964959 z + 1784435693 z - 140 z + 8691 z 32 38 40 + 1090380569232443 z - 1429872104662488 z + 1090380569232443 z 62 72 / 28 - 137080186 z + z ) / (-1 - 1464340830309338 z / 26 2 24 22 + 593258232329157 z + 167 z - 199093056643281 z + 55079988808512 z 4 6 8 10 12 - 11924 z + 493372 z - 13442385 z + 258949677 z - 3690391350 z 14 18 16 + 40144207340 z + 2300086527125 z - 340970633417 z 50 48 20 + 199093056643281 z - 593258232329157 z - 12484455829034 z 36 34 66 - 8778803866769214 z + 7346084870387621 z + 13442385 z 64 30 42 - 258949677 z + 3005004129979082 z + 5140877203835231 z 44 46 58 - 3005004129979082 z + 1464340830309338 z + 340970633417 z 56 54 52 - 2300086527125 z + 12484455829034 z - 55079988808512 z 60 70 68 32 - 40144207340 z + 11924 z - 493372 z - 5140877203835231 z 38 40 62 74 + 8778803866769214 z - 7346084870387621 z + 3690391350 z + z 72 - 167 z ) And in Maple-input format, it is: -(1+366985630439125*z^28-161321675736727*z^26-140*z^2+58697471159509*z^24-\ 17599857947458*z^22+8691*z^4-319445*z^6+7835803*z^8-137080186*z^10+1784435693*z ^12-17800594326*z^14-863839971610*z^18+139020882802*z^16-17599857947458*z^50+ 58697471159509*z^48+4323720964959*z^20+1564922171667700*z^36-1429872104662488*z ^34-319445*z^66+7835803*z^64-693313874448528*z^30-693313874448528*z^42+ 366985630439125*z^44-161321675736727*z^46-17800594326*z^58+139020882802*z^56-\ 863839971610*z^54+4323720964959*z^52+1784435693*z^60-140*z^70+8691*z^68+ 1090380569232443*z^32-1429872104662488*z^38+1090380569232443*z^40-137080186*z^ 62+z^72)/(-1-1464340830309338*z^28+593258232329157*z^26+167*z^2-199093056643281 *z^24+55079988808512*z^22-11924*z^4+493372*z^6-13442385*z^8+258949677*z^10-\ 3690391350*z^12+40144207340*z^14+2300086527125*z^18-340970633417*z^16+ 199093056643281*z^50-593258232329157*z^48-12484455829034*z^20-8778803866769214* z^36+7346084870387621*z^34+13442385*z^66-258949677*z^64+3005004129979082*z^30+ 5140877203835231*z^42-3005004129979082*z^44+1464340830309338*z^46+340970633417* z^58-2300086527125*z^56+12484455829034*z^54-55079988808512*z^52-40144207340*z^ 60+11924*z^70-493372*z^68-5140877203835231*z^32+8778803866769214*z^38-\ 7346084870387621*z^40+3690391350*z^62+z^74-167*z^72) The first , 40, terms are: [0, 27, 0, 1276, 0, 65071, 0, 3366295, 0, 174732429, 0, 9078025837, 0, 471769928204, 0, 24519371393725, 0, 1274391174916739, 0, 66237143279657687, 0, 3442726923255333881, 0, 178938767387643896668, 0, 9300508258276127385289, 0, 483402718061525412536569, 0, 25125317948681008667450067, 0, 1305912495457340014507794763, 0, 67876056660449473055534168748, 0, 3527923315594950131262387524615, 0, 183367208176474946373010299634565, 0, 9530687075843788070562517911218701] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 6}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 13543272710748 z - 8146288461612 z - 119 z 24 22 4 6 8 + 3987888282736 z - 1584126913338 z + 6148 z - 185651 z + 3708203 z 10 12 14 18 - 52448404 z + 548277456 z - 4360581838 z - 131092023652 z 16 50 48 20 + 26931055368 z - 4360581838 z + 26931055368 z + 508458626736 z 36 34 64 30 + 13543272710748 z - 18359902458110 z + z - 18359902458110 z 42 44 46 58 - 1584126913338 z + 508458626736 z - 131092023652 z - 185651 z 56 54 52 60 + 3708203 z - 52448404 z + 548277456 z + 6148 z 32 38 40 62 + 20317440844640 z - 8146288461612 z + 3987888282736 z - 119 z ) / 28 26 2 / (-1 - 60944786915931 z + 33474972146971 z + 142 z / 24 22 4 6 - 14999747762701 z + 5462860200239 z - 8545 z + 295012 z 8 10 12 14 - 6642367 z + 104764164 z - 1211279381 z + 10590648051 z 18 16 50 48 + 380483759771 z - 71600316129 z + 71600316129 z - 380483759771 z 20 36 34 66 - 1608808876821 z - 90748883019395 z + 110694824537067 z + z 64 30 42 44 - 142 z + 90748883019395 z + 14999747762701 z - 5462860200239 z 46 58 56 54 + 1608808876821 z + 6642367 z - 104764164 z + 1211279381 z 52 60 32 38 - 10590648051 z - 295012 z - 110694824537067 z + 60944786915931 z 40 62 - 33474972146971 z + 8545 z ) And in Maple-input format, it is: -(1+13543272710748*z^28-8146288461612*z^26-119*z^2+3987888282736*z^24-\ 1584126913338*z^22+6148*z^4-185651*z^6+3708203*z^8-52448404*z^10+548277456*z^12 -4360581838*z^14-131092023652*z^18+26931055368*z^16-4360581838*z^50+26931055368 *z^48+508458626736*z^20+13543272710748*z^36-18359902458110*z^34+z^64-\ 18359902458110*z^30-1584126913338*z^42+508458626736*z^44-131092023652*z^46-\ 185651*z^58+3708203*z^56-52448404*z^54+548277456*z^52+6148*z^60+20317440844640* z^32-8146288461612*z^38+3987888282736*z^40-119*z^62)/(-1-60944786915931*z^28+ 33474972146971*z^26+142*z^2-14999747762701*z^24+5462860200239*z^22-8545*z^4+ 295012*z^6-6642367*z^8+104764164*z^10-1211279381*z^12+10590648051*z^14+ 380483759771*z^18-71600316129*z^16+71600316129*z^50-380483759771*z^48-\ 1608808876821*z^20-90748883019395*z^36+110694824537067*z^34+z^66-142*z^64+ 90748883019395*z^30+14999747762701*z^42-5462860200239*z^44+1608808876821*z^46+ 6642367*z^58-104764164*z^56+1211279381*z^54-10590648051*z^52-295012*z^60-\ 110694824537067*z^32+60944786915931*z^38-33474972146971*z^40+8545*z^62) The first , 40, terms are: [0, 23, 0, 869, 0, 36224, 0, 1569315, 0, 69215397, 0, 3079661311, 0, 137630692335, 0, 6164549879293, 0, 276431237964851, 0, 12403102253567104, 0, 556681100560995477, 0, 24989143068043397583, 0, 1121842288986811190705, 0, 50365210319685621836817, 0, 2261200229904496584901055, 0, 101520166269147895537785397, 0, 4557935468908681304630549760, 0, 204637558334724573944500765779, 0, 9187623767188090006134875895533, 0, 412497588475235389754067768657807] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 2927114 z + 12369687 z + 85 z - 35806315 z + 72188276 z 4 6 8 10 12 - 2762 z + 46408 z - 461600 z + 2927114 z - 12369687 z 14 18 16 20 36 + 35806315 z + 102296904 z - 72188276 z - 102296904 z - 85 z 34 30 32 38 / 40 38 36 + 2762 z + 461600 z - 46408 z + z ) / (z - 113 z + 4503 z / 34 32 30 28 26 - 90196 z + 1056714 z - 7847884 z + 38756329 z - 131149483 z 24 22 20 18 + 309907519 z - 517175872 z + 613081140 z - 517175872 z 16 14 12 10 8 + 309907519 z - 131149483 z + 38756329 z - 7847884 z + 1056714 z 6 4 2 - 90196 z + 4503 z - 113 z + 1) And in Maple-input format, it is: -(-1-2927114*z^28+12369687*z^26+85*z^2-35806315*z^24+72188276*z^22-2762*z^4+ 46408*z^6-461600*z^8+2927114*z^10-12369687*z^12+35806315*z^14+102296904*z^18-\ 72188276*z^16-102296904*z^20-85*z^36+2762*z^34+461600*z^30-46408*z^32+z^38)/(z^ 40-113*z^38+4503*z^36-90196*z^34+1056714*z^32-7847884*z^30+38756329*z^28-\ 131149483*z^26+309907519*z^24-517175872*z^22+613081140*z^20-517175872*z^18+ 309907519*z^16-131149483*z^14+38756329*z^12-7847884*z^10+1056714*z^8-90196*z^6+ 4503*z^4-113*z^2+1) The first , 40, terms are: [0, 28, 0, 1423, 0, 78503, 0, 4393444, 0, 246641849, 0, 13857157281, 0, 778724087476, 0, 43764903725599, 0, 2459684932340903, 0, 138240993032095436, 0, 7769544592081949609, 0, 436671411854894791385, 0, 24542235893794164814764, 0, 1379347054478848767807543, 0, 77523433394499334639860975, 0, 4357049090616681491333854932, 0, 244879207013449064199007262161, 0, 13762944807578877442145436659849, 0, 773518716565607098965665737469764, 0, 43474068479264197761288266684554327] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 14 12 6 10 4 8 2 / f(z) = - (z - 42 z + 1001 z + 364 z - 364 z - 1001 z + 42 z - 1) / / 16 14 12 6 10 4 8 2 (z - 71 z + 952 z - 3976 z - 3976 z + 952 z + 6384 z - 71 z + 1 ) And in Maple-input format, it is: -(z^14-42*z^12+1001*z^6+364*z^10-364*z^4-1001*z^8+42*z^2-1)/(z^16-71*z^14+952*z ^12-3976*z^6-3976*z^10+952*z^4+6384*z^8-71*z^2+1) The first , 40, terms are: [0, 29, 0, 1471, 0, 79808, 0, 4375897, 0, 240378643, 0, 13209069847, 0, 725898384359, 0, 39891876471539, 0, 2192269974717929, 0, 120476898663671488, 0, 6620847045486150863, 0, 363850801995789860221, 0, 19995539171949615541457, 0, 1098861359580093467365169, 0, 60388283471627147242052029, 0, 3318657762342465623628311471, 0, 182377916881236755876618408128, 0, 10022637749329156449294793027529, 0, 550797317855662676767190329041971, 0, 30269245775874084451260942430868039] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 226053028 z + 642084134 z + 99 z - 1285886890 z 22 4 6 8 10 + 1819395004 z - 3916 z + 84040 z - 1107075 z + 9541737 z 12 14 18 16 - 55793836 z + 226053028 z + 1285886890 z - 642084134 z 20 36 34 30 42 - 1819395004 z - 84040 z + 1107075 z + 55793836 z + z 32 38 40 / 10 14 - 9541737 z + 3916 z - 99 z ) / (-22933518 z - 736508720 z / 30 32 12 28 - 736508720 z + 156283405 z + 156283405 z + 2437599378 z 22 8 18 38 36 - 11291842136 z + 2269267 z - 5714434908 z - 145412 z + 2269267 z 20 2 16 34 + 9523597118 z - 118 z + 2437599378 z - 22933518 z + 1 24 6 26 4 40 44 + 9523597118 z - 145412 z - 5714434908 z + 5647 z + 5647 z + z 42 - 118 z ) And in Maple-input format, it is: -(-1-226053028*z^28+642084134*z^26+99*z^2-1285886890*z^24+1819395004*z^22-3916* z^4+84040*z^6-1107075*z^8+9541737*z^10-55793836*z^12+226053028*z^14+1285886890* z^18-642084134*z^16-1819395004*z^20-84040*z^36+1107075*z^34+55793836*z^30+z^42-\ 9541737*z^32+3916*z^38-99*z^40)/(-22933518*z^10-736508720*z^14-736508720*z^30+ 156283405*z^32+156283405*z^12+2437599378*z^28-11291842136*z^22+2269267*z^8-\ 5714434908*z^18-145412*z^38+2269267*z^36+9523597118*z^20-118*z^2+2437599378*z^ 16-22933518*z^34+1+9523597118*z^24-145412*z^6-5714434908*z^26+5647*z^4+5647*z^ 40+z^44-118*z^42) The first , 40, terms are: [0, 19, 0, 511, 0, 14377, 0, 411505, 0, 11951911, 0, 352796923, 0, 10610207593, 0, 325949356057, 0, 10249851611755, 0, 330339574105879, 0, 10912480993463521, 0, 369100668387709369, 0, 12756677117319115663, 0, 449297168426031848995, 0, 16078518266065498024273, 0, 582916971377179106127025, 0, 21353118939070834724421379, 0, 788528819376581437259127151, 0, 29299356806377836829113710233, 0, 1093781650856714446730251752961] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 40335 z + 366078 z + 82 z - 2037775 z + 7222176 z 4 6 8 10 12 - 2553 z + 40335 z - 366078 z + 2037775 z - 7222176 z 14 18 16 20 34 + 16647120 z + 25222920 z - 25222920 z - 16647120 z + z 30 32 / 12 8 6 + 2553 z - 82 z ) / (25024867 z + 1 + 849138 z - 76378 z / 18 28 34 24 20 - 160963156 z + 849138 z - 102 z + 25024867 z + 131005484 z 2 10 26 30 16 - 102 z - 5777798 z - 5777798 z - 76378 z + 131005484 z 22 14 32 4 36 - 70555308 z - 70555308 z + 3930 z + 3930 z + z ) And in Maple-input format, it is: -(-1-40335*z^28+366078*z^26+82*z^2-2037775*z^24+7222176*z^22-2553*z^4+40335*z^6 -366078*z^8+2037775*z^10-7222176*z^12+16647120*z^14+25222920*z^18-25222920*z^16 -16647120*z^20+z^34+2553*z^30-82*z^32)/(25024867*z^12+1+849138*z^8-76378*z^6-\ 160963156*z^18+849138*z^28-102*z^34+25024867*z^24+131005484*z^20-102*z^2-\ 5777798*z^10-5777798*z^26-76378*z^30+131005484*z^16-70555308*z^22-70555308*z^14 +3930*z^32+3930*z^4+z^36) The first , 40, terms are: [0, 20, 0, 663, 0, 25069, 0, 995948, 0, 40461403, 0, 1662482323, 0, 68725449900, 0, 2850661560757, 0, 118467552907135, 0, 4928565306804180, 0, 205166651194663577, 0, 8543653075454570409, 0, 355849189344825577812, 0, 14823028822980971136623, 0, 617497968832012942082149, 0, 25724669130466366189595436, 0, 1071699360299059632730635235, 0, 44647917462574496408493100939, 0, 1860082971562616007273118820844, 0, 77493461491413758801379272183229] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 41364200693431 z + 26079733357409 z + 141 z 24 22 4 6 - 13040924560305 z + 5163031977015 z - 8539 z + 299051 z 8 10 12 14 - 6854893 z + 110031693 z - 1289401307 z + 11337440615 z 18 16 50 48 + 396742508001 z - 76210276549 z + 1289401307 z - 11337440615 z 20 36 34 - 1614021216503 z - 26079733357409 z + 41364200693431 z 30 42 44 + 52082462214867 z + 1614021216503 z - 396742508001 z 46 58 56 54 52 + 76210276549 z + 8539 z - 299051 z + 6854893 z - 110031693 z 60 32 38 40 - 141 z - 52082462214867 z + 13040924560305 z - 5163031977015 z 62 / 28 26 2 + z ) / (1 + 202712802316052 z - 115123978674174 z - 162 z / 24 22 4 6 + 52029122130592 z - 18665152906898 z + 11324 z - 455754 z 8 10 12 14 + 11903072 z - 215673126 z + 2829073940 z - 27658469630 z 18 16 50 - 1181428729658 z + 205718923740 z - 27658469630 z 48 20 36 + 205718923740 z + 5295137697948 z + 202712802316052 z 34 64 30 42 - 284491775901478 z + z - 284491775901478 z - 18665152906898 z 44 46 58 56 + 5295137697948 z - 1181428729658 z - 455754 z + 11903072 z 54 52 60 32 - 215673126 z + 2829073940 z + 11324 z + 318489184348742 z 38 40 62 - 115123978674174 z + 52029122130592 z - 162 z ) And in Maple-input format, it is: -(-1-41364200693431*z^28+26079733357409*z^26+141*z^2-13040924560305*z^24+ 5163031977015*z^22-8539*z^4+299051*z^6-6854893*z^8+110031693*z^10-1289401307*z^ 12+11337440615*z^14+396742508001*z^18-76210276549*z^16+1289401307*z^50-\ 11337440615*z^48-1614021216503*z^20-26079733357409*z^36+41364200693431*z^34+ 52082462214867*z^30+1614021216503*z^42-396742508001*z^44+76210276549*z^46+8539* z^58-299051*z^56+6854893*z^54-110031693*z^52-141*z^60-52082462214867*z^32+ 13040924560305*z^38-5163031977015*z^40+z^62)/(1+202712802316052*z^28-\ 115123978674174*z^26-162*z^2+52029122130592*z^24-18665152906898*z^22+11324*z^4-\ 455754*z^6+11903072*z^8-215673126*z^10+2829073940*z^12-27658469630*z^14-\ 1181428729658*z^18+205718923740*z^16-27658469630*z^50+205718923740*z^48+ 5295137697948*z^20+202712802316052*z^36-284491775901478*z^34+z^64-\ 284491775901478*z^30-18665152906898*z^42+5295137697948*z^44-1181428729658*z^46-\ 455754*z^58+11903072*z^56-215673126*z^54+2829073940*z^52+11324*z^60+ 318489184348742*z^32-115123978674174*z^38+52029122130592*z^40-162*z^62) The first , 40, terms are: [0, 21, 0, 617, 0, 18853, 0, 589933, 0, 18954913, 0, 627892365, 0, 21509631401, 0, 762935629465, 0, 27989296548445, 0, 1058782195422705, 0, 41116324753767069, 0, 1631109535416466933, 0, 65791271363010082969, 0, 2687154419110450220485, 0, 110768088853358208879249, 0, 4596431058798583278086769, 0, 191637120562398258860607845, 0, 8016516416455126825608374073, 0, 336128278171471963004544632917, 0, 14116615493329707704634436519229] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 38801987100034673 z - 10583211673372268 z - 172 z 24 22 4 6 + 2447595619450309 z - 477533259520700 z + 13293 z - 618602 z 8 10 12 14 + 19583336 z - 451320258 z + 7905857131 z - 108443534692 z 18 16 50 - 10624369827164 z + 1190238819911 z - 2438123594346751876 z 48 20 36 + 3537165331247491443 z + 78102268137104 z + 1446519749151919799 z 34 66 80 88 - 737811241083555878 z - 477533259520700 z + 19583336 z + z 84 86 82 64 + 13293 z - 172 z - 618602 z + 2447595619450309 z 30 42 - 121126876268304558 z - 4420562000694484456 z 44 46 + 4761336457335070432 z - 4420562000694484456 z 58 56 54 - 121126876268304558 z + 323012998373236376 z - 737811241083555878 z 52 60 70 + 1446519749151919799 z + 38801987100034673 z - 10624369827164 z 68 78 32 + 78102268137104 z - 451320258 z + 323012998373236376 z 38 40 - 2438123594346751876 z + 3537165331247491443 z 62 76 74 - 10583211673372268 z + 7905857131 z - 108443534692 z 72 / 28 + 1190238819911 z ) / (-1 - 133457425047719010 z / 26 2 24 + 34056846844490376 z + 196 z - 7362942068210041 z 22 4 6 8 + 1341098002093932 z - 17078 z + 887843 z - 31146362 z 10 12 14 18 + 789802306 z - 15129558769 z + 225757080846 z + 25846046406667 z 16 50 48 - 2683505402576 z + 18206601984479695588 z - 24422036356451583867 z 20 36 34 - 204390995151884 z - 6489886391967040305 z + 3095374193715554518 z 66 80 90 88 84 + 7362942068210041 z - 789802306 z + z - 196 z - 887843 z 86 82 64 + 17078 z + 31146362 z - 34056846844490376 z 30 42 + 445089159737326779 z + 24422036356451583867 z 44 46 - 28280424951989619848 z + 28280424951989619848 z 58 56 + 1268041650475554414 z - 3095374193715554518 z 54 52 + 6489886391967040305 z - 11709335307811350746 z 60 70 68 - 445089159737326779 z + 204390995151884 z - 1341098002093932 z 78 32 38 + 15129558769 z - 1268041650475554414 z + 11709335307811350746 z 40 62 76 - 18206601984479695588 z + 133457425047719010 z - 225757080846 z 74 72 + 2683505402576 z - 25846046406667 z ) And in Maple-input format, it is: -(1+38801987100034673*z^28-10583211673372268*z^26-172*z^2+2447595619450309*z^24 -477533259520700*z^22+13293*z^4-618602*z^6+19583336*z^8-451320258*z^10+ 7905857131*z^12-108443534692*z^14-10624369827164*z^18+1190238819911*z^16-\ 2438123594346751876*z^50+3537165331247491443*z^48+78102268137104*z^20+ 1446519749151919799*z^36-737811241083555878*z^34-477533259520700*z^66+19583336* z^80+z^88+13293*z^84-172*z^86-618602*z^82+2447595619450309*z^64-\ 121126876268304558*z^30-4420562000694484456*z^42+4761336457335070432*z^44-\ 4420562000694484456*z^46-121126876268304558*z^58+323012998373236376*z^56-\ 737811241083555878*z^54+1446519749151919799*z^52+38801987100034673*z^60-\ 10624369827164*z^70+78102268137104*z^68-451320258*z^78+323012998373236376*z^32-\ 2438123594346751876*z^38+3537165331247491443*z^40-10583211673372268*z^62+ 7905857131*z^76-108443534692*z^74+1190238819911*z^72)/(-1-133457425047719010*z^ 28+34056846844490376*z^26+196*z^2-7362942068210041*z^24+1341098002093932*z^22-\ 17078*z^4+887843*z^6-31146362*z^8+789802306*z^10-15129558769*z^12+225757080846* z^14+25846046406667*z^18-2683505402576*z^16+18206601984479695588*z^50-\ 24422036356451583867*z^48-204390995151884*z^20-6489886391967040305*z^36+ 3095374193715554518*z^34+7362942068210041*z^66-789802306*z^80+z^90-196*z^88-\ 887843*z^84+17078*z^86+31146362*z^82-34056846844490376*z^64+445089159737326779* z^30+24422036356451583867*z^42-28280424951989619848*z^44+28280424951989619848*z ^46+1268041650475554414*z^58-3095374193715554518*z^56+6489886391967040305*z^54-\ 11709335307811350746*z^52-445089159737326779*z^60+204390995151884*z^70-\ 1341098002093932*z^68+15129558769*z^78-1268041650475554414*z^32+ 11709335307811350746*z^38-18206601984479695588*z^40+133457425047719010*z^62-\ 225757080846*z^76+2683505402576*z^74-25846046406667*z^72) The first , 40, terms are: [0, 24, 0, 919, 0, 39493, 0, 1791152, 0, 83501415, 0, 3948614111, 0, 188122145968, 0, 8998288350677, 0, 431331226689079, 0, 20700087910030584, 0, 994070636912832041, 0, 47755366120744761465, 0, 2294659383698936557496, 0, 110272372145294516087575, 0, 5299631006978095897319445, 0, 254707844523371437917082416, 0, 12241917447050818601199060431, 0, 588386539261396284098336739383, 0, 28280016214738554016147935338800, 0, 1359248121765911076938072085696453] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 102378309871415392 z - 26994625401375352 z - 188 z 24 22 4 6 + 6005311888433461 z - 1121008063941348 z + 15956 z - 816180 z 8 10 12 14 + 28339211 z - 713206716 z + 13565067728 z - 200733771300 z 18 16 50 - 22432982099388 z + 2360924784695 z - 7138221785497674320 z 48 20 36 + 10450523524193301678 z + 174415711060724 z + 4181103532444250424 z 34 66 80 88 - 2097397654729556256 z - 1121008063941348 z + 28339211 z + z 84 86 82 64 + 15956 z - 188 z - 816180 z + 6005311888433461 z 30 42 - 329048976489914792 z - 13131618202223411456 z 44 46 + 14169479117313975968 z - 13131618202223411456 z 58 56 - 329048976489914792 z + 899497859558739418 z 54 52 - 2097397654729556256 z + 4181103532444250424 z 60 70 68 + 102378309871415392 z - 22432982099388 z + 174415711060724 z 78 32 38 - 713206716 z + 899497859558739418 z - 7138221785497674320 z 40 62 76 + 10450523524193301678 z - 26994625401375352 z + 13565067728 z 74 72 / 2 - 200733771300 z + 2360924784695 z ) / ((-1 + z ) (1 / 28 26 2 + 280086295797112848 z - 71094750615269376 z - 216 z 24 22 4 6 + 15164877247614245 z - 2703676276178920 z + 20558 z - 1160500 z 8 10 12 14 + 43968391 z - 1197295960 z + 24477597996 z - 387221097832 z 18 16 50 - 48781170825124 z + 4845762075667 z - 22164281383539516008 z 48 20 + 32835011719059797110 z + 400211172199238 z 36 34 + 12770494342992330732 z - 6273048471593251840 z 66 80 88 84 86 - 2703676276178920 z + 43968391 z + z + 20558 z - 216 z 82 64 30 - 1160500 z + 15164877247614245 z - 931304278617733472 z 42 44 - 41553863131077983200 z + 44944914275380274824 z 46 58 - 41553863131077983200 z - 931304278617733472 z 56 54 + 2622725407161076426 z - 6273048471593251840 z 52 60 70 + 12770494342992330732 z + 280086295797112848 z - 48781170825124 z 68 78 32 + 400211172199238 z - 1197295960 z + 2622725407161076426 z 38 40 - 22164281383539516008 z + 32835011719059797110 z 62 76 74 - 71094750615269376 z + 24477597996 z - 387221097832 z 72 + 4845762075667 z )) And in Maple-input format, it is: -(1+102378309871415392*z^28-26994625401375352*z^26-188*z^2+6005311888433461*z^ 24-1121008063941348*z^22+15956*z^4-816180*z^6+28339211*z^8-713206716*z^10+ 13565067728*z^12-200733771300*z^14-22432982099388*z^18+2360924784695*z^16-\ 7138221785497674320*z^50+10450523524193301678*z^48+174415711060724*z^20+ 4181103532444250424*z^36-2097397654729556256*z^34-1121008063941348*z^66+ 28339211*z^80+z^88+15956*z^84-188*z^86-816180*z^82+6005311888433461*z^64-\ 329048976489914792*z^30-13131618202223411456*z^42+14169479117313975968*z^44-\ 13131618202223411456*z^46-329048976489914792*z^58+899497859558739418*z^56-\ 2097397654729556256*z^54+4181103532444250424*z^52+102378309871415392*z^60-\ 22432982099388*z^70+174415711060724*z^68-713206716*z^78+899497859558739418*z^32 -7138221785497674320*z^38+10450523524193301678*z^40-26994625401375352*z^62+ 13565067728*z^76-200733771300*z^74+2360924784695*z^72)/(-1+z^2)/(1+ 280086295797112848*z^28-71094750615269376*z^26-216*z^2+15164877247614245*z^24-\ 2703676276178920*z^22+20558*z^4-1160500*z^6+43968391*z^8-1197295960*z^10+ 24477597996*z^12-387221097832*z^14-48781170825124*z^18+4845762075667*z^16-\ 22164281383539516008*z^50+32835011719059797110*z^48+400211172199238*z^20+ 12770494342992330732*z^36-6273048471593251840*z^34-2703676276178920*z^66+ 43968391*z^80+z^88+20558*z^84-216*z^86-1160500*z^82+15164877247614245*z^64-\ 931304278617733472*z^30-41553863131077983200*z^42+44944914275380274824*z^44-\ 41553863131077983200*z^46-931304278617733472*z^58+2622725407161076426*z^56-\ 6273048471593251840*z^54+12770494342992330732*z^52+280086295797112848*z^60-\ 48781170825124*z^70+400211172199238*z^68-1197295960*z^78+2622725407161076426*z^ 32-22164281383539516008*z^38+32835011719059797110*z^40-71094750615269376*z^62+ 24477597996*z^76-387221097832*z^74+4845762075667*z^72) The first , 40, terms are: [0, 29, 0, 1475, 0, 82507, 0, 4723371, 0, 272351435, 0, 15744230373, 0, 911052894561, 0, 52739915470769, 0, 3053566415948469, 0, 176809617640758843, 0, 10238058064257022971, 0, 592836677953133966587, 0, 34328521660492971813683, 0, 1987816469758762696946605, 0, 115106006582459318387001393, 0, 6665303450378740131803630609, 0, 385959712512150596391615749389, 0, 22349307577576616014763657288659, 0, 1294154700128605990828465755143771, 0, 74939074686811880219959962785018523] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 335545992 z + 974687988 z + 108 z - 1978578995 z 22 4 6 8 10 + 2816634016 z - 4600 z + 105051 z - 1460196 z + 13188444 z 12 14 18 16 - 80245681 z + 335545992 z + 1978578995 z - 974687988 z 20 36 34 30 42 - 2816634016 z - 105051 z + 1460196 z + 80245681 z + z 32 38 40 / 24 16 - 13188444 z + 4600 z - 108 z ) / (14634991047 z + 3731446893 z / 28 18 22 20 + 3731446893 z - 8780793434 z - 17344664660 z + 14634991047 z 2 40 36 42 34 6 - 130 z + 6787 z + 3115341 z - 130 z - 32931174 z - 188188 z 8 10 26 32 + 3115341 z - 32931174 z - 8780793434 z + 1 + 231762187 z 30 12 38 44 14 - 1115138920 z + 231762187 z - 188188 z + z - 1115138920 z 4 + 6787 z ) And in Maple-input format, it is: -(-1-335545992*z^28+974687988*z^26+108*z^2-1978578995*z^24+2816634016*z^22-4600 *z^4+105051*z^6-1460196*z^8+13188444*z^10-80245681*z^12+335545992*z^14+ 1978578995*z^18-974687988*z^16-2816634016*z^20-105051*z^36+1460196*z^34+ 80245681*z^30+z^42-13188444*z^32+4600*z^38-108*z^40)/(14634991047*z^24+ 3731446893*z^16+3731446893*z^28-8780793434*z^18-17344664660*z^22+14634991047*z^ 20-130*z^2+6787*z^40+3115341*z^36-130*z^42-32931174*z^34-188188*z^6+3115341*z^8 -32931174*z^10-8780793434*z^26+1+231762187*z^32-1115138920*z^30+231762187*z^12-\ 188188*z^38+z^44-1115138920*z^14+6787*z^4) The first , 40, terms are: [0, 22, 0, 673, 0, 21313, 0, 688030, 0, 22648321, 0, 761817793, 0, 26247390286, 0, 928059997153, 0, 33707772627073, 0, 1257202432200166, 0, 48073699635775105, 0, 1879786161055866241, 0, 74924921965736133958, 0, 3033973560853401269569, 0, 124420657668565263612961, 0, 5153000187089121562324462, 0, 215033609720821460587481665, 0, 9024472295154581642737621633, 0, 380343259682847638590130125822, 0, 16080022132673048112698980564993] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 72875222373641465 z - 19108272365828737 z - 177 z 24 22 4 6 + 4235637088322254 z - 790018031091292 z + 14190 z - 689160 z 8 10 12 14 + 22863201 z - 553586766 z + 10202124803 z - 147294374388 z 18 16 50 - 15966805594106 z + 1701202288919 z - 5257791303139340086 z 48 20 36 + 7731223594929029196 z + 123257581068269 z + 3061997885001573692 z 34 66 80 88 - 1525622762945721104 z - 790018031091292 z + 22863201 z + z 84 86 82 64 + 14190 z - 177 z - 689160 z + 4235637088322254 z 30 42 - 235833495075538174 z - 9741177359989013086 z 44 46 + 10520804050727775440 z - 9741177359989013086 z 58 56 - 235833495075538174 z + 649479992442706630 z 54 52 - 1525622762945721104 z + 3061997885001573692 z 60 70 68 + 72875222373641465 z - 15966805594106 z + 123257581068269 z 78 32 38 - 553586766 z + 649479992442706630 z - 5257791303139340086 z 40 62 76 + 7731223594929029196 z - 19108272365828737 z + 10202124803 z 74 72 / 2 - 147294374388 z + 1701202288919 z ) / ((-1 + z ) (1 / 28 26 2 + 195335577292920464 z - 49312207990708774 z - 200 z 24 22 4 6 + 10482845330665115 z - 1867751676699055 z + 17854 z - 954915 z 8 10 12 14 + 34578851 z - 907098846 z + 17995629008 z - 278105529357 z 18 16 50 - 34031149132132 z + 3420674900989 z - 16010157036395844974 z 48 20 36 + 23828541591234140925 z + 277256780017038 z + 9168791707376409917 z 34 66 80 88 - 4472037265613774134 z - 1867751676699055 z + 34578851 z + z 84 86 82 64 + 17854 z - 200 z - 954915 z + 10482845330665115 z 30 42 - 653955821650051486 z - 30243885919232551682 z 44 46 + 32744470690951713013 z - 30243885919232551682 z 58 56 - 653955821650051486 z + 1855582550715463617 z 54 52 - 4472037265613774134 z + 9168791707376409917 z 60 70 68 + 195335577292920464 z - 34031149132132 z + 277256780017038 z 78 32 38 - 907098846 z + 1855582550715463617 z - 16010157036395844974 z 40 62 76 + 23828541591234140925 z - 49312207990708774 z + 17995629008 z 74 72 - 278105529357 z + 3420674900989 z )) And in Maple-input format, it is: -(1+72875222373641465*z^28-19108272365828737*z^26-177*z^2+4235637088322254*z^24 -790018031091292*z^22+14190*z^4-689160*z^6+22863201*z^8-553586766*z^10+ 10202124803*z^12-147294374388*z^14-15966805594106*z^18+1701202288919*z^16-\ 5257791303139340086*z^50+7731223594929029196*z^48+123257581068269*z^20+ 3061997885001573692*z^36-1525622762945721104*z^34-790018031091292*z^66+22863201 *z^80+z^88+14190*z^84-177*z^86-689160*z^82+4235637088322254*z^64-\ 235833495075538174*z^30-9741177359989013086*z^42+10520804050727775440*z^44-\ 9741177359989013086*z^46-235833495075538174*z^58+649479992442706630*z^56-\ 1525622762945721104*z^54+3061997885001573692*z^52+72875222373641465*z^60-\ 15966805594106*z^70+123257581068269*z^68-553586766*z^78+649479992442706630*z^32 -5257791303139340086*z^38+7731223594929029196*z^40-19108272365828737*z^62+ 10202124803*z^76-147294374388*z^74+1701202288919*z^72)/(-1+z^2)/(1+ 195335577292920464*z^28-49312207990708774*z^26-200*z^2+10482845330665115*z^24-\ 1867751676699055*z^22+17854*z^4-954915*z^6+34578851*z^8-907098846*z^10+ 17995629008*z^12-278105529357*z^14-34031149132132*z^18+3420674900989*z^16-\ 16010157036395844974*z^50+23828541591234140925*z^48+277256780017038*z^20+ 9168791707376409917*z^36-4472037265613774134*z^34-1867751676699055*z^66+ 34578851*z^80+z^88+17854*z^84-200*z^86-954915*z^82+10482845330665115*z^64-\ 653955821650051486*z^30-30243885919232551682*z^42+32744470690951713013*z^44-\ 30243885919232551682*z^46-653955821650051486*z^58+1855582550715463617*z^56-\ 4472037265613774134*z^54+9168791707376409917*z^52+195335577292920464*z^60-\ 34031149132132*z^70+277256780017038*z^68-907098846*z^78+1855582550715463617*z^ 32-16010157036395844974*z^38+23828541591234140925*z^40-49312207990708774*z^62+ 17995629008*z^76-278105529357*z^74+3420674900989*z^72) The first , 40, terms are: [0, 24, 0, 960, 0, 43273, 0, 2041924, 0, 98314769, 0, 4778251927, 0, 233273413840, 0, 11413155713133, 0, 558995385142179, 0, 27392880037297060, 0, 1342699887832832307, 0, 65822644026904152192, 0, 3226999398111044183408, 0, 158210704304574028216523, 0, 7756744148149104692792899, 0, 380300012028625061645866224, 0, 18645533991706932623360878160, 0, 914163988280941829351513878923, 0, 44820199565266768348436112684164, 0, 2197473553174775404081438445224267] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 41343878178494592 z - 11257323706907504 z - 174 z 24 22 4 6 + 2598219221785589 z - 505723594673382 z + 13556 z - 634862 z 8 10 12 14 + 20207479 z - 467959248 z + 8233351392 z - 113389974160 z 18 16 50 - 11187336221782 z + 1249113672563 z - 2609300301990103004 z 48 20 36 + 3786579959148727974 z + 82490082985412 z + 1547401264800769224 z 34 66 80 88 - 788775059663560780 z - 505723594673382 z + 20207479 z + z 84 86 82 64 + 13556 z - 174 z - 634862 z + 2598219221785589 z 30 42 - 129241500953556528 z - 4733018016940317696 z 44 46 + 5098138365996833344 z - 4733018016940317696 z 58 56 54 - 129241500953556528 z + 345032976245768826 z - 788775059663560780 z 52 60 70 + 1547401264800769224 z + 41343878178494592 z - 11187336221782 z 68 78 32 + 82490082985412 z - 467959248 z + 345032976245768826 z 38 40 - 2609300301990103004 z + 3786579959148727974 z 62 76 74 - 11257323706907504 z + 8233351392 z - 113389974160 z 72 / 2 28 + 1249113672563 z ) / ((-1 + z ) (1 + 113045258690788208 z / 26 2 24 - 29802784083654648 z - 198 z + 6631933635236245 z 22 4 6 8 - 1239023665802806 z + 17346 z - 903058 z + 31617771 z 10 12 14 18 - 797754540 z + 15162464796 z - 223861794324 z - 24883680262290 z 16 50 48 + 2625678114551 z - 7906169465090828348 z + 11580054785645732302 z 20 36 34 + 193062379648858 z + 4628180643448777444 z - 2320086603154267044 z 66 80 88 84 86 - 1239023665802806 z + 31617771 z + z + 17346 z - 198 z 82 64 30 - 903058 z + 6631933635236245 z - 363498592791472872 z 42 44 - 14555046805394690912 z + 15706917195283509480 z 46 58 - 14555046805394690912 z - 363498592791472872 z 56 54 + 994301460211354298 z - 2320086603154267044 z 52 60 70 + 4628180643448777444 z + 113045258690788208 z - 24883680262290 z 68 78 32 + 193062379648858 z - 797754540 z + 994301460211354298 z 38 40 - 7906169465090828348 z + 11580054785645732302 z 62 76 74 - 29802784083654648 z + 15162464796 z - 223861794324 z 72 + 2625678114551 z )) And in Maple-input format, it is: -(1+41343878178494592*z^28-11257323706907504*z^26-174*z^2+2598219221785589*z^24 -505723594673382*z^22+13556*z^4-634862*z^6+20207479*z^8-467959248*z^10+ 8233351392*z^12-113389974160*z^14-11187336221782*z^18+1249113672563*z^16-\ 2609300301990103004*z^50+3786579959148727974*z^48+82490082985412*z^20+ 1547401264800769224*z^36-788775059663560780*z^34-505723594673382*z^66+20207479* z^80+z^88+13556*z^84-174*z^86-634862*z^82+2598219221785589*z^64-\ 129241500953556528*z^30-4733018016940317696*z^42+5098138365996833344*z^44-\ 4733018016940317696*z^46-129241500953556528*z^58+345032976245768826*z^56-\ 788775059663560780*z^54+1547401264800769224*z^52+41343878178494592*z^60-\ 11187336221782*z^70+82490082985412*z^68-467959248*z^78+345032976245768826*z^32-\ 2609300301990103004*z^38+3786579959148727974*z^40-11257323706907504*z^62+ 8233351392*z^76-113389974160*z^74+1249113672563*z^72)/(-1+z^2)/(1+ 113045258690788208*z^28-29802784083654648*z^26-198*z^2+6631933635236245*z^24-\ 1239023665802806*z^22+17346*z^4-903058*z^6+31617771*z^8-797754540*z^10+ 15162464796*z^12-223861794324*z^14-24883680262290*z^18+2625678114551*z^16-\ 7906169465090828348*z^50+11580054785645732302*z^48+193062379648858*z^20+ 4628180643448777444*z^36-2320086603154267044*z^34-1239023665802806*z^66+ 31617771*z^80+z^88+17346*z^84-198*z^86-903058*z^82+6631933635236245*z^64-\ 363498592791472872*z^30-14555046805394690912*z^42+15706917195283509480*z^44-\ 14555046805394690912*z^46-363498592791472872*z^58+994301460211354298*z^56-\ 2320086603154267044*z^54+4628180643448777444*z^52+113045258690788208*z^60-\ 24883680262290*z^70+193062379648858*z^68-797754540*z^78+994301460211354298*z^32 -7906169465090828348*z^38+11580054785645732302*z^40-29802784083654648*z^62+ 15162464796*z^76-223861794324*z^74+2625678114551*z^72) The first , 40, terms are: [0, 25, 0, 987, 0, 43355, 0, 2008467, 0, 95895899, 0, 4660235881, 0, 228869080153, 0, 11311606118697, 0, 561228164281705, 0, 27911377975633531, 0, 1390123604553849891, 0, 69296836445737258891, 0, 3456307594354244096827, 0, 172448233604216660748569, 0, 8605895375875282888063089, 0, 429526055003975347282106257, 0, 21439646773203384586959190681, 0, 1070205410170205171592861352443, 0, 53423197293040022145635641958699, 0, 2666863416415078869600225512021507] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 6656897793 z + 12383570158 z + 114 z - 16876841375 z 22 4 6 8 10 + 16876841375 z - 5327 z + 135717 z - 2119250 z + 21666931 z 12 14 18 16 - 151327853 z + 742869134 z + 6656897793 z - 2611987147 z 20 36 34 30 - 12383570158 z - 21666931 z + 151327853 z + 2611987147 z 42 44 46 32 38 40 + 5327 z - 114 z + z - 742869134 z + 2119250 z - 135717 z ) / 28 26 2 24 / (1 + 58961935832 z - 93090357690 z - 142 z + 108359714398 z / 22 4 6 8 10 - 93090357690 z + 7888 z - 234628 z + 4251308 z - 50329046 z 12 14 18 16 48 + 406456112 z - 2303682674 z - 27448619164 z + 9340715516 z + z 20 36 34 30 + 58961935832 z + 406456112 z - 2303682674 z - 27448619164 z 42 44 46 32 38 - 234628 z + 7888 z - 142 z + 9340715516 z - 50329046 z 40 + 4251308 z ) And in Maple-input format, it is: -(-1-6656897793*z^28+12383570158*z^26+114*z^2-16876841375*z^24+16876841375*z^22 -5327*z^4+135717*z^6-2119250*z^8+21666931*z^10-151327853*z^12+742869134*z^14+ 6656897793*z^18-2611987147*z^16-12383570158*z^20-21666931*z^36+151327853*z^34+ 2611987147*z^30+5327*z^42-114*z^44+z^46-742869134*z^32+2119250*z^38-135717*z^40 )/(1+58961935832*z^28-93090357690*z^26-142*z^2+108359714398*z^24-93090357690*z^ 22+7888*z^4-234628*z^6+4251308*z^8-50329046*z^10+406456112*z^12-2303682674*z^14 -27448619164*z^18+9340715516*z^16+z^48+58961935832*z^20+406456112*z^36-\ 2303682674*z^34-27448619164*z^30-234628*z^42+7888*z^44-142*z^46+9340715516*z^32 -50329046*z^38+4251308*z^40) The first , 40, terms are: [0, 28, 0, 1415, 0, 78977, 0, 4490740, 0, 256338615, 0, 14645825975, 0, 837001728628, 0, 47838143900641, 0, 2734226189024487, 0, 156278396246194716, 0, 8932336825837014017, 0, 510542454101494127617, 0, 29180912846238479406364, 0, 1667884526909752709807143, 0, 95330774863289146806632737, 0, 5448792645200286147801316980, 0, 311435019086741580742379285943, 0, 17800598765329454940684502472375, 0, 1017423531917059816134127329385460, 0, 58152574406715257756050174122605377] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 319216879268656 z - 158204268262566 z - 157 z 24 22 4 6 + 63843164526166 z - 20891544046432 z + 10680 z - 420989 z 8 10 12 14 + 10849813 z - 195579796 z + 2575393469 z - 25534668429 z 18 16 50 - 1164198753405 z + 194888979608 z - 1164198753405 z 48 20 36 + 5512509437353 z + 5512509437353 z + 709299758544750 z 34 66 64 30 - 783523646106136 z - 157 z + 10680 z - 526022222295542 z 42 44 46 - 158204268262566 z + 63843164526166 z - 20891544046432 z 58 56 54 52 - 195579796 z + 2575393469 z - 25534668429 z + 194888979608 z 60 68 32 38 + 10849813 z + z + 709299758544750 z - 526022222295542 z 40 62 / 28 + 319216879268656 z - 420989 z ) / (-1 - 1453859202284818 z / 26 2 24 22 + 657365456779406 z + 187 z - 242246303267914 z + 72420603319861 z 4 6 8 10 12 - 14733 z + 660531 z - 19127321 z + 384126207 z - 5601189355 z 14 18 16 + 61230245485 z + 3365626969505 z - 513668276863 z 50 48 20 + 17456505065943 z - 72420603319861 z - 17456505065943 z 36 34 66 64 - 4745376233487894 z + 4745376233487894 z + 14733 z - 660531 z 30 42 44 + 2629542648505102 z + 1453859202284818 z - 657365456779406 z 46 58 56 + 242246303267914 z + 5601189355 z - 61230245485 z 54 52 60 70 68 + 513668276863 z - 3365626969505 z - 384126207 z + z - 187 z 32 38 40 - 3898632390907290 z + 3898632390907290 z - 2629542648505102 z 62 + 19127321 z ) And in Maple-input format, it is: -(1+319216879268656*z^28-158204268262566*z^26-157*z^2+63843164526166*z^24-\ 20891544046432*z^22+10680*z^4-420989*z^6+10849813*z^8-195579796*z^10+2575393469 *z^12-25534668429*z^14-1164198753405*z^18+194888979608*z^16-1164198753405*z^50+ 5512509437353*z^48+5512509437353*z^20+709299758544750*z^36-783523646106136*z^34 -157*z^66+10680*z^64-526022222295542*z^30-158204268262566*z^42+63843164526166*z ^44-20891544046432*z^46-195579796*z^58+2575393469*z^56-25534668429*z^54+ 194888979608*z^52+10849813*z^60+z^68+709299758544750*z^32-526022222295542*z^38+ 319216879268656*z^40-420989*z^62)/(-1-1453859202284818*z^28+657365456779406*z^ 26+187*z^2-242246303267914*z^24+72420603319861*z^22-14733*z^4+660531*z^6-\ 19127321*z^8+384126207*z^10-5601189355*z^12+61230245485*z^14+3365626969505*z^18 -513668276863*z^16+17456505065943*z^50-72420603319861*z^48-17456505065943*z^20-\ 4745376233487894*z^36+4745376233487894*z^34+14733*z^66-660531*z^64+ 2629542648505102*z^30+1453859202284818*z^42-657365456779406*z^44+ 242246303267914*z^46+5601189355*z^58-61230245485*z^56+513668276863*z^54-\ 3365626969505*z^52-384126207*z^60+z^70-187*z^68-3898632390907290*z^32+ 3898632390907290*z^38-2629542648505102*z^40+19127321*z^62) The first , 40, terms are: [0, 30, 0, 1557, 0, 88711, 0, 5188098, 0, 306368711, 0, 18167818191, 0, 1079491970066, 0, 64203444405159, 0, 3820407516191781, 0, 227388850400637326, 0, 13535801528021395225, 0, 805799878109621374793, 0, 47971690458355361891310, 0, 2855948445339460442114661, 0, 170027658002397460566961287, 0, 10122569657227641214645710002, 0, 602647002605289466693086367583, 0, 35878622102101359640472772554103, 0, 2136037057482391636653213263795426, 0, 127169203140690675554347673153163751 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 7}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 14 12 6 10 4 8 2 / f(z) = - (z - 25 z + 525 z + 193 z - 193 z - 525 z + 25 z - 1) / ( / 14 12 6 10 4 8 2 16 -52 z + 578 z - 2172 z - 2172 z + 578 z + 3322 z - 52 z + z + 1 ) And in Maple-input format, it is: -(z^14-25*z^12+525*z^6+193*z^10-193*z^4-525*z^8+25*z^2-1)/(-52*z^14+578*z^12-\ 2172*z^6-2172*z^10+578*z^4+3322*z^8-52*z^2+z^16+1) The first , 40, terms are: [0, 27, 0, 1019, 0, 39029, 0, 1496373, 0, 57378187, 0, 2200206091, 0, 84368790177, 0, 3235196623073, 0, 124056528586283, 0, 4757059528527723, 0, 182413741665726517, 0, 6994819596311626357, 0, 268222672029671156763, 0, 10285240499240387170043, 0, 394396832037493808143553, 0, 15123502569852427527745857, 0, 579924358922785908703753531, 0, 22237723074975803411388698587, 0, 852725566620270562558597472373, 0, 32698531657952057112275873750069] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1045717070842146 z - 440117218523765 z - 155 z 24 22 4 6 + 152080823312700 z - 42991286169480 z + 10546 z - 422359 z 8 10 12 14 + 11257224 z - 213725262 z + 3016853576 z - 32593873758 z 18 16 50 - 1842235971386 z + 275138428345 z - 42991286169480 z 48 20 36 + 152080823312700 z + 9894561567874 z + 4831816666135398 z 34 66 64 30 - 4392250330914304 z - 422359 z + 11257224 z - 2045209995845623 z 42 44 46 - 2045209995845623 z + 1045717070842146 z - 440117218523765 z 58 56 54 - 32593873758 z + 275138428345 z - 1842235971386 z 52 60 70 68 + 9894561567874 z + 3016853576 z - 155 z + 10546 z 32 38 40 + 3298564921273693 z - 4392250330914304 z + 3298564921273693 z 62 72 / 2 28 - 213725262 z + z ) / ((-1 + z ) (1 + 3096373232485001 z / 26 2 24 - 1266453301373060 z - 180 z + 422717185156869 z 22 4 6 8 10 - 114733724530774 z + 13964 z - 628672 z + 18604687 z - 387889934 z 12 14 18 16 + 5953819794 z - 69333638246 z - 4448978014470 z + 625867684443 z 50 48 20 - 114733724530774 z + 422717185156869 z + 25199104497433 z 36 34 66 + 15065805819856771 z - 13650509734058124 z - 628672 z 64 30 42 + 18604687 z - 6193385931194656 z - 6193385931194656 z 44 46 58 + 3096373232485001 z - 1266453301373060 z - 69333638246 z 56 54 52 + 625867684443 z - 4448978014470 z + 25199104497433 z 60 70 68 32 + 5953819794 z - 180 z + 13964 z + 10151777140979965 z 38 40 62 72 - 13650509734058124 z + 10151777140979965 z - 387889934 z + z )) And in Maple-input format, it is: -(1+1045717070842146*z^28-440117218523765*z^26-155*z^2+152080823312700*z^24-\ 42991286169480*z^22+10546*z^4-422359*z^6+11257224*z^8-213725262*z^10+3016853576 *z^12-32593873758*z^14-1842235971386*z^18+275138428345*z^16-42991286169480*z^50 +152080823312700*z^48+9894561567874*z^20+4831816666135398*z^36-4392250330914304 *z^34-422359*z^66+11257224*z^64-2045209995845623*z^30-2045209995845623*z^42+ 1045717070842146*z^44-440117218523765*z^46-32593873758*z^58+275138428345*z^56-\ 1842235971386*z^54+9894561567874*z^52+3016853576*z^60-155*z^70+10546*z^68+ 3298564921273693*z^32-4392250330914304*z^38+3298564921273693*z^40-213725262*z^ 62+z^72)/(-1+z^2)/(1+3096373232485001*z^28-1266453301373060*z^26-180*z^2+ 422717185156869*z^24-114733724530774*z^22+13964*z^4-628672*z^6+18604687*z^8-\ 387889934*z^10+5953819794*z^12-69333638246*z^14-4448978014470*z^18+625867684443 *z^16-114733724530774*z^50+422717185156869*z^48+25199104497433*z^20+ 15065805819856771*z^36-13650509734058124*z^34-628672*z^66+18604687*z^64-\ 6193385931194656*z^30-6193385931194656*z^42+3096373232485001*z^44-\ 1266453301373060*z^46-69333638246*z^58+625867684443*z^56-4448978014470*z^54+ 25199104497433*z^52+5953819794*z^60-180*z^70+13964*z^68+10151777140979965*z^32-\ 13650509734058124*z^38+10151777140979965*z^40-387889934*z^62+z^72) The first , 40, terms are: [0, 26, 0, 1108, 0, 53081, 0, 2668510, 0, 136965359, 0, 7092528277, 0, 368669415144, 0, 19194841425783, 0, 1000102717735953, 0, 52124859317698078, 0, 2717124015559124287, 0, 141645951993997823332, 0, 7384367868181456451674, 0, 384972361351126355924211, 0, 20070089171675441657993003, 0, 1046335203967069246111429250, 0, 54549815751637460312103078580, 0, 2843912389490949916432655913839, 0, 148265254222169366531134491400438, 0, 7729700922461234907173765696514913] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 106710887715852627 z - 28088168210490502 z - 190 z 24 22 4 6 + 6237050059137213 z - 1162074902527932 z + 16239 z - 834376 z 8 10 12 14 + 29052684 z - 732524372 z + 13952259585 z - 206734524162 z 18 16 50 - 23172024065420 z + 2434901754287 z - 7482256129114429370 z 48 20 36 + 10959245972633328147 z + 180472705942792 z + 4379680631662647229 z 34 66 80 88 - 2195045485943483408 z - 1162074902527932 z + 29052684 z + z 84 86 82 64 + 16239 z - 190 z - 834376 z + 6237050059137213 z 30 42 - 343512517344411348 z - 13774546247538025016 z 44 46 + 14864526817389719536 z - 13774546247538025016 z 58 56 - 343512517344411348 z + 940313130191720436 z 54 52 - 2195045485943483408 z + 4379680631662647229 z 60 70 68 + 106710887715852627 z - 23172024065420 z + 180472705942792 z 78 32 38 - 732524372 z + 940313130191720436 z - 7482256129114429370 z 40 62 76 + 10959245972633328147 z - 28088168210490502 z + 13952259585 z 74 72 / 28 - 206734524162 z + 2434901754287 z ) / (-1 - 368662920740367006 z / 26 2 24 + 90571154648660320 z + 220 z - 18763525406711785 z 22 4 6 8 + 3259246451476756 z - 21282 z + 1219347 z - 46854182 z 10 12 14 18 + 1293859950 z - 26837032657 z + 431145440514 z + 56274020924507 z 16 50 48 - 5486989672056 z + 57638683357646330700 z - 77945766971856558523 z 20 36 34 - 471367386928148 z - 19967755028177767937 z + 9330409038781427658 z 66 80 90 88 84 + 18763525406711785 z - 1293859950 z + z - 220 z - 1219347 z 86 82 64 + 21282 z + 46854182 z - 90571154648660320 z 30 42 + 1271363785732951579 z + 77945766971856558523 z 44 46 - 90626839949237856984 z + 90626839949237856984 z 58 56 + 3728860135457940386 z - 9330409038781427658 z 54 52 + 19967755028177767937 z - 36619839584079755582 z 60 70 68 - 1271363785732951579 z + 471367386928148 z - 3259246451476756 z 78 32 38 + 26837032657 z - 3728860135457940386 z + 36619839584079755582 z 40 62 76 - 57638683357646330700 z + 368662920740367006 z - 431145440514 z 74 72 + 5486989672056 z - 56274020924507 z ) And in Maple-input format, it is: -(1+106710887715852627*z^28-28088168210490502*z^26-190*z^2+6237050059137213*z^ 24-1162074902527932*z^22+16239*z^4-834376*z^6+29052684*z^8-732524372*z^10+ 13952259585*z^12-206734524162*z^14-23172024065420*z^18+2434901754287*z^16-\ 7482256129114429370*z^50+10959245972633328147*z^48+180472705942792*z^20+ 4379680631662647229*z^36-2195045485943483408*z^34-1162074902527932*z^66+ 29052684*z^80+z^88+16239*z^84-190*z^86-834376*z^82+6237050059137213*z^64-\ 343512517344411348*z^30-13774546247538025016*z^42+14864526817389719536*z^44-\ 13774546247538025016*z^46-343512517344411348*z^58+940313130191720436*z^56-\ 2195045485943483408*z^54+4379680631662647229*z^52+106710887715852627*z^60-\ 23172024065420*z^70+180472705942792*z^68-732524372*z^78+940313130191720436*z^32 -7482256129114429370*z^38+10959245972633328147*z^40-28088168210490502*z^62+ 13952259585*z^76-206734524162*z^74+2434901754287*z^72)/(-1-368662920740367006*z ^28+90571154648660320*z^26+220*z^2-18763525406711785*z^24+3259246451476756*z^22 -21282*z^4+1219347*z^6-46854182*z^8+1293859950*z^10-26837032657*z^12+ 431145440514*z^14+56274020924507*z^18-5486989672056*z^16+57638683357646330700*z ^50-77945766971856558523*z^48-471367386928148*z^20-19967755028177767937*z^36+ 9330409038781427658*z^34+18763525406711785*z^66-1293859950*z^80+z^90-220*z^88-\ 1219347*z^84+21282*z^86+46854182*z^82-90571154648660320*z^64+ 1271363785732951579*z^30+77945766971856558523*z^42-90626839949237856984*z^44+ 90626839949237856984*z^46+3728860135457940386*z^58-9330409038781427658*z^56+ 19967755028177767937*z^54-36619839584079755582*z^52-1271363785732951579*z^60+ 471367386928148*z^70-3259246451476756*z^68+26837032657*z^78-3728860135457940386 *z^32+36619839584079755582*z^38-57638683357646330700*z^40+368662920740367006*z^ 62-431145440514*z^76+5486989672056*z^74-56274020924507*z^72) The first , 40, terms are: [0, 30, 0, 1557, 0, 89051, 0, 5234058, 0, 310542775, 0, 18491321895, 0, 1102680516986, 0, 65795833963051, 0, 3927006074058981, 0, 234409106258037710, 0, 13992946587021605617, 0, 835321163877454930961, 0, 49865715465588619568750, 0, 2976819636520515406407877, 0, 177706714420641099176122763, 0, 10608537930394167295240939162, 0, 633296956647584365522855489479, 0, 37805879507133616304988687368215, 0, 2256894850994282304407159671374954, 0, 134729693408295154768022986290595707 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 43892572649698784 z - 11947226553975376 z - 176 z 24 22 4 6 + 2756371896178469 z - 536240205431232 z + 13868 z - 655764 z 8 10 12 14 + 21033275 z - 489933040 z + 8657125240 z - 119594380336 z 18 16 50 - 11842384105028 z + 1320303414391 z - 2774139047593161000 z 48 20 36 + 4026522726343055422 z + 87407547048860 z + 1644768514875565112 z 34 66 80 88 - 838177329545830800 z - 536240205431232 z + 21033275 z + z 84 86 82 64 + 13868 z - 176 z - 655764 z + 2756371896178469 z 30 42 - 137252647476549392 z - 5033510384939231360 z 44 46 + 5422024225449307472 z - 5033510384939231360 z 58 56 54 - 137252647476549392 z + 366533803765820650 z - 838177329545830800 z 52 60 70 + 1644768514875565112 z + 43892572649698784 z - 11842384105028 z 68 78 32 + 87407547048860 z - 489933040 z + 366533803765820650 z 38 40 - 2774139047593161000 z + 4026522726343055422 z 62 76 74 - 11947226553975376 z + 8657125240 z - 119594380336 z 72 / 2 28 + 1320303414391 z ) / ((-1 + z ) (1 + 120403056718915920 z / 26 2 24 - 31751624080280664 z - 200 z + 7066453971880277 z 22 4 6 8 - 1319998775273536 z + 17710 z - 931388 z + 32897695 z 10 12 14 18 - 836048960 z + 15979838564 z - 236917900632 z - 26464116048772 z 16 50 48 + 2787019793723 z - 8406317451907902880 z + 12310236490293414838 z 20 36 34 + 205562294761558 z + 4922260414218404604 z - 2468310480579153560 z 66 80 88 84 86 - 1319998775273536 z + 32897695 z + z + 17710 z - 200 z 82 64 30 - 931388 z + 7066453971880277 z - 387015525711822696 z 42 44 - 15471000163806444456 z + 16694700863095038872 z 46 58 - 15471000163806444456 z - 387015525711822696 z 56 54 + 1058216734657797914 z - 2468310480579153560 z 52 60 70 + 4922260414218404604 z + 120403056718915920 z - 26464116048772 z 68 78 32 + 205562294761558 z - 836048960 z + 1058216734657797914 z 38 40 - 8406317451907902880 z + 12310236490293414838 z 62 76 74 - 31751624080280664 z + 15979838564 z - 236917900632 z 72 + 2787019793723 z )) And in Maple-input format, it is: -(1+43892572649698784*z^28-11947226553975376*z^26-176*z^2+2756371896178469*z^24 -536240205431232*z^22+13868*z^4-655764*z^6+21033275*z^8-489933040*z^10+ 8657125240*z^12-119594380336*z^14-11842384105028*z^18+1320303414391*z^16-\ 2774139047593161000*z^50+4026522726343055422*z^48+87407547048860*z^20+ 1644768514875565112*z^36-838177329545830800*z^34-536240205431232*z^66+21033275* z^80+z^88+13868*z^84-176*z^86-655764*z^82+2756371896178469*z^64-\ 137252647476549392*z^30-5033510384939231360*z^42+5422024225449307472*z^44-\ 5033510384939231360*z^46-137252647476549392*z^58+366533803765820650*z^56-\ 838177329545830800*z^54+1644768514875565112*z^52+43892572649698784*z^60-\ 11842384105028*z^70+87407547048860*z^68-489933040*z^78+366533803765820650*z^32-\ 2774139047593161000*z^38+4026522726343055422*z^40-11947226553975376*z^62+ 8657125240*z^76-119594380336*z^74+1320303414391*z^72)/(-1+z^2)/(1+ 120403056718915920*z^28-31751624080280664*z^26-200*z^2+7066453971880277*z^24-\ 1319998775273536*z^22+17710*z^4-931388*z^6+32897695*z^8-836048960*z^10+ 15979838564*z^12-236917900632*z^14-26464116048772*z^18+2787019793723*z^16-\ 8406317451907902880*z^50+12310236490293414838*z^48+205562294761558*z^20+ 4922260414218404604*z^36-2468310480579153560*z^34-1319998775273536*z^66+ 32897695*z^80+z^88+17710*z^84-200*z^86-931388*z^82+7066453971880277*z^64-\ 387015525711822696*z^30-15471000163806444456*z^42+16694700863095038872*z^44-\ 15471000163806444456*z^46-387015525711822696*z^58+1058216734657797914*z^56-\ 2468310480579153560*z^54+4922260414218404604*z^52+120403056718915920*z^60-\ 26464116048772*z^70+205562294761558*z^68-836048960*z^78+1058216734657797914*z^ 32-8406317451907902880*z^38+12310236490293414838*z^40-31751624080280664*z^62+ 15979838564*z^76-236917900632*z^74+2787019793723*z^72) The first , 40, terms are: [0, 25, 0, 983, 0, 43167, 0, 2002679, 0, 95667383, 0, 4641791961, 0, 227116694937, 0, 11163501438233, 0, 550117997813049, 0, 27147835341985623, 0, 1340827772806026455, 0, 66255166079683646463, 0, 3274838301262136085655, 0, 161895052422421887541465, 0, 8004267280674084146210433, 0, 395764123439016019029177281, 0, 19568953820430661926546541657, 0, 967628816434628956039717163671, 0, 47847154413983472843364291828863, 0, 2365958975906296711228207189150679] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 811882697754006 z - 347437735777626 z - 155 z 24 22 4 6 + 122414421587831 z - 35366793582835 z + 10489 z - 415562 z 8 10 12 14 + 10906021 z - 203115879 z + 2804650666 z - 29585341283 z 18 16 50 - 1590425555894 z + 243589424673 z - 35366793582835 z 48 20 36 + 122414421587831 z + 8333682746633 z + 3635786492467132 z 34 66 64 30 - 3311717237565006 z - 415562 z + 10906021 z - 1566715380241152 z 42 44 46 - 1566715380241152 z + 811882697754006 z - 347437735777626 z 58 56 54 - 29585341283 z + 243589424673 z - 1590425555894 z 52 60 70 68 + 8333682746633 z + 2804650666 z - 155 z + 10489 z 32 38 40 + 2502086908725922 z - 3311717237565006 z + 2502086908725922 z 62 72 / 28 - 203115879 z + z ) / (-1 - 3435867699392496 z / 26 2 24 + 1354448198310999 z + 183 z - 439959310106833 z 22 4 6 8 10 + 117202876376314 z - 14310 z + 645324 z - 19046224 z + 395044042 z 12 14 18 16 - 6027572244 z + 69828270352 z + 4469462467368 z - 628432656646 z 50 48 20 + 439959310106833 z - 1354448198310999 z - 25450036318572 z 36 34 66 - 21767656488652092 z + 18114180590590304 z + 19046224 z 64 30 42 - 395044042 z + 7207635318727960 z + 12536712621126136 z 44 46 58 - 7207635318727960 z + 3435867699392496 z + 628432656646 z 56 54 52 - 4469462467368 z + 25450036318572 z - 117202876376314 z 60 70 68 32 - 69828270352 z + 14310 z - 645324 z - 12536712621126136 z 38 40 62 74 + 21767656488652092 z - 18114180590590304 z + 6027572244 z + z 72 - 183 z ) And in Maple-input format, it is: -(1+811882697754006*z^28-347437735777626*z^26-155*z^2+122414421587831*z^24-\ 35366793582835*z^22+10489*z^4-415562*z^6+10906021*z^8-203115879*z^10+2804650666 *z^12-29585341283*z^14-1590425555894*z^18+243589424673*z^16-35366793582835*z^50 +122414421587831*z^48+8333682746633*z^20+3635786492467132*z^36-3311717237565006 *z^34-415562*z^66+10906021*z^64-1566715380241152*z^30-1566715380241152*z^42+ 811882697754006*z^44-347437735777626*z^46-29585341283*z^58+243589424673*z^56-\ 1590425555894*z^54+8333682746633*z^52+2804650666*z^60-155*z^70+10489*z^68+ 2502086908725922*z^32-3311717237565006*z^38+2502086908725922*z^40-203115879*z^ 62+z^72)/(-1-3435867699392496*z^28+1354448198310999*z^26+183*z^2-\ 439959310106833*z^24+117202876376314*z^22-14310*z^4+645324*z^6-19046224*z^8+ 395044042*z^10-6027572244*z^12+69828270352*z^14+4469462467368*z^18-628432656646 *z^16+439959310106833*z^50-1354448198310999*z^48-25450036318572*z^20-\ 21767656488652092*z^36+18114180590590304*z^34+19046224*z^66-395044042*z^64+ 7207635318727960*z^30+12536712621126136*z^42-7207635318727960*z^44+ 3435867699392496*z^46+628432656646*z^58-4469462467368*z^56+25450036318572*z^54-\ 117202876376314*z^52-69828270352*z^60+14310*z^70-645324*z^68-12536712621126136* z^32+21767656488652092*z^38-18114180590590304*z^40+6027572244*z^62+z^74-183*z^ 72) The first , 40, terms are: [0, 28, 0, 1303, 0, 67531, 0, 3641112, 0, 199445949, 0, 10994752717, 0, 607667917328, 0, 33620281226395, 0, 1860889928704519, 0, 103018422606434260, 0, 5703475048101687121, 0, 315774157391566112625, 0, 17483109277849259613412, 0, 967972054190316434813287, 0, 53592967897609021825112123, 0, 2967243193413453767165902304, 0, 164285267011779520337586949421, 0, 9095868058700625602131474948189, 0, 503604625345142283537203947282504, 0, 27882729047728762300339500936235691] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 8833034029040545088 z - 1454575755711910716 z - 224 z 24 22 4 6 + 207194808033296740 z - 25378067665284012 z + 23120 z - 1468556 z 102 8 10 12 - 2110649344 z + 64685378 z - 2110649344 z + 53215887280 z 14 18 16 - 1067667192744 z - 234919857252080 z + 17417888874091 z 50 48 - 687671736258615098459584 z + 455461347858386134946875 z 20 36 + 2653898526222384 z + 3077909080480184296512 z 34 66 - 867599600441561575592 z - 267915233120546646749072 z 80 100 90 + 214892383429025893977 z + 53215887280 z - 25378067665284012 z 88 84 94 + 207194808033296740 z + 8833034029040545088 z - 234919857252080 z 86 96 98 - 1454575755711910716 z + 17417888874091 z - 1067667192744 z 92 82 + 2653898526222384 z - 46629829483542139792 z 64 112 110 106 + 455461347858386134946875 z + z - 224 z - 1468556 z 108 30 42 + 23120 z - 46629829483542139792 z - 64736417859000562537788 z 44 46 + 139861987007011240513536 z - 267915233120546646749072 z 58 56 - 1100388736681030383886232 z + 1166924448481386029462392 z 54 52 - 1100388736681030383886232 z + 922623448387678203241296 z 60 70 + 922623448387678203241296 z - 64736417859000562537788 z 68 78 + 139861987007011240513536 z - 867599600441561575592 z 32 38 + 214892383429025893977 z - 9618820926394342299320 z 40 62 + 26535802607440948941726 z - 687671736258615098459584 z 76 74 + 3077909080480184296512 z - 9618820926394342299320 z 72 104 / + 26535802607440948941726 z + 64685378 z ) / (-1 / 28 26 2 - 25501937906033171324 z + 3974588730462557024 z + 257 z 24 22 4 6 - 535396941112823760 z + 61951324187379292 z - 29684 z + 2076088 z 102 8 10 12 + 95123021652 z - 99586034 z + 3511284462 z - 95123021652 z 14 18 16 + 2041880915260 z + 509666733567683 z - 35524456302763 z 50 48 + 3600778743432158455743947 z - 2256026199890921930084507 z 20 36 - 6112408760869224 z - 11024075090739114921348 z 34 66 + 2945569799690695550585 z + 2256026199890921930084507 z 80 100 90 - 2945569799690695550585 z - 2041880915260 z + 535396941112823760 z 88 84 - 3974588730462557024 z - 142155900905188426328 z 94 86 96 + 6112408760869224 z + 25501937906033171324 z - 509666733567683 z 98 92 82 + 35524456302763 z - 61951324187379292 z + 691452815431615322033 z 64 112 114 110 - 3600778743432158455743947 z - 257 z + z + 29684 z 106 108 30 + 99586034 z - 2076088 z + 142155900905188426328 z 42 44 + 272303131226145534598374 z - 620978981465516709238032 z 46 58 + 1256098433465710369210188 z + 7249714967039589811827344 z 56 54 - 7249714967039589811827344 z + 6452386401653438016718312 z 52 60 - 5110406044270628611520576 z - 6452386401653438016718312 z 70 68 + 620978981465516709238032 z - 1256098433465710369210188 z 78 32 + 11024075090739114921348 z - 691452815431615322033 z 38 40 + 36343631649603544334884 z - 105777868611757562390394 z 62 76 + 5110406044270628611520576 z - 36343631649603544334884 z 74 72 + 105777868611757562390394 z - 272303131226145534598374 z 104 - 3511284462 z ) And in Maple-input format, it is: -(1+8833034029040545088*z^28-1454575755711910716*z^26-224*z^2+ 207194808033296740*z^24-25378067665284012*z^22+23120*z^4-1468556*z^6-2110649344 *z^102+64685378*z^8-2110649344*z^10+53215887280*z^12-1067667192744*z^14-\ 234919857252080*z^18+17417888874091*z^16-687671736258615098459584*z^50+ 455461347858386134946875*z^48+2653898526222384*z^20+3077909080480184296512*z^36 -867599600441561575592*z^34-267915233120546646749072*z^66+214892383429025893977 *z^80+53215887280*z^100-25378067665284012*z^90+207194808033296740*z^88+ 8833034029040545088*z^84-234919857252080*z^94-1454575755711910716*z^86+ 17417888874091*z^96-1067667192744*z^98+2653898526222384*z^92-\ 46629829483542139792*z^82+455461347858386134946875*z^64+z^112-224*z^110-1468556 *z^106+23120*z^108-46629829483542139792*z^30-64736417859000562537788*z^42+ 139861987007011240513536*z^44-267915233120546646749072*z^46-\ 1100388736681030383886232*z^58+1166924448481386029462392*z^56-\ 1100388736681030383886232*z^54+922623448387678203241296*z^52+ 922623448387678203241296*z^60-64736417859000562537788*z^70+ 139861987007011240513536*z^68-867599600441561575592*z^78+214892383429025893977* z^32-9618820926394342299320*z^38+26535802607440948941726*z^40-\ 687671736258615098459584*z^62+3077909080480184296512*z^76-\ 9618820926394342299320*z^74+26535802607440948941726*z^72+64685378*z^104)/(-1-\ 25501937906033171324*z^28+3974588730462557024*z^26+257*z^2-535396941112823760*z ^24+61951324187379292*z^22-29684*z^4+2076088*z^6+95123021652*z^102-99586034*z^8 +3511284462*z^10-95123021652*z^12+2041880915260*z^14+509666733567683*z^18-\ 35524456302763*z^16+3600778743432158455743947*z^50-2256026199890921930084507*z^ 48-6112408760869224*z^20-11024075090739114921348*z^36+2945569799690695550585*z^ 34+2256026199890921930084507*z^66-2945569799690695550585*z^80-2041880915260*z^ 100+535396941112823760*z^90-3974588730462557024*z^88-142155900905188426328*z^84 +6112408760869224*z^94+25501937906033171324*z^86-509666733567683*z^96+ 35524456302763*z^98-61951324187379292*z^92+691452815431615322033*z^82-\ 3600778743432158455743947*z^64-257*z^112+z^114+29684*z^110+99586034*z^106-\ 2076088*z^108+142155900905188426328*z^30+272303131226145534598374*z^42-\ 620978981465516709238032*z^44+1256098433465710369210188*z^46+ 7249714967039589811827344*z^58-7249714967039589811827344*z^56+ 6452386401653438016718312*z^54-5110406044270628611520576*z^52-\ 6452386401653438016718312*z^60+620978981465516709238032*z^70-\ 1256098433465710369210188*z^68+11024075090739114921348*z^78-\ 691452815431615322033*z^32+36343631649603544334884*z^38-\ 105777868611757562390394*z^40+5110406044270628611520576*z^62-\ 36343631649603544334884*z^76+105777868611757562390394*z^74-\ 272303131226145534598374*z^72-3511284462*z^104) The first , 40, terms are: [0, 33, 0, 1917, 0, 120629, 0, 7707673, 0, 494277417, 0, 31729975885, 0, 2037603005709, 0, 130865600597309, 0, 8405309249481821, 0, 539872311331551369, 0, 34676260579748723593, 0, 2227281412903923874021, 0, 143060132679596185443517, 0, 9188877287393917447269425, 0, 590209788112931054532615217, 0, 37909706154402048003749729969, 0, 2434974694292049133441740345777, 0, 156400629306799755464950706820765, 0, 10045754129821394438889806619796453, 0, 645247892644206275031275450219211593] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 7429503592125924572 z + 1248066800327477349 z + 223 z 24 22 4 6 - 181084266846039569 z + 22561857973440588 z - 22872 z + 1441554 z 102 8 10 12 + 62928080 z - 62928080 z + 2032810922 z - 50693342324 z 14 18 16 + 1005030539528 z + 215343403310675 z - 16187433456375 z 50 48 + 400194030134579406099157 z - 278082305151841712895909 z 20 36 - 2397106770988388 z - 2346960166853152003092 z 34 66 + 680103355493158597459 z + 93059222821596744240934 z 80 100 90 - 38383076581293203994 z - 2032810922 z + 2397106770988388 z 88 84 94 - 22561857973440588 z - 1248066800327477349 z + 16187433456375 z 86 96 98 + 181084266846039569 z - 1005030539528 z + 50693342324 z 92 82 - 215343403310675 z + 7429503592125924572 z 64 110 106 108 - 171036996571265765469318 z + z + 22872 z - 223 z 30 42 + 38383076581293203994 z + 44754407541883567512560 z 44 46 - 93059222821596744240934 z + 171036996571265765469318 z 58 56 + 509998456353602344649612 z - 575689798540740113146278 z 54 52 + 575689798540740113146278 z - 509998456353602344649612 z 60 70 - 400194030134579406099157 z + 19005390613165498997102 z 68 78 - 44754407541883567512560 z + 172792587902751993477 z 32 38 - 172792587902751993477 z + 7117510148546044752788 z 40 62 - 19005390613165498997102 z + 278082305151841712895909 z 76 74 - 680103355493158597459 z + 2346960166853152003092 z 72 104 / - 7117510148546044752788 z - 1441554 z ) / (1 / 28 26 2 + 21656640147715613370 z - 3442149642917069026 z - 255 z 24 22 4 6 + 472095758500303665 z - 55534943870176126 z + 29252 z - 2031807 z 102 8 10 12 - 3382911335 z + 96739207 z - 3382911335 z + 90807335827 z 14 18 16 - 1929415206254 z - 470246829489008 z + 33189481175092 z 50 48 - 2130102116435891778492125 z + 1398124692989021411602575 z 20 36 + 5562593535322633 z + 8494553224053040480863 z 34 66 - 2332726013579153021468 z - 812950477502060838126293 z 80 100 90 + 561587508796437619754 z + 90807335827 z - 55534943870176126 z 88 84 94 + 472095758500303665 z + 21656640147715613370 z - 470246829489008 z 86 96 98 - 3442149642917069026 z + 33189481175092 z - 1929415206254 z 92 82 + 5562593535322633 z - 118170075959388394846 z 64 112 110 106 + 1398124692989021411602575 z + z - 255 z - 2031807 z 108 30 42 + 29252 z - 118170075959388394846 z - 190504454892383309364845 z 44 46 + 418455064551937974368420 z - 812950477502060838126293 z 58 56 - 3444029521652062059956188 z + 3657020274901682235646787 z 54 52 - 3444029521652062059956188 z + 2876438849080284487777282 z 60 70 + 2876438849080284487777282 z - 190504454892383309364845 z 68 78 + 418455064551937974368420 z - 2332726013579153021468 z 32 38 + 561587508796437619754 z - 27185090114351591797014 z 40 62 + 76619035706339968949056 z - 2130102116435891778492125 z 76 74 + 8494553224053040480863 z - 27185090114351591797014 z 72 104 + 76619035706339968949056 z + 96739207 z ) And in Maple-input format, it is: -(-1-7429503592125924572*z^28+1248066800327477349*z^26+223*z^2-\ 181084266846039569*z^24+22561857973440588*z^22-22872*z^4+1441554*z^6+62928080*z ^102-62928080*z^8+2032810922*z^10-50693342324*z^12+1005030539528*z^14+ 215343403310675*z^18-16187433456375*z^16+400194030134579406099157*z^50-\ 278082305151841712895909*z^48-2397106770988388*z^20-2346960166853152003092*z^36 +680103355493158597459*z^34+93059222821596744240934*z^66-38383076581293203994*z ^80-2032810922*z^100+2397106770988388*z^90-22561857973440588*z^88-\ 1248066800327477349*z^84+16187433456375*z^94+181084266846039569*z^86-\ 1005030539528*z^96+50693342324*z^98-215343403310675*z^92+7429503592125924572*z^ 82-171036996571265765469318*z^64+z^110+22872*z^106-223*z^108+ 38383076581293203994*z^30+44754407541883567512560*z^42-93059222821596744240934* z^44+171036996571265765469318*z^46+509998456353602344649612*z^58-\ 575689798540740113146278*z^56+575689798540740113146278*z^54-\ 509998456353602344649612*z^52-400194030134579406099157*z^60+ 19005390613165498997102*z^70-44754407541883567512560*z^68+172792587902751993477 *z^78-172792587902751993477*z^32+7117510148546044752788*z^38-\ 19005390613165498997102*z^40+278082305151841712895909*z^62-\ 680103355493158597459*z^76+2346960166853152003092*z^74-7117510148546044752788*z ^72-1441554*z^104)/(1+21656640147715613370*z^28-3442149642917069026*z^26-255*z^ 2+472095758500303665*z^24-55534943870176126*z^22+29252*z^4-2031807*z^6-\ 3382911335*z^102+96739207*z^8-3382911335*z^10+90807335827*z^12-1929415206254*z^ 14-470246829489008*z^18+33189481175092*z^16-2130102116435891778492125*z^50+ 1398124692989021411602575*z^48+5562593535322633*z^20+8494553224053040480863*z^ 36-2332726013579153021468*z^34-812950477502060838126293*z^66+ 561587508796437619754*z^80+90807335827*z^100-55534943870176126*z^90+ 472095758500303665*z^88+21656640147715613370*z^84-470246829489008*z^94-\ 3442149642917069026*z^86+33189481175092*z^96-1929415206254*z^98+ 5562593535322633*z^92-118170075959388394846*z^82+1398124692989021411602575*z^64 +z^112-255*z^110-2031807*z^106+29252*z^108-118170075959388394846*z^30-\ 190504454892383309364845*z^42+418455064551937974368420*z^44-\ 812950477502060838126293*z^46-3444029521652062059956188*z^58+ 3657020274901682235646787*z^56-3444029521652062059956188*z^54+ 2876438849080284487777282*z^52+2876438849080284487777282*z^60-\ 190504454892383309364845*z^70+418455064551937974368420*z^68-\ 2332726013579153021468*z^78+561587508796437619754*z^32-27185090114351591797014* z^38+76619035706339968949056*z^40-2130102116435891778492125*z^62+ 8494553224053040480863*z^76-27185090114351591797014*z^74+ 76619035706339968949056*z^72+96739207*z^104) The first , 40, terms are: [0, 32, 0, 1780, 0, 108089, 0, 6700832, 0, 417954981, 0, 26125150071, 0, 1634379478456, 0, 102282243687997, 0, 6401978503546179, 0, 400735326918444124, 0, 25085003953944799859, 0, 1570278163488285416452, 0, 98297312004923029900468, 0, 6153296952644799901511459, 0, 385189674876270533064909387, 0, 24112466572131856689448101628, 0, 1509415297301991598219333006580, 0, 94487836221624466807140412739311, 0, 5914841100983972006876382964025700, 0, 370262961172319546671564451530585651 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 164598051937862 z + 88224308569446 z + 155 z 24 22 4 6 - 38272434686602 z + 13389545406666 z - 10338 z + 396578 z 8 10 12 14 - 9883714 z + 171406030 z - 2162241742 z + 20461600350 z 18 16 50 + 840885959822 z - 148533634818 z + 148533634818 z 48 20 36 - 840885959822 z - 3758760777306 z - 249142386938342 z 34 66 64 30 + 306411547221204 z + z - 155 z + 249142386938342 z 42 44 46 + 38272434686602 z - 13389545406666 z + 3758760777306 z 58 56 54 52 + 9883714 z - 171406030 z + 2162241742 z - 20461600350 z 60 32 38 - 396578 z - 306411547221204 z + 164598051937862 z 40 62 / 28 - 88224308569446 z + 10338 z ) / (1 + 788646007247872 z / 26 2 24 22 - 383985099198256 z - 188 z + 151525531847984 z - 48270103315972 z 4 6 8 10 12 + 14657 z - 641892 z + 17993440 z - 347664336 z + 4856232432 z 14 18 16 - 50680547532 z - 2516280568080 z + 404681143180 z 50 48 20 - 2516280568080 z + 12345560770060 z + 12345560770060 z 36 34 66 64 + 1788938975535910 z - 1981312213870184 z - 188 z + 14657 z 30 42 44 - 1316387012495820 z - 383985099198256 z + 151525531847984 z 46 58 56 54 - 48270103315972 z - 347664336 z + 4856232432 z - 50680547532 z 52 60 68 32 + 404681143180 z + 17993440 z + z + 1788938975535910 z 38 40 62 - 1316387012495820 z + 788646007247872 z - 641892 z ) And in Maple-input format, it is: -(-1-164598051937862*z^28+88224308569446*z^26+155*z^2-38272434686602*z^24+ 13389545406666*z^22-10338*z^4+396578*z^6-9883714*z^8+171406030*z^10-2162241742* z^12+20461600350*z^14+840885959822*z^18-148533634818*z^16+148533634818*z^50-\ 840885959822*z^48-3758760777306*z^20-249142386938342*z^36+306411547221204*z^34+ z^66-155*z^64+249142386938342*z^30+38272434686602*z^42-13389545406666*z^44+ 3758760777306*z^46+9883714*z^58-171406030*z^56+2162241742*z^54-20461600350*z^52 -396578*z^60-306411547221204*z^32+164598051937862*z^38-88224308569446*z^40+ 10338*z^62)/(1+788646007247872*z^28-383985099198256*z^26-188*z^2+ 151525531847984*z^24-48270103315972*z^22+14657*z^4-641892*z^6+17993440*z^8-\ 347664336*z^10+4856232432*z^12-50680547532*z^14-2516280568080*z^18+404681143180 *z^16-2516280568080*z^50+12345560770060*z^48+12345560770060*z^20+ 1788938975535910*z^36-1981312213870184*z^34-188*z^66+14657*z^64-\ 1316387012495820*z^30-383985099198256*z^42+151525531847984*z^44-48270103315972* z^46-347664336*z^58+4856232432*z^56-50680547532*z^54+404681143180*z^52+17993440 *z^60+z^68+1788938975535910*z^32-1316387012495820*z^38+788646007247872*z^40-\ 641892*z^62) The first , 40, terms are: [0, 33, 0, 1885, 0, 116013, 0, 7254709, 0, 455923957, 0, 28710548697, 0, 1809682982281, 0, 114122552952825, 0, 7198618886416873, 0, 454134761043648837, 0, 28651765816488293797, 0, 1807735241979758252093, 0, 114058418767015266962189, 0, 7196557489873998215766289, 0, 454072216066618995461681809, 0, 28650124475998789501977145201, 0, 1807710529792506812835372989745, 0, 114059560002074580406377442641325, 0, 7196722045918889505262823511301661, 0, 454085771875434620183734911125007301 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 131973706379628489 z - 34622341450443924 z - 196 z 24 22 4 6 + 7655775623071561 z - 1418929232386372 z + 17249 z - 910158 z 8 10 12 14 + 32447196 z - 835011318 z + 16183774427 z - 243331011644 z 18 16 50 - 27882511435300 z + 2900877884635 z - 9329948148804451564 z 48 20 36 + 13673255107921009691 z + 218925206305576 z + 5456633811129901115 z 34 66 80 88 - 2731617115371888354 z - 1418929232386372 z + 32447196 z + z 84 86 82 64 + 17249 z - 196 z - 910158 z + 7655775623071561 z 30 42 - 425949455334407962 z - 17191313106099486424 z 44 46 + 18553618918389316144 z - 17191313106099486424 z 58 56 - 425949455334407962 z + 1168347520423923044 z 54 52 - 2731617115371888354 z + 5456633811129901115 z 60 70 68 + 131973706379628489 z - 27882511435300 z + 218925206305576 z 78 32 38 - 835011318 z + 1168347520423923044 z - 9329948148804451564 z 40 62 76 + 13673255107921009691 z - 34622341450443924 z + 16183774427 z 74 72 / 2 - 243331011644 z + 2900877884635 z ) / ((-1 + z ) (1 / 28 26 2 + 370885663410894439 z - 93758240585417507 z - 231 z 24 22 4 6 + 19904652360351813 z - 3529292699168156 z + 23091 z - 1352100 z 8 10 12 14 + 52673650 z - 1465552904 z + 30472852381 z - 488613839905 z 18 16 50 - 62795664470204 z + 6181912974903 z - 29719115396415245589 z 48 20 + 44074773488708066819 z + 519090612917356 z 36 34 + 17097144582527298633 z - 8381587277216959028 z 66 80 88 84 86 - 3529292699168156 z + 52673650 z + z + 23091 z - 231 z 82 64 30 - 1352100 z + 19904652360351813 z - 1237559726924379808 z 42 44 - 55814166192071650344 z + 60381870635064906600 z 46 58 - 55814166192071650344 z - 1237559726924379808 z 56 54 + 3495610404099153326 z - 8381587277216959028 z 52 60 70 + 17097144582527298633 z + 370885663410894439 z - 62795664470204 z 68 78 32 + 519090612917356 z - 1465552904 z + 3495610404099153326 z 38 40 - 29719115396415245589 z + 44074773488708066819 z 62 76 74 - 93758240585417507 z + 30472852381 z - 488613839905 z 72 + 6181912974903 z )) And in Maple-input format, it is: -(1+131973706379628489*z^28-34622341450443924*z^26-196*z^2+7655775623071561*z^ 24-1418929232386372*z^22+17249*z^4-910158*z^6+32447196*z^8-835011318*z^10+ 16183774427*z^12-243331011644*z^14-27882511435300*z^18+2900877884635*z^16-\ 9329948148804451564*z^50+13673255107921009691*z^48+218925206305576*z^20+ 5456633811129901115*z^36-2731617115371888354*z^34-1418929232386372*z^66+ 32447196*z^80+z^88+17249*z^84-196*z^86-910158*z^82+7655775623071561*z^64-\ 425949455334407962*z^30-17191313106099486424*z^42+18553618918389316144*z^44-\ 17191313106099486424*z^46-425949455334407962*z^58+1168347520423923044*z^56-\ 2731617115371888354*z^54+5456633811129901115*z^52+131973706379628489*z^60-\ 27882511435300*z^70+218925206305576*z^68-835011318*z^78+1168347520423923044*z^ 32-9329948148804451564*z^38+13673255107921009691*z^40-34622341450443924*z^62+ 16183774427*z^76-243331011644*z^74+2900877884635*z^72)/(-1+z^2)/(1+ 370885663410894439*z^28-93758240585417507*z^26-231*z^2+19904652360351813*z^24-\ 3529292699168156*z^22+23091*z^4-1352100*z^6+52673650*z^8-1465552904*z^10+ 30472852381*z^12-488613839905*z^14-62795664470204*z^18+6181912974903*z^16-\ 29719115396415245589*z^50+44074773488708066819*z^48+519090612917356*z^20+ 17097144582527298633*z^36-8381587277216959028*z^34-3529292699168156*z^66+ 52673650*z^80+z^88+23091*z^84-231*z^86-1352100*z^82+19904652360351813*z^64-\ 1237559726924379808*z^30-55814166192071650344*z^42+60381870635064906600*z^44-\ 55814166192071650344*z^46-1237559726924379808*z^58+3495610404099153326*z^56-\ 8381587277216959028*z^54+17097144582527298633*z^52+370885663410894439*z^60-\ 62795664470204*z^70+519090612917356*z^68-1465552904*z^78+3495610404099153326*z^ 32-29719115396415245589*z^38+44074773488708066819*z^40-93758240585417507*z^62+ 30472852381*z^76-488613839905*z^74+6181912974903*z^72) The first , 40, terms are: [0, 36, 0, 2279, 0, 154169, 0, 10544692, 0, 723187651, 0, 49644890323, 0, 3409313249684, 0, 234172112912257, 0, 16085651784841535, 0, 1104990886462572964, 0, 75907824316131660065, 0, 5214565084710545403905, 0, 358221292356418077957476, 0, 24608522170094829338607599, 0, 1690519039786734259182053265, 0, 116132770640114609399031611412, 0, 7977918540519024269025858755603, 0, 548055386173448380309129553963843, 0, 37649509548404448111358025417559604, 0, 2586391150046476471921607257597794921] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 117988160885864 z + 63221351352720 z + 144 z 24 22 4 6 - 27414731323126 z + 9588334128272 z - 8964 z + 324409 z 8 10 12 14 - 7723668 z + 129507780 z - 1596098865 z + 14879794004 z 18 16 50 + 603456727625 z - 107072700576 z + 107072700576 z 48 20 36 - 603456727625 z - 2692597991216 z - 178623767540694 z 34 66 64 30 + 219698885184872 z + z - 144 z + 178623767540694 z 42 44 46 58 + 27414731323126 z - 9588334128272 z + 2692597991216 z + 7723668 z 56 54 52 60 - 129507780 z + 1596098865 z - 14879794004 z - 324409 z 32 38 40 - 219698885184872 z + 117988160885864 z - 63221351352720 z 62 / 28 26 2 + 8964 z ) / (1 + 543814999939018 z - 265518654842520 z - 168 z / 24 22 4 6 + 105090120228570 z - 33578166528968 z + 12081 z - 499738 z 8 10 12 14 + 13466857 z - 253381698 z + 3478330621 z - 35901241210 z 18 16 50 - 1761871852240 z + 284669956757 z - 1761871852240 z 48 20 36 + 8614255531061 z + 8614255531061 z + 1228844833612074 z 34 66 64 30 - 1360258154052876 z - 168 z + 12081 z - 905628445851260 z 42 44 46 - 265518654842520 z + 105090120228570 z - 33578166528968 z 58 56 54 52 - 253381698 z + 3478330621 z - 35901241210 z + 284669956757 z 60 68 32 38 + 13466857 z + z + 1228844833612074 z - 905628445851260 z 40 62 + 543814999939018 z - 499738 z ) And in Maple-input format, it is: -(-1-117988160885864*z^28+63221351352720*z^26+144*z^2-27414731323126*z^24+ 9588334128272*z^22-8964*z^4+324409*z^6-7723668*z^8+129507780*z^10-1596098865*z^ 12+14879794004*z^14+603456727625*z^18-107072700576*z^16+107072700576*z^50-\ 603456727625*z^48-2692597991216*z^20-178623767540694*z^36+219698885184872*z^34+ z^66-144*z^64+178623767540694*z^30+27414731323126*z^42-9588334128272*z^44+ 2692597991216*z^46+7723668*z^58-129507780*z^56+1596098865*z^54-14879794004*z^52 -324409*z^60-219698885184872*z^32+117988160885864*z^38-63221351352720*z^40+8964 *z^62)/(1+543814999939018*z^28-265518654842520*z^26-168*z^2+105090120228570*z^ 24-33578166528968*z^22+12081*z^4-499738*z^6+13466857*z^8-253381698*z^10+ 3478330621*z^12-35901241210*z^14-1761871852240*z^18+284669956757*z^16-\ 1761871852240*z^50+8614255531061*z^48+8614255531061*z^20+1228844833612074*z^36-\ 1360258154052876*z^34-168*z^66+12081*z^64-905628445851260*z^30-265518654842520* z^42+105090120228570*z^44-33578166528968*z^46-253381698*z^58+3478330621*z^56-\ 35901241210*z^54+284669956757*z^52+13466857*z^60+z^68+1228844833612074*z^32-\ 905628445851260*z^38+543814999939018*z^40-499738*z^62) The first , 40, terms are: [0, 24, 0, 915, 0, 39105, 0, 1766048, 0, 82198179, 0, 3892677515, 0, 186259240432, 0, 8968404354761, 0, 433501895603419, 0, 21004137929863656, 0, 1019206123578271401, 0, 49501472981621926425, 0, 2405590126071006742472, 0, 116944166780474025084811, 0, 5686311089793669846014713, 0, 276529604384656829171659984, 0, 13448975329849936074854949755, 0, 654122989368522887033258447667, 0, 31815859960087228399078890336448, 0, 1547521242519314815004813236146193] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 29267836515335 z - 17394580816502 z - 134 z 24 22 4 6 8 + 8371279271198 z - 3251914358698 z + 7662 z - 252810 z + 5460366 z 10 12 14 18 - 82764007 z + 919694798 z - 7717477984 z - 253008115513 z 16 50 48 20 + 49937999446 z - 7717477984 z + 49937999446 z + 1015050226790 z 36 34 64 30 + 29267836515335 z - 39960757426524 z + z - 39960757426524 z 42 44 46 58 - 3251914358698 z + 1015050226790 z - 253008115513 z - 252810 z 56 54 52 60 + 5460366 z - 82764007 z + 919694798 z + 7662 z 32 38 40 62 + 44326064132928 z - 17394580816502 z + 8371279271198 z - 134 z ) / 2 28 26 2 / ((-1 + z ) (1 + 89932360311024 z - 52511106300600 z - 160 z / 24 22 4 6 + 24651256753385 z - 9273446734414 z + 10664 z - 401810 z 8 10 12 14 + 9736355 z - 163061088 z + 1975972104 z - 17875648056 z 18 16 50 48 - 661337025440 z + 123440060737 z - 17875648056 z + 123440060737 z 20 36 34 64 + 2782210348118 z + 89932360311024 z - 124098943387560 z + z 30 42 44 - 124098943387560 z - 9273446734414 z + 2782210348118 z 46 58 56 54 - 661337025440 z - 401810 z + 9736355 z - 163061088 z 52 60 32 38 + 1975972104 z + 10664 z + 138143239339969 z - 52511106300600 z 40 62 + 24651256753385 z - 160 z )) And in Maple-input format, it is: -(1+29267836515335*z^28-17394580816502*z^26-134*z^2+8371279271198*z^24-\ 3251914358698*z^22+7662*z^4-252810*z^6+5460366*z^8-82764007*z^10+919694798*z^12 -7717477984*z^14-253008115513*z^18+49937999446*z^16-7717477984*z^50+49937999446 *z^48+1015050226790*z^20+29267836515335*z^36-39960757426524*z^34+z^64-\ 39960757426524*z^30-3251914358698*z^42+1015050226790*z^44-253008115513*z^46-\ 252810*z^58+5460366*z^56-82764007*z^54+919694798*z^52+7662*z^60+44326064132928* z^32-17394580816502*z^38+8371279271198*z^40-134*z^62)/(-1+z^2)/(1+ 89932360311024*z^28-52511106300600*z^26-160*z^2+24651256753385*z^24-\ 9273446734414*z^22+10664*z^4-401810*z^6+9736355*z^8-163061088*z^10+1975972104*z ^12-17875648056*z^14-661337025440*z^18+123440060737*z^16-17875648056*z^50+ 123440060737*z^48+2782210348118*z^20+89932360311024*z^36-124098943387560*z^34+z ^64-124098943387560*z^30-9273446734414*z^42+2782210348118*z^44-661337025440*z^ 46-401810*z^58+9736355*z^56-163061088*z^54+1975972104*z^52+10664*z^60+ 138143239339969*z^32-52511106300600*z^38+24651256753385*z^40-160*z^62) The first , 40, terms are: [0, 27, 0, 1185, 0, 58201, 0, 3002920, 0, 158587167, 0, 8467794123, 0, 454487657057, 0, 24452939560679, 0, 1317154357894919, 0, 70986478579702993, 0, 3826699074842551803, 0, 206312056432651968527, 0, 11123698309831658091336, 0, 599770688665946433477577, 0, 32339005770545071854137089, 0, 1743695433476545489006651787, 0, 94019035958529179064756997745, 0, 5069457911106721378846215478929, 0, 273342722522775965171328714410443, 0, 14738511858722196170582396551892609] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 203566699887926217 z - 51798362959955619 z - 199 z 24 22 4 6 + 11084399002006538 z - 1984515278648184 z + 17866 z - 965972 z 8 10 12 14 + 35423665 z - 940750298 z + 18862929479 z - 293922662828 z 18 16 50 - 36259573061042 z + 3635104645775 z - 16013957402981763306 z 48 20 36 + 23712997970828999520 z + 295373011663189 z + 9232752857977231180 z 34 66 80 88 - 4539135591761791400 z - 1984515278648184 z + 35423665 z + z 84 86 82 64 + 17866 z - 199 z - 965972 z + 11084399002006538 z 30 42 - 675605853708488698 z - 30001560316303817574 z 44 46 + 32446963938281387264 z - 30001560316303817574 z 58 56 - 675605853708488698 z + 1899895673597857146 z 54 52 - 4539135591761791400 z + 9232752857977231180 z 60 70 68 + 203566699887926217 z - 36259573061042 z + 295373011663189 z 78 32 38 - 940750298 z + 1899895673597857146 z - 16013957402981763306 z 40 62 76 + 23712997970828999520 z - 51798362959955619 z + 18862929479 z 74 72 / 2 - 293922662828 z + 3635104645775 z ) / ((-1 + z ) (1 / 28 26 2 + 562637250685040000 z - 137997829630177404 z - 230 z 24 22 4 6 + 28354866056928741 z - 4855734496721301 z + 23260 z - 1392581 z 8 10 12 14 + 55859991 z - 1607654974 z + 34675179746 z - 577642021251 z 18 16 50 - 80208833126980 z + 7597217148239 z - 50052741281534440094 z 48 20 + 74980312434654329541 z + 688579661905266 z 36 34 + 28397111345338791957 z - 13678112748188087822 z 66 80 88 84 86 - 4855734496721301 z + 55859991 z + z + 23260 z - 230 z 82 64 30 - 1392581 z + 28354866056928741 z - 1929664550451022934 z 42 44 - 95530273441700043982 z + 103558649360514508857 z 46 58 - 95530273441700043982 z - 1929664550451022934 z 56 54 + 5585289585845744285 z - 13678112748188087822 z 52 60 70 + 28397111345338791957 z + 562637250685040000 z - 80208833126980 z 68 78 32 + 688579661905266 z - 1607654974 z + 5585289585845744285 z 38 40 - 50052741281534440094 z + 74980312434654329541 z 62 76 74 - 137997829630177404 z + 34675179746 z - 577642021251 z 72 + 7597217148239 z )) And in Maple-input format, it is: -(1+203566699887926217*z^28-51798362959955619*z^26-199*z^2+11084399002006538*z^ 24-1984515278648184*z^22+17866*z^4-965972*z^6+35423665*z^8-940750298*z^10+ 18862929479*z^12-293922662828*z^14-36259573061042*z^18+3635104645775*z^16-\ 16013957402981763306*z^50+23712997970828999520*z^48+295373011663189*z^20+ 9232752857977231180*z^36-4539135591761791400*z^34-1984515278648184*z^66+ 35423665*z^80+z^88+17866*z^84-199*z^86-965972*z^82+11084399002006538*z^64-\ 675605853708488698*z^30-30001560316303817574*z^42+32446963938281387264*z^44-\ 30001560316303817574*z^46-675605853708488698*z^58+1899895673597857146*z^56-\ 4539135591761791400*z^54+9232752857977231180*z^52+203566699887926217*z^60-\ 36259573061042*z^70+295373011663189*z^68-940750298*z^78+1899895673597857146*z^ 32-16013957402981763306*z^38+23712997970828999520*z^40-51798362959955619*z^62+ 18862929479*z^76-293922662828*z^74+3635104645775*z^72)/(-1+z^2)/(1+ 562637250685040000*z^28-137997829630177404*z^26-230*z^2+28354866056928741*z^24-\ 4855734496721301*z^22+23260*z^4-1392581*z^6+55859991*z^8-1607654974*z^10+ 34675179746*z^12-577642021251*z^14-80208833126980*z^18+7597217148239*z^16-\ 50052741281534440094*z^50+74980312434654329541*z^48+688579661905266*z^20+ 28397111345338791957*z^36-13678112748188087822*z^34-4855734496721301*z^66+ 55859991*z^80+z^88+23260*z^84-230*z^86-1392581*z^82+28354866056928741*z^64-\ 1929664550451022934*z^30-95530273441700043982*z^42+103558649360514508857*z^44-\ 95530273441700043982*z^46-1929664550451022934*z^58+5585289585845744285*z^56-\ 13678112748188087822*z^54+28397111345338791957*z^52+562637250685040000*z^60-\ 80208833126980*z^70+688579661905266*z^68-1607654974*z^78+5585289585845744285*z^ 32-50052741281534440094*z^38+74980312434654329541*z^40-137997829630177404*z^62+ 34675179746*z^76-577642021251*z^74+7597217148239*z^72) The first , 40, terms are: [0, 32, 0, 1768, 0, 106597, 0, 6571592, 0, 407963473, 0, 25387295603, 0, 1581218560160, 0, 98516829970993, 0, 6138807726065183, 0, 382542106655941392, 0, 23838728666297983207, 0, 1485560840641594970840, 0, 92576177601970183425496, 0, 5769107839946711419707175, 0, 359516088429860039384364519, 0, 22404131651922298010224504328, 0, 1396168875241810864354894854440, 0, 87005721027343807843827741093279, 0, 5421977098576261958973953294667296, 0, 337883941676785908502120903914830639 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 313058698375240 z - 154550004238514 z - 155 z 24 22 4 6 + 62039101904030 z - 20166856289376 z + 10364 z - 401903 z 8 10 12 14 + 10231801 z - 183176000 z + 2408176249 z - 23942175991 z 18 16 50 - 1106262511187 z + 183799083452 z - 1106262511187 z 48 20 36 + 5280642393601 z + 5280642393601 z + 698433347462638 z 34 66 64 30 - 771879950608832 z - 155 z + 10364 z - 517210783547002 z 42 44 46 - 154550004238514 z + 62039101904030 z - 20166856289376 z 58 56 54 52 - 183176000 z + 2408176249 z - 23942175991 z + 183799083452 z 60 68 32 38 + 10231801 z + z + 698433347462638 z - 517210783547002 z 40 62 / 2 + 313058698375240 z - 401903 z ) / ((-1 + z ) (1 / 28 26 2 24 + 940213935413690 z - 453928475736816 z - 180 z + 177055944867226 z 22 4 6 8 10 - 55558673258728 z + 13857 z - 610938 z + 17435925 z - 345135378 z 12 14 18 16 + 4953749401 z - 53164023034 z - 2780682491660 z + 436166025461 z 50 48 20 - 2780682491660 z + 13947109877973 z + 13947109877973 z 36 34 66 64 + 2151777549861058 z - 2385656675909548 z - 180 z + 13857 z 30 42 44 - 1578285998252092 z - 453928475736816 z + 177055944867226 z 46 58 56 54 - 55558673258728 z - 345135378 z + 4953749401 z - 53164023034 z 52 60 68 32 + 436166025461 z + 17435925 z + z + 2151777549861058 z 38 40 62 - 1578285998252092 z + 940213935413690 z - 610938 z )) And in Maple-input format, it is: -(1+313058698375240*z^28-154550004238514*z^26-155*z^2+62039101904030*z^24-\ 20166856289376*z^22+10364*z^4-401903*z^6+10231801*z^8-183176000*z^10+2408176249 *z^12-23942175991*z^14-1106262511187*z^18+183799083452*z^16-1106262511187*z^50+ 5280642393601*z^48+5280642393601*z^20+698433347462638*z^36-771879950608832*z^34 -155*z^66+10364*z^64-517210783547002*z^30-154550004238514*z^42+62039101904030*z ^44-20166856289376*z^46-183176000*z^58+2408176249*z^56-23942175991*z^54+ 183799083452*z^52+10231801*z^60+z^68+698433347462638*z^32-517210783547002*z^38+ 313058698375240*z^40-401903*z^62)/(-1+z^2)/(1+940213935413690*z^28-\ 453928475736816*z^26-180*z^2+177055944867226*z^24-55558673258728*z^22+13857*z^4 -610938*z^6+17435925*z^8-345135378*z^10+4953749401*z^12-53164023034*z^14-\ 2780682491660*z^18+436166025461*z^16-2780682491660*z^50+13947109877973*z^48+ 13947109877973*z^20+2151777549861058*z^36-2385656675909548*z^34-180*z^66+13857* z^64-1578285998252092*z^30-453928475736816*z^42+177055944867226*z^44-\ 55558673258728*z^46-345135378*z^58+4953749401*z^56-53164023034*z^54+ 436166025461*z^52+17435925*z^60+z^68+2151777549861058*z^32-1578285998252092*z^ 38+940213935413690*z^40-610938*z^62) The first , 40, terms are: [0, 26, 0, 1033, 0, 44903, 0, 2056830, 0, 97572919, 0, 4737881383, 0, 233605025070, 0, 11632933626983, 0, 583002120153193, 0, 29337860341502154, 0, 1480208425857892881, 0, 74807075348269691313, 0, 3784647177688772870634, 0, 191603546146425444578409, 0, 9704434811506559990483367, 0, 491651461474944926124606030, 0, 24912728633698158524414644871, 0, 1262508653740160133372976776023, 0, 63985092203949387406774904615006, 0, 3242972613594083176514282206232743] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 1265315909476 z - 1120874876267 z - 123 z + 778728133893 z 22 4 6 8 10 - 423562383759 z + 6373 z - 187368 z + 3536708 z - 45861813 z 12 14 18 16 + 426227660 z - 2921156960 z - 59265390098 z + 15061716733 z 50 48 20 36 - 187368 z + 3536708 z + 179792293402 z + 179792293402 z 34 30 42 44 - 423562383759 z - 1120874876267 z - 2921156960 z + 426227660 z 46 56 54 52 32 - 45861813 z + z - 123 z + 6373 z + 778728133893 z 38 40 / 2 28 - 59265390098 z + 15061716733 z ) / ((-1 + z ) (1 + 4029901952897 z / 26 2 24 22 - 3548266498424 z - 150 z + 2421021645911 z - 1278328597050 z 4 6 8 10 12 + 9095 z - 305430 z + 6473997 z - 93026428 z + 947297471 z 14 18 16 50 - 7041100478 z - 163130048056 z + 38987320993 z - 305430 z 48 20 36 34 + 6473997 z + 520946151887 z + 520946151887 z - 1278328597050 z 30 42 44 46 56 - 3548266498424 z - 7041100478 z + 947297471 z - 93026428 z + z 54 52 32 38 - 150 z + 9095 z + 2421021645911 z - 163130048056 z 40 + 38987320993 z )) And in Maple-input format, it is: -(1+1265315909476*z^28-1120874876267*z^26-123*z^2+778728133893*z^24-\ 423562383759*z^22+6373*z^4-187368*z^6+3536708*z^8-45861813*z^10+426227660*z^12-\ 2921156960*z^14-59265390098*z^18+15061716733*z^16-187368*z^50+3536708*z^48+ 179792293402*z^20+179792293402*z^36-423562383759*z^34-1120874876267*z^30-\ 2921156960*z^42+426227660*z^44-45861813*z^46+z^56-123*z^54+6373*z^52+ 778728133893*z^32-59265390098*z^38+15061716733*z^40)/(-1+z^2)/(1+4029901952897* z^28-3548266498424*z^26-150*z^2+2421021645911*z^24-1278328597050*z^22+9095*z^4-\ 305430*z^6+6473997*z^8-93026428*z^10+947297471*z^12-7041100478*z^14-\ 163130048056*z^18+38987320993*z^16-305430*z^50+6473997*z^48+520946151887*z^20+ 520946151887*z^36-1278328597050*z^34-3548266498424*z^30-7041100478*z^42+ 947297471*z^44-93026428*z^46+z^56-150*z^54+9095*z^52+2421021645911*z^32-\ 163130048056*z^38+38987320993*z^40) The first , 40, terms are: [0, 28, 0, 1356, 0, 73053, 0, 4058764, 0, 227808935, 0, 12831883479, 0, 723706815840, 0, 40835311239447, 0, 2304532876145587, 0, 130064054382962580, 0, 7340774092754034061, 0, 414314612295019580324, 0, 23384068817810091600020, 0, 1319807081657420154708621, 0, 74490524900173179664877301, 0, 4204280595538364169701104868, 0, 237291609489226172449552574708, 0, 13392852389152302050684288658149, 0, 755899027844515084374549161849924, 0, 42663304717019064059970328886138891] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 8485133371853011281 z - 1398970859121693638 z - 224 z 24 22 4 6 + 199594618785844349 z - 24497360053754776 z + 23075 z - 1460934 z 102 8 10 12 - 2082059694 z + 64090494 z - 2082059694 z + 52262185938 z 14 18 16 - 1044071372424 z - 228028564418990 z + 16966171444595 z 50 48 - 660133098953876714893650 z + 437143096566067350433735 z 20 36 + 2568283059032279 z + 2951205801705704139415 z 34 66 - 831987588053174921724 z - 257085600997359039839460 z 80 100 90 + 206136850058952029019 z + 52262185938 z - 24497360053754776 z 88 84 94 + 199594618785844349 z + 8485133371853011281 z - 228028564418990 z 86 96 98 - 1398970859121693638 z + 16966171444595 z - 1044071372424 z 92 82 + 2568283059032279 z - 44754954094367170982 z 64 112 110 106 + 437143096566067350433735 z + z - 224 z - 1460934 z 108 30 42 + 23075 z - 44754954094367170982 z - 62091982835642239733186 z 44 46 + 134178181069374563029738 z - 257085600997359039839460 z 58 56 - 1056557132517357154128462 z + 1120475002582109349655436 z 54 52 - 1056557132517357154128462 z + 885797139711757501743021 z 60 70 + 885797139711757501743021 z - 62091982835642239733186 z 68 78 + 134178181069374563029738 z - 831987588053174921724 z 32 38 + 206136850058952029019 z - 9223047948448336492000 z 40 62 + 25447080888814296728558 z - 660133098953876714893650 z 76 74 + 2951205801705704139415 z - 9223047948448336492000 z 72 104 / 2 + 25447080888814296728558 z + 64090494 z ) / ((-1 + z ) (1 / 28 26 2 + 21107541340914518498 z - 3358928891965930586 z - 255 z 24 22 4 6 + 461361234399731433 z - 54366743357314250 z + 29228 z - 2027067 z 102 8 10 12 - 3360065435 z + 96316519 z - 3360065435 z + 89960939575 z 14 18 16 - 1906392319034 z - 462309698870292 z + 32708934352784 z 50 48 - 2073224082665766309608821 z + 1360458667962478389229119 z 20 36 + 5456481949293645 z + 8259084565272373989923 z 34 66 - 2268710662021439209200 z - 790837703874890593942417 z 80 100 90 + 546441333447664560158 z + 89960939575 z - 54366743357314250 z 88 84 94 + 461361234399731433 z + 21107541340914518498 z - 462309698870292 z 86 96 98 - 3358928891965930586 z + 32708934352784 z - 1906392319034 z 92 82 + 5456481949293645 z - 115064462194597621266 z 64 112 110 106 + 1360458667962478389229119 z + z - 255 z - 2027067 z 108 30 42 + 29228 z - 115064462194597621266 z - 185234398191211511972449 z 44 46 + 406967376485922746521492 z - 790837703874890593942417 z 58 56 - 3353146173493925983281188 z + 3560671329371594042057419 z 54 52 - 3353146173493925983281188 z + 2800180278309629655458946 z 60 70 + 2800180278309629655458946 z - 185234398191211511972449 z 68 78 + 406967376485922746521492 z - 2268710662021439209200 z 32 38 + 546441333447664560158 z - 26428540096635896239594 z 40 62 + 74488999472102749051404 z - 2073224082665766309608821 z 76 74 + 8259084565272373989923 z - 26428540096635896239594 z 72 104 + 74488999472102749051404 z + 96316519 z )) And in Maple-input format, it is: -(1+8485133371853011281*z^28-1398970859121693638*z^26-224*z^2+ 199594618785844349*z^24-24497360053754776*z^22+23075*z^4-1460934*z^6-2082059694 *z^102+64090494*z^8-2082059694*z^10+52262185938*z^12-1044071372424*z^14-\ 228028564418990*z^18+16966171444595*z^16-660133098953876714893650*z^50+ 437143096566067350433735*z^48+2568283059032279*z^20+2951205801705704139415*z^36 -831987588053174921724*z^34-257085600997359039839460*z^66+206136850058952029019 *z^80+52262185938*z^100-24497360053754776*z^90+199594618785844349*z^88+ 8485133371853011281*z^84-228028564418990*z^94-1398970859121693638*z^86+ 16966171444595*z^96-1044071372424*z^98+2568283059032279*z^92-\ 44754954094367170982*z^82+437143096566067350433735*z^64+z^112-224*z^110-1460934 *z^106+23075*z^108-44754954094367170982*z^30-62091982835642239733186*z^42+ 134178181069374563029738*z^44-257085600997359039839460*z^46-\ 1056557132517357154128462*z^58+1120475002582109349655436*z^56-\ 1056557132517357154128462*z^54+885797139711757501743021*z^52+ 885797139711757501743021*z^60-62091982835642239733186*z^70+ 134178181069374563029738*z^68-831987588053174921724*z^78+206136850058952029019* z^32-9223047948448336492000*z^38+25447080888814296728558*z^40-\ 660133098953876714893650*z^62+2951205801705704139415*z^76-\ 9223047948448336492000*z^74+25447080888814296728558*z^72+64090494*z^104)/(-1+z^ 2)/(1+21107541340914518498*z^28-3358928891965930586*z^26-255*z^2+ 461361234399731433*z^24-54366743357314250*z^22+29228*z^4-2027067*z^6-3360065435 *z^102+96316519*z^8-3360065435*z^10+89960939575*z^12-1906392319034*z^14-\ 462309698870292*z^18+32708934352784*z^16-2073224082665766309608821*z^50+ 1360458667962478389229119*z^48+5456481949293645*z^20+8259084565272373989923*z^ 36-2268710662021439209200*z^34-790837703874890593942417*z^66+ 546441333447664560158*z^80+89960939575*z^100-54366743357314250*z^90+ 461361234399731433*z^88+21107541340914518498*z^84-462309698870292*z^94-\ 3358928891965930586*z^86+32708934352784*z^96-1906392319034*z^98+ 5456481949293645*z^92-115064462194597621266*z^82+1360458667962478389229119*z^64 +z^112-255*z^110-2027067*z^106+29228*z^108-115064462194597621266*z^30-\ 185234398191211511972449*z^42+406967376485922746521492*z^44-\ 790837703874890593942417*z^46-3353146173493925983281188*z^58+ 3560671329371594042057419*z^56-3353146173493925983281188*z^54+ 2800180278309629655458946*z^52+2800180278309629655458946*z^60-\ 185234398191211511972449*z^70+406967376485922746521492*z^68-\ 2268710662021439209200*z^78+546441333447664560158*z^32-26428540096635896239594* z^38+74488999472102749051404*z^40-2073224082665766309608821*z^62+ 8259084565272373989923*z^76-26428540096635896239594*z^74+ 74488999472102749051404*z^72+96316519*z^104) The first , 40, terms are: [0, 32, 0, 1784, 0, 108609, 0, 6754580, 0, 422811121, 0, 26526954523, 0, 1665765758472, 0, 104640272652449, 0, 6574321267431943, 0, 413077863227078280, 0, 25955268108125754583, 0, 1630889873583250000368, 0, 102476947784833227685428, 0, 6439153822628636818660619, 0, 404605603619504323927466211, 0, 25423491743953055555909787772, 0, 1597491647664640410192752902112, 0, 100378808573651885220723069815319, 0, 6307329156499467358122408190404224, 0, 396322714183975562378431560089214167 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 493478218164 z + 493478218164 z + 125 z - 383500204436 z 22 4 6 8 10 + 231329456839 z - 6417 z + 182701 z - 3288734 z + 40267726 z 12 14 18 16 - 350932766 z + 2242591726 z + 38889291543 z - 10722144107 z 50 48 20 36 + 6417 z - 182701 z - 108028610235 z - 38889291543 z 34 30 42 44 + 108028610235 z + 383500204436 z + 350932766 z - 40267726 z 46 54 52 32 38 + 3288734 z + z - 125 z - 231329456839 z + 10722144107 z 40 / 28 26 2 - 2242591726 z ) / (1 + 3210750952232 z - 2834963797576 z - 158 z / 24 22 4 6 8 + 1950590537051 z - 1044272312546 z + 9742 z - 322494 z + 6622859 z 10 12 14 18 - 91525708 z + 895177100 z - 6404158252 z - 139196845826 z 16 50 48 20 + 34264542233 z - 322494 z + 6622859 z + 433796654802 z 36 34 30 + 433796654802 z - 1044272312546 z - 2834963797576 z 42 44 46 56 54 52 - 6404158252 z + 895177100 z - 91525708 z + z - 158 z + 9742 z 32 38 40 + 1950590537051 z - 139196845826 z + 34264542233 z ) And in Maple-input format, it is: -(-1-493478218164*z^28+493478218164*z^26+125*z^2-383500204436*z^24+231329456839 *z^22-6417*z^4+182701*z^6-3288734*z^8+40267726*z^10-350932766*z^12+2242591726*z ^14+38889291543*z^18-10722144107*z^16+6417*z^50-182701*z^48-108028610235*z^20-\ 38889291543*z^36+108028610235*z^34+383500204436*z^30+350932766*z^42-40267726*z^ 44+3288734*z^46+z^54-125*z^52-231329456839*z^32+10722144107*z^38-2242591726*z^ 40)/(1+3210750952232*z^28-2834963797576*z^26-158*z^2+1950590537051*z^24-\ 1044272312546*z^22+9742*z^4-322494*z^6+6622859*z^8-91525708*z^10+895177100*z^12 -6404158252*z^14-139196845826*z^18+34264542233*z^16-322494*z^50+6622859*z^48+ 433796654802*z^20+433796654802*z^36-1044272312546*z^34-2834963797576*z^30-\ 6404158252*z^42+895177100*z^44-91525708*z^46+z^56-158*z^54+9742*z^52+ 1950590537051*z^32-139196845826*z^38+34264542233*z^40) The first , 40, terms are: [0, 33, 0, 1889, 0, 116769, 0, 7355041, 0, 466427681, 0, 29665589441, 0, 1889349417697, 0, 120407935415841, 0, 7676015037427329, 0, 489422512807618849, 0, 31207940406508130593, 0, 1990043092882622932897, 0, 126901807264557650393953, 0, 8092394328956730129541473, 0, 516045723182466104923474881, 0, 32907907637219765275594094657, 0, 2098518576221561376490952708833, 0, 133821407343417811503056399649121, 0, 8533721766117133663108464766879393, 0, 544191081111644274108781276797574689 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 30414745180116 z - 17951770568304 z - 132 z 24 22 4 6 8 + 8563183506432 z - 3292930762768 z + 7468 z - 244784 z + 5271296 z 10 12 14 18 - 79923280 z + 891017980 z - 7520310092 z - 250861938596 z 16 50 48 20 + 49047977820 z - 7520310092 z + 49047977820 z + 1016968959236 z 36 34 64 30 + 30414745180116 z - 41709444296076 z + z - 41709444296076 z 42 44 46 58 - 3292930762768 z + 1016968959236 z - 250861938596 z - 244784 z 56 54 52 60 + 5271296 z - 79923280 z + 891017980 z + 7468 z 32 38 40 62 + 46335532570950 z - 17951770568304 z + 8563183506432 z - 132 z ) / 2 28 26 2 / ((-1 + z ) (1 + 94148351739836 z - 54428345384498 z - 158 z / 24 22 4 6 + 25228450802720 z - 9354639583718 z + 10372 z - 385910 z 8 10 12 14 + 9276640 z - 154908274 z + 1880794708 z - 17121361850 z 18 16 50 48 - 647838303518 z + 119404150588 z - 17121361850 z + 119404150588 z 20 36 34 64 + 2764838897260 z + 94148351739836 z - 130743361253514 z + z 30 42 44 - 130743361253514 z - 9354639583718 z + 2764838897260 z 46 58 56 54 - 647838303518 z - 385910 z + 9276640 z - 154908274 z 52 60 32 38 + 1880794708 z + 10372 z + 145857054209414 z - 54428345384498 z 40 62 + 25228450802720 z - 158 z )) And in Maple-input format, it is: -(1+30414745180116*z^28-17951770568304*z^26-132*z^2+8563183506432*z^24-\ 3292930762768*z^22+7468*z^4-244784*z^6+5271296*z^8-79923280*z^10+891017980*z^12 -7520310092*z^14-250861938596*z^18+49047977820*z^16-7520310092*z^50+49047977820 *z^48+1016968959236*z^20+30414745180116*z^36-41709444296076*z^34+z^64-\ 41709444296076*z^30-3292930762768*z^42+1016968959236*z^44-250861938596*z^46-\ 244784*z^58+5271296*z^56-79923280*z^54+891017980*z^52+7468*z^60+46335532570950* z^32-17951770568304*z^38+8563183506432*z^40-132*z^62)/(-1+z^2)/(1+ 94148351739836*z^28-54428345384498*z^26-158*z^2+25228450802720*z^24-\ 9354639583718*z^22+10372*z^4-385910*z^6+9276640*z^8-154908274*z^10+1880794708*z ^12-17121361850*z^14-647838303518*z^18+119404150588*z^16-17121361850*z^50+ 119404150588*z^48+2764838897260*z^20+94148351739836*z^36-130743361253514*z^34+z ^64-130743361253514*z^30-9354639583718*z^42+2764838897260*z^44-647838303518*z^ 46-385910*z^58+9276640*z^56-154908274*z^54+1880794708*z^52+10372*z^60+ 145857054209414*z^32-54428345384498*z^38+25228450802720*z^40-158*z^62) The first , 40, terms are: [0, 27, 0, 1231, 0, 62917, 0, 3349733, 0, 181287463, 0, 9878601347, 0, 539870358225, 0, 29541363143025, 0, 1617371737424979, 0, 88571398888480215, 0, 4850907585881488165, 0, 265688455921749058693, 0, 14552287648624081768927, 0, 797065085285697623357323, 0, 43657416908728414639965665, 0, 2391239374914347717730470689, 0, 130975000698209299393002471211, 0, 7173876821025103533777934058815, 0, 392933889805178684847567262077189, 0, 21522122001772401002235477248551077] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 96801451977955631 z - 25259538469291057 z - 185 z 24 22 4 6 + 5565584197153205 z - 1030394589201948 z + 15427 z - 775674 z 8 10 12 14 + 26523776 z - 659238842 z + 12424472629 z - 182824446915 z 18 16 50 - 20409095384540 z + 2145641261335 z - 7070305153964078419 z 48 20 36 + 10405535174793073139 z + 159304281124536 z + 4112236942708503193 z 34 66 80 88 - 2045266644437276070 z - 1030394589201948 z + 26523776 z + z 84 86 82 64 + 15427 z - 185 z - 775674 z + 5565584197153205 z 30 42 - 314468426046181606 z - 13117428163539468744 z 44 46 + 14169632279609543952 z - 13117428163539468744 z 58 56 - 314468426046181606 z + 868662157778488640 z 54 52 - 2045266644437276070 z + 4112236942708503193 z 60 70 68 + 96801451977955631 z - 20409095384540 z + 159304281124536 z 78 32 38 - 659238842 z + 868662157778488640 z - 7070305153964078419 z 40 62 76 + 10405535174793073139 z - 25259538469291057 z + 12424472629 z 74 72 / 28 - 182824446915 z + 2145641261335 z ) / (-1 - 333585272509666546 z / 26 2 24 + 81295366881397697 z + 215 z - 16720385862392067 z 22 4 6 8 + 2887061917113262 z - 20292 z + 1136406 z - 42830839 z 10 12 14 18 + 1164857199 z - 23894429704 z + 381118349466 z + 49534010627109 z 16 50 48 - 4832565508047 z + 54458052709638249625 z - 73892366368801243095 z 20 36 34 - 415770232392046 z - 18658287652103458342 z + 8655483743076555161 z 66 80 90 88 84 + 16720385862392067 z - 1164857199 z + z - 215 z - 1136406 z 86 82 64 + 20292 z + 42830839 z - 81295366881397697 z 30 42 + 1160160460503471804 z + 73892366368801243095 z 44 46 - 86061301484374922212 z + 86061301484374922212 z 58 56 + 3431505165470822401 z - 8655483743076555161 z 54 52 + 18658287652103458342 z - 34430877588875506480 z 60 70 68 - 1160160460503471804 z + 415770232392046 z - 2887061917113262 z 78 32 38 + 23894429704 z - 3431505165470822401 z + 34430877588875506480 z 40 62 76 - 54458052709638249625 z + 333585272509666546 z - 381118349466 z 74 72 + 4832565508047 z - 49534010627109 z ) And in Maple-input format, it is: -(1+96801451977955631*z^28-25259538469291057*z^26-185*z^2+5565584197153205*z^24 -1030394589201948*z^22+15427*z^4-775674*z^6+26523776*z^8-659238842*z^10+ 12424472629*z^12-182824446915*z^14-20409095384540*z^18+2145641261335*z^16-\ 7070305153964078419*z^50+10405535174793073139*z^48+159304281124536*z^20+ 4112236942708503193*z^36-2045266644437276070*z^34-1030394589201948*z^66+ 26523776*z^80+z^88+15427*z^84-185*z^86-775674*z^82+5565584197153205*z^64-\ 314468426046181606*z^30-13117428163539468744*z^42+14169632279609543952*z^44-\ 13117428163539468744*z^46-314468426046181606*z^58+868662157778488640*z^56-\ 2045266644437276070*z^54+4112236942708503193*z^52+96801451977955631*z^60-\ 20409095384540*z^70+159304281124536*z^68-659238842*z^78+868662157778488640*z^32 -7070305153964078419*z^38+10405535174793073139*z^40-25259538469291057*z^62+ 12424472629*z^76-182824446915*z^74+2145641261335*z^72)/(-1-333585272509666546*z ^28+81295366881397697*z^26+215*z^2-16720385862392067*z^24+2887061917113262*z^22 -20292*z^4+1136406*z^6-42830839*z^8+1164857199*z^10-23894429704*z^12+ 381118349466*z^14+49534010627109*z^18-4832565508047*z^16+54458052709638249625*z ^50-73892366368801243095*z^48-415770232392046*z^20-18658287652103458342*z^36+ 8655483743076555161*z^34+16720385862392067*z^66-1164857199*z^80+z^90-215*z^88-\ 1136406*z^84+20292*z^86+42830839*z^82-81295366881397697*z^64+ 1160160460503471804*z^30+73892366368801243095*z^42-86061301484374922212*z^44+ 86061301484374922212*z^46+3431505165470822401*z^58-8655483743076555161*z^56+ 18658287652103458342*z^54-34430877588875506480*z^52-1160160460503471804*z^60+ 415770232392046*z^70-2887061917113262*z^68+23894429704*z^78-3431505165470822401 *z^32+34430877588875506480*z^38-54458052709638249625*z^40+333585272509666546*z^ 62-381118349466*z^76+4832565508047*z^74-49534010627109*z^72) The first , 40, terms are: [0, 30, 0, 1585, 0, 92747, 0, 5562902, 0, 335898503, 0, 20322897123, 0, 1230413515894, 0, 74510957554815, 0, 4512622639569477, 0, 273309024218238798, 0, 16553340526291752397, 0, 1002582702184334517045, 0, 60723391602961006368878, 0, 3677836227331471680204125, 0, 222755788318569808258737559, 0, 13491670291428237757314529334, 0, 817151326320279674460971644075, 0, 49492487563284545121224924287487, 0, 2997616601080242703807301326775894, 0, 181556955861244603795866800315695219 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 8443923010356559708 z - 1389722122074920812 z - 224 z 24 22 4 6 + 197949809635750428 z - 24260823933906668 z + 23040 z - 1455348 z 102 8 10 12 - 2064394488 z + 63687358 z - 2064394488 z + 51731190756 z 14 18 16 - 1032366701728 z - 225418616855720 z + 16768965247427 z 50 48 - 665805554808799254082416 z + 440699970943819550363435 z 20 36 + 2540654568056140 z + 2957137131574594797780 z 34 66 - 832340068537727669920 z - 259021757884751442406816 z 80 100 90 + 205873204925318924945 z + 51731190756 z - 24260823933906668 z 88 84 94 + 197949809635750428 z + 8443923010356559708 z - 225418616855720 z 86 96 98 - 1389722122074920812 z + 16768965247427 z - 1032366701728 z 92 82 + 2540654568056140 z - 44618134034702419240 z 64 112 110 106 + 440699970943819550363435 z + z - 224 z - 1455348 z 108 30 42 + 23040 z - 44618134034702419240 z - 62454195494703489886148 z 44 46 + 135086069920870754656136 z - 259021757884751442406816 z 58 56 - 1066168266170694452325864 z + 1130736529271809719126280 z 54 52 - 1066168266170694452325864 z + 893689665772356314166296 z 60 70 + 893689665772356314166296 z - 62454195494703489886148 z 68 78 + 135086069920870754656136 z - 832340068537727669920 z 32 38 + 205873204925318924945 z - 9254872431034990533392 z 40 62 + 25567393162208471517218 z - 665805554808799254082416 z 76 74 + 2957137131574594797780 z - 9254872431034990533392 z 72 104 / + 25567393162208471517218 z + 63687358 z ) / (-1 / 28 26 2 - 24440930894851865972 z + 3813776059281844728 z + 253 z 24 22 4 6 - 514327655655270616 z + 59580100686285668 z - 28988 z + 2018652 z 102 8 10 12 + 91983641624 z - 96586370 z + 3399807542 z - 91983641624 z 14 18 16 + 1972243656780 z + 491225409815931 z - 34275764800151 z 50 48 + 3404105426749618375289435 z - 2134682756403119873619743 z 20 36 - 5884894905198560 z - 10510500043992347579876 z 34 66 + 2812167657308959615949 z + 2134682756403119873619743 z 80 100 90 - 2812167657308959615949 z - 1972243656780 z + 514327655655270616 z 88 84 - 3813776059281844728 z - 136072836254468022416 z 94 86 96 + 5884894905198560 z + 24440930894851865972 z - 491225409815931 z 98 92 82 + 34275764800151 z - 59580100686285668 z + 661014483702855456937 z 64 112 114 110 - 3404105426749618375289435 z - 253 z + z + 28988 z 106 108 30 + 96586370 z - 2018652 z + 136072836254468022416 z 42 44 + 258558973064932716557718 z - 588885241018348902939388 z 46 58 + 1189779867277227931231316 z + 6844134126112648009349744 z 56 54 - 6844134126112648009349744 z + 6092878016024994621630696 z 52 60 - 4827951531186401511966416 z - 6092878016024994621630696 z 70 68 + 588885241018348902939388 z - 1189779867277227931231316 z 78 32 + 10510500043992347579876 z - 661014483702855456937 z 38 40 + 34602838072675334326480 z - 100573334361229991665522 z 62 76 + 4827951531186401511966416 z - 34602838072675334326480 z 74 72 + 100573334361229991665522 z - 258558973064932716557718 z 104 - 3399807542 z ) And in Maple-input format, it is: -(1+8443923010356559708*z^28-1389722122074920812*z^26-224*z^2+ 197949809635750428*z^24-24260823933906668*z^22+23040*z^4-1455348*z^6-2064394488 *z^102+63687358*z^8-2064394488*z^10+51731190756*z^12-1032366701728*z^14-\ 225418616855720*z^18+16768965247427*z^16-665805554808799254082416*z^50+ 440699970943819550363435*z^48+2540654568056140*z^20+2957137131574594797780*z^36 -832340068537727669920*z^34-259021757884751442406816*z^66+205873204925318924945 *z^80+51731190756*z^100-24260823933906668*z^90+197949809635750428*z^88+ 8443923010356559708*z^84-225418616855720*z^94-1389722122074920812*z^86+ 16768965247427*z^96-1032366701728*z^98+2540654568056140*z^92-\ 44618134034702419240*z^82+440699970943819550363435*z^64+z^112-224*z^110-1455348 *z^106+23040*z^108-44618134034702419240*z^30-62454195494703489886148*z^42+ 135086069920870754656136*z^44-259021757884751442406816*z^46-\ 1066168266170694452325864*z^58+1130736529271809719126280*z^56-\ 1066168266170694452325864*z^54+893689665772356314166296*z^52+ 893689665772356314166296*z^60-62454195494703489886148*z^70+ 135086069920870754656136*z^68-832340068537727669920*z^78+205873204925318924945* z^32-9254872431034990533392*z^38+25567393162208471517218*z^40-\ 665805554808799254082416*z^62+2957137131574594797780*z^76-\ 9254872431034990533392*z^74+25567393162208471517218*z^72+63687358*z^104)/(-1-\ 24440930894851865972*z^28+3813776059281844728*z^26+253*z^2-514327655655270616*z ^24+59580100686285668*z^22-28988*z^4+2018652*z^6+91983641624*z^102-96586370*z^8 +3399807542*z^10-91983641624*z^12+1972243656780*z^14+491225409815931*z^18-\ 34275764800151*z^16+3404105426749618375289435*z^50-2134682756403119873619743*z^ 48-5884894905198560*z^20-10510500043992347579876*z^36+2812167657308959615949*z^ 34+2134682756403119873619743*z^66-2812167657308959615949*z^80-1972243656780*z^ 100+514327655655270616*z^90-3813776059281844728*z^88-136072836254468022416*z^84 +5884894905198560*z^94+24440930894851865972*z^86-491225409815931*z^96+ 34275764800151*z^98-59580100686285668*z^92+661014483702855456937*z^82-\ 3404105426749618375289435*z^64-253*z^112+z^114+28988*z^110+96586370*z^106-\ 2018652*z^108+136072836254468022416*z^30+258558973064932716557718*z^42-\ 588885241018348902939388*z^44+1189779867277227931231316*z^46+ 6844134126112648009349744*z^58-6844134126112648009349744*z^56+ 6092878016024994621630696*z^54-4827951531186401511966416*z^52-\ 6092878016024994621630696*z^60+588885241018348902939388*z^70-\ 1189779867277227931231316*z^68+10510500043992347579876*z^78-\ 661014483702855456937*z^32+34602838072675334326480*z^38-\ 100573334361229991665522*z^40+4827951531186401511966416*z^62-\ 34602838072675334326480*z^76+100573334361229991665522*z^74-\ 258558973064932716557718*z^72-3399807542*z^104) The first , 40, terms are: [0, 29, 0, 1389, 0, 74069, 0, 4117021, 0, 232810093, 0, 13259783689, 0, 757487157389, 0, 43327420600109, 0, 2479605522517641, 0, 141938821039691141, 0, 8125725009751814789, 0, 465201699591247596765, 0, 26633508682111889876309, 0, 1524821264098238140450541, 0, 87299352971262111482495233, 0, 4998086777365462278768568737, 0, 286152053638629771035134689645, 0, 16382873681175991718739391795109, 0, 937957938622101570637853160538221, 0, 53700295441165102624546370114737749] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 9615477593711963889 z - 1580254692164934518 z - 227 z 24 22 4 6 + 224575059243029471 z - 27433108181146042 z + 23698 z - 1520106 z 102 8 10 12 - 2219619032 z + 67526515 z - 2219619032 z + 56321366945 z 14 18 16 - 1136234470290 z - 252249534950958 z + 18625434667975 z 50 48 - 755610883357695801599792 z + 500339150338338231479654 z 20 36 + 2859912301922255 z + 3369207303501197331376 z 34 66 - 948688289823865696782 z - 294219139199454949472924 z 80 100 90 + 234679444931546124267 z + 56321366945 z - 27433108181146042 z 88 84 94 + 224575059243029471 z + 9615477593711963889 z - 252249534950958 z 86 96 98 - 1580254692164934518 z + 18625434667975 z - 1136234470290 z 92 82 + 2859912301922255 z - 50848051856922762180 z 64 112 110 106 + 500339150338338231479654 z + z - 227 z - 1520106 z 108 30 42 + 23698 z - 50848051856922762180 z - 71026398142964053176256 z 44 46 + 153530354128536715155866 z - 294219139199454949472924 z 58 56 - 1209420557125766976116848 z + 1282590035457744162585274 z 54 52 - 1209420557125766976116848 z + 1013942853534073087749586 z 60 70 + 1013942853534073087749586 z - 71026398142964053176256 z 68 78 + 153530354128536715155866 z - 948688289823865696782 z 32 38 + 234679444931546124267 z - 10538740532618011595081 z 40 62 + 29095757231869306257429 z - 755610883357695801599792 z 76 74 + 3369207303501197331376 z - 10538740532618011595081 z 72 104 / + 29095757231869306257429 z + 67526515 z ) / (-1 / 28 26 2 - 27969891300135003512 z + 4352404883617912946 z + 262 z 24 22 4 6 - 585199211000139448 z + 67564553936551931 z - 30701 z + 2171620 z 102 8 10 12 + 101796415882 z - 105122610 z + 3734347972 z - 101796415882 z 14 18 16 + 2196502658732 z + 552679974131741 z - 38381436762715 z 50 48 + 3974190328249387105214181 z - 2489682356256078742278109 z 20 36 - 6648821411357861 z - 12139900375990155181600 z 34 66 + 3241320850270087232459 z + 2489682356256078742278109 z 80 100 90 - 3241320850270087232459 z - 2196502658732 z + 585199211000139448 z 88 84 - 4352404883617912946 z - 156117707715558596450 z 94 86 96 + 6648821411357861 z + 27969891300135003512 z - 552679974131741 z 98 92 82 + 38381436762715 z - 67564553936551931 z + 760192953707018393905 z 64 112 114 110 - 3974190328249387105214181 z - 262 z + z + 30701 z 106 108 30 + 105122610 z - 2171620 z + 156117707715558596450 z 42 44 + 300300350544679512550532 z - 685026289815984725536396 z 46 58 + 1385961116487492214860820 z + 8002811684243964292512297 z 56 54 - 8002811684243964292512297 z + 7122479780681687972567988 z 52 60 - 5640839361675665291875260 z - 7122479780681687972567988 z 70 68 + 685026289815984725536396 z - 1385961116487492214860820 z 78 32 + 12139900375990155181600 z - 760192953707018393905 z 38 40 + 40046129110657586548891 z - 116609624515928796024123 z 62 76 + 5640839361675665291875260 z - 40046129110657586548891 z 74 72 + 116609624515928796024123 z - 300300350544679512550532 z 104 - 3734347972 z ) And in Maple-input format, it is: -(1+9615477593711963889*z^28-1580254692164934518*z^26-227*z^2+ 224575059243029471*z^24-27433108181146042*z^22+23698*z^4-1520106*z^6-2219619032 *z^102+67526515*z^8-2219619032*z^10+56321366945*z^12-1136234470290*z^14-\ 252249534950958*z^18+18625434667975*z^16-755610883357695801599792*z^50+ 500339150338338231479654*z^48+2859912301922255*z^20+3369207303501197331376*z^36 -948688289823865696782*z^34-294219139199454949472924*z^66+234679444931546124267 *z^80+56321366945*z^100-27433108181146042*z^90+224575059243029471*z^88+ 9615477593711963889*z^84-252249534950958*z^94-1580254692164934518*z^86+ 18625434667975*z^96-1136234470290*z^98+2859912301922255*z^92-\ 50848051856922762180*z^82+500339150338338231479654*z^64+z^112-227*z^110-1520106 *z^106+23698*z^108-50848051856922762180*z^30-71026398142964053176256*z^42+ 153530354128536715155866*z^44-294219139199454949472924*z^46-\ 1209420557125766976116848*z^58+1282590035457744162585274*z^56-\ 1209420557125766976116848*z^54+1013942853534073087749586*z^52+ 1013942853534073087749586*z^60-71026398142964053176256*z^70+ 153530354128536715155866*z^68-948688289823865696782*z^78+234679444931546124267* z^32-10538740532618011595081*z^38+29095757231869306257429*z^40-\ 755610883357695801599792*z^62+3369207303501197331376*z^76-\ 10538740532618011595081*z^74+29095757231869306257429*z^72+67526515*z^104)/(-1-\ 27969891300135003512*z^28+4352404883617912946*z^26+262*z^2-585199211000139448*z ^24+67564553936551931*z^22-30701*z^4+2171620*z^6+101796415882*z^102-105122610*z ^8+3734347972*z^10-101796415882*z^12+2196502658732*z^14+552679974131741*z^18-\ 38381436762715*z^16+3974190328249387105214181*z^50-2489682356256078742278109*z^ 48-6648821411357861*z^20-12139900375990155181600*z^36+3241320850270087232459*z^ 34+2489682356256078742278109*z^66-3241320850270087232459*z^80-2196502658732*z^ 100+585199211000139448*z^90-4352404883617912946*z^88-156117707715558596450*z^84 +6648821411357861*z^94+27969891300135003512*z^86-552679974131741*z^96+ 38381436762715*z^98-67564553936551931*z^92+760192953707018393905*z^82-\ 3974190328249387105214181*z^64-262*z^112+z^114+30701*z^110+105122610*z^106-\ 2171620*z^108+156117707715558596450*z^30+300300350544679512550532*z^42-\ 685026289815984725536396*z^44+1385961116487492214860820*z^46+ 8002811684243964292512297*z^58-8002811684243964292512297*z^56+ 7122479780681687972567988*z^54-5640839361675665291875260*z^52-\ 7122479780681687972567988*z^60+685026289815984725536396*z^70-\ 1385961116487492214860820*z^68+12139900375990155181600*z^78-\ 760192953707018393905*z^32+40046129110657586548891*z^38-\ 116609624515928796024123*z^40+5640839361675665291875260*z^62-\ 40046129110657586548891*z^76+116609624515928796024123*z^74-\ 300300350544679512550532*z^72-3734347972*z^104) The first , 40, terms are: [0, 35, 0, 2167, 0, 144733, 0, 9801584, 0, 665905305, 0, 45280271199, 0, 3079803106179, 0, 209496136744321, 0, 14250917617833097, 0, 969425993109667835, 0, 65945988665591793915, 0, 4486036885539043474869, 0, 305167013437154355253376, 0, 20759287913938564737050473, 0, 1412171249437765552928077539, 0, 96064361093589922900422540723, 0, 6534874333564468575569615047681, 0, 444541372300852185744907272237089, 0, 30240372239927870297896308823669283, 0, 2057131622801578858564464063888118059] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 8346906060585502401 z - 1373680931251341006 z - 224 z 24 22 4 6 + 195671244519721321 z - 23984948943842176 z + 23007 z - 1450578 z 102 8 10 12 - 2050504206 z + 63362398 z - 2050504206 z + 51312476138 z 14 18 16 - 1022877403632 z - 223014603979702 z + 16600615492791 z 50 48 - 659465413602865749836298 z + 436453469364867818055547 z 20 36 + 2512457053971427 z + 2925224205842455880187 z 34 66 - 823177675804520153492 z - 256488939632269009952864 z 80 100 90 + 203566264140170492179 z + 51312476138 z - 23984948943842176 z 88 84 94 + 195671244519721321 z + 8346906060585502401 z - 223014603979702 z 86 96 98 - 1373680931251341006 z + 16600615492791 z - 1022877403632 z 92 82 + 2512457053971427 z - 44110767148665969734 z 64 112 110 106 + 436453469364867818055547 z + z - 224 z - 1450578 z 108 30 42 + 23007 z - 44110767148665969734 z - 61820957962034625044834 z 44 46 + 133742302224403224239070 z - 256488939632269009952864 z 58 56 - 1056156191194200106248942 z + 1120136923704071083788316 z 54 52 - 1056156191194200106248942 z + 885252870725217707534769 z 60 70 + 885252870725217707534769 z - 61820957962034625044834 z 68 78 + 133742302224403224239070 z - 823177675804520153492 z 32 38 + 203566264140170492179 z - 9157060150756059390744 z 40 62 + 25302809132077498031950 z - 659465413602865749836298 z 76 74 + 2925224205842455880187 z - 9157060150756059390744 z 72 104 / + 25302809132077498031950 z + 63362398 z ) / (-1 / 28 26 2 - 24164225759311450200 z + 3772534924123580295 z + 252 z 24 22 4 6 - 509009558467461707 z + 58989668417941227 z - 28811 z + 2003967 z 102 8 10 12 + 91200370622 z - 95822206 z + 3371674006 z - 91200370622 z 14 18 16 + 1955043007665 z + 486701008215632 z - 33969199445010 z 50 48 + 3348419457235395148442508 z - 2100333403202299438882116 z 20 36 - 5828800879647497 z - 10368824794097528439915 z 34 66 + 2775774964796388973646 z + 2100333403202299438882116 z 80 100 90 - 2775774964796388973646 z - 1955043007665 z + 509009558467461707 z 88 84 - 3772534924123580295 z - 134460073593120706136 z 94 86 96 + 5828800879647497 z + 24164225759311450200 z - 486701008215632 z 98 92 82 + 33969199445010 z - 58989668417941227 z + 652820854109351554460 z 64 112 114 110 - 3348419457235395148442508 z - 252 z + z + 28811 z 106 108 30 + 95822206 z - 2003967 z + 134460073593120706136 z 42 44 + 254690185699250925186921 z - 579823745245644790303141 z 46 58 + 1171022226522314691453941 z + 6729345747607578515819551 z 56 54 - 6729345747607578515819551 z + 5991115633038116404077546 z 52 60 - 4747984912100619093495947 z - 5991115633038116404077546 z 70 68 + 579823745245644790303141 z - 1171022226522314691453941 z 78 32 + 10368824794097528439915 z - 652820854109351554460 z 38 40 + 34118354345818083380073 z - 99115184938374330143490 z 62 76 + 4747984912100619093495947 z - 34118354345818083380073 z 74 72 + 99115184938374330143490 z - 254690185699250925186921 z 104 - 3371674006 z ) And in Maple-input format, it is: -(1+8346906060585502401*z^28-1373680931251341006*z^26-224*z^2+ 195671244519721321*z^24-23984948943842176*z^22+23007*z^4-1450578*z^6-2050504206 *z^102+63362398*z^8-2050504206*z^10+51312476138*z^12-1022877403632*z^14-\ 223014603979702*z^18+16600615492791*z^16-659465413602865749836298*z^50+ 436453469364867818055547*z^48+2512457053971427*z^20+2925224205842455880187*z^36 -823177675804520153492*z^34-256488939632269009952864*z^66+203566264140170492179 *z^80+51312476138*z^100-23984948943842176*z^90+195671244519721321*z^88+ 8346906060585502401*z^84-223014603979702*z^94-1373680931251341006*z^86+ 16600615492791*z^96-1022877403632*z^98+2512457053971427*z^92-\ 44110767148665969734*z^82+436453469364867818055547*z^64+z^112-224*z^110-1450578 *z^106+23007*z^108-44110767148665969734*z^30-61820957962034625044834*z^42+ 133742302224403224239070*z^44-256488939632269009952864*z^46-\ 1056156191194200106248942*z^58+1120136923704071083788316*z^56-\ 1056156191194200106248942*z^54+885252870725217707534769*z^52+ 885252870725217707534769*z^60-61820957962034625044834*z^70+ 133742302224403224239070*z^68-823177675804520153492*z^78+203566264140170492179* z^32-9157060150756059390744*z^38+25302809132077498031950*z^40-\ 659465413602865749836298*z^62+2925224205842455880187*z^76-\ 9157060150756059390744*z^74+25302809132077498031950*z^72+63362398*z^104)/(-1-\ 24164225759311450200*z^28+3772534924123580295*z^26+252*z^2-509009558467461707*z ^24+58989668417941227*z^22-28811*z^4+2003967*z^6+91200370622*z^102-95822206*z^8 +3371674006*z^10-91200370622*z^12+1955043007665*z^14+486701008215632*z^18-\ 33969199445010*z^16+3348419457235395148442508*z^50-2100333403202299438882116*z^ 48-5828800879647497*z^20-10368824794097528439915*z^36+2775774964796388973646*z^ 34+2100333403202299438882116*z^66-2775774964796388973646*z^80-1955043007665*z^ 100+509009558467461707*z^90-3772534924123580295*z^88-134460073593120706136*z^84 +5828800879647497*z^94+24164225759311450200*z^86-486701008215632*z^96+ 33969199445010*z^98-58989668417941227*z^92+652820854109351554460*z^82-\ 3348419457235395148442508*z^64-252*z^112+z^114+28811*z^110+95822206*z^106-\ 2003967*z^108+134460073593120706136*z^30+254690185699250925186921*z^42-\ 579823745245644790303141*z^44+1171022226522314691453941*z^46+ 6729345747607578515819551*z^58-6729345747607578515819551*z^56+ 5991115633038116404077546*z^54-4747984912100619093495947*z^52-\ 5991115633038116404077546*z^60+579823745245644790303141*z^70-\ 1171022226522314691453941*z^68+10368824794097528439915*z^78-\ 652820854109351554460*z^32+34118354345818083380073*z^38-99115184938374330143490 *z^40+4747984912100619093495947*z^62-34118354345818083380073*z^76+ 99115184938374330143490*z^74-254690185699250925186921*z^72-3371674006*z^104) The first , 40, terms are: [0, 28, 0, 1252, 0, 62185, 0, 3250516, 0, 174632713, 0, 9523090867, 0, 523589803232, 0, 28918743966269, 0, 1601293616157403, 0, 88793682495992596, 0, 4927677870394960095, 0, 273589726320458030620, 0, 15193885730215290286108, 0, 843919720078221960538651, 0, 46878018083478746039003971, 0, 2604100054969778231210656076, 0, 144663049924818811618724643068, 0, 8036447417417534193940268595007, 0, 446451523648287612757228284442324, 0, 24801995145801411257959996629038803] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 141349830799595491 z - 37034497341810046 z - 198 z 24 22 4 6 + 8175457678641953 z - 1512088228710824 z + 17587 z - 935644 z 8 10 12 14 + 33594504 z - 869855492 z + 16948272825 z - 255990001322 z 18 16 50 - 29550756609688 z + 3063894829395 z - 10014315013537764882 z 48 20 36 + 14676074138934399371 z + 232712237539248 z + 5856562753551283889 z 34 66 80 88 - 2931298150895630404 z - 1512088228710824 z + 33594504 z + z 84 86 82 64 + 17587 z - 198 z - 935644 z + 8175457678641953 z 30 42 - 456635192861186012 z - 18451667807023845312 z 44 46 + 19913603663953588640 z - 18451667807023845312 z 58 56 - 456635192861186012 z + 1253292751451506680 z 54 52 - 2931298150895630404 z + 5856562753551283889 z 60 70 68 + 141349830799595491 z - 29550756609688 z + 232712237539248 z 78 32 38 - 869855492 z + 1253292751451506680 z - 10014315013537764882 z 40 62 76 + 14676074138934399371 z - 37034497341810046 z + 16948272825 z 74 72 / 2 - 255990001322 z + 3063894829395 z ) / ((-1 + z ) (1 / 28 26 2 + 396250431045968407 z - 99988693070787509 z - 233 z 24 22 4 6 + 21183847723833577 z - 3747563697474920 z + 23495 z - 1387156 z 8 10 12 14 + 54432414 z - 1523730876 z + 31842049813 z - 512701569431 z 18 16 50 - 66329287801608 z + 6509658654107 z - 31920627328772858763 z 48 20 + 47358302589488913467 z + 549815802763876 z 36 34 + 18352896651363423693 z - 8989949930201312540 z 66 80 88 84 86 - 3747563697474920 z + 54432414 z + z + 23495 z - 233 z 82 64 30 - 1387156 z + 21183847723833577 z - 1324265547998658772 z 42 44 - 59985695572984512976 z + 64899495442464020344 z 46 58 - 59985695572984512976 z - 1324265547998658772 z 56 54 + 3745403882044603266 z - 8989949930201312540 z 52 60 70 + 18352896651363423693 z + 396250431045968407 z - 66329287801608 z 68 78 32 + 549815802763876 z - 1523730876 z + 3745403882044603266 z 38 40 - 31920627328772858763 z + 47358302589488913467 z 62 76 74 - 99988693070787509 z + 31842049813 z - 512701569431 z 72 + 6509658654107 z )) And in Maple-input format, it is: -(1+141349830799595491*z^28-37034497341810046*z^26-198*z^2+8175457678641953*z^ 24-1512088228710824*z^22+17587*z^4-935644*z^6+33594504*z^8-869855492*z^10+ 16948272825*z^12-255990001322*z^14-29550756609688*z^18+3063894829395*z^16-\ 10014315013537764882*z^50+14676074138934399371*z^48+232712237539248*z^20+ 5856562753551283889*z^36-2931298150895630404*z^34-1512088228710824*z^66+ 33594504*z^80+z^88+17587*z^84-198*z^86-935644*z^82+8175457678641953*z^64-\ 456635192861186012*z^30-18451667807023845312*z^42+19913603663953588640*z^44-\ 18451667807023845312*z^46-456635192861186012*z^58+1253292751451506680*z^56-\ 2931298150895630404*z^54+5856562753551283889*z^52+141349830799595491*z^60-\ 29550756609688*z^70+232712237539248*z^68-869855492*z^78+1253292751451506680*z^ 32-10014315013537764882*z^38+14676074138934399371*z^40-37034497341810046*z^62+ 16948272825*z^76-255990001322*z^74+3063894829395*z^72)/(-1+z^2)/(1+ 396250431045968407*z^28-99988693070787509*z^26-233*z^2+21183847723833577*z^24-\ 3747563697474920*z^22+23495*z^4-1387156*z^6+54432414*z^8-1523730876*z^10+ 31842049813*z^12-512701569431*z^14-66329287801608*z^18+6509658654107*z^16-\ 31920627328772858763*z^50+47358302589488913467*z^48+549815802763876*z^20+ 18352896651363423693*z^36-8989949930201312540*z^34-3747563697474920*z^66+ 54432414*z^80+z^88+23495*z^84-233*z^86-1387156*z^82+21183847723833577*z^64-\ 1324265547998658772*z^30-59985695572984512976*z^42+64899495442464020344*z^44-\ 59985695572984512976*z^46-1324265547998658772*z^58+3745403882044603266*z^56-\ 8989949930201312540*z^54+18352896651363423693*z^52+396250431045968407*z^60-\ 66329287801608*z^70+549815802763876*z^68-1523730876*z^78+3745403882044603266*z^ 32-31920627328772858763*z^38+47358302589488913467*z^40-99988693070787509*z^62+ 31842049813*z^76-512701569431*z^74+6509658654107*z^72) The first , 40, terms are: [0, 36, 0, 2283, 0, 155021, 0, 10662260, 0, 735950063, 0, 50859030399, 0, 3516244492980, 0, 243143540907405, 0, 16814132212928811, 0, 1162778763070613988, 0, 80412578990568132177, 0, 5560995772519243873201, 0, 384575663113371911299940, 0, 26595691705199596693187243, 0, 1839250510088320677144854669, 0, 127195140973964346325888745780, 0, 8796302841943594469425530922335, 0, 608316829078672270764602969689871, 0, 42068738806629251663960874687599860, 0, 2909304336735667967011701719451356813] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 9592703347221938551 z - 1542225999703774354 z - 221 z 24 22 4 6 + 214771738924859453 z - 25766436091165146 z + 22524 z - 1416146 z 102 8 10 12 - 2015284984 z + 61949565 z - 2015284984 z + 50863664231 z 14 18 16 - 1025638772306 z - 230536361817754 z + 16881604296125 z 50 48 - 939245382339732672414460 z + 615168241196332308660386 z 20 36 + 2645389463673097 z + 3696167819975164516490 z 34 66 - 1016247963532164547522 z - 356812276622172490383840 z 80 100 90 + 245407785341073485389 z + 50863664231 z - 25766436091165146 z 88 84 94 + 214771738924859453 z + 9592703347221938551 z - 230536361817754 z 86 96 98 - 1542225999703774354 z + 16881604296125 z - 1025638772306 z 92 82 + 2645389463673097 z - 51917618071048978344 z 64 112 110 106 + 615168241196332308660386 z + z - 221 z - 1416146 z 108 30 42 + 22524 z - 51917618071048978344 z - 83193305618937854848332 z 44 46 + 183189508210963971388694 z - 356812276622172490383840 z 58 56 - 1522699835367417700116140 z + 1617442611727225312806606 z 54 52 - 1522699835367417700116140 z + 1270430832993389081402342 z 60 70 + 1270430832993389081402342 z - 83193305618937854848332 z 68 78 + 183189508210963971388694 z - 1016247963532164547522 z 32 38 + 245407785341073485389 z - 11831755854324549002707 z 40 62 + 33391015601168319731545 z - 939245382339732672414460 z 76 74 + 3696167819975164516490 z - 11831755854324549002707 z 72 104 / 2 + 33391015601168319731545 z + 61949565 z ) / ((-1 + z ) (1 / 28 26 2 + 23594685185725586646 z - 3658095138393055122 z - 251 z 24 22 4 6 + 490088073355565992 z - 56421954071397812 z + 28396 z - 1951324 z 102 8 10 12 - 3218111978 z + 92272578 z - 3218111978 z + 86520327900 z 14 18 16 - 1848877549052 z - 461031451734962 z + 32111833682939 z 50 48 - 2948725933553464739766966 z + 1912654077600287217040691 z 20 36 + 5543895596591251 z + 10272694386109482056142 z 34 66 - 2748883254332786584266 z - 1095739249659225110139190 z 80 100 90 + 644577269377646262553 z + 86520327900 z - 56421954071397812 z 88 84 94 + 490088073355565992 z + 23594685185725586646 z - 461031451734962 z 86 96 98 - 3658095138393055122 z + 32111833682939 z - 1848877549052 z 92 82 + 5543895596591251 z - 132111937299653061028 z 64 112 110 106 + 1912654077600287217040691 z + z - 251 z - 1951324 z 108 30 42 + 28396 z - 132111937299653061028 z - 247302556176987526117150 z 44 46 + 554188633707421355159434 z - 1095739249659225110139190 z 58 56 - 4834158641597738783952670 z + 5142150020502936271323455 z 54 52 - 4834158641597738783952670 z + 4016369744839240449524802 z 60 70 + 4016369744839240449524802 z - 247302556176987526117150 z 68 78 + 554188633707421355159434 z - 2748883254332786584266 z 32 38 + 644577269377646262553 z - 33709433685180884235751 z 40 62 + 97291696440999467263338 z - 2948725933553464739766966 z 76 74 + 10272694386109482056142 z - 33709433685180884235751 z 72 104 + 97291696440999467263338 z + 92272578 z )) And in Maple-input format, it is: -(1+9592703347221938551*z^28-1542225999703774354*z^26-221*z^2+ 214771738924859453*z^24-25766436091165146*z^22+22524*z^4-1416146*z^6-2015284984 *z^102+61949565*z^8-2015284984*z^10+50863664231*z^12-1025638772306*z^14-\ 230536361817754*z^18+16881604296125*z^16-939245382339732672414460*z^50+ 615168241196332308660386*z^48+2645389463673097*z^20+3696167819975164516490*z^36 -1016247963532164547522*z^34-356812276622172490383840*z^66+ 245407785341073485389*z^80+50863664231*z^100-25766436091165146*z^90+ 214771738924859453*z^88+9592703347221938551*z^84-230536361817754*z^94-\ 1542225999703774354*z^86+16881604296125*z^96-1025638772306*z^98+ 2645389463673097*z^92-51917618071048978344*z^82+615168241196332308660386*z^64+z ^112-221*z^110-1416146*z^106+22524*z^108-51917618071048978344*z^30-\ 83193305618937854848332*z^42+183189508210963971388694*z^44-\ 356812276622172490383840*z^46-1522699835367417700116140*z^58+ 1617442611727225312806606*z^56-1522699835367417700116140*z^54+ 1270430832993389081402342*z^52+1270430832993389081402342*z^60-\ 83193305618937854848332*z^70+183189508210963971388694*z^68-\ 1016247963532164547522*z^78+245407785341073485389*z^32-11831755854324549002707* z^38+33391015601168319731545*z^40-939245382339732672414460*z^62+ 3696167819975164516490*z^76-11831755854324549002707*z^74+ 33391015601168319731545*z^72+61949565*z^104)/(-1+z^2)/(1+23594685185725586646*z ^28-3658095138393055122*z^26-251*z^2+490088073355565992*z^24-56421954071397812* z^22+28396*z^4-1951324*z^6-3218111978*z^102+92272578*z^8-3218111978*z^10+ 86520327900*z^12-1848877549052*z^14-461031451734962*z^18+32111833682939*z^16-\ 2948725933553464739766966*z^50+1912654077600287217040691*z^48+5543895596591251* z^20+10272694386109482056142*z^36-2748883254332786584266*z^34-\ 1095739249659225110139190*z^66+644577269377646262553*z^80+86520327900*z^100-\ 56421954071397812*z^90+490088073355565992*z^88+23594685185725586646*z^84-\ 461031451734962*z^94-3658095138393055122*z^86+32111833682939*z^96-1848877549052 *z^98+5543895596591251*z^92-132111937299653061028*z^82+ 1912654077600287217040691*z^64+z^112-251*z^110-1951324*z^106+28396*z^108-\ 132111937299653061028*z^30-247302556176987526117150*z^42+ 554188633707421355159434*z^44-1095739249659225110139190*z^46-\ 4834158641597738783952670*z^58+5142150020502936271323455*z^56-\ 4834158641597738783952670*z^54+4016369744839240449524802*z^52+ 4016369744839240449524802*z^60-247302556176987526117150*z^70+ 554188633707421355159434*z^68-2748883254332786584266*z^78+644577269377646262553 *z^32-33709433685180884235751*z^38+97291696440999467263338*z^40-\ 2948725933553464739766966*z^62+10272694386109482056142*z^76-\ 33709433685180884235751*z^74+97291696440999467263338*z^72+92272578*z^104) The first , 40, terms are: [0, 31, 0, 1689, 0, 101145, 0, 6200740, 0, 382991355, 0, 23722977191, 0, 1471196313969, 0, 91285918975055, 0, 5665554599534711, 0, 351665341143218701, 0, 21829264767426412591, 0, 1355061529855881037871, 0, 84116990217218206418436, 0, 5221684871940492097635517, 0, 324144475040527528179461393, 0, 20121811043030642320447779171, 0, 1249095750038351085911625067361, 0, 77539768081897357606526908498001, 0, 4813415058840767326983755538357499, 0, 298801070456728855510871604812898913 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 87408102182918039 z - 23156323603561089 z - 189 z 24 22 4 6 + 5184664056090961 z - 975980586533184 z + 15975 z - 807864 z 8 10 12 14 + 27599870 z - 681768976 z + 12719541637 z - 184751252795 z 18 16 50 - 19993212898112 z + 2136122019507 z - 6047973597220306375 z 48 20 36 + 8853792266363061883 z + 153463876724332 z + 3543490741692898941 z 34 66 80 88 - 1778735125733975224 z - 975980586533184 z + 27599870 z + z 84 86 82 64 + 15975 z - 189 z - 807864 z + 5184664056090961 z 30 42 - 280012382681262720 z - 11125402813851012864 z 44 46 + 12004867811497941096 z - 11125402813851012864 z 58 56 - 280012382681262720 z + 763811109546676642 z 54 52 - 1778735125733975224 z + 3543490741692898941 z 60 70 68 + 87408102182918039 z - 19993212898112 z + 153463876724332 z 78 32 38 - 681768976 z + 763811109546676642 z - 6047973597220306375 z 40 62 76 + 8853792266363061883 z - 23156323603561089 z + 12719541637 z 74 72 / 2 - 184751252795 z + 2136122019507 z ) / ((-1 + z ) (1 / 28 26 2 + 245321302502464052 z - 62794881705191282 z - 222 z 24 22 4 6 + 13532581282407443 z - 2442347416863532 z + 21314 z - 1198484 z 8 10 12 14 + 44869711 z - 1201768728 z + 24110630008 z - 374064807630 z 18 16 50 - 45423423356364 z + 4593118280773 z - 19020110886777149922 z 48 20 + 28142203414471355463 z + 366705440140386 z 36 34 + 10980087737529810354 z - 5408700440648544452 z 66 80 88 84 86 - 2442347416863532 z + 44869711 z + z + 21314 z - 222 z 82 64 30 - 1198484 z + 13532581282407443 z - 810277588375334672 z 42 44 - 35590529636437735816 z + 38486517109659111388 z 46 58 - 35590529636437735816 z - 810277588375334672 z 56 54 + 2270126666936033793 z - 5408700440648544452 z 52 60 70 + 10980087737529810354 z + 245321302502464052 z - 45423423356364 z 68 78 32 + 366705440140386 z - 1201768728 z + 2270126666936033793 z 38 40 - 19020110886777149922 z + 28142203414471355463 z 62 76 74 - 62794881705191282 z + 24110630008 z - 374064807630 z 72 + 4593118280773 z )) And in Maple-input format, it is: -(1+87408102182918039*z^28-23156323603561089*z^26-189*z^2+5184664056090961*z^24 -975980586533184*z^22+15975*z^4-807864*z^6+27599870*z^8-681768976*z^10+ 12719541637*z^12-184751252795*z^14-19993212898112*z^18+2136122019507*z^16-\ 6047973597220306375*z^50+8853792266363061883*z^48+153463876724332*z^20+ 3543490741692898941*z^36-1778735125733975224*z^34-975980586533184*z^66+27599870 *z^80+z^88+15975*z^84-189*z^86-807864*z^82+5184664056090961*z^64-\ 280012382681262720*z^30-11125402813851012864*z^42+12004867811497941096*z^44-\ 11125402813851012864*z^46-280012382681262720*z^58+763811109546676642*z^56-\ 1778735125733975224*z^54+3543490741692898941*z^52+87408102182918039*z^60-\ 19993212898112*z^70+153463876724332*z^68-681768976*z^78+763811109546676642*z^32 -6047973597220306375*z^38+8853792266363061883*z^40-23156323603561089*z^62+ 12719541637*z^76-184751252795*z^74+2136122019507*z^72)/(-1+z^2)/(1+ 245321302502464052*z^28-62794881705191282*z^26-222*z^2+13532581282407443*z^24-\ 2442347416863532*z^22+21314*z^4-1198484*z^6+44869711*z^8-1201768728*z^10+ 24110630008*z^12-374064807630*z^14-45423423356364*z^18+4593118280773*z^16-\ 19020110886777149922*z^50+28142203414471355463*z^48+366705440140386*z^20+ 10980087737529810354*z^36-5408700440648544452*z^34-2442347416863532*z^66+ 44869711*z^80+z^88+21314*z^84-222*z^86-1198484*z^82+13532581282407443*z^64-\ 810277588375334672*z^30-35590529636437735816*z^42+38486517109659111388*z^44-\ 35590529636437735816*z^46-810277588375334672*z^58+2270126666936033793*z^56-\ 5408700440648544452*z^54+10980087737529810354*z^52+245321302502464052*z^60-\ 45423423356364*z^70+366705440140386*z^68-1201768728*z^78+2270126666936033793*z^ 32-19020110886777149922*z^38+28142203414471355463*z^40-62794881705191282*z^62+ 24110630008*z^76-374064807630*z^74+4593118280773*z^72) The first , 40, terms are: [0, 34, 0, 2021, 0, 130393, 0, 8558190, 0, 564095313, 0, 37226223305, 0, 2457642944270, 0, 162275872257321, 0, 10715601599788101, 0, 707606098579702530, 0, 46727488649767424041, 0, 3085717472592254091529, 0, 203770485066866505044930, 0, 13456344392673294217466677, 0, 888614210510090863964497113, 0, 58681281206185638757020592334, 0, 3875127595602302605658764251497, 0, 255901286883881570349013049459729, 0, 16898920171025979629102115315606702, 0, 1115951832618564580043438104560832009] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {4, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 196630077542584 z + 104637839669432 z + 154 z 24 22 4 6 - 44941464944682 z + 15520961531596 z - 10274 z + 396941 z 8 10 12 14 - 10019760 z + 176751748 z - 2274482897 z + 21987347286 z 18 16 50 + 942047363333 z - 163058424670 z + 163058424670 z 48 20 36 - 942047363333 z - 4288886607012 z - 299003741218706 z 34 66 64 30 + 368560118949308 z + z - 154 z + 299003741218706 z 42 44 46 + 44941464944682 z - 15520961531596 z + 4288886607012 z 58 56 54 52 + 10019760 z - 176751748 z + 2274482897 z - 21987347286 z 60 32 38 - 396941 z - 368560118949308 z + 196630077542584 z 40 62 / 28 - 104637839669432 z + 10274 z ) / (1 + 917215606740494 z / 26 2 24 22 - 445084433598620 z - 182 z + 174658105351994 z - 55171423221792 z 4 6 8 10 12 + 14091 z - 623414 z + 17824857 z - 352884224 z + 5056639021 z 14 18 16 - 54086471918 z - 2798686299854 z + 441599675671 z 50 48 20 - 2798686299854 z + 13944970592661 z + 13944970592661 z 36 34 66 64 + 2086066840397898 z - 2310921312063296 z - 182 z + 14091 z 30 42 44 - 1533762558307836 z - 445084433598620 z + 174658105351994 z 46 58 56 54 - 55171423221792 z - 352884224 z + 5056639021 z - 54086471918 z 52 60 68 32 + 441599675671 z + 17824857 z + z + 2086066840397898 z 38 40 62 - 1533762558307836 z + 917215606740494 z - 623414 z ) And in Maple-input format, it is: -(-1-196630077542584*z^28+104637839669432*z^26+154*z^2-44941464944682*z^24+ 15520961531596*z^22-10274*z^4+396941*z^6-10019760*z^8+176751748*z^10-2274482897 *z^12+21987347286*z^14+942047363333*z^18-163058424670*z^16+163058424670*z^50-\ 942047363333*z^48-4288886607012*z^20-299003741218706*z^36+368560118949308*z^34+ z^66-154*z^64+299003741218706*z^30+44941464944682*z^42-15520961531596*z^44+ 4288886607012*z^46+10019760*z^58-176751748*z^56+2274482897*z^54-21987347286*z^ 52-396941*z^60-368560118949308*z^32+196630077542584*z^38-104637839669432*z^40+ 10274*z^62)/(1+917215606740494*z^28-445084433598620*z^26-182*z^2+ 174658105351994*z^24-55171423221792*z^22+14091*z^4-623414*z^6+17824857*z^8-\ 352884224*z^10+5056639021*z^12-54086471918*z^14-2798686299854*z^18+441599675671 *z^16-2798686299854*z^50+13944970592661*z^48+13944970592661*z^20+ 2086066840397898*z^36-2310921312063296*z^34-182*z^66+14091*z^64-\ 1533762558307836*z^30-445084433598620*z^42+174658105351994*z^44-55171423221792* z^46-352884224*z^58+5056639021*z^56-54086471918*z^54+441599675671*z^52+17824857 *z^60+z^68+2086066840397898*z^32-1533762558307836*z^38+917215606740494*z^40-\ 623414*z^62) The first , 40, terms are: [0, 28, 0, 1279, 0, 64703, 0, 3404052, 0, 182190477, 0, 9829536169, 0, 532393676948, 0, 28892115555387, 0, 1569481970681083, 0, 85301054568465692, 0, 4637315468315124989, 0, 252137848503137471333, 0, 13710085523863830347868, 0, 745518214577696874824291, 0, 40540086127818612411999395, 0, 2204526914783219486971099732, 0, 119880459889025869590371319889, 0, 6519023875768657536135426539557, 0, 354500911751960125745810980603348, 0, 19277577664061254771782731431262615] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 349431203158424 z - 172804058242334 z - 161 z 24 22 4 6 + 69538046991982 z - 22676717367072 z + 11116 z - 441845 z 8 10 12 14 + 11445165 z - 207108684 z + 2737363677 z - 27249208213 z 18 16 50 - 1253254709249 z + 208870468924 z - 1253254709249 z 48 20 36 + 5959845375953 z + 5959845375953 z + 778299247569590 z 34 66 64 30 - 859994230949288 z - 161 z + 11116 z - 576681359928886 z 42 44 46 - 172804058242334 z + 69538046991982 z - 22676717367072 z 58 56 54 52 - 207108684 z + 2737363677 z - 27249208213 z + 208870468924 z 60 68 32 38 + 11445165 z + z + 778299247569590 z - 576681359928886 z 40 62 / 2 + 349431203158424 z - 441845 z ) / ((-1 + z ) (1 / 28 26 2 + 1074784044520138 z - 518948413638184 z - 192 z 24 22 4 6 + 202562329680802 z - 63659805120088 z + 15373 z - 693722 z 8 10 12 14 + 20059873 z - 399611474 z + 5746170093 z - 61603400490 z 18 16 50 - 3203361027784 z + 504074092761 z - 3203361027784 z 48 20 36 + 16018751488525 z + 16018751488525 z + 2461041554764258 z 34 66 64 30 - 2728864484162572 z - 192 z + 15373 z - 1804610697701036 z 42 44 46 - 518948413638184 z + 202562329680802 z - 63659805120088 z 58 56 54 52 - 399611474 z + 5746170093 z - 61603400490 z + 504074092761 z 60 68 32 38 + 20059873 z + z + 2461041554764258 z - 1804610697701036 z 40 62 + 1074784044520138 z - 693722 z )) And in Maple-input format, it is: -(1+349431203158424*z^28-172804058242334*z^26-161*z^2+69538046991982*z^24-\ 22676717367072*z^22+11116*z^4-441845*z^6+11445165*z^8-207108684*z^10+2737363677 *z^12-27249208213*z^14-1253254709249*z^18+208870468924*z^16-1253254709249*z^50+ 5959845375953*z^48+5959845375953*z^20+778299247569590*z^36-859994230949288*z^34 -161*z^66+11116*z^64-576681359928886*z^30-172804058242334*z^42+69538046991982*z ^44-22676717367072*z^46-207108684*z^58+2737363677*z^56-27249208213*z^54+ 208870468924*z^52+11445165*z^60+z^68+778299247569590*z^32-576681359928886*z^38+ 349431203158424*z^40-441845*z^62)/(-1+z^2)/(1+1074784044520138*z^28-\ 518948413638184*z^26-192*z^2+202562329680802*z^24-63659805120088*z^22+15373*z^4 -693722*z^6+20059873*z^8-399611474*z^10+5746170093*z^12-61603400490*z^14-\ 3203361027784*z^18+504074092761*z^16-3203361027784*z^50+16018751488525*z^48+ 16018751488525*z^20+2461041554764258*z^36-2728864484162572*z^34-192*z^66+15373* z^64-1804610697701036*z^30-518948413638184*z^42+202562329680802*z^44-\ 63659805120088*z^46-399611474*z^58+5746170093*z^56-61603400490*z^54+ 504074092761*z^52+20059873*z^60+z^68+2461041554764258*z^32-1804610697701036*z^ 38+1074784044520138*z^40-693722*z^62) The first , 40, terms are: [0, 32, 0, 1727, 0, 102481, 0, 6280688, 0, 390110707, 0, 24380829075, 0, 1528389913680, 0, 95963299274697, 0, 6030316289585895, 0, 379115135449687168, 0, 23840133788960679761, 0, 1499354666098617488401, 0, 94304376116338554459584, 0, 5931666414270501478212663, 0, 373105049288902884740975673, 0, 23468793276024918638863145360, 0, 1476227384370582552832500072307, 0, 92857572438121153388783624099411, 0, 5840933414558981942101518231342256, 0, 367407195244951650674528272002021281 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 656718832484 z + 656718832484 z + 126 z - 509294898169 z 22 4 6 8 10 + 305800849490 z - 6623 z + 195601 z - 3670843 z + 46821280 z 12 14 18 16 - 423219132 z + 2787994512 z + 50387975061 z - 13650735795 z 50 48 20 36 + 6623 z - 195601 z - 141685390087 z - 50387975061 z 34 30 42 44 + 141685390087 z + 509294898169 z + 423219132 z - 46821280 z 46 54 52 32 38 + 3670843 z + z - 126 z - 305800849490 z + 13650735795 z 40 / 28 26 2 - 2787994512 z ) / (1 + 4202450787399 z - 3711837633246 z - 158 z / 24 22 4 6 8 + 2555423504057 z - 1367595402040 z + 9887 z - 339462 z + 7305409 z 10 12 14 18 - 105847268 z + 1079297205 z - 7980701666 z - 180288799494 z 16 50 48 20 + 43720571515 z - 339462 z + 7305409 z + 566352642121 z 36 34 30 + 566352642121 z - 1367595402040 z - 3711837633246 z 42 44 46 56 54 - 7980701666 z + 1079297205 z - 105847268 z + z - 158 z 52 32 38 40 + 9887 z + 2555423504057 z - 180288799494 z + 43720571515 z ) And in Maple-input format, it is: -(-1-656718832484*z^28+656718832484*z^26+126*z^2-509294898169*z^24+305800849490 *z^22-6623*z^4+195601*z^6-3670843*z^8+46821280*z^10-423219132*z^12+2787994512*z ^14+50387975061*z^18-13650735795*z^16+6623*z^50-195601*z^48-141685390087*z^20-\ 50387975061*z^36+141685390087*z^34+509294898169*z^30+423219132*z^42-46821280*z^ 44+3670843*z^46+z^54-126*z^52-305800849490*z^32+13650735795*z^38-2787994512*z^ 40)/(1+4202450787399*z^28-3711837633246*z^26-158*z^2+2555423504057*z^24-\ 1367595402040*z^22+9887*z^4-339462*z^6+7305409*z^8-105847268*z^10+1079297205*z^ 12-7980701666*z^14-180288799494*z^18+43720571515*z^16-339462*z^50+7305409*z^48+ 566352642121*z^20+566352642121*z^36-1367595402040*z^34-3711837633246*z^30-\ 7980701666*z^42+1079297205*z^44-105847268*z^46+z^56-158*z^54+9887*z^52+ 2555423504057*z^32-180288799494*z^38+43720571515*z^40) The first , 40, terms are: [0, 32, 0, 1792, 0, 110613, 0, 6987568, 0, 443973817, 0, 28250430051, 0, 1798272922928, 0, 114479613501045, 0, 7288054652621367, 0, 463978593088275616, 0, 29538267509140078739, 0, 1880495483310161672752, 0, 119718046978006075494880, 0, 7621614289257356187466607, 0, 485215106345945370012013679, 0, 30890266952645263233734476544, 0, 1966568189189246809941668050000, 0, 125197702220990873130558002623811, 0, 7970465874725438654464286205251936, 0, 507424059181165779046228799040672055 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 24 22 4 6 8 10 f(z) = - (1 - 80 z + z - 80 z + 1914 z - 18912 z + 90143 z - 225040 z 12 14 18 16 20 / 24 + 304460 z - 225040 z - 18912 z + 90143 z + 1914 z ) / ((z / 22 20 18 16 14 12 - 116 z + 3778 z - 46772 z + 261823 z - 725720 z + 1020924 z 10 8 6 4 2 2 - 725720 z + 261823 z - 46772 z + 3778 z - 116 z + 1) (-1 + z )) And in Maple-input format, it is: -(1-80*z^2+z^24-80*z^22+1914*z^4-18912*z^6+90143*z^8-225040*z^10+304460*z^12-\ 225040*z^14-18912*z^18+90143*z^16+1914*z^20)/(z^24-116*z^22+3778*z^20-46772*z^ 18+261823*z^16-725720*z^14+1020924*z^12-725720*z^10+261823*z^8-46772*z^6+3778*z ^4-116*z^2+1)/(-1+z^2) The first , 40, terms are: [0, 37, 0, 2349, 0, 162393, 0, 11504873, 0, 821798237, 0, 58869591669, 0, 4221374186993, 0, 302810475448081, 0, 21724131555850965, 0, 1558594751664492029, 0, 111822913032403360969, 0, 8022889301674099655801, 0, 575614398410513294897805, 0, 41298359307012842802509253, 0, 2963016393626597743996479713, 0, 212586336499823981076146457377, 0, 15252346205688863615038155566725, 0, 1094303947863476870123212315321293, 0, 78512585456540493250971449834079289, 0, 5633010915692101822610271398632984841] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 11973439756 z - 17897611692 z - 122 z + 20452835796 z 22 4 6 8 10 - 17897611692 z + 5748 z - 142470 z + 2130102 z - 20799166 z 12 14 18 16 48 + 139671280 z - 668480034 z - 6093655764 z + 2339024495 z + z 20 36 34 30 + 11973439756 z + 139671280 z - 668480034 z - 6093655764 z 42 44 46 32 38 - 142470 z + 5748 z - 122 z + 2339024495 z - 20799166 z 40 / 28 26 2 + 2130102 z ) / (-1 - 94094080008 z + 121961667724 z + 157 z / 24 22 4 6 8 - 121961667724 z + 94094080008 z - 9174 z + 274114 z - 4807378 z 10 12 14 18 + 53935858 z - 410565494 z + 2210151570 z + 25301984823 z 16 50 48 20 36 - 8667735803 z + z - 157 z - 55826298584 z - 2210151570 z 34 30 42 44 46 + 8667735803 z + 55826298584 z + 4807378 z - 274114 z + 9174 z 32 38 40 - 25301984823 z + 410565494 z - 53935858 z ) And in Maple-input format, it is: -(1+11973439756*z^28-17897611692*z^26-122*z^2+20452835796*z^24-17897611692*z^22 +5748*z^4-142470*z^6+2130102*z^8-20799166*z^10+139671280*z^12-668480034*z^14-\ 6093655764*z^18+2339024495*z^16+z^48+11973439756*z^20+139671280*z^36-668480034* z^34-6093655764*z^30-142470*z^42+5748*z^44-122*z^46+2339024495*z^32-20799166*z^ 38+2130102*z^40)/(-1-94094080008*z^28+121961667724*z^26+157*z^2-121961667724*z^ 24+94094080008*z^22-9174*z^4+274114*z^6-4807378*z^8+53935858*z^10-410565494*z^ 12+2210151570*z^14+25301984823*z^18-8667735803*z^16+z^50-157*z^48-55826298584*z ^20-2210151570*z^36+8667735803*z^34+55826298584*z^30+4807378*z^42-274114*z^44+ 9174*z^46-25301984823*z^32+410565494*z^38-53935858*z^40) The first , 40, terms are: [0, 35, 0, 2069, 0, 135387, 0, 9191467, 0, 633040309, 0, 43846678107, 0, 3043835137433, 0, 211494827234665, 0, 14700721662616251, 0, 1021982032124472197, 0, 71051780713501701643, 0, 4939897133337461295259, 0, 343451577782734198502469, 0, 23878941651519893523714947, 0, 1660219256148167471145610577, 0, 115429325960327830882369013809, 0, 8025406052088778751780626973443, 0, 557979097827240743202910563698597, 0, 38794384925892577942057375274571291, 0, 2697241427232939184570598986293581515] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 20742753 z - 64247402 z - 102 z + 141994837 z 22 4 6 8 10 - 227275250 z + 3753 z - 67720 z + 705770 z - 4667430 z 12 14 18 16 + 20742753 z - 64247402 z - 227275250 z + 141994837 z 20 36 34 30 32 + 265621644 z + 3753 z - 67720 z - 4667430 z + 705770 z 38 40 / 2 40 38 36 34 - 102 z + z ) / ((-1 + z ) (z - 141 z + 6405 z - 136236 z / 32 30 28 26 + 1628222 z - 12059672 z + 58745739 z - 195433847 z 24 22 20 18 + 454759775 z - 750975628 z + 886932604 z - 750975628 z 16 14 12 10 8 + 454759775 z - 195433847 z + 58745739 z - 12059672 z + 1628222 z 6 4 2 - 136236 z + 6405 z - 141 z + 1)) And in Maple-input format, it is: -(1+20742753*z^28-64247402*z^26-102*z^2+141994837*z^24-227275250*z^22+3753*z^4-\ 67720*z^6+705770*z^8-4667430*z^10+20742753*z^12-64247402*z^14-227275250*z^18+ 141994837*z^16+265621644*z^20+3753*z^36-67720*z^34-4667430*z^30+705770*z^32-102 *z^38+z^40)/(-1+z^2)/(z^40-141*z^38+6405*z^36-136236*z^34+1628222*z^32-12059672 *z^30+58745739*z^28-195433847*z^26+454759775*z^24-750975628*z^22+886932604*z^20 -750975628*z^18+454759775*z^16-195433847*z^14+58745739*z^12-12059672*z^10+ 1628222*z^8-136236*z^6+6405*z^4-141*z^2+1) The first , 40, terms are: [0, 40, 0, 2887, 0, 223035, 0, 17419620, 0, 1363845641, 0, 106854646793, 0, 8373717070732, 0, 656259547550275, 0, 51433263742718063, 0, 4031032925421977376, 0, 315929310537293747217, 0, 24760758595575346973457, 0, 1940609388426276602518192, 0, 152094098783913943347365823, 0, 11920284425987649862503331539, 0, 934245207519909542221077347452, 0, 73220913276434167952481930207337, 0, 5738645600981774275590715148194793, 0, 449762941758180909224780885418974068, 0, 35249903528838149307224370361495163499] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 49576314581218112 z - 13438992051178704 z - 180 z 24 22 4 6 + 3085913081246517 z - 597159667121940 z + 14396 z - 688316 z 8 10 12 14 + 22281823 z - 523347692 z + 9320265808 z - 129724878748 z 18 16 50 - 13026756105228 z + 1442471745691 z - 3170546544552020248 z 48 20 36 + 4606041451308891798 z + 96765823818444 z + 1877311535195096056 z 34 66 80 88 - 954968399933539928 z - 597159667121940 z + 22281823 z + z 84 86 82 64 + 14396 z - 180 z - 688316 z + 3085913081246517 z 30 42 - 155566954581073392 z - 5760972366775930552 z 44 46 + 6206697931533503200 z - 5760972366775930552 z 58 56 54 - 155566954581073392 z + 416641176994586426 z - 954968399933539928 z 52 60 70 + 1877311535195096056 z + 49576314581218112 z - 13026756105228 z 68 78 32 + 96765823818444 z - 523347692 z + 416641176994586426 z 38 40 - 3170546544552020248 z + 4606041451308891798 z 62 76 74 - 13438992051178704 z + 9320265808 z - 129724878748 z 72 / 2 28 + 1442471745691 z ) / ((-1 + z ) (1 + 137470670361206320 z / 26 2 24 - 36199375491211336 z - 206 z + 8039840700417237 z 22 4 6 8 - 1497799177596446 z + 18658 z - 998118 z + 35720051 z 10 12 14 18 - 917237720 z + 17679937484 z - 263961016672 z - 29804800499830 z 16 50 48 + 3123401252127 z - 9612382692527460836 z + 14073770648265791838 z 20 36 34 + 232465784058778 z + 5629333829584181380 z - 2822960638155710676 z 66 80 88 84 86 - 1497799177596446 z + 35720051 z + z + 18658 z - 206 z 82 64 30 - 998118 z + 8039840700417237 z - 442299699557172984 z 42 44 - 17684780901985134216 z + 19082557633004856840 z 46 58 - 17684780901985134216 z - 442299699557172984 z 56 54 + 1210021764259323706 z - 2822960638155710676 z 52 60 70 + 5629333829584181380 z + 137470670361206320 z - 29804800499830 z 68 78 32 + 232465784058778 z - 917237720 z + 1210021764259323706 z 38 40 - 9612382692527460836 z + 14073770648265791838 z 62 76 74 - 36199375491211336 z + 17679937484 z - 263961016672 z 72 + 3123401252127 z )) And in Maple-input format, it is: -(1+49576314581218112*z^28-13438992051178704*z^26-180*z^2+3085913081246517*z^24 -597159667121940*z^22+14396*z^4-688316*z^6+22281823*z^8-523347692*z^10+ 9320265808*z^12-129724878748*z^14-13026756105228*z^18+1442471745691*z^16-\ 3170546544552020248*z^50+4606041451308891798*z^48+96765823818444*z^20+ 1877311535195096056*z^36-954968399933539928*z^34-597159667121940*z^66+22281823* z^80+z^88+14396*z^84-180*z^86-688316*z^82+3085913081246517*z^64-\ 155566954581073392*z^30-5760972366775930552*z^42+6206697931533503200*z^44-\ 5760972366775930552*z^46-155566954581073392*z^58+416641176994586426*z^56-\ 954968399933539928*z^54+1877311535195096056*z^52+49576314581218112*z^60-\ 13026756105228*z^70+96765823818444*z^68-523347692*z^78+416641176994586426*z^32-\ 3170546544552020248*z^38+4606041451308891798*z^40-13438992051178704*z^62+ 9320265808*z^76-129724878748*z^74+1442471745691*z^72)/(-1+z^2)/(1+ 137470670361206320*z^28-36199375491211336*z^26-206*z^2+8039840700417237*z^24-\ 1497799177596446*z^22+18658*z^4-998118*z^6+35720051*z^8-917237720*z^10+ 17679937484*z^12-263961016672*z^14-29804800499830*z^18+3123401252127*z^16-\ 9612382692527460836*z^50+14073770648265791838*z^48+232465784058778*z^20+ 5629333829584181380*z^36-2822960638155710676*z^34-1497799177596446*z^66+ 35720051*z^80+z^88+18658*z^84-206*z^86-998118*z^82+8039840700417237*z^64-\ 442299699557172984*z^30-17684780901985134216*z^42+19082557633004856840*z^44-\ 17684780901985134216*z^46-442299699557172984*z^58+1210021764259323706*z^56-\ 2822960638155710676*z^54+5629333829584181380*z^52+137470670361206320*z^60-\ 29804800499830*z^70+232465784058778*z^68-917237720*z^78+1210021764259323706*z^ 32-9612382692527460836*z^38+14073770648265791838*z^40-36199375491211336*z^62+ 17679937484*z^76-263961016672*z^74+3123401252127*z^72) The first , 40, terms are: [0, 27, 0, 1121, 0, 51179, 0, 2464115, 0, 122656561, 0, 6236304643, 0, 321433522577, 0, 16714952447217, 0, 874248622735731, 0, 45900176043857905, 0, 2415890123919427523, 0, 127365579971013739867, 0, 6721946390485482385761, 0, 355014145380595579636267, 0, 18758514342526503393952929, 0, 991480211116714289672439393, 0, 52415160218871726990802354955, 0, 2771322679439912032925554027233, 0, 146539570008890435187935458667771, 0, 7749031866616289599492932717725091] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {3, 6}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 26 2 24 22 4 6 8 f(z) = - (-1 + z + 97 z - 97 z + 3060 z - 3060 z + 41204 z - 268652 z 10 12 14 18 16 + 897240 z - 1614686 z + 1614686 z + 268652 z - 897240 z 20 / 14 20 18 - 41204 z ) / (1 - 10964066 z + 819644 z - 3512360 z / 16 4 22 28 26 24 + 8267414 z + 5785 z - 99464 z + z - 139 z + 5785 z 12 10 8 6 2 + 8267414 z - 3512360 z + 819644 z - 99464 z - 139 z ) And in Maple-input format, it is: -(-1+z^26+97*z^2-97*z^24+3060*z^22-3060*z^4+41204*z^6-268652*z^8+897240*z^10-\ 1614686*z^12+1614686*z^14+268652*z^18-897240*z^16-41204*z^20)/(1-10964066*z^14+ 819644*z^20-3512360*z^18+8267414*z^16+5785*z^4-99464*z^22+z^28-139*z^26+5785*z^ 24+8267414*z^12-3512360*z^10+819644*z^8-99464*z^6-139*z^2) The first , 40, terms are: [0, 42, 0, 3113, 0, 247997, 0, 20089374, 0, 1635581845, 0, 133384936093, 0, 10884161480742, 0, 888327739697093, 0, 72507630696537089, 0, 5918420126175485826, 0, 483094438711156155865, 0, 39432997140988688170729, 0, 3218756161569042344635026, 0, 262734172669750961958339857, 0, 21445941597607308083031514229, 0, 1750546695744318892029575795670, 0, 142890149375154759471559859536717, 0, 11663553444588564705686464428081253, 0, 952049388434959270376094639710157294, 0, 77711997738668630446401709967562044813] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {3, 6}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 10 6 12 2 8 4 -23 z - 314 z + z - 23 z + 1 + 155 z + 155 z f(z) = - ----------------------------------------------------------------- 2 10 6 12 2 8 4 (-1 + z ) (-50 z - 1022 z + z - 50 z + 1 + 446 z + 446 z ) And in Maple-input format, it is: -(-23*z^10-314*z^6+z^12-23*z^2+1+155*z^8+155*z^4)/(-1+z^2)/(-50*z^10-1022*z^6+z ^12-50*z^2+1+446*z^8+446*z^4) The first , 40, terms are: [0, 28, 0, 1087, 0, 42703, 0, 1678492, 0, 65977489, 0, 2593426033, 0, 101941769116, 0, 4007102965423, 0, 157510258572895, 0, 6191376123287836, 0, 243369153647841313, 0, 9566297342239640545, 0, 376029761660927960860, 0, 14780889261190451486239, 0, 581003712011286584998639, 0, 22837957000142523652162972, 0, 897709032072826191581334193, 0, 35286935090564573682866173009, 0, 1387050529290770125915281115804, 0, 54521855351507782538591799933583] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {3, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : f(z) = - ( 16 14 12 10 8 6 4 2 z - 60 z + 656 z - 2548 z + 3998 z - 2548 z + 656 z - 60 z + 1) / 18 16 14 12 10 8 6 / (z - 105 z + 1696 z - 9776 z + 23014 z - 23014 z + 9776 z / 4 2 - 1696 z + 105 z - 1) And in Maple-input format, it is: -(z^16-60*z^14+656*z^12-2548*z^10+3998*z^8-2548*z^6+656*z^4-60*z^2+1)/(z^18-105 *z^16+1696*z^14-9776*z^12+23014*z^10-23014*z^8+9776*z^6-1696*z^4+105*z^2-1) The first , 40, terms are: [0, 45, 0, 3685, 0, 317833, 0, 27543609, 0, 2388043573, 0, 207053969629, 0, 17952580992465, 0, 1556576375730481, 0, 134962774048511101, 0, 11701931723287191189, 0, 1014614637950284504537, 0, 87972044953058563588585, 0, 7627605993247561898103045, 0, 661350696341146067730658957, 0, 57342335712954454155053175457, 0, 4971860592584999133954223974753, 0, 431084598224958755142632779922893, 0, 37377140281029933363391641203355589, 0, 3240780629464169442712601136190651689, 0, 280991510033757344337922849404247056025] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {3, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 127263615546770229 z - 33328712874492260 z - 196 z 24 22 4 6 + 7358606365841617 z - 1362346917890780 z + 17157 z - 898642 z 8 10 12 14 + 31794036 z - 812642618 z + 15663236623 z - 234543755532 z 18 16 50 - 26766497143484 z + 2788668524771 z - 9067342882585252940 z 48 20 36 + 13299660657598910475 z + 210091384785336 z + 5296943320414422903 z 34 66 80 88 - 2647914817700676830 z - 1362346917890780 z + 31794036 z + z 84 86 82 64 + 17157 z - 196 z - 898642 z + 7358606365841617 z 30 42 - 411494092189071958 z - 16730247711833783080 z 44 46 + 18059157533182041104 z - 16730247711833783080 z 58 56 - 411494092189071958 z + 1130699703692537420 z 54 52 - 2647914817700676830 z + 5296943320414422903 z 60 70 68 + 127263615546770229 z - 26766497143484 z + 210091384785336 z 78 32 38 - 812642618 z + 1130699703692537420 z - 9067342882585252940 z 40 62 76 + 13299660657598910475 z - 33328712874492260 z + 15663236623 z 74 72 / 2 - 234543755532 z + 2788668524771 z ) / ((-1 + z ) (1 / 28 26 2 + 354701152631580867 z - 89867583604672095 z - 227 z 24 22 4 6 + 19120323994010909 z - 3397149552713764 z + 22503 z - 1312712 z 8 10 12 14 + 51058978 z - 1419705292 z + 29507099041 z - 472866731669 z 18 16 50 - 60647615783044 z + 5977601902351 z - 28161466392871857769 z 48 20 + 41723156116448157139 z + 500565871358668 z 36 34 + 16222905777642460085 z - 7966196877313308880 z 66 80 88 84 86 - 3397149552713764 z + 51058978 z + z + 22503 z - 227 z 82 64 30 - 1312712 z + 19120323994010909 z - 1180947082671510684 z 42 44 - 52803960017923850328 z + 57113574319208042920 z 46 58 - 52803960017923850328 z - 1180947082671510684 z 56 54 + 3328719122544291454 z - 7966196877313308880 z 52 60 70 + 16222905777642460085 z + 354701152631580867 z - 60647615783044 z 68 78 32 + 500565871358668 z - 1419705292 z + 3328719122544291454 z 38 40 - 28161466392871857769 z + 41723156116448157139 z 62 76 74 - 89867583604672095 z + 29507099041 z - 472866731669 z 72 + 5977601902351 z )) And in Maple-input format, it is: -(1+127263615546770229*z^28-33328712874492260*z^26-196*z^2+7358606365841617*z^ 24-1362346917890780*z^22+17157*z^4-898642*z^6+31794036*z^8-812642618*z^10+ 15663236623*z^12-234543755532*z^14-26766497143484*z^18+2788668524771*z^16-\ 9067342882585252940*z^50+13299660657598910475*z^48+210091384785336*z^20+ 5296943320414422903*z^36-2647914817700676830*z^34-1362346917890780*z^66+ 31794036*z^80+z^88+17157*z^84-196*z^86-898642*z^82+7358606365841617*z^64-\ 411494092189071958*z^30-16730247711833783080*z^42+18059157533182041104*z^44-\ 16730247711833783080*z^46-411494092189071958*z^58+1130699703692537420*z^56-\ 2647914817700676830*z^54+5296943320414422903*z^52+127263615546770229*z^60-\ 26766497143484*z^70+210091384785336*z^68-812642618*z^78+1130699703692537420*z^ 32-9067342882585252940*z^38+13299660657598910475*z^40-33328712874492260*z^62+ 15663236623*z^76-234543755532*z^74+2788668524771*z^72)/(-1+z^2)/(1+ 354701152631580867*z^28-89867583604672095*z^26-227*z^2+19120323994010909*z^24-\ 3397149552713764*z^22+22503*z^4-1312712*z^6+51058978*z^8-1419705292*z^10+ 29507099041*z^12-472866731669*z^14-60647615783044*z^18+5977601902351*z^16-\ 28161466392871857769*z^50+41723156116448157139*z^48+500565871358668*z^20+ 16222905777642460085*z^36-7966196877313308880*z^34-3397149552713764*z^66+ 51058978*z^80+z^88+22503*z^84-227*z^86-1312712*z^82+19120323994010909*z^64-\ 1180947082671510684*z^30-52803960017923850328*z^42+57113574319208042920*z^44-\ 52803960017923850328*z^46-1180947082671510684*z^58+3328719122544291454*z^56-\ 7966196877313308880*z^54+16222905777642460085*z^52+354701152631580867*z^60-\ 60647615783044*z^70+500565871358668*z^68-1419705292*z^78+3328719122544291454*z^ 32-28161466392871857769*z^38+41723156116448157139*z^40-89867583604672095*z^62+ 29507099041*z^76-472866731669*z^74+5977601902351*z^72) The first , 40, terms are: [0, 32, 0, 1723, 0, 102057, 0, 6254432, 0, 389057903, 0, 24374466839, 0, 1532505334384, 0, 96529426824249, 0, 6085928997348027, 0, 383890389985717264, 0, 24221383707632680921, 0, 1528441771074991492553, 0, 96456018214838684847472, 0, 6087314267943450104678555, 0, 384176217998468390482172217, 0, 24245972538513997917201171472, 0, 1530209996465112101459285887399, 0, 96574765480036646886623773366943, 0, 6095045037750183458369400662126912, 0, 384671938217507344036064024430709993 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 213376124909664327 z - 54246961533922537 z - 201 z 24 22 4 6 + 11596067221705829 z - 2073527397417264 z + 18187 z - 989200 z 8 10 12 14 + 36439842 z - 971127608 z + 19526780109 z - 304984225531 z 18 16 50 - 37770023992608 z + 3779644297751 z - 16820972826085926675 z 48 20 36 + 24910950462680472387 z + 308177737134220 z + 9696195433460968097 z 34 66 80 88 - 4765636890022942448 z - 2073527397417264 z + 36439842 z + z 84 86 82 64 + 18187 z - 201 z - 989200 z + 11596067221705829 z 30 42 - 708655854309980552 z - 31519216529106073456 z 44 46 + 34089007148855659304 z - 31519216529106073456 z 58 56 - 708655854309980552 z + 1993907024795058302 z 54 52 - 4765636890022942448 z + 9696195433460968097 z 60 70 68 + 213376124909664327 z - 37770023992608 z + 308177737134220 z 78 32 38 - 971127608 z + 1993907024795058302 z - 16820972826085926675 z 40 62 76 + 24910950462680472387 z - 54246961533922537 z + 19526780109 z 74 72 / 28 - 304984225531 z + 3779644297751 z ) / (-1 - 739459131289353998 z / 26 2 24 + 175522420909460877 z + 235 z - 35029740896765555 z 22 4 6 8 + 5845651108117986 z - 24180 z + 1469114 z - 59723791 z 10 12 14 18 + 1741449479 z - 38072431256 z + 643560860214 z + 92471482693861 z 16 50 - 8601892940787 z + 132110363609356669237 z 48 20 - 180179086213273418439 z - 810162219753666 z 36 34 - 44443825931706897418 z + 20344128453316024977 z 66 80 90 88 84 + 35029740896765555 z - 1741449479 z + z - 235 z - 1469114 z 86 82 64 + 24180 z + 59723791 z - 175522420909460877 z 30 42 + 2631036706391117764 z + 180179086213273418439 z 44 46 - 210388285757362540668 z + 210388285757362540668 z 58 56 + 7934988090040690809 z - 20344128453316024977 z 54 52 + 44443825931706897418 z - 82879457426131955432 z 60 70 68 - 2631036706391117764 z + 810162219753666 z - 5845651108117986 z 78 32 38 + 38072431256 z - 7934988090040690809 z + 82879457426131955432 z 40 62 76 - 132110363609356669237 z + 739459131289353998 z - 643560860214 z 74 72 + 8601892940787 z - 92471482693861 z ) And in Maple-input format, it is: -(1+213376124909664327*z^28-54246961533922537*z^26-201*z^2+11596067221705829*z^ 24-2073527397417264*z^22+18187*z^4-989200*z^6+36439842*z^8-971127608*z^10+ 19526780109*z^12-304984225531*z^14-37770023992608*z^18+3779644297751*z^16-\ 16820972826085926675*z^50+24910950462680472387*z^48+308177737134220*z^20+ 9696195433460968097*z^36-4765636890022942448*z^34-2073527397417264*z^66+ 36439842*z^80+z^88+18187*z^84-201*z^86-989200*z^82+11596067221705829*z^64-\ 708655854309980552*z^30-31519216529106073456*z^42+34089007148855659304*z^44-\ 31519216529106073456*z^46-708655854309980552*z^58+1993907024795058302*z^56-\ 4765636890022942448*z^54+9696195433460968097*z^52+213376124909664327*z^60-\ 37770023992608*z^70+308177737134220*z^68-971127608*z^78+1993907024795058302*z^ 32-16820972826085926675*z^38+24910950462680472387*z^40-54246961533922537*z^62+ 19526780109*z^76-304984225531*z^74+3779644297751*z^72)/(-1-739459131289353998*z ^28+175522420909460877*z^26+235*z^2-35029740896765555*z^24+5845651108117986*z^ 22-24180*z^4+1469114*z^6-59723791*z^8+1741449479*z^10-38072431256*z^12+ 643560860214*z^14+92471482693861*z^18-8601892940787*z^16+132110363609356669237* z^50-180179086213273418439*z^48-810162219753666*z^20-44443825931706897418*z^36+ 20344128453316024977*z^34+35029740896765555*z^66-1741449479*z^80+z^90-235*z^88-\ 1469114*z^84+24180*z^86+59723791*z^82-175522420909460877*z^64+ 2631036706391117764*z^30+180179086213273418439*z^42-210388285757362540668*z^44+ 210388285757362540668*z^46+7934988090040690809*z^58-20344128453316024977*z^56+ 44443825931706897418*z^54-82879457426131955432*z^52-2631036706391117764*z^60+ 810162219753666*z^70-5845651108117986*z^68+38072431256*z^78-7934988090040690809 *z^32+82879457426131955432*z^38-132110363609356669237*z^40+739459131289353998*z ^62-643560860214*z^76+8601892940787*z^74-92471482693861*z^72) The first , 40, terms are: [0, 34, 0, 1997, 0, 127089, 0, 8244382, 0, 537951385, 0, 35172868373, 0, 2301448303046, 0, 150633805710037, 0, 9860397294482449, 0, 645485888918571914, 0, 42255896646443906109, 0, 2766248731598387260741, 0, 181090842957089134073466, 0, 11855020388061514123551113, 0, 776083312798161096910294157, 0, 50805938445406915582387261046, 0, 3325987768064462212690138157341, 0, 217734291810249305857555098416913, 0, 14253877593641322451808735525956078, 0, 933123698570721336762038946240289849] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 430324572106288 z - 212090606726422 z - 163 z 24 22 4 6 + 84946674837826 z - 27527172465184 z + 11480 z - 467225 z 8 10 12 14 + 12405791 z - 229932328 z + 3106296339 z - 31519748213 z 18 16 50 - 1492930061303 z + 245542320376 z - 1492930061303 z 48 20 36 + 7175107654381 z + 7175107654381 z + 961900702611870 z 34 66 64 30 - 1063318891356528 z - 163 z + 11480 z - 711793813312706 z 42 44 46 - 212090606726422 z + 84946674837826 z - 27527172465184 z 58 56 54 52 - 229932328 z + 3106296339 z - 31519748213 z + 245542320376 z 60 68 32 38 + 12405791 z + z + 961900702611870 z - 711793813312706 z 40 62 / 2 + 430324572106288 z - 467225 z ) / ((-1 + z ) (1 / 28 26 2 + 1342328873979698 z - 644284827074032 z - 196 z 24 22 4 6 + 249541890229538 z - 77670921560872 z + 16009 z - 736910 z 8 10 12 14 + 21744437 z - 442126534 z + 6486711457 z - 70884579406 z 18 16 50 - 3810708414524 z + 590302478917 z - 3810708414524 z 48 20 36 + 19318292500861 z + 19318292500861 z + 3094395062782346 z 34 66 64 30 - 3434009707493412 z - 196 z + 16009 z - 2263338299703860 z 42 44 46 - 644284827074032 z + 249541890229538 z - 77670921560872 z 58 56 54 52 - 442126534 z + 6486711457 z - 70884579406 z + 590302478917 z 60 68 32 38 + 21744437 z + z + 3094395062782346 z - 2263338299703860 z 40 62 + 1342328873979698 z - 736910 z )) And in Maple-input format, it is: -(1+430324572106288*z^28-212090606726422*z^26-163*z^2+84946674837826*z^24-\ 27527172465184*z^22+11480*z^4-467225*z^6+12405791*z^8-229932328*z^10+3106296339 *z^12-31519748213*z^14-1492930061303*z^18+245542320376*z^16-1492930061303*z^50+ 7175107654381*z^48+7175107654381*z^20+961900702611870*z^36-1063318891356528*z^ 34-163*z^66+11480*z^64-711793813312706*z^30-212090606726422*z^42+84946674837826 *z^44-27527172465184*z^46-229932328*z^58+3106296339*z^56-31519748213*z^54+ 245542320376*z^52+12405791*z^60+z^68+961900702611870*z^32-711793813312706*z^38+ 430324572106288*z^40-467225*z^62)/(-1+z^2)/(1+1342328873979698*z^28-\ 644284827074032*z^26-196*z^2+249541890229538*z^24-77670921560872*z^22+16009*z^4 -736910*z^6+21744437*z^8-442126534*z^10+6486711457*z^12-70884579406*z^14-\ 3810708414524*z^18+590302478917*z^16-3810708414524*z^50+19318292500861*z^48+ 19318292500861*z^20+3094395062782346*z^36-3434009707493412*z^34-196*z^66+16009* z^64-2263338299703860*z^30-644284827074032*z^42+249541890229538*z^44-\ 77670921560872*z^46-442126534*z^58+6486711457*z^56-70884579406*z^54+ 590302478917*z^52+21744437*z^60+z^68+3094395062782346*z^32-2263338299703860*z^ 38+1342328873979698*z^40-736910*z^62) The first , 40, terms are: [0, 34, 0, 1973, 0, 123405, 0, 7862010, 0, 504119977, 0, 32415106345, 0, 2087146280074, 0, 134482331860653, 0, 8668453467705717, 0, 558865173817192658, 0, 36034721408616488225, 0, 2323603558247839373985, 0, 149836497907265628309618, 0, 9662317475168743069876661, 0, 623088107383564639909751597, 0, 40180939425051548241941341546, 0, 2591147108062997222629223154921, 0, 167095519076828990074991545551977, 0, 10775512902214141352167585664056346, 0, 694882423952313957490639498016819981] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 250318185462081545 z - 63466450321969821 z - 205 z 24 22 4 6 + 13519868650383912 z - 2406886661573188 z + 18916 z - 1048396 z 8 10 12 14 + 39304329 z - 1064282726 z + 21703350693 z - 343124710288 z 18 16 50 - 43298411835962 z + 4296188456477 z - 19864436954441967638 z 48 20 + 29432026150972272920 z + 355740225072725 z 36 34 + 11442650281411817344 z - 5618698672555233236 z 66 80 88 84 86 - 2406886661573188 z + 39304329 z + z + 18916 z - 205 z 82 64 30 - 1048396 z + 13519868650383912 z - 833101402560245322 z 42 44 - 37249853452745982722 z + 40290504821759288240 z 46 58 - 37249853452745982722 z - 833101402560245322 z 56 54 + 2347880506370932666 z - 5618698672555233236 z 52 60 70 + 11442650281411817344 z + 250318185462081545 z - 43298411835962 z 68 78 32 + 355740225072725 z - 1064282726 z + 2347880506370932666 z 38 40 - 19864436954441967638 z + 29432026150972272920 z 62 76 74 - 63466450321969821 z + 21703350693 z - 343124710288 z 72 / 2 28 + 4296188456477 z ) / ((-1 + z ) (1 + 704652631127497676 z / 26 2 24 - 172062564183746868 z - 242 z + 35177542748825529 z 22 4 6 8 - 5990091263907145 z + 25384 z - 1561677 z + 63974419 z 10 12 14 18 - 1872390710 z + 40946421350 z - 690052961119 z - 97590758231068 z 16 50 48 + 9165531221555 z - 63636407597202875658 z + 95471512259010089237 z 20 36 + 843990878770426 z + 36030190584338248105 z 34 66 80 88 - 17310634159795273918 z - 5990091263907145 z + 63974419 z + z 84 86 82 64 + 25384 z - 242 z - 1561677 z + 35177542748825529 z 30 42 - 2426295834507129178 z - 121748670379669518902 z 44 46 + 132021031876816968625 z - 121748670379669518902 z 58 56 - 2426295834507129178 z + 7047256757329452265 z 54 52 - 17310634159795273918 z + 36030190584338248105 z 60 70 68 + 704652631127497676 z - 97590758231068 z + 843990878770426 z 78 32 38 - 1872390710 z + 7047256757329452265 z - 63636407597202875658 z 40 62 76 + 95471512259010089237 z - 172062564183746868 z + 40946421350 z 74 72 - 690052961119 z + 9165531221555 z )) And in Maple-input format, it is: -(1+250318185462081545*z^28-63466450321969821*z^26-205*z^2+13519868650383912*z^ 24-2406886661573188*z^22+18916*z^4-1048396*z^6+39304329*z^8-1064282726*z^10+ 21703350693*z^12-343124710288*z^14-43298411835962*z^18+4296188456477*z^16-\ 19864436954441967638*z^50+29432026150972272920*z^48+355740225072725*z^20+ 11442650281411817344*z^36-5618698672555233236*z^34-2406886661573188*z^66+ 39304329*z^80+z^88+18916*z^84-205*z^86-1048396*z^82+13519868650383912*z^64-\ 833101402560245322*z^30-37249853452745982722*z^42+40290504821759288240*z^44-\ 37249853452745982722*z^46-833101402560245322*z^58+2347880506370932666*z^56-\ 5618698672555233236*z^54+11442650281411817344*z^52+250318185462081545*z^60-\ 43298411835962*z^70+355740225072725*z^68-1064282726*z^78+2347880506370932666*z^ 32-19864436954441967638*z^38+29432026150972272920*z^40-63466450321969821*z^62+ 21703350693*z^76-343124710288*z^74+4296188456477*z^72)/(-1+z^2)/(1+ 704652631127497676*z^28-172062564183746868*z^26-242*z^2+35177542748825529*z^24-\ 5990091263907145*z^22+25384*z^4-1561677*z^6+63974419*z^8-1872390710*z^10+ 40946421350*z^12-690052961119*z^14-97590758231068*z^18+9165531221555*z^16-\ 63636407597202875658*z^50+95471512259010089237*z^48+843990878770426*z^20+ 36030190584338248105*z^36-17310634159795273918*z^34-5990091263907145*z^66+ 63974419*z^80+z^88+25384*z^84-242*z^86-1561677*z^82+35177542748825529*z^64-\ 2426295834507129178*z^30-121748670379669518902*z^42+132021031876816968625*z^44-\ 121748670379669518902*z^46-2426295834507129178*z^58+7047256757329452265*z^56-\ 17310634159795273918*z^54+36030190584338248105*z^52+704652631127497676*z^60-\ 97590758231068*z^70+843990878770426*z^68-1872390710*z^78+7047256757329452265*z^ 32-63636407597202875658*z^38+95471512259010089237*z^40-172062564183746868*z^62+ 40946421350*z^76-690052961119*z^74+9165531221555*z^72) The first , 40, terms are: [0, 38, 0, 2524, 0, 178209, 0, 12701314, 0, 907088187, 0, 64820417857, 0, 4633103024776, 0, 331187350977495, 0, 23675238819177605, 0, 1692480798292632242, 0, 120992164866240961571, 0, 8649533601852380198684, 0, 618342439352113280955182, 0, 44204436349927481804917947, 0, 3160114855111301102819088995, 0, 225912351629599496266690791478, 0, 16150171267277895943194216304204, 0, 1154554159963120423198438733994171, 0, 82537535904687715771012787249749402, 0, 5900498346931950302403070870721032757] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 9227094429971855091 z - 1516473012338343478 z - 227 z 24 22 4 6 + 215632680818046365 z - 26371927006690658 z + 23636 z - 1509662 z 102 8 10 12 - 2181212776 z + 66718281 z - 2181212776 z + 55057627987 z 14 18 16 - 1105455302798 z - 243582870992906 z + 18046402384445 z 50 48 - 733599438981827693371436 z + 485377980895361216841098 z 20 36 + 2754438429047153 z + 3244262823066576088338 z 34 66 - 912392289462523902790 z - 285140665493998171998360 z 80 100 90 + 225466988846561219513 z + 55057627987 z - 26371927006690658 z 88 84 94 + 215632680818046365 z + 9227094429971855091 z - 243582870992906 z 86 96 98 - 1516473012338343478 z + 18046402384445 z - 1105455302798 z 92 82 + 2754438429047153 z - 48814429719982075664 z 64 112 110 106 + 485377980895361216841098 z + z - 227 z - 1509662 z 108 30 42 + 23636 z - 48814429719982075664 z - 68669149653355103888300 z 44 46 + 148622944904889281010862 z - 285140665493998171998360 z 58 56 - 1175292918725581361322820 z + 1246546917651971095267022 z 54 52 - 1175292918725581361322820 z + 984980404079758194384566 z 60 70 + 984980404079758194384566 z - 68669149653355103888300 z 68 78 + 148622944904889281010862 z - 912392289462523902790 z 32 38 + 225466988846561219513 z - 10161409383150468714901 z 40 62 + 28092368029167216302153 z - 733599438981827693371436 z 76 74 + 3244262823066576088338 z - 10161409383150468714901 z 72 104 / + 28092368029167216302153 z + 66718281 z ) / (-1 / 28 26 2 - 26890765337242923800 z + 4189872127873995114 z + 260 z 24 22 4 6 - 564228908053031804 z + 65263531748442127 z - 30343 z + 2140208 z 102 8 10 12 + 99544982878 z - 103337710 z + 3661478132 z - 99544982878 z 14 18 16 + 2142156097032 z + 536231503144861 z - 37333064086775 z 50 48 + 3813068119587809699994009 z - 2388295138935548006382441 z 20 36 - 6435905659075597 z - 11640673741166812738976 z 34 66 + 3109062267200237793123 z + 2388295138935548006382441 z 80 100 90 - 3109062267200237793123 z - 2142156097032 z + 564228908053031804 z 88 84 - 4189872127873995114 z - 149941862774636946674 z 94 86 96 + 6435905659075597 z + 26890765337242923800 z - 536231503144861 z 98 92 82 + 37333064086775 z - 65263531748442127 z + 729563354624552699221 z 64 112 114 110 - 3813068119587809699994009 z - 260 z + z + 30343 z 106 108 30 + 103337710 z - 2140208 z + 149941862774636946674 z 42 44 + 287918253498491733278184 z - 656876941777911075895848 z 46 58 + 1329256086619907821178352 z + 7680911138460310468231069 z 56 54 - 7680911138460310468231069 z + 6835582429358720432663968 z 52 60 - 5413006072304826453835416 z - 6835582429358720432663968 z 70 68 + 656876941777911075895848 z - 1329256086619907821178352 z 78 32 + 11640673741166812738976 z - 729563354624552699221 z 38 40 + 38393059291451981916133 z - 111793067257086050756425 z 62 76 + 5413006072304826453835416 z - 38393059291451981916133 z 74 72 + 111793067257086050756425 z - 287918253498491733278184 z 104 - 3661478132 z ) And in Maple-input format, it is: -(1+9227094429971855091*z^28-1516473012338343478*z^26-227*z^2+ 215632680818046365*z^24-26371927006690658*z^22+23636*z^4-1509662*z^6-2181212776 *z^102+66718281*z^8-2181212776*z^10+55057627987*z^12-1105455302798*z^14-\ 243582870992906*z^18+18046402384445*z^16-733599438981827693371436*z^50+ 485377980895361216841098*z^48+2754438429047153*z^20+3244262823066576088338*z^36 -912392289462523902790*z^34-285140665493998171998360*z^66+225466988846561219513 *z^80+55057627987*z^100-26371927006690658*z^90+215632680818046365*z^88+ 9227094429971855091*z^84-243582870992906*z^94-1516473012338343478*z^86+ 18046402384445*z^96-1105455302798*z^98+2754438429047153*z^92-\ 48814429719982075664*z^82+485377980895361216841098*z^64+z^112-227*z^110-1509662 *z^106+23636*z^108-48814429719982075664*z^30-68669149653355103888300*z^42+ 148622944904889281010862*z^44-285140665493998171998360*z^46-\ 1175292918725581361322820*z^58+1246546917651971095267022*z^56-\ 1175292918725581361322820*z^54+984980404079758194384566*z^52+ 984980404079758194384566*z^60-68669149653355103888300*z^70+ 148622944904889281010862*z^68-912392289462523902790*z^78+225466988846561219513* z^32-10161409383150468714901*z^38+28092368029167216302153*z^40-\ 733599438981827693371436*z^62+3244262823066576088338*z^76-\ 10161409383150468714901*z^74+28092368029167216302153*z^72+66718281*z^104)/(-1-\ 26890765337242923800*z^28+4189872127873995114*z^26+260*z^2-564228908053031804*z ^24+65263531748442127*z^22-30343*z^4+2140208*z^6+99544982878*z^102-103337710*z^ 8+3661478132*z^10-99544982878*z^12+2142156097032*z^14+536231503144861*z^18-\ 37333064086775*z^16+3813068119587809699994009*z^50-2388295138935548006382441*z^ 48-6435905659075597*z^20-11640673741166812738976*z^36+3109062267200237793123*z^ 34+2388295138935548006382441*z^66-3109062267200237793123*z^80-2142156097032*z^ 100+564228908053031804*z^90-4189872127873995114*z^88-149941862774636946674*z^84 +6435905659075597*z^94+26890765337242923800*z^86-536231503144861*z^96+ 37333064086775*z^98-65263531748442127*z^92+729563354624552699221*z^82-\ 3813068119587809699994009*z^64-260*z^112+z^114+30343*z^110+103337710*z^106-\ 2140208*z^108+149941862774636946674*z^30+287918253498491733278184*z^42-\ 656876941777911075895848*z^44+1329256086619907821178352*z^46+ 7680911138460310468231069*z^58-7680911138460310468231069*z^56+ 6835582429358720432663968*z^54-5413006072304826453835416*z^52-\ 6835582429358720432663968*z^60+656876941777911075895848*z^70-\ 1329256086619907821178352*z^68+11640673741166812738976*z^78-\ 729563354624552699221*z^32+38393059291451981916133*z^38-\ 111793067257086050756425*z^40+5413006072304826453835416*z^62-\ 38393059291451981916133*z^76+111793067257086050756425*z^74-\ 287918253498491733278184*z^72-3661478132*z^104) The first , 40, terms are: [0, 33, 0, 1873, 0, 116207, 0, 7388816, 0, 473753669, 0, 30474153743, 0, 1962875142967, 0, 126504483373919, 0, 8155133856709095, 0, 525783099415702967, 0, 33900413138058833219, 0, 2185816906775197976625, 0, 140937754244856702646624, 0, 9087472521346697019325931, 0, 585949096170451437225456421, 0, 37781320804581958789049765513, 0, 2436097142080145274078946010433, 0, 157076845031010540526913217177441, 0, 10128142063965878869103891188670489, 0, 653051484966162891749543171653292845] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1074443724038016 z - 461419171601880 z - 164 z 24 22 4 6 + 163096019059228 z - 47225899362652 z + 11710 z - 486792 z 8 10 12 14 + 13301665 z - 255776704 z + 3617100992 z - 38797327256 z 18 16 50 - 2120255761828 z + 322854386588 z - 47225899362652 z 48 20 36 + 163096019059228 z + 11133728922488 z + 4771923426395412 z 34 66 64 30 - 4349171690055956 z - 486792 z + 13301665 z - 2066556573837456 z 42 44 46 - 2066556573837456 z + 1074443724038016 z - 461419171601880 z 58 56 54 - 38797327256 z + 322854386588 z - 2120255761828 z 52 60 70 68 + 11133728922488 z + 3617100992 z - 164 z + 11710 z 32 38 40 + 3291570974247590 z - 4349171690055956 z + 3291570974247590 z 62 72 / 28 - 255776704 z + z ) / (-1 - 4737071637765556 z / 26 2 24 + 1865097195652148 z + 203 z - 604747899927168 z 22 4 6 8 10 + 160694432482848 z - 16928 z + 796984 z - 24247727 z + 514148705 z 12 14 18 16 - 7973412012 z + 93488323436 z + 6077122815912 z - 848887286100 z 50 48 20 + 604747899927168 z - 1865097195652148 z - 34771613706032 z 36 34 66 - 30063571688038424 z + 25014486789720094 z + 24247727 z 64 30 42 - 514148705 z + 9945339780674676 z + 17307374144361814 z 44 46 58 - 9945339780674676 z + 4737071637765556 z + 848887286100 z 56 54 52 - 6077122815912 z + 34771613706032 z - 160694432482848 z 60 70 68 32 - 93488323436 z + 16928 z - 796984 z - 17307374144361814 z 38 40 62 74 + 30063571688038424 z - 25014486789720094 z + 7973412012 z + z 72 - 203 z ) And in Maple-input format, it is: -(1+1074443724038016*z^28-461419171601880*z^26-164*z^2+163096019059228*z^24-\ 47225899362652*z^22+11710*z^4-486792*z^6+13301665*z^8-255776704*z^10+3617100992 *z^12-38797327256*z^14-2120255761828*z^18+322854386588*z^16-47225899362652*z^50 +163096019059228*z^48+11133728922488*z^20+4771923426395412*z^36-\ 4349171690055956*z^34-486792*z^66+13301665*z^64-2066556573837456*z^30-\ 2066556573837456*z^42+1074443724038016*z^44-461419171601880*z^46-38797327256*z^ 58+322854386588*z^56-2120255761828*z^54+11133728922488*z^52+3617100992*z^60-164 *z^70+11710*z^68+3291570974247590*z^32-4349171690055956*z^38+3291570974247590*z ^40-255776704*z^62+z^72)/(-1-4737071637765556*z^28+1865097195652148*z^26+203*z^ 2-604747899927168*z^24+160694432482848*z^22-16928*z^4+796984*z^6-24247727*z^8+ 514148705*z^10-7973412012*z^12+93488323436*z^14+6077122815912*z^18-848887286100 *z^16+604747899927168*z^50-1865097195652148*z^48-34771613706032*z^20-\ 30063571688038424*z^36+25014486789720094*z^34+24247727*z^66-514148705*z^64+ 9945339780674676*z^30+17307374144361814*z^42-9945339780674676*z^44+ 4737071637765556*z^46+848887286100*z^58-6077122815912*z^56+34771613706032*z^54-\ 160694432482848*z^52-93488323436*z^60+16928*z^70-796984*z^68-17307374144361814* z^32+30063571688038424*z^38-25014486789720094*z^40+7973412012*z^62+z^74-203*z^ 72) The first , 40, terms are: [0, 39, 0, 2699, 0, 197897, 0, 14620733, 0, 1081778847, 0, 80072953667, 0, 5927810394045, 0, 438859699557205, 0, 32491221277640363, 0, 2405525190760988375, 0, 178096422664730998661, 0, 13185635028888070438881, 0, 976218725001793348615891, 0, 72275866851212460322258943, 0, 5351056320317468838492043913, 0, 396173798919803407309745408825, 0, 29331345396344436903659834023247, 0, 2171591937758781793811078598092099, 0, 160777198978423405943650528165819505, 0, 11903390911642309834823102455941610549] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1390314272117252 z - 587953308656693 z - 165 z 24 22 4 6 + 204100171371196 z - 57910146705556 z + 11900 z - 501515 z 8 10 12 14 + 13937636 z - 273302118 z + 3949459806 z - 43349956232 z 18 16 50 - 2484664303032 z + 369424135055 z - 57910146705556 z 48 20 36 + 204100171371196 z + 13354488862536 z + 6354653971956354 z 34 66 64 30 - 5781004446072928 z - 501515 z + 13937636 z - 2707311892472601 z 42 44 46 - 2707311892472601 z + 1390314272117252 z - 587953308656693 z 58 56 54 - 43349956232 z + 369424135055 z - 2484664303032 z 52 60 70 68 + 13354488862536 z + 3949459806 z - 165 z + 11900 z 32 38 40 + 4351238776817281 z - 5781004446072928 z + 4351238776817281 z 62 72 / 2 28 - 273302118 z + z ) / ((-1 + z ) (1 + 4256348864913049 z / 26 2 24 - 1743505218128054 z - 198 z + 582544072789135 z 22 4 6 8 10 - 158139618206230 z + 16416 z - 775656 z + 23783097 z - 508833932 z 12 14 18 16 + 7956067482 z - 93852508618 z - 6107177078552 z + 854529686253 z 50 48 20 - 158139618206230 z + 582544072789135 z + 34693046977509 z 36 34 66 + 20620167812567775 z - 18689466546378194 z - 775656 z 64 30 42 + 23783097 z - 8499837502963788 z - 8499837502963788 z 44 46 58 + 4256348864913049 z - 1743505218128054 z - 93852508618 z 56 54 52 + 854529686253 z - 6107177078552 z + 34693046977509 z 60 70 68 32 + 7956067482 z - 198 z + 16416 z + 13912618033649761 z 38 40 62 72 - 18689466546378194 z + 13912618033649761 z - 508833932 z + z )) And in Maple-input format, it is: -(1+1390314272117252*z^28-587953308656693*z^26-165*z^2+204100171371196*z^24-\ 57910146705556*z^22+11900*z^4-501515*z^6+13937636*z^8-273302118*z^10+3949459806 *z^12-43349956232*z^14-2484664303032*z^18+369424135055*z^16-57910146705556*z^50 +204100171371196*z^48+13354488862536*z^20+6354653971956354*z^36-\ 5781004446072928*z^34-501515*z^66+13937636*z^64-2707311892472601*z^30-\ 2707311892472601*z^42+1390314272117252*z^44-587953308656693*z^46-43349956232*z^ 58+369424135055*z^56-2484664303032*z^54+13354488862536*z^52+3949459806*z^60-165 *z^70+11900*z^68+4351238776817281*z^32-5781004446072928*z^38+4351238776817281*z ^40-273302118*z^62+z^72)/(-1+z^2)/(1+4256348864913049*z^28-1743505218128054*z^ 26-198*z^2+582544072789135*z^24-158139618206230*z^22+16416*z^4-775656*z^6+ 23783097*z^8-508833932*z^10+7956067482*z^12-93852508618*z^14-6107177078552*z^18 +854529686253*z^16-158139618206230*z^50+582544072789135*z^48+34693046977509*z^ 20+20620167812567775*z^36-18689466546378194*z^34-775656*z^66+23783097*z^64-\ 8499837502963788*z^30-8499837502963788*z^42+4256348864913049*z^44-\ 1743505218128054*z^46-93852508618*z^58+854529686253*z^56-6107177078552*z^54+ 34693046977509*z^52+7956067482*z^60-198*z^70+16416*z^68+13912618033649761*z^32-\ 18689466546378194*z^38+13912618033649761*z^40-508833932*z^62+z^72) The first , 40, terms are: [0, 34, 0, 2052, 0, 134029, 0, 8889174, 0, 591836873, 0, 39450395201, 0, 2630713674968, 0, 175452226085949, 0, 11702220263053001, 0, 780525567686159590, 0, 52060676851619768737, 0, 3472434335215950956612, 0, 231610858100223737682298, 0, 15448419592640277108457377, 0, 1030408209215093741289146737, 0, 68728143150840798191528591370, 0, 4584161729841896718410304316196, 0, 305763231973264396223074363806649, 0, 20394384037443213281968905138648742, 0, 1360303850361950102850903612580689369] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 19586656024044 z + 12656109462127 z + 137 z 24 22 4 6 8 - 6559578448079 z + 2719315367861 z - 7959 z + 264659 z - 5713857 z 10 12 14 18 + 85878774 z - 938909806 z + 7691814790 z + 234775324223 z 16 50 48 20 - 48215179502 z + 938909806 z - 7691814790 z - 897919599453 z 36 34 30 - 12656109462127 z + 19586656024044 z + 24356296442708 z 42 44 46 58 + 897919599453 z - 234775324223 z + 48215179502 z + 7959 z 56 54 52 60 32 - 264659 z + 5713857 z - 85878774 z - 137 z - 24356296442708 z 38 40 62 / + 6559578448079 z - 2719315367861 z + z ) / (1 / 28 26 2 24 + 100550791319428 z - 58876176846836 z - 168 z + 27738951557793 z 22 4 6 8 10 - 10477558925252 z + 11564 z - 445616 z + 10962165 z - 185300118 z 12 14 18 16 + 2256097690 z - 20436594186 z - 753128976086 z + 140964312179 z 50 48 20 - 20436594186 z + 140964312179 z + 3156529103328 z 36 34 64 30 + 100550791319428 z - 138504619201876 z + z - 138504619201876 z 42 44 46 58 - 10477558925252 z + 3156529103328 z - 753128976086 z - 445616 z 56 54 52 60 + 10962165 z - 185300118 z + 2256097690 z + 11564 z 32 38 40 62 + 154085206579117 z - 58876176846836 z + 27738951557793 z - 168 z ) And in Maple-input format, it is: -(-1-19586656024044*z^28+12656109462127*z^26+137*z^2-6559578448079*z^24+ 2719315367861*z^22-7959*z^4+264659*z^6-5713857*z^8+85878774*z^10-938909806*z^12 +7691814790*z^14+234775324223*z^18-48215179502*z^16+938909806*z^50-7691814790*z ^48-897919599453*z^20-12656109462127*z^36+19586656024044*z^34+24356296442708*z^ 30+897919599453*z^42-234775324223*z^44+48215179502*z^46+7959*z^58-264659*z^56+ 5713857*z^54-85878774*z^52-137*z^60-24356296442708*z^32+6559578448079*z^38-\ 2719315367861*z^40+z^62)/(1+100550791319428*z^28-58876176846836*z^26-168*z^2+ 27738951557793*z^24-10477558925252*z^22+11564*z^4-445616*z^6+10962165*z^8-\ 185300118*z^10+2256097690*z^12-20436594186*z^14-753128976086*z^18+140964312179* z^16-20436594186*z^50+140964312179*z^48+3156529103328*z^20+100550791319428*z^36 -138504619201876*z^34+z^64-138504619201876*z^30-10477558925252*z^42+ 3156529103328*z^44-753128976086*z^46-445616*z^58+10962165*z^56-185300118*z^54+ 2256097690*z^52+11564*z^60+154085206579117*z^32-58876176846836*z^38+ 27738951557793*z^40-168*z^62) The first , 40, terms are: [0, 31, 0, 1603, 0, 91777, 0, 5447232, 0, 327742425, 0, 19821001463, 0, 1201055800951, 0, 72831675742861, 0, 4417724265322085, 0, 267992725972551543, 0, 16257920454549532279, 0, 986310625992345932321, 0, 59836338708600723713664, 0, 3630089269695479163499337, 0, 220226705308424887383823891, 0, 13360502915710699657136981903, 0, 810542306834159957041231031961, 0, 49173213689196701427389629857321, 0, 2983194057090825241704418123226383, 0, 180981599203525007122797871515928051 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 5}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 8923959376349901582 z + 1491219185230330233 z + 229 z 24 22 4 6 - 215072057730743237 z + 26615855832595286 z - 24036 z + 1545660 z 102 8 10 12 + 68668934 z - 68668934 z + 2252844322 z - 56954629458 z 14 18 16 + 1142970438298 z + 249956602165063 z - 18609279547159 z 50 48 + 493498533698807831091761 z - 342681319128219038575431 z 20 36 - 2806356856726350 z - 2862437847408543510626 z 34 66 + 826956903863243910379 z + 114430015670644433689252 z 80 100 90 - 46318372913503628828 z - 2252844322 z + 2806356856726350 z 88 84 94 - 26615855832595286 z - 1491219185230330233 z + 18609279547159 z 86 96 98 + 215072057730743237 z - 1142970438298 z + 56954629458 z 92 82 - 249956602165063 z + 8923959376349901582 z 64 110 106 108 - 210571009046315077001082 z + z + 24036 z - 229 z 30 42 + 46318372913503628828 z + 54948774572751179211238 z 44 46 - 114430015670644433689252 z + 210571009046315077001082 z 58 56 + 629185819907997429933504 z - 710386648123192839518334 z 54 52 + 710386648123192839518334 z - 629185819907997429933504 z 60 70 - 493498533698807831091761 z + 23291509669845004114046 z 68 78 - 54948774572751179211238 z + 209363547292384914265 z 32 38 - 209363547292384914265 z + 8703426474333212256650 z 40 62 - 23291509669845004114046 z + 342681319128219038575431 z 76 74 - 826956903863243910379 z + 2862437847408543510626 z 72 104 / - 8703426474333212256650 z - 1545660 z ) / (1 / 28 26 2 + 26435090235883364130 z - 4183970629558023886 z - 265 z 24 22 4 6 + 571052986130140449 z - 66800056222734898 z + 31312 z - 2226765 z 102 8 10 12 - 3840924577 z + 108085283 z - 3840924577 z + 104512739843 z 14 18 16 - 2246543195910 z - 557814042374452 z + 39032175268296 z 50 48 - 2653966190699645498156763 z + 1740944140515521199738827 z 20 36 + 6647828051894769 z + 10492461555758995826691 z 34 66 - 2874663497837989248480 z - 1011501298041073149954095 z 80 100 90 + 690177557457614561098 z + 104512739843 z - 66800056222734898 z 88 84 94 + 571052986130140449 z + 26435090235883364130 z - 557814042374452 z 86 96 98 - 4183970629558023886 z + 39032175268296 z - 2246543195910 z 92 82 + 6647828051894769 z - 144771097473204006130 z 64 112 110 106 + 1740944140515521199738827 z + z - 265 z - 2226765 z 108 30 42 + 31312 z - 144771097473204006130 z - 236519560804551952863467 z 44 46 + 520149384287953495551472 z - 1011501298041073149954095 z 58 56 - 4293866085689884856832788 z + 4559785342452124570142491 z 54 52 - 4293866085689884856832788 z + 3585337168372378899236186 z 60 70 + 3585337168372378899236186 z - 236519560804551952863467 z 68 78 + 520149384287953495551472 z - 2874663497837989248480 z 32 38 + 690177557457614561098 z - 33646081933090607180518 z 40 62 + 94989886748988138284568 z - 2653966190699645498156763 z 76 74 + 10492461555758995826691 z - 33646081933090607180518 z 72 104 + 94989886748988138284568 z + 108085283 z ) And in Maple-input format, it is: -(-1-8923959376349901582*z^28+1491219185230330233*z^26+229*z^2-\ 215072057730743237*z^24+26615855832595286*z^22-24036*z^4+1545660*z^6+68668934*z ^102-68668934*z^8+2252844322*z^10-56954629458*z^12+1142970438298*z^14+ 249956602165063*z^18-18609279547159*z^16+493498533698807831091761*z^50-\ 342681319128219038575431*z^48-2806356856726350*z^20-2862437847408543510626*z^36 +826956903863243910379*z^34+114430015670644433689252*z^66-46318372913503628828* z^80-2252844322*z^100+2806356856726350*z^90-26615855832595286*z^88-\ 1491219185230330233*z^84+18609279547159*z^94+215072057730743237*z^86-\ 1142970438298*z^96+56954629458*z^98-249956602165063*z^92+8923959376349901582*z^ 82-210571009046315077001082*z^64+z^110+24036*z^106-229*z^108+ 46318372913503628828*z^30+54948774572751179211238*z^42-114430015670644433689252 *z^44+210571009046315077001082*z^46+629185819907997429933504*z^58-\ 710386648123192839518334*z^56+710386648123192839518334*z^54-\ 629185819907997429933504*z^52-493498533698807831091761*z^60+ 23291509669845004114046*z^70-54948774572751179211238*z^68+209363547292384914265 *z^78-209363547292384914265*z^32+8703426474333212256650*z^38-\ 23291509669845004114046*z^40+342681319128219038575431*z^62-\ 826956903863243910379*z^76+2862437847408543510626*z^74-8703426474333212256650*z ^72-1545660*z^104)/(1+26435090235883364130*z^28-4183970629558023886*z^26-265*z^ 2+571052986130140449*z^24-66800056222734898*z^22+31312*z^4-2226765*z^6-\ 3840924577*z^102+108085283*z^8-3840924577*z^10+104512739843*z^12-2246543195910* z^14-557814042374452*z^18+39032175268296*z^16-2653966190699645498156763*z^50+ 1740944140515521199738827*z^48+6647828051894769*z^20+10492461555758995826691*z^ 36-2874663497837989248480*z^34-1011501298041073149954095*z^66+ 690177557457614561098*z^80+104512739843*z^100-66800056222734898*z^90+ 571052986130140449*z^88+26435090235883364130*z^84-557814042374452*z^94-\ 4183970629558023886*z^86+39032175268296*z^96-2246543195910*z^98+ 6647828051894769*z^92-144771097473204006130*z^82+1740944140515521199738827*z^64 +z^112-265*z^110-2226765*z^106+31312*z^108-144771097473204006130*z^30-\ 236519560804551952863467*z^42+520149384287953495551472*z^44-\ 1011501298041073149954095*z^46-4293866085689884856832788*z^58+ 4559785342452124570142491*z^56-4293866085689884856832788*z^54+ 3585337168372378899236186*z^52+3585337168372378899236186*z^60-\ 236519560804551952863467*z^70+520149384287953495551472*z^68-\ 2874663497837989248480*z^78+690177557457614561098*z^32-33646081933090607180518* z^38+94989886748988138284568*z^40-2653966190699645498156763*z^62+ 10492461555758995826691*z^76-33646081933090607180518*z^74+ 94989886748988138284568*z^72+108085283*z^104) The first , 40, terms are: [0, 36, 0, 2264, 0, 153833, 0, 10622568, 0, 736567651, 0, 51136612219, 0, 3551642608296, 0, 246712253090095, 0, 17138656138648615, 0, 1190618633774992340, 0, 82712792755285815265, 0, 5746116749456077432808, 0, 399187534336131201102096, 0, 27731913379054250135013461, 0, 1926561385944090391553543849, 0, 133840002805428159483642007256, 0, 9297989560352804451288403910160, 0, 645940010031599489751985598738289, 0, 44874055868758915198400652445398620, 0, 3117442600285624171957684962349442755] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 10575293100080838008 z - 1696765943158318504 z - 224 z 24 22 4 6 + 235695486927327420 z - 28189002171036088 z + 23112 z - 1469304 z 102 8 10 12 - 2130630680 z + 64916950 z - 2130630680 z + 54199412416 z 14 18 16 - 1100489198460 z - 250155857834284 z + 18223188804395 z 50 48 - 1039560702597932965285864 z + 680987742512406367955611 z 20 36 + 2883300366431192 z + 4091623885551237207824 z 34 66 - 1124358351454048166812 z - 395060121436770120842952 z 80 100 90 + 271288046677460023097 z + 54199412416 z - 28189002171036088 z 88 84 94 + 235695486927327420 z + 10575293100080838008 z - 250155857834284 z 86 96 98 - 1696765943158318504 z + 18223188804395 z - 1100489198460 z 92 82 + 2883300366431192 z - 57325422421280692268 z 64 112 110 106 + 680987742512406367955611 z + z - 224 z - 1469304 z 108 30 42 + 23112 z - 57325422421280692268 z - 92134983903980800939288 z 44 46 + 202858222122954276891368 z - 395060121436770120842952 z 58 56 - 1684940595421350636322848 z + 1789721492241531297823944 z 54 52 - 1684940595421350636322848 z + 1405921030156692398572464 z 60 70 + 1405921030156692398572464 z - 92134983903980800939288 z 68 78 + 202858222122954276891368 z - 1124358351454048166812 z 32 38 + 271288046677460023097 z - 13101793385638104016856 z 40 62 + 36980138990542539959818 z - 1039560702597932965285864 z 76 74 + 4091623885551237207824 z - 13101793385638104016856 z 72 104 / 2 + 36980138990542539959818 z + 64916950 z ) / ((-1 + z ) (1 / 28 26 2 + 26278360798013779440 z - 4068647601892559772 z - 256 z 24 22 4 6 + 544072580813210044 z - 62482516801814524 z + 29424 z - 2048716 z 102 8 10 12 - 3449129072 z + 97961726 z - 3449129072 z + 93486419456 z 14 18 16 - 2011546147584 z - 506944474769720 z + 35140176219747 z 50 48 - 3283405931882513168235840 z + 2130324327792114231043851 z 20 36 + 6120045116016336 z + 11461626394856949316096 z 34 66 - 3066907131032084702800 z - 1220836735745347640490800 z 80 100 90 + 718956479277021167505 z + 93486419456 z - 62482516801814524 z 88 84 94 + 544072580813210044 z + 26278360798013779440 z - 506944474769720 z 86 96 98 - 4068647601892559772 z + 35140176219747 z - 2011546147584 z 92 82 + 6120045116016336 z - 147274054066716927496 z 64 112 110 106 + 2130324327792114231043851 z + z - 256 z - 2048716 z 108 30 42 + 29424 z - 147274054066716927496 z - 275731468370146193486892 z 44 46 + 617677063814497488778768 z - 1220836735745347640490800 z 58 56 - 5381026240133187622734120 z + 5723609399700127068434248 z 54 52 - 5381026240133187622734120 z + 4471300323337321453748800 z 60 70 + 4471300323337321453748800 z - 275731468370146193486892 z 68 78 + 617677063814497488778768 z - 3066907131032084702800 z 32 38 + 718956479277021167505 z - 37605971939186564190680 z 40 62 + 108510834637841760752290 z - 3283405931882513168235840 z 76 74 + 11461626394856949316096 z - 37605971939186564190680 z 72 104 + 108510834637841760752290 z + 97961726 z )) And in Maple-input format, it is: -(1+10575293100080838008*z^28-1696765943158318504*z^26-224*z^2+ 235695486927327420*z^24-28189002171036088*z^22+23112*z^4-1469304*z^6-2130630680 *z^102+64916950*z^8-2130630680*z^10+54199412416*z^12-1100489198460*z^14-\ 250155857834284*z^18+18223188804395*z^16-1039560702597932965285864*z^50+ 680987742512406367955611*z^48+2883300366431192*z^20+4091623885551237207824*z^36 -1124358351454048166812*z^34-395060121436770120842952*z^66+ 271288046677460023097*z^80+54199412416*z^100-28189002171036088*z^90+ 235695486927327420*z^88+10575293100080838008*z^84-250155857834284*z^94-\ 1696765943158318504*z^86+18223188804395*z^96-1100489198460*z^98+ 2883300366431192*z^92-57325422421280692268*z^82+680987742512406367955611*z^64+z ^112-224*z^110-1469304*z^106+23112*z^108-57325422421280692268*z^30-\ 92134983903980800939288*z^42+202858222122954276891368*z^44-\ 395060121436770120842952*z^46-1684940595421350636322848*z^58+ 1789721492241531297823944*z^56-1684940595421350636322848*z^54+ 1405921030156692398572464*z^52+1405921030156692398572464*z^60-\ 92134983903980800939288*z^70+202858222122954276891368*z^68-\ 1124358351454048166812*z^78+271288046677460023097*z^32-13101793385638104016856* z^38+36980138990542539959818*z^40-1039560702597932965285864*z^62+ 4091623885551237207824*z^76-13101793385638104016856*z^74+ 36980138990542539959818*z^72+64916950*z^104)/(-1+z^2)/(1+26278360798013779440*z ^28-4068647601892559772*z^26-256*z^2+544072580813210044*z^24-62482516801814524* z^22+29424*z^4-2048716*z^6-3449129072*z^102+97961726*z^8-3449129072*z^10+ 93486419456*z^12-2011546147584*z^14-506944474769720*z^18+35140176219747*z^16-\ 3283405931882513168235840*z^50+2130324327792114231043851*z^48+6120045116016336* z^20+11461626394856949316096*z^36-3066907131032084702800*z^34-\ 1220836735745347640490800*z^66+718956479277021167505*z^80+93486419456*z^100-\ 62482516801814524*z^90+544072580813210044*z^88+26278360798013779440*z^84-\ 506944474769720*z^94-4068647601892559772*z^86+35140176219747*z^96-2011546147584 *z^98+6120045116016336*z^92-147274054066716927496*z^82+ 2130324327792114231043851*z^64+z^112-256*z^110-2048716*z^106+29424*z^108-\ 147274054066716927496*z^30-275731468370146193486892*z^42+ 617677063814497488778768*z^44-1220836735745347640490800*z^46-\ 5381026240133187622734120*z^58+5723609399700127068434248*z^56-\ 5381026240133187622734120*z^54+4471300323337321453748800*z^52+ 4471300323337321453748800*z^60-275731468370146193486892*z^70+ 617677063814497488778768*z^68-3066907131032084702800*z^78+718956479277021167505 *z^32-37605971939186564190680*z^38+108510834637841760752290*z^40-\ 3283405931882513168235840*z^62+11461626394856949316096*z^76-\ 37605971939186564190680*z^74+108510834637841760752290*z^72+97961726*z^104) The first , 40, terms are: [0, 33, 0, 1913, 0, 121037, 0, 7813797, 0, 507365021, 0, 33009031293, 0, 2149182411977, 0, 139975149907361, 0, 9117776529137405, 0, 593956031880109645, 0, 38692987389824939725, 0, 2520670609526207765221, 0, 164211164763289725490953, 0, 10697703477881617679272865, 0, 696913755988416394282177417, 0, 45401249323530334919663841241, 0, 2957717595839048054062163374017, 0, 192684007255533992665673932701449, 0, 12552628168134453139513971276814677, 0, 817755875448299983492157631323645917] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 106439494967911853 z - 27874893210314997 z - 189 z 24 22 4 6 + 6158729266942084 z - 1142097301513158 z + 16060 z - 820574 z 8 10 12 14 + 28435471 z - 714401072 z + 13578124685 z - 201068995308 z 18 16 50 - 22605634960068 z + 2370216177637 z - 7611838705436669142 z 48 20 36 + 11171241812282248876 z + 176639769427091 z + 4443331765156282684 z 34 66 80 88 - 2219296875696117776 z - 1142097301513158 z + 28435471 z + z 84 86 82 64 + 16060 z - 189 z - 820574 z + 6158729266942084 z 30 42 - 344326486923550266 z - 14057919074178310950 z 44 46 + 15176438406071797208 z - 14057919074178310950 z 58 56 - 344326486923550266 z + 946850911864646534 z 54 52 - 2219296875696117776 z + 4443331765156282684 z 60 70 68 + 106439494967911853 z - 22605634960068 z + 176639769427091 z 78 32 38 - 714401072 z + 946850911864646534 z - 7611838705436669142 z 40 62 76 + 11171241812282248876 z - 27874893210314997 z + 13578124685 z 74 72 / 2 - 201068995308 z + 2370216177637 z ) / ((-1 + z ) (1 / 28 26 2 + 296288596413823100 z - 75093773101881972 z - 218 z 24 22 4 6 + 15989923464081357 z - 2845188153400083 z + 20920 z - 1189075 z 8 10 12 14 + 45301675 z - 1239139126 z + 25426837610 z - 403509078265 z 18 16 50 - 51104598811164 z + 5063708572399 z - 23538709095175615674 z 48 20 + 34878775586377455245 z + 420253260600298 z 36 34 + 13557518925057616889 z - 6656036538327347274 z 66 80 88 84 86 - 2845188153400083 z + 45301675 z + z + 20920 z - 218 z 82 64 30 - 1189075 z + 15989923464081357 z - 986420374009094594 z 42 44 - 44145358719611684518 z + 47749563624253275725 z 46 58 - 44145358719611684518 z - 986420374009094594 z 56 54 + 2780735011874374593 z - 6656036538327347274 z 52 60 70 + 13557518925057616889 z + 296288596413823100 z - 51104598811164 z 68 78 32 + 420253260600298 z - 1239139126 z + 2780735011874374593 z 38 40 - 23538709095175615674 z + 34878775586377455245 z 62 76 74 - 75093773101881972 z + 25426837610 z - 403509078265 z 72 + 5063708572399 z )) And in Maple-input format, it is: -(1+106439494967911853*z^28-27874893210314997*z^26-189*z^2+6158729266942084*z^ 24-1142097301513158*z^22+16060*z^4-820574*z^6+28435471*z^8-714401072*z^10+ 13578124685*z^12-201068995308*z^14-22605634960068*z^18+2370216177637*z^16-\ 7611838705436669142*z^50+11171241812282248876*z^48+176639769427091*z^20+ 4443331765156282684*z^36-2219296875696117776*z^34-1142097301513158*z^66+ 28435471*z^80+z^88+16060*z^84-189*z^86-820574*z^82+6158729266942084*z^64-\ 344326486923550266*z^30-14057919074178310950*z^42+15176438406071797208*z^44-\ 14057919074178310950*z^46-344326486923550266*z^58+946850911864646534*z^56-\ 2219296875696117776*z^54+4443331765156282684*z^52+106439494967911853*z^60-\ 22605634960068*z^70+176639769427091*z^68-714401072*z^78+946850911864646534*z^32 -7611838705436669142*z^38+11171241812282248876*z^40-27874893210314997*z^62+ 13578124685*z^76-201068995308*z^74+2370216177637*z^72)/(-1+z^2)/(1+ 296288596413823100*z^28-75093773101881972*z^26-218*z^2+15989923464081357*z^24-\ 2845188153400083*z^22+20920*z^4-1189075*z^6+45301675*z^8-1239139126*z^10+ 25426837610*z^12-403509078265*z^14-51104598811164*z^18+5063708572399*z^16-\ 23538709095175615674*z^50+34878775586377455245*z^48+420253260600298*z^20+ 13557518925057616889*z^36-6656036538327347274*z^34-2845188153400083*z^66+ 45301675*z^80+z^88+20920*z^84-218*z^86-1189075*z^82+15989923464081357*z^64-\ 986420374009094594*z^30-44145358719611684518*z^42+47749563624253275725*z^44-\ 44145358719611684518*z^46-986420374009094594*z^58+2780735011874374593*z^56-\ 6656036538327347274*z^54+13557518925057616889*z^52+296288596413823100*z^60-\ 51104598811164*z^70+420253260600298*z^68-1239139126*z^78+2780735011874374593*z^ 32-23538709095175615674*z^38+34878775586377455245*z^40-75093773101881972*z^62+ 25426837610*z^76-403509078265*z^74+5063708572399*z^72) The first , 40, terms are: [0, 30, 0, 1492, 0, 82029, 0, 4671026, 0, 269655461, 0, 15654251205, 0, 910905545976, 0, 53057843298989, 0, 3091818815589685, 0, 180202338166143234, 0, 10503707874137179533, 0, 612266577714814229364, 0, 35689902979431800214190, 0, 2080430508421181758633625, 0, 121272517918840068864204385, 0, 7069230837173215868292042766, 0, 412080620534416649567234078740, 0, 24021069110435174891910257316805, 0, 1400240145897263767655196362868578, 0, 81623035084205614899069678587098237] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1374924671543832 z - 580508632150433 z - 165 z 24 22 4 6 + 201165600452726 z - 56979325119022 z + 11856 z - 497311 z 8 10 12 14 + 13761514 z - 268955140 z + 3878152446 z - 42520176168 z 18 16 50 - 2438144668912 z + 362292278421 z - 56979325119022 z 48 20 36 + 201165600452726 z + 13119686362588 z + 6304852799700438 z 34 66 64 30 - 5734448644017584 z - 497311 z + 13761514 z - 2681007017424625 z 42 44 46 - 2681007017424625 z + 1374924671543832 z - 580508632150433 z 58 56 54 - 42520176168 z + 362292278421 z - 2438144668912 z 52 60 70 68 + 13119686362588 z + 3878152446 z - 165 z + 11856 z 32 38 40 + 4313422011210541 z - 5734448644017584 z + 4313422011210541 z 62 72 / 28 - 268955140 z + z ) / (-1 - 5877682257386085 z / 26 2 24 + 2281603299863925 z + 197 z - 727561396535431 z 22 4 6 8 10 + 189690460379171 z - 16384 z + 780144 z - 24192403 z + 525049669 z 12 14 18 16 - 8350629480 z + 100470690920 z + 6864732041485 z - 935783568481 z 50 48 20 + 727561396535431 z - 2281603299863925 z - 40188981555915 z 36 34 66 - 38405262474194607 z + 31860710170465497 z + 24192403 z 64 30 42 - 525049669 z + 12483095309099969 z + 21914754257106405 z 44 46 58 - 12483095309099969 z + 5877682257386085 z + 935783568481 z 56 54 52 - 6864732041485 z + 40188981555915 z - 189690460379171 z 60 70 68 32 - 100470690920 z + 16384 z - 780144 z - 21914754257106405 z 38 40 62 74 + 38405262474194607 z - 31860710170465497 z + 8350629480 z + z 72 - 197 z ) And in Maple-input format, it is: -(1+1374924671543832*z^28-580508632150433*z^26-165*z^2+201165600452726*z^24-\ 56979325119022*z^22+11856*z^4-497311*z^6+13761514*z^8-268955140*z^10+3878152446 *z^12-42520176168*z^14-2438144668912*z^18+362292278421*z^16-56979325119022*z^50 +201165600452726*z^48+13119686362588*z^20+6304852799700438*z^36-\ 5734448644017584*z^34-497311*z^66+13761514*z^64-2681007017424625*z^30-\ 2681007017424625*z^42+1374924671543832*z^44-580508632150433*z^46-42520176168*z^ 58+362292278421*z^56-2438144668912*z^54+13119686362588*z^52+3878152446*z^60-165 *z^70+11856*z^68+4313422011210541*z^32-5734448644017584*z^38+4313422011210541*z ^40-268955140*z^62+z^72)/(-1-5877682257386085*z^28+2281603299863925*z^26+197*z^ 2-727561396535431*z^24+189690460379171*z^22-16384*z^4+780144*z^6-24192403*z^8+ 525049669*z^10-8350629480*z^12+100470690920*z^14+6864732041485*z^18-\ 935783568481*z^16+727561396535431*z^50-2281603299863925*z^48-40188981555915*z^ 20-38405262474194607*z^36+31860710170465497*z^34+24192403*z^66-525049669*z^64+ 12483095309099969*z^30+21914754257106405*z^42-12483095309099969*z^44+ 5877682257386085*z^46+935783568481*z^58-6864732041485*z^56+40188981555915*z^54-\ 189690460379171*z^52-100470690920*z^60+16384*z^70-780144*z^68-21914754257106405 *z^32+38405262474194607*z^38-31860710170465497*z^40+8350629480*z^62+z^74-197*z^ 72) The first , 40, terms are: [0, 32, 0, 1776, 0, 108417, 0, 6793884, 0, 429564397, 0, 27257467447, 0, 1732296673392, 0, 110172968103365, 0, 7009374506151091, 0, 446022904890675924, 0, 28383842088333104083, 0, 1806354470567275966240, 0, 114959148591855924123680, 0, 7316248096374767430458575, 0, 465624093892310973403417391, 0, 29633538994641996569903994496, 0, 1885958433039297877850465293184, 0, 120027557922543260824131489894147, 0, 7638884624333748483756341849932356, 0, 486159743210651528487320834577318035 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 51969 z + 494976 z + 92 z - 2857003 z + 10379860 z 4 6 8 10 12 - 3095 z + 51969 z - 494976 z + 2857003 z - 10379860 z 14 18 16 20 34 + 24286948 z + 37056488 z - 37056488 z - 24286948 z + z 30 32 / 36 34 32 30 + 3095 z - 92 z ) / (z - 128 z + 5392 z - 110230 z / 28 26 24 22 20 + 1266004 z - 8778676 z + 38333275 z - 108202852 z + 200591988 z 18 16 14 12 - 246225548 z + 200591988 z - 108202852 z + 38333275 z 10 8 6 4 2 - 8778676 z + 1266004 z - 110230 z + 5392 z - 128 z + 1) And in Maple-input format, it is: -(-1-51969*z^28+494976*z^26+92*z^2-2857003*z^24+10379860*z^22-3095*z^4+51969*z^ 6-494976*z^8+2857003*z^10-10379860*z^12+24286948*z^14+37056488*z^18-37056488*z^ 16-24286948*z^20+z^34+3095*z^30-92*z^32)/(z^36-128*z^34+5392*z^32-110230*z^30+ 1266004*z^28-8778676*z^26+38333275*z^24-108202852*z^22+200591988*z^20-246225548 *z^18+200591988*z^16-108202852*z^14+38333275*z^12-8778676*z^10+1266004*z^8-\ 110230*z^6+5392*z^4-128*z^2+1) The first , 40, terms are: [0, 36, 0, 2311, 0, 159957, 0, 11210836, 0, 787585923, 0, 55356574219, 0, 3891233891908, 0, 273537277293901, 0, 19228634261662495, 0, 1351702320345690964, 0, 95019745671385580777, 0, 6679542494052917154393, 0, 469547555337321838048884, 0, 33007486476877971485892015, 0, 2320306332312389218731414173, 0, 163109102016722196804027749092, 0, 11465977054266716381840507358395, 0, 806016514035647444512603541956179, 0, 56660031485461993510115224509610164, 0, 3982994283666255526729775924171500037] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {4, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 475752099300816 z - 235489077564606 z - 169 z 24 22 4 6 + 94759605763974 z - 30849712444064 z + 12296 z - 513225 z 8 10 12 14 + 13858597 z - 259349668 z + 3519211069 z - 35743341737 z 18 16 50 - 1686357289625 z + 278095215400 z - 1686357289625 z 48 20 36 + 8075498216249 z + 8075498216249 z + 1057344065092510 z 34 66 64 30 - 1167902062274168 z - 169 z + 12296 z - 784200715032014 z 42 44 46 - 235489077564606 z + 94759605763974 z - 30849712444064 z 58 56 54 52 - 259349668 z + 3519211069 z - 35743341737 z + 278095215400 z 60 68 32 38 + 13858597 z + z + 1057344065092510 z - 784200715032014 z 40 62 / 28 + 475752099300816 z - 513225 z ) / (-1 - 2271783461012146 z / 26 2 24 + 1021954711830062 z + 211 z - 374035194772746 z 22 4 6 8 10 + 110857483304757 z - 18061 z + 858163 z - 25887065 z + 535531071 z 12 14 18 16 - 7988397899 z + 88957227261 z + 5036963544513 z - 758170629183 z 50 48 20 + 26444364715423 z - 110857483304757 z - 26444364715423 z 36 34 66 64 - 7465669875771958 z + 7465669875771958 z + 18061 z - 858163 z 30 42 44 + 4123513324879630 z + 2271783461012146 z - 1021954711830062 z 46 58 56 + 374035194772746 z + 7988397899 z - 88957227261 z 54 52 60 70 68 + 758170629183 z - 5036963544513 z - 535531071 z + z - 211 z 32 38 40 - 6127119856430410 z + 6127119856430410 z - 4123513324879630 z 62 + 25887065 z ) And in Maple-input format, it is: -(1+475752099300816*z^28-235489077564606*z^26-169*z^2+94759605763974*z^24-\ 30849712444064*z^22+12296*z^4-513225*z^6+13858597*z^8-259349668*z^10+3519211069 *z^12-35743341737*z^14-1686357289625*z^18+278095215400*z^16-1686357289625*z^50+ 8075498216249*z^48+8075498216249*z^20+1057344065092510*z^36-1167902062274168*z^ 34-169*z^66+12296*z^64-784200715032014*z^30-235489077564606*z^42+94759605763974 *z^44-30849712444064*z^46-259349668*z^58+3519211069*z^56-35743341737*z^54+ 278095215400*z^52+13858597*z^60+z^68+1057344065092510*z^32-784200715032014*z^38 +475752099300816*z^40-513225*z^62)/(-1-2271783461012146*z^28+1021954711830062*z ^26+211*z^2-374035194772746*z^24+110857483304757*z^22-18061*z^4+858163*z^6-\ 25887065*z^8+535531071*z^10-7988397899*z^12+88957227261*z^14+5036963544513*z^18 -758170629183*z^16+26444364715423*z^50-110857483304757*z^48-26444364715423*z^20 -7465669875771958*z^36+7465669875771958*z^34+18061*z^66-858163*z^64+ 4123513324879630*z^30+2271783461012146*z^42-1021954711830062*z^44+ 374035194772746*z^46+7988397899*z^58-88957227261*z^56+758170629183*z^54-\ 5036963544513*z^52-535531071*z^60+z^70-211*z^68-6127119856430410*z^32+ 6127119856430410*z^38-4123513324879630*z^40+25887065*z^62) The first , 40, terms are: [0, 42, 0, 3097, 0, 239843, 0, 18686334, 0, 1457667535, 0, 113749237767, 0, 8877485833502, 0, 692865968027459, 0, 54077259633271849, 0, 4220678779973834474, 0, 329420555465267687497, 0, 25711023281893339695577, 0, 2006726218040682764649386, 0, 156623499190380618026914985, 0, 12224348781194093335031728995, 0, 954101430707842235424559944606, 0, 74466915151705299487933202356087, 0, 5812087982780892992998091205723935, 0, 453629194491703129957366484825742974, 0, 35405425166176061979959004584946815619] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {4, 5}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 6334062 z + 29727419 z + 101 z - 93165727 z 22 4 6 8 10 + 197948136 z - 3882 z + 76466 z - 879598 z + 6334062 z 12 14 18 16 - 29727419 z + 93165727 z + 287856608 z - 197948136 z 20 36 34 30 32 38 / - 287856608 z - 101 z + 3882 z + 879598 z - 76466 z + z ) / ( / 40 38 36 34 32 30 z - 133 z + 6439 z - 155208 z + 2132138 z - 18034248 z 28 26 24 22 + 98677369 z - 360678415 z + 899026207 z - 1547880496 z 20 18 16 14 + 1853826292 z - 1547880496 z + 899026207 z - 360678415 z 12 10 8 6 4 2 + 98677369 z - 18034248 z + 2132138 z - 155208 z + 6439 z - 133 z + 1) And in Maple-input format, it is: -(-1-6334062*z^28+29727419*z^26+101*z^2-93165727*z^24+197948136*z^22-3882*z^4+ 76466*z^6-879598*z^8+6334062*z^10-29727419*z^12+93165727*z^14+287856608*z^18-\ 197948136*z^16-287856608*z^20-101*z^36+3882*z^34+879598*z^30-76466*z^32+z^38)/( z^40-133*z^38+6439*z^36-155208*z^34+2132138*z^32-18034248*z^30+98677369*z^28-\ 360678415*z^26+899026207*z^24-1547880496*z^22+1853826292*z^20-1547880496*z^18+ 899026207*z^16-360678415*z^14+98677369*z^12-18034248*z^10+2132138*z^8-155208*z^ 6+6439*z^4-133*z^2+1) The first , 40, terms are: [0, 32, 0, 1699, 0, 98661, 0, 5896168, 0, 356082327, 0, 21592143751, 0, 1311464615288, 0, 79710826905653, 0, 4846258739263603, 0, 294680731755105712, 0, 17919313083434470257, 0, 1089686953865500953297, 0, 66265412100341145875600, 0, 4029713395181714659787539, 0, 245054310461579387769782485, 0, 14902219144782288902115917400, 0, 906232697757874277922113227751, 0, 55109769061098001682520462725559, 0, 3351332284003326759948711467723528, 0, 203801044356166580767181188753124421 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 96756 z + 1050762 z + 115 z - 6650750 z + 25592708 z 4 6 8 10 12 - 4832 z + 96756 z - 1050762 z + 6650750 z - 25592708 z 14 18 16 20 34 + 61642712 z + 95074132 z - 95074132 z - 61642712 z + z 30 32 / 20 16 + 4832 z - 115 z ) / (529158656 z + 1 + 529158656 z / 26 32 4 2 14 28 - 20420664 z + 8511 z + 8511 z - 162 z - 279489828 z + 2684242 z 18 6 30 12 10 - 653678636 z - 205932 z - 205932 z + 95098222 z - 20420664 z 24 36 34 8 22 + 95098222 z + z - 162 z + 2684242 z - 279489828 z ) And in Maple-input format, it is: -(-1-96756*z^28+1050762*z^26+115*z^2-6650750*z^24+25592708*z^22-4832*z^4+96756* z^6-1050762*z^8+6650750*z^10-25592708*z^12+61642712*z^14+95074132*z^18-95074132 *z^16-61642712*z^20+z^34+4832*z^30-115*z^32)/(529158656*z^20+1+529158656*z^16-\ 20420664*z^26+8511*z^32+8511*z^4-162*z^2-279489828*z^14+2684242*z^28-653678636* z^18-205932*z^6-205932*z^30+95098222*z^12-20420664*z^10+95098222*z^24+z^36-162* z^34+2684242*z^8-279489828*z^22) The first , 40, terms are: [0, 47, 0, 3935, 0, 346629, 0, 30708437, 0, 2722560335, 0, 241405043615, 0, 21405423326033, 0, 1898030555943025, 0, 168299603130855679, 0, 14923240724218268975, 0, 1323254109661657859445, 0, 117333865167652981925861, 0, 10404075851366011299004159, 0, 922536684197925057672759119, 0, 81801973334303535184147415649, 0, 7253438221700573366780187372705, 0, 643167443335620912198936188795407, 0, 57030107317442384684092869459513023, 0, 5056899528178186479460516246308642085, 0, 448398820223883782143078896372699427125] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 4 6 f(z) = - (1 + z - 84 z - 84 z + 2267 z - 28612 z + 2267 z - 28612 z 8 10 12 14 18 + 193208 z - 736916 z + 1623900 z - 2107592 z - 736916 z 16 20 / 2 22 18 + 1623900 z + 193208 z ) / ((-1 + z ) (-65996 z - 2219676 z / 20 12 14 24 26 28 + 518688 z + 5292646 z - 7079686 z + 1 + 4293 z - 123 z + z 4 8 2 6 16 10 + 4293 z + 518688 z - 123 z - 65996 z + 5292646 z - 2219676 z )) And in Maple-input format, it is: -(1+z^28-84*z^26-84*z^2+2267*z^24-28612*z^22+2267*z^4-28612*z^6+193208*z^8-\ 736916*z^10+1623900*z^12-2107592*z^14-736916*z^18+1623900*z^16+193208*z^20)/(-1 +z^2)/(-65996*z^22-2219676*z^18+518688*z^20+5292646*z^12-7079686*z^14+1+4293*z^ 24-123*z^26+z^28+4293*z^4+518688*z^8-123*z^2-65996*z^6+5292646*z^16-2219676*z^ 10) The first , 40, terms are: [0, 40, 0, 2811, 0, 213601, 0, 16493232, 0, 1278095219, 0, 99123594747, 0, 7689068271552, 0, 596471661555769, 0, 46271171195656115, 0, 3589485906427632952, 0, 278454518281400133417, 0, 21601123849258012394649, 0, 1675708370463474119460376, 0, 129993170014041373730043779, 0, 10084227410850290846764169289, 0, 782284503932851203527945623712, 0, 60685764037793159043707836585003, 0, 4707701531129316272039739854140867, 0, 365200208940729631574804063400485840, 0, 28330426584743177677191082876628757457] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 7}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 31897907726025 z - 19072441819085 z - 144 z 24 22 4 6 8 + 9253513246539 z - 3630235067864 z + 8653 z - 294005 z + 6439477 z 10 12 14 18 - 97919919 z + 1083861344 z - 9019315383 z - 289038361778 z 16 50 48 20 + 57732389262 z - 9019315383 z + 57732389262 z + 1145866977279 z 36 34 64 30 + 31897907726025 z - 43390047902194 z + z - 43390047902194 z 42 44 46 58 - 3630235067864 z + 1145866977279 z - 289038361778 z - 294005 z 56 54 52 60 + 6439477 z - 97919919 z + 1083861344 z + 8653 z 32 38 40 62 + 48069540079136 z - 19072441819085 z + 9253513246539 z - 144 z ) / 28 26 2 / (-1 - 167161932874657 z + 90576206744657 z + 188 z / 24 22 4 6 - 39866767667679 z + 14202958320961 z - 13445 z + 522058 z 8 10 12 14 - 12811749 z + 216073969 z - 2638546083 z + 24158299889 z 18 16 50 + 935011260279 z - 169954342509 z + 169954342509 z 48 20 36 - 935011260279 z - 4074962130601 z - 251203115573239 z 34 66 64 30 + 307834463427237 z + z - 188 z + 251203115573239 z 42 44 46 + 39866767667679 z - 14202958320961 z + 4074962130601 z 58 56 54 52 + 12811749 z - 216073969 z + 2638546083 z - 24158299889 z 60 32 38 - 522058 z - 307834463427237 z + 167161932874657 z 40 62 - 90576206744657 z + 13445 z ) And in Maple-input format, it is: -(1+31897907726025*z^28-19072441819085*z^26-144*z^2+9253513246539*z^24-\ 3630235067864*z^22+8653*z^4-294005*z^6+6439477*z^8-97919919*z^10+1083861344*z^ 12-9019315383*z^14-289038361778*z^18+57732389262*z^16-9019315383*z^50+ 57732389262*z^48+1145866977279*z^20+31897907726025*z^36-43390047902194*z^34+z^ 64-43390047902194*z^30-3630235067864*z^42+1145866977279*z^44-289038361778*z^46-\ 294005*z^58+6439477*z^56-97919919*z^54+1083861344*z^52+8653*z^60+48069540079136 *z^32-19072441819085*z^38+9253513246539*z^40-144*z^62)/(-1-167161932874657*z^28 +90576206744657*z^26+188*z^2-39866767667679*z^24+14202958320961*z^22-13445*z^4+ 522058*z^6-12811749*z^8+216073969*z^10-2638546083*z^12+24158299889*z^14+ 935011260279*z^18-169954342509*z^16+169954342509*z^50-935011260279*z^48-\ 4074962130601*z^20-251203115573239*z^36+307834463427237*z^34+z^66-188*z^64+ 251203115573239*z^30+39866767667679*z^42-14202958320961*z^44+4074962130601*z^46 +12811749*z^58-216073969*z^56+2638546083*z^54-24158299889*z^52-522058*z^60-\ 307834463427237*z^32+167161932874657*z^38-90576206744657*z^40+13445*z^62) The first , 40, terms are: [0, 44, 0, 3480, 0, 290713, 0, 24463724, 0, 2061742761, 0, 173829600619, 0, 14657754700656, 0, 1236029228852661, 0, 104230739116650463, 0, 8789513044034951124, 0, 741198365490030242211, 0, 62503495340252334279144, 0, 5270771975519560236713044, 0, 444471762695375540598131195, 0, 37481255571925689476116312319, 0, 3160705911658735268235083027444, 0, 266534877986064166260340549054328, 0, 22476257904955680069913392770662879, 0, 1895369841816369678587794109348928740, 0, 159832070477943091889271260274983079299] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1279689616299458 z - 544792159588993 z - 167 z 24 22 4 6 + 190616756694110 z - 54571717904158 z + 12102 z - 509399 z 8 10 12 14 + 14082482 z - 273985376 z + 3922039864 z - 42603253714 z 18 16 50 - 2389525131806 z + 359149084591 z - 54571717904158 z 48 20 36 + 190616756694110 z + 12709160212402 z + 5777400440808434 z 34 66 64 30 - 5260003351831032 z - 509399 z + 14082482 z - 2478682907141035 z 42 44 46 - 2478682907141035 z + 1279689616299458 z - 544792159588993 z 58 56 54 - 42603253714 z + 359149084591 z - 2389525131806 z 52 60 70 68 + 12709160212402 z + 3922039864 z - 167 z + 12102 z 32 38 40 + 3968403700797061 z - 5260003351831032 z + 3968403700797061 z 62 72 / 28 - 273985376 z + z ) / (-1 - 5560001489955839 z / 26 2 24 + 2176550423294029 z + 203 z - 700959805316201 z 22 4 6 8 10 + 184807388762279 z - 17150 z + 821432 z - 25448861 z + 549149489 z 12 14 18 16 - 8654750510 z + 102959893800 z + 6857444030053 z - 947019820991 z 50 48 20 + 700959805316201 z - 2176550423294029 z - 39634381004713 z 36 34 66 - 35708445857995037 z + 29675131151078199 z + 25448861 z 64 30 42 - 549149489 z + 11727872005192833 z + 20482502324963889 z 44 46 58 - 11727872005192833 z + 5560001489955839 z + 947019820991 z 56 54 52 - 6857444030053 z + 39634381004713 z - 184807388762279 z 60 70 68 32 - 102959893800 z + 17150 z - 821432 z - 20482502324963889 z 38 40 62 74 + 35708445857995037 z - 29675131151078199 z + 8654750510 z + z 72 - 203 z ) And in Maple-input format, it is: -(1+1279689616299458*z^28-544792159588993*z^26-167*z^2+190616756694110*z^24-\ 54571717904158*z^22+12102*z^4-509399*z^6+14082482*z^8-273985376*z^10+3922039864 *z^12-42603253714*z^14-2389525131806*z^18+359149084591*z^16-54571717904158*z^50 +190616756694110*z^48+12709160212402*z^20+5777400440808434*z^36-\ 5260003351831032*z^34-509399*z^66+14082482*z^64-2478682907141035*z^30-\ 2478682907141035*z^42+1279689616299458*z^44-544792159588993*z^46-42603253714*z^ 58+359149084591*z^56-2389525131806*z^54+12709160212402*z^52+3922039864*z^60-167 *z^70+12102*z^68+3968403700797061*z^32-5260003351831032*z^38+3968403700797061*z ^40-273985376*z^62+z^72)/(-1-5560001489955839*z^28+2176550423294029*z^26+203*z^ 2-700959805316201*z^24+184807388762279*z^22-17150*z^4+821432*z^6-25448861*z^8+ 549149489*z^10-8654750510*z^12+102959893800*z^14+6857444030053*z^18-\ 947019820991*z^16+700959805316201*z^50-2176550423294029*z^48-39634381004713*z^ 20-35708445857995037*z^36+29675131151078199*z^34+25448861*z^66-549149489*z^64+ 11727872005192833*z^30+20482502324963889*z^42-11727872005192833*z^44+ 5560001489955839*z^46+947019820991*z^58-6857444030053*z^56+39634381004713*z^54-\ 184807388762279*z^52-102959893800*z^60+17150*z^70-821432*z^68-20482502324963889 *z^32+35708445857995037*z^38-29675131151078199*z^40+8654750510*z^62+z^74-203*z^ 72) The first , 40, terms are: [0, 36, 0, 2260, 0, 153413, 0, 10589012, 0, 733977923, 0, 50936555083, 0, 3536229961856, 0, 245533041157479, 0, 17049141538735799, 0, 1183872624915514240, 0, 82207584781760735189, 0, 5708484965814468955340, 0, 396397407762889965809296, 0, 27525879971526583477063057, 0, 1911401129765332694818117865, 0, 132728006188688643702608959000, 0, 9216655471057293821871285601356, 0, 640006169459146110366801165630685, 0, 44442141644339502167355071850203384, 0, 3086070218706631303395451660914083447] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 36087762216423 z - 21520649039910 z - 142 z 24 22 4 6 + 10400132880320 z - 4057774096316 z + 8560 z - 294636 z 8 10 12 14 + 6564858 z - 101644603 z + 1144636282 z - 9674267976 z 18 16 50 48 - 317813111485 z + 62765482870 z - 9674267976 z + 62765482870 z 20 36 34 64 + 1271586120274 z + 36087762216423 z - 49163677679600 z + z 30 42 44 - 49163677679600 z - 4057774096316 z + 1271586120274 z 46 58 56 54 - 317813111485 z - 294636 z + 6564858 z - 101644603 z 52 60 32 38 + 1144636282 z + 8560 z + 54492584493808 z - 21520649039910 z 40 62 / 28 + 10400132880320 z - 142 z ) / (-1 - 186739589831210 z / 26 2 24 22 + 101015131491025 z + 183 z - 44337097599227 z + 15727257660028 z 4 6 8 10 12 - 13088 z + 515142 z - 12885429 z + 221749229 z - 2759885168 z 14 18 16 + 25696576780 z + 1020039053585 z - 183334006829 z 50 48 20 + 183334006829 z - 1020039053585 z - 4484024133812 z 36 34 66 64 - 280875563613344 z + 344327282442905 z + z - 183 z 30 42 44 + 280875563613344 z + 44337097599227 z - 15727257660028 z 46 58 56 54 + 4484024133812 z + 12885429 z - 221749229 z + 2759885168 z 52 60 32 38 - 25696576780 z - 515142 z - 344327282442905 z + 186739589831210 z 40 62 - 101015131491025 z + 13088 z ) And in Maple-input format, it is: -(1+36087762216423*z^28-21520649039910*z^26-142*z^2+10400132880320*z^24-\ 4057774096316*z^22+8560*z^4-294636*z^6+6564858*z^8-101644603*z^10+1144636282*z^ 12-9674267976*z^14-317813111485*z^18+62765482870*z^16-9674267976*z^50+ 62765482870*z^48+1271586120274*z^20+36087762216423*z^36-49163677679600*z^34+z^ 64-49163677679600*z^30-4057774096316*z^42+1271586120274*z^44-317813111485*z^46-\ 294636*z^58+6564858*z^56-101644603*z^54+1144636282*z^52+8560*z^60+ 54492584493808*z^32-21520649039910*z^38+10400132880320*z^40-142*z^62)/(-1-\ 186739589831210*z^28+101015131491025*z^26+183*z^2-44337097599227*z^24+ 15727257660028*z^22-13088*z^4+515142*z^6-12885429*z^8+221749229*z^10-2759885168 *z^12+25696576780*z^14+1020039053585*z^18-183334006829*z^16+183334006829*z^50-\ 1020039053585*z^48-4484024133812*z^20-280875563613344*z^36+344327282442905*z^34 +z^66-183*z^64+280875563613344*z^30+44337097599227*z^42-15727257660028*z^44+ 4484024133812*z^46+12885429*z^58-221749229*z^56+2759885168*z^54-25696576780*z^ 52-515142*z^60-344327282442905*z^32+186739589831210*z^38-101015131491025*z^40+ 13088*z^62) The first , 40, terms are: [0, 41, 0, 2975, 0, 228323, 0, 17646560, 0, 1365378543, 0, 105667181183, 0, 8178035164849, 0, 632941487230317, 0, 48986881361498925, 0, 3791373004397053809, 0, 293435995906580350951, 0, 22710689028567058713671, 0, 1757710103099177167770144, 0, 136039238549729590481239259, 0, 10528854822306491569343303335, 0, 814888301261221994546797791161, 0, 63068866938511988942667310967761, 0, 4881260377999107669557276507198769, 0, 377788662385636940891338179176585785, 0, 29239225605030662155977383569629292215] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 147089772865511040 z - 38472893737926888 z - 200 z 24 22 4 6 + 8478337602325589 z - 1565424841594224 z + 17880 z - 955192 z 8 10 12 14 + 34390971 z - 892238324 z + 17412099464 z - 263378278700 z 18 16 50 - 30493728264864 z + 3156894947079 z - 10490709865674916136 z 48 20 36 + 15385086438044685742 z + 240520370024584 z + 6129293389483238688 z 34 66 80 88 - 3064179786950879944 z - 1565424841594224 z + 34390971 z + z 84 86 82 64 + 17880 z - 200 z - 955192 z + 8478337602325589 z 30 42 - 475955180705150840 z - 19351440993932898432 z 44 46 + 20887705169530629360 z - 19351440993932898432 z 58 56 - 475955180705150840 z + 1308310686370935098 z 54 52 - 3064179786950879944 z + 6129293389483238688 z 60 70 68 + 147089772865511040 z - 30493728264864 z + 240520370024584 z 78 32 38 - 892238324 z + 1308310686370935098 z - 10490709865674916136 z 40 62 76 + 15385086438044685742 z - 38472893737926888 z + 17412099464 z 74 72 / 28 - 263378278700 z + 3156894947079 z ) / (-1 - 519733877331400896 z / 26 2 24 + 126930799175059989 z + 237 z - 26120243091779681 z 22 4 6 8 + 4502636863400434 z - 24266 z + 1453198 z - 57817559 z 10 12 14 18 + 1641244319 z - 34800354564 z + 569101958596 z + 76282692377587 z 16 50 48 - 7348703464619 z + 83031606538421067718 z - 112433560963254113382 z 20 36 - 645514228915766 z - 28632350267948048964 z 34 66 80 90 + 13335381652204995106 z + 26120243091779681 z - 1641244319 z + z 88 84 86 82 - 237 z - 1453198 z + 24266 z + 57817559 z 64 30 - 126930799175059989 z + 1801733974956317248 z 42 44 + 112433560963254113382 z - 130812015276618209336 z 46 58 + 130812015276618209336 z + 5308586817767498042 z 56 54 - 13335381652204995106 z + 28632350267948048964 z 52 60 70 - 52648566899498557500 z - 1801733974956317248 z + 645514228915766 z 68 78 32 - 4502636863400434 z + 34800354564 z - 5308586817767498042 z 38 40 + 52648566899498557500 z - 83031606538421067718 z 62 76 74 + 519733877331400896 z - 569101958596 z + 7348703464619 z 72 - 76282692377587 z ) And in Maple-input format, it is: -(1+147089772865511040*z^28-38472893737926888*z^26-200*z^2+8478337602325589*z^ 24-1565424841594224*z^22+17880*z^4-955192*z^6+34390971*z^8-892238324*z^10+ 17412099464*z^12-263378278700*z^14-30493728264864*z^18+3156894947079*z^16-\ 10490709865674916136*z^50+15385086438044685742*z^48+240520370024584*z^20+ 6129293389483238688*z^36-3064179786950879944*z^34-1565424841594224*z^66+ 34390971*z^80+z^88+17880*z^84-200*z^86-955192*z^82+8478337602325589*z^64-\ 475955180705150840*z^30-19351440993932898432*z^42+20887705169530629360*z^44-\ 19351440993932898432*z^46-475955180705150840*z^58+1308310686370935098*z^56-\ 3064179786950879944*z^54+6129293389483238688*z^52+147089772865511040*z^60-\ 30493728264864*z^70+240520370024584*z^68-892238324*z^78+1308310686370935098*z^ 32-10490709865674916136*z^38+15385086438044685742*z^40-38472893737926888*z^62+ 17412099464*z^76-263378278700*z^74+3156894947079*z^72)/(-1-519733877331400896*z ^28+126930799175059989*z^26+237*z^2-26120243091779681*z^24+4502636863400434*z^ 22-24266*z^4+1453198*z^6-57817559*z^8+1641244319*z^10-34800354564*z^12+ 569101958596*z^14+76282692377587*z^18-7348703464619*z^16+83031606538421067718*z ^50-112433560963254113382*z^48-645514228915766*z^20-28632350267948048964*z^36+ 13335381652204995106*z^34+26120243091779681*z^66-1641244319*z^80+z^90-237*z^88-\ 1453198*z^84+24266*z^86+57817559*z^82-126930799175059989*z^64+ 1801733974956317248*z^30+112433560963254113382*z^42-130812015276618209336*z^44+ 130812015276618209336*z^46+5308586817767498042*z^58-13335381652204995106*z^56+ 28632350267948048964*z^54-52648566899498557500*z^52-1801733974956317248*z^60+ 645514228915766*z^70-4502636863400434*z^68+34800354564*z^78-5308586817767498042 *z^32+52648566899498557500*z^38-83031606538421067718*z^40+519733877331400896*z^ 62-569101958596*z^76+7348703464619*z^74-76282692377587*z^72) The first , 40, terms are: [0, 37, 0, 2383, 0, 164935, 0, 11605455, 0, 820907271, 0, 58178805933, 0, 4126321842665, 0, 292747447938505, 0, 20771936688371933, 0, 1473951311026621463, 0, 104592028526565823695, 0, 7421948733372939977223, 0, 526670478057291161762463, 0, 37373236518228404030957077, 0, 2652056354011547581153542433, 0, 188193629763141356374688368417, 0, 13354485054004363317908930782901, 0, 947653124640509431197464807923711, 0, 67246806112997737776501159889759687, 0, 4771928530170980657024515926299574159] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 4 6 f(z) = - (1 + z - 86 z - 86 z + 2329 z - 27724 z + 2329 z - 27724 z 8 10 12 14 18 + 171536 z - 601108 z + 1252550 z - 1595060 z - 601108 z 16 20 / 2 28 26 24 + 1252550 z + 171536 z ) / ((-1 + z ) (z - 129 z + 4345 z / 22 20 18 16 14 - 62572 z + 457692 z - 1844924 z + 4237790 z - 5596166 z 12 10 8 6 4 2 + 4237790 z - 1844924 z + 457692 z - 62572 z + 4345 z - 129 z + 1) ) And in Maple-input format, it is: -(1+z^28-86*z^26-86*z^2+2329*z^24-27724*z^22+2329*z^4-27724*z^6+171536*z^8-\ 601108*z^10+1252550*z^12-1595060*z^14-601108*z^18+1252550*z^16+171536*z^20)/(-1 +z^2)/(z^28-129*z^26+4345*z^24-62572*z^22+457692*z^20-1844924*z^18+4237790*z^16 -5596166*z^14+4237790*z^12-1844924*z^10+457692*z^8-62572*z^6+4345*z^4-129*z^2+1 ) The first , 40, terms are: [0, 44, 0, 3575, 0, 307087, 0, 26522380, 0, 2292040329, 0, 198089996569, 0, 17120115365196, 0, 1479623957862847, 0, 127878078272296327, 0, 11051999599509214700, 0, 955180883625318665649, 0, 82552529355650794038481, 0, 7134690634244081842973292, 0, 616623268188515008690886631, 0, 53292325395283491713588040415, 0, 4605846215283818588590868721868, 0, 398065184836727191257422143935097, 0, 34403209306749851387166880891461161, 0, 2973334156539022382723996970481178508, 0, 256973584284391717018697464863903943023] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 838431644 z - 1824439590 z - 106 z + 2900969655 z 22 4 6 8 10 - 3384501504 z + 4372 z - 96070 z + 1282867 z - 11192400 z 12 14 18 16 + 66700133 z - 279412570 z - 1824439590 z + 838431644 z 20 36 34 30 42 + 2900969655 z + 1282867 z - 11192400 z - 279412570 z - 106 z 44 32 38 40 / 2 44 42 + z + 66700133 z - 96070 z + 4372 z ) / ((-1 + z ) (z - 141 z / 40 38 36 34 32 + 7024 z - 179365 z + 2721833 z - 26523432 z + 173871955 z 30 28 26 24 - 789478567 z + 2528741536 z - 5776689343 z + 9467281403 z 22 20 18 16 - 11159672400 z + 9467281403 z - 5776689343 z + 2528741536 z 14 12 10 8 6 - 789478567 z + 173871955 z - 26523432 z + 2721833 z - 179365 z 4 2 + 7024 z - 141 z + 1)) And in Maple-input format, it is: -(1+838431644*z^28-1824439590*z^26-106*z^2+2900969655*z^24-3384501504*z^22+4372 *z^4-96070*z^6+1282867*z^8-11192400*z^10+66700133*z^12-279412570*z^14-\ 1824439590*z^18+838431644*z^16+2900969655*z^20+1282867*z^36-11192400*z^34-\ 279412570*z^30-106*z^42+z^44+66700133*z^32-96070*z^38+4372*z^40)/(-1+z^2)/(z^44 -141*z^42+7024*z^40-179365*z^38+2721833*z^36-26523432*z^34+173871955*z^32-\ 789478567*z^30+2528741536*z^28-5776689343*z^26+9467281403*z^24-11159672400*z^22 +9467281403*z^20-5776689343*z^18+2528741536*z^16-789478567*z^14+173871955*z^12-\ 26523432*z^10+2721833*z^8-179365*z^6+7024*z^4-141*z^2+1) The first , 40, terms are: [0, 36, 0, 2319, 0, 161677, 0, 11434172, 0, 811082547, 0, 57573949771, 0, 4087530787644, 0, 290211860288245, 0, 20605083028276823, 0, 1462968471123847780, 0, 103871390864898980281, 0, 7374915250127938596809, 0, 523622320202590003925348, 0, 37177422413741584902111431, 0, 2639613881123585947126226597, 0, 187413785028521288189126021820, 0, 13306463905654191753326438059803, 0, 944764984429449380238240585932387, 0, 67078743245672172001499686299842108, 0, 4762621254618881829119278561503311325] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 26 24 22 20 f(z) = - (1992823416 z - 4048118903 z - 1 + 5758149852 z - 5758149852 z 2 4 6 8 10 12 + 132 z - 6607 z + 171151 z - 2614823 z + 25220869 z - 159739817 z 16 14 18 34 - 1992823416 z + 681355709 z + 4048118903 z + 2614823 z 32 28 40 42 30 - 25220869 z - 681355709 z - 132 z + z + 159739817 z 36 38 / 24 38 - 171151 z + 6607 z ) / (32444073587 z - 320114 z / 20 28 32 4 + 32444073587 z + 8228527454 z + 493082043 z + 1 + 9994 z 6 8 30 40 22 - 320114 z + 5891961 z - 2431453186 z + 9994 z - 38457128232 z 26 18 16 44 2 - 19444416212 z - 19444416212 z + 8228527454 z + z - 160 z 10 14 34 12 42 - 66969492 z - 2431453186 z - 66969492 z + 493082043 z - 160 z 36 + 5891961 z ) And in Maple-input format, it is: -(1992823416*z^26-4048118903*z^24-1+5758149852*z^22-5758149852*z^20+132*z^2-\ 6607*z^4+171151*z^6-2614823*z^8+25220869*z^10-159739817*z^12-1992823416*z^16+ 681355709*z^14+4048118903*z^18+2614823*z^34-25220869*z^32-681355709*z^28-132*z^ 40+z^42+159739817*z^30-171151*z^36+6607*z^38)/(32444073587*z^24-320114*z^38+ 32444073587*z^20+8228527454*z^28+493082043*z^32+1+9994*z^4-320114*z^6+5891961*z ^8-2431453186*z^30+9994*z^40-38457128232*z^22-19444416212*z^26-19444416212*z^18 +8228527454*z^16+z^44-160*z^2-66969492*z^10-2431453186*z^14-66969492*z^34+ 493082043*z^12-160*z^42+5891961*z^36) The first , 40, terms are: [0, 28, 0, 1093, 0, 44011, 0, 1804372, 0, 75511903, 0, 3239438143, 0, 143079242164, 0, 6529162433131, 0, 308387775595909, 0, 15070445079415804, 0, 760058212518731809, 0, 39399172931321048161, 0, 2089091377491988599100, 0, 112765892098638107106949, 0, 6170115212110914866079211, 0, 341013646594164673408674292, 0, 18985147367344804431634788607, 0, 1062469409335809952590020085151, 0, 59678314401256335021411922026004, 0, 3360759662630118157742046566699371] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {3, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 4 f(z) = - (-1 - 100 z + 3333 z + 100 z - 51293 z + 416550 z - 3333 z 6 8 10 12 14 + 51293 z - 416550 z + 1912849 z - 5164777 z + 8426238 z 18 16 20 30 / 22 + 5164777 z - 8426238 z - 1912849 z + z ) / (-6815584 z / 30 14 32 12 10 18 - 128 z - 44982376 z + z + 22379552 z - 6815584 z - 44982376 z 16 4 24 2 8 6 + 56635923 z + 5836 z + 1217498 z - 128 z + 1217498 z - 119480 z 26 28 20 - 119480 z + 5836 z + 1 + 22379552 z ) And in Maple-input format, it is: -(-1-100*z^28+3333*z^26+100*z^2-51293*z^24+416550*z^22-3333*z^4+51293*z^6-\ 416550*z^8+1912849*z^10-5164777*z^12+8426238*z^14+5164777*z^18-8426238*z^16-\ 1912849*z^20+z^30)/(-6815584*z^22-128*z^30-44982376*z^14+z^32+22379552*z^12-\ 6815584*z^10-44982376*z^18+56635923*z^16+5836*z^4+1217498*z^24-128*z^2+1217498* z^8-119480*z^6-119480*z^26+5836*z^28+1+22379552*z^20) The first , 40, terms are: [0, 28, 0, 1081, 0, 43147, 0, 1758592, 0, 73264555, 0, 3127429927, 0, 137101848616, 0, 6181013361439, 0, 286576501163581, 0, 13644552460856068, 0, 665380847572694893, 0, 33121565897499644725, 0, 1676938165245308633140, 0, 86058966114335513438677, 0, 4463053923097002324178711, 0, 233308054669103068977711064, 0, 12269154409644056509729009423, 0, 648052258505763482653043454787, 0, 34340256897957565261858114712272, 0, 1823952325441476221179042502441107] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 197203965050 z - 224260923874 z - 132 z + 197203965050 z 22 4 6 8 10 - 133957183078 z + 6917 z - 195949 z + 3436469 z - 40238482 z 12 14 18 16 + 329796937 z - 1953042550 z - 28076383374 z + 8545608654 z 50 48 20 36 34 - 132 z + 6917 z + 70074686674 z + 8545608654 z - 28076383374 z 30 42 44 46 52 - 133957183078 z - 40238482 z + 3436469 z - 195949 z + z 32 38 40 / + 70074686674 z - 1953042550 z + 329796937 z ) / (-1 / 28 26 2 24 - 1414629021963 z + 1414629021963 z + 161 z - 1099868945427 z 22 4 6 8 10 + 663452266843 z - 10270 z + 348636 z - 7186251 z + 97248057 z 12 14 18 16 - 909470206 z + 6091250114 z + 110471694835 z - 29977315695 z 50 48 20 36 + 10270 z - 348636 z - 309120874023 z - 110471694835 z 34 30 42 44 + 309120874023 z + 1099868945427 z + 909470206 z - 97248057 z 46 54 52 32 38 + 7186251 z + z - 161 z - 663452266843 z + 29977315695 z 40 - 6091250114 z ) And in Maple-input format, it is: -(1+197203965050*z^28-224260923874*z^26-132*z^2+197203965050*z^24-133957183078* z^22+6917*z^4-195949*z^6+3436469*z^8-40238482*z^10+329796937*z^12-1953042550*z^ 14-28076383374*z^18+8545608654*z^16-132*z^50+6917*z^48+70074686674*z^20+ 8545608654*z^36-28076383374*z^34-133957183078*z^30-40238482*z^42+3436469*z^44-\ 195949*z^46+z^52+70074686674*z^32-1953042550*z^38+329796937*z^40)/(-1-\ 1414629021963*z^28+1414629021963*z^26+161*z^2-1099868945427*z^24+663452266843*z ^22-10270*z^4+348636*z^6-7186251*z^8+97248057*z^10-909470206*z^12+6091250114*z^ 14+110471694835*z^18-29977315695*z^16+10270*z^50-348636*z^48-309120874023*z^20-\ 110471694835*z^36+309120874023*z^34+1099868945427*z^30+909470206*z^42-97248057* z^44+7186251*z^46+z^54-161*z^52-663452266843*z^32+29977315695*z^38-6091250114*z ^40) The first , 40, terms are: [0, 29, 0, 1316, 0, 66733, 0, 3589355, 0, 199951517, 0, 11378458643, 0, 655990058332, 0, 38121926593499, 0, 2226246063634447, 0, 130397207275390143, 0, 7651653260900172507, 0, 449495374396299521596, 0, 26423445683534306388515, 0, 1553935058922168406084733, 0, 91408239213310692771909995, 0, 5377794229661926421875936285, 0, 316419606683977081377958413764, 0, 18618605305393256754637725552061, 0, 1095584395049616147231143390486545, 0, 64469397541296042372832991177891633] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 575129864793388 z - 278448345713506 z - 167 z 24 22 4 6 + 109174162667638 z - 34548901012120 z + 11972 z - 494901 z 8 10 12 14 + 13346243 z - 251535668 z + 3462214335 z - 35873412017 z 18 16 50 - 1781805627123 z + 285953334552 z - 1781805627123 z 48 20 36 + 8782672242445 z + 8782672242445 z + 1313980982165034 z 34 66 64 30 - 1456661429387640 z - 167 z + 11972 z - 964211218886670 z 42 44 46 - 278448345713506 z + 109174162667638 z - 34548901012120 z 58 56 54 52 - 251535668 z + 3462214335 z - 35873412017 z + 285953334552 z 60 68 32 38 + 13346243 z + z + 1313980982165034 z - 964211218886670 z 40 62 / 2 + 575129864793388 z - 494901 z ) / ((-1 + z ) (1 / 28 26 2 + 1896066306738914 z - 888746293878504 z - 198 z 24 22 4 6 + 334642019154910 z - 100937632500404 z + 16363 z - 764720 z 8 10 12 14 + 22988989 z - 477824574 z + 7190133617 z - 80834882960 z 18 16 50 - 4632113859314 z + 694365426295 z - 4632113859314 z 48 20 36 + 24285494427453 z + 24285494427453 z + 4500193961337806 z 34 66 64 30 - 5013163359533260 z - 198 z + 16363 z - 3254910414394048 z 42 44 46 - 888746293878504 z + 334642019154910 z - 100937632500404 z 58 56 54 52 - 477824574 z + 7190133617 z - 80834882960 z + 694365426295 z 60 68 32 38 + 22988989 z + z + 4500193961337806 z - 3254910414394048 z 40 62 + 1896066306738914 z - 764720 z )) And in Maple-input format, it is: -(1+575129864793388*z^28-278448345713506*z^26-167*z^2+109174162667638*z^24-\ 34548901012120*z^22+11972*z^4-494901*z^6+13346243*z^8-251535668*z^10+3462214335 *z^12-35873412017*z^14-1781805627123*z^18+285953334552*z^16-1781805627123*z^50+ 8782672242445*z^48+8782672242445*z^20+1313980982165034*z^36-1456661429387640*z^ 34-167*z^66+11972*z^64-964211218886670*z^30-278448345713506*z^42+ 109174162667638*z^44-34548901012120*z^46-251535668*z^58+3462214335*z^56-\ 35873412017*z^54+285953334552*z^52+13346243*z^60+z^68+1313980982165034*z^32-\ 964211218886670*z^38+575129864793388*z^40-494901*z^62)/(-1+z^2)/(1+ 1896066306738914*z^28-888746293878504*z^26-198*z^2+334642019154910*z^24-\ 100937632500404*z^22+16363*z^4-764720*z^6+22988989*z^8-477824574*z^10+ 7190133617*z^12-80834882960*z^14-4632113859314*z^18+694365426295*z^16-\ 4632113859314*z^50+24285494427453*z^48+24285494427453*z^20+4500193961337806*z^ 36-5013163359533260*z^34-198*z^66+16363*z^64-3254910414394048*z^30-\ 888746293878504*z^42+334642019154910*z^44-100937632500404*z^46-477824574*z^58+ 7190133617*z^56-80834882960*z^54+694365426295*z^52+22988989*z^60+z^68+ 4500193961337806*z^32-3254910414394048*z^38+1896066306738914*z^40-764720*z^62) The first , 40, terms are: [0, 32, 0, 1779, 0, 110251, 0, 7065120, 0, 458797933, 0, 29972960029, 0, 1963765702384, 0, 128845097173791, 0, 8459710118410323, 0, 555646954888134544, 0, 36502384099582981413, 0, 2398189422361238956501, 0, 157567272754694841103984, 0, 10352824609022940216539155, 0, 680231801566619386104483151, 0, 44694869101030790853875022608, 0, 2936701206384180624579851033533, 0, 192957890364574089776310951833869, 0, 12678435724572533588375650996801856, 0, 833045998153657033577704816998460955] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 18 16 14 12 10 8 6 f(z) = - (z - 65 z + 1023 z - 6023 z + 14936 z - 14936 z + 6023 z 4 2 / 20 18 16 14 - 1023 z + 65 z - 1) / (z - 97 z + 2232 z - 19639 z / 12 10 8 6 4 2 + 71719 z - 108816 z + 71719 z - 19639 z + 2232 z - 97 z + 1) And in Maple-input format, it is: -(z^18-65*z^16+1023*z^14-6023*z^12+14936*z^10-14936*z^8+6023*z^6-1023*z^4+65*z^ 2-1)/(z^20-97*z^18+2232*z^16-19639*z^14+71719*z^12-108816*z^10+71719*z^8-19639* z^6+2232*z^4-97*z^2+1) The first , 40, terms are: [0, 32, 0, 1895, 0, 126007, 0, 8564704, 0, 584543441, 0, 39926454833, 0, 2727534214240, 0, 186334144522135, 0, 12729671404762631, 0, 869645870605355936, 0, 59411125993407684769, 0, 4058757903714928294753, 0, 277279980695458291999904, 0, 18942787339260377314782407, 0, 1294104217001014557252814039, 0, 88408621951187112484699034464, 0, 6039764288483208616427813346161, 0, 412615329313476659468593159463825, 0, 28188419589354580916046656411433184, 0, 1925733104166679143652011986143606519] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 15939933 z + 84674251 z + 127 z - 290546828 z 22 4 6 8 10 + 653984461 z - 5908 z + 139497 z - 1902419 z + 15939933 z 12 14 18 16 - 84674251 z + 290546828 z + 977736831 z - 653984461 z 20 36 34 30 32 38 - 977736831 z - 127 z + 5908 z + 1902419 z - 139497 z + z ) / 14 2 16 36 34 / (-1138370576 z - 156 z + 3024190196 z + 9268 z - 274840 z / 22 26 28 12 - 5396976996 z - 1138370576 z + 282952740 z + 282952740 z 18 32 10 20 - 5396976996 z + 4560492 z - 45386364 z + 6538650070 z 30 38 40 24 6 - 45386364 z + 1 - 156 z + z + 3024190196 z - 274840 z 8 4 + 4560492 z + 9268 z ) And in Maple-input format, it is: -(-1-15939933*z^28+84674251*z^26+127*z^2-290546828*z^24+653984461*z^22-5908*z^4 +139497*z^6-1902419*z^8+15939933*z^10-84674251*z^12+290546828*z^14+977736831*z^ 18-653984461*z^16-977736831*z^20-127*z^36+5908*z^34+1902419*z^30-139497*z^32+z^ 38)/(-1138370576*z^14-156*z^2+3024190196*z^16+9268*z^36-274840*z^34-5396976996* z^22-1138370576*z^26+282952740*z^28+282952740*z^12-5396976996*z^18+4560492*z^32 -45386364*z^10+6538650070*z^20-45386364*z^30+1-156*z^38+z^40+3024190196*z^24-\ 274840*z^6+4560492*z^8+9268*z^4) The first , 40, terms are: [0, 29, 0, 1164, 0, 48155, 0, 2036515, 0, 88501723, 0, 3976281347, 0, 185643133692, 0, 9031322425925, 0, 457621704030313, 0, 24067032120945337, 0, 1306369611040301621, 0, 72721289608815320412, 0, 4126330783397381188787, 0, 237425924601747936248395, 0, 13797211842073265240772979, 0, 807313612205421235082021483, 0, 47461381617381896111140902828, 0, 2799158871251945470539913432493, 0, 165444000882913356210922064092657, 0, 9792707801738213649570609570281617] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 205181645340603416 z - 54071123158157438 z - 219 z 24 22 4 6 + 11991769030590024 z - 2224283444578530 z + 21062 z - 1191233 z 8 10 12 14 + 44797793 z - 1200637936 z + 23989322266 z - 368813894332 z 18 16 50 - 43366989195848 z + 4466109910086 z - 14117525443302794208 z 48 20 36 + 20618390158166472350 z + 342461370433960 z + 8294253123034048054 z 34 66 80 88 - 4174476764637780020 z - 2224283444578530 z + 44797793 z + z 84 86 82 64 + 21062 z - 219 z - 1191233 z + 11991769030590024 z 30 42 - 658618920342733544 z - 25867763647436649348 z 44 46 + 27897262671296494276 z - 25867763647436649348 z 58 56 - 658618920342733544 z + 1795928936593766954 z 54 52 - 4174476764637780020 z + 8294253123034048054 z 60 70 68 + 205181645340603416 z - 43366989195848 z + 342461370433960 z 78 32 38 - 1200637936 z + 1795928936593766954 z - 14117525443302794208 z 40 62 76 + 20618390158166472350 z - 54071123158157438 z + 23989322266 z 74 72 / 2 - 368813894332 z + 4466109910086 z ) / ((-1 + z ) (1 / 28 26 2 + 618973718378131476 z - 157766020818522846 z - 255 z 24 22 4 6 + 33687451435583324 z - 5986838177966418 z + 27995 z - 1781069 z 8 10 12 14 + 74410833 z - 2191072504 z + 47620398326 z - 789299110236 z 18 16 50 - 105422924873272 z + 10222898393134 z - 47309227542585757608 z 48 20 + 69747166491128960150 z + 878525147150420 z 36 34 + 27430433702463292066 z - 13572530059242689908 z 66 80 88 84 86 - 5986838177966418 z + 74410833 z + z + 27995 z - 255 z 82 64 30 - 1781069 z + 33687451435583324 z - 2045417020577820648 z 42 44 - 87999059354875458980 z + 95081494078419500162 z 46 58 - 87999059354875458980 z - 2045417020577820648 z 56 54 + 5718195852348577706 z - 13572530059242689908 z 52 60 70 + 27430433702463292066 z + 618973718378131476 z - 105422924873272 z 68 78 32 + 878525147150420 z - 2191072504 z + 5718195852348577706 z 38 40 - 47309227542585757608 z + 69747166491128960150 z 62 76 74 - 157766020818522846 z + 47620398326 z - 789299110236 z 72 + 10222898393134 z )) And in Maple-input format, it is: -(1+205181645340603416*z^28-54071123158157438*z^26-219*z^2+11991769030590024*z^ 24-2224283444578530*z^22+21062*z^4-1191233*z^6+44797793*z^8-1200637936*z^10+ 23989322266*z^12-368813894332*z^14-43366989195848*z^18+4466109910086*z^16-\ 14117525443302794208*z^50+20618390158166472350*z^48+342461370433960*z^20+ 8294253123034048054*z^36-4174476764637780020*z^34-2224283444578530*z^66+ 44797793*z^80+z^88+21062*z^84-219*z^86-1191233*z^82+11991769030590024*z^64-\ 658618920342733544*z^30-25867763647436649348*z^42+27897262671296494276*z^44-\ 25867763647436649348*z^46-658618920342733544*z^58+1795928936593766954*z^56-\ 4174476764637780020*z^54+8294253123034048054*z^52+205181645340603416*z^60-\ 43366989195848*z^70+342461370433960*z^68-1200637936*z^78+1795928936593766954*z^ 32-14117525443302794208*z^38+20618390158166472350*z^40-54071123158157438*z^62+ 23989322266*z^76-368813894332*z^74+4466109910086*z^72)/(-1+z^2)/(1+ 618973718378131476*z^28-157766020818522846*z^26-255*z^2+33687451435583324*z^24-\ 5986838177966418*z^22+27995*z^4-1781069*z^6+74410833*z^8-2191072504*z^10+ 47620398326*z^12-789299110236*z^14-105422924873272*z^18+10222898393134*z^16-\ 47309227542585757608*z^50+69747166491128960150*z^48+878525147150420*z^20+ 27430433702463292066*z^36-13572530059242689908*z^34-5986838177966418*z^66+ 74410833*z^80+z^88+27995*z^84-255*z^86-1781069*z^82+33687451435583324*z^64-\ 2045417020577820648*z^30-87999059354875458980*z^42+95081494078419500162*z^44-\ 87999059354875458980*z^46-2045417020577820648*z^58+5718195852348577706*z^56-\ 13572530059242689908*z^54+27430433702463292066*z^52+618973718378131476*z^60-\ 105422924873272*z^70+878525147150420*z^68-2191072504*z^78+5718195852348577706*z ^32-47309227542585757608*z^38+69747166491128960150*z^40-157766020818522846*z^62 +47620398326*z^76-789299110236*z^74+10222898393134*z^72) The first , 40, terms are: [0, 37, 0, 2284, 0, 157285, 0, 11283219, 0, 822850017, 0, 60415729579, 0, 4448580918148, 0, 327967862808523, 0, 24192474627169339, 0, 1784999325823809587, 0, 131718353314974867147, 0, 9720271290127414600132, 0, 717334865798042625802315, 0, 52938432623804150513377513, 0, 3906816141496077039249275355, 0, 288320978371910921947049872421, 0, 21277970635549257617706640922348, 0, 1570306828126768137814632152714949, 0, 115888145440078170295894817830296121, 0, 8552510168107636987809586735946401993] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 34906492858753 z + 27255278909654 z + 182 z 24 22 4 6 - 16587212571625 z + 7839744942147 z - 13772 z + 576355 z 8 10 12 14 - 15002972 z + 259554846 z - 3118010533 z + 26840050878 z 18 16 50 48 + 800032157105 z - 169519625631 z + 15002972 z - 259554846 z 20 36 34 - 2861368972650 z - 7839744942147 z + 16587212571625 z 30 42 44 46 + 34906492858753 z + 169519625631 z - 26840050878 z + 3118010533 z 58 56 54 52 32 + z - 182 z + 13772 z - 576355 z - 27255278909654 z 38 40 / 28 + 2861368972650 z - 800032157105 z ) / (1 + 218874961478514 z / 26 2 24 22 - 151813719349864 z - 226 z + 82283963649294 z - 34701821793084 z 4 6 8 10 12 + 20323 z - 987584 z + 29434843 z - 577780126 z + 7827102833 z 14 18 16 50 - 75674230956 z - 2828058806840 z + 535503330776 z - 577780126 z 48 20 36 + 7827102833 z + 11315498824568 z + 82283963649294 z 34 30 42 - 151813719349864 z - 247195863927664 z - 2828058806840 z 44 46 58 56 54 + 535503330776 z - 75674230956 z - 226 z + 20323 z - 987584 z 52 60 32 38 + 29434843 z + z + 218874961478514 z - 34701821793084 z 40 + 11315498824568 z ) And in Maple-input format, it is: -(-1-34906492858753*z^28+27255278909654*z^26+182*z^2-16587212571625*z^24+ 7839744942147*z^22-13772*z^4+576355*z^6-15002972*z^8+259554846*z^10-3118010533* z^12+26840050878*z^14+800032157105*z^18-169519625631*z^16+15002972*z^50-\ 259554846*z^48-2861368972650*z^20-7839744942147*z^36+16587212571625*z^34+ 34906492858753*z^30+169519625631*z^42-26840050878*z^44+3118010533*z^46+z^58-182 *z^56+13772*z^54-576355*z^52-27255278909654*z^32+2861368972650*z^38-\ 800032157105*z^40)/(1+218874961478514*z^28-151813719349864*z^26-226*z^2+ 82283963649294*z^24-34701821793084*z^22+20323*z^4-987584*z^6+29434843*z^8-\ 577780126*z^10+7827102833*z^12-75674230956*z^14-2828058806840*z^18+535503330776 *z^16-577780126*z^50+7827102833*z^48+11315498824568*z^20+82283963649294*z^36-\ 151813719349864*z^34-247195863927664*z^30-2828058806840*z^42+535503330776*z^44-\ 75674230956*z^46-226*z^58+20323*z^56-987584*z^54+29434843*z^52+z^60+ 218874961478514*z^32-34701821793084*z^38+11315498824568*z^40) The first , 40, terms are: [0, 44, 0, 3393, 0, 283835, 0, 24212596, 0, 2077632691, 0, 178624115243, 0, 15367504233828, 0, 1322421448995515, 0, 113808164666582185, 0, 9794681032022439100, 0, 842969791245942948497, 0, 72549677284986628530129, 0, 6243952648149671746034972, 0, 537383035497915242813479033, 0, 46249643563536882566327267227, 0, 3980456296346843518868040952196, 0, 342576321435063366284621420016027, 0, 29483689422722103163075167994480067, 0, 2537501546076026716253242321959848788, 0, 218389022258251979835654590036432178139] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 456587327069904 z - 225530435463178 z - 173 z 24 22 4 6 + 90570121598650 z - 29441457950216 z + 12586 z - 518901 z 8 10 12 14 + 13795665 z - 254507916 z + 3414464357 z - 34397679681 z 18 16 50 - 1609892534381 z + 266228320038 z - 1609892534381 z 48 20 36 + 7702609578697 z + 7702609578697 z + 1017809963378598 z 34 66 64 30 - 1124712630435832 z - 173 z + 12586 z - 753966924358870 z 42 44 46 - 225530435463178 z + 90570121598650 z - 29441457950216 z 58 56 54 52 - 254507916 z + 3414464357 z - 34397679681 z + 266228320038 z 60 68 32 38 + 13795665 z + z + 1017809963378598 z - 753966924358870 z 40 62 / 28 + 456587327069904 z - 518901 z ) / (-1 - 2205176154453046 z / 26 2 24 + 1001243605712578 z + 205 z - 370274431970242 z 22 4 6 8 10 + 110914473113409 z - 17603 z + 848081 z - 25970259 z + 543783999 z 12 14 18 16 - 8173189103 z + 91267221819 z + 5131415131123 z - 776556932569 z 50 48 20 + 26721347762149 z - 110914473113409 z - 26721347762149 z 36 34 66 64 - 7135106317589694 z + 7135106317589694 z + 17603 z - 848081 z 30 42 44 + 3972593026469578 z + 2205176154453046 z - 1001243605712578 z 46 58 56 + 370274431970242 z + 8173189103 z - 91267221819 z 54 52 60 70 68 + 776556932569 z - 5131415131123 z - 543783999 z + z - 205 z 32 38 40 - 5871767007865262 z + 5871767007865262 z - 3972593026469578 z 62 + 25970259 z ) And in Maple-input format, it is: -(1+456587327069904*z^28-225530435463178*z^26-173*z^2+90570121598650*z^24-\ 29441457950216*z^22+12586*z^4-518901*z^6+13795665*z^8-254507916*z^10+3414464357 *z^12-34397679681*z^14-1609892534381*z^18+266228320038*z^16-1609892534381*z^50+ 7702609578697*z^48+7702609578697*z^20+1017809963378598*z^36-1124712630435832*z^ 34-173*z^66+12586*z^64-753966924358870*z^30-225530435463178*z^42+90570121598650 *z^44-29441457950216*z^46-254507916*z^58+3414464357*z^56-34397679681*z^54+ 266228320038*z^52+13795665*z^60+z^68+1017809963378598*z^32-753966924358870*z^38 +456587327069904*z^40-518901*z^62)/(-1-2205176154453046*z^28+1001243605712578*z ^26+205*z^2-370274431970242*z^24+110914473113409*z^22-17603*z^4+848081*z^6-\ 25970259*z^8+543783999*z^10-8173189103*z^12+91267221819*z^14+5131415131123*z^18 -776556932569*z^16+26721347762149*z^50-110914473113409*z^48-26721347762149*z^20 -7135106317589694*z^36+7135106317589694*z^34+17603*z^66-848081*z^64+ 3972593026469578*z^30+2205176154453046*z^42-1001243605712578*z^44+ 370274431970242*z^46+8173189103*z^58-91267221819*z^56+776556932569*z^54-\ 5131415131123*z^52-543783999*z^60+z^70-205*z^68-5871767007865262*z^32+ 5871767007865262*z^38-3972593026469578*z^40+25970259*z^62) The first , 40, terms are: [0, 32, 0, 1543, 0, 82199, 0, 4653364, 0, 273807401, 0, 16499014417, 0, 1008552751924, 0, 62180442141723, 0, 3853182010298351, 0, 239499030940488432, 0, 14913445722471180533, 0, 929668355022751616437, 0, 57991741845538627384592, 0, 3618923968157212918152223, 0, 225891429079871286625036235, 0, 14102158801916710370799309844, 0, 880464247151878365050120107137, 0, 54974672483605872612963896273897, 0, 3432644865544051221541570270666996, 0, 214340652484327423700622825601842839 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 21054066485296978868 z - 3321890313189238464 z - 248 z 24 22 4 6 + 452396980688475120 z - 52868372316615048 z + 27962 z - 1921140 z 102 8 10 12 - 3166129692 z + 90862887 z - 3166129692 z + 84892078174 z 14 18 16 - 1805267788888 z - 442807327970296 z + 31131779292541 z 50 48 - 2203534681346137141437648 z + 1442415939590879847050666 z 20 36 + 5264314114304304 z + 8501247162436625616988 z 34 66 - 2318466178073574886824 z - 835821194554457563585064 z 80 100 90 + 554136920658339454692 z + 84892078174 z - 52868372316615048 z 88 84 94 + 452396980688475120 z + 21054066485296978868 z - 442807327970296 z 86 96 98 - 3321890313189238464 z + 31131779292541 z - 1805267788888 z 92 82 + 5264314114304304 z - 115741673944354624984 z 64 112 110 106 + 1442415939590879847050666 z + z - 248 z - 1921140 z 108 30 42 + 27962 z - 115741673944354624984 z - 194113380262898736084056 z 44 46 + 428444591724038426384784 z - 835821194554457563585064 z 58 56 - 3573890755760563233187184 z + 3796400661420286658801514 z 54 52 - 3573890755760563233187184 z + 2981397936950209497160408 z 60 70 + 2981397936950209497160408 z - 194113380262898736084056 z 68 78 + 428444591724038426384784 z - 2318466178073574886824 z 32 38 + 554136920658339454692 z - 27384250311980620829600 z 40 62 + 77646150473995786894312 z - 2203534681346137141437648 z 76 74 + 8501247162436625616988 z - 27384250311980620829600 z 72 104 / + 77646150473995786894312 z + 90862887 z ) / (-1 / 28 26 2 - 64276018614013184825 z + 9598202172844350551 z + 280 z 24 22 4 6 - 1235025505744056905 z + 136077299446524587 z - 35291 z + 2687603 z 102 8 10 12 + 155017854124 z - 139836302 z + 5323905925 z - 155017854124 z 14 18 16 + 3560474348958 z + 1004891194861132 z - 65999991924873 z 50 48 + 12021823510838764362048723 z - 7446917800440303826418087 z 20 36 - 12742420461745133 z - 32024396074848103627313 z 34 66 + 8293866318770826194523 z + 7446917800440303826418087 z 80 100 - 8293866318770826194523 z - 3560474348958 z 90 88 + 1235025505744056905 z - 9598202172844350551 z 84 94 - 372849677063662265779 z + 12742420461745133 z 86 96 98 + 64276018614013184825 z - 1004891194861132 z + 65999991924873 z 92 82 - 136077299446524587 z + 1881756798618163160297 z 64 112 114 110 - 12021823510838764362048723 z - 280 z + z + 35291 z 106 108 30 + 139836302 z - 2687603 z + 372849677063662265779 z 42 44 + 854021678881699702625155 z - 1986685005098044640235377 z 46 58 + 4087725049745055368672823 z + 24621270671506564365709673 z 56 54 - 24621270671506564365709673 z + 21851118599484386136680253 z 52 60 - 17208246336680475982820699 z - 21851118599484386136680253 z 70 68 + 1986685005098044640235377 z - 4087725049745055368672823 z 78 32 + 32024396074848103627313 z - 1881756798618163160297 z 38 40 + 108614475410272470285235 z - 324299720777806142464373 z 62 76 + 17208246336680475982820699 z - 108614475410272470285235 z 74 72 + 324299720777806142464373 z - 854021678881699702625155 z 104 - 5323905925 z ) And in Maple-input format, it is: -(1+21054066485296978868*z^28-3321890313189238464*z^26-248*z^2+ 452396980688475120*z^24-52868372316615048*z^22+27962*z^4-1921140*z^6-3166129692 *z^102+90862887*z^8-3166129692*z^10+84892078174*z^12-1805267788888*z^14-\ 442807327970296*z^18+31131779292541*z^16-2203534681346137141437648*z^50+ 1442415939590879847050666*z^48+5264314114304304*z^20+8501247162436625616988*z^ 36-2318466178073574886824*z^34-835821194554457563585064*z^66+ 554136920658339454692*z^80+84892078174*z^100-52868372316615048*z^90+ 452396980688475120*z^88+21054066485296978868*z^84-442807327970296*z^94-\ 3321890313189238464*z^86+31131779292541*z^96-1805267788888*z^98+ 5264314114304304*z^92-115741673944354624984*z^82+1442415939590879847050666*z^64 +z^112-248*z^110-1921140*z^106+27962*z^108-115741673944354624984*z^30-\ 194113380262898736084056*z^42+428444591724038426384784*z^44-\ 835821194554457563585064*z^46-3573890755760563233187184*z^58+ 3796400661420286658801514*z^56-3573890755760563233187184*z^54+ 2981397936950209497160408*z^52+2981397936950209497160408*z^60-\ 194113380262898736084056*z^70+428444591724038426384784*z^68-\ 2318466178073574886824*z^78+554136920658339454692*z^32-27384250311980620829600* z^38+77646150473995786894312*z^40-2203534681346137141437648*z^62+ 8501247162436625616988*z^76-27384250311980620829600*z^74+ 77646150473995786894312*z^72+90862887*z^104)/(-1-64276018614013184825*z^28+ 9598202172844350551*z^26+280*z^2-1235025505744056905*z^24+136077299446524587*z^ 22-35291*z^4+2687603*z^6+155017854124*z^102-139836302*z^8+5323905925*z^10-\ 155017854124*z^12+3560474348958*z^14+1004891194861132*z^18-65999991924873*z^16+ 12021823510838764362048723*z^50-7446917800440303826418087*z^48-\ 12742420461745133*z^20-32024396074848103627313*z^36+8293866318770826194523*z^34 +7446917800440303826418087*z^66-8293866318770826194523*z^80-3560474348958*z^100 +1235025505744056905*z^90-9598202172844350551*z^88-372849677063662265779*z^84+ 12742420461745133*z^94+64276018614013184825*z^86-1004891194861132*z^96+ 65999991924873*z^98-136077299446524587*z^92+1881756798618163160297*z^82-\ 12021823510838764362048723*z^64-280*z^112+z^114+35291*z^110+139836302*z^106-\ 2687603*z^108+372849677063662265779*z^30+854021678881699702625155*z^42-\ 1986685005098044640235377*z^44+4087725049745055368672823*z^46+ 24621270671506564365709673*z^58-24621270671506564365709673*z^56+ 21851118599484386136680253*z^54-17208246336680475982820699*z^52-\ 21851118599484386136680253*z^60+1986685005098044640235377*z^70-\ 4087725049745055368672823*z^68+32024396074848103627313*z^78-\ 1881756798618163160297*z^32+108614475410272470285235*z^38-\ 324299720777806142464373*z^40+17208246336680475982820699*z^62-\ 108614475410272470285235*z^76+324299720777806142464373*z^74-\ 854021678881699702625155*z^72-5323905925*z^104) The first , 40, terms are: [0, 32, 0, 1631, 0, 93831, 0, 5742940, 0, 363128441, 0, 23348550121, 0, 1514116734184, 0, 98612861767107, 0, 6436612450143051, 0, 420593469686663548, 0, 27498686714134403425, 0, 1798396477459463093793, 0, 117631054324751049219572, 0, 7694680681372378156717331, 0, 503356320843212650826603627, 0, 32928257678574768199869943280, 0, 2154101687792941413502880544761, 0, 140917782511103923031552258403081, 0, 9218632335000613368314317576736404, 0, 603070028761728443601911146676891295 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 102679602088787574 z - 27072088397552308 z - 197 z 24 22 4 6 + 6027005663994072 z - 1127179020456748 z + 17038 z - 873761 z 8 10 12 14 + 30151047 z - 751120846 z + 14126821904 z - 206852795412 z 18 16 50 - 22748618837570 z + 2411123526376 z - 7187896112035273090 z 48 20 36 + 10528989225215710152 z + 175965942620378 z + 4207102693628413536 z 34 66 80 88 - 2108586897601773564 z - 1127179020456748 z + 30151047 z + z 84 86 82 64 + 17038 z - 197 z - 873761 z + 6027005663994072 z 30 42 - 330211858051752462 z - 13234615374613290490 z 44 46 + 14282208731595434580 z - 13234615374613290490 z 58 56 - 330211858051752462 z + 903451182326516584 z 54 52 - 2108586897601773564 z + 4207102693628413536 z 60 70 68 + 102679602088787574 z - 22748618837570 z + 175965942620378 z 78 32 38 - 751120846 z + 903451182326516584 z - 7187896112035273090 z 40 62 76 + 10528989225215710152 z - 27072088397552308 z + 14126821904 z 74 72 / 2 - 206852795412 z + 2411123526376 z ) / ((-1 + z ) (1 / 28 26 2 + 304968412106213320 z - 77903654134664718 z - 227 z 24 22 4 6 + 16726350368934780 z - 3001827695190898 z + 22395 z - 1293649 z 8 10 12 14 + 49659413 z - 1360413716 z + 27844267786 z - 439565707492 z 18 16 50 - 54848880671580 z + 5478067751882 z - 23545300345090415420 z 48 20 + 34788448761136770466 z + 447239819821936 z 36 34 + 13614821763759287358 z - 6717168888115234956 z 66 80 88 84 86 - 3001827695190898 z + 49659413 z + z + 22395 z - 227 z 82 64 30 - 1293649 z + 16726350368934780 z - 1007935601526127284 z 42 44 - 43954095094348010380 z + 47514773628847530626 z 46 58 - 43954095094348010380 z - 1007935601526127284 z 56 54 + 2822655853151862206 z - 6717168888115234956 z 52 60 70 + 13614821763759287358 z + 304968412106213320 z - 54848880671580 z 68 78 32 + 447239819821936 z - 1360413716 z + 2822655853151862206 z 38 40 - 23545300345090415420 z + 34788448761136770466 z 62 76 74 - 77903654134664718 z + 27844267786 z - 439565707492 z 72 + 5478067751882 z )) And in Maple-input format, it is: -(1+102679602088787574*z^28-27072088397552308*z^26-197*z^2+6027005663994072*z^ 24-1127179020456748*z^22+17038*z^4-873761*z^6+30151047*z^8-751120846*z^10+ 14126821904*z^12-206852795412*z^14-22748618837570*z^18+2411123526376*z^16-\ 7187896112035273090*z^50+10528989225215710152*z^48+175965942620378*z^20+ 4207102693628413536*z^36-2108586897601773564*z^34-1127179020456748*z^66+ 30151047*z^80+z^88+17038*z^84-197*z^86-873761*z^82+6027005663994072*z^64-\ 330211858051752462*z^30-13234615374613290490*z^42+14282208731595434580*z^44-\ 13234615374613290490*z^46-330211858051752462*z^58+903451182326516584*z^56-\ 2108586897601773564*z^54+4207102693628413536*z^52+102679602088787574*z^60-\ 22748618837570*z^70+175965942620378*z^68-751120846*z^78+903451182326516584*z^32 -7187896112035273090*z^38+10528989225215710152*z^40-27072088397552308*z^62+ 14126821904*z^76-206852795412*z^74+2411123526376*z^72)/(-1+z^2)/(1+ 304968412106213320*z^28-77903654134664718*z^26-227*z^2+16726350368934780*z^24-\ 3001827695190898*z^22+22395*z^4-1293649*z^6+49659413*z^8-1360413716*z^10+ 27844267786*z^12-439565707492*z^14-54848880671580*z^18+5478067751882*z^16-\ 23545300345090415420*z^50+34788448761136770466*z^48+447239819821936*z^20+ 13614821763759287358*z^36-6717168888115234956*z^34-3001827695190898*z^66+ 49659413*z^80+z^88+22395*z^84-227*z^86-1293649*z^82+16726350368934780*z^64-\ 1007935601526127284*z^30-43954095094348010380*z^42+47514773628847530626*z^44-\ 43954095094348010380*z^46-1007935601526127284*z^58+2822655853151862206*z^56-\ 6717168888115234956*z^54+13614821763759287358*z^52+304968412106213320*z^60-\ 54848880671580*z^70+447239819821936*z^68-1360413716*z^78+2822655853151862206*z^ 32-23545300345090415420*z^38+34788448761136770466*z^40-77903654134664718*z^62+ 27844267786*z^76-439565707492*z^74+5478067751882*z^72) The first , 40, terms are: [0, 31, 0, 1484, 0, 79353, 0, 4516785, 0, 267120071, 0, 16176768843, 0, 994283096436, 0, 61680406751705, 0, 3848500730323147, 0, 240988597879018315, 0, 15124138950667416385, 0, 950485972742446920404, 0, 59785075890145052917611, 0, 3762443243976654041111375, 0, 236858707233597179610178313, 0, 14914085257094925399452228329, 0, 939200164408991368705452960300, 0, 59149790692857676981407967611543, 0, 3725366090635892477193872188925937, 0, 234637529706299564982594870294019441 ] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 628919018706516 z - 303820696113302 z - 171 z 24 22 4 6 + 118774295760178 z - 37451336881464 z + 12420 z - 517325 z 8 10 12 14 + 14022187 z - 265404792 z + 3668620107 z - 38182901069 z 18 16 50 - 1914589097499 z + 305800726208 z - 1914589097499 z 48 20 36 + 9480647856033 z + 9480647856033 z + 1440282630662662 z 34 66 64 30 - 1597127292720240 z - 171 z + 12420 z - 1055977359398154 z 42 44 46 - 303820696113302 z + 118774295760178 z - 37451336881464 z 58 56 54 52 - 265404792 z + 3668620107 z - 38182901069 z + 305800726208 z 60 68 32 38 + 14022187 z + z + 1440282630662662 z - 1055977359398154 z 40 62 / 28 + 628919018706516 z - 517325 z ) / (-1 - 3051257440164942 z / 26 2 24 + 1341873819435874 z + 205 z - 478233535009450 z 22 4 6 8 10 + 137584633163129 z - 17415 z + 833237 z - 25601287 z + 543756647 z 12 14 18 16 - 8368597811 z + 96401346875 z + 5852234699031 z - 850780524777 z 50 48 20 + 31781618328389 z - 137584633163129 z - 31781618328389 z 36 34 66 64 - 10400922856228954 z + 10400922856228954 z + 17415 z - 833237 z 30 42 44 + 5638204164454750 z + 3051257440164942 z - 1341873819435874 z 46 58 56 + 478233535009450 z + 8368597811 z - 96401346875 z 54 52 60 70 68 + 850780524777 z - 5852234699031 z - 543756647 z + z - 205 z 32 38 40 - 8482160346976226 z + 8482160346976226 z - 5638204164454750 z 62 + 25601287 z ) And in Maple-input format, it is: -(1+628919018706516*z^28-303820696113302*z^26-171*z^2+118774295760178*z^24-\ 37451336881464*z^22+12420*z^4-517325*z^6+14022187*z^8-265404792*z^10+3668620107 *z^12-38182901069*z^14-1914589097499*z^18+305800726208*z^16-1914589097499*z^50+ 9480647856033*z^48+9480647856033*z^20+1440282630662662*z^36-1597127292720240*z^ 34-171*z^66+12420*z^64-1055977359398154*z^30-303820696113302*z^42+ 118774295760178*z^44-37451336881464*z^46-265404792*z^58+3668620107*z^56-\ 38182901069*z^54+305800726208*z^52+14022187*z^60+z^68+1440282630662662*z^32-\ 1055977359398154*z^38+628919018706516*z^40-517325*z^62)/(-1-3051257440164942*z^ 28+1341873819435874*z^26+205*z^2-478233535009450*z^24+137584633163129*z^22-\ 17415*z^4+833237*z^6-25601287*z^8+543756647*z^10-8368597811*z^12+96401346875*z^ 14+5852234699031*z^18-850780524777*z^16+31781618328389*z^50-137584633163129*z^ 48-31781618328389*z^20-10400922856228954*z^36+10400922856228954*z^34+17415*z^66 -833237*z^64+5638204164454750*z^30+3051257440164942*z^42-1341873819435874*z^44+ 478233535009450*z^46+8368597811*z^58-96401346875*z^56+850780524777*z^54-\ 5852234699031*z^52-543756647*z^60+z^70-205*z^68-8482160346976226*z^32+ 8482160346976226*z^38-5638204164454750*z^40+25601287*z^62) The first , 40, terms are: [0, 34, 0, 1975, 0, 128677, 0, 8735118, 0, 603340407, 0, 42006347383, 0, 2935860233334, 0, 205579755353593, 0, 14409132022180455, 0, 1010422130053814546, 0, 70871657053768712397, 0, 4971588968794053324189, 0, 348774381112441270189842, 0, 24468506000883751095885271, 0, 1716631838390296248326593433, 0, 120434337799906381389818432982, 0, 8449387784981030000277031607111, 0, 592790233456816434114406575922967, 0, 41588885662644478582449639710549774, 0, 2917788136852372643051437365212163733] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 241085468002271178 z - 61677475770016121 z - 213 z 24 22 4 6 + 13267387838479858 z - 2386680885327211 z + 20035 z - 1118052 z 8 10 12 14 + 41869343 z - 1126818165 z + 22769817276 z - 356128643591 z 18 16 50 - 43909914857079 z + 4407978336120 z - 18542268146666678523 z 48 20 + 27389242992743409937 z + 356657600701954 z 36 34 + 10726085981119196747 z - 5294849408600565514 z 66 80 88 84 86 - 2386680885327211 z + 41869343 z + z + 20035 z - 213 z 82 64 30 - 1118052 z + 13267387838479858 z - 795844571042420257 z 42 44 - 34600278969435529994 z + 37401408106671015956 z 46 58 - 34600278969435529994 z - 795844571042420257 z 56 54 + 2226602548154694533 z - 5294849408600565514 z 52 60 70 + 10726085981119196747 z + 241085468002271178 z - 43909914857079 z 68 78 32 + 356657600701954 z - 1126818165 z + 2226602548154694533 z 38 40 - 18542268146666678523 z + 27389242992743409937 z 62 76 74 - 61677475770016121 z + 22769817276 z - 356128643591 z 72 / 28 + 4407978336120 z ) / (-1 - 900295455723068242 z / 26 2 24 + 215241513227306047 z + 249 z - 43230029146870591 z 22 4 6 8 + 7250596807334016 z - 26828 z + 1689748 z - 70615970 z 10 12 14 + 2101822258 z - 46627844142 z + 795789191626 z 18 16 50 + 115183141084441 z - 10694217005949 z + 154627102208441395667 z 48 20 - 210233596100249420807 z - 1008167154981804 z 36 34 - 52552033465851998212 z + 24216197469277590122 z 66 80 90 88 84 + 43230029146870591 z - 2101822258 z + z - 249 z - 1689748 z 86 82 64 + 26828 z + 70615970 z - 215241513227306047 z 30 42 + 3179001976569390810 z + 210233596100249420807 z 44 46 - 245092022238389211020 z + 245092022238389211020 z 58 56 + 9514577889579028538 z - 24216197469277590122 z 54 52 + 52552033465851998212 z - 97442891281754164768 z 60 70 68 - 3179001976569390810 z + 1008167154981804 z - 7250596807334016 z 78 32 38 + 46627844142 z - 9514577889579028538 z + 97442891281754164768 z 40 62 76 - 154627102208441395667 z + 900295455723068242 z - 795789191626 z 74 72 + 10694217005949 z - 115183141084441 z ) And in Maple-input format, it is: -(1+241085468002271178*z^28-61677475770016121*z^26-213*z^2+13267387838479858*z^ 24-2386680885327211*z^22+20035*z^4-1118052*z^6+41869343*z^8-1126818165*z^10+ 22769817276*z^12-356128643591*z^14-43909914857079*z^18+4407978336120*z^16-\ 18542268146666678523*z^50+27389242992743409937*z^48+356657600701954*z^20+ 10726085981119196747*z^36-5294849408600565514*z^34-2386680885327211*z^66+ 41869343*z^80+z^88+20035*z^84-213*z^86-1118052*z^82+13267387838479858*z^64-\ 795844571042420257*z^30-34600278969435529994*z^42+37401408106671015956*z^44-\ 34600278969435529994*z^46-795844571042420257*z^58+2226602548154694533*z^56-\ 5294849408600565514*z^54+10726085981119196747*z^52+241085468002271178*z^60-\ 43909914857079*z^70+356657600701954*z^68-1126818165*z^78+2226602548154694533*z^ 32-18542268146666678523*z^38+27389242992743409937*z^40-61677475770016121*z^62+ 22769817276*z^76-356128643591*z^74+4407978336120*z^72)/(-1-900295455723068242*z ^28+215241513227306047*z^26+249*z^2-43230029146870591*z^24+7250596807334016*z^ 22-26828*z^4+1689748*z^6-70615970*z^8+2101822258*z^10-46627844142*z^12+ 795789191626*z^14+115183141084441*z^18-10694217005949*z^16+ 154627102208441395667*z^50-210233596100249420807*z^48-1008167154981804*z^20-\ 52552033465851998212*z^36+24216197469277590122*z^34+43230029146870591*z^66-\ 2101822258*z^80+z^90-249*z^88-1689748*z^84+26828*z^86+70615970*z^82-\ 215241513227306047*z^64+3179001976569390810*z^30+210233596100249420807*z^42-\ 245092022238389211020*z^44+245092022238389211020*z^46+9514577889579028538*z^58-\ 24216197469277590122*z^56+52552033465851998212*z^54-97442891281754164768*z^52-\ 3179001976569390810*z^60+1008167154981804*z^70-7250596807334016*z^68+ 46627844142*z^78-9514577889579028538*z^32+97442891281754164768*z^38-\ 154627102208441395667*z^40+900295455723068242*z^62-795789191626*z^76+ 10694217005949*z^74-115183141084441*z^72) The first , 40, terms are: [0, 36, 0, 2171, 0, 146467, 0, 10310996, 0, 739293409, 0, 53453282021, 0, 3880293518636, 0, 282227734404699, 0, 20547226742118319, 0, 1496635274836707196, 0, 109039545300421672529, 0, 7945208208351865120257, 0, 578966558283881361058348, 0, 42190565601531173792128303, 0, 3074568407353863956879675931, 0, 224055961024855512790277999164, 0, 16327911598085951751007555819285, 0, 1189886921294191043522856588610929, 0, 86712399893964133159714458431577828, 0, 6319125165031806105060070525774123779] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 4}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 362610770321175 z + 226134378217590 z + 196 z 24 22 4 6 - 110916996211393 z + 42579350821137 z - 16361 z + 771913 z 8 10 12 14 - 23075248 z + 465371581 z - 6595290195 z + 67619026076 z 18 16 50 48 + 2923057140183 z - 512628647309 z + 6595290195 z - 67619026076 z 20 36 34 - 12707575004806 z - 226134378217590 z + 362610770321175 z 30 42 44 + 458804775938501 z + 12707575004806 z - 2923057140183 z 46 58 56 54 52 + 512628647309 z + 16361 z - 771913 z + 23075248 z - 465371581 z 60 32 38 40 - 196 z - 458804775938501 z + 110916996211393 z - 42579350821137 z 62 / 28 26 2 + z ) / (1 + 2049735880590056 z - 1144767737592268 z - 240 z / 24 22 4 6 + 503794289468688 z - 173760995754324 z + 23600 z - 1282984 z 8 10 12 14 + 43601324 z - 991119004 z + 15741636312 z - 180182477680 z 18 16 50 - 9648316090456 z + 1521314674298 z - 180182477680 z 48 20 36 + 1521314674298 z + 46632250774920 z + 2049735880590056 z 34 64 30 42 - 2902941031149448 z + z - 2902941031149448 z - 173760995754324 z 44 46 58 56 + 46632250774920 z - 9648316090456 z - 1282984 z + 43601324 z 54 52 60 32 - 991119004 z + 15741636312 z + 23600 z + 3259199471546499 z 38 40 62 - 1144767737592268 z + 503794289468688 z - 240 z ) And in Maple-input format, it is: -(-1-362610770321175*z^28+226134378217590*z^26+196*z^2-110916996211393*z^24+ 42579350821137*z^22-16361*z^4+771913*z^6-23075248*z^8+465371581*z^10-6595290195 *z^12+67619026076*z^14+2923057140183*z^18-512628647309*z^16+6595290195*z^50-\ 67619026076*z^48-12707575004806*z^20-226134378217590*z^36+362610770321175*z^34+ 458804775938501*z^30+12707575004806*z^42-2923057140183*z^44+512628647309*z^46+ 16361*z^58-771913*z^56+23075248*z^54-465371581*z^52-196*z^60-458804775938501*z^ 32+110916996211393*z^38-42579350821137*z^40+z^62)/(1+2049735880590056*z^28-\ 1144767737592268*z^26-240*z^2+503794289468688*z^24-173760995754324*z^22+23600*z ^4-1282984*z^6+43601324*z^8-991119004*z^10+15741636312*z^12-180182477680*z^14-\ 9648316090456*z^18+1521314674298*z^16-180182477680*z^50+1521314674298*z^48+ 46632250774920*z^20+2049735880590056*z^36-2902941031149448*z^34+z^64-\ 2902941031149448*z^30-173760995754324*z^42+46632250774920*z^44-9648316090456*z^ 46-1282984*z^58+43601324*z^56-991119004*z^54+15741636312*z^52+23600*z^60+ 3259199471546499*z^32-1144767737592268*z^38+503794289468688*z^40-240*z^62) The first , 40, terms are: [0, 44, 0, 3321, 0, 269711, 0, 22280260, 0, 1850161831, 0, 153922494119, 0, 12814477441596, 0, 1067139982780727, 0, 88877390044987381, 0, 7402553318888955124, 0, 616566985288732394633, 0, 51354977686671603845589, 0, 4277463656680677681379220, 0, 356279433318937977156670777, 0, 29675322848481710858288484667, 0, 2471725686087266998968897166748, 0, 205875724506731004805311149285259, 0, 17147864089489149497399248548926411, 0, 1428285195795618615090070153327502308, 0, 118965173165848875129503210237719564387] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 23277002202414501256 z - 3665319918562314871 z - 251 z 24 22 4 6 + 497952623736874300 z - 58018976391384701 z + 28656 z - 1992573 z 102 8 10 12 - 3353031923 z + 95285132 z - 3353031923 z + 90680001116 z 14 18 16 - 1942681775533 z - 482061095806399 z + 33712891641568 z 50 48 - 2457155223236300957157067 z + 1608002167995338686340860 z 20 36 + 5756241678914848 z + 9446056917069871319784 z 34 66 - 2573737641314003566859 z - 931452911341376713969805 z 80 100 90 + 614465942226106327692 z + 90680001116 z - 58018976391384701 z 88 84 94 + 497952623736874300 z + 23277002202414501256 z - 482061095806399 z 86 96 98 - 3665319918562314871 z + 33712891641568 z - 1942681775533 z 92 82 + 5756241678914848 z - 128170585743081889241 z 64 112 110 106 + 1608002167995338686340860 z + z - 251 z - 1992573 z 108 30 42 + 28656 z - 128170585743081889241 z - 216126606714594429628539 z 44 46 + 477267091330510938210332 z - 931452911341376713969805 z 58 56 - 3986473138969456265298949 z + 4234836204899074445541374 z 54 52 - 3986473138969456265298949 z + 3325191815466756058977880 z 60 70 + 3325191815466756058977880 z - 216126606714594429628539 z 68 78 + 477267091330510938210332 z - 2573737641314003566859 z 32 38 + 614465942226106327692 z - 30451664831410418830097 z 40 62 + 86401696296789788356536 z - 2457155223236300957157067 z 76 74 + 9446056917069871319784 z - 30451664831410418830097 z 72 104 / 2 + 86401696296789788356536 z + 95285132 z ) / ((-1 + z ) (1 / 28 26 2 + 62033737272985388068 z - 9412099468184018184 z - 287 z 24 22 4 6 + 1228421476675427378 z - 137066964723088706 z + 36700 z - 2816688 z 102 8 10 12 - 5596189950 z + 147014754 z - 5596189950 z + 162505212748 z 14 18 16 - 3714788891375 z - 1033308181757649 z + 68415640182648 z 50 48 - 8307114083026313408761833 z + 5387159975771535361898684 z 20 36 + 12977970282587584 z + 28428870483714985449888 z 34 66 - 7543015407004582743783 z - 3084485118606806869173163 z 80 100 90 + 1749261018114106309708 z + 162505212748 z - 137066964723088706 z 88 84 + 1228421476675427378 z + 62033737272985388068 z 94 86 96 - 1033308181757649 z - 9412099468184018184 z + 68415640182648 z 98 92 82 - 3714788891375 z + 12977970282587584 z - 353507377916216647265 z 64 112 110 106 + 5387159975771535361898684 z + z - 287 z - 2816688 z 108 30 42 + 36700 z - 353507377916216647265 z - 693987479386715797428722 z 44 46 + 1558247070846120103811172 z - 3084485118606806869173163 z 58 56 - 13618598513475893462534458 z + 14485963631407922856824628 z 54 52 - 13618598513475893462534458 z + 11315158743896887795691940 z 60 70 + 11315158743896887795691940 z - 693987479386715797428722 z 68 78 + 1558247070846120103811172 z - 7543015407004582743783 z 32 38 + 1749261018114106309708 z - 93879197028732430316616 z 40 62 + 272181451587273528202838 z - 8307114083026313408761833 z 76 74 + 28428870483714985449888 z - 93879197028732430316616 z 72 104 + 272181451587273528202838 z + 147014754 z )) And in Maple-input format, it is: -(1+23277002202414501256*z^28-3665319918562314871*z^26-251*z^2+ 497952623736874300*z^24-58018976391384701*z^22+28656*z^4-1992573*z^6-3353031923 *z^102+95285132*z^8-3353031923*z^10+90680001116*z^12-1942681775533*z^14-\ 482061095806399*z^18+33712891641568*z^16-2457155223236300957157067*z^50+ 1608002167995338686340860*z^48+5756241678914848*z^20+9446056917069871319784*z^ 36-2573737641314003566859*z^34-931452911341376713969805*z^66+ 614465942226106327692*z^80+90680001116*z^100-58018976391384701*z^90+ 497952623736874300*z^88+23277002202414501256*z^84-482061095806399*z^94-\ 3665319918562314871*z^86+33712891641568*z^96-1942681775533*z^98+ 5756241678914848*z^92-128170585743081889241*z^82+1608002167995338686340860*z^64 +z^112-251*z^110-1992573*z^106+28656*z^108-128170585743081889241*z^30-\ 216126606714594429628539*z^42+477267091330510938210332*z^44-\ 931452911341376713969805*z^46-3986473138969456265298949*z^58+ 4234836204899074445541374*z^56-3986473138969456265298949*z^54+ 3325191815466756058977880*z^52+3325191815466756058977880*z^60-\ 216126606714594429628539*z^70+477267091330510938210332*z^68-\ 2573737641314003566859*z^78+614465942226106327692*z^32-30451664831410418830097* z^38+86401696296789788356536*z^40-2457155223236300957157067*z^62+ 9446056917069871319784*z^76-30451664831410418830097*z^74+ 86401696296789788356536*z^72+95285132*z^104)/(-1+z^2)/(1+62033737272985388068*z ^28-9412099468184018184*z^26-287*z^2+1228421476675427378*z^24-\ 137066964723088706*z^22+36700*z^4-2816688*z^6-5596189950*z^102+147014754*z^8-\ 5596189950*z^10+162505212748*z^12-3714788891375*z^14-1033308181757649*z^18+ 68415640182648*z^16-8307114083026313408761833*z^50+5387159975771535361898684*z^ 48+12977970282587584*z^20+28428870483714985449888*z^36-7543015407004582743783*z ^34-3084485118606806869173163*z^66+1749261018114106309708*z^80+162505212748*z^ 100-137066964723088706*z^90+1228421476675427378*z^88+62033737272985388068*z^84-\ 1033308181757649*z^94-9412099468184018184*z^86+68415640182648*z^96-\ 3714788891375*z^98+12977970282587584*z^92-353507377916216647265*z^82+ 5387159975771535361898684*z^64+z^112-287*z^110-2816688*z^106+36700*z^108-\ 353507377916216647265*z^30-693987479386715797428722*z^42+ 1558247070846120103811172*z^44-3084485118606806869173163*z^46-\ 13618598513475893462534458*z^58+14485963631407922856824628*z^56-\ 13618598513475893462534458*z^54+11315158743896887795691940*z^52+ 11315158743896887795691940*z^60-693987479386715797428722*z^70+ 1558247070846120103811172*z^68-7543015407004582743783*z^78+ 1749261018114106309708*z^32-93879197028732430316616*z^38+ 272181451587273528202838*z^40-8307114083026313408761833*z^62+ 28428870483714985449888*z^76-93879197028732430316616*z^74+ 272181451587273528202838*z^72+147014754*z^104) The first , 40, terms are: [0, 37, 0, 2325, 0, 161896, 0, 11660319, 0, 850661047, 0, 62381336147, 0, 4584539207251, 0, 337240877152467, 0, 24817605504120211, 0, 1826654122800285016, 0, 134458006920933569409, 0, 9897649628331533085329, 0, 728591674446020281140393, 0, 53633896115795716517340121, 0, 3948170012261510465628616857, 0, 290638375887241067852801487257, 0, 21394904322670443701301482008888, 0, 1574953972043890908549025871903835, 0, 115937903127653139136365000276140155, 0, 8534597481601388945845321365996431667] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 573504582746281 z - 284666447452474 z - 180 z 24 22 4 6 + 114972120405560 z - 37594683924843 z + 13742 z - 594787 z 8 10 12 14 + 16491843 z - 314241615 z + 4310674398 z - 44004612609 z 18 16 50 - 2072770505903 z + 342605248644 z - 2072770505903 z 48 20 36 + 9886407087807 z + 9886407087807 z + 1270880560643802 z 34 66 64 30 - 1403295149592650 z - 180 z + 13742 z - 943564807369966 z 42 44 46 - 284666447452474 z + 114972120405560 z - 37594683924843 z 58 56 54 52 - 314241615 z + 4310674398 z - 44004612609 z + 342605248644 z 60 68 32 38 + 16491843 z + z + 1270880560643802 z - 943564807369966 z 40 62 / 28 + 573504582746281 z - 594787 z ) / (-1 - 2895114865920650 z / 26 2 24 + 1310730877449590 z + 223 z - 483036503499526 z 22 4 6 8 10 + 144084435313941 z - 20043 z + 995175 z - 31194848 z + 665892802 z 12 14 18 - 10169641178 z + 115046697886 z + 6589322136951 z 16 50 48 - 989085222311 z + 34534539364103 z - 144084435313941 z 20 36 34 - 34534539364103 z - 9410884400943337 z + 9410884400943337 z 66 64 30 42 + 20043 z - 995175 z + 5227262007305204 z + 2895114865920650 z 44 46 58 - 1310730877449590 z + 483036503499526 z + 10169641178 z 56 54 52 60 - 115046697886 z + 989085222311 z - 6589322136951 z - 665892802 z 70 68 32 38 + z - 223 z - 7738356689169997 z + 7738356689169997 z 40 62 - 5227262007305204 z + 31194848 z ) And in Maple-input format, it is: -(1+573504582746281*z^28-284666447452474*z^26-180*z^2+114972120405560*z^24-\ 37594683924843*z^22+13742*z^4-594787*z^6+16491843*z^8-314241615*z^10+4310674398 *z^12-44004612609*z^14-2072770505903*z^18+342605248644*z^16-2072770505903*z^50+ 9886407087807*z^48+9886407087807*z^20+1270880560643802*z^36-1403295149592650*z^ 34-180*z^66+13742*z^64-943564807369966*z^30-284666447452474*z^42+ 114972120405560*z^44-37594683924843*z^46-314241615*z^58+4310674398*z^56-\ 44004612609*z^54+342605248644*z^52+16491843*z^60+z^68+1270880560643802*z^32-\ 943564807369966*z^38+573504582746281*z^40-594787*z^62)/(-1-2895114865920650*z^ 28+1310730877449590*z^26+223*z^2-483036503499526*z^24+144084435313941*z^22-\ 20043*z^4+995175*z^6-31194848*z^8+665892802*z^10-10169641178*z^12+115046697886* z^14+6589322136951*z^18-989085222311*z^16+34534539364103*z^50-144084435313941*z ^48-34534539364103*z^20-9410884400943337*z^36+9410884400943337*z^34+20043*z^66-\ 995175*z^64+5227262007305204*z^30+2895114865920650*z^42-1310730877449590*z^44+ 483036503499526*z^46+10169641178*z^58-115046697886*z^56+989085222311*z^54-\ 6589322136951*z^52-665892802*z^60+z^70-223*z^68-7738356689169997*z^32+ 7738356689169997*z^38-5227262007305204*z^40+31194848*z^62) The first , 40, terms are: [0, 43, 0, 3288, 0, 271763, 0, 22791285, 0, 1917918869, 0, 161547689539, 0, 13611201496280, 0, 1146919604789751, 0, 96645874484456445, 0, 8144013395890756125, 0, 686270530360299432095, 0, 57829952965592378034200, 0, 4873158703220592732134739, 0, 410646717582180902530126045, 0, 34603990815238292976053571181, 0, 2915976586361742541744306414931, 0, 245720777954677454243800815413112, 0, 20706167961914261823523732401323571, 0, 1744847936486142596122743092172421793, 0, 147033209090888131749344049838779787297] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 514677913787640 z + 268264852096462 z + 181 z 24 22 4 6 - 112161478103618 z + 37494273948308 z - 13784 z + 593572 z 8 10 12 14 - 16405782 z + 312776738 z - 4307601552 z + 44225992496 z 18 16 50 + 2098671704125 z - 346133230973 z + 346133230973 z 48 20 36 - 2098671704125 z - 9975236660972 z - 793791006981452 z 34 66 64 30 + 985495508677704 z + z - 181 z + 793791006981452 z 42 44 46 + 112161478103618 z - 37494273948308 z + 9975236660972 z 58 56 54 52 + 16405782 z - 312776738 z + 4307601552 z - 44225992496 z 60 32 38 - 593572 z - 985495508677704 z + 514677913787640 z 40 62 / 28 - 268264852096462 z + 13784 z ) / (1 + 2626802116100922 z / 26 2 24 - 1243363808934236 z - 221 z + 473131846784620 z 22 4 6 8 10 - 144167762250304 z + 19931 z - 994202 z + 31300122 z - 671093616 z 12 14 18 + 10295858200 z - 116957241984 z - 6712345460359 z 16 50 48 + 1008103796587 z - 6712345460359 z + 34986246633939 z 20 36 34 + 34986246633939 z + 6153456660319502 z - 6842586531054496 z 66 64 30 42 - 221 z + 19931 z - 4473740492780966 z - 1243363808934236 z 44 46 58 + 473131846784620 z - 144167762250304 z - 671093616 z 56 54 52 60 + 10295858200 z - 116957241984 z + 1008103796587 z + 31300122 z 68 32 38 40 + z + 6153456660319502 z - 4473740492780966 z + 2626802116100922 z 62 - 994202 z ) And in Maple-input format, it is: -(-1-514677913787640*z^28+268264852096462*z^26+181*z^2-112161478103618*z^24+ 37494273948308*z^22-13784*z^4+593572*z^6-16405782*z^8+312776738*z^10-4307601552 *z^12+44225992496*z^14+2098671704125*z^18-346133230973*z^16+346133230973*z^50-\ 2098671704125*z^48-9975236660972*z^20-793791006981452*z^36+985495508677704*z^34 +z^66-181*z^64+793791006981452*z^30+112161478103618*z^42-37494273948308*z^44+ 9975236660972*z^46+16405782*z^58-312776738*z^56+4307601552*z^54-44225992496*z^ 52-593572*z^60-985495508677704*z^32+514677913787640*z^38-268264852096462*z^40+ 13784*z^62)/(1+2626802116100922*z^28-1243363808934236*z^26-221*z^2+ 473131846784620*z^24-144167762250304*z^22+19931*z^4-994202*z^6+31300122*z^8-\ 671093616*z^10+10295858200*z^12-116957241984*z^14-6712345460359*z^18+ 1008103796587*z^16-6712345460359*z^50+34986246633939*z^48+34986246633939*z^20+ 6153456660319502*z^36-6842586531054496*z^34-221*z^66+19931*z^64-\ 4473740492780966*z^30-1243363808934236*z^42+473131846784620*z^44-\ 144167762250304*z^46-671093616*z^58+10295858200*z^56-116957241984*z^54+ 1008103796587*z^52+31300122*z^60+z^68+6153456660319502*z^32-4473740492780966*z^ 38+2626802116100922*z^40-994202*z^62) The first , 40, terms are: [0, 40, 0, 2693, 0, 198543, 0, 15077560, 0, 1158678211, 0, 89513143403, 0, 6932661482872, 0, 537592966602455, 0, 41714085499338413, 0, 3237835912095042536, 0, 251363324846460279049, 0, 19515899188655768970553, 0, 1515291548005138378613864, 0, 117656245892021440289389981, 0, 9135655407675729786707066887, 0, 709361471184320103203505693688, 0, 55080410635422313276632208395739, 0, 4276885684989790396121562853608659, 0, 332092138970644468035058858696786488, 0, 25786345941189714335444634398633503711] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 205087718459383527 z - 52087420114408797 z - 205 z 24 22 4 6 + 11137608251758973 z - 1995160672581160 z + 18659 z - 1012190 z 8 10 12 14 + 37010496 z - 976835818 z + 19441269245 z - 300703154679 z 18 16 50 - 36663384182296 z + 3694758450159 z - 16387477634346676063 z 48 20 36 + 24321668800466821843 z + 297572308612056 z + 9420463598736696057 z 34 66 80 88 - 4616052311754749898 z - 1995160672581160 z + 37010496 z + z 84 86 82 64 + 18659 z - 205 z - 1012190 z + 11137608251758973 z 30 42 - 682497190145137166 z - 30816409712472940768 z 44 46 + 33344845270831325232 z - 30816409712472940768 z 58 56 - 682497190145137166 z + 1925474746807996688 z 54 52 - 4616052311754749898 z + 9420463598736696057 z 60 70 68 + 205087718459383527 z - 36663384182296 z + 297572308612056 z 78 32 38 - 976835818 z + 1925474746807996688 z - 16387477634346676063 z 40 62 76 + 24321668800466821843 z - 52087420114408797 z + 19441269245 z 74 72 / 28 - 300703154679 z + 3694758450159 z ) / (-1 - 760083802408935584 z / 26 2 24 + 179800498310048409 z + 243 z - 35790590941171867 z 22 4 6 8 + 5963519702622742 z - 25314 z + 1540468 z - 62404565 z 10 12 14 18 + 1809588541 z - 39332817286 z + 661449793188 z + 94418588491437 z 16 50 - 8805813096111 z + 138988767022689975221 z 48 20 - 189946984814356375015 z - 826281238056278 z 36 34 - 46454254557333946440 z + 21177577051171048395 z 66 80 90 88 84 + 35790590941171867 z - 1809588541 z + z - 243 z - 1540468 z 86 82 64 + 25314 z + 62404565 z - 179800498310048409 z 30 42 + 2715270189091393166 z + 189946984814356375015 z 44 46 - 222026856606183552932 z + 222026856606183552932 z 58 56 + 8224363326718723027 z - 21177577051171048395 z 54 52 + 46454254557333946440 z - 86941478799626972202 z 60 70 68 - 2715270189091393166 z + 826281238056278 z - 5963519702622742 z 78 32 38 + 39332817286 z - 8224363326718723027 z + 86941478799626972202 z 40 62 76 - 138988767022689975221 z + 760083802408935584 z - 661449793188 z 74 72 + 8805813096111 z - 94418588491437 z ) And in Maple-input format, it is: -(1+205087718459383527*z^28-52087420114408797*z^26-205*z^2+11137608251758973*z^ 24-1995160672581160*z^22+18659*z^4-1012190*z^6+37010496*z^8-976835818*z^10+ 19441269245*z^12-300703154679*z^14-36663384182296*z^18+3694758450159*z^16-\ 16387477634346676063*z^50+24321668800466821843*z^48+297572308612056*z^20+ 9420463598736696057*z^36-4616052311754749898*z^34-1995160672581160*z^66+ 37010496*z^80+z^88+18659*z^84-205*z^86-1012190*z^82+11137608251758973*z^64-\ 682497190145137166*z^30-30816409712472940768*z^42+33344845270831325232*z^44-\ 30816409712472940768*z^46-682497190145137166*z^58+1925474746807996688*z^56-\ 4616052311754749898*z^54+9420463598736696057*z^52+205087718459383527*z^60-\ 36663384182296*z^70+297572308612056*z^68-976835818*z^78+1925474746807996688*z^ 32-16387477634346676063*z^38+24321668800466821843*z^40-52087420114408797*z^62+ 19441269245*z^76-300703154679*z^74+3694758450159*z^72)/(-1-760083802408935584*z ^28+179800498310048409*z^26+243*z^2-35790590941171867*z^24+5963519702622742*z^ 22-25314*z^4+1540468*z^6-62404565*z^8+1809588541*z^10-39332817286*z^12+ 661449793188*z^14+94418588491437*z^18-8805813096111*z^16+138988767022689975221* z^50-189946984814356375015*z^48-826281238056278*z^20-46454254557333946440*z^36+ 21177577051171048395*z^34+35790590941171867*z^66-1809588541*z^80+z^90-243*z^88-\ 1540468*z^84+25314*z^86+62404565*z^82-179800498310048409*z^64+ 2715270189091393166*z^30+189946984814356375015*z^42-222026856606183552932*z^44+ 222026856606183552932*z^46+8224363326718723027*z^58-21177577051171048395*z^56+ 46454254557333946440*z^54-86941478799626972202*z^52-2715270189091393166*z^60+ 826281238056278*z^70-5963519702622742*z^68+39332817286*z^78-8224363326718723027 *z^32+86941478799626972202*z^38-138988767022689975221*z^40+760083802408935584*z ^62-661449793188*z^76+8805813096111*z^74-94418588491437*z^72) The first , 40, terms are: [0, 38, 0, 2579, 0, 193043, 0, 14768358, 0, 1136266717, 0, 87574605325, 0, 6753620318966, 0, 520948840995835, 0, 40187810207244731, 0, 3100351116214914614, 0, 239185507754779220993, 0, 18452791836318297021681, 0, 1423608980556044042059766, 0, 109829745225012738371335515, 0, 8473239960389955407111745819, 0, 653701008351564298736399531222, 0, 50432309484443545933176647640477, 0, 3890797080965037882590564192950317, 0, 300170712654493585741902500330631494, 0, 23157840356363176399884052837029605395] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 24832070845393474335 z - 3902375971501242909 z - 254 z 24 22 4 6 + 529072531886601567 z - 61514555622828356 z + 29279 z - 2051833 z 102 8 10 12 - 3492314320 z + 98738855 z - 3492314320 z + 94840836789 z 14 18 16 - 2038879237510 z - 508761961301466 z + 35487344315946 z 50 48 - 2664920585722599012135020 z + 1742617920273074552171818 z 20 36 + 6089522802013925 z + 10153104338511040338520 z 34 66 - 2761445625846854001988 z - 1008448985538903908523834 z 80 100 90 + 658057528756040819273 z + 94840836789 z - 61514555622828356 z 88 84 94 + 529072531886601567 z + 24832070845393474335 z - 508761961301466 z 86 96 98 - 3902375971501242909 z + 35487344315946 z - 2038879237510 z 92 82 + 6089522802013925 z - 137001169217957344450 z 64 112 110 106 + 1742617920273074552171818 z + z - 254 z - 2051833 z 108 30 42 + 29279 z - 137001169217957344450 z - 233412983104379513810592 z 44 46 + 516121984306367799791218 z - 1008448985538903908523834 z 58 56 - 4327459074271865559077648 z + 4597592075708293997464732 z 54 52 - 4327459074271865559077648 z + 3608382841953795892416742 z 60 70 + 3608382841953795892416742 z - 233412983104379513810592 z 68 78 + 516121984306367799791218 z - 2761445625846854001988 z 32 38 + 658057528756040819273 z - 32786704309348735636780 z 40 62 + 93175430620361757347158 z - 2664920585722599012135020 z 76 74 + 10153104338511040338520 z - 32786704309348735636780 z 72 104 / 2 + 93175430620361757347158 z + 98738855 z ) / ((-1 + z ) (1 / 28 26 2 + 66770386274749326686 z - 10099834453079495016 z - 296 z 24 22 4 6 + 1314616012236552099 z - 146351018403742296 z + 38482 z - 2978256 z 102 8 10 12 - 5948234032 z + 156040043 z - 5948234032 z + 172793161648 z 14 18 16 - 3950412991628 z - 1099971451869652 z + 72776940030437 z 50 48 - 9241399972105340649061784 z + 5982979430696902523483482 z 20 36 + 13832406196959156 z + 31029522618880410438340 z 34 66 - 8203698155328185645976 z - 3418421533747214159652200 z 80 100 90 + 1895691931663111574977 z + 172793161648 z - 146351018403742296 z 88 84 + 1314616012236552099 z + 66770386274749326686 z 94 86 96 - 1099971451869652 z - 10099834453079495016 z + 72776940030437 z 98 92 82 - 3950412991628 z + 13832406196959156 z - 381764859139752545040 z 64 112 110 106 + 5982979430696902523483482 z + z - 296 z - 2978256 z 108 30 42 + 38482 z - 381764859139752545040 z - 765038787801724465700904 z 44 46 + 1722650246723773008814456 z - 3418421533747214159652200 z 58 56 - 15180091233902908888889344 z + 16150935327367403846193342 z 54 52 - 15180091233902908888889344 z + 12603166965333277057296932 z 60 70 + 12603166965333277057296932 z - 765038787801724465700904 z 68 78 + 1722650246723773008814456 z - 8203698155328185645976 z 32 38 + 1895691931663111574977 z - 102825429931632462927368 z 40 62 + 299115047896920677148614 z - 9241399972105340649061784 z 76 74 + 31029522618880410438340 z - 102825429931632462927368 z 72 104 + 299115047896920677148614 z + 156040043 z )) And in Maple-input format, it is: -(1+24832070845393474335*z^28-3902375971501242909*z^26-254*z^2+ 529072531886601567*z^24-61514555622828356*z^22+29279*z^4-2051833*z^6-3492314320 *z^102+98738855*z^8-3492314320*z^10+94840836789*z^12-2038879237510*z^14-\ 508761961301466*z^18+35487344315946*z^16-2664920585722599012135020*z^50+ 1742617920273074552171818*z^48+6089522802013925*z^20+10153104338511040338520*z^ 36-2761445625846854001988*z^34-1008448985538903908523834*z^66+ 658057528756040819273*z^80+94840836789*z^100-61514555622828356*z^90+ 529072531886601567*z^88+24832070845393474335*z^84-508761961301466*z^94-\ 3902375971501242909*z^86+35487344315946*z^96-2038879237510*z^98+ 6089522802013925*z^92-137001169217957344450*z^82+1742617920273074552171818*z^64 +z^112-254*z^110-2051833*z^106+29279*z^108-137001169217957344450*z^30-\ 233412983104379513810592*z^42+516121984306367799791218*z^44-\ 1008448985538903908523834*z^46-4327459074271865559077648*z^58+ 4597592075708293997464732*z^56-4327459074271865559077648*z^54+ 3608382841953795892416742*z^52+3608382841953795892416742*z^60-\ 233412983104379513810592*z^70+516121984306367799791218*z^68-\ 2761445625846854001988*z^78+658057528756040819273*z^32-32786704309348735636780* z^38+93175430620361757347158*z^40-2664920585722599012135020*z^62+ 10153104338511040338520*z^76-32786704309348735636780*z^74+ 93175430620361757347158*z^72+98738855*z^104)/(-1+z^2)/(1+66770386274749326686*z ^28-10099834453079495016*z^26-296*z^2+1314616012236552099*z^24-\ 146351018403742296*z^22+38482*z^4-2978256*z^6-5948234032*z^102+156040043*z^8-\ 5948234032*z^10+172793161648*z^12-3950412991628*z^14-1099971451869652*z^18+ 72776940030437*z^16-9241399972105340649061784*z^50+5982979430696902523483482*z^ 48+13832406196959156*z^20+31029522618880410438340*z^36-8203698155328185645976*z ^34-3418421533747214159652200*z^66+1895691931663111574977*z^80+172793161648*z^ 100-146351018403742296*z^90+1314616012236552099*z^88+66770386274749326686*z^84-\ 1099971451869652*z^94-10099834453079495016*z^86+72776940030437*z^96-\ 3950412991628*z^98+13832406196959156*z^92-381764859139752545040*z^82+ 5982979430696902523483482*z^64+z^112-296*z^110-2978256*z^106+38482*z^108-\ 381764859139752545040*z^30-765038787801724465700904*z^42+ 1722650246723773008814456*z^44-3418421533747214159652200*z^46-\ 15180091233902908888889344*z^58+16150935327367403846193342*z^56-\ 15180091233902908888889344*z^54+12603166965333277057296932*z^52+ 12603166965333277057296932*z^60-765038787801724465700904*z^70+ 1722650246723773008814456*z^68-8203698155328185645976*z^78+ 1895691931663111574977*z^32-102825429931632462927368*z^38+ 299115047896920677148614*z^40-9241399972105340649061784*z^62+ 31029522618880410438340*z^76-102825429931632462927368*z^74+ 299115047896920677148614*z^72+156040043*z^104) The first , 40, terms are: [0, 43, 0, 3272, 0, 269235, 0, 22521469, 0, 1893421097, 0, 159495348363, 0, 13446746456128, 0, 1134103000412371, 0, 95667353606868769, 0, 8070677235964340161, 0, 680882740281226749055, 0, 57443662572315385116464, 0, 4846356580673043021467599, 0, 408874665627792537903410529, 0, 34495764702323794004408542773, 0, 2910326363297921830841036825351, 0, 245537465261953291858462338793032, 0, 20715428270606643436563277454639079, 0, 1747712926535821823639967155989917137, 0, 147450515833949829646419308182760328129] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 300032475543638242 z - 75404859256698536 z - 213 z 24 22 4 6 + 15921500015160936 z - 2809843385551288 z + 20174 z - 1138781 z 8 10 12 14 + 43267187 z - 1183758742 z + 24348970012 z - 387978095772 z 18 16 50 - 49718491391690 z + 4894987645340 z - 24666291385887913674 z 48 20 + 36686779465431751068 z + 411807323451118 z 36 34 + 14135467902788282724 z - 6897139466825445236 z 66 80 88 84 86 - 2809843385551288 z + 43267187 z + z + 20174 z - 213 z 82 64 30 - 1138781 z + 15921500015160936 z - 1007111418772860086 z 42 44 - 46540979119008607342 z + 50380018270348288948 z 46 58 - 46540979119008607342 z - 1007111418772860086 z 56 54 + 2861158995166342884 z - 6897139466825445236 z 52 60 70 + 14135467902788282724 z + 300032475543638242 z - 49718491391690 z 68 78 32 + 411807323451118 z - 1183758742 z + 2861158995166342884 z 38 40 - 24666291385887913674 z + 36686779465431751068 z 62 76 74 - 75404859256698536 z + 24348970012 z - 387978095772 z 72 / 28 + 4894987645340 z ) / (-1 - 1106754988589257434 z / 26 2 24 + 258529947574717342 z + 252 z - 50731061469786586 z 22 4 6 8 + 8317591007943678 z - 27314 z + 1729068 z - 72708810 z 10 12 14 + 2182103865 z - 48932097162 z + 846325949442 z 18 16 50 + 126731199807346 z - 11554965305714 z + 211470798828845341262 z 48 20 - 289740258242930148114 z - 1131762266753148 z 36 34 - 70045269634508724502 z + 31723976123681602214 z 66 80 90 88 84 + 50731061469786586 z - 2182103865 z + z - 252 z - 1729068 z 86 82 64 + 27314 z + 72708810 z - 258529947574717342 z 30 42 + 3997326314719748684 z + 289740258242930148114 z 44 46 - 339103684972419183322 z + 339103684972419183322 z 58 56 + 12222393770899602470 z - 31723976123681602214 z 54 52 + 70045269634508724502 z - 131773402035624328590 z 60 70 68 - 3997326314719748684 z + 1131762266753148 z - 8317591007943678 z 78 32 38 + 48932097162 z - 12222393770899602470 z + 131773402035624328590 z 40 62 76 - 211470798828845341262 z + 1106754988589257434 z - 846325949442 z 74 72 + 11554965305714 z - 126731199807346 z ) And in Maple-input format, it is: -(1+300032475543638242*z^28-75404859256698536*z^26-213*z^2+15921500015160936*z^ 24-2809843385551288*z^22+20174*z^4-1138781*z^6+43267187*z^8-1183758742*z^10+ 24348970012*z^12-387978095772*z^14-49718491391690*z^18+4894987645340*z^16-\ 24666291385887913674*z^50+36686779465431751068*z^48+411807323451118*z^20+ 14135467902788282724*z^36-6897139466825445236*z^34-2809843385551288*z^66+ 43267187*z^80+z^88+20174*z^84-213*z^86-1138781*z^82+15921500015160936*z^64-\ 1007111418772860086*z^30-46540979119008607342*z^42+50380018270348288948*z^44-\ 46540979119008607342*z^46-1007111418772860086*z^58+2861158995166342884*z^56-\ 6897139466825445236*z^54+14135467902788282724*z^52+300032475543638242*z^60-\ 49718491391690*z^70+411807323451118*z^68-1183758742*z^78+2861158995166342884*z^ 32-24666291385887913674*z^38+36686779465431751068*z^40-75404859256698536*z^62+ 24348970012*z^76-387978095772*z^74+4894987645340*z^72)/(-1-1106754988589257434* z^28+258529947574717342*z^26+252*z^2-50731061469786586*z^24+8317591007943678*z^ 22-27314*z^4+1729068*z^6-72708810*z^8+2182103865*z^10-48932097162*z^12+ 846325949442*z^14+126731199807346*z^18-11554965305714*z^16+ 211470798828845341262*z^50-289740258242930148114*z^48-1131762266753148*z^20-\ 70045269634508724502*z^36+31723976123681602214*z^34+50731061469786586*z^66-\ 2182103865*z^80+z^90-252*z^88-1729068*z^84+27314*z^86+72708810*z^82-\ 258529947574717342*z^64+3997326314719748684*z^30+289740258242930148114*z^42-\ 339103684972419183322*z^44+339103684972419183322*z^46+12222393770899602470*z^58 -31723976123681602214*z^56+70045269634508724502*z^54-131773402035624328590*z^52 -3997326314719748684*z^60+1131762266753148*z^70-8317591007943678*z^68+ 48932097162*z^78-12222393770899602470*z^32+131773402035624328590*z^38-\ 211470798828845341262*z^40+1106754988589257434*z^62-846325949442*z^76+ 11554965305714*z^74-126731199807346*z^72) The first , 40, terms are: [0, 39, 0, 2688, 0, 202417, 0, 15581081, 0, 1208050791, 0, 93917552559, 0, 7309264561704, 0, 569102921260173, 0, 44318725227427259, 0, 3451574444102412667, 0, 268819990643838983405, 0, 20936884169660981303544, 0, 1630666735895570775819111, 0, 127004624921170408736110879, 0, 9891778147860053192196578273, 0, 770423320645082618684993805721, 0, 60004603496770201729345105261776, 0, 4673473267256165650219947643741943, 0, 363994627837135200848231818552323569, 0, 28349812658945999008995721692499643889] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 617003137194840 z + 322480284001662 z + 185 z 24 22 4 6 - 135213593685514 z + 45303140240188 z - 14500 z + 644624 z 8 10 12 14 - 18367658 z + 359434150 z - 5052421324 z + 52639113764 z 18 16 50 + 2533996313445 z - 415862625001 z + 415862625001 z 48 20 36 - 2533996313445 z - 12062984909076 z - 949608154129588 z 34 66 64 30 + 1177606342152992 z + z - 185 z + 949608154129588 z 42 44 46 + 135213593685514 z - 45303140240188 z + 12062984909076 z 58 56 54 52 + 18367658 z - 359434150 z + 5052421324 z - 52639113764 z 60 32 38 - 644624 z - 1177606342152992 z + 617003137194840 z 40 62 / 28 - 322480284001662 z + 14500 z ) / (1 + 3249537332402514 z / 26 2 24 - 1535542108109892 z - 233 z + 582712004349880 z 22 4 6 8 - 176812384314112 z + 21555 z - 1095786 z + 35126190 z 10 12 14 18 - 766691112 z + 11964647856 z - 138028473128 z - 8114431858899 z 16 50 48 + 1205599085819 z - 8114431858899 z + 42647941976423 z 20 36 34 + 42647941976423 z + 7623184998874114 z - 8478124306307760 z 66 64 30 42 - 233 z + 21555 z - 5539630008603630 z - 1535542108109892 z 44 46 58 + 582712004349880 z - 176812384314112 z - 766691112 z 56 54 52 60 + 11964647856 z - 138028473128 z + 1205599085819 z + 35126190 z 68 32 38 40 + z + 7623184998874114 z - 5539630008603630 z + 3249537332402514 z 62 - 1095786 z ) And in Maple-input format, it is: -(-1-617003137194840*z^28+322480284001662*z^26+185*z^2-135213593685514*z^24+ 45303140240188*z^22-14500*z^4+644624*z^6-18367658*z^8+359434150*z^10-5052421324 *z^12+52639113764*z^14+2533996313445*z^18-415862625001*z^16+415862625001*z^50-\ 2533996313445*z^48-12062984909076*z^20-949608154129588*z^36+1177606342152992*z^ 34+z^66-185*z^64+949608154129588*z^30+135213593685514*z^42-45303140240188*z^44+ 12062984909076*z^46+18367658*z^58-359434150*z^56+5052421324*z^54-52639113764*z^ 52-644624*z^60-1177606342152992*z^32+617003137194840*z^38-322480284001662*z^40+ 14500*z^62)/(1+3249537332402514*z^28-1535542108109892*z^26-233*z^2+ 582712004349880*z^24-176812384314112*z^22+21555*z^4-1095786*z^6+35126190*z^8-\ 766691112*z^10+11964647856*z^12-138028473128*z^14-8114431858899*z^18+ 1205599085819*z^16-8114431858899*z^50+42647941976423*z^48+42647941976423*z^20+ 7623184998874114*z^36-8478124306307760*z^34-233*z^66+21555*z^64-\ 5539630008603630*z^30-1535542108109892*z^42+582712004349880*z^44-\ 176812384314112*z^46-766691112*z^58+11964647856*z^56-138028473128*z^54+ 1205599085819*z^52+35126190*z^60+z^68+7623184998874114*z^32-5539630008603630*z^ 38+3249537332402514*z^40-1095786*z^62) The first , 40, terms are: [0, 48, 0, 4129, 0, 378579, 0, 35047508, 0, 3251499255, 0, 301844767903, 0, 28027049060876, 0, 2602587150381603, 0, 241682933462743553, 0, 22443548943206101128, 0, 2084197771275844884697, 0, 193547260617015396691145, 0, 17973613625778311934076120, 0, 1669105845967083293704766177, 0, 155000248679843233273005665635, 0, 14393981065020599058324481226268, 0, 1336686202066929690544196724165807, 0, 124130357273254658591593435273204391, 0, 11527272148206136847544974644148824644, 0, 1070471447930629940072767802043454945107] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {3, 7}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 666824567103492 z + 344084687093300 z + 178 z 24 22 4 6 - 141906893011290 z + 46619986241632 z - 13538 z + 590807 z 8 10 12 14 - 16722492 z + 328455484 z - 4671864323 z + 49538528890 z 18 16 50 + 2492622077365 z - 399769210298 z + 399769210298 z 48 20 36 - 2492622077365 z - 12144022399008 z - 1035261624991510 z 34 66 64 30 + 1289483921176584 z + z - 178 z + 1035261624991510 z 42 44 46 + 141906893011290 z - 46619986241632 z + 12144022399008 z 58 56 54 52 + 16722492 z - 328455484 z + 4671864323 z - 49538528890 z 60 32 38 - 590807 z - 1289483921176584 z + 666824567103492 z 40 62 / 28 - 344084687093300 z + 13538 z ) / (1 + 3364154887191622 z / 26 2 24 - 1573431450022724 z - 214 z + 589267841796674 z 22 4 6 8 10 - 175981359976744 z + 19043 z - 955000 z + 30673565 z - 677370758 z 12 14 18 + 10756589033 z - 126653966944 z - 7771882976662 z 16 50 48 + 1130343328599 z - 7771882976662 z + 41678859824349 z 20 36 34 + 41678859824349 z + 7984892177149018 z - 8893288370242212 z 66 64 30 42 - 214 z + 19043 z - 5777178688152952 z - 1573431450022724 z 44 46 58 + 589267841796674 z - 175981359976744 z - 677370758 z 56 54 52 60 + 10756589033 z - 126653966944 z + 1130343328599 z + 30673565 z 68 32 38 40 + z + 7984892177149018 z - 5777178688152952 z + 3364154887191622 z 62 - 955000 z ) And in Maple-input format, it is: -(-1-666824567103492*z^28+344084687093300*z^26+178*z^2-141906893011290*z^24+ 46619986241632*z^22-13538*z^4+590807*z^6-16722492*z^8+328455484*z^10-4671864323 *z^12+49538528890*z^14+2492622077365*z^18-399769210298*z^16+399769210298*z^50-\ 2492622077365*z^48-12144022399008*z^20-1035261624991510*z^36+1289483921176584*z ^34+z^66-178*z^64+1035261624991510*z^30+141906893011290*z^42-46619986241632*z^ 44+12144022399008*z^46+16722492*z^58-328455484*z^56+4671864323*z^54-49538528890 *z^52-590807*z^60-1289483921176584*z^32+666824567103492*z^38-344084687093300*z^ 40+13538*z^62)/(1+3364154887191622*z^28-1573431450022724*z^26-214*z^2+ 589267841796674*z^24-175981359976744*z^22+19043*z^4-955000*z^6+30673565*z^8-\ 677370758*z^10+10756589033*z^12-126653966944*z^14-7771882976662*z^18+ 1130343328599*z^16-7771882976662*z^50+41678859824349*z^48+41678859824349*z^20+ 7984892177149018*z^36-8893288370242212*z^34-214*z^66+19043*z^64-\ 5777178688152952*z^30-1573431450022724*z^42+589267841796674*z^44-\ 175981359976744*z^46-677370758*z^58+10756589033*z^56-126653966944*z^54+ 1130343328599*z^52+30673565*z^60+z^68+7984892177149018*z^32-5777178688152952*z^ 38+3364154887191622*z^40-955000*z^62) The first , 40, terms are: [0, 36, 0, 2199, 0, 149231, 0, 10488804, 0, 747510057, 0, 53593915769, 0, 3853041530308, 0, 277364987660799, 0, 19978720287996935, 0, 1439505910064134276, 0, 103734316228226192753, 0, 7475880666629261781009, 0, 538787399139211238296388, 0, 38831122388616233451272999, 0, 2798634068037469700171417567, 0, 201703804225295891136412371332, 0, 14537273050278568899352808994905, 0, 1047736906628308861215001149420873, 0, 75513004691860834037792105844713252, 0, 5442411381237548715678163934342968591] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 499689457296680 z - 247166112647890 z - 177 z 24 22 4 6 + 99363319862466 z - 32308412638888 z + 13022 z - 541965 z 8 10 12 14 + 14547805 z - 270963524 z + 3667738121 z - 37233326105 z 18 16 50 - 1760611254273 z + 289919855498 z - 1760611254273 z 48 20 36 + 8444717440193 z + 8444717440193 z + 1111358779632294 z 34 66 64 30 - 1227671681846600 z - 177 z + 13022 z - 824041394090534 z 42 44 46 - 247166112647890 z + 99363319862466 z - 32308412638888 z 58 56 54 52 - 270963524 z + 3667738121 z - 37233326105 z + 289919855498 z 60 68 32 38 + 14547805 z + z + 1111358779632294 z - 824041394090534 z 40 62 / 2 + 499689457296680 z - 541965 z ) / ((-1 + z ) (1 / 28 26 2 + 1626223403041230 z - 789289728294644 z - 210 z 24 22 4 6 + 309497712775242 z - 97531223625336 z + 18259 z - 882828 z 8 10 12 14 + 26976189 z - 560936574 z + 8332302437 z - 91475024388 z 18 16 50 - 4883233510306 z + 760803222715 z - 4883233510306 z 48 20 36 + 24532541545609 z + 24532541545609 z + 3694579368994786 z 34 66 64 30 - 4091986897830780 z - 210 z + 18259 z - 2717812517717968 z 42 44 46 - 789289728294644 z + 309497712775242 z - 97531223625336 z 58 56 54 52 - 560936574 z + 8332302437 z - 91475024388 z + 760803222715 z 60 68 32 38 + 26976189 z + z + 3694579368994786 z - 2717812517717968 z 40 62 + 1626223403041230 z - 882828 z )) And in Maple-input format, it is: -(1+499689457296680*z^28-247166112647890*z^26-177*z^2+99363319862466*z^24-\ 32308412638888*z^22+13022*z^4-541965*z^6+14547805*z^8-270963524*z^10+3667738121 *z^12-37233326105*z^14-1760611254273*z^18+289919855498*z^16-1760611254273*z^50+ 8444717440193*z^48+8444717440193*z^20+1111358779632294*z^36-1227671681846600*z^ 34-177*z^66+13022*z^64-824041394090534*z^30-247166112647890*z^42+99363319862466 *z^44-32308412638888*z^46-270963524*z^58+3667738121*z^56-37233326105*z^54+ 289919855498*z^52+14547805*z^60+z^68+1111358779632294*z^32-824041394090534*z^38 +499689457296680*z^40-541965*z^62)/(-1+z^2)/(1+1626223403041230*z^28-\ 789289728294644*z^26-210*z^2+309497712775242*z^24-97531223625336*z^22+18259*z^4 -882828*z^6+26976189*z^8-560936574*z^10+8332302437*z^12-91475024388*z^14-\ 4883233510306*z^18+760803222715*z^16-4883233510306*z^50+24532541545609*z^48+ 24532541545609*z^20+3694579368994786*z^36-4091986897830780*z^34-210*z^66+18259* z^64-2717812517717968*z^30-789289728294644*z^42+309497712775242*z^44-\ 97531223625336*z^46-560936574*z^58+8332302437*z^56-91475024388*z^54+ 760803222715*z^52+26976189*z^60+z^68+3694579368994786*z^32-2717812517717968*z^ 38+1626223403041230*z^40-882828*z^62) The first , 40, terms are: [0, 34, 0, 1727, 0, 95573, 0, 5595686, 0, 341471919, 0, 21474448719, 0, 1379623102686, 0, 89955530397473, 0, 5924727310419135, 0, 392862036052180594, 0, 26167100090842363141, 0, 1748036684755011326741, 0, 116999296903529617864946, 0, 7840840430844120756303759, 0, 525893553455102240863064209, 0, 35291034236507514388133938558, 0, 2369087578098192006094094422431, 0, 159072570917450946695186769530975, 0, 10682494862695867525661399541483590, 0, 717449056755943645600715592705064165] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {4, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2718042266555740 z - 1121839094669004 z - 182 z 24 22 4 6 + 378519493727119 z - 104019445009502 z + 14174 z - 636280 z 8 10 12 14 + 18675706 z - 384750050 z + 5821963418 z - 66755725210 z 18 16 50 - 4150758814552 z + 593105954898 z - 104019445009502 z 48 20 36 + 378519493727119 z + 23162803939110 z + 13006802706985900 z 34 66 64 30 - 11797433499866276 z - 636280 z + 18675706 z - 5397674642172208 z 42 44 46 - 5397674642172208 z + 2718042266555740 z - 1121839094669004 z 58 56 54 - 66755725210 z + 593105954898 z - 4150758814552 z 52 60 70 68 + 23162803939110 z + 5821963418 z - 182 z + 14174 z 32 38 40 + 8801459725729236 z - 11797433499866276 z + 8801459725729236 z 62 72 / 28 - 384750050 z + z ) / (-1 - 12391490740613910 z / 26 2 24 + 4691290276446393 z + 218 z - 1453013390911100 z 22 4 6 8 + 366564250883443 z - 19783 z + 1016077 z - 33672775 z 10 12 14 18 + 775473693 z - 13017122981 z + 164597952833 z + 12293429985555 z 16 50 48 - 1605460471521 z + 1453013390911100 z - 4691290276446393 z 20 36 34 - 74885464689329 z - 85323525725353482 z + 70402594375037234 z 66 64 30 + 33672775 z - 775473693 z + 26863812132765170 z 42 44 46 + 47910259300298486 z - 26863812132765170 z + 12391490740613910 z 58 56 54 + 1605460471521 z - 12293429985555 z + 74885464689329 z 52 60 70 68 - 366564250883443 z - 164597952833 z + 19783 z - 1016077 z 32 38 40 - 47910259300298486 z + 85323525725353482 z - 70402594375037234 z 62 74 72 + 13017122981 z + z - 218 z ) And in Maple-input format, it is: -(1+2718042266555740*z^28-1121839094669004*z^26-182*z^2+378519493727119*z^24-\ 104019445009502*z^22+14174*z^4-636280*z^6+18675706*z^8-384750050*z^10+ 5821963418*z^12-66755725210*z^14-4150758814552*z^18+593105954898*z^16-\ 104019445009502*z^50+378519493727119*z^48+23162803939110*z^20+13006802706985900 *z^36-11797433499866276*z^34-636280*z^66+18675706*z^64-5397674642172208*z^30-\ 5397674642172208*z^42+2718042266555740*z^44-1121839094669004*z^46-66755725210*z ^58+593105954898*z^56-4150758814552*z^54+23162803939110*z^52+5821963418*z^60-\ 182*z^70+14174*z^68+8801459725729236*z^32-11797433499866276*z^38+ 8801459725729236*z^40-384750050*z^62+z^72)/(-1-12391490740613910*z^28+ 4691290276446393*z^26+218*z^2-1453013390911100*z^24+366564250883443*z^22-19783* z^4+1016077*z^6-33672775*z^8+775473693*z^10-13017122981*z^12+164597952833*z^14+ 12293429985555*z^18-1605460471521*z^16+1453013390911100*z^50-4691290276446393*z ^48-74885464689329*z^20-85323525725353482*z^36+70402594375037234*z^34+33672775* z^66-775473693*z^64+26863812132765170*z^30+47910259300298486*z^42-\ 26863812132765170*z^44+12391490740613910*z^46+1605460471521*z^58-12293429985555 *z^56+74885464689329*z^54-366564250883443*z^52-164597952833*z^60+19783*z^70-\ 1016077*z^68-47910259300298486*z^32+85323525725353482*z^38-70402594375037234*z^ 40+13017122981*z^62+z^74-218*z^72) The first , 40, terms are: [0, 36, 0, 2239, 0, 155711, 0, 11232564, 0, 821768385, 0, 60474610225, 0, 4461860930932, 0, 329582441584175, 0, 24358145633125039, 0, 1800662655944209892, 0, 133128533260958033457, 0, 9843144036552386549521, 0, 727792977899808749261796, 0, 53813007666139660066618127, 0, 3978956905420220925082385551, 0, 294206611827415801724726591348, 0, 21753853826147078045870770209169, 0, 1608497027513408757753920548583905, 0, 118933569371499024814897723523667508, 0, 8794045473755711640712945416741016671] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 216702891840189775 z - 55278392133048787 z - 207 z 24 22 4 6 + 11862325686451365 z - 2130299551753728 z + 19019 z - 1041772 z 8 10 12 14 + 38455826 z - 1023840272 z + 20527756357 z - 319341084993 z 18 16 50 - 39175106786032 z + 3939371683607 z - 16923996128015971005 z 48 20 36 + 25046927577236404699 z + 318092988076700 z + 9765298739660549881 z 34 66 80 88 - 4806249087863204236 z - 2130299551753728 z + 38455826 z + z 84 86 82 64 + 19019 z - 207 z - 1041772 z + 11862325686451365 z 30 42 - 717664616947792208 z - 31679344204069168472 z 44 46 + 34258020553220006232 z - 31679344204069168472 z 58 56 - 717664616947792208 z + 2014578360122194038 z 54 52 - 4806249087863204236 z + 9765298739660549881 z 60 70 68 + 216702891840189775 z - 39175106786032 z + 318092988076700 z 78 32 38 - 1023840272 z + 2014578360122194038 z - 16923996128015971005 z 40 62 76 + 25046927577236404699 z - 55278392133048787 z + 20527756357 z 74 72 / 2 - 319341084993 z + 3939371683607 z ) / ((-1 + z ) (1 / 28 26 2 + 651382258584087504 z - 160076489521716986 z - 246 z 24 22 4 6 + 32948219592570823 z - 5650024934248652 z + 25778 z - 1574876 z 8 10 12 14 + 63943531 z - 1854238984 z + 40184389636 z - 671375047862 z 18 16 50 - 93429198658028 z + 8844172431353 z - 57429093758086050042 z 48 20 + 85943325993370889455 z + 801904095058694 z 36 34 + 32626527144578248602 z - 15741465939269321356 z 66 80 88 84 86 - 5650024934248652 z + 63943531 z + z + 25778 z - 246 z 82 64 30 - 1574876 z + 32948219592570823 z - 2229459596249405936 z 42 44 - 109429966425833315272 z + 118601650199685846740 z 46 58 - 109429966425833315272 z - 2229459596249405936 z 56 54 + 6439986832173941109 z - 15741465939269321356 z 52 60 70 + 32626527144578248602 z + 651382258584087504 z - 93429198658028 z 68 78 32 + 801904095058694 z - 1854238984 z + 6439986832173941109 z 38 40 - 57429093758086050042 z + 85943325993370889455 z 62 76 74 - 160076489521716986 z + 40184389636 z - 671375047862 z 72 + 8844172431353 z )) And in Maple-input format, it is: -(1+216702891840189775*z^28-55278392133048787*z^26-207*z^2+11862325686451365*z^ 24-2130299551753728*z^22+19019*z^4-1041772*z^6+38455826*z^8-1023840272*z^10+ 20527756357*z^12-319341084993*z^14-39175106786032*z^18+3939371683607*z^16-\ 16923996128015971005*z^50+25046927577236404699*z^48+318092988076700*z^20+ 9765298739660549881*z^36-4806249087863204236*z^34-2130299551753728*z^66+ 38455826*z^80+z^88+19019*z^84-207*z^86-1041772*z^82+11862325686451365*z^64-\ 717664616947792208*z^30-31679344204069168472*z^42+34258020553220006232*z^44-\ 31679344204069168472*z^46-717664616947792208*z^58+2014578360122194038*z^56-\ 4806249087863204236*z^54+9765298739660549881*z^52+216702891840189775*z^60-\ 39175106786032*z^70+318092988076700*z^68-1023840272*z^78+2014578360122194038*z^ 32-16923996128015971005*z^38+25046927577236404699*z^40-55278392133048787*z^62+ 20527756357*z^76-319341084993*z^74+3939371683607*z^72)/(-1+z^2)/(1+ 651382258584087504*z^28-160076489521716986*z^26-246*z^2+32948219592570823*z^24-\ 5650024934248652*z^22+25778*z^4-1574876*z^6+63943531*z^8-1854238984*z^10+ 40184389636*z^12-671375047862*z^14-93429198658028*z^18+8844172431353*z^16-\ 57429093758086050042*z^50+85943325993370889455*z^48+801904095058694*z^20+ 32626527144578248602*z^36-15741465939269321356*z^34-5650024934248652*z^66+ 63943531*z^80+z^88+25778*z^84-246*z^86-1574876*z^82+32948219592570823*z^64-\ 2229459596249405936*z^30-109429966425833315272*z^42+118601650199685846740*z^44-\ 109429966425833315272*z^46-2229459596249405936*z^58+6439986832173941109*z^56-\ 15741465939269321356*z^54+32626527144578248602*z^52+651382258584087504*z^60-\ 93429198658028*z^70+801904095058694*z^68-1854238984*z^78+6439986832173941109*z^ 32-57429093758086050042*z^38+85943325993370889455*z^40-160076489521716986*z^62+ 40184389636*z^76-671375047862*z^74+8844172431353*z^72) The first , 40, terms are: [0, 40, 0, 2875, 0, 228047, 0, 18472188, 0, 1503421521, 0, 122500246125, 0, 9984272518524, 0, 813819159048663, 0, 66335804760909067, 0, 5407175655214108712, 0, 440751328586088037577, 0, 35926671056938046401693, 0, 2928467324315437473508968, 0, 238706257898151279481700607, 0, 19457508628862725037974128427, 0, 1586027307066015544823633288684, 0, 129280817471272775111380607871569, 0, 10537983612981619040255840338382429, 0, 858975838882776623855429438838681388, 0, 70017141694808508850938637444946152955] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 237872221126082314 z - 62416304799421300 z - 225 z 24 22 4 6 + 13775219274025224 z - 2541211590267532 z + 22066 z - 1267573 z 8 10 12 14 + 48295003 z - 1309057262 z + 26416845368 z - 409755664508 z 18 16 50 - 48923303639490 z + 5001792883536 z - 16587977365629061842 z 48 20 36 + 24254605246778072596 z + 388907286594726 z + 9729642428427850536 z 34 66 80 88 - 4886392401665969428 z - 2541211590267532 z + 48295003 z + z 84 86 82 64 + 22066 z - 225 z - 1267573 z + 13775219274025224 z 30 42 - 766424976004173470 z - 30450778625294602026 z 44 46 + 32847374392820990636 z - 30450778625294602026 z 58 56 - 766424976004173470 z + 2096603971792499408 z 54 52 - 4886392401665969428 z + 9729642428427850536 z 60 70 68 + 237872221126082314 z - 48923303639490 z + 388907286594726 z 78 32 38 - 1309057262 z + 2096603971792499408 z - 16587977365629061842 z 40 62 76 + 24254605246778072596 z - 62416304799421300 z + 26416845368 z 74 72 / 2 - 409755664508 z + 5001792883536 z ) / ((-1 + z ) (1 / 28 26 2 + 721696174531605112 z - 183476061083491958 z - 263 z 24 22 4 6 + 39061423730633884 z - 6918271964324394 z + 29611 z - 1922341 z 8 10 12 14 + 81620693 z - 2434529908 z + 53457936418 z - 893379454956 z 18 16 50 - 120786090815500 z + 11648416396010 z - 55595763091130197340 z 48 20 + 82019230433057850146 z + 1011211126185984 z 36 34 + 32204185310830196710 z - 15914631468944426244 z 66 80 88 84 86 - 6918271964324394 z + 81620693 z + z + 29611 z - 263 z 82 64 30 - 1922341 z + 39061423730633884 z - 2390150492126769284 z 42 44 - 103524085626701180660 z + 111870906141129203042 z 46 58 - 103524085626701180660 z - 2390150492126769284 z 56 54 + 6694497368070410462 z - 15914631468944426244 z 52 60 70 + 32204185310830196710 z + 721696174531605112 z - 120786090815500 z 68 78 32 + 1011211126185984 z - 2434529908 z + 6694497368070410462 z 38 40 - 55595763091130197340 z + 82019230433057850146 z 62 76 74 - 183476061083491958 z + 53457936418 z - 893379454956 z 72 + 11648416396010 z )) And in Maple-input format, it is: -(1+237872221126082314*z^28-62416304799421300*z^26-225*z^2+13775219274025224*z^ 24-2541211590267532*z^22+22066*z^4-1267573*z^6+48295003*z^8-1309057262*z^10+ 26416845368*z^12-409755664508*z^14-48923303639490*z^18+5001792883536*z^16-\ 16587977365629061842*z^50+24254605246778072596*z^48+388907286594726*z^20+ 9729642428427850536*z^36-4886392401665969428*z^34-2541211590267532*z^66+ 48295003*z^80+z^88+22066*z^84-225*z^86-1267573*z^82+13775219274025224*z^64-\ 766424976004173470*z^30-30450778625294602026*z^42+32847374392820990636*z^44-\ 30450778625294602026*z^46-766424976004173470*z^58+2096603971792499408*z^56-\ 4886392401665969428*z^54+9729642428427850536*z^52+237872221126082314*z^60-\ 48923303639490*z^70+388907286594726*z^68-1309057262*z^78+2096603971792499408*z^ 32-16587977365629061842*z^38+24254605246778072596*z^40-62416304799421300*z^62+ 26416845368*z^76-409755664508*z^74+5001792883536*z^72)/(-1+z^2)/(1+ 721696174531605112*z^28-183476061083491958*z^26-263*z^2+39061423730633884*z^24-\ 6918271964324394*z^22+29611*z^4-1922341*z^6+81620693*z^8-2434529908*z^10+ 53457936418*z^12-893379454956*z^14-120786090815500*z^18+11648416396010*z^16-\ 55595763091130197340*z^50+82019230433057850146*z^48+1011211126185984*z^20+ 32204185310830196710*z^36-15914631468944426244*z^34-6918271964324394*z^66+ 81620693*z^80+z^88+29611*z^84-263*z^86-1922341*z^82+39061423730633884*z^64-\ 2390150492126769284*z^30-103524085626701180660*z^42+111870906141129203042*z^44-\ 103524085626701180660*z^46-2390150492126769284*z^58+6694497368070410462*z^56-\ 15914631468944426244*z^54+32204185310830196710*z^52+721696174531605112*z^60-\ 120786090815500*z^70+1011211126185984*z^68-2434529908*z^78+6694497368070410462* z^32-55595763091130197340*z^38+82019230433057850146*z^40-183476061083491958*z^ 62+53457936418*z^76-893379454956*z^74+11648416396010*z^72) The first , 40, terms are: [0, 39, 0, 2488, 0, 176125, 0, 13048585, 0, 988639779, 0, 75774203255, 0, 5843417861584, 0, 452126004688217, 0, 35046897915388839, 0, 2719461708224388079, 0, 211136757989880185489, 0, 16397715680143138812736, 0, 1273739385933375750736503, 0, 98951281689836424597416771, 0, 7687528892435843685679009433, 0, 597263332036565593841311329037, 0, 46403706410834723061309319515304, 0, 3605320084045998637978284384843119, 0, 280115682144521152225704048244991513, 0, 21763682833583605063717580298234793289] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 14865590883 z + 27590040066 z + 150 z - 37528776431 z 22 4 6 8 10 + 37528776431 z - 8533 z + 248167 z - 4229070 z + 45712727 z 12 14 18 16 - 330008953 z + 1648013898 z + 14865590883 z - 5831908657 z 20 36 34 30 - 27590040066 z - 45712727 z + 330008953 z + 5831908657 z 42 44 46 32 38 40 + 8533 z - 150 z + z - 1648013898 z + 4229070 z - 248167 z ) / 28 26 2 24 / (1 + 145953677052 z - 229387340156 z - 196 z + 266582108830 z / 22 4 6 8 10 - 229387340156 z + 13900 z - 487400 z + 9773752 z - 122344964 z 12 14 18 16 + 1013688912 z - 5788144444 z - 68386218760 z + 23416369360 z 48 20 36 34 + z + 145953677052 z + 1013688912 z - 5788144444 z 30 42 44 46 32 - 68386218760 z - 487400 z + 13900 z - 196 z + 23416369360 z 38 40 - 122344964 z + 9773752 z ) And in Maple-input format, it is: -(-1-14865590883*z^28+27590040066*z^26+150*z^2-37528776431*z^24+37528776431*z^ 22-8533*z^4+248167*z^6-4229070*z^8+45712727*z^10-330008953*z^12+1648013898*z^14 +14865590883*z^18-5831908657*z^16-27590040066*z^20-45712727*z^36+330008953*z^34 +5831908657*z^30+8533*z^42-150*z^44+z^46-1648013898*z^32+4229070*z^38-248167*z^ 40)/(1+145953677052*z^28-229387340156*z^26-196*z^2+266582108830*z^24-\ 229387340156*z^22+13900*z^4-487400*z^6+9773752*z^8-122344964*z^10+1013688912*z^ 12-5788144444*z^14-68386218760*z^18+23416369360*z^16+z^48+145953677052*z^20+ 1013688912*z^36-5788144444*z^34-68386218760*z^30-487400*z^42+13900*z^44-196*z^ 46+23416369360*z^32-122344964*z^38+9773752*z^40) The first , 40, terms are: [0, 46, 0, 3649, 0, 315037, 0, 27901870, 0, 2495314465, 0, 224074443277, 0, 20157944962198, 0, 1814925673332769, 0, 163469696982584053, 0, 14726274488586699766, 0, 1326736303138580014621, 0, 119534489077622017822885, 0, 10769852691357837784485862, 0, 970353569634075426699340045, 0, 87428291713233878547177807577, 0, 7877252985672045951697633171654, 0, 709737866980452167718102776897845, 0, 63947171544292239352468945846153081, 0, 5761622327335522585220874962080921726, 0, 519120612500548447753782036458965431013] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 7}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 8 10 4 6 2 -26 z + z - 175 z + 175 z + 26 z - 1 f(z) = - ---------------------------------------------------- 8 10 4 12 6 2 642 z - 64 z + 642 z + z - 1212 z - 64 z + 1 And in Maple-input format, it is: -(-26*z^8+z^10-175*z^4+175*z^6+26*z^2-1)/(642*z^8-64*z^10+642*z^4+z^12-1212*z^6 -64*z^2+1) The first , 40, terms are: [0, 38, 0, 1965, 0, 102401, 0, 5337574, 0, 278220541, 0, 14502243029, 0, 755929490502, 0, 39402830492393, 0, 2053872861077029, 0, 107058139836054406, 0, 5580406424385240713, 0, 290878731041342868921, 0, 15162056262320327583302, 0, 790322308128999286737013, 0, 41195556850611032921688409, 0, 2147318741703845280631361798, 0, 111929007178944132531458346597, 0, 5834300425340399623666623559661, 0, 304112957945816075020091159429158, 0, 15851890449257714793826931761286161] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1125756200196 z - 1295095250268 z - 162 z 24 22 4 6 8 + 1125756200196 z - 738712701638 z + 10587 z - 372538 z + 8007568 z 10 12 14 18 - 112922454 z + 1093032030 z - 7487390672 z - 134966637994 z 16 50 48 20 + 37090321346 z - 162 z + 10587 z + 364879816720 z 36 34 30 42 + 37090321346 z - 134966637994 z - 738712701638 z - 112922454 z 44 46 52 32 38 + 8007568 z - 372538 z + z + 364879816720 z - 7487390672 z 40 / 28 26 2 + 1093032030 z ) / ((1 + 3900449534034 z - 4517994886726 z - 201 z / 24 22 4 6 + 3900449534034 z - 2507756693566 z + 15781 z - 651386 z 8 10 12 14 + 16079086 z - 255633490 z + 2743786712 z - 20523662892 z 18 16 50 48 - 422783271014 z + 109428877472 z - 201 z + 15781 z 20 36 34 + 1197674406250 z + 109428877472 z - 422783271014 z 30 42 44 46 52 - 2507756693566 z - 255633490 z + 16079086 z - 651386 z + z 32 38 40 2 + 1197674406250 z - 20523662892 z + 2743786712 z ) (-1 + z )) And in Maple-input format, it is: -(1+1125756200196*z^28-1295095250268*z^26-162*z^2+1125756200196*z^24-\ 738712701638*z^22+10587*z^4-372538*z^6+8007568*z^8-112922454*z^10+1093032030*z^ 12-7487390672*z^14-134966637994*z^18+37090321346*z^16-162*z^50+10587*z^48+ 364879816720*z^20+37090321346*z^36-134966637994*z^34-738712701638*z^30-\ 112922454*z^42+8007568*z^44-372538*z^46+z^52+364879816720*z^32-7487390672*z^38+ 1093032030*z^40)/(1+3900449534034*z^28-4517994886726*z^26-201*z^2+3900449534034 *z^24-2507756693566*z^22+15781*z^4-651386*z^6+16079086*z^8-255633490*z^10+ 2743786712*z^12-20523662892*z^14-422783271014*z^18+109428877472*z^16-201*z^50+ 15781*z^48+1197674406250*z^20+109428877472*z^36-422783271014*z^34-2507756693566 *z^30-255633490*z^42+16079086*z^44-651386*z^46+z^52+1197674406250*z^32-\ 20523662892*z^38+2743786712*z^40)/(-1+z^2) The first , 40, terms are: [0, 40, 0, 2685, 0, 197719, 0, 14991344, 0, 1149221067, 0, 88503385347, 0, 6829982073792, 0, 527612516199967, 0, 40778448556703493, 0, 3152542617983388408, 0, 243754249074061845881, 0, 18848476433279472192393, 0, 1457532081287910521914712, 0, 112711883895483140248669429, 0, 8716188353293608590039146895, 0, 674041154382772060126144684064, 0, 52125209775459349104274454573459, 0, 4030974889583685446096894375566907, 0, 311725873013836849559519443637990288, 0, 24106594937338078894025333276440968935] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 9463038704928 z - 8299501032366 z - 165 z 24 22 4 6 + 5596040077926 z - 2895799871296 z + 11184 z - 417493 z 8 10 12 14 + 9749857 z - 152937152 z + 1684118204 z - 13404562524 z 18 16 50 48 - 345484511868 z + 78682573760 z - 417493 z + 9749857 z 20 36 34 + 1146408270716 z + 1146408270716 z - 2895799871296 z 30 42 44 46 - 8299501032366 z - 13404562524 z + 1684118204 z - 152937152 z 56 54 52 32 38 + z - 165 z + 11184 z + 5596040077926 z - 345484511868 z 40 / 2 28 + 78682573760 z ) / ((-1 + z ) (1 + 33486004394542 z / 26 2 24 22 - 29178831170092 z - 208 z + 19299950263966 z - 9679468396580 z 4 6 8 10 12 + 16821 z - 726756 z + 19247545 z - 337071324 z + 4089702796 z 14 18 16 50 - 35447657892 z - 1049025075836 z + 224130044980 z - 726756 z 48 20 36 34 + 19247545 z + 3671945036828 z + 3671945036828 z - 9679468396580 z 30 42 44 46 - 29178831170092 z - 35447657892 z + 4089702796 z - 337071324 z 56 54 52 32 38 + z - 208 z + 16821 z + 19299950263966 z - 1049025075836 z 40 + 224130044980 z )) And in Maple-input format, it is: -(1+9463038704928*z^28-8299501032366*z^26-165*z^2+5596040077926*z^24-\ 2895799871296*z^22+11184*z^4-417493*z^6+9749857*z^8-152937152*z^10+1684118204*z ^12-13404562524*z^14-345484511868*z^18+78682573760*z^16-417493*z^50+9749857*z^ 48+1146408270716*z^20+1146408270716*z^36-2895799871296*z^34-8299501032366*z^30-\ 13404562524*z^42+1684118204*z^44-152937152*z^46+z^56-165*z^54+11184*z^52+ 5596040077926*z^32-345484511868*z^38+78682573760*z^40)/(-1+z^2)/(1+ 33486004394542*z^28-29178831170092*z^26-208*z^2+19299950263966*z^24-\ 9679468396580*z^22+16821*z^4-726756*z^6+19247545*z^8-337071324*z^10+4089702796* z^12-35447657892*z^14-1049025075836*z^18+224130044980*z^16-726756*z^50+19247545 *z^48+3671945036828*z^20+3671945036828*z^36-9679468396580*z^34-29178831170092*z ^30-35447657892*z^42+4089702796*z^44-337071324*z^46+z^56-208*z^54+16821*z^52+ 19299950263966*z^32-1049025075836*z^38+224130044980*z^40) The first , 40, terms are: [0, 44, 0, 3351, 0, 277167, 0, 23356668, 0, 1977905769, 0, 167738104377, 0, 14232149777564, 0, 1207783705710751, 0, 102503888960302183, 0, 8699723421508652300, 0, 738374504277020264561, 0, 62668700946748277144081, 0, 5318950360444575701607500, 0, 451441777658579198882793031, 0, 38315792688897829522987915583, 0, 3252025927281441554251773732444, 0, 276013444870241929656026697013529, 0, 23426450996295815021491932100933769, 0, 1988303941448774556277129758798277948, 0, 168755933639395630775974330534279541519] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4694806344383932 z - 2172798063871766 z - 211 z 24 22 4 6 + 802823767511002 z - 235824636474752 z + 19146 z - 994605 z 8 10 12 14 + 33266751 z - 764875344 z + 12612181995 z - 153645158697 z 18 16 50 - 9979061731799 z + 1413336365242 z - 9979061731799 z 48 20 36 + 54762189072069 z + 54762189072069 z + 11278945089697398 z 34 66 64 30 - 12581232799386208 z - 211 z + 19146 z - 8123390818571626 z 42 44 46 - 2172798063871766 z + 802823767511002 z - 235824636474752 z 58 56 54 52 - 764875344 z + 12612181995 z - 153645158697 z + 1413336365242 z 60 68 32 38 + 33266751 z + z + 11278945089697398 z - 8123390818571626 z 40 62 / 2 + 4694806344383932 z - 994605 z ) / ((-1 + z ) (1 / 28 26 2 + 16228628017867970 z - 7269595184789400 z - 260 z 24 22 4 6 + 2578763813648582 z - 721811592345520 z + 27547 z - 1621736 z 8 10 12 14 + 60383357 z - 1527770948 z + 27490513989 z - 362945677944 z 18 16 50 - 27172787720324 z + 3595535749443 z - 27172787720324 z 48 20 36 + 158612763510073 z + 158612763510073 z + 40517448810840382 z 34 66 64 30 - 45418040526293208 z - 260 z + 27547 z - 28759112729739200 z 42 44 46 - 7269595184789400 z + 2578763813648582 z - 721811592345520 z 58 56 54 52 - 1527770948 z + 27490513989 z - 362945677944 z + 3595535749443 z 60 68 32 38 + 60383357 z + z + 40517448810840382 z - 28759112729739200 z 40 62 + 16228628017867970 z - 1621736 z )) And in Maple-input format, it is: -(1+4694806344383932*z^28-2172798063871766*z^26-211*z^2+802823767511002*z^24-\ 235824636474752*z^22+19146*z^4-994605*z^6+33266751*z^8-764875344*z^10+ 12612181995*z^12-153645158697*z^14-9979061731799*z^18+1413336365242*z^16-\ 9979061731799*z^50+54762189072069*z^48+54762189072069*z^20+11278945089697398*z^ 36-12581232799386208*z^34-211*z^66+19146*z^64-8123390818571626*z^30-\ 2172798063871766*z^42+802823767511002*z^44-235824636474752*z^46-764875344*z^58+ 12612181995*z^56-153645158697*z^54+1413336365242*z^52+33266751*z^60+z^68+ 11278945089697398*z^32-8123390818571626*z^38+4694806344383932*z^40-994605*z^62) /(-1+z^2)/(1+16228628017867970*z^28-7269595184789400*z^26-260*z^2+ 2578763813648582*z^24-721811592345520*z^22+27547*z^4-1621736*z^6+60383357*z^8-\ 1527770948*z^10+27490513989*z^12-362945677944*z^14-27172787720324*z^18+ 3595535749443*z^16-27172787720324*z^50+158612763510073*z^48+158612763510073*z^ 20+40517448810840382*z^36-45418040526293208*z^34-260*z^66+27547*z^64-\ 28759112729739200*z^30-7269595184789400*z^42+2578763813648582*z^44-\ 721811592345520*z^46-1527770948*z^58+27490513989*z^56-362945677944*z^54+ 3595535749443*z^52+60383357*z^60+z^68+40517448810840382*z^32-28759112729739200* z^38+16228628017867970*z^40-1621736*z^62) The first , 40, terms are: [0, 50, 0, 4389, 0, 409857, 0, 38653562, 0, 3653413481, 0, 345532761669, 0, 32687309454818, 0, 3092484874500869, 0, 292584568662700233, 0, 27682259189449313978, 0, 2619112901550447437469, 0, 247803804196444206985637, 0, 23445643434180228417064106, 0, 2218280814771396920118012481, 0, 209879956305525257958041112893, 0, 19857539632007267594421785779090, 0, 1878797283938689592504112762940877, 0, 177760152700679214187729793451979361, 0, 16818563832434266768640493430646896330, 0, 1591268266154577299878545387180536608073] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 4}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 72172 z + 769978 z + 105 z - 5006071 z + 20347973 z 4 6 8 10 12 - 3883 z + 72172 z - 769978 z + 5006071 z - 20347973 z 14 18 16 20 34 + 52044069 z + 83494266 z - 83494266 z - 52044069 z + z 30 32 / 36 34 32 30 + 3883 z - 105 z ) / (z - 140 z + 6831 z - 160072 z / 28 26 24 22 + 2079337 z - 16108684 z + 77443703 z - 235876496 z 20 18 16 14 + 459508017 z - 573948948 z + 459508017 z - 235876496 z 12 10 8 6 4 2 + 77443703 z - 16108684 z + 2079337 z - 160072 z + 6831 z - 140 z + 1) And in Maple-input format, it is: -(-1-72172*z^28+769978*z^26+105*z^2-5006071*z^24+20347973*z^22-3883*z^4+72172*z ^6-769978*z^8+5006071*z^10-20347973*z^12+52044069*z^14+83494266*z^18-83494266*z ^16-52044069*z^20+z^34+3883*z^30-105*z^32)/(z^36-140*z^34+6831*z^32-160072*z^30 +2079337*z^28-16108684*z^26+77443703*z^24-235876496*z^22+459508017*z^20-\ 573948948*z^18+459508017*z^16-235876496*z^14+77443703*z^12-16108684*z^10+ 2079337*z^8-160072*z^6+6831*z^4-140*z^2+1) The first , 40, terms are: [0, 35, 0, 1952, 0, 122095, 0, 8052349, 0, 544084277, 0, 37158035987, 0, 2549481754096, 0, 175273512249907, 0, 12060110516159129, 0, 830127474374749925, 0, 57148656906669079543, 0, 3934560346680037921840, 0, 270893607840391765296767, 0, 18651191682159775572452465, 0, 1284152701372837669443876313, 0, 88415359610914286995202400475, 0, 6087502846519763292246614896160, 0, 419131993286013401627719222640327, 0, 28857753763402388505263199809139085, 0, 1986892022400116834619359183251778677] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 4}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 393313481349120023 z - 100424323211662825 z - 229 z 24 22 4 6 + 21510260771965725 z - 3841658420495444 z + 23179 z - 1384918 z 8 10 12 14 + 55083168 z - 1560313726 z + 32893018157 z - 532355703779 z 18 16 50 - 68844944492548 z + 6769412446815 z - 30072861373735911727 z 48 20 + 44355106372527452995 z + 567859067407200 z 36 34 + 17427758351537223985 z - 8619169879551813242 z 66 80 88 84 86 - 3841658420495444 z + 55083168 z + z + 23179 z - 229 z 82 64 30 - 1384918 z + 21510260771965725 z - 1298716515720265266 z 42 44 - 55978315868471818280 z + 60489680884068165984 z 46 58 - 55978315868471818280 z - 1298716515720265266 z 56 54 + 3630295428326932976 z - 8619169879551813242 z 52 60 70 + 17427758351537223985 z + 393313481349120023 z - 68844944492548 z 68 78 32 + 567859067407200 z - 1560313726 z + 3630295428326932976 z 38 40 - 30072861373735911727 z + 44355106372527452995 z 62 76 74 - 100424323211662825 z + 32893018157 z - 532355703779 z 72 / 28 + 6769412446815 z ) / (-1 - 1506318218120911700 z / 26 2 24 + 358730398238512569 z + 275 z - 71636961786290203 z 22 4 6 8 + 11918103139890086 z - 32190 z + 2173848 z - 96272601 z 10 12 14 + 3005341681 z - 69302301930 z + 1220073727200 z 18 16 50 + 184505610386493 z - 16804437547231 z + 259971776862589899045 z 48 20 - 353413004145493647671 z - 1638915962381110 z 36 34 - 88365341535040332596 z + 40707896867843067191 z 66 80 90 88 84 + 71636961786290203 z - 3005341681 z + z - 275 z - 2173848 z 86 82 64 + 32190 z + 96272601 z - 358730398238512569 z 30 42 + 5332171184553594874 z + 353413004145493647671 z 44 46 - 411976206265607893140 z + 411976206265607893140 z 58 56 + 15982274214394755727 z - 40707896867843067191 z 54 52 + 88365341535040332596 z - 163848931190845971518 z 60 70 68 - 5332171184553594874 z + 1638915962381110 z - 11918103139890086 z 78 32 38 + 69302301930 z - 15982274214394755727 z + 163848931190845971518 z 40 62 76 - 259971776862589899045 z + 1506318218120911700 z - 1220073727200 z 74 72 + 16804437547231 z - 184505610386493 z ) And in Maple-input format, it is: -(1+393313481349120023*z^28-100424323211662825*z^26-229*z^2+21510260771965725*z ^24-3841658420495444*z^22+23179*z^4-1384918*z^6+55083168*z^8-1560313726*z^10+ 32893018157*z^12-532355703779*z^14-68844944492548*z^18+6769412446815*z^16-\ 30072861373735911727*z^50+44355106372527452995*z^48+567859067407200*z^20+ 17427758351537223985*z^36-8619169879551813242*z^34-3841658420495444*z^66+ 55083168*z^80+z^88+23179*z^84-229*z^86-1384918*z^82+21510260771965725*z^64-\ 1298716515720265266*z^30-55978315868471818280*z^42+60489680884068165984*z^44-\ 55978315868471818280*z^46-1298716515720265266*z^58+3630295428326932976*z^56-\ 8619169879551813242*z^54+17427758351537223985*z^52+393313481349120023*z^60-\ 68844944492548*z^70+567859067407200*z^68-1560313726*z^78+3630295428326932976*z^ 32-30072861373735911727*z^38+44355106372527452995*z^40-100424323211662825*z^62+ 32893018157*z^76-532355703779*z^74+6769412446815*z^72)/(-1-1506318218120911700* z^28+358730398238512569*z^26+275*z^2-71636961786290203*z^24+11918103139890086*z ^22-32190*z^4+2173848*z^6-96272601*z^8+3005341681*z^10-69302301930*z^12+ 1220073727200*z^14+184505610386493*z^18-16804437547231*z^16+ 259971776862589899045*z^50-353413004145493647671*z^48-1638915962381110*z^20-\ 88365341535040332596*z^36+40707896867843067191*z^34+71636961786290203*z^66-\ 3005341681*z^80+z^90-275*z^88-2173848*z^84+32190*z^86+96272601*z^82-\ 358730398238512569*z^64+5332171184553594874*z^30+353413004145493647671*z^42-\ 411976206265607893140*z^44+411976206265607893140*z^46+15982274214394755727*z^58 -40707896867843067191*z^56+88365341535040332596*z^54-163848931190845971518*z^52 -5332171184553594874*z^60+1638915962381110*z^70-11918103139890086*z^68+ 69302301930*z^78-15982274214394755727*z^32+163848931190845971518*z^38-\ 259971776862589899045*z^40+1506318218120911700*z^62-1220073727200*z^76+ 16804437547231*z^74-184505610386493*z^72) The first , 40, terms are: [0, 46, 0, 3639, 0, 308915, 0, 26619790, 0, 2303589581, 0, 199630788109, 0, 17309495343390, 0, 1501185747321067, 0, 130203402937135735, 0, 11293428211941393726, 0, 979570446928264778569, 0, 84966593201692905474761, 0, 7369904040836182650443774, 0, 639257622126123398089303495, 0, 55448548839022194836268801611, 0, 4809550973002082263624536662462, 0, 417175611609835926959126645673597, 0, 36185393861564410115673077513487805, 0, 3138684786653938602714303171265564206, 0, 272246373201556230247992714008023388083] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 56530549582572 z - 33306626809042 z - 153 z 24 22 4 6 + 15834967128282 z - 6055322828848 z + 9690 z - 346707 z 8 10 12 14 + 7996982 z - 127962623 z + 1487808010 z - 12970533893 z 18 16 50 48 - 451466897464 z + 86691637061 z - 12970533893 z + 86691637061 z 20 36 34 64 + 1853845383616 z + 56530549582572 z - 77587566867478 z + z 30 42 44 - 77587566867478 z - 6055322828848 z + 1853845383616 z 46 58 56 54 - 451466897464 z - 346707 z + 7996982 z - 127962623 z 52 60 32 38 + 1487808010 z + 9690 z + 86213065576116 z - 33306626809042 z 40 62 / 28 + 15834967128282 z - 153 z ) / (-1 - 298203409256198 z / 26 2 24 22 + 160082952553990 z + 187 z - 69525470020738 z + 24324446854800 z 4 6 8 10 12 - 14159 z + 593843 z - 15769986 z + 286134274 z - 3726611895 z 14 18 16 + 36052738611 z + 1517207926701 z - 265578575255 z 50 48 20 + 265578575255 z - 1517207926701 z - 6815291202216 z 36 34 66 64 - 450760420893670 z + 553942014296868 z + z - 187 z 30 42 44 + 450760420893670 z + 69525470020738 z - 24324446854800 z 46 58 56 54 + 6815291202216 z + 15769986 z - 286134274 z + 3726611895 z 52 60 32 38 - 36052738611 z - 593843 z - 553942014296868 z + 298203409256198 z 40 62 - 160082952553990 z + 14159 z ) And in Maple-input format, it is: -(1+56530549582572*z^28-33306626809042*z^26-153*z^2+15834967128282*z^24-\ 6055322828848*z^22+9690*z^4-346707*z^6+7996982*z^8-127962623*z^10+1487808010*z^ 12-12970533893*z^14-451466897464*z^18+86691637061*z^16-12970533893*z^50+ 86691637061*z^48+1853845383616*z^20+56530549582572*z^36-77587566867478*z^34+z^ 64-77587566867478*z^30-6055322828848*z^42+1853845383616*z^44-451466897464*z^46-\ 346707*z^58+7996982*z^56-127962623*z^54+1487808010*z^52+9690*z^60+ 86213065576116*z^32-33306626809042*z^38+15834967128282*z^40-153*z^62)/(-1-\ 298203409256198*z^28+160082952553990*z^26+187*z^2-69525470020738*z^24+ 24324446854800*z^22-14159*z^4+593843*z^6-15769986*z^8+286134274*z^10-3726611895 *z^12+36052738611*z^14+1517207926701*z^18-265578575255*z^16+265578575255*z^50-\ 1517207926701*z^48-6815291202216*z^20-450760420893670*z^36+553942014296868*z^34 +z^66-187*z^64+450760420893670*z^30+69525470020738*z^42-24324446854800*z^44+ 6815291202216*z^46+15769986*z^58-286134274*z^56+3726611895*z^54-36052738611*z^ 52-593843*z^60-553942014296868*z^32+298203409256198*z^38-160082952553990*z^40+ 14159*z^62) The first , 40, terms are: [0, 34, 0, 1889, 0, 118973, 0, 7919258, 0, 540124093, 0, 37225972485, 0, 2577118249898, 0, 178750481570349, 0, 12408301181058241, 0, 861645082576550994, 0, 59842458080175712313, 0, 4156410424298139315337, 0, 288695170901993296028850, 0, 20052376310922564555604641, 0, 1392818222264032279569486893, 0, 96743994898073050387243585738, 0, 6719763995220023756101827869685, 0, 466749873768844570373703640432269, 0, 32420109569876429410089380650561274, 0, 2251877607583543955309265348841241917] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 25424550675913678195 z - 3990561400749211013 z - 254 z 24 22 4 6 + 540190075491824935 z - 62691269746437280 z + 29299 z - 2055537 z 102 8 10 12 - 3510445660 z + 99066479 z - 3510445660 z + 95539101969 z 14 18 16 - 2058728538578 z - 516215958849022 z + 35920332775962 z 50 48 - 2725730792748889518021236 z + 1783152103419529825228202 z 20 36 + 6192969392401257 z + 10414853466158604159480 z 34 66 - 2832480661790165629868 z - 1032420726783120998421978 z 80 100 90 + 674776079611142480065 z + 95539101969 z - 62691269746437280 z 88 84 94 + 540190075491824935 z + 25424550675913678195 z - 516215958849022 z 86 96 98 - 3990561400749211013 z + 35920332775962 z - 2058728538578 z 92 82 + 6192969392401257 z - 140397728428635117874 z 64 112 110 106 + 1783152103419529825228202 z + z - 254 z - 2055537 z 108 30 42 + 29299 z - 140397728428635117874 z - 239208774108650992878848 z 44 46 + 528668722589892378723898 z - 1032420726783120998421978 z 58 56 - 4423808030803705499301328 z + 4699619575546014469069948 z 54 52 - 4423808030803705499301328 z + 3689494394868969731119670 z 60 70 + 3689494394868969731119670 z - 239208774108650992878848 z 68 78 + 528668722589892378723898 z - 2832480661790165629868 z 32 38 + 674776079611142480065 z - 33626616654294958604564 z 40 62 + 95531264092927991968174 z - 2725730792748889518021236 z 76 74 + 10414853466158604159480 z - 33626616654294958604564 z 72 104 / 2 + 95531264092927991968174 z + 99066479 z ) / ((-1 + z ) (1 / 28 26 2 + 68370836165636280662 z - 10344463406914586608 z - 292 z 24 22 4 6 + 1345845695977596207 z - 149640269426956412 z + 37834 z - 2934440 z 102 8 10 12 - 5925420996 z + 154504695 z - 5925420996 z + 173195370976 z 14 18 16 - 3982301363376 z - 1118854286411912 z + 73728331677973 z 50 48 - 9280312956780775671012736 z + 6016321131104581182012162 z 20 36 + 14113162765549444 z + 31587494312025453050660 z 34 66 - 8367799211402106005848 z - 3443196198037246125561224 z 80 100 90 + 1936920961593053056177 z + 173195370976 z - 149640269426956412 z 88 84 + 1345845695977596207 z + 68370836165636280662 z 94 86 96 - 1118854286411912 z - 10344463406914586608 z + 73728331677973 z 98 92 82 - 3982301363376 z + 14113162765549444 z - 390586764458195729292 z 64 112 110 106 + 6016321131104581182012162 z + z - 292 z - 2934440 z 108 30 42 + 37834 z - 390586764458195729292 z - 773683150345078035270416 z 44 46 + 1738461726465207494504776 z - 3443196198037246125561224 z 58 56 - 15219402638257074378800520 z + 16189409607139561543357278 z 54 52 - 15219402638257074378800520 z + 12643568445428609552702020 z 60 70 + 12643568445428609552702020 z - 773683150345078035270416 z 68 78 + 1738461726465207494504776 z - 8367799211402106005848 z 32 38 + 1936920961593053056177 z - 104449171525486419536872 z 40 62 + 303162743437385862758710 z - 9280312956780775671012736 z 76 74 + 31587494312025453050660 z - 104449171525486419536872 z 72 104 + 303162743437385862758710 z + 154504695 z )) And in Maple-input format, it is: -(1+25424550675913678195*z^28-3990561400749211013*z^26-254*z^2+ 540190075491824935*z^24-62691269746437280*z^22+29299*z^4-2055537*z^6-3510445660 *z^102+99066479*z^8-3510445660*z^10+95539101969*z^12-2058728538578*z^14-\ 516215958849022*z^18+35920332775962*z^16-2725730792748889518021236*z^50+ 1783152103419529825228202*z^48+6192969392401257*z^20+10414853466158604159480*z^ 36-2832480661790165629868*z^34-1032420726783120998421978*z^66+ 674776079611142480065*z^80+95539101969*z^100-62691269746437280*z^90+ 540190075491824935*z^88+25424550675913678195*z^84-516215958849022*z^94-\ 3990561400749211013*z^86+35920332775962*z^96-2058728538578*z^98+ 6192969392401257*z^92-140397728428635117874*z^82+1783152103419529825228202*z^64 +z^112-254*z^110-2055537*z^106+29299*z^108-140397728428635117874*z^30-\ 239208774108650992878848*z^42+528668722589892378723898*z^44-\ 1032420726783120998421978*z^46-4423808030803705499301328*z^58+ 4699619575546014469069948*z^56-4423808030803705499301328*z^54+ 3689494394868969731119670*z^52+3689494394868969731119670*z^60-\ 239208774108650992878848*z^70+528668722589892378723898*z^68-\ 2832480661790165629868*z^78+674776079611142480065*z^32-33626616654294958604564* z^38+95531264092927991968174*z^40-2725730792748889518021236*z^62+ 10414853466158604159480*z^76-33626616654294958604564*z^74+ 95531264092927991968174*z^72+99066479*z^104)/(-1+z^2)/(1+68370836165636280662*z ^28-10344463406914586608*z^26-292*z^2+1345845695977596207*z^24-\ 149640269426956412*z^22+37834*z^4-2934440*z^6-5925420996*z^102+154504695*z^8-\ 5925420996*z^10+173195370976*z^12-3982301363376*z^14-1118854286411912*z^18+ 73728331677973*z^16-9280312956780775671012736*z^50+6016321131104581182012162*z^ 48+14113162765549444*z^20+31587494312025453050660*z^36-8367799211402106005848*z ^34-3443196198037246125561224*z^66+1936920961593053056177*z^80+173195370976*z^ 100-149640269426956412*z^90+1345845695977596207*z^88+68370836165636280662*z^84-\ 1118854286411912*z^94-10344463406914586608*z^86+73728331677973*z^96-\ 3982301363376*z^98+14113162765549444*z^92-390586764458195729292*z^82+ 6016321131104581182012162*z^64+z^112-292*z^110-2934440*z^106+37834*z^108-\ 390586764458195729292*z^30-773683150345078035270416*z^42+ 1738461726465207494504776*z^44-3443196198037246125561224*z^46-\ 15219402638257074378800520*z^58+16189409607139561543357278*z^56-\ 15219402638257074378800520*z^54+12643568445428609552702020*z^52+ 12643568445428609552702020*z^60-773683150345078035270416*z^70+ 1738461726465207494504776*z^68-8367799211402106005848*z^78+ 1936920961593053056177*z^32-104449171525486419536872*z^38+ 303162743437385862758710*z^40-9280312956780775671012736*z^62+ 31587494312025453050660*z^76-104449171525486419536872*z^74+ 303162743437385862758710*z^72+154504695*z^104) The first , 40, terms are: [0, 39, 0, 2600, 0, 191623, 0, 14563969, 0, 1118690585, 0, 86260180959, 0, 6661046912144, 0, 514660958262467, 0, 39773930509107953, 0, 3074084964822112057, 0, 237601797172860255543, 0, 18364978891621025498544, 0, 1419495493025870211922107, 0, 109718235076503986354281849, 0, 8480551991738373878860597057, 0, 655495399095026248858081670419, 0, 50665842422648743002975323696952, 0, 3916164522528202513788240061263387, 0, 302695947602345382355296520638624009, 0, 23396575339742947992347158960870929257] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 25116998038926546204 z - 3942586566794791056 z - 254 z 24 22 4 6 + 533812331508866368 z - 61974150201637236 z + 29264 z - 2049872 z 102 8 10 12 - 3489985548 z + 98636209 z - 3489985548 z + 94854283028 z 14 18 16 - 2041540335582 z - 510939206928036 z + 35583850279773 z 50 48 - 2701381996268614319969492 z + 1766794086212307304758426 z 20 36 + 6125391416785704 z + 10296974139616225984084 z 34 66 - 2799525006920463230724 z - 1022643856633467114970092 z 80 100 90 + 666756877048564205996 z + 94854283028 z - 61974150201637236 z 88 84 94 + 533812331508866368 z + 25116998038926546204 z - 510939206928036 z 86 96 98 - 3942586566794791056 z + 35583850279773 z - 2041540335582 z 92 82 + 6125391416785704 z - 138706659231116026860 z 64 112 110 106 + 1766794086212307304758426 z + z - 254 z - 2049872 z 108 30 42 + 29264 z - 138706659231116026860 z - 236772774526675682755708 z 44 46 + 523481982982256573296984 z - 1022643856633467114970092 z 58 56 - 4385554658097407935307316 z + 4659152761998605874114886 z 54 52 - 4385554658097407935307316 z + 3657190523107720802191020 z 60 70 + 3657190523107720802191020 z - 236772774526675682755708 z 68 78 + 523481982982256573296984 z - 2799525006920463230724 z 32 38 + 666756877048564205996 z - 33258178302664232310080 z 40 62 + 94521339278991106569800 z - 2701381996268614319969492 z 76 74 + 10296974139616225984084 z - 33258178302664232310080 z 72 104 / 2 + 94521339278991106569800 z + 98636209 z ) / ((-1 + z ) (1 / 28 26 2 + 67567944238874976102 z - 10228705559182169311 z - 291 z 24 22 4 6 + 1331546832026450532 z - 148135217828013381 z + 37636 z - 2916167 z 102 8 10 12 - 5882376882 z + 153449575 z - 5882376882 z + 171870070386 z 14 18 16 - 3950232607496 z - 1108808212265083 z + 73102151881413 z 50 48 - 9134842095632412344827217 z + 5923024595092648036851450 z 20 36 + 13979024407081642 z + 31155529247639878634490 z 34 66 - 8256988242289564375203 z - 3390526786879853356584833 z 80 100 90 + 1912177451543793362736 z + 171870070386 z - 148135217828013381 z 88 84 + 1331546832026450532 z + 67567944238874976102 z 94 86 96 - 1108808212265083 z - 10228705559182169311 z + 73102151881413 z 98 92 82 - 3950232607496 z + 13979024407081642 z - 385792886872647898697 z 64 112 110 106 + 5923024595092648036851450 z + z - 291 z - 2916167 z 108 30 42 + 37636 z - 385792886872647898697 z - 762265336025140640239103 z 44 46 + 1712304608145170821243814 z - 3390526786879853356584833 z 58 56 - 14977833379886000154925915 z + 15932037684406723885409714 z 54 52 - 14977833379886000154925915 z + 12443828210973569315324618 z 60 70 + 12443828210973569315324618 z - 762265336025140640239103 z 68 78 + 1712304608145170821243814 z - 8256988242289564375203 z 32 38 + 1912177451543793362736 z - 102979366908703788407013 z 40 62 + 298787247757547552475324 z - 9134842095632412344827217 z 76 74 + 31155529247639878634490 z - 102979366908703788407013 z 72 104 + 298787247757547552475324 z + 153449575 z )) And in Maple-input format, it is: -(1+25116998038926546204*z^28-3942586566794791056*z^26-254*z^2+ 533812331508866368*z^24-61974150201637236*z^22+29264*z^4-2049872*z^6-3489985548 *z^102+98636209*z^8-3489985548*z^10+94854283028*z^12-2041540335582*z^14-\ 510939206928036*z^18+35583850279773*z^16-2701381996268614319969492*z^50+ 1766794086212307304758426*z^48+6125391416785704*z^20+10296974139616225984084*z^ 36-2799525006920463230724*z^34-1022643856633467114970092*z^66+ 666756877048564205996*z^80+94854283028*z^100-61974150201637236*z^90+ 533812331508866368*z^88+25116998038926546204*z^84-510939206928036*z^94-\ 3942586566794791056*z^86+35583850279773*z^96-2041540335582*z^98+ 6125391416785704*z^92-138706659231116026860*z^82+1766794086212307304758426*z^64 +z^112-254*z^110-2049872*z^106+29264*z^108-138706659231116026860*z^30-\ 236772774526675682755708*z^42+523481982982256573296984*z^44-\ 1022643856633467114970092*z^46-4385554658097407935307316*z^58+ 4659152761998605874114886*z^56-4385554658097407935307316*z^54+ 3657190523107720802191020*z^52+3657190523107720802191020*z^60-\ 236772774526675682755708*z^70+523481982982256573296984*z^68-\ 2799525006920463230724*z^78+666756877048564205996*z^32-33258178302664232310080* z^38+94521339278991106569800*z^40-2701381996268614319969492*z^62+ 10296974139616225984084*z^76-33258178302664232310080*z^74+ 94521339278991106569800*z^72+98636209*z^104)/(-1+z^2)/(1+67567944238874976102*z ^28-10228705559182169311*z^26-291*z^2+1331546832026450532*z^24-\ 148135217828013381*z^22+37636*z^4-2916167*z^6-5882376882*z^102+153449575*z^8-\ 5882376882*z^10+171870070386*z^12-3950232607496*z^14-1108808212265083*z^18+ 73102151881413*z^16-9134842095632412344827217*z^50+5923024595092648036851450*z^ 48+13979024407081642*z^20+31155529247639878634490*z^36-8256988242289564375203*z ^34-3390526786879853356584833*z^66+1912177451543793362736*z^80+171870070386*z^ 100-148135217828013381*z^90+1331546832026450532*z^88+67567944238874976102*z^84-\ 1108808212265083*z^94-10228705559182169311*z^86+73102151881413*z^96-\ 3950232607496*z^98+13979024407081642*z^92-385792886872647898697*z^82+ 5923024595092648036851450*z^64+z^112-291*z^110-2916167*z^106+37636*z^108-\ 385792886872647898697*z^30-762265336025140640239103*z^42+ 1712304608145170821243814*z^44-3390526786879853356584833*z^46-\ 14977833379886000154925915*z^58+15932037684406723885409714*z^56-\ 14977833379886000154925915*z^54+12443828210973569315324618*z^52+ 12443828210973569315324618*z^60-762265336025140640239103*z^70+ 1712304608145170821243814*z^68-8256988242289564375203*z^78+ 1912177451543793362736*z^32-102979366908703788407013*z^38+ 298787247757547552475324*z^40-9134842095632412344827217*z^62+ 31155529247639878634490*z^76-102979366908703788407013*z^74+ 298787247757547552475324*z^72+153449575*z^104) The first , 40, terms are: [0, 38, 0, 2433, 0, 173141, 0, 12795762, 0, 960189209, 0, 72520179717, 0, 5492770710882, 0, 416555829916081, 0, 31608432573382981, 0, 2399084345067837486, 0, 182112444713392118701, 0, 13824752243511674267749, 0, 1049508444159460220381070, 0, 79674529062371683119800141, 0, 6048607014300577338812954649, 0, 459189860641926084777285225682, 0, 34860185870740411076691531098413, 0, 2646472172401259746567941755788913, 0, 200911626186458823633015509172041922, 0, 15252563926415962024436730783867473485] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 28341463208088988715 z - 4440775703799157974 z - 257 z 24 22 4 6 + 599742345039332095 z - 69389747366667291 z + 30013 z - 2132094 z 102 8 10 12 - 3727376603 z + 104008619 z - 3727376603 z + 102475769082 z 14 18 16 - 2227763114927 z - 566279969660857 z + 39161178659026 z 50 48 - 3049422365213816026480168 z + 1994901231797137456969893 z 20 36 + 6827622413222512 z + 11645384700836813032541 z 34 66 - 3165930919572359833397 z - 1155010077761930943987479 z 80 100 90 + 753771721285134258248 z + 102475769082 z - 69389747366667291 z 88 84 94 + 599742345039332095 z + 28341463208088988715 z - 566279969660857 z 86 96 98 - 4440775703799157974 z + 39161178659026 z - 2227763114927 z 92 82 + 6827622413222512 z - 156698578861815167011 z 64 112 110 106 + 1994901231797137456969893 z + z - 257 z - 2132094 z 108 30 42 + 30013 z - 156698578861815167011 z - 267595566152856386144629 z 44 46 + 591430580822429454318704 z - 1155010077761930943987479 z 58 56 - 4949163068519856415005247 z + 5257729763066748288136922 z 54 52 - 4949163068519856415005247 z + 4127642769795549191516257 z 60 70 + 4127642769795549191516257 z - 267595566152856386144629 z 68 78 + 591430580822429454318704 z - 3165930919572359833397 z 32 38 + 753771721285134258248 z - 37608668973256592693506 z 40 62 + 106859318411064659690025 z - 3049422365213816026480168 z 76 74 + 11645384700836813032541 z - 37608668973256592693506 z 72 104 / 2 + 106859318411064659690025 z + 104008619 z ) / ((-1 + z ) (1 / 28 26 2 + 76844007186275071616 z - 11598776253176444556 z - 300 z 24 22 4 6 + 1504743274339909804 z - 166744142135751700 z + 39536 z - 3104116 z 102 8 10 12 - 6383106404 z + 165046326 z - 6383106404 z + 187967625728 z 14 18 16 - 4350843020232 z - 1236144679887168 z + 81031208109521 z 50 48 - 10530832735266390692988156 z + 6825550248108633562213558 z 20 36 + 15664246299763152 z + 35717178368123601132736 z 34 66 - 9451451813500201857364 z - 3905198932955463624420296 z 80 100 90 + 2184832734548166979320 z + 187967625728 z - 166744142135751700 z 88 84 + 1504743274339909804 z + 76844007186275071616 z 94 86 96 - 1236144679887168 z - 11598776253176444556 z + 81031208109521 z 98 92 82 - 4350843020232 z + 15664246299763152 z - 439861118334857526516 z 64 112 110 106 + 6825550248108633562213558 z + z - 300 z - 3104116 z 108 30 42 + 39536 z - 439861118334857526516 z - 876757196614773977897268 z 44 46 + 1970992757385702606490192 z - 3905198932955463624420296 z 58 56 - 17274300278472632838148016 z + 18375807253037712900686596 z 54 52 - 17274300278472632838148016 z + 14349424067958684543986096 z 60 70 + 14349424067958684543986096 z - 876757196614773977897268 z 68 78 + 1970992757385702606490192 z - 9451451813500201857364 z 32 38 + 2184832734548166979320 z - 118209825110555278906212 z 40 62 + 343350972542125772798876 z - 10530832735266390692988156 z 76 74 + 35717178368123601132736 z - 118209825110555278906212 z 72 104 + 343350972542125772798876 z + 165046326 z )) And in Maple-input format, it is: -(1+28341463208088988715*z^28-4440775703799157974*z^26-257*z^2+ 599742345039332095*z^24-69389747366667291*z^22+30013*z^4-2132094*z^6-3727376603 *z^102+104008619*z^8-3727376603*z^10+102475769082*z^12-2227763114927*z^14-\ 566279969660857*z^18+39161178659026*z^16-3049422365213816026480168*z^50+ 1994901231797137456969893*z^48+6827622413222512*z^20+11645384700836813032541*z^ 36-3165930919572359833397*z^34-1155010077761930943987479*z^66+ 753771721285134258248*z^80+102475769082*z^100-69389747366667291*z^90+ 599742345039332095*z^88+28341463208088988715*z^84-566279969660857*z^94-\ 4440775703799157974*z^86+39161178659026*z^96-2227763114927*z^98+ 6827622413222512*z^92-156698578861815167011*z^82+1994901231797137456969893*z^64 +z^112-257*z^110-2132094*z^106+30013*z^108-156698578861815167011*z^30-\ 267595566152856386144629*z^42+591430580822429454318704*z^44-\ 1155010077761930943987479*z^46-4949163068519856415005247*z^58+ 5257729763066748288136922*z^56-4949163068519856415005247*z^54+ 4127642769795549191516257*z^52+4127642769795549191516257*z^60-\ 267595566152856386144629*z^70+591430580822429454318704*z^68-\ 3165930919572359833397*z^78+753771721285134258248*z^32-37608668973256592693506* z^38+106859318411064659690025*z^40-3049422365213816026480168*z^62+ 11645384700836813032541*z^76-37608668973256592693506*z^74+ 106859318411064659690025*z^72+104008619*z^104)/(-1+z^2)/(1+76844007186275071616 *z^28-11598776253176444556*z^26-300*z^2+1504743274339909804*z^24-\ 166744142135751700*z^22+39536*z^4-3104116*z^6-6383106404*z^102+165046326*z^8-\ 6383106404*z^10+187967625728*z^12-4350843020232*z^14-1236144679887168*z^18+ 81031208109521*z^16-10530832735266390692988156*z^50+6825550248108633562213558*z ^48+15664246299763152*z^20+35717178368123601132736*z^36-9451451813500201857364* z^34-3905198932955463624420296*z^66+2184832734548166979320*z^80+187967625728*z^ 100-166744142135751700*z^90+1504743274339909804*z^88+76844007186275071616*z^84-\ 1236144679887168*z^94-11598776253176444556*z^86+81031208109521*z^96-\ 4350843020232*z^98+15664246299763152*z^92-439861118334857526516*z^82+ 6825550248108633562213558*z^64+z^112-300*z^110-3104116*z^106+39536*z^108-\ 439861118334857526516*z^30-876757196614773977897268*z^42+ 1970992757385702606490192*z^44-3905198932955463624420296*z^46-\ 17274300278472632838148016*z^58+18375807253037712900686596*z^56-\ 17274300278472632838148016*z^54+14349424067958684543986096*z^52+ 14349424067958684543986096*z^60-876757196614773977897268*z^70+ 1970992757385702606490192*z^68-9451451813500201857364*z^78+ 2184832734548166979320*z^32-118209825110555278906212*z^38+ 343350972542125772798876*z^40-10530832735266390692988156*z^62+ 35717178368123601132736*z^76-118209825110555278906212*z^74+ 343350972542125772798876*z^72+165046326*z^104) The first , 40, terms are: [0, 44, 0, 3421, 0, 288495, 0, 24736904, 0, 2129911455, 0, 183613018939, 0, 15834867754432, 0, 1365788298821411, 0, 117807535595586117, 0, 10161794193364431764, 0, 876537698570947982245, 0, 75608724232560735961577, 0, 6521892756136344770421460, 0, 562568702177018249230421105, 0, 48526341410113362166653931711, 0, 4185810464416310004531767430176, 0, 361061831346874422996142189586951, 0, 31144660884267536836895544182480955, 0, 2686492510772009777502439918633810968, 0, 231732881931488776458903703775807660859] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 7895811403804 z - 6939406415296 z - 164 z 24 22 4 6 + 4707308205544 z - 2459002610456 z + 10992 z - 404455 z 8 10 12 14 + 9285198 z - 142972857 z + 1545115656 z - 12076733540 z 18 16 50 48 - 301193630172 z + 69686267823 z - 404455 z + 9285198 z 20 36 34 + 985368312624 z + 985368312624 z - 2459002610456 z 30 42 44 46 - 6939406415296 z - 12076733540 z + 1545115656 z - 142972857 z 56 54 52 32 38 + z - 164 z + 10992 z + 4707308205544 z - 301193630172 z 40 / 28 26 + 69686267823 z ) / (-1 - 52048786393730 z + 40411924948654 z / 2 24 22 4 6 + 207 z - 24327665297306 z + 11321930379902 z - 16705 z + 720795 z 8 10 12 14 - 19073553 z + 334307969 z - 4075141443 z + 35706474057 z 18 16 50 48 + 1110003826257 z - 230167187839 z + 19073553 z - 334307969 z 20 36 34 - 4054705367042 z - 11321930379902 z + 24327665297306 z 30 42 44 46 + 52048786393730 z + 230167187839 z - 35706474057 z + 4075141443 z 58 56 54 52 32 + z - 207 z + 16705 z - 720795 z - 40411924948654 z 38 40 + 4054705367042 z - 1110003826257 z ) And in Maple-input format, it is: -(1+7895811403804*z^28-6939406415296*z^26-164*z^2+4707308205544*z^24-\ 2459002610456*z^22+10992*z^4-404455*z^6+9285198*z^8-142972857*z^10+1545115656*z ^12-12076733540*z^14-301193630172*z^18+69686267823*z^16-404455*z^50+9285198*z^ 48+985368312624*z^20+985368312624*z^36-2459002610456*z^34-6939406415296*z^30-\ 12076733540*z^42+1545115656*z^44-142972857*z^46+z^56-164*z^54+10992*z^52+ 4707308205544*z^32-301193630172*z^38+69686267823*z^40)/(-1-52048786393730*z^28+ 40411924948654*z^26+207*z^2-24327665297306*z^24+11321930379902*z^22-16705*z^4+ 720795*z^6-19073553*z^8+334307969*z^10-4075141443*z^12+35706474057*z^14+ 1110003826257*z^18-230167187839*z^16+19073553*z^50-334307969*z^48-4054705367042 *z^20-11321930379902*z^36+24327665297306*z^34+52048786393730*z^30+230167187839* z^42-35706474057*z^44+4075141443*z^46+z^58-207*z^56+16705*z^54-720795*z^52-\ 40411924948654*z^32+4054705367042*z^38-1110003826257*z^40) The first , 40, terms are: [0, 43, 0, 3188, 0, 257941, 0, 21344077, 0, 1778386327, 0, 148534476415, 0, 12417718135852, 0, 1038558892659153, 0, 86875564219590579, 0, 7267734066875626123, 0, 608017912279836475865, 0, 50867578668118145076780, 0, 4255681811603990508729399, 0, 356040008047850581414954255, 0, 29787165720728807645617745253, 0, 2492068051493006958884147944173, 0, 208492659577474143181448193134580, 0, 17443021389717208711515405424209811, 0, 1459327245691231134902328277501018361, 0, 122091009262630132770551973875539323849] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3309414689721984 z - 1378793437431966 z - 189 z 24 22 4 6 + 469415808880239 z - 129959472179445 z + 15312 z - 713737 z 8 10 12 14 + 21668735 z - 459269482 z + 7105316000 z - 82759853550 z 18 16 50 - 5215104254757 z + 742308447359 z - 129959472179445 z 48 20 36 + 469415808880239 z + 29075769836272 z + 15496050030882112 z 34 66 64 30 - 14077253048762152 z - 713737 z + 21668735 z - 6515317536614402 z 42 44 46 - 6515317536614402 z + 3309414689721984 z - 1378793437431966 z 58 56 54 - 82759853550 z + 742308447359 z - 5215104254757 z 52 60 70 68 + 29075769836272 z + 7105316000 z - 189 z + 15312 z 32 38 40 + 10549941099859218 z - 14077253048762152 z + 10549941099859218 z 62 72 / 2 28 - 459269482 z + z ) / ((-1 + z ) (1 + 11065990075473302 z / 26 2 24 - 4468506554066244 z - 232 z + 1465053694679815 z 22 4 6 8 - 388148822068780 z + 21931 z - 1159856 z + 39250783 z 10 12 14 18 - 915727132 z + 15440836606 z - 194378322404 z - 13996658646676 z 16 50 48 + 1870203449551 z - 388148822068780 z + 1465053694679815 z 20 36 34 + 82582799259671 z + 54846298851642836 z - 49645098177378312 z 66 64 30 - 1159856 z + 39250783 z - 22329400638770012 z 42 44 46 - 22329400638770012 z + 11065990075473302 z - 4468506554066244 z 58 56 54 - 194378322404 z + 1870203449551 z - 13996658646676 z 52 60 70 68 + 82582799259671 z + 15440836606 z - 232 z + 21931 z 32 38 40 + 36806684483086986 z - 49645098177378312 z + 36806684483086986 z 62 72 - 915727132 z + z )) And in Maple-input format, it is: -(1+3309414689721984*z^28-1378793437431966*z^26-189*z^2+469415808880239*z^24-\ 129959472179445*z^22+15312*z^4-713737*z^6+21668735*z^8-459269482*z^10+ 7105316000*z^12-82759853550*z^14-5215104254757*z^18+742308447359*z^16-\ 129959472179445*z^50+469415808880239*z^48+29075769836272*z^20+15496050030882112 *z^36-14077253048762152*z^34-713737*z^66+21668735*z^64-6515317536614402*z^30-\ 6515317536614402*z^42+3309414689721984*z^44-1378793437431966*z^46-82759853550*z ^58+742308447359*z^56-5215104254757*z^54+29075769836272*z^52+7105316000*z^60-\ 189*z^70+15312*z^68+10549941099859218*z^32-14077253048762152*z^38+ 10549941099859218*z^40-459269482*z^62+z^72)/(-1+z^2)/(1+11065990075473302*z^28-\ 4468506554066244*z^26-232*z^2+1465053694679815*z^24-388148822068780*z^22+21931* z^4-1159856*z^6+39250783*z^8-915727132*z^10+15440836606*z^12-194378322404*z^14-\ 13996658646676*z^18+1870203449551*z^16-388148822068780*z^50+1465053694679815*z^ 48+82582799259671*z^20+54846298851642836*z^36-49645098177378312*z^34-1159856*z^ 66+39250783*z^64-22329400638770012*z^30-22329400638770012*z^42+ 11065990075473302*z^44-4468506554066244*z^46-194378322404*z^58+1870203449551*z^ 56-13996658646676*z^54+82582799259671*z^52+15440836606*z^60-232*z^70+21931*z^68 +36806684483086986*z^32-49645098177378312*z^38+36806684483086986*z^40-915727132 *z^62+z^72) The first , 40, terms are: [0, 44, 0, 3401, 0, 285311, 0, 24357824, 0, 2088923203, 0, 179384681027, 0, 15411236048864, 0, 1324209166903607, 0, 113789155459739033, 0, 9778112219336269164, 0, 840258924751731853337, 0, 72205932751962921904569, 0, 6204878889577164740276700, 0, 533204762074297273052432713, 0, 45819975132404593753078433959, 0, 3937456140143527508451197320976, 0, 338358141169831766643937361802307, 0, 29076192848850003764242455727900963, 0, 2498609890280836303237792040466204848, 0, 214713509412733930836959066206799519151] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 26720998550602203630 z - 4191442438461703573 z - 257 z 24 22 4 6 + 567072362053583464 z - 65777189655247423 z + 29930 z - 2115879 z 102 8 10 12 - 3650561553 z + 102574396 z - 3650561553 z + 99660720582 z 14 18 16 - 2152145946159 z - 540950648886749 z + 37604634673428 z 50 48 - 2890923188178674639450985 z + 1890203634595857067928352 z 20 36 + 6494219933427038 z + 10982748033267521656966 z 34 66 - 2984102683669603596785 z - 1093670589257179537311223 z 80 100 90 + 710265094818926977748 z + 99660720582 z - 65777189655247423 z 88 84 94 + 567072362053583464 z + 26720998550602203630 z - 540950648886749 z 86 96 98 - 4191442438461703573 z + 37604634673428 z - 2152145946159 z 92 82 + 6494219933427038 z - 147662366375286838803 z 64 112 110 106 + 1890203634595857067928352 z + z - 257 z - 2115879 z 108 30 42 + 29930 z - 147662366375286838803 z - 252981088199450493550257 z 44 46 + 559594945827349885016510 z - 1093670589257179537311223 z 58 56 - 4694878621602604395242263 z + 4987991200107739891930210 z 54 52 - 4694878621602604395242263 z + 3914633079857013027770522 z 60 70 + 3914633079857013027770522 z - 252981088199450493550257 z 68 78 + 559594945827349885016510 z - 2984102683669603596785 z 32 38 + 710265094818926977748 z - 35494786989599364746075 z 40 62 + 100936728975276223986344 z - 2890923188178674639450985 z 76 74 + 10982748033267521656966 z - 35494786989599364746075 z 72 104 / + 100936728975276223986344 z + 102574396 z ) / (-1 / 28 26 2 - 83210824935858606380 z + 12335603937008722450 z + 304 z 24 22 4 6 - 1576439092774088060 z + 172593339557122970 z - 40307 z + 3169068 z 102 8 10 12 + 190833168682 z - 168285186 z + 6493467368 z - 190833168682 z 14 18 16 + 4413794149299 z + 1259981526500361 z - 82299759480999 z 50 48 + 16790250216330327924063053 z - 10357285327245737286304407 z 20 36 - 16066525097127657 z - 42788353096238782328599 z 34 66 + 10993082619818050251171 z + 10357285327245737286304407 z 80 100 - 10993082619818050251171 z - 4413794149299 z 90 88 + 1576439092774088060 z - 12335603937008722450 z 84 94 - 486387385049710066829 z + 16066525097127657 z 86 96 98 + 83210824935858606380 z - 1259981526500361 z + 82299759480999 z 92 82 - 172593339557122970 z + 2474233821628092029677 z 64 112 114 110 - 16790250216330327924063053 z - 304 z + z + 40307 z 106 108 30 + 168285186 z - 3169068 z + 486387385049710066829 z 42 44 + 1167136058325488055782400 z - 2732980268847001828886694 z 46 58 + 5656452002295734700302039 z + 34610784368478984167204374 z 56 54 - 34610784368478984167204374 z + 30683028121144655440455566 z 52 60 - 24111110215456673534816693 z - 30683028121144655440455566 z 70 68 + 2732980268847001828886694 z - 5656452002295734700302039 z 78 32 + 42788353096238782328599 z - 2474233821628092029677 z 38 40 + 146268033043272727918488 z - 440045845786480594461174 z 62 76 + 24111110215456673534816693 z - 146268033043272727918488 z 74 72 + 440045845786480594461174 z - 1167136058325488055782400 z 104 - 6493467368 z ) And in Maple-input format, it is: -(1+26720998550602203630*z^28-4191442438461703573*z^26-257*z^2+ 567072362053583464*z^24-65777189655247423*z^22+29930*z^4-2115879*z^6-3650561553 *z^102+102574396*z^8-3650561553*z^10+99660720582*z^12-2152145946159*z^14-\ 540950648886749*z^18+37604634673428*z^16-2890923188178674639450985*z^50+ 1890203634595857067928352*z^48+6494219933427038*z^20+10982748033267521656966*z^ 36-2984102683669603596785*z^34-1093670589257179537311223*z^66+ 710265094818926977748*z^80+99660720582*z^100-65777189655247423*z^90+ 567072362053583464*z^88+26720998550602203630*z^84-540950648886749*z^94-\ 4191442438461703573*z^86+37604634673428*z^96-2152145946159*z^98+ 6494219933427038*z^92-147662366375286838803*z^82+1890203634595857067928352*z^64 +z^112-257*z^110-2115879*z^106+29930*z^108-147662366375286838803*z^30-\ 252981088199450493550257*z^42+559594945827349885016510*z^44-\ 1093670589257179537311223*z^46-4694878621602604395242263*z^58+ 4987991200107739891930210*z^56-4694878621602604395242263*z^54+ 3914633079857013027770522*z^52+3914633079857013027770522*z^60-\ 252981088199450493550257*z^70+559594945827349885016510*z^68-\ 2984102683669603596785*z^78+710265094818926977748*z^32-35494786989599364746075* z^38+100936728975276223986344*z^40-2890923188178674639450985*z^62+ 10982748033267521656966*z^76-35494786989599364746075*z^74+ 100936728975276223986344*z^72+102574396*z^104)/(-1-83210824935858606380*z^28+ 12335603937008722450*z^26+304*z^2-1576439092774088060*z^24+172593339557122970*z ^22-40307*z^4+3169068*z^6+190833168682*z^102-168285186*z^8+6493467368*z^10-\ 190833168682*z^12+4413794149299*z^14+1259981526500361*z^18-82299759480999*z^16+ 16790250216330327924063053*z^50-10357285327245737286304407*z^48-\ 16066525097127657*z^20-42788353096238782328599*z^36+10993082619818050251171*z^ 34+10357285327245737286304407*z^66-10993082619818050251171*z^80-4413794149299*z ^100+1576439092774088060*z^90-12335603937008722450*z^88-486387385049710066829*z ^84+16066525097127657*z^94+83210824935858606380*z^86-1259981526500361*z^96+ 82299759480999*z^98-172593339557122970*z^92+2474233821628092029677*z^82-\ 16790250216330327924063053*z^64-304*z^112+z^114+40307*z^110+168285186*z^106-\ 3169068*z^108+486387385049710066829*z^30+1167136058325488055782400*z^42-\ 2732980268847001828886694*z^44+5656452002295734700302039*z^46+ 34610784368478984167204374*z^58-34610784368478984167204374*z^56+ 30683028121144655440455566*z^54-24111110215456673534816693*z^52-\ 30683028121144655440455566*z^60+2732980268847001828886694*z^70-\ 5656452002295734700302039*z^68+42788353096238782328599*z^78-\ 2474233821628092029677*z^32+146268033043272727918488*z^38-\ 440045845786480594461174*z^40+24111110215456673534816693*z^62-\ 146268033043272727918488*z^76+440045845786480594461174*z^74-\ 1167136058325488055782400*z^72-6493467368*z^104) The first , 40, terms are: [0, 47, 0, 3911, 0, 347704, 0, 31296745, 0, 2827032373, 0, 255694716299, 0, 23138520818435, 0, 2094311928876493, 0, 189576969622333361, 0, 17161132642118313288, 0, 1553506707136531903983, 0, 140631689308517302575311, 0, 12730764081131880063599873, 0, 1152461060809297410188563985, 0, 104327372086732914352724244767, 0, 9444312905990365738221359647407, 0, 854953522452380197927686532771720, 0, 77395312981248187068723097071043697, 0, 7006269241798259200007235538666933677, 0, 634247826268684626441771882675063352931] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 289446136696407406 z - 73805024004951841 z - 219 z 24 22 4 6 + 15810533000102586 z - 2829711552838331 z + 21125 z - 1205206 z 8 10 12 14 + 46007375 z - 1258901001 z + 25804974180 z - 408572712255 z 18 16 50 - 51356636613883 z + 5110162522412 z - 22437772085794418805 z 48 20 + 33159155473085805861 z + 420254140701950 z 36 34 + 12969919476937336557 z - 6395652712821258148 z 66 80 88 84 86 - 2829711552838331 z + 46007375 z + z + 21125 z - 219 z 82 64 30 - 1205206 z + 15810533000102586 z - 957969955260251569 z 42 44 - 41900348422627361894 z + 45296286890168672060 z 46 58 - 41900348422627361894 z - 957969955260251569 z 56 54 + 2685529223573491141 z - 6395652712821258148 z 52 60 70 + 12969919476937336557 z + 289446136696407406 z - 51356636613883 z 68 78 32 + 420254140701950 z - 1258901001 z + 2685529223573491141 z 38 40 - 22437772085794418805 z + 33159155473085805861 z 62 76 74 - 73805024004951841 z + 25804974180 z - 408572712255 z 72 / 28 + 5110162522412 z ) / (-1 - 1098058627883812474 z / 26 2 24 + 261423606037342887 z + 261 z - 52265927898920431 z 22 4 6 8 + 8722401026385640 z - 29060 z + 1874928 z - 79804810 z 10 12 14 + 2409811086 z - 54086288218 z + 932003643314 z 18 16 50 + 136955905039189 z - 12626920042805 z + 191811528197605154407 z 48 20 - 261089254965091015571 z - 1206148283871008 z 36 34 - 64937306681406871708 z + 29843995473239618774 z 66 80 90 88 84 + 52265927898920431 z - 2409811086 z + z - 261 z - 1874928 z 86 82 64 + 29060 z + 79804810 z - 261423606037342887 z 30 42 + 3892215722681856866 z + 261089254965091015571 z 44 46 - 304556536124972661768 z + 304556536124972661768 z 58 56 + 11689678713078977042 z - 29843995473239618774 z 54 52 + 64937306681406871708 z - 120672074555081491928 z 60 70 68 - 3892215722681856866 z + 1206148283871008 z - 8722401026385640 z 78 32 38 + 54086288218 z - 11689678713078977042 z + 120672074555081491928 z 40 62 76 - 191811528197605154407 z + 1098058627883812474 z - 932003643314 z 74 72 + 12626920042805 z - 136955905039189 z ) And in Maple-input format, it is: -(1+289446136696407406*z^28-73805024004951841*z^26-219*z^2+15810533000102586*z^ 24-2829711552838331*z^22+21125*z^4-1205206*z^6+46007375*z^8-1258901001*z^10+ 25804974180*z^12-408572712255*z^14-51356636613883*z^18+5110162522412*z^16-\ 22437772085794418805*z^50+33159155473085805861*z^48+420254140701950*z^20+ 12969919476937336557*z^36-6395652712821258148*z^34-2829711552838331*z^66+ 46007375*z^80+z^88+21125*z^84-219*z^86-1205206*z^82+15810533000102586*z^64-\ 957969955260251569*z^30-41900348422627361894*z^42+45296286890168672060*z^44-\ 41900348422627361894*z^46-957969955260251569*z^58+2685529223573491141*z^56-\ 6395652712821258148*z^54+12969919476937336557*z^52+289446136696407406*z^60-\ 51356636613883*z^70+420254140701950*z^68-1258901001*z^78+2685529223573491141*z^ 32-22437772085794418805*z^38+33159155473085805861*z^40-73805024004951841*z^62+ 25804974180*z^76-408572712255*z^74+5110162522412*z^72)/(-1-1098058627883812474* z^28+261423606037342887*z^26+261*z^2-52265927898920431*z^24+8722401026385640*z^ 22-29060*z^4+1874928*z^6-79804810*z^8+2409811086*z^10-54086288218*z^12+ 932003643314*z^14+136955905039189*z^18-12626920042805*z^16+ 191811528197605154407*z^50-261089254965091015571*z^48-1206148283871008*z^20-\ 64937306681406871708*z^36+29843995473239618774*z^34+52265927898920431*z^66-\ 2409811086*z^80+z^90-261*z^88-1874928*z^84+29060*z^86+79804810*z^82-\ 261423606037342887*z^64+3892215722681856866*z^30+261089254965091015571*z^42-\ 304556536124972661768*z^44+304556536124972661768*z^46+11689678713078977042*z^58 -29843995473239618774*z^56+64937306681406871708*z^54-120672074555081491928*z^52 -3892215722681856866*z^60+1206148283871008*z^70-8722401026385640*z^68+ 54086288218*z^78-11689678713078977042*z^32+120672074555081491928*z^38-\ 191811528197605154407*z^40+1098058627883812474*z^62-932003643314*z^76+ 12626920042805*z^74-136955905039189*z^72) The first , 40, terms are: [0, 42, 0, 3027, 0, 239249, 0, 19428910, 0, 1592884691, 0, 131075020527, 0, 10802242601102, 0, 890821952820473, 0, 73484081659569631, 0, 6062508438508403474, 0, 500192847393112366233, 0, 41270018631308260415001, 0, 3405159673479519886240930, 0, 280958991269789351314816127, 0, 23181931982119056602797949657, 0, 1912744238746713366487707995790, 0, 157820877055025177579754219340927, 0, 13021833322044441557401398430067779, 0, 1074434321350759359990777900466073678, 0, 88651817223629138221069793117356543729] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 7595594171048 z - 6686829085654 z - 166 z 24 22 4 6 + 4558795829796 z - 2401003330652 z + 11274 z - 419831 z 8 10 12 14 + 9705276 z - 149478835 z + 1605699218 z - 12416116158 z 18 16 50 48 - 301263423888 z + 70690379589 z - 419831 z + 9705276 z 20 36 34 + 972763157818 z + 972763157818 z - 2401003330652 z 30 42 44 46 - 6686829085654 z - 12416116158 z + 1605699218 z - 149478835 z 56 54 52 32 38 + z - 166 z + 11274 z + 4558795829796 z - 301263423888 z 40 / 2 28 + 70690379589 z ) / ((-1 + z ) (1 + 27613398122734 z / 26 2 24 22 - 24122563162444 z - 212 z + 16074144349862 z - 8157457413676 z 4 6 8 10 12 + 17065 z - 727172 z + 18889709 z - 323342840 z + 3827846905 z 14 18 16 50 - 32364563208 z - 915219893828 z + 199832260977 z - 727172 z 48 20 36 34 + 18889709 z + 3143414203030 z + 3143414203030 z - 8157457413676 z 30 42 44 46 - 24122563162444 z - 32364563208 z + 3827846905 z - 323342840 z 56 54 52 32 38 + z - 212 z + 17065 z + 16074144349862 z - 915219893828 z 40 + 199832260977 z )) And in Maple-input format, it is: -(1+7595594171048*z^28-6686829085654*z^26-166*z^2+4558795829796*z^24-\ 2401003330652*z^22+11274*z^4-419831*z^6+9705276*z^8-149478835*z^10+1605699218*z ^12-12416116158*z^14-301263423888*z^18+70690379589*z^16-419831*z^50+9705276*z^ 48+972763157818*z^20+972763157818*z^36-2401003330652*z^34-6686829085654*z^30-\ 12416116158*z^42+1605699218*z^44-149478835*z^46+z^56-166*z^54+11274*z^52+ 4558795829796*z^32-301263423888*z^38+70690379589*z^40)/(-1+z^2)/(1+ 27613398122734*z^28-24122563162444*z^26-212*z^2+16074144349862*z^24-\ 8157457413676*z^22+17065*z^4-727172*z^6+18889709*z^8-323342840*z^10+3827846905* z^12-32364563208*z^14-915219893828*z^18+199832260977*z^16-727172*z^50+18889709* z^48+3143414203030*z^20+3143414203030*z^36-8157457413676*z^34-24122563162444*z^ 30-32364563208*z^42+3827846905*z^44-323342840*z^46+z^56-212*z^54+17065*z^52+ 16074144349862*z^32-915219893828*z^38+199832260977*z^40) The first , 40, terms are: [0, 47, 0, 4008, 0, 366091, 0, 33798701, 0, 3127831309, 0, 289661359435, 0, 26831346720360, 0, 2485610063351551, 0, 230270434900228689, 0, 21332855393065144209, 0, 1976341398672803189247, 0, 183094705298881442303208, 0, 16962502636046579095784939, 0, 1571463132024410947563342253, 0, 145585631174630514669077871181, 0, 13487543229516658005416315585451, 0, 1249531454226738158102149253181736, 0, 115760805266028697149066130866537583, 0, 10724471180353106450373571697052535425, 0, 993551158761379288101786642340285804801] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 4}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 30847316612 z + 58839117195 z + 158 z - 81122352971 z 22 4 6 8 10 + 81122352971 z - 9857 z + 320463 z - 6106117 z + 73124197 z 12 14 18 16 - 576765000 z + 3097888923 z + 30847316612 z - 11606027541 z 20 36 34 30 - 58839117195 z - 73124197 z + 576765000 z + 11606027541 z 42 44 46 32 38 40 + 9857 z - 158 z + z - 3097888923 z + 6106117 z - 320463 z ) / 28 26 2 24 / (1 + 320015352266 z - 516139325720 z - 210 z + 605073706153 z / 22 4 6 8 10 - 516139325720 z + 16186 z - 622556 z + 13751389 z - 189605072 z 12 14 18 16 + 1722258190 z - 10687863978 z - 143718955526 z + 46448171478 z 48 20 36 34 + z + 320015352266 z + 1722258190 z - 10687863978 z 30 42 44 46 32 - 143718955526 z - 622556 z + 16186 z - 210 z + 46448171478 z 38 40 - 189605072 z + 13751389 z ) And in Maple-input format, it is: -(-1-30847316612*z^28+58839117195*z^26+158*z^2-81122352971*z^24+81122352971*z^ 22-9857*z^4+320463*z^6-6106117*z^8+73124197*z^10-576765000*z^12+3097888923*z^14 +30847316612*z^18-11606027541*z^16-58839117195*z^20-73124197*z^36+576765000*z^ 34+11606027541*z^30+9857*z^42-158*z^44+z^46-3097888923*z^32+6106117*z^38-320463 *z^40)/(1+320015352266*z^28-516139325720*z^26-210*z^2+605073706153*z^24-\ 516139325720*z^22+16186*z^4-622556*z^6+13751389*z^8-189605072*z^10+1722258190*z ^12-10687863978*z^14-143718955526*z^18+46448171478*z^16+z^48+320015352266*z^20+ 1722258190*z^36-10687863978*z^34-143718955526*z^30-622556*z^42+16186*z^44-210*z ^46+46448171478*z^32-189605072*z^38+13751389*z^40) The first , 40, terms are: [0, 52, 0, 4591, 0, 424531, 0, 39569224, 0, 3697641517, 0, 345912923797, 0, 32376364730920, 0, 3031054399774135, 0, 283798182846535543, 0, 26573563293382713172, 0, 2488294407980378793493, 0, 233001896739555426215653, 0, 21818250790900621395971764, 0, 2043062975715960834781941463, 0, 191312890936890535172800580311, 0, 17914596648473342301948253189096, 0, 1677528817602239219157928922831605, 0, 157084386423440211568083466836669373, 0, 14709438241510508822154485879244197224, 0, 1377397125105758442562264955473798704163] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 4}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 445529787879704 z + 277472170052966 z + 203 z 24 22 4 6 - 135804190146218 z + 51968575477500 z - 17494 z + 848756 z 8 10 12 14 - 25989084 z + 534905794 z - 7710030319 z + 80137585397 z 18 16 50 48 + 3530309786688 z - 614088047380 z + 7710030319 z - 80137585397 z 20 36 34 - 15441084452720 z - 277472170052966 z + 445529787879704 z 30 42 44 + 564083383681536 z + 15441084452720 z - 3530309786688 z 46 58 56 54 52 + 614088047380 z + 17494 z - 848756 z + 25989084 z - 534905794 z 60 32 38 40 - 203 z - 564083383681536 z + 135804190146218 z - 51968575477500 z 62 / 28 26 2 + z ) / (1 + 2638760827508370 z - 1464394485861054 z - 259 z / 24 22 4 6 + 639150316218866 z - 218337103127848 z + 26625 z - 1485106 z 8 10 12 14 + 51244710 z - 1176722722 z + 18846962797 z - 217553506303 z 18 16 50 - 11874602818824 z + 1853782071181 z - 217553506303 z 48 20 36 + 1853782071181 z + 57993200735128 z + 2638760827508370 z 34 64 30 42 - 3752140049350460 z + z - 3752140049350460 z - 218337103127848 z 44 46 58 56 + 57993200735128 z - 11874602818824 z - 1485106 z + 51244710 z 54 52 60 32 - 1176722722 z + 18846962797 z + 26625 z + 4218375897085892 z 38 40 62 - 1464394485861054 z + 639150316218866 z - 259 z ) And in Maple-input format, it is: -(-1-445529787879704*z^28+277472170052966*z^26+203*z^2-135804190146218*z^24+ 51968575477500*z^22-17494*z^4+848756*z^6-25989084*z^8+534905794*z^10-7710030319 *z^12+80137585397*z^14+3530309786688*z^18-614088047380*z^16+7710030319*z^50-\ 80137585397*z^48-15441084452720*z^20-277472170052966*z^36+445529787879704*z^34+ 564083383681536*z^30+15441084452720*z^42-3530309786688*z^44+614088047380*z^46+ 17494*z^58-848756*z^56+25989084*z^54-534905794*z^52-203*z^60-564083383681536*z^ 32+135804190146218*z^38-51968575477500*z^40+z^62)/(1+2638760827508370*z^28-\ 1464394485861054*z^26-259*z^2+639150316218866*z^24-218337103127848*z^22+26625*z ^4-1485106*z^6+51244710*z^8-1176722722*z^10+18846962797*z^12-217553506303*z^14-\ 11874602818824*z^18+1853782071181*z^16-217553506303*z^50+1853782071181*z^48+ 57993200735128*z^20+2638760827508370*z^36-3752140049350460*z^34+z^64-\ 3752140049350460*z^30-218337103127848*z^42+57993200735128*z^44-11874602818824*z ^46-1485106*z^58+51244710*z^56-1176722722*z^54+18846962797*z^52+26625*z^60+ 4218375897085892*z^32-1464394485861054*z^38+639150316218866*z^40-259*z^62) The first , 40, terms are: [0, 56, 0, 5373, 0, 536957, 0, 53926048, 0, 5421954013, 0, 545364836933, 0, 54864176052720, 0, 5519762633395493, 0, 555347410007547301, 0, 55874626402869667048, 0, 5621690211496093598489, 0, 565614098105898828228681, 0, 56908087758141297554022536, 0, 5725691635512922843339488085, 0, 576078944198619061049143364245, 0, 57961028855728742975117922275792, 0, 5831633014168111667142451778531349, 0, 586738103639159665644177729609822029, 0, 59033482356658102971117147119402356672, 0, 5939535932718622008514894675707080023693] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 96537219668935076 z + 26279395124953118 z + 202 z 24 22 4 6 - 6014888050153492 z + 1151800199903212 z - 17846 z + 930027 z 8 10 12 14 - 32440404 z + 812795942 z - 15301877016 z + 223282159000 z 18 16 50 + 24082301468940 z - 2582668647418 z + 3318051563514185138 z 48 20 36 - 5352582185715455996 z - 183377365811378 z - 3318051563514185138 z 34 66 80 84 + 1751250881802749736 z + 183377365811378 z - 930027 z - 202 z 86 82 64 30 + z + 17846 z - 1151800199903212 z + 299342642475368110 z 42 44 + 8626917254690540288 z - 8626917254690540288 z 46 58 56 + 7358688771946950596 z + 96537219668935076 z - 299342642475368110 z 54 52 60 + 785911839735496360 z - 1751250881802749736 z - 26279395124953118 z 70 68 78 + 2582668647418 z - 24082301468940 z + 32440404 z 32 38 - 785911839735496360 z + 5352582185715455996 z 40 62 76 - 7358688771946950596 z + 6014888050153492 z - 812795942 z 74 72 / 28 + 15301877016 z - 223282159000 z ) / (1 + 371420716969618264 z / 26 2 24 - 94347250090109704 z - 241 z + 20146531591253804 z 22 4 6 8 - 3596814208956768 z + 24701 z - 1461661 z + 56987949 z 10 12 14 18 - 1577563676 z + 32530793436 z - 516548140952 z - 65085291596116 z 16 50 48 + 6469796917348 z - 29388633886589343476 z + 43545582589428874506 z 20 36 34 + 533237230211568 z + 16929515009123928020 z - 8314642335613018920 z 66 80 88 84 86 - 3596814208956768 z + 56987949 z + z + 24701 z - 241 z 82 64 30 - 1461661 z + 20146531591253804 z - 1234499986560437532 z 42 44 - 55115536917286104970 z + 59616011671976143406 z 46 58 - 55115536917286104970 z - 1234499986560437532 z 56 54 + 3476112371072941164 z - 8314642335613018920 z 52 60 70 + 16929515009123928020 z + 371420716969618264 z - 65085291596116 z 68 78 32 + 533237230211568 z - 1577563676 z + 3476112371072941164 z 38 40 - 29388633886589343476 z + 43545582589428874506 z 62 76 74 - 94347250090109704 z + 32530793436 z - 516548140952 z 72 + 6469796917348 z ) And in Maple-input format, it is: -(-1-96537219668935076*z^28+26279395124953118*z^26+202*z^2-6014888050153492*z^ 24+1151800199903212*z^22-17846*z^4+930027*z^6-32440404*z^8+812795942*z^10-\ 15301877016*z^12+223282159000*z^14+24082301468940*z^18-2582668647418*z^16+ 3318051563514185138*z^50-5352582185715455996*z^48-183377365811378*z^20-\ 3318051563514185138*z^36+1751250881802749736*z^34+183377365811378*z^66-930027*z ^80-202*z^84+z^86+17846*z^82-1151800199903212*z^64+299342642475368110*z^30+ 8626917254690540288*z^42-8626917254690540288*z^44+7358688771946950596*z^46+ 96537219668935076*z^58-299342642475368110*z^56+785911839735496360*z^54-\ 1751250881802749736*z^52-26279395124953118*z^60+2582668647418*z^70-\ 24082301468940*z^68+32440404*z^78-785911839735496360*z^32+5352582185715455996*z ^38-7358688771946950596*z^40+6014888050153492*z^62-812795942*z^76+15301877016*z ^74-223282159000*z^72)/(1+371420716969618264*z^28-94347250090109704*z^26-241*z^ 2+20146531591253804*z^24-3596814208956768*z^22+24701*z^4-1461661*z^6+56987949*z ^8-1577563676*z^10+32530793436*z^12-516548140952*z^14-65085291596116*z^18+ 6469796917348*z^16-29388633886589343476*z^50+43545582589428874506*z^48+ 533237230211568*z^20+16929515009123928020*z^36-8314642335613018920*z^34-\ 3596814208956768*z^66+56987949*z^80+z^88+24701*z^84-241*z^86-1461661*z^82+ 20146531591253804*z^64-1234499986560437532*z^30-55115536917286104970*z^42+ 59616011671976143406*z^44-55115536917286104970*z^46-1234499986560437532*z^58+ 3476112371072941164*z^56-8314642335613018920*z^54+16929515009123928020*z^52+ 371420716969618264*z^60-65085291596116*z^70+533237230211568*z^68-1577563676*z^ 78+3476112371072941164*z^32-29388633886589343476*z^38+43545582589428874506*z^40 -94347250090109704*z^62+32530793436*z^76-516548140952*z^74+6469796917348*z^72) The first , 40, terms are: [0, 39, 0, 2544, 0, 181399, 0, 13335049, 0, 993713417, 0, 74558456575, 0, 5614524046688, 0, 423635297862607, 0, 31999861877422993, 0, 2418625617651669297, 0, 182867401448683374991, 0, 13828840928960657433216, 0, 1045877486350341872110047, 0, 79104501191539558067547465, 0, 5983229838584230573969305481, 0, 452561940009095007368361312855, 0, 34231406090239064640795374997072, 0, 2589249384641492306527384534162503, 0, 195850361847263419782920471912213025, 0, 14814112147963319277363623440852915681] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 62451697395918 z - 36813559162310 z - 157 z 24 22 4 6 + 17513348218626 z - 6701576132588 z + 10182 z - 370985 z 8 10 12 14 + 8670145 z - 140004250 z + 1637573662 z - 14327624596 z 18 16 50 48 - 499958532676 z + 95940162472 z - 14327624596 z + 95940162472 z 20 36 34 64 + 2052751315520 z + 62451697395918 z - 85687654306482 z + z 30 42 44 - 85687654306482 z - 6701576132588 z + 2052751315520 z 46 58 56 54 - 499958532676 z - 370985 z + 8670145 z - 140004250 z 52 60 32 38 + 1637573662 z + 10182 z + 95203666050620 z - 36813559162310 z 40 62 / 28 + 17513348218626 z - 157 z ) / (-1 - 341165923275475 z / 26 2 24 22 + 182469566235907 z + 200 z - 78900885248389 z + 27475252667211 z 4 6 8 10 12 - 15583 z + 659630 z - 17548981 z + 318352716 z - 4146196973 z 14 18 16 + 40151434107 z + 1698900674615 z - 296414069877 z 50 48 20 + 296414069877 z - 1698900674615 z - 7662810853401 z 36 34 66 64 - 517084598456867 z + 636341599754627 z + z - 200 z 30 42 44 + 517084598456867 z + 78900885248389 z - 27475252667211 z 46 58 56 54 + 7662810853401 z + 17548981 z - 318352716 z + 4146196973 z 52 60 32 38 - 40151434107 z - 659630 z - 636341599754627 z + 341165923275475 z 40 62 - 182469566235907 z + 15583 z ) And in Maple-input format, it is: -(1+62451697395918*z^28-36813559162310*z^26-157*z^2+17513348218626*z^24-\ 6701576132588*z^22+10182*z^4-370985*z^6+8670145*z^8-140004250*z^10+1637573662*z ^12-14327624596*z^14-499958532676*z^18+95940162472*z^16-14327624596*z^50+ 95940162472*z^48+2052751315520*z^20+62451697395918*z^36-85687654306482*z^34+z^ 64-85687654306482*z^30-6701576132588*z^42+2052751315520*z^44-499958532676*z^46-\ 370985*z^58+8670145*z^56-140004250*z^54+1637573662*z^52+10182*z^60+ 95203666050620*z^32-36813559162310*z^38+17513348218626*z^40-157*z^62)/(-1-\ 341165923275475*z^28+182469566235907*z^26+200*z^2-78900885248389*z^24+ 27475252667211*z^22-15583*z^4+659630*z^6-17548981*z^8+318352716*z^10-4146196973 *z^12+40151434107*z^14+1698900674615*z^18-296414069877*z^16+296414069877*z^50-\ 1698900674615*z^48-7662810853401*z^20-517084598456867*z^36+636341599754627*z^34 +z^66-200*z^64+517084598456867*z^30+78900885248389*z^42-27475252667211*z^44+ 7662810853401*z^46+17548981*z^58-318352716*z^56+4146196973*z^54-40151434107*z^ 52-659630*z^60-636341599754627*z^32+341165923275475*z^38-182469566235907*z^40+ 15583*z^62) The first , 40, terms are: [0, 43, 0, 3199, 0, 258376, 0, 21310437, 0, 1769712845, 0, 147335943367, 0, 12278469131375, 0, 1023654105711477, 0, 85355843182738605, 0, 7117746631972001224, 0, 593558962408294441079, 0, 49498290613658808593747, 0, 4127799556080782690353657, 0, 344229315319057311737214057, 0, 28706315881884074613493743619, 0, 2393906655588268726289797764231, 0, 199635156867020146770343026867272, 0, 16648183863829848479004485517909597, 0, 1388342799563533827407791771646108517, 0, 115778139158787530587865143036277929663] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 30858384144786429292 z - 4824333493960128988 z - 260 z 24 22 4 6 + 649839720589486680 z - 74957994681695688 z + 30686 z - 2200624 z 102 8 10 12 - 3908681544 z + 108257879 z - 3908681544 z + 108172500298 z 14 18 16 - 2365405000448 z - 607149039659792 z + 41796410138917 z 50 48 - 3350913914726220405368604 z + 2191767947795281773648930 z 20 36 + 7349755731520600 z + 12754673152413790733428 z 34 66 - 3463733914979314474960 z - 1268691787446011707816480 z 80 100 90 + 823583509588536879556 z + 108172500298 z - 74957994681695688 z 88 84 94 + 649839720589486680 z + 30858384144786429292 z - 607149039659792 z 86 96 98 - 4824333493960128988 z + 41796410138917 z - 2365405000448 z 92 82 + 7349755731520600 z - 170938730992474986176 z 64 112 110 106 + 2191767947795281773648930 z + z - 260 z - 2200624 z 108 30 42 + 30686 z - 170938730992474986176 z - 293713340603547322218696 z 44 46 + 649432582358440374521240 z - 1268691787446011707816480 z 58 56 - 5439415245674944024270968 z + 5778663967920338907746538 z 54 52 - 5439415245674944024270968 z + 4536234968479541330285784 z 60 70 + 4536234968479541330285784 z - 293713340603547322218696 z 68 78 + 649432582358440374521240 z - 3463733914979314474960 z 32 38 + 823583509588536879556 z - 41227439725018731362756 z 40 62 + 117224045851762153503744 z - 3350913914726220405368604 z 76 74 + 12754673152413790733428 z - 41227439725018731362756 z 72 104 / 2 + 117224045851762153503744 z + 108257879 z ) / ((-1 + z ) (1 / 28 26 2 + 84166151828393273422 z - 12674692786247909811 z - 307 z 24 22 4 6 + 1640460635755500852 z - 181345324782687481 z + 41104 z - 3263427 z 102 8 10 12 - 6806110946 z + 174924091 z - 6806110946 z + 201367455814 z 14 18 16 - 4678818203568 z - 1337486636221779 z + 87422506684037 z 50 48 - 11757591913777822733525281 z + 7613772332356511170980682 z 20 36 + 16993336258308658 z + 39462616075631445001218 z 34 66 - 10420931153398366601923 z - 4351224852735003870459457 z 80 100 90 + 2403760171927740693648 z + 201367455814 z - 181345324782687481 z 88 84 + 1640460635755500852 z + 84166151828393273422 z 94 86 96 - 1337486636221779 z - 12674692786247909811 z + 87422506684037 z 98 92 82 - 4678818203568 z + 16993336258308658 z - 482865879550453899745 z 64 112 110 106 + 7613772332356511170980682 z + z - 307 z - 3263427 z 108 30 42 + 41104 z - 482865879550453899745 z - 974075572667145510459267 z 44 46 + 2193144988886979465961182 z - 4351224852735003870459457 z 58 56 - 19307062886927424540388871 z + 20540948164100919302676906 z 54 52 - 19307062886927424540388871 z + 16031581148251201433886906 z 60 70 + 16031581148251201433886906 z - 974075572667145510459267 z 68 78 + 2193144988886979465961182 z - 10420931153398366601923 z 32 38 + 2403760171927740693648 z - 130863924293761129638553 z 40 62 + 380811053328320884815292 z - 11757591913777822733525281 z 76 74 + 39462616075631445001218 z - 130863924293761129638553 z 72 104 + 380811053328320884815292 z + 174924091 z )) And in Maple-input format, it is: -(1+30858384144786429292*z^28-4824333493960128988*z^26-260*z^2+ 649839720589486680*z^24-74957994681695688*z^22+30686*z^4-2200624*z^6-3908681544 *z^102+108257879*z^8-3908681544*z^10+108172500298*z^12-2365405000448*z^14-\ 607149039659792*z^18+41796410138917*z^16-3350913914726220405368604*z^50+ 2191767947795281773648930*z^48+7349755731520600*z^20+12754673152413790733428*z^ 36-3463733914979314474960*z^34-1268691787446011707816480*z^66+ 823583509588536879556*z^80+108172500298*z^100-74957994681695688*z^90+ 649839720589486680*z^88+30858384144786429292*z^84-607149039659792*z^94-\ 4824333493960128988*z^86+41796410138917*z^96-2365405000448*z^98+ 7349755731520600*z^92-170938730992474986176*z^82+2191767947795281773648930*z^64 +z^112-260*z^110-2200624*z^106+30686*z^108-170938730992474986176*z^30-\ 293713340603547322218696*z^42+649432582358440374521240*z^44-\ 1268691787446011707816480*z^46-5439415245674944024270968*z^58+ 5778663967920338907746538*z^56-5439415245674944024270968*z^54+ 4536234968479541330285784*z^52+4536234968479541330285784*z^60-\ 293713340603547322218696*z^70+649432582358440374521240*z^68-\ 3463733914979314474960*z^78+823583509588536879556*z^32-41227439725018731362756* z^38+117224045851762153503744*z^40-3350913914726220405368604*z^62+ 12754673152413790733428*z^76-41227439725018731362756*z^74+ 117224045851762153503744*z^72+108257879*z^104)/(-1+z^2)/(1+84166151828393273422 *z^28-12674692786247909811*z^26-307*z^2+1640460635755500852*z^24-\ 181345324782687481*z^22+41104*z^4-3263427*z^6-6806110946*z^102+174924091*z^8-\ 6806110946*z^10+201367455814*z^12-4678818203568*z^14-1337486636221779*z^18+ 87422506684037*z^16-11757591913777822733525281*z^50+7613772332356511170980682*z ^48+16993336258308658*z^20+39462616075631445001218*z^36-10420931153398366601923 *z^34-4351224852735003870459457*z^66+2403760171927740693648*z^80+201367455814*z ^100-181345324782687481*z^90+1640460635755500852*z^88+84166151828393273422*z^84 -1337486636221779*z^94-12674692786247909811*z^86+87422506684037*z^96-\ 4678818203568*z^98+16993336258308658*z^92-482865879550453899745*z^82+ 7613772332356511170980682*z^64+z^112-307*z^110-3263427*z^106+41104*z^108-\ 482865879550453899745*z^30-974075572667145510459267*z^42+ 2193144988886979465961182*z^44-4351224852735003870459457*z^46-\ 19307062886927424540388871*z^58+20540948164100919302676906*z^56-\ 19307062886927424540388871*z^54+16031581148251201433886906*z^52+ 16031581148251201433886906*z^60-974075572667145510459267*z^70+ 2193144988886979465961182*z^68-10420931153398366601923*z^78+ 2403760171927740693648*z^32-130863924293761129638553*z^38+ 380811053328320884815292*z^40-11757591913777822733525281*z^62+ 39462616075631445001218*z^76-130863924293761129638553*z^74+ 380811053328320884815292*z^72+174924091*z^104) The first , 40, terms are: [0, 48, 0, 4059, 0, 366351, 0, 33436708, 0, 3059988761, 0, 280272739533, 0, 25678742581160, 0, 2352978666224359, 0, 215616979793842595, 0, 19758613488276604372, 0, 1810645920881697549861, 0, 165925100224594715703133, 0, 15205169879198217729255532, 0, 1393383717673175612573389123, 0, 127688063065900922512970544583, 0, 11701186864655385330902542374640, 0, 1072283322371656770639635281752853, 0, 98262813832509377686585425133024081, 0, 9004691636395852156424169511540604796, 0, 825179623903019700477340848259686294063] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 23675528170108929968 z - 3782139931450826896 z - 260 z 24 22 4 6 + 520986191666452464 z - 61497194129154808 z + 30474 z - 2157912 z 102 8 10 12 - 3693148448 z + 104365239 z - 3693148448 z + 99970161034 z 14 18 16 - 2135580384764 z - 522409084823680 z + 36841675526741 z 50 48 - 2195516397486638140409360 z + 1445613024038046286243266 z 20 36 + 6174005996343012 z + 9065469583068800382432 z 34 66 - 2505203443882071432976 z - 843873373254198476716080 z 80 100 90 + 606866372507232590988 z + 99970161034 z - 61497194129154808 z 88 84 94 + 520986191666452464 z + 23675528170108929968 z - 522409084823680 z 86 96 98 - 3782139931450826896 z + 36841675526741 z - 2135580384764 z 92 82 + 6174005996343012 z - 128464132117952324256 z 64 112 110 106 + 1445613024038046286243266 z + z - 260 z - 2157912 z 108 30 42 + 30474 z - 128464132117952324256 z - 199702216273459320610520 z 44 46 + 436382765993155178290716 z - 843873373254198476716080 z 58 56 - 3536643864698680408412376 z + 3753590046679255800990842 z 54 52 - 3536643864698680408412376 z + 2957943506481952275136604 z 60 70 + 2957943506481952275136604 z - 199702216273459320610520 z 68 78 + 436382765993155178290716 z - 2505203443882071432976 z 32 38 + 606866372507232590988 z - 28832378735113183921888 z 40 62 + 80774883652097150937336 z - 2195516397486638140409360 z 76 74 + 9065469583068800382432 z - 28832378735113183921888 z 72 104 / + 80774883652097150937336 z + 104365239 z ) / (-1 / 28 26 2 - 74909238762216411317 z + 11343058581833654903 z + 300 z 24 22 4 6 - 1478705139822854665 z + 164850195665425775 z - 39807 z + 3147287 z 102 8 10 12 + 191434749836 z - 168103738 z + 6510615137 z - 191434749836 z 14 18 16 + 4413987149182 z + 1238963924303972 z - 81751483971461 z 50 48 + 12278624498750184871544675 z - 7656890705056043167181039 z 20 36 - 15591237265836897 z - 35217456884638791900053 z 34 66 + 9251989514876365435107 z + 7656890705056043167181039 z 80 100 - 9251989514876365435107 z - 4413987149182 z 90 88 + 1478705139822854665 z - 11343058581833654903 z 84 94 - 428286917534445601247 z + 15591237265836897 z 86 96 98 + 74909238762216411317 z - 1238963924303972 z + 81751483971461 z 92 82 - 164850195665425775 z + 2130035844338217041921 z 64 112 114 110 - 12278624498750184871544675 z - 300 z + z + 39807 z 106 108 30 + 168103738 z - 3147287 z + 428286917534445601247 z 42 44 + 903581934228873908271267 z - 2079289573533250688019389 z 46 58 + 4237487550818566076101427 z + 24891524773796823978781889 z 56 54 - 24891524773796823978781889 z + 22129023973308565915795177 z 52 60 - 17486822077780077110447367 z - 22129023973308565915795177 z 70 68 + 2079289573533250688019389 z - 4237487550818566076101427 z 78 32 + 35217456884638791900053 z - 2130035844338217041921 z 38 40 + 117817237294523675668871 z - 347254397097577591733301 z 62 76 + 17486822077780077110447367 z - 117817237294523675668871 z 74 72 + 347254397097577591733301 z - 903581934228873908271267 z 104 - 6510615137 z ) And in Maple-input format, it is: -(1+23675528170108929968*z^28-3782139931450826896*z^26-260*z^2+ 520986191666452464*z^24-61497194129154808*z^22+30474*z^4-2157912*z^6-3693148448 *z^102+104365239*z^8-3693148448*z^10+99970161034*z^12-2135580384764*z^14-\ 522409084823680*z^18+36841675526741*z^16-2195516397486638140409360*z^50+ 1445613024038046286243266*z^48+6174005996343012*z^20+9065469583068800382432*z^ 36-2505203443882071432976*z^34-843873373254198476716080*z^66+ 606866372507232590988*z^80+99970161034*z^100-61497194129154808*z^90+ 520986191666452464*z^88+23675528170108929968*z^84-522409084823680*z^94-\ 3782139931450826896*z^86+36841675526741*z^96-2135580384764*z^98+ 6174005996343012*z^92-128464132117952324256*z^82+1445613024038046286243266*z^64 +z^112-260*z^110-2157912*z^106+30474*z^108-128464132117952324256*z^30-\ 199702216273459320610520*z^42+436382765993155178290716*z^44-\ 843873373254198476716080*z^46-3536643864698680408412376*z^58+ 3753590046679255800990842*z^56-3536643864698680408412376*z^54+ 2957943506481952275136604*z^52+2957943506481952275136604*z^60-\ 199702216273459320610520*z^70+436382765993155178290716*z^68-\ 2505203443882071432976*z^78+606866372507232590988*z^32-28832378735113183921888* z^38+80774883652097150937336*z^40-2195516397486638140409360*z^62+ 9065469583068800382432*z^76-28832378735113183921888*z^74+ 80774883652097150937336*z^72+104365239*z^104)/(-1-74909238762216411317*z^28+ 11343058581833654903*z^26+300*z^2-1478705139822854665*z^24+164850195665425775*z ^22-39807*z^4+3147287*z^6+191434749836*z^102-168103738*z^8+6510615137*z^10-\ 191434749836*z^12+4413987149182*z^14+1238963924303972*z^18-81751483971461*z^16+ 12278624498750184871544675*z^50-7656890705056043167181039*z^48-\ 15591237265836897*z^20-35217456884638791900053*z^36+9251989514876365435107*z^34 +7656890705056043167181039*z^66-9251989514876365435107*z^80-4413987149182*z^100 +1478705139822854665*z^90-11343058581833654903*z^88-428286917534445601247*z^84+ 15591237265836897*z^94+74909238762216411317*z^86-1238963924303972*z^96+ 81751483971461*z^98-164850195665425775*z^92+2130035844338217041921*z^82-\ 12278624498750184871544675*z^64-300*z^112+z^114+39807*z^110+168103738*z^106-\ 3147287*z^108+428286917534445601247*z^30+903581934228873908271267*z^42-\ 2079289573533250688019389*z^44+4237487550818566076101427*z^46+ 24891524773796823978781889*z^58-24891524773796823978781889*z^56+ 22129023973308565915795177*z^54-17486822077780077110447367*z^52-\ 22129023973308565915795177*z^60+2079289573533250688019389*z^70-\ 4237487550818566076101427*z^68+35217456884638791900053*z^78-\ 2130035844338217041921*z^32+117817237294523675668871*z^38-\ 347254397097577591733301*z^40+17486822077780077110447367*z^62-\ 117817237294523675668871*z^76+347254397097577591733301*z^74-\ 903581934228873908271267*z^72-6510615137*z^104) The first , 40, terms are: [0, 40, 0, 2667, 0, 197195, 0, 15146212, 0, 1181253833, 0, 92707496213, 0, 7295164386960, 0, 574705944548387, 0, 45296868221387027, 0, 3570944410114272004, 0, 281539052759545326893, 0, 22197915879493462364181, 0, 1750224602618405948287564, 0, 137999963339958034496222019, 0, 10880924243907649163241296387, 0, 857932942076346734984719016808, 0, 67645862880137953850480024907677, 0, 5333708726842124102181177500321697, 0, 420549792155432984924814887795142892, 0, 33159317600049196831306205718298048683] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 27114085063208720116 z - 4251654011233707523 z - 257 z 24 22 4 6 + 574922172440018292 z - 66639711863445057 z + 29944 z - 2118829 z 102 8 10 12 - 3666557539 z + 102855708 z - 3666557539 z + 100273588256 z 14 18 16 - 2169148098323 z - 546876337461497 z + 37962809031680 z 50 48 - 2933753330783457133452897 z + 1918182424758641749665584 z 20 36 + 6573145695692208 z + 11147402852829588943616 z 34 66 - 3028930514254333137549 z - 1109854893660898960919971 z 80 100 90 + 720923002742980899624 z + 100273588256 z - 66639711863445057 z 88 84 94 + 574922172440018292 z + 27114085063208720116 z - 546876337461497 z 86 96 98 - 4251654011233707523 z + 37962809031680 z - 2169148098323 z 92 82 + 6573145695692208 z - 149864407058713572451 z 64 112 110 106 + 1918182424758641749665584 z + z - 257 z - 2118829 z 108 30 42 + 29944 z - 149864407058713572451 z - 256735009883911205748315 z 44 46 + 567882248948107835606512 z - 1109854893660898960919971 z 58 56 - 4764570311644730041427317 z + 5062057560679475264273170 z 54 52 - 4764570311644730041427317 z + 3972693137745132138759272 z 60 70 + 3972693137745132138759272 z - 256735009883911205748315 z 68 78 + 567882248948107835606512 z - 3028930514254333137549 z 32 38 + 720923002742980899624 z - 36025076225038747685341 z 40 62 + 102439122696490276099052 z - 2933753330783457133452897 z 76 74 + 11147402852829588943616 z - 36025076225038747685341 z 72 104 / 2 + 102439122696490276099052 z + 102855708 z ) / ((-1 + z ) (1 / 28 26 2 + 73628090054640831812 z - 11119679005472166276 z - 301 z 24 22 4 6 + 1444418113013740162 z - 160386885159894434 z + 39660 z - 3104436 z 102 8 10 12 - 6316205790 z + 164264626 z - 6316205790 z + 184875712276 z 14 18 16 - 4254401884077 z - 1196897877608491 z + 78817007562080 z 50 48 - 10225281650258217742191555 z + 6620148231290654373372924 z 20 36 + 15110435032076048 z + 34320294138492825167152 z 34 66 - 9070126475157454945093 z - 3782578588715197299963945 z 80 100 90 + 2094650023204046803596 z + 184875712276 z - 160386885159894434 z 88 84 + 1444418113013740162 z + 73628090054640831812 z 94 86 96 - 1196897877608491 z - 11119679005472166276 z + 78817007562080 z 98 92 82 - 4254401884077 z + 15110435032076048 z - 421470768182801085827 z 64 112 110 106 + 6620148231290654373372924 z + z - 301 z - 3104436 z 108 30 42 + 39660 z - 421470768182801085827 z - 846542028452915681770370 z 44 46 + 1906191856129944796892700 z - 3782578588715197299963945 z 58 56 - 16795564808033001179608362 z + 17869630787793019051907940 z 54 52 - 16795564808033001179608362 z + 13944620552470371568262132 z 60 70 + 13944620552470371568262132 z - 846542028452915681770370 z 68 78 + 1906191856129944796892700 z - 9070126475157454945093 z 32 38 + 2094650023204046803596 z - 113758120934292419104044 z 40 62 + 330962009257398023945822 z - 10225281650258217742191555 z 76 74 + 34320294138492825167152 z - 113758120934292419104044 z 72 104 + 330962009257398023945822 z + 164264626 z )) And in Maple-input format, it is: -(1+27114085063208720116*z^28-4251654011233707523*z^26-257*z^2+ 574922172440018292*z^24-66639711863445057*z^22+29944*z^4-2118829*z^6-3666557539 *z^102+102855708*z^8-3666557539*z^10+100273588256*z^12-2169148098323*z^14-\ 546876337461497*z^18+37962809031680*z^16-2933753330783457133452897*z^50+ 1918182424758641749665584*z^48+6573145695692208*z^20+11147402852829588943616*z^ 36-3028930514254333137549*z^34-1109854893660898960919971*z^66+ 720923002742980899624*z^80+100273588256*z^100-66639711863445057*z^90+ 574922172440018292*z^88+27114085063208720116*z^84-546876337461497*z^94-\ 4251654011233707523*z^86+37962809031680*z^96-2169148098323*z^98+ 6573145695692208*z^92-149864407058713572451*z^82+1918182424758641749665584*z^64 +z^112-257*z^110-2118829*z^106+29944*z^108-149864407058713572451*z^30-\ 256735009883911205748315*z^42+567882248948107835606512*z^44-\ 1109854893660898960919971*z^46-4764570311644730041427317*z^58+ 5062057560679475264273170*z^56-4764570311644730041427317*z^54+ 3972693137745132138759272*z^52+3972693137745132138759272*z^60-\ 256735009883911205748315*z^70+567882248948107835606512*z^68-\ 3028930514254333137549*z^78+720923002742980899624*z^32-36025076225038747685341* z^38+102439122696490276099052*z^40-2933753330783457133452897*z^62+ 11147402852829588943616*z^76-36025076225038747685341*z^74+ 102439122696490276099052*z^72+102855708*z^104)/(-1+z^2)/(1+73628090054640831812 *z^28-11119679005472166276*z^26-301*z^2+1444418113013740162*z^24-\ 160386885159894434*z^22+39660*z^4-3104436*z^6-6316205790*z^102+164264626*z^8-\ 6316205790*z^10+184875712276*z^12-4254401884077*z^14-1196897877608491*z^18+ 78817007562080*z^16-10225281650258217742191555*z^50+6620148231290654373372924*z ^48+15110435032076048*z^20+34320294138492825167152*z^36-9070126475157454945093* z^34-3782578588715197299963945*z^66+2094650023204046803596*z^80+184875712276*z^ 100-160386885159894434*z^90+1444418113013740162*z^88+73628090054640831812*z^84-\ 1196897877608491*z^94-11119679005472166276*z^86+78817007562080*z^96-\ 4254401884077*z^98+15110435032076048*z^92-421470768182801085827*z^82+ 6620148231290654373372924*z^64+z^112-301*z^110-3104436*z^106+39660*z^108-\ 421470768182801085827*z^30-846542028452915681770370*z^42+ 1906191856129944796892700*z^44-3782578588715197299963945*z^46-\ 16795564808033001179608362*z^58+17869630787793019051907940*z^56-\ 16795564808033001179608362*z^54+13944620552470371568262132*z^52+ 13944620552470371568262132*z^60-846542028452915681770370*z^70+ 1906191856129944796892700*z^68-9070126475157454945093*z^78+ 2094650023204046803596*z^32-113758120934292419104044*z^38+ 330962009257398023945822*z^40-10225281650258217742191555*z^62+ 34320294138492825167152*z^76-113758120934292419104044*z^74+ 330962009257398023945822*z^72+164264626*z^104) The first , 40, terms are: [0, 45, 0, 3573, 0, 306068, 0, 26622849, 0, 2325477145, 0, 203418783813, 0, 17803773482897, 0, 1558598484127761, 0, 136458427436678837, 0, 11947744942760770852, 0, 1046116994246022943401, 0, 91596423953019744199253, 0, 8020077619751373649497949, 0, 702230098693832755667588293, 0, 61486628598389190429675703877, 0, 5383715369511475027759928080449, 0, 471393488667550010235934052827844, 0, 41274811922656131416746365665793101, 0, 3613987473096690878578587734161972001, 0, 316437678833626397912995989029273723161] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 224168016126059667 z - 57003743150238441 z - 209 z 24 22 4 6 + 12196869172271889 z - 2184841654866184 z + 19331 z - 1062838 z 8 10 12 14 + 39296184 z - 1046508938 z + 20975902413 z - 326207813359 z 18 16 50 - 40045352233248 z + 4024280027207 z - 17738738882036307831 z 48 20 + 26288655179971440503 z + 325591075351788 z 36 34 + 10216171602307401101 z - 5016365826285035794 z 66 80 88 84 86 - 2184841654866184 z + 39296184 z + z + 19331 z - 209 z 82 64 30 - 1062838 z + 12196869172271889 z - 744737722258366318 z 42 44 - 33277437612437988456 z + 35996175035510240952 z 46 58 - 33277437612437988456 z - 744737722258366318 z 56 54 + 2096900178735539824 z - 5016365826285035794 z 52 60 70 + 10216171602307401101 z + 224168016126059667 z - 40045352233248 z 68 78 32 + 325591075351788 z - 1046508938 z + 2096900178735539824 z 38 40 - 17738738882036307831 z + 26288655179971440503 z 62 76 74 - 57003743150238441 z + 20975902413 z - 326207813359 z 72 / 28 + 4024280027207 z ) / (-1 - 840830881184248984 z / 26 2 24 + 199632987973397603 z + 247 z - 39851021543288641 z 22 4 6 8 + 6651607977228648 z - 26220 z + 1624680 z - 66866529 z 10 12 14 + 1964523051 z - 43141793192 z + 731061382020 z 18 16 50 + 105225520795483 z - 9783494379425 z + 149996084445665184845 z 48 20 - 204539494809161985387 z - 921976517728604 z 36 34 - 50484612772218797276 z + 23115430868795946707 z 66 80 90 88 84 + 39851021543288641 z - 1964523051 z + z - 247 z - 1624680 z 86 82 64 + 26220 z + 66866529 z - 199632987973397603 z 30 42 + 2990986798740192428 z + 204539494809161985387 z 44 46 - 238812426515937581396 z + 238812426515937581396 z 58 56 + 9018275438652759705 z - 23115430868795946707 z 54 52 + 50484612772218797276 z - 94120624549757222928 z 60 70 68 - 2990986798740192428 z + 921976517728604 z - 6651607977228648 z 78 32 38 + 43141793192 z - 9018275438652759705 z + 94120624549757222928 z 40 62 76 - 149996084445665184845 z + 840830881184248984 z - 731061382020 z 74 72 + 9783494379425 z - 105225520795483 z ) And in Maple-input format, it is: -(1+224168016126059667*z^28-57003743150238441*z^26-209*z^2+12196869172271889*z^ 24-2184841654866184*z^22+19331*z^4-1062838*z^6+39296184*z^8-1046508938*z^10+ 20975902413*z^12-326207813359*z^14-40045352233248*z^18+4024280027207*z^16-\ 17738738882036307831*z^50+26288655179971440503*z^48+325591075351788*z^20+ 10216171602307401101*z^36-5016365826285035794*z^34-2184841654866184*z^66+ 39296184*z^80+z^88+19331*z^84-209*z^86-1062838*z^82+12196869172271889*z^64-\ 744737722258366318*z^30-33277437612437988456*z^42+35996175035510240952*z^44-\ 33277437612437988456*z^46-744737722258366318*z^58+2096900178735539824*z^56-\ 5016365826285035794*z^54+10216171602307401101*z^52+224168016126059667*z^60-\ 40045352233248*z^70+325591075351788*z^68-1046508938*z^78+2096900178735539824*z^ 32-17738738882036307831*z^38+26288655179971440503*z^40-57003743150238441*z^62+ 20975902413*z^76-326207813359*z^74+4024280027207*z^72)/(-1-840830881184248984*z ^28+199632987973397603*z^26+247*z^2-39851021543288641*z^24+6651607977228648*z^ 22-26220*z^4+1624680*z^6-66866529*z^8+1964523051*z^10-43141793192*z^12+ 731061382020*z^14+105225520795483*z^18-9783494379425*z^16+149996084445665184845 *z^50-204539494809161985387*z^48-921976517728604*z^20-50484612772218797276*z^36 +23115430868795946707*z^34+39851021543288641*z^66-1964523051*z^80+z^90-247*z^88 -1624680*z^84+26220*z^86+66866529*z^82-199632987973397603*z^64+ 2990986798740192428*z^30+204539494809161985387*z^42-238812426515937581396*z^44+ 238812426515937581396*z^46+9018275438652759705*z^58-23115430868795946707*z^56+ 50484612772218797276*z^54-94120624549757222928*z^52-2990986798740192428*z^60+ 921976517728604*z^70-6651607977228648*z^68+43141793192*z^78-9018275438652759705 *z^32+94120624549757222928*z^38-149996084445665184845*z^40+840830881184248984*z ^62-731061382020*z^76+9783494379425*z^74-105225520795483*z^72) The first , 40, terms are: [0, 38, 0, 2497, 0, 182241, 0, 13709682, 0, 1041844405, 0, 79471276121, 0, 6071147428770, 0, 464094179796601, 0, 35486301026081617, 0, 2713739614146399614, 0, 207538811241304777457, 0, 15872348972858549871805, 0, 1213914118365156943752230, 0, 92840388560445008156512589, 0, 7100467689796748607454282837, 0, 543047012164066278513239044810, 0, 41532504008627607139405379644093, 0, 3176427161345232853884308204226697, 0, 242934800089124278053001861177351674, 0, 18579780649265570916249090459208503325] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 739246249545586410 z - 182548889779029952 z - 241 z 24 22 4 6 + 37715464644081400 z - 6481707555039728 z + 25678 z - 1614761 z 8 10 12 14 + 67559947 z - 2011358402 z + 44514059164 z - 755336328964 z 18 16 50 - 106891969965150 z + 10055460724124 z - 63523015176535965086 z 48 20 + 94762900691728372788 z + 919982807907926 z 36 34 + 36239721879937298788 z - 17571739729773048396 z 66 80 88 84 86 - 6481707555039728 z + 67559947 z + z + 25678 z - 241 z 82 64 30 - 1614761 z + 37715464644081400 z - 2516461544073964802 z 42 44 - 120422384249981885270 z + 130428132901023254676 z 46 58 - 120422384249981885270 z - 2516461544073964802 z 56 54 + 7228158127855596548 z - 17571739729773048396 z 52 60 70 + 36239721879937298788 z + 739246249545586410 z - 106891969965150 z 68 78 32 + 919982807907926 z - 2011358402 z + 7228158127855596548 z 38 40 - 63523015176535965086 z + 94762900691728372788 z 62 76 74 - 182548889779029952 z + 44514059164 z - 755336328964 z 72 / 2 28 + 10055460724124 z ) / ((-1 + z ) (1 + 2282781595298632284 z / 26 2 24 - 541896322031199982 z - 293 z + 107216861135183692 z 22 4 6 8 - 17580230416614594 z + 36023 z - 2536971 z + 116700033 z 10 12 14 - 3773874880 z + 89957244230 z - 1633678709476 z 18 16 50 - 260986427166024 z + 23158032946150 z - 224323050484389394496 z 48 20 + 338960869898322305894 z + 2371927698891020 z 36 34 + 125724432816244954258 z - 59604635815329217484 z 66 80 88 84 86 - 17580230416614594 z + 116700033 z + z + 36023 z - 293 z 82 64 30 - 2536971 z + 107216861135183692 z - 8051679681980016376 z 42 44 - 434096068074986654528 z + 471386244066390588842 z 46 58 - 434096068074986654528 z - 8051679681980016376 z 56 54 + 23864354856352734386 z - 59604635815329217484 z 52 60 + 125724432816244954258 z + 2282781595298632284 z 70 68 78 - 260986427166024 z + 2371927698891020 z - 3773874880 z 32 38 + 23864354856352734386 z - 224323050484389394496 z 40 62 76 + 338960869898322305894 z - 541896322031199982 z + 89957244230 z 74 72 - 1633678709476 z + 23158032946150 z )) And in Maple-input format, it is: -(1+739246249545586410*z^28-182548889779029952*z^26-241*z^2+37715464644081400*z ^24-6481707555039728*z^22+25678*z^4-1614761*z^6+67559947*z^8-2011358402*z^10+ 44514059164*z^12-755336328964*z^14-106891969965150*z^18+10055460724124*z^16-\ 63523015176535965086*z^50+94762900691728372788*z^48+919982807907926*z^20+ 36239721879937298788*z^36-17571739729773048396*z^34-6481707555039728*z^66+ 67559947*z^80+z^88+25678*z^84-241*z^86-1614761*z^82+37715464644081400*z^64-\ 2516461544073964802*z^30-120422384249981885270*z^42+130428132901023254676*z^44-\ 120422384249981885270*z^46-2516461544073964802*z^58+7228158127855596548*z^56-\ 17571739729773048396*z^54+36239721879937298788*z^52+739246249545586410*z^60-\ 106891969965150*z^70+919982807907926*z^68-2011358402*z^78+7228158127855596548*z ^32-63523015176535965086*z^38+94762900691728372788*z^40-182548889779029952*z^62 +44514059164*z^76-755336328964*z^74+10055460724124*z^72)/(-1+z^2)/(1+ 2282781595298632284*z^28-541896322031199982*z^26-293*z^2+107216861135183692*z^ 24-17580230416614594*z^22+36023*z^4-2536971*z^6+116700033*z^8-3773874880*z^10+ 89957244230*z^12-1633678709476*z^14-260986427166024*z^18+23158032946150*z^16-\ 224323050484389394496*z^50+338960869898322305894*z^48+2371927698891020*z^20+ 125724432816244954258*z^36-59604635815329217484*z^34-17580230416614594*z^66+ 116700033*z^80+z^88+36023*z^84-293*z^86-2536971*z^82+107216861135183692*z^64-\ 8051679681980016376*z^30-434096068074986654528*z^42+471386244066390588842*z^44-\ 434096068074986654528*z^46-8051679681980016376*z^58+23864354856352734386*z^56-\ 59604635815329217484*z^54+125724432816244954258*z^52+2282781595298632284*z^60-\ 260986427166024*z^70+2371927698891020*z^68-3773874880*z^78+23864354856352734386 *z^32-224323050484389394496*z^38+338960869898322305894*z^40-541896322031199982* z^62+89957244230*z^76-1633678709476*z^74+23158032946150*z^72) The first , 40, terms are: [0, 53, 0, 4944, 0, 487021, 0, 48329495, 0, 4802754529, 0, 477453664647, 0, 47470672345784, 0, 4719966268736139, 0, 469309944900694059, 0, 46664161425627201643, 0, 4639896441602964886083, 0, 461353210676999675503304, 0, 45873194029827989610947735, 0, 4561256350789382035335960057, 0, 453534166044186981292366741503, 0, 45095743158662915498476949109933, 0, 4483953411589144906927515066869344, 0, 445847808861674101050409394212015805, 0, 44331475091582127625540675325889340305, 0, 4407960846635033166436251408961457973137] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 617245205663905306 z - 152457492191665425 z - 237 z 24 22 4 6 + 31543162478867186 z - 5437126990376871 z + 24727 z - 1519452 z 8 10 12 14 + 62116623 z - 1809783681 z + 39297476112 z - 656219128543 z 18 16 50 - 90765731055583 z + 8623844016532 z - 53351319351310798659 z 48 20 + 79669754572174862941 z + 775534551995710 z 36 34 + 30397510863411992891 z - 14717896753421017742 z 66 80 88 84 86 - 5437126990376871 z + 62116623 z + z + 24727 z - 237 z 82 64 30 - 1519452 z + 31543162478867186 z - 2102433641877006961 z 42 44 - 101307807125846366398 z + 109749642797987982796 z 46 58 - 101307807125846366398 z - 2102433641877006961 z 56 54 + 6045783514972421165 z - 14717896753421017742 z 52 60 70 + 30397510863411992891 z + 617245205663905306 z - 90765731055583 z 68 78 32 + 775534551995710 z - 1809783681 z + 6045783514972421165 z 38 40 - 53351319351310798659 z + 79669754572174862941 z 62 76 74 - 152457492191665425 z + 39297476112 z - 656219128543 z 72 / 28 + 8623844016532 z ) / (-1 - 2328126084565276026 z / 26 2 24 + 537546073173198711 z + 281 z - 103919568189195451 z 22 4 6 8 + 16720281551286556 z - 33788 z + 2348848 z - 107201638 z 10 12 14 + 3451940826 z - 82180215806 z + 1494894222578 z 18 16 50 + 241804905862669 z - 21289019537021 z + 457176924950398949131 z 48 20 - 626909964775166099159 z - 2222447999098560 z 36 34 - 150871175707867364056 z + 68101342922850708702 z 66 80 90 88 84 + 103919568189195451 z - 3451940826 z + z - 281 z - 2348848 z 86 82 64 + 33788 z + 107201638 z - 537546073173198711 z 30 42 + 8483607288190405786 z + 626909964775166099159 z 44 46 - 733995778925261012908 z + 733995778925261012908 z 58 56 + 26112084909718775690 z - 68101342922850708702 z 54 52 + 150871175707867364056 z - 284470204525581679568 z 60 70 68 - 8483607288190405786 z + 2222447999098560 z - 16720281551286556 z 78 32 38 + 82180215806 z - 26112084909718775690 z + 284470204525581679568 z 40 62 76 - 457176924950398949131 z + 2328126084565276026 z - 1494894222578 z 74 72 + 21289019537021 z - 241804905862669 z ) And in Maple-input format, it is: -(1+617245205663905306*z^28-152457492191665425*z^26-237*z^2+31543162478867186*z ^24-5437126990376871*z^22+24727*z^4-1519452*z^6+62116623*z^8-1809783681*z^10+ 39297476112*z^12-656219128543*z^14-90765731055583*z^18+8623844016532*z^16-\ 53351319351310798659*z^50+79669754572174862941*z^48+775534551995710*z^20+ 30397510863411992891*z^36-14717896753421017742*z^34-5437126990376871*z^66+ 62116623*z^80+z^88+24727*z^84-237*z^86-1519452*z^82+31543162478867186*z^64-\ 2102433641877006961*z^30-101307807125846366398*z^42+109749642797987982796*z^44-\ 101307807125846366398*z^46-2102433641877006961*z^58+6045783514972421165*z^56-\ 14717896753421017742*z^54+30397510863411992891*z^52+617245205663905306*z^60-\ 90765731055583*z^70+775534551995710*z^68-1809783681*z^78+6045783514972421165*z^ 32-53351319351310798659*z^38+79669754572174862941*z^40-152457492191665425*z^62+ 39297476112*z^76-656219128543*z^74+8623844016532*z^72)/(-1-2328126084565276026* z^28+537546073173198711*z^26+281*z^2-103919568189195451*z^24+16720281551286556* z^22-33788*z^4+2348848*z^6-107201638*z^8+3451940826*z^10-82180215806*z^12+ 1494894222578*z^14+241804905862669*z^18-21289019537021*z^16+ 457176924950398949131*z^50-626909964775166099159*z^48-2222447999098560*z^20-\ 150871175707867364056*z^36+68101342922850708702*z^34+103919568189195451*z^66-\ 3451940826*z^80+z^90-281*z^88-2348848*z^84+33788*z^86+107201638*z^82-\ 537546073173198711*z^64+8483607288190405786*z^30+626909964775166099159*z^42-\ 733995778925261012908*z^44+733995778925261012908*z^46+26112084909718775690*z^58 -68101342922850708702*z^56+150871175707867364056*z^54-284470204525581679568*z^ 52-8483607288190405786*z^60+2222447999098560*z^70-16720281551286556*z^68+ 82180215806*z^78-26112084909718775690*z^32+284470204525581679568*z^38-\ 457176924950398949131*z^40+2328126084565276026*z^62-1494894222578*z^76+ 21289019537021*z^74-241804905862669*z^72) The first , 40, terms are: [0, 44, 0, 3303, 0, 270867, 0, 22776160, 0, 1931576781, 0, 164353238933, 0, 14004001794128, 0, 1193981639811839, 0, 101828215863599455, 0, 8685543582742708508, 0, 740889337812660155213, 0, 63200812268116459569437, 0, 5391357288376070832685596, 0, 459913835859091593474750479, 0, 39233424188131791938870168639, 0, 3346852754421884906408664128416, 0, 285507359572072638154416125021765, 0, 24355562050346052994786509323411133, 0, 2077681986250494511940427906183334576, 0, 177239299814422309332468695873095082707] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 29808482174990337832 z - 4662088754887875358 z - 260 z 24 22 4 6 + 628474289393400408 z - 72582757595236878 z + 30632 z - 2189886 z 102 8 10 12 - 3857160576 z + 107299477 z - 3857160576 z + 106283767384 z 14 18 16 - 2314765853214 z - 590319361222570 z + 40757575784501 z 50 48 - 3245959305366864381515990 z + 2122731052441353976191746 z 20 36 + 7129358521731700 z + 12327893892643972572552 z 34 66 - 3346686544757142632890 z - 1228438065425683166250062 z 80 100 90 + 795552671074783140972 z + 106283767384 z - 72582757595236878 z 88 84 94 + 628474289393400408 z + 29808482174990337832 z - 590319361222570 z 86 96 98 - 4662088754887875358 z + 40757575784501 z - 2314765853214 z 92 82 + 7129358521731700 z - 165103929166447540542 z 64 112 110 106 + 2122731052441353976191746 z + z - 260 z - 2189886 z 108 30 42 + 30632 z - 165103929166447540542 z - 284214956491585831696378 z 44 46 + 628643919145498769983564 z - 1228438065425683166250062 z 58 56 - 5270125934152488812414794 z + 5598957484654960084442238 z 54 52 - 5270125934152488812414794 z + 4394720142422839380750928 z 60 70 + 4394720142422839380750928 z - 284214956491585831696378 z 68 78 + 628643919145498769983564 z - 3346686544757142632890 z 32 38 + 795552671074783140972 z - 39863396285495696145898 z 40 62 + 113390436432616167869552 z - 3245959305366864381515990 z 76 74 + 12327893892643972572552 z - 39863396285495696145898 z 72 104 / + 113390436432616167869552 z + 107299477 z ) / (-1 / 28 26 2 - 93522917446626737515 z + 13823486138809192469 z + 310 z 24 22 4 6 - 1760327838804529303 z + 191921163824140329 z - 41709 z + 3319109 z 102 8 10 12 + 205640808004 z - 178115136 z + 6937823757 z - 205640808004 z 14 18 16 + 4793279405546 z + 1386578176780210 z - 90003184961093 z 50 48 + 19022088969590386406072953 z - 11735613257882218909304001 z 20 36 - 17779209752717859 z - 48418669067498814205771 z 34 66 + 12426503118944715654901 z + 11735613257882218909304001 z 80 100 - 12426503118944715654901 z - 4793279405546 z 90 88 + 1760327838804529303 z - 13823486138809192469 z 84 94 - 547975842543296410361 z + 17779209752717859 z 86 96 98 + 93522917446626737515 z - 1386578176780210 z + 90003184961093 z 92 82 - 191921163824140329 z + 2792816468708582150991 z 64 112 114 110 - 19022088969590386406072953 z - 310 z + z + 41709 z 106 108 30 + 178115136 z - 3319109 z + 547975842543296410361 z 42 44 + 1322564376143019602532437 z - 3097147014602514838393171 z 46 58 + 6409896874722289274754241 z + 39200219636346486554769871 z 56 54 - 39200219636346486554769871 z + 34753531851485228379570067 z 52 60 - 27312512883031455708613085 z - 34753531851485228379570067 z 70 68 + 3097147014602514838393171 z - 6409896874722289274754241 z 78 32 + 48418669067498814205771 z - 2792816468708582150991 z 38 40 + 165634214251407359682369 z - 498531879656925811508055 z 62 76 + 27312512883031455708613085 z - 165634214251407359682369 z 74 72 + 498531879656925811508055 z - 1322564376143019602532437 z 104 - 6937823757 z ) And in Maple-input format, it is: -(1+29808482174990337832*z^28-4662088754887875358*z^26-260*z^2+ 628474289393400408*z^24-72582757595236878*z^22+30632*z^4-2189886*z^6-3857160576 *z^102+107299477*z^8-3857160576*z^10+106283767384*z^12-2314765853214*z^14-\ 590319361222570*z^18+40757575784501*z^16-3245959305366864381515990*z^50+ 2122731052441353976191746*z^48+7129358521731700*z^20+12327893892643972572552*z^ 36-3346686544757142632890*z^34-1228438065425683166250062*z^66+ 795552671074783140972*z^80+106283767384*z^100-72582757595236878*z^90+ 628474289393400408*z^88+29808482174990337832*z^84-590319361222570*z^94-\ 4662088754887875358*z^86+40757575784501*z^96-2314765853214*z^98+ 7129358521731700*z^92-165103929166447540542*z^82+2122731052441353976191746*z^64 +z^112-260*z^110-2189886*z^106+30632*z^108-165103929166447540542*z^30-\ 284214956491585831696378*z^42+628643919145498769983564*z^44-\ 1228438065425683166250062*z^46-5270125934152488812414794*z^58+ 5598957484654960084442238*z^56-5270125934152488812414794*z^54+ 4394720142422839380750928*z^52+4394720142422839380750928*z^60-\ 284214956491585831696378*z^70+628643919145498769983564*z^68-\ 3346686544757142632890*z^78+795552671074783140972*z^32-39863396285495696145898* z^38+113390436432616167869552*z^40-3245959305366864381515990*z^62+ 12327893892643972572552*z^76-39863396285495696145898*z^74+ 113390436432616167869552*z^72+107299477*z^104)/(-1-93522917446626737515*z^28+ 13823486138809192469*z^26+310*z^2-1760327838804529303*z^24+191921163824140329*z ^22-41709*z^4+3319109*z^6+205640808004*z^102-178115136*z^8+6937823757*z^10-\ 205640808004*z^12+4793279405546*z^14+1386578176780210*z^18-90003184961093*z^16+ 19022088969590386406072953*z^50-11735613257882218909304001*z^48-\ 17779209752717859*z^20-48418669067498814205771*z^36+12426503118944715654901*z^ 34+11735613257882218909304001*z^66-12426503118944715654901*z^80-4793279405546*z ^100+1760327838804529303*z^90-13823486138809192469*z^88-547975842543296410361*z ^84+17779209752717859*z^94+93522917446626737515*z^86-1386578176780210*z^96+ 90003184961093*z^98-191921163824140329*z^92+2792816468708582150991*z^82-\ 19022088969590386406072953*z^64-310*z^112+z^114+41709*z^110+178115136*z^106-\ 3319109*z^108+547975842543296410361*z^30+1322564376143019602532437*z^42-\ 3097147014602514838393171*z^44+6409896874722289274754241*z^46+ 39200219636346486554769871*z^58-39200219636346486554769871*z^56+ 34753531851485228379570067*z^54-27312512883031455708613085*z^52-\ 34753531851485228379570067*z^60+3097147014602514838393171*z^70-\ 6409896874722289274754241*z^68+48418669067498814205771*z^78-\ 2792816468708582150991*z^32+165634214251407359682369*z^38-\ 498531879656925811508055*z^40+27312512883031455708613085*z^62-\ 165634214251407359682369*z^76+498531879656925811508055*z^74-\ 1322564376143019602532437*z^72-6937823757*z^104) The first , 40, terms are: [0, 50, 0, 4423, 0, 414903, 0, 39280814, 0, 3727188601, 0, 353904177313, 0, 33612252240182, 0, 3192639522933895, 0, 303261809694613327, 0, 28806577659412353826, 0, 2736327221259074102353, 0, 259923391419871548500513, 0, 24690113624018393783730498, 0, 2345313857015618978809859695, 0, 222781392571326418497484650119, 0, 21162008526530776245187755088006, 0, 2010179623175842394276636795940401, 0, 190947005454078334168873790344077545, 0, 18138060210817035513222727603179354686, 0, 1722934737700926821140850450756658711511] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2812212680856675 z - 1155127433872725 z - 182 z 24 22 4 6 + 387634746629299 z - 105906687527112 z + 14141 z - 632943 z 8 10 12 14 + 18542581 z - 381919188 z + 5787716257 z - 66562349314 z 18 16 50 - 4177493667606 z + 593886130950 z - 105906687527112 z 48 20 36 + 387634746629299 z + 23444412218139 z + 13584251431848124 z 34 66 64 30 - 12313595999985760 z - 632943 z + 18542581 z - 5606929292987766 z 42 44 46 - 5606929292987766 z + 2812212680856675 z - 1155127433872725 z 58 56 54 - 66562349314 z + 593886130950 z - 4177493667606 z 52 60 70 68 + 23444412218139 z + 5787716257 z - 182 z + 14141 z 32 38 40 + 9169786614428443 z - 12313595999985760 z + 9169786614428443 z 62 72 / 28 - 381919188 z + z ) / (-1 - 13184357943702186 z / 26 2 24 + 4949238695993173 z + 227 z - 1519296698781197 z 22 4 6 8 + 379959497456228 z - 20760 z + 1059836 z - 34788897 z 10 12 14 18 + 794433633 z - 13261909554 z + 167341935012 z + 12563968817749 z 16 50 48 - 1634177300473 z + 1519296698781197 z - 4949238695993173 z 20 36 34 - 77015504188722 z - 92620703366910206 z + 76256187168760949 z 66 64 30 + 34788897 z - 794433633 z + 28798356003651302 z 42 44 46 + 51673192416639603 z - 28798356003651302 z + 13184357943702186 z 58 56 54 + 1634177300473 z - 12563968817749 z + 77015504188722 z 52 60 70 68 - 379959497456228 z - 167341935012 z + 20760 z - 1059836 z 32 38 40 - 51673192416639603 z + 92620703366910206 z - 76256187168760949 z 62 74 72 + 13261909554 z + z - 227 z ) And in Maple-input format, it is: -(1+2812212680856675*z^28-1155127433872725*z^26-182*z^2+387634746629299*z^24-\ 105906687527112*z^22+14141*z^4-632943*z^6+18542581*z^8-381919188*z^10+ 5787716257*z^12-66562349314*z^14-4177493667606*z^18+593886130950*z^16-\ 105906687527112*z^50+387634746629299*z^48+23444412218139*z^20+13584251431848124 *z^36-12313595999985760*z^34-632943*z^66+18542581*z^64-5606929292987766*z^30-\ 5606929292987766*z^42+2812212680856675*z^44-1155127433872725*z^46-66562349314*z ^58+593886130950*z^56-4177493667606*z^54+23444412218139*z^52+5787716257*z^60-\ 182*z^70+14141*z^68+9169786614428443*z^32-12313595999985760*z^38+ 9169786614428443*z^40-381919188*z^62+z^72)/(-1-13184357943702186*z^28+ 4949238695993173*z^26+227*z^2-1519296698781197*z^24+379959497456228*z^22-20760* z^4+1059836*z^6-34788897*z^8+794433633*z^10-13261909554*z^12+167341935012*z^14+ 12563968817749*z^18-1634177300473*z^16+1519296698781197*z^50-4949238695993173*z ^48-77015504188722*z^20-92620703366910206*z^36+76256187168760949*z^34+34788897* z^66-794433633*z^64+28798356003651302*z^30+51673192416639603*z^42-\ 28798356003651302*z^44+13184357943702186*z^46+1634177300473*z^58-12563968817749 *z^56+77015504188722*z^54-379959497456228*z^52-167341935012*z^60+20760*z^70-\ 1059836*z^68-51673192416639603*z^32+92620703366910206*z^38-76256187168760949*z^ 40+13261909554*z^62+z^74-227*z^72) The first , 40, terms are: [0, 45, 0, 3596, 0, 308985, 0, 26932939, 0, 2357432889, 0, 206657325199, 0, 18126934043028, 0, 1590401304446375, 0, 139551639874366483, 0, 12245671266200121111, 0, 1074579379318097608355, 0, 94297008839650434458452, 0, 8274825107819226953458715, 0, 726140006768417867465927541, 0, 63720940365779112969062738295, 0, 5591703083506974456511296710725, 0, 490688723553002145702120405610796, 0, 43059409870175446361919142324156025, 0, 3778592674169812472026557244242051925, 0, 331582870364260008185179884346700189757] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 29496632020235248817 z - 4614262163250884913 z - 260 z 24 22 4 6 + 622240658590910201 z - 71897177758547638 z + 30609 z - 2185821 z 102 8 10 12 - 3840819390 z + 106971361 z - 3840819390 z + 105715567549 z 14 18 16 - 2300021710870 z - 585521766895746 z + 40460084322374 z 50 48 - 3223669414461027758005804 z + 2107366956414570305961398 z 20 36 + 7066295451705041 z + 12204839323427670017464 z 34 66 - 3312294473611214878828 z - 1219006612106638158568706 z 80 100 90 + 787228397058208128469 z + 105715567549 z - 71897177758547638 z 88 84 94 + 622240658590910201 z + 29496632020235248817 z - 585521766895746 z 86 96 98 - 4614262163250884913 z + 40460084322374 z - 2300021710870 z 92 82 + 7066295451705041 z - 163365633252501641524 z 64 112 110 106 + 2107366956414570305961398 z + z - 260 z - 2185821 z 108 30 42 + 30609 z - 163365633252501641524 z - 281753265161639410901640 z 44 46 + 623513427374503721091342 z - 1219006612106638158568706 z 58 56 - 5236341702690278488045368 z + 5563398834608552953291940 z 54 52 - 5236341702690278488045368 z + 4365777171352549428937650 z 60 70 + 4365777171352549428937650 z - 281753265161639410901640 z 68 78 + 623513427374503721091342 z - 3312294473611214878828 z 32 38 + 787228397058208128469 z - 39480764176662822216804 z 40 62 + 112352939304352296591946 z - 3223669414461027758005804 z 76 74 + 12204839323427670017464 z - 39480764176662822216804 z 72 104 / + 112352939304352296591946 z + 106971361 z ) / (-1 / 28 26 2 - 92703818813296533106 z + 13710581657540284627 z + 309 z 24 22 4 6 - 1746974031358289099 z + 190569134408872848 z - 41518 z + 3301686 z 102 8 10 12 + 204467729740 z - 177126299 z + 6898554675 z - 204467729740 z 14 18 16 + 4765589108448 z + 1377983635278989 z - 89469177739181 z 50 48 + 18808664449284471043834218 z - 11603157030655744298534202 z 20 36 - 17662331859748660 z - 47901206354136619370020 z 34 66 + 12298384053959959132513 z + 11603157030655744298534202 z 80 100 - 12298384053959959132513 z - 4765589108448 z 90 88 + 1746974031358289099 z - 13710581657540284627 z 84 94 - 542864059881531860890 z + 17662331859748660 z 86 96 98 + 92703818813296533106 z - 1377983635278989 z + 89469177739181 z 92 82 - 190569134408872848 z + 2765300108258394073205 z 64 112 114 110 - 18808664449284471043834218 z - 309 z + z + 41518 z 106 108 30 + 177126299 z - 3301686 z + 542864059881531860890 z 42 44 + 1307624495523339804413598 z - 3062015494466532515165920 z 46 58 + 6337246659832720212481088 z + 38766431351166329707365198 z 56 54 - 38766431351166329707365198 z + 34367894404173855017846324 z 52 60 - 27007942337616272445421164 z - 34367894404173855017846324 z 70 68 + 3062015494466532515165920 z - 6337246659832720212481088 z 78 32 + 47901206354136619370020 z - 2765300108258394073205 z 38 40 + 163816112396687467792676 z - 492959458905043844934182 z 62 76 + 27007942337616272445421164 z - 163816112396687467792676 z 74 72 + 492959458905043844934182 z - 1307624495523339804413598 z 104 - 6898554675 z ) And in Maple-input format, it is: -(1+29496632020235248817*z^28-4614262163250884913*z^26-260*z^2+ 622240658590910201*z^24-71897177758547638*z^22+30609*z^4-2185821*z^6-3840819390 *z^102+106971361*z^8-3840819390*z^10+105715567549*z^12-2300021710870*z^14-\ 585521766895746*z^18+40460084322374*z^16-3223669414461027758005804*z^50+ 2107366956414570305961398*z^48+7066295451705041*z^20+12204839323427670017464*z^ 36-3312294473611214878828*z^34-1219006612106638158568706*z^66+ 787228397058208128469*z^80+105715567549*z^100-71897177758547638*z^90+ 622240658590910201*z^88+29496632020235248817*z^84-585521766895746*z^94-\ 4614262163250884913*z^86+40460084322374*z^96-2300021710870*z^98+ 7066295451705041*z^92-163365633252501641524*z^82+2107366956414570305961398*z^64 +z^112-260*z^110-2185821*z^106+30609*z^108-163365633252501641524*z^30-\ 281753265161639410901640*z^42+623513427374503721091342*z^44-\ 1219006612106638158568706*z^46-5236341702690278488045368*z^58+ 5563398834608552953291940*z^56-5236341702690278488045368*z^54+ 4365777171352549428937650*z^52+4365777171352549428937650*z^60-\ 281753265161639410901640*z^70+623513427374503721091342*z^68-\ 3312294473611214878828*z^78+787228397058208128469*z^32-39480764176662822216804* z^38+112352939304352296591946*z^40-3223669414461027758005804*z^62+ 12204839323427670017464*z^76-39480764176662822216804*z^74+ 112352939304352296591946*z^72+106971361*z^104)/(-1-92703818813296533106*z^28+ 13710581657540284627*z^26+309*z^2-1746974031358289099*z^24+190569134408872848*z ^22-41518*z^4+3301686*z^6+204467729740*z^102-177126299*z^8+6898554675*z^10-\ 204467729740*z^12+4765589108448*z^14+1377983635278989*z^18-89469177739181*z^16+ 18808664449284471043834218*z^50-11603157030655744298534202*z^48-\ 17662331859748660*z^20-47901206354136619370020*z^36+12298384053959959132513*z^ 34+11603157030655744298534202*z^66-12298384053959959132513*z^80-4765589108448*z ^100+1746974031358289099*z^90-13710581657540284627*z^88-542864059881531860890*z ^84+17662331859748660*z^94+92703818813296533106*z^86-1377983635278989*z^96+ 89469177739181*z^98-190569134408872848*z^92+2765300108258394073205*z^82-\ 18808664449284471043834218*z^64-309*z^112+z^114+41518*z^110+177126299*z^106-\ 3301686*z^108+542864059881531860890*z^30+1307624495523339804413598*z^42-\ 3062015494466532515165920*z^44+6337246659832720212481088*z^46+ 38766431351166329707365198*z^58-38766431351166329707365198*z^56+ 34367894404173855017846324*z^54-27007942337616272445421164*z^52-\ 34367894404173855017846324*z^60+3062015494466532515165920*z^70-\ 6337246659832720212481088*z^68+47901206354136619370020*z^78-\ 2765300108258394073205*z^32+163816112396687467792676*z^38-\ 492959458905043844934182*z^40+27007942337616272445421164*z^62-\ 163816112396687467792676*z^76+492959458905043844934182*z^74-\ 1307624495523339804413598*z^72-6898554675*z^104) The first , 40, terms are: [0, 49, 0, 4232, 0, 389171, 0, 36177339, 0, 3372477959, 0, 314683890551, 0, 29373909088840, 0, 2742312000417545, 0, 256036192602567125, 0, 23905551294015467729, 0, 2232039520933298858725, 0, 208404712688414669538384, 0, 19458722368810254856717379, 0, 1816860621456228559140573427, 0, 169640337386782064067359551143, 0, 15839327675045683145903073780719, 0, 1478919052890131319094144289671824, 0, 138086774307274461894893078943168381, 0, 12893171937942810631326307145425960813, 0, 1203836391892287648001922988344614544165] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 29142320672673623632 z - 4559366658843422028 z - 260 z 24 22 4 6 + 614992858991231232 z - 71087980021994064 z + 30574 z - 2180048 z 102 8 10 12 - 3819176592 z + 106524275 z - 3819176592 z + 104979874718 z 14 18 16 - 2281314159104 z - 579660507399352 z + 40089817038501 z 50 48 - 3194489771532518727803724 z + 2087748375254180686952978 z 20 36 + 6990610727183164 z + 12065552652162912930288 z 34 66 - 3273579298964369681608 z - 1207279083364720551266920 z 80 100 90 + 777866686475527856476 z + 104979874718 z - 71087980021994064 z 88 84 94 + 614992858991231232 z + 29142320672673623632 z - 579660507399352 z 86 96 98 - 4559366658843422028 z + 40089817038501 z - 2281314159104 z 92 82 + 6990610727183164 z - 161404366606653608504 z 64 112 110 106 + 2087748375254180686952978 z + z - 260 z - 2180048 z 108 30 42 + 30574 z - 161404366606653608504 z - 278839064941822147630288 z 44 46 + 617296664456028432396244 z - 1207279083364720551266920 z 58 56 - 5190579200383839612127392 z + 5515002727437178918520130 z 54 52 - 5190579200383839612127392 z + 4327101856617767815690580 z 60 70 + 4327101856617767815690580 z - 278839064941822147630288 z 68 78 + 617296664456028432396244 z - 3273579298964369681608 z 32 38 + 777866686475527856476 z - 39043189049448320062244 z 40 62 + 111148480244965896087848 z - 3194489771532518727803724 z 76 74 + 12065552652162912930288 z - 39043189049448320062244 z 72 104 / 2 + 111148480244965896087848 z + 106524275 z ) / ((-1 + z ) (1 / 28 26 2 + 79633804504250824122 z - 12003665726432473075 z - 307 z 24 22 4 6 + 1556245541240053700 z - 172465736338887273 z + 41008 z - 3241403 z 102 8 10 12 - 6677866426 z + 172740119 z - 6677866426 z + 196277698674 z 14 18 16 - 4531918352040 z - 1281622684129835 z + 84192564984909 z 50 48 - 11254158669636590451805721 z + 7279900712673503457308274 z 20 36 + 16215352561053766 z + 37402811958701310236470 z 34 66 - 9866337845524247089547 z - 4154994131049751137415993 z 80 100 90 + 2274201775392000775496 z + 196277698674 z - 172465736338887273 z 88 84 + 1556245541240053700 z + 79633804504250824122 z 94 86 96 - 1281622684129835 z - 12003665726432473075 z + 84192564984909 z 98 92 82 - 4531918352040 z + 16215352561053766 z - 456723850636758599657 z 64 112 110 106 + 7279900712673503457308274 z + z - 307 z - 3241403 z 108 30 42 + 41008 z - 456723850636758599657 z - 927329523337538775017091 z 44 46 + 2091163539584162540828970 z - 4154994131049751137415993 z 58 56 - 18504525986285151745554087 z + 19690443038728111936777794 z 54 52 - 18504525986285151745554087 z + 15357535541478031450515214 z 60 70 + 15357535541478031450515214 z - 927329523337538775017091 z 68 78 + 2091163539584162540828970 z - 9866337845524247089547 z 32 38 + 2274201775392000775496 z - 124199877749219045283881 z 40 62 + 361963893988496126890748 z - 11254158669636590451805721 z 76 74 + 37402811958701310236470 z - 124199877749219045283881 z 72 104 + 361963893988496126890748 z + 172740119 z )) And in Maple-input format, it is: -(1+29142320672673623632*z^28-4559366658843422028*z^26-260*z^2+ 614992858991231232*z^24-71087980021994064*z^22+30574*z^4-2180048*z^6-3819176592 *z^102+106524275*z^8-3819176592*z^10+104979874718*z^12-2281314159104*z^14-\ 579660507399352*z^18+40089817038501*z^16-3194489771532518727803724*z^50+ 2087748375254180686952978*z^48+6990610727183164*z^20+12065552652162912930288*z^ 36-3273579298964369681608*z^34-1207279083364720551266920*z^66+ 777866686475527856476*z^80+104979874718*z^100-71087980021994064*z^90+ 614992858991231232*z^88+29142320672673623632*z^84-579660507399352*z^94-\ 4559366658843422028*z^86+40089817038501*z^96-2281314159104*z^98+ 6990610727183164*z^92-161404366606653608504*z^82+2087748375254180686952978*z^64 +z^112-260*z^110-2180048*z^106+30574*z^108-161404366606653608504*z^30-\ 278839064941822147630288*z^42+617296664456028432396244*z^44-\ 1207279083364720551266920*z^46-5190579200383839612127392*z^58+ 5515002727437178918520130*z^56-5190579200383839612127392*z^54+ 4327101856617767815690580*z^52+4327101856617767815690580*z^60-\ 278839064941822147630288*z^70+617296664456028432396244*z^68-\ 3273579298964369681608*z^78+777866686475527856476*z^32-39043189049448320062244* z^38+111148480244965896087848*z^40-3194489771532518727803724*z^62+ 12065552652162912930288*z^76-39043189049448320062244*z^74+ 111148480244965896087848*z^72+106524275*z^104)/(-1+z^2)/(1+79633804504250824122 *z^28-12003665726432473075*z^26-307*z^2+1556245541240053700*z^24-\ 172465736338887273*z^22+41008*z^4-3241403*z^6-6677866426*z^102+172740119*z^8-\ 6677866426*z^10+196277698674*z^12-4531918352040*z^14-1281622684129835*z^18+ 84192564984909*z^16-11254158669636590451805721*z^50+7279900712673503457308274*z ^48+16215352561053766*z^20+37402811958701310236470*z^36-9866337845524247089547* z^34-4154994131049751137415993*z^66+2274201775392000775496*z^80+196277698674*z^ 100-172465736338887273*z^90+1556245541240053700*z^88+79633804504250824122*z^84-\ 1281622684129835*z^94-12003665726432473075*z^86+84192564984909*z^96-\ 4531918352040*z^98+16215352561053766*z^92-456723850636758599657*z^82+ 7279900712673503457308274*z^64+z^112-307*z^110-3241403*z^106+41008*z^108-\ 456723850636758599657*z^30-927329523337538775017091*z^42+ 2091163539584162540828970*z^44-4154994131049751137415993*z^46-\ 18504525986285151745554087*z^58+19690443038728111936777794*z^56-\ 18504525986285151745554087*z^54+15357535541478031450515214*z^52+ 15357535541478031450515214*z^60-927329523337538775017091*z^70+ 2091163539584162540828970*z^68-9866337845524247089547*z^78+ 2274201775392000775496*z^32-124199877749219045283881*z^38+ 361963893988496126890748*z^40-11254158669636590451805721*z^62+ 37402811958701310236470*z^76-124199877749219045283881*z^74+ 361963893988496126890748*z^72+172740119*z^104) The first , 40, terms are: [0, 48, 0, 4043, 0, 364487, 0, 33323932, 0, 3060095221, 0, 281487345977, 0, 25912149484384, 0, 2386112706427783, 0, 219757161753946171, 0, 20240648860701563284, 0, 1864314393611468142741, 0, 171719628458078810980845, 0, 15816977211523019314833932, 0, 1456895140803347342160331163, 0, 134194177578681586360091462151, 0, 12360593153159850015919113818568, 0, 1138531548833041875153523822480577, 0, 104869906556315546251488075719419565, 0, 9659546048926915359566973359626588372, 0, 889738871159004681196058039922002185191] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 34634794144445890375 z - 5324708112708863498 z - 257 z 24 22 4 6 + 705171656762435351 z - 79975631576328267 z + 30105 z - 2152594 z 102 8 10 12 - 3848932859 z + 106043643 z - 3848932859 z + 107430531050 z 14 18 16 - 2375570782307 z - 627098828731133 z + 42535702128050 z 50 48 - 4267070847603097839831444 z + 2776415630332746032331533 z 20 36 + 7712822194863576 z + 15217812063513350183033 z 34 66 - 4074909180381330372101 z - 1596405459577719487968035 z 80 100 90 + 954507611251303662016 z + 107430531050 z - 79975631576328267 z 88 84 94 + 705171656762435351 z + 34634794144445890375 z - 627098828731133 z 86 96 98 - 5324708112708863498 z + 42535702128050 z - 2375570782307 z 92 82 + 7712822194863576 z - 195019415733401876579 z 64 112 110 106 + 2776415630332746032331533 z + z - 257 z - 2152594 z 108 30 42 + 30105 z - 195019415733401876579 z - 363165457770995574551061 z 44 46 + 810607377780527923819528 z - 1596405459577719487968035 z 58 56 - 6968525959172211890835627 z + 7408758332251032551935082 z 54 52 - 6968525959172211890835627 z + 5798269923466993511617885 z 60 70 + 5798269923466993511617885 z - 363165457770995574551061 z 68 78 + 810607377780527923819528 z - 4074909180381330372101 z 32 38 + 954507611251303662016 z - 49835099369279072672990 z 40 62 + 143398392300834217885369 z - 4267070847603097839831444 z 76 74 + 15217812063513350183033 z - 49835099369279072672990 z 72 104 / + 143398392300834217885369 z + 106043643 z ) / (-1 / 28 26 2 - 107786651531359311256 z + 15656700953737017124 z + 305 z 24 22 4 6 - 1957826791684760868 z + 209478345759008544 z - 40616 z + 3222816 z 102 8 10 12 + 204963958136 z - 173607526 z + 6824548806 z - 204963958136 z 14 18 16 + 4854332532400 z + 1456477142436265 z - 92782194597841 z 50 48 + 24742854642819750287489638 z - 15187610038852077813003806 z 20 36 - 19037362572032408 z - 59249448891112180110512 z 34 66 + 15003896016374268075408 z + 15187610038852077813003806 z 80 100 - 15003896016374268075408 z - 4854332532400 z 90 88 + 1957826791684760868 z - 15656700953737017124 z 84 94 - 642068892092198581296 z + 19037362572032408 z 86 96 98 + 107786651531359311256 z - 1456477142436265 z + 92782194597841 z 92 82 - 209478345759008544 z + 3323590154895896647104 z 64 112 114 110 - 24742854642819750287489638 z - 305 z + z + 40616 z 106 108 30 + 173607526 z - 3222816 z + 642068892092198581296 z 42 44 + 1673278584957979810786404 z - 3952919967402590207668848 z 46 58 + 8243035938902415986268696 z + 51381864165203984452320164 z 56 54 - 51381864165203984452320164 z + 45495012296804612911400840 z 52 60 - 35662793846113737448621592 z - 45495012296804612911400840 z 70 68 + 3952919967402590207668848 z - 8243035938902415986268696 z 78 32 + 59249448891112180110512 z - 3323590154895896647104 z 38 40 + 205187141423563245653560 z - 624486162518113407513988 z 62 76 + 35662793846113737448621592 z - 205187141423563245653560 z 74 72 + 624486162518113407513988 z - 1673278584957979810786404 z 104 - 6824548806 z ) And in Maple-input format, it is: -(1+34634794144445890375*z^28-5324708112708863498*z^26-257*z^2+ 705171656762435351*z^24-79975631576328267*z^22+30105*z^4-2152594*z^6-3848932859 *z^102+106043643*z^8-3848932859*z^10+107430531050*z^12-2375570782307*z^14-\ 627098828731133*z^18+42535702128050*z^16-4267070847603097839831444*z^50+ 2776415630332746032331533*z^48+7712822194863576*z^20+15217812063513350183033*z^ 36-4074909180381330372101*z^34-1596405459577719487968035*z^66+ 954507611251303662016*z^80+107430531050*z^100-79975631576328267*z^90+ 705171656762435351*z^88+34634794144445890375*z^84-627098828731133*z^94-\ 5324708112708863498*z^86+42535702128050*z^96-2375570782307*z^98+ 7712822194863576*z^92-195019415733401876579*z^82+2776415630332746032331533*z^64 +z^112-257*z^110-2152594*z^106+30105*z^108-195019415733401876579*z^30-\ 363165457770995574551061*z^42+810607377780527923819528*z^44-\ 1596405459577719487968035*z^46-6968525959172211890835627*z^58+ 7408758332251032551935082*z^56-6968525959172211890835627*z^54+ 5798269923466993511617885*z^52+5798269923466993511617885*z^60-\ 363165457770995574551061*z^70+810607377780527923819528*z^68-\ 4074909180381330372101*z^78+954507611251303662016*z^32-49835099369279072672990* z^38+143398392300834217885369*z^40-4267070847603097839831444*z^62+ 15217812063513350183033*z^76-49835099369279072672990*z^74+ 143398392300834217885369*z^72+106043643*z^104)/(-1-107786651531359311256*z^28+ 15656700953737017124*z^26+305*z^2-1957826791684760868*z^24+209478345759008544*z ^22-40616*z^4+3222816*z^6+204963958136*z^102-173607526*z^8+6824548806*z^10-\ 204963958136*z^12+4854332532400*z^14+1456477142436265*z^18-92782194597841*z^16+ 24742854642819750287489638*z^50-15187610038852077813003806*z^48-\ 19037362572032408*z^20-59249448891112180110512*z^36+15003896016374268075408*z^ 34+15187610038852077813003806*z^66-15003896016374268075408*z^80-4854332532400*z ^100+1957826791684760868*z^90-15656700953737017124*z^88-642068892092198581296*z ^84+19037362572032408*z^94+107786651531359311256*z^86-1456477142436265*z^96+ 92782194597841*z^98-209478345759008544*z^92+3323590154895896647104*z^82-\ 24742854642819750287489638*z^64-305*z^112+z^114+40616*z^110+173607526*z^106-\ 3222816*z^108+642068892092198581296*z^30+1673278584957979810786404*z^42-\ 3952919967402590207668848*z^44+8243035938902415986268696*z^46+ 51381864165203984452320164*z^58-51381864165203984452320164*z^56+ 45495012296804612911400840*z^54-35662793846113737448621592*z^52-\ 45495012296804612911400840*z^60+3952919967402590207668848*z^70-\ 8243035938902415986268696*z^68+59249448891112180110512*z^78-\ 3323590154895896647104*z^32+205187141423563245653560*z^38-\ 624486162518113407513988*z^40+35662793846113737448621592*z^62-\ 205187141423563245653560*z^76+624486162518113407513988*z^74-\ 1673278584957979810786404*z^72-6824548806*z^104) The first , 40, terms are: [0, 48, 0, 4129, 0, 379999, 0, 35327516, 0, 3290314959, 0, 306569970571, 0, 28566859961532, 0, 2661990413647711, 0, 248058288456247041, 0, 23115432258779787072, 0, 2154024430008924554969, 0, 200724016354403856693141, 0, 18704585488484551043717168, 0, 1742997845750395011051184317, 0, 162422285217749390467063227043, 0, 15135416767524747501586797993228, 0, 1410402769983146230766782092855847, 0, 131429216973390863107223415069842283, 0, 12247309381460886977190373858829279244, 0, 1141272774395154106884136239890843905843] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 870891749129572 z - 424238258269934 z - 183 z 24 22 4 6 + 167289135877146 z - 53151235371496 z + 14240 z - 631269 z 8 10 12 14 + 18022479 z - 355220016 z + 5057366031 z - 53685489245 z 18 16 50 - 2734278056455 z + 434839088148 z - 2734278056455 z 48 20 36 + 13524046409129 z + 13524046409129 z + 1971826412800206 z 34 66 64 30 - 2183124114837152 z - 183 z + 14240 z - 1452231388253810 z 42 44 46 - 424238258269934 z + 167289135877146 z - 53151235371496 z 58 56 54 52 - 355220016 z + 5057366031 z - 53685489245 z + 434839088148 z 60 68 32 38 + 18022479 z + z + 1971826412800206 z - 1452231388253810 z 40 62 / 2 + 870891749129572 z - 631269 z ) / ((-1 + z ) (1 / 28 26 2 + 3010084989371570 z - 1423665541114836 z - 228 z 24 22 4 6 + 540938022746998 z - 164422389074404 z + 20891 z - 1055634 z 8 10 12 14 + 33656653 z - 730616090 z + 11339807273 z - 130148567266 z 18 16 50 - 7586864907824 z + 1131571234183 z - 7586864907824 z 48 20 36 + 39752678677701 z + 39752678677701 z + 7054660584081798 z 34 66 64 30 - 7844918283423748 z - 228 z + 20891 z - 5128314406235748 z 42 44 46 - 1423665541114836 z + 540938022746998 z - 164422389074404 z 58 56 54 52 - 730616090 z + 11339807273 z - 130148567266 z + 1131571234183 z 60 68 32 38 + 33656653 z + z + 7054660584081798 z - 5128314406235748 z 40 62 + 3010084989371570 z - 1055634 z )) And in Maple-input format, it is: -(1+870891749129572*z^28-424238258269934*z^26-183*z^2+167289135877146*z^24-\ 53151235371496*z^22+14240*z^4-631269*z^6+18022479*z^8-355220016*z^10+5057366031 *z^12-53685489245*z^14-2734278056455*z^18+434839088148*z^16-2734278056455*z^50+ 13524046409129*z^48+13524046409129*z^20+1971826412800206*z^36-2183124114837152* z^34-183*z^66+14240*z^64-1452231388253810*z^30-424238258269934*z^42+ 167289135877146*z^44-53151235371496*z^46-355220016*z^58+5057366031*z^56-\ 53685489245*z^54+434839088148*z^52+18022479*z^60+z^68+1971826412800206*z^32-\ 1452231388253810*z^38+870891749129572*z^40-631269*z^62)/(-1+z^2)/(1+ 3010084989371570*z^28-1423665541114836*z^26-228*z^2+540938022746998*z^24-\ 164422389074404*z^22+20891*z^4-1055634*z^6+33656653*z^8-730616090*z^10+ 11339807273*z^12-130148567266*z^14-7586864907824*z^18+1131571234183*z^16-\ 7586864907824*z^50+39752678677701*z^48+39752678677701*z^20+7054660584081798*z^ 36-7844918283423748*z^34-228*z^66+20891*z^64-5128314406235748*z^30-\ 1423665541114836*z^42+540938022746998*z^44-164422389074404*z^46-730616090*z^58+ 11339807273*z^56-130148567266*z^54+1131571234183*z^52+33656653*z^60+z^68+ 7054660584081798*z^32-5128314406235748*z^38+3010084989371570*z^40-1055634*z^62) The first , 40, terms are: [0, 46, 0, 3655, 0, 310777, 0, 26808330, 0, 2322794507, 0, 201584110619, 0, 17505701460930, 0, 1520601291462813, 0, 132098362705740055, 0, 11476213763792892990, 0, 997028922883185957997, 0, 86620400383693108445581, 0, 7525476323572724625757406, 0, 653805208000227908760599319, 0, 56801917682653459849038489421, 0, 4934892778881663054489182189346, 0, 428738502043511222062857943973723, 0, 37248369688146472801397979971432955, 0, 3236100936621064761482640429263150538, 0, 281149200202057785764156605280635780313] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}, {4, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 27314106544055885432 z - 4205101757565976591 z - 251 z 24 22 4 6 + 558912049021074300 z - 63774105281407113 z + 28624 z - 1990849 z 102 8 10 12 - 3378122283 z + 95462592 z - 3378122283 z + 92138097196 z 14 18 16 - 1996259724789 z - 510341085749135 z + 35123293263440 z 50 48 - 3527410089419630541490383 z + 2286358160205548404040776 z 20 36 + 6204420750500476 z + 12131745824645253780156 z 34 66 - 3233771898664378858791 z - 1308518735389198200622445 z 80 100 90 + 754812387083486418396 z + 92138097196 z - 63774105281407113 z 88 84 94 + 558912049021074300 z + 27314106544055885432 z - 510341085749135 z 86 96 98 - 4205101757565976591 z + 35123293263440 z - 1996259724789 z 92 82 + 6204420750500476 z - 153881963167516896217 z 64 112 110 106 + 2286358160205548404040776 z + z - 251 z - 1990849 z 108 30 42 + 28624 z - 153881963167516896217 z - 294426836948864423462379 z 44 46 + 660924940709375750197868 z - 1308518735389198200622445 z 58 56 - 5787265248072144259424989 z + 6156536712722060989912014 z 54 52 - 5787265248072144259424989 z + 4806905579103108334502168 z 60 70 + 4806905579103108334502168 z - 294426836948864423462379 z 68 78 + 660924940709375750197868 z - 3233771898664378858791 z 32 38 + 754812387083486418396 z - 39939253506139041727769 z 40 62 + 115580612737043466291416 z - 3527410089419630541490383 z 76 74 + 12131745824645253780156 z - 39939253506139041727769 z 72 104 / + 115580612737043466291416 z + 95462592 z ) / (-1 / 28 26 2 - 83479398794125002752 z + 12157224577673974330 z + 288 z 24 22 4 6 - 1526508641792476852 z + 164280815180625362 z - 36927 z + 2847280 z 102 8 10 12 + 170262104190 z - 149802182 z + 5770554064 z - 170262104190 z 14 18 16 + 3970325379711 z + 1161934197011897 z - 74871487314487 z 50 48 + 19585490770645348548322297 z - 11992675537035563021804783 z 20 36 - 15043859422322789 z - 45966044082408827265391 z 34 66 + 11618553519529146185451 z + 11992675537035563021804783 z 80 100 - 11618553519529146185451 z - 3970325379711 z 90 88 + 1526508641792476852 z - 12157224577673974330 z 84 94 - 496680495687501125401 z + 15043859422322789 z 86 96 98 + 83479398794125002752 z - 1161934197011897 z + 74871487314487 z 92 82 - 164280815180625362 z + 2571035779863053896293 z 64 112 114 110 - 19585490770645348548322297 z - 288 z + z + 36927 z 106 108 30 + 149802182 z - 2847280 z + 496680495687501125401 z 42 44 + 1309265703365226261465488 z - 3102915724480615702835642 z 46 58 + 6490581742331639231547235 z + 40832695061689890986759110 z 56 54 - 40832695061689890986759110 z + 36129808700561342159063842 z 52 60 - 28283786801326975591604785 z - 36129808700561342159063842 z 70 68 + 3102915724480615702835642 z - 6490581742331639231547235 z 78 32 + 45966044082408827265391 z - 2571035779863053896293 z 38 40 + 159580355971380632652764 z - 487096751307326516477030 z 62 76 + 28283786801326975591604785 z - 159580355971380632652764 z 74 72 + 487096751307326516477030 z - 1309265703365226261465488 z 104 - 5770554064 z ) And in Maple-input format, it is: -(1+27314106544055885432*z^28-4205101757565976591*z^26-251*z^2+ 558912049021074300*z^24-63774105281407113*z^22+28624*z^4-1990849*z^6-3378122283 *z^102+95462592*z^8-3378122283*z^10+92138097196*z^12-1996259724789*z^14-\ 510341085749135*z^18+35123293263440*z^16-3527410089419630541490383*z^50+ 2286358160205548404040776*z^48+6204420750500476*z^20+12131745824645253780156*z^ 36-3233771898664378858791*z^34-1308518735389198200622445*z^66+ 754812387083486418396*z^80+92138097196*z^100-63774105281407113*z^90+ 558912049021074300*z^88+27314106544055885432*z^84-510341085749135*z^94-\ 4205101757565976591*z^86+35123293263440*z^96-1996259724789*z^98+ 6204420750500476*z^92-153881963167516896217*z^82+2286358160205548404040776*z^64 +z^112-251*z^110-1990849*z^106+28624*z^108-153881963167516896217*z^30-\ 294426836948864423462379*z^42+660924940709375750197868*z^44-\ 1308518735389198200622445*z^46-5787265248072144259424989*z^58+ 6156536712722060989912014*z^56-5787265248072144259424989*z^54+ 4806905579103108334502168*z^52+4806905579103108334502168*z^60-\ 294426836948864423462379*z^70+660924940709375750197868*z^68-\ 3233771898664378858791*z^78+754812387083486418396*z^32-39939253506139041727769* z^38+115580612737043466291416*z^40-3527410089419630541490383*z^62+ 12131745824645253780156*z^76-39939253506139041727769*z^74+ 115580612737043466291416*z^72+95462592*z^104)/(-1-83479398794125002752*z^28+ 12157224577673974330*z^26+288*z^2-1526508641792476852*z^24+164280815180625362*z ^22-36927*z^4+2847280*z^6+170262104190*z^102-149802182*z^8+5770554064*z^10-\ 170262104190*z^12+3970325379711*z^14+1161934197011897*z^18-74871487314487*z^16+ 19585490770645348548322297*z^50-11992675537035563021804783*z^48-\ 15043859422322789*z^20-45966044082408827265391*z^36+11618553519529146185451*z^ 34+11992675537035563021804783*z^66-11618553519529146185451*z^80-3970325379711*z ^100+1526508641792476852*z^90-12157224577673974330*z^88-496680495687501125401*z ^84+15043859422322789*z^94+83479398794125002752*z^86-1161934197011897*z^96+ 74871487314487*z^98-164280815180625362*z^92+2571035779863053896293*z^82-\ 19585490770645348548322297*z^64-288*z^112+z^114+36927*z^110+149802182*z^106-\ 2847280*z^108+496680495687501125401*z^30+1309265703365226261465488*z^42-\ 3102915724480615702835642*z^44+6490581742331639231547235*z^46+ 40832695061689890986759110*z^58-40832695061689890986759110*z^56+ 36129808700561342159063842*z^54-28283786801326975591604785*z^52-\ 36129808700561342159063842*z^60+3102915724480615702835642*z^70-\ 6490581742331639231547235*z^68+45966044082408827265391*z^78-\ 2571035779863053896293*z^32+159580355971380632652764*z^38-\ 487096751307326516477030*z^40+28283786801326975591604785*z^62-\ 159580355971380632652764*z^76+487096751307326516477030*z^74-\ 1309265703365226261465488*z^72-5770554064*z^104) The first , 40, terms are: [0, 37, 0, 2353, 0, 167796, 0, 12445787, 0, 937584651, 0, 71102956947, 0, 5408378184231, 0, 411957005230839, 0, 31399547440879027, 0, 2394043539530692404, 0, 182560464435579988721, 0, 13922374838507505593785, 0, 1061782000267040974304901, 0, 80977596583347172476457805, 0, 6175868483583747033191898321, 0, 471013069675827277234578017937, 0, 35922680272544325982416276643028, 0, 2739712070701550684352399526568483, 0, 208949489847262685138297241853522463, 0, 15935944372360570944778502163981571919] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 48672427 z - 167365006 z - 114 z + 398691903 z 22 4 6 8 10 - 667461220 z + 4757 z - 100842 z + 1242724 z - 9596210 z 12 14 18 16 + 48672427 z - 167365006 z - 667461220 z + 398691903 z 20 36 34 30 32 + 791823672 z + 4757 z - 100842 z - 9596210 z + 1242724 z 38 40 / 2 40 38 36 34 - 114 z + z ) / ((-1 + z ) (z - 149 z + 7717 z - 198014 z / 32 30 28 26 + 2875218 z - 25336014 z + 142290595 z - 527778811 z 24 22 20 18 + 1325139479 z - 2288392916 z + 2742792252 z - 2288392916 z 16 14 12 10 + 1325139479 z - 527778811 z + 142290595 z - 25336014 z 8 6 4 2 + 2875218 z - 198014 z + 7717 z - 149 z + 1)) And in Maple-input format, it is: -(1+48672427*z^28-167365006*z^26-114*z^2+398691903*z^24-667461220*z^22+4757*z^4 -100842*z^6+1242724*z^8-9596210*z^10+48672427*z^12-167365006*z^14-667461220*z^ 18+398691903*z^16+791823672*z^20+4757*z^36-100842*z^34-9596210*z^30+1242724*z^ 32-114*z^38+z^40)/(-1+z^2)/(z^40-149*z^38+7717*z^36-198014*z^34+2875218*z^32-\ 25336014*z^30+142290595*z^28-527778811*z^26+1325139479*z^24-2288392916*z^22+ 2742792252*z^20-2288392916*z^18+1325139479*z^16-527778811*z^14+142290595*z^12-\ 25336014*z^10+2875218*z^8-198014*z^6+7717*z^4-149*z^2+1) The first , 40, terms are: [0, 36, 0, 2291, 0, 165363, 0, 12359252, 0, 932450833, 0, 70525919729, 0, 5337812195956, 0, 404069868054771, 0, 30589401584786675, 0, 2315748115919808772, 0, 175312650062869183777, 0, 13271977288052705955553, 0, 1004750258418394608941380, 0, 76064262188269031111113971, 0, 5758418133028976612361760499, 0, 435939015782545042691410904116, 0, 33002609642296388246310897087025, 0, 2498450939887243428038340936927697, 0, 189144348507794291827841548998471828, 0, 14319106292069932229533349183824682995] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 269026697454607114 z - 70462320752390469 z - 229 z 24 22 4 6 + 15517722962115954 z - 2855421176901159 z + 22895 z - 1338584 z 8 10 12 14 + 51771519 z - 1420572585 z + 28947602412 z - 452451386555 z 18 16 50 - 54600121288091 z + 5556029507336 z - 18864431126541731611 z 48 20 + 27596320652277116337 z + 435663455150610 z 36 34 + 11057431766707348911 z - 5548347246729775046 z 66 80 88 84 86 - 2855421176901159 z + 51771519 z + z + 22895 z - 229 z 82 64 30 - 1338584 z + 15517722962115954 z - 868156199134521405 z 42 44 - 34655989412078650234 z + 37387062701899755124 z 46 58 - 34655989412078650234 z - 868156199134521405 z 56 54 + 2378027341574346677 z - 5548347246729775046 z 52 60 70 + 11057431766707348911 z + 269026697454607114 z - 54600121288091 z 68 78 32 + 435663455150610 z - 1420572585 z + 2378027341574346677 z 38 40 - 18864431126541731611 z + 27596320652277116337 z 62 76 74 - 70462320752390469 z + 28947602412 z - 452451386555 z 72 / 28 + 5556029507336 z ) / (-1 - 1037504915411308790 z / 26 2 24 + 254124360039292963 z + 273 z - 52303455146931171 z 22 4 6 8 + 8984376285606676 z - 31612 z + 2100296 z - 91004766 z 10 12 14 + 2766126278 z - 61879227906 z + 1054219611270 z 18 16 50 + 148825524834293 z - 14032158826553 z + 160231123638098506731 z 48 20 - 216262911138340159911 z - 1277412592271528 z 36 34 - 55825893139113827472 z + 26165266290391961758 z 66 80 90 88 84 + 52303455146931171 z - 2766126278 z + z - 273 z - 2100296 z 86 82 64 + 31612 z + 91004766 z - 254124360039292963 z 30 42 + 3579331551713745766 z + 216262911138340159911 z 44 46 - 251189601550518836716 z + 251189601550518836716 z 58 56 + 10483239209665375494 z - 26165266290391961758 z 54 52 + 55825893139113827472 z - 102070425383472097472 z 60 70 68 - 3579331551713745766 z + 1277412592271528 z - 8984376285606676 z 78 32 38 + 61879227906 z - 10483239209665375494 z + 102070425383472097472 z 40 62 76 - 160231123638098506731 z + 1037504915411308790 z - 1054219611270 z 74 72 + 14032158826553 z - 148825524834293 z ) And in Maple-input format, it is: -(1+269026697454607114*z^28-70462320752390469*z^26-229*z^2+15517722962115954*z^ 24-2855421176901159*z^22+22895*z^4-1338584*z^6+51771519*z^8-1420572585*z^10+ 28947602412*z^12-452451386555*z^14-54600121288091*z^18+5556029507336*z^16-\ 18864431126541731611*z^50+27596320652277116337*z^48+435663455150610*z^20+ 11057431766707348911*z^36-5548347246729775046*z^34-2855421176901159*z^66+ 51771519*z^80+z^88+22895*z^84-229*z^86-1338584*z^82+15517722962115954*z^64-\ 868156199134521405*z^30-34655989412078650234*z^42+37387062701899755124*z^44-\ 34655989412078650234*z^46-868156199134521405*z^58+2378027341574346677*z^56-\ 5548347246729775046*z^54+11057431766707348911*z^52+269026697454607114*z^60-\ 54600121288091*z^70+435663455150610*z^68-1420572585*z^78+2378027341574346677*z^ 32-18864431126541731611*z^38+27596320652277116337*z^40-70462320752390469*z^62+ 28947602412*z^76-452451386555*z^74+5556029507336*z^72)/(-1-1037504915411308790* z^28+254124360039292963*z^26+273*z^2-52303455146931171*z^24+8984376285606676*z^ 22-31612*z^4+2100296*z^6-91004766*z^8+2766126278*z^10-61879227906*z^12+ 1054219611270*z^14+148825524834293*z^18-14032158826553*z^16+ 160231123638098506731*z^50-216262911138340159911*z^48-1277412592271528*z^20-\ 55825893139113827472*z^36+26165266290391961758*z^34+52303455146931171*z^66-\ 2766126278*z^80+z^90-273*z^88-2100296*z^84+31612*z^86+91004766*z^82-\ 254124360039292963*z^64+3579331551713745766*z^30+216262911138340159911*z^42-\ 251189601550518836716*z^44+251189601550518836716*z^46+10483239209665375494*z^58 -26165266290391961758*z^56+55825893139113827472*z^54-102070425383472097472*z^52 -3579331551713745766*z^60+1277412592271528*z^70-8984376285606676*z^68+ 61879227906*z^78-10483239209665375494*z^32+102070425383472097472*z^38-\ 160231123638098506731*z^40+1037504915411308790*z^62-1054219611270*z^76+ 14032158826553*z^74-148825524834293*z^72) The first , 40, terms are: [0, 44, 0, 3295, 0, 270319, 0, 22815324, 0, 1945078533, 0, 166435558413, 0, 14261169539964, 0, 1222629819539075, 0, 104839770365602055, 0, 8990716158274704700, 0, 771042132642521827925, 0, 66125456197134612236741, 0, 5671034029681016126024380, 0, 486359264093999546243760071, 0, 41711207988310283565387143283, 0, 3577244647892877923954289346332, 0, 306792445444928777680668188871885, 0, 26311205653492015335568671916964037, 0, 2256507964691560745344321938932038844, 0, 193523187603890385017215344721283450591] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 1153872428 z - 2485257265 z - 123 z + 3923859931 z 22 4 6 8 10 - 4566394672 z + 5588 z - 129945 z + 1786071 z - 15741288 z 12 14 18 16 + 93626957 z - 388826555 z - 2485257265 z + 1153872428 z 20 36 34 30 42 + 3923859931 z + 1786071 z - 15741288 z - 388826555 z - 123 z 44 32 38 40 / 2 44 + z + 93626957 z - 129945 z + 5588 z ) / ((-1 + z ) (z / 42 40 38 36 34 - 170 z + 9499 z - 260072 z + 4109289 z - 40821438 z 32 30 28 26 + 268578855 z - 1211588580 z + 3836419093 z - 8659579050 z 24 22 20 18 + 14066804175 z - 16527413796 z + 14066804175 z - 8659579050 z 16 14 12 10 + 3836419093 z - 1211588580 z + 268578855 z - 40821438 z 8 6 4 2 + 4109289 z - 260072 z + 9499 z - 170 z + 1)) And in Maple-input format, it is: -(1+1153872428*z^28-2485257265*z^26-123*z^2+3923859931*z^24-4566394672*z^22+ 5588*z^4-129945*z^6+1786071*z^8-15741288*z^10+93626957*z^12-388826555*z^14-\ 2485257265*z^18+1153872428*z^16+3923859931*z^20+1786071*z^36-15741288*z^34-\ 388826555*z^30-123*z^42+z^44+93626957*z^32-129945*z^38+5588*z^40)/(-1+z^2)/(z^ 44-170*z^42+9499*z^40-260072*z^38+4109289*z^36-40821438*z^34+268578855*z^32-\ 1211588580*z^30+3836419093*z^28-8659579050*z^26+14066804175*z^24-16527413796*z^ 22+14066804175*z^20-8659579050*z^18+3836419093*z^16-1211588580*z^14+268578855*z ^12-40821438*z^10+4109289*z^8-260072*z^6+9499*z^4-170*z^2+1) The first , 40, terms are: [0, 48, 0, 4127, 0, 381231, 0, 35642656, 0, 3340751265, 0, 313316996065, 0, 29389642140160, 0, 2756920044086735, 0, 258618479164438143, 0, 24260318757407843664, 0, 2275798983651388619969, 0, 213486994500681592141889, 0, 20026681170004178880988176, 0, 1878652932766554189145187775, 0, 176231739852136384789077226639, 0, 16531859424736605188877159605056, 0, 1550812449688176717486494877292129, 0, 145477843278529561514215547783265569, 0, 13646913197144455242193154456280483296, 0, 1280182848574924286538403516715022474863] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 575704085528148953 z - 140743531152919997 z - 225 z 24 22 4 6 + 28803096387110705 z - 4908907025728004 z + 22537 z - 1346042 z 8 10 12 14 + 54099202 z - 1564176586 z + 33944156827 z - 569326592527 z 18 16 50 - 80140601147236 z + 7540049779427 z - 51595918352959198611 z 48 20 + 77320225938313840171 z + 692273688095548 z 36 34 + 29254741636354384067 z - 14077879333323933814 z 66 80 88 84 86 - 4908907025728004 z + 54099202 z + z + 22537 z - 225 z 82 64 30 - 1346042 z + 28803096387110705 z - 1979519211503399910 z 42 44 - 98530079522462530488 z + 106816724906505280168 z 46 58 - 98530079522462530488 z - 1979519211503399910 z 56 54 + 5740592512504909678 z - 14077879333323933814 z 52 60 70 + 29254741636354384067 z + 575704085528148953 z - 80140601147236 z 68 78 32 + 692273688095548 z - 1564176586 z + 5740592512504909678 z 38 40 - 51595918352959198611 z + 77320225938313840171 z 62 76 74 - 140743531152919997 z + 33944156827 z - 569326592527 z 72 / 2 28 + 7540049779427 z ) / ((-1 + z ) (1 + 1735475805792015138 z / 26 2 24 - 407848157561036558 z - 266 z + 79908864580580659 z 22 4 6 8 - 12985817951256344 z + 30408 z - 2028276 z + 89724689 z 10 12 14 - 2825511340 z + 66247107346 z - 1192674613890 z 18 16 50 - 190371233935800 z + 16861291311237 z - 177622267920977156854 z 48 20 + 269551368769467637095 z + 1739053336653918 z 36 34 + 98969349860858815108 z - 46581934085569102140 z 66 80 88 84 86 - 12985817951256344 z + 89724689 z + z + 30408 z - 266 z 82 64 30 - 2028276 z + 79908864580580659 z - 6182041357135785396 z 42 44 - 346114643541298927104 z + 376179622819589680580 z 46 58 - 346114643541298927104 z - 6182041357135785396 z 56 54 + 18494241760577610015 z - 46581934085569102140 z 52 60 70 + 98969349860858815108 z + 1735475805792015138 z - 190371233935800 z 68 78 32 + 1739053336653918 z - 2825511340 z + 18494241760577610015 z 38 40 - 177622267920977156854 z + 269551368769467637095 z 62 76 74 - 407848157561036558 z + 66247107346 z - 1192674613890 z 72 + 16861291311237 z )) And in Maple-input format, it is: -(1+575704085528148953*z^28-140743531152919997*z^26-225*z^2+28803096387110705*z ^24-4908907025728004*z^22+22537*z^4-1346042*z^6+54099202*z^8-1564176586*z^10+ 33944156827*z^12-569326592527*z^14-80140601147236*z^18+7540049779427*z^16-\ 51595918352959198611*z^50+77320225938313840171*z^48+692273688095548*z^20+ 29254741636354384067*z^36-14077879333323933814*z^34-4908907025728004*z^66+ 54099202*z^80+z^88+22537*z^84-225*z^86-1346042*z^82+28803096387110705*z^64-\ 1979519211503399910*z^30-98530079522462530488*z^42+106816724906505280168*z^44-\ 98530079522462530488*z^46-1979519211503399910*z^58+5740592512504909678*z^56-\ 14077879333323933814*z^54+29254741636354384067*z^52+575704085528148953*z^60-\ 80140601147236*z^70+692273688095548*z^68-1564176586*z^78+5740592512504909678*z^ 32-51595918352959198611*z^38+77320225938313840171*z^40-140743531152919997*z^62+ 33944156827*z^76-569326592527*z^74+7540049779427*z^72)/(-1+z^2)/(1+ 1735475805792015138*z^28-407848157561036558*z^26-266*z^2+79908864580580659*z^24 -12985817951256344*z^22+30408*z^4-2028276*z^6+89724689*z^8-2825511340*z^10+ 66247107346*z^12-1192674613890*z^14-190371233935800*z^18+16861291311237*z^16-\ 177622267920977156854*z^50+269551368769467637095*z^48+1739053336653918*z^20+ 98969349860858815108*z^36-46581934085569102140*z^34-12985817951256344*z^66+ 89724689*z^80+z^88+30408*z^84-266*z^86-2028276*z^82+79908864580580659*z^64-\ 6182041357135785396*z^30-346114643541298927104*z^42+376179622819589680580*z^44-\ 346114643541298927104*z^46-6182041357135785396*z^58+18494241760577610015*z^56-\ 46581934085569102140*z^54+98969349860858815108*z^52+1735475805792015138*z^60-\ 190371233935800*z^70+1739053336653918*z^68-2825511340*z^78+18494241760577610015 *z^32-177622267920977156854*z^38+269551368769467637095*z^40-407848157561036558* z^62+66247107346*z^76-1192674613890*z^74+16861291311237*z^72) The first , 40, terms are: [0, 42, 0, 3077, 0, 245893, 0, 20080498, 0, 1650976665, 0, 136065136769, 0, 11224047822130, 0, 926213259016981, 0, 76443058440902061, 0, 6309467707895008938, 0, 520785826569095802065, 0, 42986357401100388588593, 0, 3548169156851795248897066, 0, 292872727985239740417991197, 0, 24174302853652572386432500245, 0, 1995396301166226742746539916050, 0, 164704112989745087822895471853681, 0, 13595017152184594538385302572750825, 0, 1122160790345474794161823798996603538, 0, 92625470331373341936394339437760709701] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1117196466477236 z - 536323166238558 z - 183 z 24 22 4 6 + 207656384568098 z - 64569957887408 z + 14250 z - 635253 z 8 10 12 14 + 18363819 z - 368862876 z + 5377878463 z - 58632960913 z 18 16 50 - 3157227348219 z + 488363645218 z - 3157227348219 z 48 20 36 + 16034634553229 z + 16034634553229 z + 2573221389527014 z 34 66 64 30 - 2855166908087912 z - 183 z + 14250 z - 1882909827398122 z 42 44 46 - 536323166238558 z + 207656384568098 z - 64569957887408 z 58 56 54 52 - 368862876 z + 5377878463 z - 58632960913 z + 488363645218 z 60 68 32 38 + 18363819 z + z + 2573221389527014 z - 1882909827398122 z 40 62 / 2 + 1117196466477236 z - 635253 z ) / ((-1 + z ) (1 / 28 26 2 + 3750523247751966 z - 1754067373902404 z - 222 z 24 22 4 6 + 656764995078114 z - 196024695318840 z + 20167 z - 1024792 z 8 10 12 14 + 33230157 z - 739333278 z + 11811797817 z - 139754546720 z 18 16 50 - 8631475516654 z + 1251940003931 z - 8631475516654 z 48 20 36 + 46373982531197 z + 46373982531197 z + 8901190008667546 z 34 66 64 30 - 9913640430496980 z - 222 z + 20167 z - 6440431771100984 z 42 44 46 - 1754067373902404 z + 656764995078114 z - 196024695318840 z 58 56 54 52 - 739333278 z + 11811797817 z - 139754546720 z + 1251940003931 z 60 68 32 38 + 33230157 z + z + 8901190008667546 z - 6440431771100984 z 40 62 + 3750523247751966 z - 1024792 z )) And in Maple-input format, it is: -(1+1117196466477236*z^28-536323166238558*z^26-183*z^2+207656384568098*z^24-\ 64569957887408*z^22+14250*z^4-635253*z^6+18363819*z^8-368862876*z^10+5377878463 *z^12-58632960913*z^14-3157227348219*z^18+488363645218*z^16-3157227348219*z^50+ 16034634553229*z^48+16034634553229*z^20+2573221389527014*z^36-2855166908087912* z^34-183*z^66+14250*z^64-1882909827398122*z^30-536323166238558*z^42+ 207656384568098*z^44-64569957887408*z^46-368862876*z^58+5377878463*z^56-\ 58632960913*z^54+488363645218*z^52+18363819*z^60+z^68+2573221389527014*z^32-\ 1882909827398122*z^38+1117196466477236*z^40-635253*z^62)/(-1+z^2)/(1+ 3750523247751966*z^28-1754067373902404*z^26-222*z^2+656764995078114*z^24-\ 196024695318840*z^22+20167*z^4-1024792*z^6+33230157*z^8-739333278*z^10+ 11811797817*z^12-139754546720*z^14-8631475516654*z^18+1251940003931*z^16-\ 8631475516654*z^50+46373982531197*z^48+46373982531197*z^20+8901190008667546*z^ 36-9913640430496980*z^34-222*z^66+20167*z^64-6440431771100984*z^30-\ 1754067373902404*z^42+656764995078114*z^44-196024695318840*z^46-739333278*z^58+ 11811797817*z^56-139754546720*z^54+1251940003931*z^52+33230157*z^60+z^68+ 8901190008667546*z^32-6440431771100984*z^38+3750523247751966*z^40-1024792*z^62) The first , 40, terms are: [0, 40, 0, 2781, 0, 214309, 0, 16996328, 0, 1360168521, 0, 109189838521, 0, 8775567113384, 0, 705613761880597, 0, 56746717775924013, 0, 4564035977311462248, 0, 367089746616615012945, 0, 29525816542420924951729, 0, 2374840292113587930993896, 0, 191015294551714811306029709, 0, 15363933651868214644956508469, 0, 1235767998440447188773728725288, 0, 99396609898820595287333519110105, 0, 7994775021662882023927162855656233, 0, 643044373042075529401788972930424680, 0, 51722040118398313169471168538398476805] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 414374887084882923 z - 105466269381604209 z - 233 z 24 22 4 6 + 22514626529742153 z - 4007380354229556 z + 23799 z - 1426898 z 8 10 12 14 + 56791372 z - 1608555530 z + 33917708077 z - 549477960735 z 18 16 50 - 71364598277220 z + 6999699170871 z - 32031751121653912039 z 48 20 + 47292056932730792335 z + 590408069092792 z 36 34 + 18536701756624483953 z - 9150952132597960182 z 66 80 88 84 86 - 4007380354229556 z + 56791372 z + z + 23799 z - 233 z 82 64 30 - 1426898 z + 22514626529742153 z - 1372250272763726798 z 42 44 - 59721001707701292840 z + 64546990123476434960 z 46 58 - 59721001707701292840 z - 1372250272763726798 z 56 54 + 3845760093689363364 z - 9150952132597960182 z 52 60 70 + 18536701756624483953 z + 414374887084882923 z - 71364598277220 z 68 78 32 + 590408069092792 z - 1608555530 z + 3845760093689363364 z 38 40 - 32031751121653912039 z + 47292056932730792335 z 62 76 74 - 105466269381604209 z + 33917708077 z - 549477960735 z 72 / 2 28 + 6999699170871 z ) / ((-1 + z ) (1 + 1265072355013134876 z / 26 2 24 - 311591189678121744 z - 274 z + 64095397562474123 z 22 4 6 8 - 10942359765071398 z + 32134 z - 2174510 z + 96308551 z 10 12 14 - 2998303656 z + 68741719200 z - 1199454993596 z 18 16 50 - 176471463420278 z + 16321551978441 z - 108656818513237843994 z 48 20 + 161975523913481653203 z + 1538494014198078 z 36 34 + 62039323725399341814 z - 30103116141016521022 z 66 80 88 84 86 - 10942359765071398 z + 96308551 z + z + 32134 z - 274 z 82 64 30 - 2174510 z + 64095397562474123 z - 4311679477899695796 z 42 44 - 205733539725889860204 z + 222789150440991981020 z 46 58 - 205733539725889860204 z - 4311679477899695796 z 56 54 + 12387719585127886117 z - 30103116141016521022 z 52 60 70 + 62039323725399341814 z + 1265072355013134876 z - 176471463420278 z 68 78 32 + 1538494014198078 z - 2998303656 z + 12387719585127886117 z 38 40 - 108656818513237843994 z + 161975523913481653203 z 62 76 74 - 311591189678121744 z + 68741719200 z - 1199454993596 z 72 + 16321551978441 z )) And in Maple-input format, it is: -(1+414374887084882923*z^28-105466269381604209*z^26-233*z^2+22514626529742153*z ^24-4007380354229556*z^22+23799*z^4-1426898*z^6+56791372*z^8-1608555530*z^10+ 33917708077*z^12-549477960735*z^14-71364598277220*z^18+6999699170871*z^16-\ 32031751121653912039*z^50+47292056932730792335*z^48+590408069092792*z^20+ 18536701756624483953*z^36-9150952132597960182*z^34-4007380354229556*z^66+ 56791372*z^80+z^88+23799*z^84-233*z^86-1426898*z^82+22514626529742153*z^64-\ 1372250272763726798*z^30-59721001707701292840*z^42+64546990123476434960*z^44-\ 59721001707701292840*z^46-1372250272763726798*z^58+3845760093689363364*z^56-\ 9150952132597960182*z^54+18536701756624483953*z^52+414374887084882923*z^60-\ 71364598277220*z^70+590408069092792*z^68-1608555530*z^78+3845760093689363364*z^ 32-32031751121653912039*z^38+47292056932730792335*z^40-105466269381604209*z^62+ 33917708077*z^76-549477960735*z^74+6999699170871*z^72)/(-1+z^2)/(1+ 1265072355013134876*z^28-311591189678121744*z^26-274*z^2+64095397562474123*z^24 -10942359765071398*z^22+32134*z^4-2174510*z^6+96308551*z^8-2998303656*z^10+ 68741719200*z^12-1199454993596*z^14-176471463420278*z^18+16321551978441*z^16-\ 108656818513237843994*z^50+161975523913481653203*z^48+1538494014198078*z^20+ 62039323725399341814*z^36-30103116141016521022*z^34-10942359765071398*z^66+ 96308551*z^80+z^88+32134*z^84-274*z^86-2174510*z^82+64095397562474123*z^64-\ 4311679477899695796*z^30-205733539725889860204*z^42+222789150440991981020*z^44-\ 205733539725889860204*z^46-4311679477899695796*z^58+12387719585127886117*z^56-\ 30103116141016521022*z^54+62039323725399341814*z^52+1265072355013134876*z^60-\ 176471463420278*z^70+1538494014198078*z^68-2998303656*z^78+12387719585127886117 *z^32-108656818513237843994*z^38+161975523913481653203*z^40-311591189678121744* z^62+68741719200*z^76-1199454993596*z^74+16321551978441*z^72) The first , 40, terms are: [0, 42, 0, 2941, 0, 227385, 0, 18206306, 0, 1477149189, 0, 120456523581, 0, 9842250432138, 0, 804817446859417, 0, 65831801567590769, 0, 5385532796715437234, 0, 440599258555030199089, 0, 36046897159366725751597, 0, 2949142217264911239154970, 0, 241282079477417598314942469, 0, 19740358971744878885157537461, 0, 1615047442124813469558076551474, 0, 132134318881880313681709253060681, 0, 10810505978666322649416837592805233, 0, 884456404256701467500574227897188250, 0, 72361381303047433147118964017759322773] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 441290055564046441 z - 112525062022838851 z - 235 z 24 22 4 6 + 24063129920236905 z - 4288944415788696 z + 24257 z - 1471170 z 8 10 12 14 + 59204554 z - 1692940518 z + 35962861451 z - 585659555337 z 18 16 50 - 76422911719784 z + 7484540630987 z - 33875381749833170985 z 48 20 + 49982450260826920459 z + 632374696907116 z 36 34 + 19621246572614107595 z - 9697706200367738934 z 66 80 88 84 86 - 4288944415788696 z + 59204554 z + z + 24257 z - 235 z 82 64 30 - 1471170 z + 24063129920236905 z - 1458737816137998882 z 42 44 - 63094935381522839936 z + 68185171264282502728 z 46 58 - 63094935381522839936 z - 1458737816137998882 z 56 54 + 4081376618249487942 z - 9697706200367738934 z 52 60 70 + 19621246572614107595 z + 441290055564046441 z - 76422911719784 z 68 78 32 + 632374696907116 z - 1692940518 z + 4081376618249487942 z 38 40 - 33875381749833170985 z + 49982450260826920459 z 62 76 74 - 112525062022838851 z + 35962861451 z - 585659555337 z 72 / 2 28 + 7484540630987 z ) / ((-1 + z ) (1 + 1364507276681384288 z / 26 2 24 - 335642217778088440 z - 284 z + 68959233592002639 z 22 4 6 8 - 11759889517083616 z + 33866 z - 2309900 z + 102705199 z 10 12 14 - 3204162396 z + 73556066036 z - 1284658383808 z 18 16 50 - 189309900830288 z + 17495018871233 z - 117941700966523927516 z 48 20 + 175954789963557077327 z + 1651841646122578 z 36 34 + 67271315078294639138 z - 32602195740512361060 z 66 80 88 84 86 - 11759889517083616 z + 102705199 z + z + 33866 z - 284 z 82 64 30 - 2309900 z + 68959233592002639 z - 4656964173027736164 z 42 44 - 223599454872397520704 z + 242176821218887464220 z 46 58 - 223599454872397520704 z - 4656964173027736164 z 56 54 + 13398256885966103009 z - 32602195740512361060 z 52 60 70 + 67271315078294639138 z + 1364507276681384288 z - 189309900830288 z 68 78 32 + 1651841646122578 z - 3204162396 z + 13398256885966103009 z 38 40 - 117941700966523927516 z + 175954789963557077327 z 62 76 74 - 335642217778088440 z + 73556066036 z - 1284658383808 z 72 + 17495018871233 z )) And in Maple-input format, it is: -(1+441290055564046441*z^28-112525062022838851*z^26-235*z^2+24063129920236905*z ^24-4288944415788696*z^22+24257*z^4-1471170*z^6+59204554*z^8-1692940518*z^10+ 35962861451*z^12-585659555337*z^14-76422911719784*z^18+7484540630987*z^16-\ 33875381749833170985*z^50+49982450260826920459*z^48+632374696907116*z^20+ 19621246572614107595*z^36-9697706200367738934*z^34-4288944415788696*z^66+ 59204554*z^80+z^88+24257*z^84-235*z^86-1471170*z^82+24063129920236905*z^64-\ 1458737816137998882*z^30-63094935381522839936*z^42+68185171264282502728*z^44-\ 63094935381522839936*z^46-1458737816137998882*z^58+4081376618249487942*z^56-\ 9697706200367738934*z^54+19621246572614107595*z^52+441290055564046441*z^60-\ 76422911719784*z^70+632374696907116*z^68-1692940518*z^78+4081376618249487942*z^ 32-33875381749833170985*z^38+49982450260826920459*z^40-112525062022838851*z^62+ 35962861451*z^76-585659555337*z^74+7484540630987*z^72)/(-1+z^2)/(1+ 1364507276681384288*z^28-335642217778088440*z^26-284*z^2+68959233592002639*z^24 -11759889517083616*z^22+33866*z^4-2309900*z^6+102705199*z^8-3204162396*z^10+ 73556066036*z^12-1284658383808*z^14-189309900830288*z^18+17495018871233*z^16-\ 117941700966523927516*z^50+175954789963557077327*z^48+1651841646122578*z^20+ 67271315078294639138*z^36-32602195740512361060*z^34-11759889517083616*z^66+ 102705199*z^80+z^88+33866*z^84-284*z^86-2309900*z^82+68959233592002639*z^64-\ 4656964173027736164*z^30-223599454872397520704*z^42+242176821218887464220*z^44-\ 223599454872397520704*z^46-4656964173027736164*z^58+13398256885966103009*z^56-\ 32602195740512361060*z^54+67271315078294639138*z^52+1364507276681384288*z^60-\ 189309900830288*z^70+1651841646122578*z^68-3204162396*z^78+13398256885966103009 *z^32-117941700966523927516*z^38+175954789963557077327*z^40-335642217778088440* z^62+73556066036*z^76-1284658383808*z^74+17495018871233*z^72) The first , 40, terms are: [0, 50, 0, 4357, 0, 406841, 0, 38535890, 0, 3664069089, 0, 348794376497, 0, 33215664366098, 0, 3163547225818761, 0, 301318860049974309, 0, 28700259540076952722, 0, 2733682858626998196785, 0, 260382322529943018973361, 0, 24801347277448893829838994, 0, 2362322638238174691998410821, 0, 225010722659629621588101354505, 0, 21432224170511747909149060711090, 0, 2041414910062517605008230890934001, 0, 194444349038368700447117570093188321, 0, 18520784210197082287207265217761394738, 0, 1764100885757288081184871218192189120441] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6728303757056 z - 5919740708374 z - 163 z 24 22 4 6 + 4028554628362 z - 2115440079824 z + 10656 z - 382831 z 8 10 12 14 + 8620429 z - 130687536 z + 1393922976 z - 10768576408 z 18 16 50 48 - 263158555968 z + 61478836448 z - 382831 z + 8620429 z 20 36 34 + 853651155144 z + 853651155144 z - 2115440079824 z 30 42 44 46 - 5919740708374 z - 10768576408 z + 1393922976 z - 130687536 z 56 54 52 32 38 + z - 163 z + 10656 z + 4028554628362 z - 263158555968 z 40 / 2 28 + 61478836448 z ) / ((-1 + z ) (1 + 22672661134858 z / 26 2 24 22 - 19857960253152 z - 198 z + 13331589952998 z - 6843402568732 z 4 6 8 10 12 + 15647 z - 664074 z + 17212145 z - 293210404 z + 3442536388 z 14 18 16 50 - 28777932300 z - 790687376068 z + 175267743900 z - 664074 z 48 20 36 34 + 17212145 z + 2674474860772 z + 2674474860772 z - 6843402568732 z 30 42 44 46 - 19857960253152 z - 28777932300 z + 3442536388 z - 293210404 z 56 54 52 32 38 + z - 198 z + 15647 z + 13331589952998 z - 790687376068 z 40 + 175267743900 z )) And in Maple-input format, it is: -(1+6728303757056*z^28-5919740708374*z^26-163*z^2+4028554628362*z^24-\ 2115440079824*z^22+10656*z^4-382831*z^6+8620429*z^8-130687536*z^10+1393922976*z ^12-10768576408*z^14-263158555968*z^18+61478836448*z^16-382831*z^50+8620429*z^ 48+853651155144*z^20+853651155144*z^36-2115440079824*z^34-5919740708374*z^30-\ 10768576408*z^42+1393922976*z^44-130687536*z^46+z^56-163*z^54+10656*z^52+ 4028554628362*z^32-263158555968*z^38+61478836448*z^40)/(-1+z^2)/(1+ 22672661134858*z^28-19857960253152*z^26-198*z^2+13331589952998*z^24-\ 6843402568732*z^22+15647*z^4-664074*z^6+17212145*z^8-293210404*z^10+3442536388* z^12-28777932300*z^14-790687376068*z^18+175267743900*z^16-664074*z^50+17212145* z^48+2674474860772*z^20+2674474860772*z^36-6843402568732*z^34-19857960253152*z^ 30-28777932300*z^42+3442536388*z^44-293210404*z^46+z^56-198*z^54+15647*z^52+ 13331589952998*z^32-790687376068*z^38+175267743900*z^40) The first , 40, terms are: [0, 36, 0, 1975, 0, 119495, 0, 7699796, 0, 517501233, 0, 35730594065, 0, 2508224971604, 0, 177831935045511, 0, 12682379969638711, 0, 907570785800647140, 0, 65076625114520308257, 0, 4671647744851188618721, 0, 335586350066523570559460, 0, 24116001168772333864107191, 0, 1733414694369197998496311943, 0, 124610624316589706811119615700, 0, 8958588744882106367843508935505, 0, 644084017259806234696123128756977, 0, 46308001617771077619816814998970580, 0, 3329474245986681574391645554072457415] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1638984664001960 z - 786824696560498 z - 197 z 24 22 4 6 + 304555207907818 z - 94604619992096 z + 16500 z - 784703 z 8 10 12 14 + 23915019 z - 500163560 z + 7508400663 z - 83523577083 z 18 16 50 - 4593847656057 z + 704873755876 z - 4593847656057 z 48 20 36 + 23438098955845 z + 23438098955845 z + 3774302950506670 z 34 66 64 30 - 4187708814783984 z - 197 z + 16500 z - 2762031732710422 z 42 44 46 - 786824696560498 z + 304555207907818 z - 94604619992096 z 58 56 54 52 - 500163560 z + 7508400663 z - 83523577083 z + 704873755876 z 60 68 32 38 + 23915019 z + z + 3774302950506670 z - 2762031732710422 z 40 62 / 2 + 1638984664001960 z - 784703 z ) / ((-1 + z ) (1 / 28 26 2 + 5780004699637462 z - 2683569369109596 z - 250 z 24 22 4 6 + 996463713147218 z - 294786830518232 z + 24651 z - 1328840 z 8 10 12 14 + 44946957 z - 1030102090 z + 16803216585 z - 201837891840 z 18 16 50 - 12741806741178 z + 1829530886879 z - 12741806741178 z 48 20 36 + 69104245116877 z + 69104245116877 z + 13847396760423946 z 34 66 64 30 - 15441967442768828 z - 250 z + 24651 z - 9982531253234216 z 42 44 46 - 2683569369109596 z + 996463713147218 z - 294786830518232 z 58 56 54 52 - 1030102090 z + 16803216585 z - 201837891840 z + 1829530886879 z 60 68 32 38 + 44946957 z + z + 13847396760423946 z - 9982531253234216 z 40 62 + 5780004699637462 z - 1328840 z )) And in Maple-input format, it is: -(1+1638984664001960*z^28-786824696560498*z^26-197*z^2+304555207907818*z^24-\ 94604619992096*z^22+16500*z^4-784703*z^6+23915019*z^8-500163560*z^10+7508400663 *z^12-83523577083*z^14-4593847656057*z^18+704873755876*z^16-4593847656057*z^50+ 23438098955845*z^48+23438098955845*z^20+3774302950506670*z^36-4187708814783984* z^34-197*z^66+16500*z^64-2762031732710422*z^30-786824696560498*z^42+ 304555207907818*z^44-94604619992096*z^46-500163560*z^58+7508400663*z^56-\ 83523577083*z^54+704873755876*z^52+23915019*z^60+z^68+3774302950506670*z^32-\ 2762031732710422*z^38+1638984664001960*z^40-784703*z^62)/(-1+z^2)/(1+ 5780004699637462*z^28-2683569369109596*z^26-250*z^2+996463713147218*z^24-\ 294786830518232*z^22+24651*z^4-1328840*z^6+44946957*z^8-1030102090*z^10+ 16803216585*z^12-201837891840*z^14-12741806741178*z^18+1829530886879*z^16-\ 12741806741178*z^50+69104245116877*z^48+69104245116877*z^20+13847396760423946*z ^36-15441967442768828*z^34-250*z^66+24651*z^64-9982531253234216*z^30-\ 2683569369109596*z^42+996463713147218*z^44-294786830518232*z^46-1030102090*z^58 +16803216585*z^56-201837891840*z^54+1829530886879*z^52+44946957*z^60+z^68+ 13847396760423946*z^32-9982531253234216*z^38+5780004699637462*z^40-1328840*z^62 ) The first , 40, terms are: [0, 54, 0, 5153, 0, 517537, 0, 52314670, 0, 5294324905, 0, 535938173737, 0, 54256341071646, 0, 5492825898046545, 0, 556088916626418545, 0, 56298094013407486918, 0, 5699588680991031577281, 0, 577023463855946729487873, 0, 58417569188285936652062758, 0, 5914165975344274279329062673, 0, 598747264550922203304293234353, 0, 60616879965263076070475159138750, 0, 6136823269035384514012331563634793, 0, 621288985687803913043590625747451625, 0, 62898993004568947243271029135434655374, 0, 6367863285275496477720046134502092961217] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 521574955285173603 z - 126553114560606707 z - 223 z 24 22 4 6 + 25743985508846917 z - 4370456420555680 z + 22023 z - 1293166 z 8 10 12 14 + 51054016 z - 1451200338 z + 31025562241 z - 514164436817 z 18 16 50 - 71385594361616 z + 6750955612855 z - 48668329211372890077 z 48 20 + 73299504956283497283 z + 615550951778448 z 36 34 + 27412894500571861717 z - 13088620030111479002 z 66 80 88 84 86 - 4370456420555680 z + 51054016 z + z + 22023 z - 223 z 82 64 30 - 1293166 z + 25743985508846917 z - 1808661024899641254 z 42 44 - 93698796175670972944 z + 101687013492013553888 z 46 58 - 93698796175670972944 z - 1808661024899641254 z 56 54 + 5291554438172671808 z - 13088620030111479002 z 52 60 70 + 27412894500571861717 z + 521574955285173603 z - 71385594361616 z 68 78 32 + 615550951778448 z - 1451200338 z + 5291554438172671808 z 38 40 - 48668329211372890077 z + 73299504956283497283 z 62 76 74 - 126553114560606707 z + 31025562241 z - 514164436817 z 72 / 2 28 + 6750955612855 z ) / ((-1 + z ) (1 + 1567012368986938962 z / 26 2 24 - 365190218389796442 z - 266 z + 71106155944954679 z 22 4 6 8 - 11515231149842400 z + 30104 z - 1975556 z + 85729193 z 10 12 14 - 2647526604 z + 60973232034 z - 1081371260334 z 18 16 50 - 169379828005344 z + 15114496586281 z - 168400380323520603950 z 48 20 + 257190473657773899775 z + 1541842966741234 z 36 34 + 93053521934664256148 z - 43374064433163906028 z 66 80 88 84 86 - 11515231149842400 z + 85729193 z + z + 30104 z - 266 z 82 64 30 - 1975556 z + 71106155944954679 z - 5636910831047666308 z 42 44 - 331566971541330947712 z + 360860252811115638204 z 46 58 - 331566971541330947712 z - 5636910831047666308 z 56 54 + 17041405516230107719 z - 43374064433163906028 z 52 60 70 + 93053521934664256148 z + 1567012368986938962 z - 169379828005344 z 68 78 32 + 1541842966741234 z - 2647526604 z + 17041405516230107719 z 38 40 - 168400380323520603950 z + 257190473657773899775 z 62 76 74 - 365190218389796442 z + 60973232034 z - 1081371260334 z 72 + 15114496586281 z )) And in Maple-input format, it is: -(1+521574955285173603*z^28-126553114560606707*z^26-223*z^2+25743985508846917*z ^24-4370456420555680*z^22+22023*z^4-1293166*z^6+51054016*z^8-1451200338*z^10+ 31025562241*z^12-514164436817*z^14-71385594361616*z^18+6750955612855*z^16-\ 48668329211372890077*z^50+73299504956283497283*z^48+615550951778448*z^20+ 27412894500571861717*z^36-13088620030111479002*z^34-4370456420555680*z^66+ 51054016*z^80+z^88+22023*z^84-223*z^86-1293166*z^82+25743985508846917*z^64-\ 1808661024899641254*z^30-93698796175670972944*z^42+101687013492013553888*z^44-\ 93698796175670972944*z^46-1808661024899641254*z^58+5291554438172671808*z^56-\ 13088620030111479002*z^54+27412894500571861717*z^52+521574955285173603*z^60-\ 71385594361616*z^70+615550951778448*z^68-1451200338*z^78+5291554438172671808*z^ 32-48668329211372890077*z^38+73299504956283497283*z^40-126553114560606707*z^62+ 31025562241*z^76-514164436817*z^74+6750955612855*z^72)/(-1+z^2)/(1+ 1567012368986938962*z^28-365190218389796442*z^26-266*z^2+71106155944954679*z^24 -11515231149842400*z^22+30104*z^4-1975556*z^6+85729193*z^8-2647526604*z^10+ 60973232034*z^12-1081371260334*z^14-169379828005344*z^18+15114496586281*z^16-\ 168400380323520603950*z^50+257190473657773899775*z^48+1541842966741234*z^20+ 93053521934664256148*z^36-43374064433163906028*z^34-11515231149842400*z^66+ 85729193*z^80+z^88+30104*z^84-266*z^86-1975556*z^82+71106155944954679*z^64-\ 5636910831047666308*z^30-331566971541330947712*z^42+360860252811115638204*z^44-\ 331566971541330947712*z^46-5636910831047666308*z^58+17041405516230107719*z^56-\ 43374064433163906028*z^54+93053521934664256148*z^52+1567012368986938962*z^60-\ 169379828005344*z^70+1541842966741234*z^68-2647526604*z^78+17041405516230107719 *z^32-168400380323520603950*z^38+257190473657773899775*z^40-365190218389796442* z^62+60973232034*z^76-1081371260334*z^74+15114496586281*z^72) The first , 40, terms are: [0, 44, 0, 3401, 0, 284281, 0, 24212964, 0, 2075543581, 0, 178377657229, 0, 15347425192020, 0, 1321132621256585, 0, 113750665474316313, 0, 9795010282896720988, 0, 843481149007185845025, 0, 72636458073206108508897, 0, 6255152404556472386590588, 0, 538670156112952855311664281, 0, 46388329917613890065802051529, 0, 3994798893709497797065627466484, 0, 344018079794069579069912196214893, 0, 29625636321649698748774488796646333, 0, 2551256592994661532461569683738723652, 0, 219705337480821179414554484991817932537] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 30415653661622788935 z - 4676006643676490017 z - 254 z 24 22 4 6 + 620178315414820231 z - 70555766115651436 z + 29347 z - 2067597 z 102 8 10 12 - 3590021516 z + 100345811 z - 3590021516 z + 98873814461 z 14 18 16 - 2160385164922 z - 559661896652178 z + 38286237310706 z 50 48 - 3908114599866786169904232 z + 2534706028860486921844598 z 20 36 + 6837552795595761 z + 13515788110675214986456 z 34 66 - 3604326571625018229536 z - 1451752355104565910937058 z 80 100 90 + 841419936342228524149 z + 98873814461 z - 70555766115651436 z 88 84 94 + 620178315414820231 z + 30415653661622788935 z - 559661896652178 z 86 96 98 - 4676006643676490017 z + 38286237310706 z - 2160385164922 z 92 82 + 6837552795595761 z - 171490953709704088950 z 64 112 110 106 + 2534706028860486921844598 z + z - 254 z - 2067597 z 108 30 42 + 29347 z - 171490953709704088950 z - 327229552341735507124172 z 44 46 + 733897351673462141787878 z - 1451752355104565910937058 z 58 56 - 6407036145063854813745732 z + 6815191799231115537458700 z 54 52 - 6407036145063854813745732 z + 5323218259222819602377874 z 60 70 + 5323218259222819602377874 z - 327229552341735507124172 z 68 78 + 733897351673462141787878 z - 3604326571625018229536 z 32 38 + 841419936342228524149 z - 44465444658316720708528 z 40 62 + 128572871557649744543402 z - 3908114599866786169904232 z 76 74 + 13515788110675214986456 z - 44465444658316720708528 z 72 104 / 2 + 128572871557649744543402 z + 100345811 z ) / ((-1 + z ) (1 / 28 26 2 + 81341432532926065466 z - 12035346013072387224 z - 296 z 24 22 4 6 + 1532494927559192959 z - 166952445601574536 z + 38526 z - 2994464 z 102 8 10 12 - 6093187872 z + 158108279 z - 6093187872 z + 179431418000 z 14 18 16 - 4167771754916 z - 1204006765346236 z + 78150946055125 z 50 48 - 13547274309306773376442200 z + 8695671805613467932603858 z 20 36 + 15450382057901668 z + 41140249402627123107940 z 34 66 - 10659592617349705703784 z - 4914946381371818619529384 z 80 100 90 + 2412050584060015405569 z + 179431418000 z - 166952445601574536 z 88 84 + 1532494927559192959 z + 81341432532926065466 z 94 86 96 - 1204006765346236 z - 12035346013072387224 z + 78150946055125 z 98 92 82 - 4167771754916 z + 15450382057901668 z - 475387632190556967440 z 64 112 110 106 + 8695671805613467932603858 z + z - 296 z - 2994464 z 108 30 42 + 38526 z - 475387632190556967440 z - 1070010354222074062642056 z 44 46 + 2445138926290027213318160 z - 4914946381371818619529384 z 58 56 - 22476959377367299351125600 z + 23944802224352681052354462 z 54 52 - 22476959377367299351125600 z + 18590915679033783761187156 z 60 70 + 18590915679033783761187156 z - 1070010354222074062642056 z 68 78 + 2445138926290027213318160 z - 10659592617349705703784 z 32 38 + 2412050584060015405569 z - 138964925183871489213464 z 40 62 + 411539818652588737962758 z - 13547274309306773376442200 z 76 74 + 41140249402627123107940 z - 138964925183871489213464 z 72 104 + 411539818652588737962758 z + 158108279 z )) And in Maple-input format, it is: -(1+30415653661622788935*z^28-4676006643676490017*z^26-254*z^2+ 620178315414820231*z^24-70555766115651436*z^22+29347*z^4-2067597*z^6-3590021516 *z^102+100345811*z^8-3590021516*z^10+98873814461*z^12-2160385164922*z^14-\ 559661896652178*z^18+38286237310706*z^16-3908114599866786169904232*z^50+ 2534706028860486921844598*z^48+6837552795595761*z^20+13515788110675214986456*z^ 36-3604326571625018229536*z^34-1451752355104565910937058*z^66+ 841419936342228524149*z^80+98873814461*z^100-70555766115651436*z^90+ 620178315414820231*z^88+30415653661622788935*z^84-559661896652178*z^94-\ 4676006643676490017*z^86+38286237310706*z^96-2160385164922*z^98+ 6837552795595761*z^92-171490953709704088950*z^82+2534706028860486921844598*z^64 +z^112-254*z^110-2067597*z^106+29347*z^108-171490953709704088950*z^30-\ 327229552341735507124172*z^42+733897351673462141787878*z^44-\ 1451752355104565910937058*z^46-6407036145063854813745732*z^58+ 6815191799231115537458700*z^56-6407036145063854813745732*z^54+ 5323218259222819602377874*z^52+5323218259222819602377874*z^60-\ 327229552341735507124172*z^70+733897351673462141787878*z^68-\ 3604326571625018229536*z^78+841419936342228524149*z^32-44465444658316720708528* z^38+128572871557649744543402*z^40-3908114599866786169904232*z^62+ 13515788110675214986456*z^76-44465444658316720708528*z^74+ 128572871557649744543402*z^72+100345811*z^104)/(-1+z^2)/(1+81341432532926065466 *z^28-12035346013072387224*z^26-296*z^2+1532494927559192959*z^24-\ 166952445601574536*z^22+38526*z^4-2994464*z^6-6093187872*z^102+158108279*z^8-\ 6093187872*z^10+179431418000*z^12-4167771754916*z^14-1204006765346236*z^18+ 78150946055125*z^16-13547274309306773376442200*z^50+8695671805613467932603858*z ^48+15450382057901668*z^20+41140249402627123107940*z^36-10659592617349705703784 *z^34-4914946381371818619529384*z^66+2412050584060015405569*z^80+179431418000*z ^100-166952445601574536*z^90+1532494927559192959*z^88+81341432532926065466*z^84 -1204006765346236*z^94-12035346013072387224*z^86+78150946055125*z^96-\ 4167771754916*z^98+15450382057901668*z^92-475387632190556967440*z^82+ 8695671805613467932603858*z^64+z^112-296*z^110-2994464*z^106+38526*z^108-\ 475387632190556967440*z^30-1070010354222074062642056*z^42+ 2445138926290027213318160*z^44-4914946381371818619529384*z^46-\ 22476959377367299351125600*z^58+23944802224352681052354462*z^56-\ 22476959377367299351125600*z^54+18590915679033783761187156*z^52+ 18590915679033783761187156*z^60-1070010354222074062642056*z^70+ 2445138926290027213318160*z^68-10659592617349705703784*z^78+ 2412050584060015405569*z^32-138964925183871489213464*z^38+ 411539818652588737962758*z^40-13547274309306773376442200*z^62+ 41140249402627123107940*z^76-138964925183871489213464*z^74+ 411539818652588737962758*z^72+158108279*z^104) The first , 40, terms are: [0, 43, 0, 3296, 0, 274959, 0, 23367149, 0, 1996176681, 0, 170813215343, 0, 14625166049256, 0, 1252494075664319, 0, 107272260393948705, 0, 9187847757845020349, 0, 786948005973223490623, 0, 67403227613597253782456, 0, 5773196472704188903750343, 0, 494484168822703503864778597, 0, 42353432856974006453010438905, 0, 3627646103178677458511514111127, 0, 310714297765635151625417806478144, 0, 26613229307212142668058173094561755, 0, 2279470200429618151094913433869027885, 0, 195240658718524051901353199996243316917] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2944237118636 z - 2604189273606 z - 147 z 24 22 4 6 8 + 1800536600298 z - 970601353976 z + 8770 z - 289511 z + 6015029 z 10 12 14 18 - 84407480 z + 836346792 z - 6031205496 z - 131083127048 z 16 50 48 20 + 32346530200 z - 289511 z + 6015029 z + 406145601208 z 36 34 30 42 + 406145601208 z - 970601353976 z - 2604189273606 z - 6031205496 z 44 46 56 54 52 + 836346792 z - 84407480 z + z - 147 z + 8770 z 32 38 40 / 2 + 1800536600298 z - 131083127048 z + 32346530200 z ) / ((-1 + z ) ( / 28 26 2 24 1 + 10677007743274 z - 9379489684378 z - 194 z + 6354455324898 z 22 4 6 8 10 - 3313281818834 z + 13795 z - 521372 z + 12152077 z - 188925606 z 12 14 18 16 + 2054427116 z - 16121403634 z - 404076853190 z + 93279290456 z 50 48 20 36 - 521372 z + 12152077 z + 1324854278852 z + 1324854278852 z 34 30 42 44 - 3313281818834 z - 9379489684378 z - 16121403634 z + 2054427116 z 46 56 54 52 32 - 188925606 z + z - 194 z + 13795 z + 6354455324898 z 38 40 - 404076853190 z + 93279290456 z )) And in Maple-input format, it is: -(1+2944237118636*z^28-2604189273606*z^26-147*z^2+1800536600298*z^24-\ 970601353976*z^22+8770*z^4-289511*z^6+6015029*z^8-84407480*z^10+836346792*z^12-\ 6031205496*z^14-131083127048*z^18+32346530200*z^16-289511*z^50+6015029*z^48+ 406145601208*z^20+406145601208*z^36-970601353976*z^34-2604189273606*z^30-\ 6031205496*z^42+836346792*z^44-84407480*z^46+z^56-147*z^54+8770*z^52+ 1800536600298*z^32-131083127048*z^38+32346530200*z^40)/(-1+z^2)/(1+ 10677007743274*z^28-9379489684378*z^26-194*z^2+6354455324898*z^24-3313281818834 *z^22+13795*z^4-521372*z^6+12152077*z^8-188925606*z^10+2054427116*z^12-\ 16121403634*z^14-404076853190*z^18+93279290456*z^16-521372*z^50+12152077*z^48+ 1324854278852*z^20+1324854278852*z^36-3313281818834*z^34-9379489684378*z^30-\ 16121403634*z^42+2054427116*z^44-188925606*z^46+z^56-194*z^54+13795*z^52+ 6354455324898*z^32-404076853190*z^38+93279290456*z^40) The first , 40, terms are: [0, 48, 0, 4141, 0, 381679, 0, 35528552, 0, 3313231307, 0, 309097166875, 0, 28838800414216, 0, 2690721371475231, 0, 251051363401488509, 0, 23423782374119854864, 0, 2185504039164460067313, 0, 203913622894386429199377, 0, 19025710226649731029222480, 0, 1775151883574339103600677149, 0, 165626627183352046152230758847, 0, 15453426772655857577672480801736, 0, 1441847866518330650168328004745659, 0, 134528431837479255410448762275255851, 0, 12551878317428498244205031271870118824, 0, 1171125294065438192745747725997312158415] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2193465918639349 z - 936099610618537 z - 192 z 24 22 4 6 + 328104845186181 z - 93963113563946 z + 15571 z - 715107 z 8 10 12 14 + 21102875 z - 430667722 z + 6379639789 z - 70971573046 z 18 16 50 - 4084492720842 z + 607887036034 z - 93963113563946 z 48 20 36 + 328104845186181 z + 21839690942743 z + 9837039980960020 z 34 66 64 30 - 8960597374090296 z - 715107 z + 21102875 z - 4237796260883012 z 42 44 46 - 4237796260883012 z + 2193465918639349 z - 936099610618537 z 58 56 54 - 70971573046 z + 607887036034 z - 4084492720842 z 52 60 70 68 + 21839690942743 z + 6379639789 z - 192 z + 15571 z 32 38 40 + 6770067522019747 z - 8960597374090296 z + 6770067522019747 z 62 72 / 2 28 - 430667722 z + z ) / ((-1 + z ) (1 + 7419295392325960 z / 26 2 24 - 3070302704568766 z - 238 z + 1036932781099111 z 22 4 6 8 - 284380917909478 z + 22674 z - 1185922 z + 39066819 z 10 12 14 18 - 877402802 z + 14148237656 z - 169823308940 z - 11132267640884 z 16 50 48 + 1557636577481 z - 284380917909478 z + 1036932781099111 z 20 36 34 + 62916203310514 z + 35185838513577026 z - 31937596048067200 z 66 64 30 - 1185922 z + 39066819 z - 14686060358076234 z 42 44 46 - 14686060358076234 z + 7419295392325960 z - 3070302704568766 z 58 56 54 - 169823308940 z + 1557636577481 z - 11132267640884 z 52 60 70 68 + 62916203310514 z + 14148237656 z - 238 z + 22674 z 32 38 40 + 23876007546745897 z - 31937596048067200 z + 23876007546745897 z 62 72 - 877402802 z + z )) And in Maple-input format, it is: -(1+2193465918639349*z^28-936099610618537*z^26-192*z^2+328104845186181*z^24-\ 93963113563946*z^22+15571*z^4-715107*z^6+21102875*z^8-430667722*z^10+6379639789 *z^12-70971573046*z^14-4084492720842*z^18+607887036034*z^16-93963113563946*z^50 +328104845186181*z^48+21839690942743*z^20+9837039980960020*z^36-\ 8960597374090296*z^34-715107*z^66+21102875*z^64-4237796260883012*z^30-\ 4237796260883012*z^42+2193465918639349*z^44-936099610618537*z^46-70971573046*z^ 58+607887036034*z^56-4084492720842*z^54+21839690942743*z^52+6379639789*z^60-192 *z^70+15571*z^68+6770067522019747*z^32-8960597374090296*z^38+6770067522019747*z ^40-430667722*z^62+z^72)/(-1+z^2)/(1+7419295392325960*z^28-3070302704568766*z^ 26-238*z^2+1036932781099111*z^24-284380917909478*z^22+22674*z^4-1185922*z^6+ 39066819*z^8-877402802*z^10+14148237656*z^12-169823308940*z^14-11132267640884*z ^18+1557636577481*z^16-284380917909478*z^50+1036932781099111*z^48+ 62916203310514*z^20+35185838513577026*z^36-31937596048067200*z^34-1185922*z^66+ 39066819*z^64-14686060358076234*z^30-14686060358076234*z^42+7419295392325960*z^ 44-3070302704568766*z^46-169823308940*z^58+1557636577481*z^56-11132267640884*z^ 54+62916203310514*z^52+14148237656*z^60-238*z^70+22674*z^68+23876007546745897*z ^32-31937596048067200*z^38+23876007546745897*z^40-877402802*z^62+z^72) The first , 40, terms are: [0, 47, 0, 3892, 0, 346813, 0, 31368949, 0, 2848778059, 0, 259053804707, 0, 23568563495292, 0, 2144672365228921, 0, 195175068677058811, 0, 17762458707567905535, 0, 1616547895427509088053, 0, 147121815941225472224748, 0, 13389579053490359149656415, 0, 1218589346049558404445481143, 0, 110904228150153149075579456481, 0, 10093434156605403174190145451017, 0, 918607233726390351600604831292036, 0, 83602794267742233992944835844718987, 0, 7608722355230974386942449795168627781, 0, 692472746674895384391855689274264440941] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 222630734521286792 z + 60386011870498730 z + 230 z 24 22 4 6 - 13719504968389708 z + 2595138413832692 z - 23066 z + 1350055 z 8 10 12 14 - 52125160 z + 1423250618 z - 28764533812 z + 444450342492 z 18 16 50 + 51917778413208 z - 5378167840502 z + 7613676184974681326 z 48 20 36 - 12251915357454653688 z - 405656135036310 z - 7613676184974681326 z 34 66 80 84 + 4028929531388723644 z + 405656135036310 z - 1350055 z - 230 z 86 82 64 30 + z + 23066 z - 2595138413832692 z + 690904999008091362 z 42 44 + 19689316385001289408 z - 19689316385001289408 z 46 58 + 16811993415071739288 z + 222630734521286792 z 56 54 - 690904999008091362 z + 1812006689406126004 z 52 60 70 - 4028929531388723644 z - 60386011870498730 z + 5378167840502 z 68 78 32 - 51917778413208 z + 52125160 z - 1812006689406126004 z 38 40 + 12251915357454653688 z - 16811993415071739288 z 62 76 74 + 13719504968389708 z - 1423250618 z + 28764533812 z 72 / 28 - 444450342492 z ) / (1 + 879509352081026452 z / 26 2 24 - 222077695852823636 z - 281 z + 46970510233560620 z 22 4 6 8 - 8268426777781796 z + 32853 z - 2181185 z + 93875577 z 10 12 14 - 2824138040 z + 62380182728 z - 1047361912112 z 18 16 50 - 142853594202472 z + 13715040721000 z - 69764589375130653224 z 48 20 + 103258598407403533102 z + 1201887203016132 z 36 34 + 40234497633334902904 z - 19777004925251865296 z 66 80 88 84 86 - 8268426777781796 z + 93875577 z + z + 32853 z - 281 z 82 64 30 - 2181185 z + 46970510233560620 z - 2932783129971180344 z 42 44 - 130594381646369520942 z + 141219551013442012926 z 46 58 - 130594381646369520942 z - 2932783129971180344 z 56 54 + 8268682028648721752 z - 19777004925251865296 z 52 60 70 + 40234497633334902904 z + 879509352081026452 z - 142853594202472 z 68 78 32 + 1201887203016132 z - 2824138040 z + 8268682028648721752 z 38 40 - 69764589375130653224 z + 103258598407403533102 z 62 76 74 - 222077695852823636 z + 62380182728 z - 1047361912112 z 72 + 13715040721000 z ) And in Maple-input format, it is: -(-1-222630734521286792*z^28+60386011870498730*z^26+230*z^2-13719504968389708*z ^24+2595138413832692*z^22-23066*z^4+1350055*z^6-52125160*z^8+1423250618*z^10-\ 28764533812*z^12+444450342492*z^14+51917778413208*z^18-5378167840502*z^16+ 7613676184974681326*z^50-12251915357454653688*z^48-405656135036310*z^20-\ 7613676184974681326*z^36+4028929531388723644*z^34+405656135036310*z^66-1350055* z^80-230*z^84+z^86+23066*z^82-2595138413832692*z^64+690904999008091362*z^30+ 19689316385001289408*z^42-19689316385001289408*z^44+16811993415071739288*z^46+ 222630734521286792*z^58-690904999008091362*z^56+1812006689406126004*z^54-\ 4028929531388723644*z^52-60386011870498730*z^60+5378167840502*z^70-\ 51917778413208*z^68+52125160*z^78-1812006689406126004*z^32+12251915357454653688 *z^38-16811993415071739288*z^40+13719504968389708*z^62-1423250618*z^76+ 28764533812*z^74-444450342492*z^72)/(1+879509352081026452*z^28-\ 222077695852823636*z^26-281*z^2+46970510233560620*z^24-8268426777781796*z^22+ 32853*z^4-2181185*z^6+93875577*z^8-2824138040*z^10+62380182728*z^12-\ 1047361912112*z^14-142853594202472*z^18+13715040721000*z^16-\ 69764589375130653224*z^50+103258598407403533102*z^48+1201887203016132*z^20+ 40234497633334902904*z^36-19777004925251865296*z^34-8268426777781796*z^66+ 93875577*z^80+z^88+32853*z^84-281*z^86-2181185*z^82+46970510233560620*z^64-\ 2932783129971180344*z^30-130594381646369520942*z^42+141219551013442012926*z^44-\ 130594381646369520942*z^46-2932783129971180344*z^58+8268682028648721752*z^56-\ 19777004925251865296*z^54+40234497633334902904*z^52+879509352081026452*z^60-\ 142853594202472*z^70+1201887203016132*z^68-2824138040*z^78+8268682028648721752* z^32-69764589375130653224*z^38+103258598407403533102*z^40-222077695852823636*z^ 62+62380182728*z^76-1047361912112*z^74+13715040721000*z^72) The first , 40, terms are: [0, 51, 0, 4544, 0, 432491, 0, 41735957, 0, 4043714729, 0, 392320094599, 0, 38081693291472, 0, 3697203932573183, 0, 358973004936925013, 0, 34854771246253386957, 0, 3384288380942533069911, 0, 328605108469102534099568, 0, 31906705913358506201204063, 0, 3098060584593011370572544801, 0, 300813936225096311870765437853, 0, 29208283923838284602077381012403, 0, 2836051725858092226787248990493216, 0, 275373573346610341950518682025569339, 0, 26738089671574505634185755280785187081, 0, 2596202069510346265450361616555992095353] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 57581601541988 z - 34544445627598 z - 165 z 24 22 4 6 + 16815463461378 z - 6609651889570 z + 11158 z - 417309 z 8 10 12 14 + 9860003 z - 158962482 z + 1838194030 z - 15781835046 z 18 16 50 48 - 522749986900 z + 103140070224 z - 15781835046 z + 103140070224 z 20 36 34 64 + 2084232383488 z + 57581601541988 z - 78138703126574 z + z 30 42 44 - 78138703126574 z - 6609651889570 z + 2084232383488 z 46 58 56 54 - 522749986900 z - 417309 z + 9860003 z - 158962482 z 52 60 32 38 + 1838194030 z + 11158 z + 86490412667356 z - 34544445627598 z 40 62 / 2 28 + 16815463461378 z - 165 z ) / ((-1 + z ) (1 + 205929370623746 z / 26 2 24 22 - 120908871441341 z - 217 z + 57116485904054 z - 21606892284867 z 4 6 8 10 12 + 17462 z - 744720 z + 19640579 z - 348804397 z + 4399297808 z 14 18 16 50 - 40844821095 z - 1544772503933 z + 286358549762 z - 40844821095 z 48 20 36 + 286358549762 z + 6503949257380 z + 205929370623746 z 34 64 30 42 - 283096385223649 z + z - 283096385223649 z - 21606892284867 z 44 46 58 56 + 6503949257380 z - 1544772503933 z - 744720 z + 19640579 z 54 52 60 32 - 348804397 z + 4399297808 z + 17462 z + 314715065392266 z 38 40 62 - 120908871441341 z + 57116485904054 z - 217 z )) And in Maple-input format, it is: -(1+57581601541988*z^28-34544445627598*z^26-165*z^2+16815463461378*z^24-\ 6609651889570*z^22+11158*z^4-417309*z^6+9860003*z^8-158962482*z^10+1838194030*z ^12-15781835046*z^14-522749986900*z^18+103140070224*z^16-15781835046*z^50+ 103140070224*z^48+2084232383488*z^20+57581601541988*z^36-78138703126574*z^34+z^ 64-78138703126574*z^30-6609651889570*z^42+2084232383488*z^44-522749986900*z^46-\ 417309*z^58+9860003*z^56-158962482*z^54+1838194030*z^52+11158*z^60+ 86490412667356*z^32-34544445627598*z^38+16815463461378*z^40-165*z^62)/(-1+z^2)/ (1+205929370623746*z^28-120908871441341*z^26-217*z^2+57116485904054*z^24-\ 21606892284867*z^22+17462*z^4-744720*z^6+19640579*z^8-348804397*z^10+4399297808 *z^12-40844821095*z^14-1544772503933*z^18+286358549762*z^16-40844821095*z^50+ 286358549762*z^48+6503949257380*z^20+205929370623746*z^36-283096385223649*z^34+ z^64-283096385223649*z^30-21606892284867*z^42+6503949257380*z^44-1544772503933* z^46-744720*z^58+19640579*z^56-348804397*z^54+4399297808*z^52+17462*z^60+ 314715065392266*z^32-120908871441341*z^38+57116485904054*z^40-217*z^62) The first , 40, terms are: [0, 53, 0, 5033, 0, 505080, 0, 50999383, 0, 5153679827, 0, 520865460475, 0, 52643566068003, 0, 5320692118800667, 0, 537764226791012127, 0, 54352059122211988920, 0, 5493387081778051017201, 0, 555219152095725764157741, 0, 56116256013422626105137369, 0, 5671695958192587389445184745, 0, 573240936592099776920326027677, 0, 57937726952892912752540236115009, 0, 5855792898958772439860122769419960, 0, 591847700694469276834600351147188527, 0, 59818321255432523075359248955974352747, 0, 6045865437481344778042154941495238929875] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 28474875319976759636 z - 4456277285951458216 z - 260 z 24 22 4 6 + 601443227573261804 z - 69585903811807304 z + 30538 z - 2173132 z 102 8 10 12 - 3786825284 z + 105917507 z - 3786825284 z + 103797768918 z 14 18 16 - 2249630837628 z - 569086563527336 z + 39438858936073 z 50 48 - 3134754309885487350314208 z + 2047988634609563430038574 z 20 36 + 6851689802469540 z + 11799265594248463746620 z 34 66 - 3199946636760618632664 z - 1183776329415204078665304 z 80 100 90 + 760130691677915136428 z + 103797768918 z - 69585903811807304 z 88 84 94 + 601443227573261804 z + 28474875319976759636 z - 569086563527336 z 86 96 98 - 4456277285951458216 z + 39438858936073 z - 2249630837628 z 92 82 + 6851689802469540 z - 157699833938181836040 z 64 112 110 106 + 2047988634609563430038574 z + z - 260 z - 2173132 z 108 30 42 + 30538 z - 157699833938181836040 z - 273127708683827190997688 z 44 46 + 604979414776796105906268 z - 1183776329415204078665304 z 58 56 - 5095648134578199824257552 z + 5414427852029801192991266 z 54 52 - 5095648134578199824257552 z + 4247288282849506005083216 z 60 70 + 4247288282849506005083216 z - 273127708683827190997688 z 68 78 + 604979414776796105906268 z - 3199946636760618632664 z 32 38 + 760130691677915136428 z - 38200996032382522074872 z 40 62 + 108810913753858663868636 z - 3134754309885487350314208 z 76 74 + 11799265594248463746620 z - 38200996032382522074872 z 72 104 / 2 + 108810913753858663868636 z + 105917507 z ) / ((-1 + z ) (1 / 28 26 2 + 77541985090923721310 z - 11693609472237565059 z - 307 z 24 22 4 6 + 1517385304786396468 z - 168384811738862689 z + 40972 z - 3233107 z 102 8 10 12 - 6625328426 z + 171882551 z - 6625328426 z + 194118222242 z 14 18 16 - 4467880602832 z - 1256396823123259 z + 82755634299117 z 50 48 - 11052142113736182050366689 z + 7143884070433677253912322 z 20 36 + 15860311196785794 z + 36478741136965613645522 z 34 66 - 9614884838293444116075 z - 4073676765088510340179145 z 80 100 90 + 2214971214683467142344 z + 194118222242 z - 168384811738862689 z 88 84 + 1517385304786396468 z + 77541985090923721310 z 94 86 96 - 1256396823123259 z - 11693609472237565059 z + 82755634299117 z 98 92 82 - 4467880602832 z + 15860311196785794 z - 444701339346588332137 z 64 112 110 106 + 7143884070433677253912322 z + z - 307 z - 3233107 z 108 30 42 + 40972 z - 444701339346588332137 z - 907262150307683666969499 z 44 46 + 2048144338986564130018654 z - 4073676765088510340179145 z 58 56 - 18188754403685709558093999 z + 19356685882812076331832898 z 54 52 - 18188754403685709558093999 z + 15090272066796823358727394 z 60 70 + 15090272066796823358727394 z - 907262150307683666969499 z 68 78 + 2048144338986564130018654 z - 9614884838293444116075 z 32 38 + 2214971214683467142344 z - 121248297132886904581785 z 40 62 + 353738974095438022418204 z - 11052142113736182050366689 z 76 74 + 36478741136965613645522 z - 121248297132886904581785 z 72 104 + 353738974095438022418204 z + 171882551 z )) And in Maple-input format, it is: -(1+28474875319976759636*z^28-4456277285951458216*z^26-260*z^2+ 601443227573261804*z^24-69585903811807304*z^22+30538*z^4-2173132*z^6-3786825284 *z^102+105917507*z^8-3786825284*z^10+103797768918*z^12-2249630837628*z^14-\ 569086563527336*z^18+39438858936073*z^16-3134754309885487350314208*z^50+ 2047988634609563430038574*z^48+6851689802469540*z^20+11799265594248463746620*z^ 36-3199946636760618632664*z^34-1183776329415204078665304*z^66+ 760130691677915136428*z^80+103797768918*z^100-69585903811807304*z^90+ 601443227573261804*z^88+28474875319976759636*z^84-569086563527336*z^94-\ 4456277285951458216*z^86+39438858936073*z^96-2249630837628*z^98+ 6851689802469540*z^92-157699833938181836040*z^82+2047988634609563430038574*z^64 +z^112-260*z^110-2173132*z^106+30538*z^108-157699833938181836040*z^30-\ 273127708683827190997688*z^42+604979414776796105906268*z^44-\ 1183776329415204078665304*z^46-5095648134578199824257552*z^58+ 5414427852029801192991266*z^56-5095648134578199824257552*z^54+ 4247288282849506005083216*z^52+4247288282849506005083216*z^60-\ 273127708683827190997688*z^70+604979414776796105906268*z^68-\ 3199946636760618632664*z^78+760130691677915136428*z^32-38200996032382522074872* z^38+108810913753858663868636*z^40-3134754309885487350314208*z^62+ 11799265594248463746620*z^76-38200996032382522074872*z^74+ 108810913753858663868636*z^72+105917507*z^104)/(-1+z^2)/(1+77541985090923721310 *z^28-11693609472237565059*z^26-307*z^2+1517385304786396468*z^24-\ 168384811738862689*z^22+40972*z^4-3233107*z^6-6625328426*z^102+171882551*z^8-\ 6625328426*z^10+194118222242*z^12-4467880602832*z^14-1256396823123259*z^18+ 82755634299117*z^16-11052142113736182050366689*z^50+7143884070433677253912322*z ^48+15860311196785794*z^20+36478741136965613645522*z^36-9614884838293444116075* z^34-4073676765088510340179145*z^66+2214971214683467142344*z^80+194118222242*z^ 100-168384811738862689*z^90+1517385304786396468*z^88+77541985090923721310*z^84-\ 1256396823123259*z^94-11693609472237565059*z^86+82755634299117*z^96-\ 4467880602832*z^98+15860311196785794*z^92-444701339346588332137*z^82+ 7143884070433677253912322*z^64+z^112-307*z^110-3233107*z^106+40972*z^108-\ 444701339346588332137*z^30-907262150307683666969499*z^42+ 2048144338986564130018654*z^44-4073676765088510340179145*z^46-\ 18188754403685709558093999*z^58+19356685882812076331832898*z^56-\ 18188754403685709558093999*z^54+15090272066796823358727394*z^52+ 15090272066796823358727394*z^60-907262150307683666969499*z^70+ 2048144338986564130018654*z^68-9614884838293444116075*z^78+ 2214971214683467142344*z^32-121248297132886904581785*z^38+ 353738974095438022418204*z^40-11052142113736182050366689*z^62+ 36478741136965613645522*z^76-121248297132886904581785*z^74+ 353738974095438022418204*z^72+171882551*z^104) The first , 40, terms are: [0, 48, 0, 4043, 0, 364799, 0, 33424736, 0, 3078216273, 0, 284059419713, 0, 26235622569232, 0, 2424012463247383, 0, 224000421697449939, 0, 20701115518147759456, 0, 1913164185740925710321, 0, 176814021860978711733905, 0, 16341193999895655289490176, 0, 1510261049078531597818492483, 0, 139579219111351246537543004391, 0, 12900000575284182055403206663472, 0, 1192226560893297008365860187577185, 0, 110186375973176413633614713138336561, 0, 10183498990586579669664029776950251584, 0, 941165845261445816451887760162570698543] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 5}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2996290522103248 z - 1236192734917860 z - 186 z 24 22 4 6 + 416736223048975 z - 114347359073186 z + 14720 z - 668614 z 8 10 12 14 + 19810156 z - 411404838 z + 6268908038 z - 72315327454 z 18 16 50 - 4537968058462 z + 645771754764 z - 114347359073186 z 48 20 36 + 416736223048975 z + 25404131056272 z + 14336065343795220 z 34 66 64 30 - 13003703919781948 z - 668614 z + 19810156 z - 5950703835396668 z 42 44 46 - 5950703835396668 z + 2996290522103248 z - 1236192734917860 z 58 56 54 - 72315327454 z + 645771754764 z - 4537968058462 z 52 60 70 68 + 25404131056272 z + 6268908038 z - 186 z + 14720 z 32 38 40 + 9702442938745448 z - 13003703919781948 z + 9702442938745448 z 62 72 / 28 - 411404838 z + z ) / (-1 - 13803095786836134 z / 26 2 24 + 5231186954721217 z + 226 z - 1621420355803292 z 22 4 6 8 + 409127674617383 z - 20947 z + 1090681 z - 36515511 z 10 12 14 18 + 847860865 z - 14327338145 z + 182120556357 z + 13692008312379 z 16 50 48 - 1783350149245 z + 1621420355803292 z - 5231186954721217 z 20 36 34 - 83532974782165 z - 94719705757356122 z + 78188860317045410 z 66 64 30 + 36515511 z - 847860865 z + 29889816987371250 z 42 44 46 + 53251145776164942 z - 29889816987371250 z + 13803095786836134 z 58 56 54 + 1783350149245 z - 13692008312379 z + 83532974782165 z 52 60 70 68 - 409127674617383 z - 182120556357 z + 20947 z - 1090681 z 32 38 40 - 53251145776164942 z + 94719705757356122 z - 78188860317045410 z 62 74 72 + 14327338145 z + z - 226 z ) And in Maple-input format, it is: -(1+2996290522103248*z^28-1236192734917860*z^26-186*z^2+416736223048975*z^24-\ 114347359073186*z^22+14720*z^4-668614*z^6+19810156*z^8-411404838*z^10+ 6268908038*z^12-72315327454*z^14-4537968058462*z^18+645771754764*z^16-\ 114347359073186*z^50+416736223048975*z^48+25404131056272*z^20+14336065343795220 *z^36-13003703919781948*z^34-668614*z^66+19810156*z^64-5950703835396668*z^30-\ 5950703835396668*z^42+2996290522103248*z^44-1236192734917860*z^46-72315327454*z ^58+645771754764*z^56-4537968058462*z^54+25404131056272*z^52+6268908038*z^60-\ 186*z^70+14720*z^68+9702442938745448*z^32-13003703919781948*z^38+ 9702442938745448*z^40-411404838*z^62+z^72)/(-1-13803095786836134*z^28+ 5231186954721217*z^26+226*z^2-1621420355803292*z^24+409127674617383*z^22-20947* z^4+1090681*z^6-36515511*z^8+847860865*z^10-14327338145*z^12+182120556357*z^14+ 13692008312379*z^18-1783350149245*z^16+1621420355803292*z^50-5231186954721217*z ^48-83532974782165*z^20-94719705757356122*z^36+78188860317045410*z^34+36515511* z^66-847860865*z^64+29889816987371250*z^30+53251145776164942*z^42-\ 29889816987371250*z^44+13803095786836134*z^46+1783350149245*z^58-13692008312379 *z^56+83532974782165*z^54-409127674617383*z^52-182120556357*z^60+20947*z^70-\ 1090681*z^68-53251145776164942*z^32+94719705757356122*z^38-78188860317045410*z^ 40+14327338145*z^62+z^74-226*z^72) The first , 40, terms are: [0, 40, 0, 2813, 0, 219925, 0, 17701024, 0, 1437583689, 0, 117116454961, 0, 9552494404720, 0, 779518874710637, 0, 63624945430955477, 0, 5193600993837665624, 0, 423963001263399083209, 0, 34609524261861452690617, 0, 2825316436891337483104120, 0, 230643013096952270518828965, 0, 18828438717265231950737872701, 0, 1537052432599349911249885720016, 0, 125476739444040501651189927716705, 0, 10243251708295154916964525672192537, 0, 836204500126503919428943380735775808, 0, 68263283220641474480417240613761731141] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 90127206804408 z - 51765709358266 z - 153 z 24 22 4 6 + 23791973313988 z - 8735646398228 z + 9738 z - 354003 z 8 10 12 14 + 8384995 z - 139013802 z + 1686064480 z - 15405432392 z 18 16 50 48 - 592496559458 z + 108199657044 z - 15405432392 z + 108199657044 z 20 36 34 64 + 2554928913024 z + 90127206804408 z - 125672929167176 z + z 30 42 44 - 125672929167176 z - 8735646398228 z + 2554928913024 z 46 58 56 54 - 592496559458 z - 354003 z + 8384995 z - 139013802 z 52 60 32 38 + 1686064480 z + 9738 z + 140396648775472 z - 51765709358266 z 40 62 / 2 28 + 23791973313988 z - 153 z ) / ((-1 + z ) (1 + 313371903095134 z / 26 2 24 22 - 175634313439263 z - 195 z + 78061037290778 z - 27481931876009 z 4 6 8 10 12 + 14580 z - 601744 z + 15885079 z - 289986855 z + 3837427004 z 14 18 16 50 - 37955597565 z - 1673852436299 z + 286481361550 z - 37955597565 z 48 20 36 + 286481361550 z + 7645620948632 z + 313371903095134 z 34 64 30 42 - 443543106733891 z + z - 443543106733891 z - 27481931876009 z 44 46 58 56 + 7645620948632 z - 1673852436299 z - 601744 z + 15885079 z 54 52 60 32 - 289986855 z + 3837427004 z + 14580 z + 498005175520510 z 38 40 62 - 175634313439263 z + 78061037290778 z - 195 z )) And in Maple-input format, it is: -(1+90127206804408*z^28-51765709358266*z^26-153*z^2+23791973313988*z^24-\ 8735646398228*z^22+9738*z^4-354003*z^6+8384995*z^8-139013802*z^10+1686064480*z^ 12-15405432392*z^14-592496559458*z^18+108199657044*z^16-15405432392*z^50+ 108199657044*z^48+2554928913024*z^20+90127206804408*z^36-125672929167176*z^34+z ^64-125672929167176*z^30-8735646398228*z^42+2554928913024*z^44-592496559458*z^ 46-354003*z^58+8384995*z^56-139013802*z^54+1686064480*z^52+9738*z^60+ 140396648775472*z^32-51765709358266*z^38+23791973313988*z^40-153*z^62)/(-1+z^2) /(1+313371903095134*z^28-175634313439263*z^26-195*z^2+78061037290778*z^24-\ 27481931876009*z^22+14580*z^4-601744*z^6+15885079*z^8-289986855*z^10+3837427004 *z^12-37955597565*z^14-1673852436299*z^18+286481361550*z^16-37955597565*z^50+ 286481361550*z^48+7645620948632*z^20+313371903095134*z^36-443543106733891*z^34+ z^64-443543106733891*z^30-27481931876009*z^42+7645620948632*z^44-1673852436299* z^46-601744*z^58+15885079*z^56-289986855*z^54+3837427004*z^52+14580*z^60+ 498005175520510*z^32-175634313439263*z^38+78061037290778*z^40-195*z^62) The first , 40, terms are: [0, 43, 0, 3391, 0, 291632, 0, 25457951, 0, 2228775023, 0, 195242806241, 0, 17105820609833, 0, 1498742028727087, 0, 131314670066711903, 0, 11505365826027218928, 0, 1008063336659607829807, 0, 88323293909433182715979, 0, 7738605487022923292218729, 0, 678031951183412054010727129, 0, 59406998896192708826078023339, 0, 5205051934817407732162268750319, 0, 456050063972905322495765988053552, 0, 39957653345203921328694910386028991, 0, 3500962256110709273380341290366239631, 0, 306743156632420702307608260154335039897] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 28298325472083762983 z - 4499530714451950413 z - 266 z 24 22 4 6 + 616364369127089595 z - 72280276785517884 z + 31843 z - 2298729 z 102 8 10 12 - 4067522176 z + 113138071 z - 4067522176 z + 111688386377 z 14 18 16 - 2416663294306 z - 603995314863994 z + 42168766740678 z 50 48 - 2668950279287119784305316 z + 1757011221062124444146846 z 20 36 + 7201371616231105 z + 10962625276698653902008 z 34 66 - 3023318859016209405424 z - 1025347578966164124981786 z 80 100 90 + 730500316476640420817 z + 111688386377 z - 72280276785517884 z 88 84 94 + 616364369127089595 z + 28298325472083762983 z - 603995314863994 z 86 96 98 - 4499530714451950413 z + 42168766740678 z - 2416663294306 z 92 82 + 7201371616231105 z - 154144765237389714262 z 64 112 110 106 + 1757011221062124444146846 z + z - 266 z - 2298729 z 108 30 42 + 31843 z - 154144765237389714262 z - 242384415054410291681504 z 44 46 + 529990461350693000887846 z - 1025347578966164124981786 z 58 56 - 4299991328273981800148116 z + 4563841874485280813129892 z 54 52 - 4299991328273981800148116 z + 3596181000493255610342166 z 60 70 + 3596181000493255610342166 z - 242384415054410291681504 z 68 78 + 529990461350693000887846 z - 3023318859016209405424 z 32 38 + 730500316476640420817 z - 34921423839727314939200 z 40 62 + 97951393787682255989934 z - 2668950279287119784305316 z 76 74 + 10962625276698653902008 z - 34921423839727314939200 z 72 104 / 2 + 97951393787682255989934 z + 113138071 z ) / ((-1 + z ) (1 / 28 26 2 + 78722933533180917474 z - 12086467168547800764 z - 312 z 24 22 4 6 + 1593876532672542923 z - 179349191496129296 z + 42422 z - 3406564 z 102 8 10 12 - 7163292120 z + 183787747 z - 7163292120 z + 211376603616 z 14 18 16 - 4879799532496 z - 1364432949949096 z + 90297604842213 z 50 48 - 9259615671055393323407672 z + 6045078889149644860950458 z 20 36 + 17083989476726116 z + 34163317413554534718340 z 34 66 - 9192297615068885278872 z - 3490220033623602349563136 z 80 100 90 + 2161736797601561620433 z + 211376603616 z - 179349191496129296 z 88 84 + 1593876532672542923 z + 78722933533180917474 z 94 86 96 - 1364432949949096 z - 12086467168547800764 z + 90297604842213 z 98 92 82 - 4879799532496 z + 17083989476726116 z - 442855922682324594288 z 64 112 110 106 + 6045078889149644860950458 z + z - 312 z - 3406564 z 108 30 42 + 42422 z - 442855922682324594288 z - 801981052176345735860728 z 44 46 + 1780702951238106302922256 z - 3490220033623602349563136 z 58 56 - 15061806099072901910616040 z + 16005189619119392030253118 z 54 52 - 15061806099072901910616040 z + 12551345354394412729047204 z 60 70 + 12551345354394412729047204 z - 801981052176345735860728 z 68 78 + 1780702951238106302922256 z - 9192297615068885278872 z 32 38 + 2161736797601561620433 z - 111279860644192867702344 z 40 62 + 318422744758292016099542 z - 9259615671055393323407672 z 76 74 + 34163317413554534718340 z - 111279860644192867702344 z 72 104 + 318422744758292016099542 z + 183787747 z )) And in Maple-input format, it is: -(1+28298325472083762983*z^28-4499530714451950413*z^26-266*z^2+ 616364369127089595*z^24-72280276785517884*z^22+31843*z^4-2298729*z^6-4067522176 *z^102+113138071*z^8-4067522176*z^10+111688386377*z^12-2416663294306*z^14-\ 603995314863994*z^18+42168766740678*z^16-2668950279287119784305316*z^50+ 1757011221062124444146846*z^48+7201371616231105*z^20+10962625276698653902008*z^ 36-3023318859016209405424*z^34-1025347578966164124981786*z^66+ 730500316476640420817*z^80+111688386377*z^100-72280276785517884*z^90+ 616364369127089595*z^88+28298325472083762983*z^84-603995314863994*z^94-\ 4499530714451950413*z^86+42168766740678*z^96-2416663294306*z^98+ 7201371616231105*z^92-154144765237389714262*z^82+1757011221062124444146846*z^64 +z^112-266*z^110-2298729*z^106+31843*z^108-154144765237389714262*z^30-\ 242384415054410291681504*z^42+529990461350693000887846*z^44-\ 1025347578966164124981786*z^46-4299991328273981800148116*z^58+ 4563841874485280813129892*z^56-4299991328273981800148116*z^54+ 3596181000493255610342166*z^52+3596181000493255610342166*z^60-\ 242384415054410291681504*z^70+529990461350693000887846*z^68-\ 3023318859016209405424*z^78+730500316476640420817*z^32-34921423839727314939200* z^38+97951393787682255989934*z^40-2668950279287119784305316*z^62+ 10962625276698653902008*z^76-34921423839727314939200*z^74+ 97951393787682255989934*z^72+113138071*z^104)/(-1+z^2)/(1+78722933533180917474* z^28-12086467168547800764*z^26-312*z^2+1593876532672542923*z^24-\ 179349191496129296*z^22+42422*z^4-3406564*z^6-7163292120*z^102+183787747*z^8-\ 7163292120*z^10+211376603616*z^12-4879799532496*z^14-1364432949949096*z^18+ 90297604842213*z^16-9259615671055393323407672*z^50+6045078889149644860950458*z^ 48+17083989476726116*z^20+34163317413554534718340*z^36-9192297615068885278872*z ^34-3490220033623602349563136*z^66+2161736797601561620433*z^80+211376603616*z^ 100-179349191496129296*z^90+1593876532672542923*z^88+78722933533180917474*z^84-\ 1364432949949096*z^94-12086467168547800764*z^86+90297604842213*z^96-\ 4879799532496*z^98+17083989476726116*z^92-442855922682324594288*z^82+ 6045078889149644860950458*z^64+z^112-312*z^110-3406564*z^106+42422*z^108-\ 442855922682324594288*z^30-801981052176345735860728*z^42+ 1780702951238106302922256*z^44-3490220033623602349563136*z^46-\ 15061806099072901910616040*z^58+16005189619119392030253118*z^56-\ 15061806099072901910616040*z^54+12551345354394412729047204*z^52+ 12551345354394412729047204*z^60-801981052176345735860728*z^70+ 1780702951238106302922256*z^68-9192297615068885278872*z^78+ 2161736797601561620433*z^32-111279860644192867702344*z^38+ 318422744758292016099542*z^40-9259615671055393323407672*z^62+ 34163317413554534718340*z^76-111279860644192867702344*z^74+ 318422744758292016099542*z^72+183787747*z^104) The first , 40, terms are: [0, 47, 0, 3820, 0, 337419, 0, 30414369, 0, 2756985545, 0, 250341214243, 0, 22744578268708, 0, 2066877826792839, 0, 187840260807391409, 0, 17071772614472113601, 0, 1551586107614817645735, 0, 141018664623049889757044, 0, 12816781117541468851038819, 0, 1164882548398442427504804217, 0, 105873117434924378532186288081, 0, 9622533999720693408046108968379, 0, 874567433017706878071329874348476, 0, 79487199091175133857349387359741855, 0, 7224388776516154944556913490640410449, 0, 656606285308803978685765420775820385905] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 32189507482060278303 z - 5024519689258481562 z - 263 z 24 22 4 6 + 675787642973514699 z - 77841702875707953 z + 31289 z - 2255794 z 102 8 10 12 - 4029478237 z + 111353883 z - 4029478237 z + 111687360966 z 14 18 16 - 2445163383741 z - 628964131267507 z + 43251614727770 z 50 48 - 3542573095982716247116316 z + 2315828190298053098269425 z 20 36 + 7622711260501848 z + 13389033332096976452165 z 34 66 - 3630518811459714713703 z - 1339537549687174030742353 z 80 100 90 + 861881492113131671220 z + 111687360966 z - 77841702875707953 z 88 84 94 + 675787642973514699 z + 32189507482060278303 z - 628964131267507 z 86 96 98 - 5024519689258481562 z + 43251614727770 z - 2445163383741 z 92 82 + 7622711260501848 z - 178599841088645560917 z 64 112 110 106 + 2315828190298053098269425 z + z - 263 z - 2255794 z 108 30 42 + 31289 z - 178599841088645560917 z - 309524069863166368821203 z 44 46 + 685096525340290462385888 z - 1339537549687174030742353 z 58 56 - 5754218037304317678424845 z + 6113590321013282507074322 z 54 52 - 5754218037304317678424845 z + 4797613086792477958912101 z 60 70 + 4797613086792477958912101 z - 309524069863166368821203 z 68 78 + 685096525340290462385888 z - 3630518811459714713703 z 32 38 + 861881492113131671220 z - 43339298108124813320102 z 40 62 + 123389521867982980235665 z - 3542573095982716247116316 z 76 74 + 13389033332096976452165 z - 43339298108124813320102 z 72 104 / + 123389521867982980235665 z + 111353883 z ) / (-1 / 28 26 2 - 101902474598815893552 z + 15036769953042126580 z + 315 z 24 22 4 6 - 1910753016359527966 z + 207771831641087002 z - 42854 z + 3440656 z 102 8 10 12 + 217724524946 z - 186076066 z + 7298901988 z - 217724524946 z 14 18 16 + 5104091042334 z + 1490612664002823 z - 96327703216031 z 50 48 + 20807552119063951586988070 z - 12837588903212290415632656 z 20 36 - 19186046401136058 z - 52936522089675148610918 z 34 66 + 13579652411154058610646 z + 12837588903212290415632656 z 80 100 - 13579652411154058610646 z - 5104091042334 z 90 88 + 1910753016359527966 z - 15036769953042126580 z 84 94 - 597843825058862065474 z + 19186046401136058 z 86 96 98 + 101902474598815893552 z - 1490612664002823 z + 96327703216031 z 92 82 - 207771831641087002 z + 3049893067593917304242 z 64 112 114 110 - 20807552119063951586988070 z - 315 z + z + 42854 z 106 108 30 + 186076066 z - 3440656 z + 597843825058862065474 z 42 44 + 1446773843393687402464730 z - 3388102514270168216590926 z 46 58 + 7011988159889105119275806 z + 42876770389614720976363890 z 56 54 - 42876770389614720976363890 z + 38013502203500367283923286 z 52 60 - 29875196182260153824474364 z - 38013502203500367283923286 z 70 68 + 3388102514270168216590926 z - 7011988159889105119275806 z 78 32 + 52936522089675148610918 z - 3049893067593917304242 z 38 40 + 181143274036327734514032 z - 545305939583941961923708 z 62 76 + 29875196182260153824474364 z - 181143274036327734514032 z 74 72 + 545305939583941961923708 z - 1446773843393687402464730 z 104 - 7298901988 z ) And in Maple-input format, it is: -(1+32189507482060278303*z^28-5024519689258481562*z^26-263*z^2+ 675787642973514699*z^24-77841702875707953*z^22+31289*z^4-2255794*z^6-4029478237 *z^102+111353883*z^8-4029478237*z^10+111687360966*z^12-2445163383741*z^14-\ 628964131267507*z^18+43251614727770*z^16-3542573095982716247116316*z^50+ 2315828190298053098269425*z^48+7622711260501848*z^20+13389033332096976452165*z^ 36-3630518811459714713703*z^34-1339537549687174030742353*z^66+ 861881492113131671220*z^80+111687360966*z^100-77841702875707953*z^90+ 675787642973514699*z^88+32189507482060278303*z^84-628964131267507*z^94-\ 5024519689258481562*z^86+43251614727770*z^96-2445163383741*z^98+ 7622711260501848*z^92-178599841088645560917*z^82+2315828190298053098269425*z^64 +z^112-263*z^110-2255794*z^106+31289*z^108-178599841088645560917*z^30-\ 309524069863166368821203*z^42+685096525340290462385888*z^44-\ 1339537549687174030742353*z^46-5754218037304317678424845*z^58+ 6113590321013282507074322*z^56-5754218037304317678424845*z^54+ 4797613086792477958912101*z^52+4797613086792477958912101*z^60-\ 309524069863166368821203*z^70+685096525340290462385888*z^68-\ 3630518811459714713703*z^78+861881492113131671220*z^32-43339298108124813320102* z^38+123389521867982980235665*z^40-3542573095982716247116316*z^62+ 13389033332096976452165*z^76-43339298108124813320102*z^74+ 123389521867982980235665*z^72+111353883*z^104)/(-1-101902474598815893552*z^28+ 15036769953042126580*z^26+315*z^2-1910753016359527966*z^24+207771831641087002*z ^22-42854*z^4+3440656*z^6+217724524946*z^102-186076066*z^8+7298901988*z^10-\ 217724524946*z^12+5104091042334*z^14+1490612664002823*z^18-96327703216031*z^16+ 20807552119063951586988070*z^50-12837588903212290415632656*z^48-\ 19186046401136058*z^20-52936522089675148610918*z^36+13579652411154058610646*z^ 34+12837588903212290415632656*z^66-13579652411154058610646*z^80-5104091042334*z ^100+1910753016359527966*z^90-15036769953042126580*z^88-597843825058862065474*z ^84+19186046401136058*z^94+101902474598815893552*z^86-1490612664002823*z^96+ 96327703216031*z^98-207771831641087002*z^92+3049893067593917304242*z^82-\ 20807552119063951586988070*z^64-315*z^112+z^114+42854*z^110+186076066*z^106-\ 3440656*z^108+597843825058862065474*z^30+1446773843393687402464730*z^42-\ 3388102514270168216590926*z^44+7011988159889105119275806*z^46+ 42876770389614720976363890*z^58-42876770389614720976363890*z^56+ 38013502203500367283923286*z^54-29875196182260153824474364*z^52-\ 38013502203500367283923286*z^60+3388102514270168216590926*z^70-\ 7011988159889105119275806*z^68+52936522089675148610918*z^78-\ 3049893067593917304242*z^32+181143274036327734514032*z^38-\ 545305939583941961923708*z^40+29875196182260153824474364*z^62-\ 181143274036327734514032*z^76+545305939583941961923708*z^74-\ 1446773843393687402464730*z^72-7298901988*z^104) The first , 40, terms are: [0, 52, 0, 4815, 0, 473179, 0, 46901304, 0, 4656524853, 0, 462492494109, 0, 45939851209104, 0, 4563370026232147, 0, 453299077832119627, 0, 45028224958053353284, 0, 4472857413931326326089, 0, 444309242402121493401109, 0, 44135257032939908290588292, 0, 4384155792780989511523557671, 0, 435498134233947250324813550431, 0, 43260010392451532986490788806096, 0, 4297213587852715364068202953339337, 0, 426861770365636649314848437925945721, 0, 42402121114831495437444262818834312232, 0, 4211995544899616595119766436727246411311] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 25874366170282257577 z + 4059242675995069998 z + 253 z 24 22 4 6 - 548314194299085535 z + 63418386697238560 z - 29054 z + 2031931 z 102 8 10 12 + 97805720 z - 97805720 z + 3467969944 z - 94596070045 z 14 18 16 + 2045539432337 z + 517558853422985 z - 35844513370301 z 50 48 + 2235664974624354217478780 z - 1526537042017032191273792 z 20 36 - 6238248923847732 z - 10360340248478929545936 z 34 66 + 2846524222473946098536 z + 485038182941876235268524 z 80 100 90 - 142653974756071147017 z - 3467969944 z + 6238248923847732 z 88 84 94 - 63418386697238560 z - 4059242675995069998 z + 35844513370301 z 86 96 98 + 548314194299085535 z - 2045539432337 z + 94596070045 z 92 82 - 517558853422985 z + 25874366170282257577 z 64 110 106 108 - 917410048944689655975006 z + z + 29054 z - 253 z 30 42 + 142653974756071147017 z + 225462099357958470710834 z 44 46 - 485038182941876235268524 z + 917410048944689655975006 z 58 56 + 2882707286929244504162034 z - 3273228675917200864271114 z 54 52 + 3273228675917200864271114 z - 2882707286929244504162034 z 60 70 - 2235664974624354217478780 z + 92065996741178842570430 z 68 78 - 225462099357958470710834 z + 682839015996740826216 z 32 38 - 682839015996740826216 z + 32990837442838926756816 z 40 62 - 92065996741178842570430 z + 1526537042017032191273792 z 76 74 - 2846524222473946098536 z + 10360340248478929545936 z 72 104 / - 32990837442838926756816 z - 2031931 z ) / (1 / 28 26 2 + 80777028393274371442 z - 11985943783450614916 z - 292 z 24 22 4 6 + 1529318353566774399 z - 166800262514408828 z + 37846 z - 2942596 z 102 8 10 12 - 6022430700 z + 155755111 z - 6022430700 z + 177950608808 z 14 18 16 - 4145870669988 z - 1202400245117460 z + 77924759012597 z 50 48 - 12796912881875430539666168 z + 8240562047064853320116866 z 20 36 + 15440146346800348 z + 40144873054538305442452 z 34 66 - 10454455376953290159056 z - 4676115066424689105105080 z 80 100 90 + 2376768793585074760449 z + 177950608808 z - 166800262514408828 z 88 84 + 1529318353566774399 z + 80777028393274371442 z 94 86 96 - 1202400245117460 z - 11985943783450614916 z + 77924759012597 z 98 92 82 - 4145870669988 z + 15440146346800348 z - 470404076939984892852 z 64 112 110 106 + 8240562047064853320116866 z + z - 292 z - 2942596 z 108 30 42 + 37846 z - 470404076939984892852 z - 1027716563726571898847752 z 44 46 + 2336876729332872310539040 z - 4676115066424689105105080 z 58 56 - 21150160746306177413551432 z + 22520175809552276616375358 z 54 52 - 21150160746306177413551432 z + 17519255925065632221947332 z 60 70 + 17519255925065632221947332 z - 1027716563726571898847752 z 68 78 + 2336876729332872310539040 z - 10454455376953290159056 z 32 38 + 2376768793585074760449 z - 134889213978321398655984 z 40 62 + 397344915763124821022918 z - 12796912881875430539666168 z 76 74 + 40144873054538305442452 z - 134889213978321398655984 z 72 104 + 397344915763124821022918 z + 155755111 z ) And in Maple-input format, it is: -(-1-25874366170282257577*z^28+4059242675995069998*z^26+253*z^2-\ 548314194299085535*z^24+63418386697238560*z^22-29054*z^4+2031931*z^6+97805720*z ^102-97805720*z^8+3467969944*z^10-94596070045*z^12+2045539432337*z^14+ 517558853422985*z^18-35844513370301*z^16+2235664974624354217478780*z^50-\ 1526537042017032191273792*z^48-6238248923847732*z^20-10360340248478929545936*z^ 36+2846524222473946098536*z^34+485038182941876235268524*z^66-\ 142653974756071147017*z^80-3467969944*z^100+6238248923847732*z^90-\ 63418386697238560*z^88-4059242675995069998*z^84+35844513370301*z^94+ 548314194299085535*z^86-2045539432337*z^96+94596070045*z^98-517558853422985*z^ 92+25874366170282257577*z^82-917410048944689655975006*z^64+z^110+29054*z^106-\ 253*z^108+142653974756071147017*z^30+225462099357958470710834*z^42-\ 485038182941876235268524*z^44+917410048944689655975006*z^46+ 2882707286929244504162034*z^58-3273228675917200864271114*z^56+ 3273228675917200864271114*z^54-2882707286929244504162034*z^52-\ 2235664974624354217478780*z^60+92065996741178842570430*z^70-\ 225462099357958470710834*z^68+682839015996740826216*z^78-682839015996740826216* z^32+32990837442838926756816*z^38-92065996741178842570430*z^40+ 1526537042017032191273792*z^62-2846524222473946098536*z^76+ 10360340248478929545936*z^74-32990837442838926756816*z^72-2031931*z^104)/(1+ 80777028393274371442*z^28-11985943783450614916*z^26-292*z^2+1529318353566774399 *z^24-166800262514408828*z^22+37846*z^4-2942596*z^6-6022430700*z^102+155755111* z^8-6022430700*z^10+177950608808*z^12-4145870669988*z^14-1202400245117460*z^18+ 77924759012597*z^16-12796912881875430539666168*z^50+8240562047064853320116866*z ^48+15440146346800348*z^20+40144873054538305442452*z^36-10454455376953290159056 *z^34-4676115066424689105105080*z^66+2376768793585074760449*z^80+177950608808*z ^100-166800262514408828*z^90+1529318353566774399*z^88+80777028393274371442*z^84 -1202400245117460*z^94-11985943783450614916*z^86+77924759012597*z^96-\ 4145870669988*z^98+15440146346800348*z^92-470404076939984892852*z^82+ 8240562047064853320116866*z^64+z^112-292*z^110-2942596*z^106+37846*z^108-\ 470404076939984892852*z^30-1027716563726571898847752*z^42+ 2336876729332872310539040*z^44-4676115066424689105105080*z^46-\ 21150160746306177413551432*z^58+22520175809552276616375358*z^56-\ 21150160746306177413551432*z^54+17519255925065632221947332*z^52+ 17519255925065632221947332*z^60-1027716563726571898847752*z^70+ 2336876729332872310539040*z^68-10454455376953290159056*z^78+ 2376768793585074760449*z^32-134889213978321398655984*z^38+ 397344915763124821022918*z^40-12796912881875430539666168*z^62+ 40144873054538305442452*z^76-134889213978321398655984*z^74+ 397344915763124821022918*z^72+155755111*z^104) The first , 40, terms are: [0, 39, 0, 2596, 0, 192703, 0, 14832913, 0, 1157163501, 0, 90752384263, 0, 7132967244404, 0, 561163772372491, 0, 44166047082409841, 0, 3476706664024800849, 0, 273705941947189858715, 0, 21548508120560352612884, 0, 1696515909631913657552087, 0, 133567946468781563305124429, 0, 10515942975656868778478401489, 0, 827932568856309889545537342239, 0, 65184159496306081438070701314452, 0, 5132031809294782554490178841063767, 0, 404051465797852409239908865796454113, 0, 31811494283904656848514691117686676337] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 248606745326291182 z - 65109285009727465 z - 227 z 24 22 4 6 + 14339268285751606 z - 2639207621923539 z + 22377 z - 1289774 z 8 10 12 14 + 49273507 z - 1338946521 z + 27088246876 z - 421250893471 z 18 16 50 - 50557689687539 z + 5155502997664 z - 17439822418799208149 z 48 20 + 25512999918173082293 z + 402921205333174 z 36 34 + 10221934966391764657 z - 5128775519451992916 z 66 80 88 84 86 - 2639207621923539 z + 49273507 z + z + 22377 z - 227 z 82 64 30 - 1289774 z + 14339268285751606 z - 802347532802388137 z 42 44 - 32040182592959347446 z + 34565272551195541004 z 46 58 - 32040182592959347446 z - 802347532802388137 z 56 54 + 2198002259305446577 z - 5128775519451992916 z 52 60 70 + 10221934966391764657 z + 248606745326291182 z - 50557689687539 z 68 78 32 + 402921205333174 z - 1338946521 z + 2198002259305446577 z 38 40 - 17439822418799208149 z + 25512999918173082293 z 62 76 74 - 65109285009727465 z + 27088246876 z - 421250893471 z 72 / 2 28 + 5155502997664 z ) / ((-1 + z ) (1 + 758427687832757302 z / 26 2 24 - 192726205076286850 z - 266 z + 41003205128141447 z 22 4 6 8 - 7255510814587698 z + 30200 z - 1972732 z + 84165902 z 10 12 14 - 2520384442 z + 55525773822 z - 930495864022 z 18 16 50 - 126329690953458 z + 12160153173005 z - 58444652338757164942 z 48 20 + 86215727429038859011 z + 1059230325901732 z 36 34 + 33856840897946276924 z - 16731986103681426152 z 66 80 88 84 86 - 7255510814587698 z + 84165902 z + z + 30200 z - 266 z 82 64 30 - 1972732 z + 41003205128141447 z - 2512495896498791610 z 42 44 - 108814972675498045524 z + 117586062262099085640 z 46 58 - 108814972675498045524 z - 2512495896498791610 z 56 54 + 7038114195968919658 z - 16731986103681426152 z 52 60 70 + 33856840897946276924 z + 758427687832757302 z - 126329690953458 z 68 78 32 + 1059230325901732 z - 2520384442 z + 7038114195968919658 z 38 40 - 58444652338757164942 z + 86215727429038859011 z 62 76 74 - 192726205076286850 z + 55525773822 z - 930495864022 z 72 + 12160153173005 z )) And in Maple-input format, it is: -(1+248606745326291182*z^28-65109285009727465*z^26-227*z^2+14339268285751606*z^ 24-2639207621923539*z^22+22377*z^4-1289774*z^6+49273507*z^8-1338946521*z^10+ 27088246876*z^12-421250893471*z^14-50557689687539*z^18+5155502997664*z^16-\ 17439822418799208149*z^50+25512999918173082293*z^48+402921205333174*z^20+ 10221934966391764657*z^36-5128775519451992916*z^34-2639207621923539*z^66+ 49273507*z^80+z^88+22377*z^84-227*z^86-1289774*z^82+14339268285751606*z^64-\ 802347532802388137*z^30-32040182592959347446*z^42+34565272551195541004*z^44-\ 32040182592959347446*z^46-802347532802388137*z^58+2198002259305446577*z^56-\ 5128775519451992916*z^54+10221934966391764657*z^52+248606745326291182*z^60-\ 50557689687539*z^70+402921205333174*z^68-1338946521*z^78+2198002259305446577*z^ 32-17439822418799208149*z^38+25512999918173082293*z^40-65109285009727465*z^62+ 27088246876*z^76-421250893471*z^74+5155502997664*z^72)/(-1+z^2)/(1+ 758427687832757302*z^28-192726205076286850*z^26-266*z^2+41003205128141447*z^24-\ 7255510814587698*z^22+30200*z^4-1972732*z^6+84165902*z^8-2520384442*z^10+ 55525773822*z^12-930495864022*z^14-126329690953458*z^18+12160153173005*z^16-\ 58444652338757164942*z^50+86215727429038859011*z^48+1059230325901732*z^20+ 33856840897946276924*z^36-16731986103681426152*z^34-7255510814587698*z^66+ 84165902*z^80+z^88+30200*z^84-266*z^86-1972732*z^82+41003205128141447*z^64-\ 2512495896498791610*z^30-108814972675498045524*z^42+117586062262099085640*z^44-\ 108814972675498045524*z^46-2512495896498791610*z^58+7038114195968919658*z^56-\ 16731986103681426152*z^54+33856840897946276924*z^52+758427687832757302*z^60-\ 126329690953458*z^70+1059230325901732*z^68-2520384442*z^78+7038114195968919658* z^32-58444652338757164942*z^38+86215727429038859011*z^40-192726205076286850*z^ 62+55525773822*z^76-930495864022*z^74+12160153173005*z^72) The first , 40, terms are: [0, 40, 0, 2591, 0, 186315, 0, 14060852, 0, 1087629969, 0, 85234461949, 0, 6727191884300, 0, 533048321037539, 0, 42331560476438811, 0, 3365962545810912112, 0, 267834204678044604273, 0, 21320668079169895187537, 0, 1697607760751319922973168, 0, 135186125892010239910181739, 0, 10766144867806955243051018851, 0, 857447216010506282812949341468, 0, 68291317727391776186947094468365, 0, 5439135086358194892716232096063025, 0, 433209306972855225656694674139865540, 0, 34503866279949078295626336141032572091] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 33840546890798182428 z - 5201801764416083695 z - 257 z 24 22 4 6 + 689001951190310556 z - 78181029804906433 z + 30040 z - 2141509 z 102 8 10 12 - 3805073903 z + 105160092 z - 3805073903 z + 105906581700 z 14 18 16 - 2336074299023 z - 614349410082449 z + 41741154843748 z 50 48 - 4207114836305206735159117 z + 2735742419638433049620200 z 20 36 + 7546378729921568 z + 14910624131661477548768 z 34 66 - 3988927169453823179089 z - 1571839218857217333502171 z 80 100 90 + 933608903600892675272 z + 105906581700 z - 78181029804906433 z 88 84 94 + 689001951190310556 z + 33840546890798182428 z - 614349410082449 z 86 96 98 - 5201801764416083695 z + 41741154843748 z - 2336074299023 z 92 82 + 7546378729921568 z - 190627357105827403851 z 64 112 110 106 + 2735742419638433049620200 z + z - 257 z - 2141509 z 108 30 42 + 30040 z - 190627357105827403851 z - 356923381264641349687619 z 44 46 + 797439703076592475781868 z - 1571839218857217333502171 z 58 56 - 6875557244187412198865129 z + 7310587571321499952512074 z 54 52 - 6875557244187412198865129 z + 5719352565369829574475208 z 60 70 + 5719352565369829574475208 z - 356923381264641349687619 z 68 78 + 797439703076592475781868 z - 3988927169453823179089 z 32 38 + 933608903600892675272 z - 48878294949015926549833 z 40 62 + 140790566771606254351344 z - 4207114836305206735159117 z 76 74 + 14910624131661477548768 z - 48878294949015926549833 z 72 104 / 2 + 140790566771606254351344 z + 105160092 z ) / ((-1 + z ) (1 / 28 26 2 + 91736410068994490624 z - 13584683494107029672 z - 301 z 24 22 4 6 + 1728722007445952666 z - 187913544525658814 z + 39704 z - 3126392 z 102 8 10 12 - 6528071874 z + 167242522 z - 6528071874 z + 194560578688 z 14 18 16 - 4568283785429 z - 1342109709347595 z + 86459515943312 z 50 48 - 14587887890487493903599619 z + 9394642823225578098581588 z 20 36 + 17321801567188684 z + 45760541567653801413100 z 34 66 - 11911572691786801639469 z - 5331476453162748391143945 z 80 100 90 + 2706112289766676679764 z + 194560578688 z - 187913544525658814 z 88 84 + 1728722007445952666 z + 91736410068994490624 z 94 86 96 - 1342109709347595 z - 13584683494107029672 z + 86459515943312 z 98 92 82 - 4568283785429 z + 17321801567188684 z - 535021854265867291139 z 64 112 110 106 + 9394642823225578098581588 z + z - 301 z - 3126392 z 108 30 42 + 39704 z - 535021854265867291139 z - 1171903350655405442706014 z 44 46 + 2664605135493655116637896 z - 5331476453162748391143945 z 58 56 - 24107573844077080936316046 z + 25668782267909219772220708 z 54 52 - 24107573844077080936316046 z + 19969811636932739153355296 z 60 70 + 19969811636932739153355296 z - 1171903350655405442706014 z 68 78 + 2664605135493655116637896 z - 11911572691786801639469 z 32 38 + 2706112289766676679764 z - 153796198192992717181840 z 40 62 + 453087239061126716410958 z - 14587887890487493903599619 z 76 74 + 45760541567653801413100 z - 153796198192992717181840 z 72 104 + 453087239061126716410958 z + 167242522 z )) And in Maple-input format, it is: -(1+33840546890798182428*z^28-5201801764416083695*z^26-257*z^2+ 689001951190310556*z^24-78181029804906433*z^22+30040*z^4-2141509*z^6-3805073903 *z^102+105160092*z^8-3805073903*z^10+105906581700*z^12-2336074299023*z^14-\ 614349410082449*z^18+41741154843748*z^16-4207114836305206735159117*z^50+ 2735742419638433049620200*z^48+7546378729921568*z^20+14910624131661477548768*z^ 36-3988927169453823179089*z^34-1571839218857217333502171*z^66+ 933608903600892675272*z^80+105906581700*z^100-78181029804906433*z^90+ 689001951190310556*z^88+33840546890798182428*z^84-614349410082449*z^94-\ 5201801764416083695*z^86+41741154843748*z^96-2336074299023*z^98+ 7546378729921568*z^92-190627357105827403851*z^82+2735742419638433049620200*z^64 +z^112-257*z^110-2141509*z^106+30040*z^108-190627357105827403851*z^30-\ 356923381264641349687619*z^42+797439703076592475781868*z^44-\ 1571839218857217333502171*z^46-6875557244187412198865129*z^58+ 7310587571321499952512074*z^56-6875557244187412198865129*z^54+ 5719352565369829574475208*z^52+5719352565369829574475208*z^60-\ 356923381264641349687619*z^70+797439703076592475781868*z^68-\ 3988927169453823179089*z^78+933608903600892675272*z^32-48878294949015926549833* z^38+140790566771606254351344*z^40-4207114836305206735159117*z^62+ 14910624131661477548768*z^76-48878294949015926549833*z^74+ 140790566771606254351344*z^72+105160092*z^104)/(-1+z^2)/(1+91736410068994490624 *z^28-13584683494107029672*z^26-301*z^2+1728722007445952666*z^24-\ 187913544525658814*z^22+39704*z^4-3126392*z^6-6528071874*z^102+167242522*z^8-\ 6528071874*z^10+194560578688*z^12-4568283785429*z^14-1342109709347595*z^18+ 86459515943312*z^16-14587887890487493903599619*z^50+9394642823225578098581588*z ^48+17321801567188684*z^20+45760541567653801413100*z^36-11911572691786801639469 *z^34-5331476453162748391143945*z^66+2706112289766676679764*z^80+194560578688*z ^100-187913544525658814*z^90+1728722007445952666*z^88+91736410068994490624*z^84 -1342109709347595*z^94-13584683494107029672*z^86+86459515943312*z^96-\ 4568283785429*z^98+17321801567188684*z^92-535021854265867291139*z^82+ 9394642823225578098581588*z^64+z^112-301*z^110-3126392*z^106+39704*z^108-\ 535021854265867291139*z^30-1171903350655405442706014*z^42+ 2664605135493655116637896*z^44-5331476453162748391143945*z^46-\ 24107573844077080936316046*z^58+25668782267909219772220708*z^56-\ 24107573844077080936316046*z^54+19969811636932739153355296*z^52+ 19969811636932739153355296*z^60-1171903350655405442706014*z^70+ 2664605135493655116637896*z^68-11911572691786801639469*z^78+ 2706112289766676679764*z^32-153796198192992717181840*z^38+ 453087239061126716410958*z^40-14587887890487493903599619*z^62+ 45760541567653801413100*z^76-153796198192992717181840*z^74+ 453087239061126716410958*z^72+167242522*z^104) The first , 40, terms are: [0, 45, 0, 3625, 0, 319112, 0, 28619197, 0, 2577659297, 0, 232401124169, 0, 20958535058401, 0, 1890216053158085, 0, 170478307442748737, 0, 15375479311918613160, 0, 1386719713374608454733, 0, 125068760076156167466913, 0, 11279998274397886349624521, 0, 1017347286751364363847153209, 0, 91754934922919387031919810777, 0, 8275412147639547275233204910197, 0, 746362539495468250548289575045320, 0, 67314718648753294252344829883466793, 0, 6071139837687472698274976988209110109, 0, 547558389441721026891669600568267600241] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 367119447846261142 z - 91380598542655315 z - 221 z 24 22 4 6 + 19102870971031162 z - 3337254707936297 z + 21497 z - 1237790 z 8 10 12 14 + 47756339 z - 1323026351 z + 27511305504 z - 442811249933 z 18 16 50 - 57873398691545 z + 5641968250484 z - 31282371336642554179 z 48 20 + 46694842149072633649 z + 484173069175286 z 36 34 + 17838448572336609777 z - 8650147445825927824 z 66 80 88 84 86 - 3337254707936297 z + 47756339 z + z + 21497 z - 221 z 82 64 30 - 1237790 z + 19102870971031162 z - 1243520722561726627 z 42 44 - 59366862373980704894 z + 64310983026400404180 z 46 58 - 59366862373980704894 z - 1243520722561726627 z 56 54 + 3562183653964304301 z - 8650147445825927824 z 52 60 70 + 17838448572336609777 z + 367119447846261142 z - 57873398691545 z 68 78 32 + 484173069175286 z - 1323026351 z + 3562183653964304301 z 38 40 - 31282371336642554179 z + 46694842149072633649 z 62 76 74 - 91380598542655315 z + 27511305504 z - 442811249933 z 72 / 2 28 + 5641968250484 z ) / ((-1 + z ) (1 + 1118650950908892650 z / 26 2 24 - 267502089497192484 z - 268 z + 53535067390683583 z 22 4 6 8 - 8923203944295072 z + 29992 z - 1929956 z + 81751918 z 10 12 14 - 2457543940 z + 54990585738 z - 946455178828 z 18 16 50 - 139418725940872 z + 12829805575473 z - 109611806268124224964 z 48 20 + 165872561961958724175 z + 1230977883041196 z 36 34 + 61338144320707806116 z - 29047001278737696772 z 66 80 88 84 86 - 8923203944295072 z + 81751918 z + z + 29992 z - 268 z 82 64 30 - 1929956 z + 53535067390683583 z - 3929655871871032948 z 42 44 - 212648244540434342712 z + 231001257788920590376 z 46 58 - 212648244540434342712 z - 3929655871871032948 z 56 54 + 11628508568628503378 z - 29047001278737696772 z 52 60 70 + 61338144320707806116 z + 1118650950908892650 z - 139418725940872 z 68 78 32 + 1230977883041196 z - 2457543940 z + 11628508568628503378 z 38 40 - 109611806268124224964 z + 165872561961958724175 z 62 76 74 - 267502089497192484 z + 54990585738 z - 946455178828 z 72 + 12829805575473 z )) And in Maple-input format, it is: -(1+367119447846261142*z^28-91380598542655315*z^26-221*z^2+19102870971031162*z^ 24-3337254707936297*z^22+21497*z^4-1237790*z^6+47756339*z^8-1323026351*z^10+ 27511305504*z^12-442811249933*z^14-57873398691545*z^18+5641968250484*z^16-\ 31282371336642554179*z^50+46694842149072633649*z^48+484173069175286*z^20+ 17838448572336609777*z^36-8650147445825927824*z^34-3337254707936297*z^66+ 47756339*z^80+z^88+21497*z^84-221*z^86-1237790*z^82+19102870971031162*z^64-\ 1243520722561726627*z^30-59366862373980704894*z^42+64310983026400404180*z^44-\ 59366862373980704894*z^46-1243520722561726627*z^58+3562183653964304301*z^56-\ 8650147445825927824*z^54+17838448572336609777*z^52+367119447846261142*z^60-\ 57873398691545*z^70+484173069175286*z^68-1323026351*z^78+3562183653964304301*z^ 32-31282371336642554179*z^38+46694842149072633649*z^40-91380598542655315*z^62+ 27511305504*z^76-442811249933*z^74+5641968250484*z^72)/(-1+z^2)/(1+ 1118650950908892650*z^28-267502089497192484*z^26-268*z^2+53535067390683583*z^24 -8923203944295072*z^22+29992*z^4-1929956*z^6+81751918*z^8-2457543940*z^10+ 54990585738*z^12-946455178828*z^14-139418725940872*z^18+12829805575473*z^16-\ 109611806268124224964*z^50+165872561961958724175*z^48+1230977883041196*z^20+ 61338144320707806116*z^36-29047001278737696772*z^34-8923203944295072*z^66+ 81751918*z^80+z^88+29992*z^84-268*z^86-1929956*z^82+53535067390683583*z^64-\ 3929655871871032948*z^30-212648244540434342712*z^42+231001257788920590376*z^44-\ 212648244540434342712*z^46-3929655871871032948*z^58+11628508568628503378*z^56-\ 29047001278737696772*z^54+61338144320707806116*z^52+1118650950908892650*z^60-\ 139418725940872*z^70+1230977883041196*z^68-2457543940*z^78+11628508568628503378 *z^32-109611806268124224964*z^38+165872561961958724175*z^40-267502089497192484* z^62+54990585738*z^76-946455178828*z^74+12829805575473*z^72) The first , 40, terms are: [0, 48, 0, 4149, 0, 385759, 0, 36372400, 0, 3442472067, 0, 326217024339, 0, 30926814207536, 0, 2932480559271511, 0, 278075027886824197, 0, 26369325928950732784, 0, 2500575311876219695129, 0, 237127718589444753878665, 0, 22486676265052785791526768, 0, 2132398780703035687812450629, 0, 202214207091179977442132364167, 0, 19175863742921632565512856726448, 0, 1818436822810764726848490220525395, 0, 172441385539495843832526739303143555, 0, 16352523877034040431859962649376006960, 0, 1550701049597153806938411511440763177519] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1341793563064 z - 1539929727684 z - 174 z 24 22 4 6 8 + 1341793563064 z - 886493273646 z + 12135 z - 448946 z + 9950960 z 10 12 14 18 - 142286286 z + 1379684322 z - 9395005520 z - 165600139858 z 16 50 48 20 + 46064272254 z - 174 z + 12135 z + 442348519152 z 36 34 30 42 + 46064272254 z - 165600139858 z - 886493273646 z - 142286286 z 44 46 52 32 38 + 9950960 z - 448946 z + z + 442348519152 z - 9395005520 z 40 / 2 28 + 1379684322 z ) / ((-1 + z ) (1 + 4942182523966 z / 26 2 24 22 - 5721828384670 z - 229 z + 4942182523966 z - 3181854793062 z 4 6 8 10 12 + 19289 z - 826258 z + 20725634 z - 330692874 z + 3540233764 z 14 18 16 50 - 26361258420 z - 538898874094 z + 139954133620 z - 229 z 48 20 36 34 + 19289 z + 1522661795494 z + 139954133620 z - 538898874094 z 30 42 44 46 52 - 3181854793062 z - 330692874 z + 20725634 z - 826258 z + z 32 38 40 + 1522661795494 z - 26361258420 z + 3540233764 z )) And in Maple-input format, it is: -(1+1341793563064*z^28-1539929727684*z^26-174*z^2+1341793563064*z^24-\ 886493273646*z^22+12135*z^4-448946*z^6+9950960*z^8-142286286*z^10+1379684322*z^ 12-9395005520*z^14-165600139858*z^18+46064272254*z^16-174*z^50+12135*z^48+ 442348519152*z^20+46064272254*z^36-165600139858*z^34-886493273646*z^30-\ 142286286*z^42+9950960*z^44-448946*z^46+z^52+442348519152*z^32-9395005520*z^38+ 1379684322*z^40)/(-1+z^2)/(1+4942182523966*z^28-5721828384670*z^26-229*z^2+ 4942182523966*z^24-3181854793062*z^22+19289*z^4-826258*z^6+20725634*z^8-\ 330692874*z^10+3540233764*z^12-26361258420*z^14-538898874094*z^18+139954133620* z^16-229*z^50+19289*z^48+1522661795494*z^20+139954133620*z^36-538898874094*z^34 -3181854793062*z^30-330692874*z^42+20725634*z^44-826258*z^46+z^52+1522661795494 *z^32-26361258420*z^38+3540233764*z^40) The first , 40, terms are: [0, 56, 0, 5497, 0, 567903, 0, 59076944, 0, 6153564495, 0, 641162162607, 0, 66810456365504, 0, 6961957329240367, 0, 725473472838592329, 0, 75598436111708507816, 0, 7877791368278314530401, 0, 820911393748273898911905, 0, 85543723073782498630822856, 0, 8914151986161575684485371561, 0, 928906346527129165236528430735, 0, 96797430558288418375920363467488, 0, 10086853890124187221221152060101615, 0, 1051108701531385279155643900967758095, 0, 109531625531777595952414923760486970416, 0, 11413830916907604903126182533090483785471] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {4, 6}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 251135495274269565 z - 64079059587628301 z - 213 z 24 22 4 6 + 13741466702438569 z - 2463306889432672 z + 20041 z - 1119850 z 8 10 12 14 + 42034370 z - 1134839206 z + 23018296719 z - 361506731167 z 18 16 50 - 44958098284496 z + 4493849404979 z - 19451796584932428335 z 48 20 + 28746245235399361515 z + 366687037593572 z 36 34 + 11244148143597950211 z - 5544959704779575670 z 66 80 88 84 86 - 2463306889432672 z + 42034370 z + z + 20041 z - 213 z 82 64 30 - 1119850 z + 13741466702438569 z - 830839291894041426 z 42 44 - 36324144584292304712 z + 39268166167742163336 z 46 58 - 36324144584292304712 z - 830839291894041426 z 56 54 + 2328603224325478774 z - 5544959704779575670 z 52 60 70 + 11244148143597950211 z + 251135495274269565 z - 44958098284496 z 68 78 32 + 366687037593572 z - 1134839206 z + 2328603224325478774 z 38 40 - 19451796584932428335 z + 28746245235399361515 z 62 76 74 - 64079059587628301 z + 23018296719 z - 361506731167 z 72 / 28 + 4493849404979 z ) / (-1 - 954804241355546518 z / 26 2 24 + 227842146451389341 z + 255 z - 45664518172779175 z 22 4 6 8 + 7641256967613182 z - 27680 z + 1748042 z - 73173019 z 10 12 14 + 2181872027 z - 48509578700 z + 829965533182 z 18 16 50 + 120771677634913 z - 11183067166371 z + 165107154250135519665 z 48 20 - 224572902110068338407 z - 1059851183997238 z 36 34 - 56034296969651802138 z + 25794490297192002349 z 66 80 90 88 84 + 45664518172779175 z - 2181872027 z + z - 255 z - 1748042 z 86 82 64 + 27680 z + 73173019 z - 227842146451389341 z 30 42 + 3377114183634055568 z + 224572902110068338407 z 44 46 - 261861545862957403788 z + 261861545862957403788 z 58 56 + 10122193165503090301 z - 25794490297192002349 z 54 52 + 56034296969651802138 z - 103984157354476913676 z 60 70 68 - 3377114183634055568 z + 1059851183997238 z - 7641256967613182 z 78 32 38 + 48509578700 z - 10122193165503090301 z + 103984157354476913676 z 40 62 76 - 165107154250135519665 z + 954804241355546518 z - 829965533182 z 74 72 + 11183067166371 z - 120771677634913 z ) And in Maple-input format, it is: -(1+251135495274269565*z^28-64079059587628301*z^26-213*z^2+13741466702438569*z^ 24-2463306889432672*z^22+20041*z^4-1119850*z^6+42034370*z^8-1134839206*z^10+ 23018296719*z^12-361506731167*z^14-44958098284496*z^18+4493849404979*z^16-\ 19451796584932428335*z^50+28746245235399361515*z^48+366687037593572*z^20+ 11244148143597950211*z^36-5544959704779575670*z^34-2463306889432672*z^66+ 42034370*z^80+z^88+20041*z^84-213*z^86-1119850*z^82+13741466702438569*z^64-\ 830839291894041426*z^30-36324144584292304712*z^42+39268166167742163336*z^44-\ 36324144584292304712*z^46-830839291894041426*z^58+2328603224325478774*z^56-\ 5544959704779575670*z^54+11244148143597950211*z^52+251135495274269565*z^60-\ 44958098284496*z^70+366687037593572*z^68-1134839206*z^78+2328603224325478774*z^ 32-19451796584932428335*z^38+28746245235399361515*z^40-64079059587628301*z^62+ 23018296719*z^76-361506731167*z^74+4493849404979*z^72)/(-1-954804241355546518*z ^28+227842146451389341*z^26+255*z^2-45664518172779175*z^24+7641256967613182*z^ 22-27680*z^4+1748042*z^6-73173019*z^8+2181872027*z^10-48509578700*z^12+ 829965533182*z^14+120771677634913*z^18-11183067166371*z^16+ 165107154250135519665*z^50-224572902110068338407*z^48-1059851183997238*z^20-\ 56034296969651802138*z^36+25794490297192002349*z^34+45664518172779175*z^66-\ 2181872027*z^80+z^90-255*z^88-1748042*z^84+27680*z^86+73173019*z^82-\ 227842146451389341*z^64+3377114183634055568*z^30+224572902110068338407*z^42-\ 261861545862957403788*z^44+261861545862957403788*z^46+10122193165503090301*z^58 -25794490297192002349*z^56+56034296969651802138*z^54-103984157354476913676*z^52 -3377114183634055568*z^60+1059851183997238*z^70-7641256967613182*z^68+ 48509578700*z^78-10122193165503090301*z^32+103984157354476913676*z^38-\ 165107154250135519665*z^40+954804241355546518*z^62-829965533182*z^76+ 11183067166371*z^74-120771677634913*z^72) The first , 40, terms are: [0, 42, 0, 3071, 0, 248737, 0, 20701770, 0, 1735914195, 0, 145868850883, 0, 12264763756434, 0, 1031413597986241, 0, 86742036876565363, 0, 7295136646591345874, 0, 613535127522391112297, 0, 51599574677753710952541, 0, 4339633258557024021668090, 0, 364972385310875461983857287, 0, 30694955317053211399377694941, 0, 2581511193016793566050014475866, 0, 217110597085199692721184621399887, 0, 18259464293357123636753629330170039, 0, 1535659893871644827976969820762302722, 0, 129152272608852373456010877606327314853] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}, {3, 7}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 16046422502 z + 30293975724 z + 152 z - 41593016548 z 22 4 6 8 10 + 41593016548 z - 8648 z + 249583 z - 4227934 z + 45800444 z 12 14 18 16 - 334483456 z + 1702107572 z + 16046422502 z - 6159816024 z 20 36 34 30 - 30293975724 z - 45800444 z + 334483456 z + 6159816024 z 42 44 46 32 38 40 + 8648 z - 152 z + z - 1702107572 z + 4227934 z - 249583 z ) / 28 26 2 24 / (1 + 167896517366 z - 268229328146 z - 213 z + 313537453042 z / 22 4 6 8 10 - 268229328146 z + 15089 z - 517045 z + 10168621 z - 126558736 z 12 14 18 16 + 1057766680 z - 6160599800 z - 76799975592 z + 25583858640 z 48 20 36 34 + z + 167896517366 z + 1057766680 z - 6160599800 z 30 42 44 46 32 - 76799975592 z - 517045 z + 15089 z - 213 z + 25583858640 z 38 40 - 126558736 z + 10168621 z ) And in Maple-input format, it is: -(-1-16046422502*z^28+30293975724*z^26+152*z^2-41593016548*z^24+41593016548*z^ 22-8648*z^4+249583*z^6-4227934*z^8+45800444*z^10-334483456*z^12+1702107572*z^14 +16046422502*z^18-6159816024*z^16-30293975724*z^20-45800444*z^36+334483456*z^34 +6159816024*z^30+8648*z^42-152*z^44+z^46-1702107572*z^32+4227934*z^38-249583*z^ 40)/(1+167896517366*z^28-268229328146*z^26-213*z^2+313537453042*z^24-\ 268229328146*z^22+15089*z^4-517045*z^6+10168621*z^8-126558736*z^10+1057766680*z ^12-6160599800*z^14-76799975592*z^18+25583858640*z^16+z^48+167896517366*z^20+ 1057766680*z^36-6160599800*z^34-76799975592*z^30-517045*z^42+15089*z^44-213*z^ 46+25583858640*z^32-126558736*z^38+10168621*z^40) The first , 40, terms are: [0, 61, 0, 6552, 0, 742609, 0, 84911647, 0, 9729104861, 0, 1115401759095, 0, 127899092053552, 0, 14666569672089127, 0, 1681890930737041239, 0, 192872285105985514575, 0, 22117840911317621809463, 0, 2536389445376336255449632, 0, 290863513724169164266603519, 0, 33355126280860702820449837085, 0, 3825039669650769808227415619663, 0, 438641079638337158146958218646697, 0, 50301699833680399897902082202051848, 0, 5768408674240745359168416563897622589, 0, 661499288376508295120054700192247686761, 0, 75858236349552722249130549169740753277241] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}, {3, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1133598934238 z - 1302522267956 z - 166 z 24 22 4 6 8 + 1133598934238 z - 746431159256 z + 10977 z - 388040 z + 8348728 z 10 12 14 18 - 117584168 z + 1134749552 z - 7738774848 z - 137893675704 z 16 50 48 20 + 38125204304 z - 166 z + 10977 z + 370575451976 z 36 34 30 42 + 38125204304 z - 137893675704 z - 746431159256 z - 117584168 z 44 46 52 32 38 + 8348728 z - 388040 z + z + 370575451976 z - 7738774848 z 40 / 2 28 + 1134749552 z ) / ((-1 + z ) (1 + 3922187658882 z / 26 2 24 22 - 4534469553154 z - 207 z + 3922187658882 z - 2535430594776 z 4 6 8 10 12 + 16605 z - 693288 z + 17174700 z - 272434392 z + 2905873116 z 14 18 16 50 - 21548816752 z - 435079245096 z + 113765711044 z - 207 z 48 20 36 34 + 16605 z + 1220688993300 z + 113765711044 z - 435079245096 z 30 42 44 46 52 - 2535430594776 z - 272434392 z + 17174700 z - 693288 z + z 32 38 40 + 1220688993300 z - 21548816752 z + 2905873116 z )) And in Maple-input format, it is: -(1+1133598934238*z^28-1302522267956*z^26-166*z^2+1133598934238*z^24-\ 746431159256*z^22+10977*z^4-388040*z^6+8348728*z^8-117584168*z^10+1134749552*z^ 12-7738774848*z^14-137893675704*z^18+38125204304*z^16-166*z^50+10977*z^48+ 370575451976*z^20+38125204304*z^36-137893675704*z^34-746431159256*z^30-\ 117584168*z^42+8348728*z^44-388040*z^46+z^52+370575451976*z^32-7738774848*z^38+ 1134749552*z^40)/(-1+z^2)/(1+3922187658882*z^28-4534469553154*z^26-207*z^2+ 3922187658882*z^24-2535430594776*z^22+16605*z^4-693288*z^6+17174700*z^8-\ 272434392*z^10+2905873116*z^12-21548816752*z^14-435079245096*z^18+113765711044* z^16-207*z^50+16605*z^48+1220688993300*z^20+113765711044*z^36-435079245096*z^34 -2535430594776*z^30-272434392*z^42+17174700*z^44-693288*z^46+z^52+1220688993300 *z^32-21548816752*z^38+2905873116*z^40) The first , 40, terms are: [0, 42, 0, 2901, 0, 219157, 0, 17109290, 0, 1355233857, 0, 108110269697, 0, 8655998028522, 0, 694439673171093, 0, 55774362955572181, 0, 4482365213472921706, 0, 360358608419588228033, 0, 28976847045004419065281, 0, 2330333379167828529145834, 0, 187419183038731752783403861, 0, 15073936712068905145265884693, 0, 1212408277239265990459748304746, 0, 97516157777859150010097854588673, 0, 7843455127510490444460854905166657, 0, 630870262559525422613361587004818602, 0, 50742718736744735472100421016203464597] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 18 16 14 12 10 8 f(z) = - (z - 107 z + 2971 z - 30179 z + 112194 z - 112194 z 6 4 2 / 20 18 16 + 30179 z - 2971 z + 107 z - 1) / (z - 159 z + 6666 z / 14 12 10 8 6 4 - 93015 z + 476837 z - 864000 z + 476837 z - 93015 z + 6666 z 2 - 159 z + 1) And in Maple-input format, it is: -(z^18-107*z^16+2971*z^14-30179*z^12+112194*z^10-112194*z^8+30179*z^6-2971*z^4+ 107*z^2-1)/(z^20-159*z^18+6666*z^16-93015*z^14+476837*z^12-864000*z^10+476837*z ^8-93015*z^6+6666*z^4-159*z^2+1) The first , 40, terms are: [0, 52, 0, 4573, 0, 443311, 0, 44474968, 0, 4517722663, 0, 460946245135, 0, 47104692132640, 0, 4816367887161199, 0, 492561529382503309, 0, 50376903696302158300, 0, 5152441839262483324729, 0, 526985279722185405409321, 0, 53899557234068377888672780, 0, 5512801812557293687928005645, 0, 563844991779272711430213183631, 0, 57669625780658431029154684939408, 0, 5898404604435549556214699599210975, 0, 603284259094126998943403653626971287, 0, 61703447598887784696327508302924954088, 0, 6310980928409882658134549369911049458639] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 22950315509 z - 34882580296 z - 140 z + 40085689722 z 22 4 6 8 10 - 34882580296 z + 7197 z - 191348 z + 3053593 z - 31757960 z 12 14 18 16 48 + 226454186 z - 1144803756 z - 11362569644 z + 4199228085 z + z 20 36 34 30 + 22950315509 z + 226454186 z - 1144803756 z - 11362569644 z 42 44 46 32 38 - 191348 z + 7197 z - 140 z + 4199228085 z - 31757960 z 40 / 28 26 2 + 3053593 z ) / ((1 + 83720648051 z - 130820995155 z - 203 z / 24 22 4 6 8 + 151751035474 z - 130820995155 z + 12667 z - 388540 z + 6995927 z 10 12 14 18 - 80958803 z + 635193386 z - 3495431059 z - 39636169092 z 16 48 20 36 34 + 13795609391 z + z + 83720648051 z + 635193386 z - 3495431059 z 30 42 44 46 32 - 39636169092 z - 388540 z + 12667 z - 203 z + 13795609391 z 38 40 2 - 80958803 z + 6995927 z ) (-1 + z )) And in Maple-input format, it is: -(1+22950315509*z^28-34882580296*z^26-140*z^2+40085689722*z^24-34882580296*z^22 +7197*z^4-191348*z^6+3053593*z^8-31757960*z^10+226454186*z^12-1144803756*z^14-\ 11362569644*z^18+4199228085*z^16+z^48+22950315509*z^20+226454186*z^36-\ 1144803756*z^34-11362569644*z^30-191348*z^42+7197*z^44-140*z^46+4199228085*z^32 -31757960*z^38+3053593*z^40)/(1+83720648051*z^28-130820995155*z^26-203*z^2+ 151751035474*z^24-130820995155*z^22+12667*z^4-388540*z^6+6995927*z^8-80958803*z ^10+635193386*z^12-3495431059*z^14-39636169092*z^18+13795609391*z^16+z^48+ 83720648051*z^20+635193386*z^36-3495431059*z^34-39636169092*z^30-388540*z^42+ 12667*z^44-203*z^46+13795609391*z^32-80958803*z^38+6995927*z^40)/(-1+z^2) The first , 40, terms are: [0, 64, 0, 7383, 0, 892311, 0, 108358608, 0, 13166815625, 0, 1600076406773, 0, 194450182646744, 0, 23630748825906463, 0, 2871751902752395843, 0, 348992757697976282840, 0, 42411723860266593306865, 0, 5154130832088600934906209, 0, 626361351380904466828587560, 0, 76119244039845560707532738947, 0, 9250473869107123133165190198239, 0, 1124173891682481116148242182900264, 0, 136616454100226868432136047352367237, 0, 16602463078908282935240029983142460025, 0, 2017632371605315306852025530070359475040, 0, 245194967011936199037361492772036294361847] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2170347168565319 z - 925806624829409 z - 192 z 24 22 4 6 + 324342778127083 z - 92843703891760 z + 15509 z - 709651 z 8 10 12 14 + 20889409 z - 425672380 z + 6300634695 z - 70069187626 z 18 16 50 - 4033049149090 z + 600134958210 z - 92843703891760 z 48 20 36 + 324342778127083 z + 21571032795293 z + 9743095833761796 z 34 66 64 30 - 8874403668348060 z - 709651 z + 20889409 z - 4194839245820332 z 42 44 46 - 4194839245820332 z + 2170347168565319 z - 925806624829409 z 58 56 54 - 70069187626 z + 600134958210 z - 4033049149090 z 52 60 70 68 + 21571032795293 z + 6300634695 z - 192 z + 15509 z 32 38 40 + 6703581434785603 z - 8874403668348060 z + 6703581434785603 z 62 72 / 28 - 425672380 z + z ) / (-1 - 10284580779566770 z / 26 2 24 + 4033803055749993 z + 237 z - 1300160338252695 z 22 4 6 8 + 342415632871558 z - 22658 z + 1194552 z - 39811857 z 10 12 14 18 + 907392891 z - 14888806068 z + 182331755776 z + 12555799520057 z 16 50 48 - 1711179803217 z + 1300160338252695 z - 4033803055749993 z 20 36 34 - 73146709394170 z - 65576067469516890 z + 54546816226577865 z 66 64 30 + 39811857 z - 907392891 z + 21642714212016736 z 42 44 46 + 37714222708411709 z - 21642714212016736 z + 10284580779566770 z 58 56 54 + 1711179803217 z - 12555799520057 z + 73146709394170 z 52 60 70 68 - 342415632871558 z - 182331755776 z + 22658 z - 1194552 z 32 38 40 - 37714222708411709 z + 65576067469516890 z - 54546816226577865 z 62 74 72 + 14888806068 z + z - 237 z ) And in Maple-input format, it is: -(1+2170347168565319*z^28-925806624829409*z^26-192*z^2+324342778127083*z^24-\ 92843703891760*z^22+15509*z^4-709651*z^6+20889409*z^8-425672380*z^10+6300634695 *z^12-70069187626*z^14-4033049149090*z^18+600134958210*z^16-92843703891760*z^50 +324342778127083*z^48+21571032795293*z^20+9743095833761796*z^36-\ 8874403668348060*z^34-709651*z^66+20889409*z^64-4194839245820332*z^30-\ 4194839245820332*z^42+2170347168565319*z^44-925806624829409*z^46-70069187626*z^ 58+600134958210*z^56-4033049149090*z^54+21571032795293*z^52+6300634695*z^60-192 *z^70+15509*z^68+6703581434785603*z^32-8874403668348060*z^38+6703581434785603*z ^40-425672380*z^62+z^72)/(-1-10284580779566770*z^28+4033803055749993*z^26+237*z ^2-1300160338252695*z^24+342415632871558*z^22-22658*z^4+1194552*z^6-39811857*z^ 8+907392891*z^10-14888806068*z^12+182331755776*z^14+12555799520057*z^18-\ 1711179803217*z^16+1300160338252695*z^50-4033803055749993*z^48-73146709394170*z ^20-65576067469516890*z^36+54546816226577865*z^34+39811857*z^66-907392891*z^64+ 21642714212016736*z^30+37714222708411709*z^42-21642714212016736*z^44+ 10284580779566770*z^46+1711179803217*z^58-12555799520057*z^56+73146709394170*z^ 54-342415632871558*z^52-182331755776*z^60+22658*z^70-1194552*z^68-\ 37714222708411709*z^32+65576067469516890*z^38-54546816226577865*z^40+ 14888806068*z^62+z^74-237*z^72) The first , 40, terms are: [0, 45, 0, 3516, 0, 298583, 0, 25931035, 0, 2270593459, 0, 199524198079, 0, 17561614073236, 0, 1546939489239601, 0, 136315785592831553, 0, 12014294912300323885, 0, 1058982656181698003757, 0, 93346497469866466777780, 0, 8228415855707068308190619, 0, 725335298095107806671562871, 0, 63938657578220176150226260991, 0, 5636236952687963376598544123427, 0, 496838761436633892319874018746428, 0, 43796755573978692995717546744721977, 0, 3860721958311865482208315944753481645, 0, 340326030827840301624670634613398882101] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {3, 6}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 20256479 z + 104586475 z + 141 z - 348755676 z 22 4 6 8 10 + 768282159 z - 7164 z + 177761 z - 2454393 z + 20256479 z 12 14 18 16 - 104586475 z + 348755676 z + 1135662231 z - 768282159 z 20 36 34 30 32 38 - 1135662231 z - 141 z + 7164 z + 2454393 z - 177761 z + z ) / 40 38 36 34 32 30 / (z - 188 z + 12232 z - 374352 z + 6211920 z - 60656844 z / 28 26 24 22 + 368716680 z - 1451293152 z + 3797305112 z - 6717410428 z 20 18 16 14 + 8114980342 z - 6717410428 z + 3797305112 z - 1451293152 z 12 10 8 6 4 + 368716680 z - 60656844 z + 6211920 z - 374352 z + 12232 z 2 - 188 z + 1) And in Maple-input format, it is: -(-1-20256479*z^28+104586475*z^26+141*z^2-348755676*z^24+768282159*z^22-7164*z^ 4+177761*z^6-2454393*z^8+20256479*z^10-104586475*z^12+348755676*z^14+1135662231 *z^18-768282159*z^16-1135662231*z^20-141*z^36+7164*z^34+2454393*z^30-177761*z^ 32+z^38)/(z^40-188*z^38+12232*z^36-374352*z^34+6211920*z^32-60656844*z^30+ 368716680*z^28-1451293152*z^26+3797305112*z^24-6717410428*z^22+8114980342*z^20-\ 6717410428*z^18+3797305112*z^16-1451293152*z^14+368716680*z^12-60656844*z^10+ 6211920*z^8-374352*z^6+12232*z^4-188*z^2+1) The first , 40, terms are: [0, 47, 0, 3768, 0, 330071, 0, 29800189, 0, 2724005521, 0, 250340091995, 0, 23061615310104, 0, 2126741120222099, 0, 196222840184128309, 0, 18108372543828146317, 0, 1671291613568433498251, 0, 154256859955606060825944, 0, 14237886920248351304910995, 0, 1314167056466348484371119273, 0, 121299054752015675757270151525, 0, 11196056299114433394736595839391, 0, 1033411048646501690126538438306936, 0, 95385265898500784265738847573362359, 0, 8804193210687192816901611273668335657, 0, 812639389172001884184011644027291055833] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {3, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 10394410 z + 48746782 z + 126 z - 151489867 z 22 4 6 8 10 + 318911308 z - 5578 z + 118777 z - 1421791 z + 10394410 z 12 14 18 16 - 48746782 z + 151489867 z + 461141824 z - 318911308 z 20 36 34 30 32 38 - 461141824 z - 126 z + 5578 z + 1421791 z - 118777 z + z ) / 14 30 32 22 / (-654876880 z - 32217836 z + 3718620 z - 2805918876 z / 8 40 2 4 28 18 + 3718620 z + z - 176 z + 9957 z + 178431655 z - 2805918876 z 6 10 34 24 38 - 259464 z - 32217836 z - 259464 z + 1 + 1632041443 z - 176 z 26 12 36 20 - 654876880 z + 178431655 z + 9957 z + 3358143880 z 16 + 1632041443 z ) And in Maple-input format, it is: -(-1-10394410*z^28+48746782*z^26+126*z^2-151489867*z^24+318911308*z^22-5578*z^4 +118777*z^6-1421791*z^8+10394410*z^10-48746782*z^12+151489867*z^14+461141824*z^ 18-318911308*z^16-461141824*z^20-126*z^36+5578*z^34+1421791*z^30-118777*z^32+z^ 38)/(-654876880*z^14-32217836*z^30+3718620*z^32-2805918876*z^22+3718620*z^8+z^ 40-176*z^2+9957*z^4+178431655*z^28-2805918876*z^18-259464*z^6-32217836*z^10-\ 259464*z^34+1+1632041443*z^24-176*z^38-654876880*z^26+178431655*z^12+9957*z^36+ 3358143880*z^20+1632041443*z^16) The first , 40, terms are: [0, 50, 0, 4421, 0, 420933, 0, 40740682, 0, 3962112921, 0, 385935051241, 0, 37613277963226, 0, 3666512303084741, 0, 357433701240740357, 0, 34845664028180821666, 0, 3397080997425647539697, 0, 331180275459536973547025, 0, 32286690329970784100139714, 0, 3147623467766802181271637637, 0, 306861275564150108275097813701, 0, 29915855078956097377833903116730, 0, 2916491827602319089564751046420041, 0, 284328314100451243938808439113725753, 0, 27719121183707802573146390568470786922, 0, 2702332626686894814925907525188199572037] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {4, 5}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4765217057530400 z - 2222357551044962 z - 217 z 24 22 4 6 + 828591650029018 z - 245808344610280 z + 20134 z - 1062645 z 8 10 12 14 + 35888501 z - 828695748 z + 13661811889 z - 165804548921 z 18 16 50 - 10608766301705 z + 1515282545962 z - 10608766301705 z 48 20 36 + 57659492755945 z + 57659492755945 z + 11339965823794278 z 34 66 64 30 - 12633573283556232 z - 217 z + 20134 z - 8197235066504542 z 42 44 46 - 2222357551044962 z + 828591650029018 z - 245808344610280 z 58 56 54 52 - 828695748 z + 13661811889 z - 165804548921 z + 1515282545962 z 60 68 32 38 + 35888501 z + z + 11339965823794278 z - 8197235066504542 z 40 62 / 28 + 4765217057530400 z - 1062645 z ) / (-1 - 24102753071783618 z / 26 2 24 + 10193586716076910 z + 271 z - 3452057869246066 z 22 4 6 8 + 931238573910673 z - 29681 z + 1793835 z - 68201669 z 10 12 14 18 + 1755463163 z - 32066667475 z + 429571229309 z + 33279313928353 z 16 50 48 - 4322893818451 z + 198766701018159 z - 931238573910673 z 20 36 34 - 198766701018159 z - 86848751727654062 z + 86848751727654062 z 66 64 30 42 + 29681 z - 1793835 z + 45813026288959534 z + 24102753071783618 z 44 46 58 - 10193586716076910 z + 3452057869246066 z + 32066667475 z 56 54 52 - 429571229309 z + 4322893818451 z - 33279313928353 z 60 70 68 32 - 1755463163 z + z - 271 z - 70193598324865106 z 38 40 62 + 70193598324865106 z - 45813026288959534 z + 68201669 z ) And in Maple-input format, it is: -(1+4765217057530400*z^28-2222357551044962*z^26-217*z^2+828591650029018*z^24-\ 245808344610280*z^22+20134*z^4-1062645*z^6+35888501*z^8-828695748*z^10+ 13661811889*z^12-165804548921*z^14-10608766301705*z^18+1515282545962*z^16-\ 10608766301705*z^50+57659492755945*z^48+57659492755945*z^20+11339965823794278*z ^36-12633573283556232*z^34-217*z^66+20134*z^64-8197235066504542*z^30-\ 2222357551044962*z^42+828591650029018*z^44-245808344610280*z^46-828695748*z^58+ 13661811889*z^56-165804548921*z^54+1515282545962*z^52+35888501*z^60+z^68+ 11339965823794278*z^32-8197235066504542*z^38+4765217057530400*z^40-1062645*z^62 )/(-1-24102753071783618*z^28+10193586716076910*z^26+271*z^2-3452057869246066*z^ 24+931238573910673*z^22-29681*z^4+1793835*z^6-68201669*z^8+1755463163*z^10-\ 32066667475*z^12+429571229309*z^14+33279313928353*z^18-4322893818451*z^16+ 198766701018159*z^50-931238573910673*z^48-198766701018159*z^20-\ 86848751727654062*z^36+86848751727654062*z^34+29681*z^66-1793835*z^64+ 45813026288959534*z^30+24102753071783618*z^42-10193586716076910*z^44+ 3452057869246066*z^46+32066667475*z^58-429571229309*z^56+4322893818451*z^54-\ 33279313928353*z^52-1755463163*z^60+z^70-271*z^68-70193598324865106*z^32+ 70193598324865106*z^38-45813026288959534*z^40+68201669*z^62) The first , 40, terms are: [0, 54, 0, 5087, 0, 506993, 0, 50961778, 0, 5131698539, 0, 517003824419, 0, 52095552095722, 0, 5249719872323637, 0, 529033565040332535, 0, 53313249965288080502, 0, 5372657122277423395933, 0, 541432045537055374276877, 0, 54563114997864583061389494, 0, 5498630198988514907980792647, 0, 554127797735526827531566316005, 0, 55842569059915857001416260462698, 0, 5627569345628167412834161222772467, 0, 567121779327009843606295053473956507, 0, 57152048162319126361716509350962867058, 0, 5759533023763379554947287059464300054785] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 592505257505797937 z - 143678653072190503 z - 227 z 24 22 4 6 + 29193490660319545 z - 4946255239897900 z + 22817 z - 1362428 z 8 10 12 14 + 54623176 z - 1574031156 z + 34048464207 z - 569762089521 z 18 16 50 - 80194458387820 z + 7539089113319 z - 55267327150788398309 z 48 20 + 83226980769324006127 z + 694533011305656 z 36 34 + 31135325044459574687 z - 14868757902587022116 z 66 80 88 84 86 - 4946255239897900 z + 54623176 z + z + 22817 z - 227 z 82 64 30 - 1362428 z + 29193490660319545 z - 2055035958115558508 z 42 44 - 106380196166427217416 z + 115446316213954530288 z 46 58 - 106380196166427217416 z - 2055035958115558508 z 56 54 + 6012158077365944248 z - 14868757902587022116 z 52 60 70 + 31135325044459574687 z + 592505257505797937 z - 80194458387820 z 68 78 32 + 694533011305656 z - 1574031156 z + 6012158077365944248 z 38 40 - 55267327150788398309 z + 83226980769324006127 z 62 76 74 - 143678653072190503 z + 34048464207 z - 569762089521 z 72 / 28 + 7539089113319 z ) / (-1 - 2216521680509475068 z / 26 2 24 + 499659586414104655 z + 275 z - 94421096734577553 z 22 4 6 8 + 14881926430729004 z - 31964 z + 2146984 z - 95194901 z 10 12 14 + 3001168207 z - 70534989788 z + 1276695198012 z 18 16 50 + 208669214600039 z - 18220359478881 z + 490811778745837886909 z 48 20 - 679218851266881093991 z - 1943763544087732 z 36 34 - 157155066547754510920 z + 69540255158974257627 z 66 80 90 88 84 + 94421096734577553 z - 3001168207 z + z - 275 z - 2146984 z 86 82 64 + 31964 z + 95194901 z - 499659586414104655 z 30 42 + 8275110244653873732 z + 679218851266881093991 z 44 46 - 798959477218163330416 z + 798959477218163330416 z 58 56 + 26079341001460166929 z - 69540255158974257627 z 54 52 + 157155066547754510920 z - 301370104971275146076 z 60 70 68 - 8275110244653873732 z + 1943763544087732 z - 14881926430729004 z 78 32 38 + 70534989788 z - 26079341001460166929 z + 301370104971275146076 z 40 62 76 - 490811778745837886909 z + 2216521680509475068 z - 1276695198012 z 74 72 + 18220359478881 z - 208669214600039 z ) And in Maple-input format, it is: -(1+592505257505797937*z^28-143678653072190503*z^26-227*z^2+29193490660319545*z ^24-4946255239897900*z^22+22817*z^4-1362428*z^6+54623176*z^8-1574031156*z^10+ 34048464207*z^12-569762089521*z^14-80194458387820*z^18+7539089113319*z^16-\ 55267327150788398309*z^50+83226980769324006127*z^48+694533011305656*z^20+ 31135325044459574687*z^36-14868757902587022116*z^34-4946255239897900*z^66+ 54623176*z^80+z^88+22817*z^84-227*z^86-1362428*z^82+29193490660319545*z^64-\ 2055035958115558508*z^30-106380196166427217416*z^42+115446316213954530288*z^44-\ 106380196166427217416*z^46-2055035958115558508*z^58+6012158077365944248*z^56-\ 14868757902587022116*z^54+31135325044459574687*z^52+592505257505797937*z^60-\ 80194458387820*z^70+694533011305656*z^68-1574031156*z^78+6012158077365944248*z^ 32-55267327150788398309*z^38+83226980769324006127*z^40-143678653072190503*z^62+ 34048464207*z^76-569762089521*z^74+7539089113319*z^72)/(-1-2216521680509475068* z^28+499659586414104655*z^26+275*z^2-94421096734577553*z^24+14881926430729004*z ^22-31964*z^4+2146984*z^6-95194901*z^8+3001168207*z^10-70534989788*z^12+ 1276695198012*z^14+208669214600039*z^18-18220359478881*z^16+ 490811778745837886909*z^50-679218851266881093991*z^48-1943763544087732*z^20-\ 157155066547754510920*z^36+69540255158974257627*z^34+94421096734577553*z^66-\ 3001168207*z^80+z^90-275*z^88-2146984*z^84+31964*z^86+95194901*z^82-\ 499659586414104655*z^64+8275110244653873732*z^30+679218851266881093991*z^42-\ 798959477218163330416*z^44+798959477218163330416*z^46+26079341001460166929*z^58 -69540255158974257627*z^56+157155066547754510920*z^54-301370104971275146076*z^ 52-8275110244653873732*z^60+1943763544087732*z^70-14881926430729004*z^68+ 70534989788*z^78-26079341001460166929*z^32+301370104971275146076*z^38-\ 490811778745837886909*z^40+2216521680509475068*z^62-1276695198012*z^76+ 18220359478881*z^74-208669214600039*z^72) The first , 40, terms are: [0, 48, 0, 4053, 0, 364859, 0, 33269640, 0, 3046305879, 0, 279394393623, 0, 25642993853408, 0, 2354255823500987, 0, 216171005289642321, 0, 19850305329988023000, 0, 1822839392538572448401, 0, 167391995297469740083981, 0, 15371746256468120501541864, 0, 1411603472608666252143577461, 0, 129629148190701681375651107407, 0, 11903997036742843721652024062096, 0, 1093158254749533086923954643580235, 0, 100386035923869855490226249576601627, 0, 9218570644735807385007268277472418680, 0, 846552465189640234155585013886680174959] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 9643789973808 z - 8475663780550 z - 171 z 24 22 4 6 + 5749411778474 z - 3003243593296 z + 11944 z - 455543 z 8 10 12 14 + 10762349 z - 169255600 z + 1855485568 z - 14630428664 z 18 16 50 48 - 367450441248 z + 84826232256 z - 455543 z + 10762349 z 20 36 34 + 1203158773352 z + 1203158773352 z - 3003243593296 z 30 42 44 46 - 8475663780550 z - 14630428664 z + 1855485568 z - 169255600 z 56 54 52 32 38 + z - 171 z + 11944 z + 5749411778474 z - 367450441248 z 40 / 28 26 + 84826232256 z ) / (-1 - 66380014472794 z + 51420794361334 z / 2 24 22 4 6 + 223 z - 30816775949250 z + 14249984912704 z - 18669 z + 823281 z 8 10 12 14 - 22143731 z + 393747541 z - 4864875176 z + 43170602128 z 18 16 50 48 + 1372616578688 z - 281580581448 z + 22143731 z - 393747541 z 20 36 34 - 5062059481512 z - 14249984912704 z + 30816775949250 z 30 42 44 46 + 66380014472794 z + 281580581448 z - 43170602128 z + 4864875176 z 58 56 54 52 32 + z - 223 z + 18669 z - 823281 z - 51420794361334 z 38 40 + 5062059481512 z - 1372616578688 z ) And in Maple-input format, it is: -(1+9643789973808*z^28-8475663780550*z^26-171*z^2+5749411778474*z^24-\ 3003243593296*z^22+11944*z^4-455543*z^6+10762349*z^8-169255600*z^10+1855485568* z^12-14630428664*z^14-367450441248*z^18+84826232256*z^16-455543*z^50+10762349*z ^48+1203158773352*z^20+1203158773352*z^36-3003243593296*z^34-8475663780550*z^30 -14630428664*z^42+1855485568*z^44-169255600*z^46+z^56-171*z^54+11944*z^52+ 5749411778474*z^32-367450441248*z^38+84826232256*z^40)/(-1-66380014472794*z^28+ 51420794361334*z^26+223*z^2-30816775949250*z^24+14249984912704*z^22-18669*z^4+ 823281*z^6-22143731*z^8+393747541*z^10-4864875176*z^12+43170602128*z^14+ 1372616578688*z^18-281580581448*z^16+22143731*z^50-393747541*z^48-5062059481512 *z^20-14249984912704*z^36+30816775949250*z^34+66380014472794*z^30+281580581448* z^42-43170602128*z^44+4864875176*z^46+z^58-223*z^56+18669*z^54-823281*z^52-\ 51420794361334*z^32+5062059481512*z^38-1372616578688*z^40) The first , 40, terms are: [0, 52, 0, 4871, 0, 483183, 0, 48242340, 0, 4820718073, 0, 481782637065, 0, 48150580838948, 0, 4812323099888063, 0, 480959989326666103, 0, 48068819060065272180, 0, 4804166840970339710833, 0, 480145384159028530776465, 0, 47987425800335826195333748, 0, 4796032929248720510038150679, 0, 479332483947397273463159809183, 0, 47906182908651486887717993959076, 0, 4787913275185767329310844064898281, 0, 478520978870778574717359299835333913, 0, 47825078294552268115134200599740645668, 0, 4779807396133327322525615129761391594383] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1434630640058008 z - 689913791734598 z - 191 z 24 22 4 6 + 267627123996698 z - 83348065489728 z + 15588 z - 726821 z 8 10 12 14 + 21837507 z - 452200968 z + 6741307983 z - 74606309209 z 18 16 50 - 4072075544875 z + 627036697732 z - 4072075544875 z 48 20 36 + 20709479947381 z + 20709479947381 z + 3297121734020702 z 34 66 64 30 - 3657347775542640 z - 191 z + 15588 z - 2414639899287634 z 42 44 46 - 689913791734598 z + 267627123996698 z - 83348065489728 z 58 56 54 52 - 452200968 z + 6741307983 z - 74606309209 z + 627036697732 z 60 68 32 38 + 21837507 z + z + 3297121734020702 z - 2414639899287634 z 40 62 / 2 + 1434630640058008 z - 726821 z ) / ((-1 + z ) (1 / 28 26 2 + 4981311776027094 z - 2322327280167412 z - 238 z 24 22 4 6 + 866812546629730 z - 257974468500168 z + 22859 z - 1214864 z 8 10 12 14 + 40777637 z - 930300942 z + 15119474353 z - 180871403736 z 18 16 50 - 11294605926862 z + 1631296649183 z - 11294605926862 z 48 20 36 + 60864905676285 z + 60864905676285 z + 11877670450790730 z 34 66 64 30 - 13237685204544596 z - 238 z + 22859 z - 8577685538980584 z 42 44 46 - 2322327280167412 z + 866812546629730 z - 257974468500168 z 58 56 54 52 - 930300942 z + 15119474353 z - 180871403736 z + 1631296649183 z 60 68 32 38 + 40777637 z + z + 11877670450790730 z - 8577685538980584 z 40 62 + 4981311776027094 z - 1214864 z )) And in Maple-input format, it is: -(1+1434630640058008*z^28-689913791734598*z^26-191*z^2+267627123996698*z^24-\ 83348065489728*z^22+15588*z^4-726821*z^6+21837507*z^8-452200968*z^10+6741307983 *z^12-74606309209*z^14-4072075544875*z^18+627036697732*z^16-4072075544875*z^50+ 20709479947381*z^48+20709479947381*z^20+3297121734020702*z^36-3657347775542640* z^34-191*z^66+15588*z^64-2414639899287634*z^30-689913791734598*z^42+ 267627123996698*z^44-83348065489728*z^46-452200968*z^58+6741307983*z^56-\ 74606309209*z^54+627036697732*z^52+21837507*z^60+z^68+3297121734020702*z^32-\ 2414639899287634*z^38+1434630640058008*z^40-726821*z^62)/(-1+z^2)/(1+ 4981311776027094*z^28-2322327280167412*z^26-238*z^2+866812546629730*z^24-\ 257974468500168*z^22+22859*z^4-1214864*z^6+40777637*z^8-930300942*z^10+ 15119474353*z^12-180871403736*z^14-11294605926862*z^18+1631296649183*z^16-\ 11294605926862*z^50+60864905676285*z^48+60864905676285*z^20+11877670450790730*z ^36-13237685204544596*z^34-238*z^66+22859*z^64-8577685538980584*z^30-\ 2322327280167412*z^42+866812546629730*z^44-257974468500168*z^46-930300942*z^58+ 15119474353*z^56-180871403736*z^54+1631296649183*z^52+40777637*z^60+z^68+ 11877670450790730*z^32-8577685538980584*z^38+4981311776027094*z^40-1214864*z^62 ) The first , 40, terms are: [0, 48, 0, 3963, 0, 349403, 0, 31229616, 0, 2802050945, 0, 251730887489, 0, 22625399373968, 0, 2033910302770875, 0, 182850848268238875, 0, 16438944620425543824, 0, 1477935902245440204545, 0, 132873741222019433414913, 0, 11946027033058060990909008, 0, 1074009524629250972933335899, 0, 96559030783560184308032982139, 0, 8681159061100079773060928907088, 0, 780481386417932949154499642707649, 0, 70169340518602081282409242479134721, 0, 6308589125740214599094351058982211824, 0, 567175016739574812465472688629902772251] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 31541382770454813830 z - 4922859463116767761 z - 263 z 24 22 4 6 + 662246509115863296 z - 76322680983301663 z + 31246 z - 2247763 z 102 8 10 12 - 3993864981 z + 110669124 z - 3993864981 z + 110412426410 z 14 18 16 - 2411555667001 z - 618032132743315 z + 42570551982656 z 50 48 - 3500746462501352307609771 z + 2287152334982842021623732 z 20 36 + 7480662440021842 z + 13152859328789469482514 z 34 66 - 3563392061598734920831 z - 1322000998772952173188353 z 80 100 90 + 845316828320943559144 z + 110412426410 z - 76322680983301663 z 88 84 94 + 662246509115863296 z + 31541382770454813830 z - 618032132743315 z 86 96 98 - 4922859463116767761 z + 42570551982656 z - 2411555667001 z 92 82 + 7480662440021842 z - 175066688325463127077 z 64 112 110 106 + 2287152334982842021623732 z + z - 263 z - 2247763 z 108 30 42 + 31246 z - 175066688325463127077 z - 304940087075170451825557 z 44 46 + 675566739555103777021210 z - 1322000998772952173188353 z 58 56 - 5690213766035664893824607 z + 6046123143173379442985294 z 54 52 - 5690213766035664893824607 z + 4743003388998042864576614 z 60 70 + 4743003388998042864576614 z - 304940087075170451825557 z 68 78 + 675566739555103777021210 z - 3563392061598734920831 z 32 38 + 845316828320943559144 z - 42614930352667915509491 z 40 62 + 121445280857912429614680 z - 3500746462501352307609771 z 76 74 + 13152859328789469482514 z - 42614930352667915509491 z 72 104 / 2 + 121445280857912429614680 z + 110669124 z ) / ((-1 + z ) (1 / 28 26 2 + 86656016849629887196 z - 13032375847757168096 z - 313 z 24 22 4 6 + 1685561688954026326 z - 186323203280393322 z + 42368 z - 3381724 z 102 8 10 12 - 7061294530 z + 181573330 z - 7061294530 z + 208565051696 z 14 18 16 - 4835582481221 z - 1376832763848831 z + 90157588045532 z 50 48 - 12449281619760401031147667 z + 8047557102560694418055724 z 20 36 + 17470912786012872 z + 41028058877958237009704 z 34 66 - 10802919883424840328117 z - 4589179344734675287159565 z 80 100 90 + 2485243295603229105784 z + 208565051696 z - 186323203280393322 z 88 84 + 1685561688954026326 z + 86656016849629887196 z 94 86 96 - 1376832763848831 z - 13032375847757168096 z + 90157588045532 z 98 92 82 - 4835582481221 z + 17470912786012872 z - 498079054348869321199 z 64 112 110 106 + 8047557102560694418055724 z + z - 313 z - 3381724 z 108 30 42 + 42368 z - 498079054348869321199 z - 1021918027241400881955074 z 44 46 + 2307275309709557428459584 z - 4589179344734675287159565 z 58 56 - 20485439890159917822137478 z + 21800423598469089441343796 z 54 52 - 20485439890159917822137478 z + 16996623260921679599876768 z 60 70 + 16996623260921679599876768 z - 1021918027241400881955074 z 68 78 + 2307275309709557428459584 z - 10802919883424840328117 z 32 38 + 2485243295603229105784 z - 136468101011119958755904 z 40 62 + 398332068186217350255522 z - 12449281619760401031147667 z 76 74 + 41028058877958237009704 z - 136468101011119958755904 z 72 104 + 398332068186217350255522 z + 181573330 z )) And in Maple-input format, it is: -(1+31541382770454813830*z^28-4922859463116767761*z^26-263*z^2+ 662246509115863296*z^24-76322680983301663*z^22+31246*z^4-2247763*z^6-3993864981 *z^102+110669124*z^8-3993864981*z^10+110412426410*z^12-2411555667001*z^14-\ 618032132743315*z^18+42570551982656*z^16-3500746462501352307609771*z^50+ 2287152334982842021623732*z^48+7480662440021842*z^20+13152859328789469482514*z^ 36-3563392061598734920831*z^34-1322000998772952173188353*z^66+ 845316828320943559144*z^80+110412426410*z^100-76322680983301663*z^90+ 662246509115863296*z^88+31541382770454813830*z^84-618032132743315*z^94-\ 4922859463116767761*z^86+42570551982656*z^96-2411555667001*z^98+ 7480662440021842*z^92-175066688325463127077*z^82+2287152334982842021623732*z^64 +z^112-263*z^110-2247763*z^106+31246*z^108-175066688325463127077*z^30-\ 304940087075170451825557*z^42+675566739555103777021210*z^44-\ 1322000998772952173188353*z^46-5690213766035664893824607*z^58+ 6046123143173379442985294*z^56-5690213766035664893824607*z^54+ 4743003388998042864576614*z^52+4743003388998042864576614*z^60-\ 304940087075170451825557*z^70+675566739555103777021210*z^68-\ 3563392061598734920831*z^78+845316828320943559144*z^32-42614930352667915509491* z^38+121445280857912429614680*z^40-3500746462501352307609771*z^62+ 13152859328789469482514*z^76-42614930352667915509491*z^74+ 121445280857912429614680*z^72+110669124*z^104)/(-1+z^2)/(1+86656016849629887196 *z^28-13032375847757168096*z^26-313*z^2+1685561688954026326*z^24-\ 186323203280393322*z^22+42368*z^4-3381724*z^6-7061294530*z^102+181573330*z^8-\ 7061294530*z^10+208565051696*z^12-4835582481221*z^14-1376832763848831*z^18+ 90157588045532*z^16-12449281619760401031147667*z^50+8047557102560694418055724*z ^48+17470912786012872*z^20+41028058877958237009704*z^36-10802919883424840328117 *z^34-4589179344734675287159565*z^66+2485243295603229105784*z^80+208565051696*z ^100-186323203280393322*z^90+1685561688954026326*z^88+86656016849629887196*z^84 -1376832763848831*z^94-13032375847757168096*z^86+90157588045532*z^96-\ 4835582481221*z^98+17470912786012872*z^92-498079054348869321199*z^82+ 8047557102560694418055724*z^64+z^112-313*z^110-3381724*z^106+42368*z^108-\ 498079054348869321199*z^30-1021918027241400881955074*z^42+ 2307275309709557428459584*z^44-4589179344734675287159565*z^46-\ 20485439890159917822137478*z^58+21800423598469089441343796*z^56-\ 20485439890159917822137478*z^54+16996623260921679599876768*z^52+ 16996623260921679599876768*z^60-1021918027241400881955074*z^70+ 2307275309709557428459584*z^68-10802919883424840328117*z^78+ 2485243295603229105784*z^32-136468101011119958755904*z^38+ 398332068186217350255522*z^40-12449281619760401031147667*z^62+ 41028058877958237009704*z^76-136468101011119958755904*z^74+ 398332068186217350255522*z^72+181573330*z^104) The first , 40, terms are: [0, 51, 0, 4579, 0, 437404, 0, 42251319, 0, 4093286435, 0, 396938080297, 0, 38506061957885, 0, 3735918837896683, 0, 362486128451075235, 0, 35171928265998796188, 0, 3412758808973004138319, 0, 331144089875882847793699, 0, 32131372946906414756921741, 0, 3117754621733148767009516389, 0, 302520453554170544785516664947, 0, 29354022623009172123953501402791, 0, 2848265943568652459448107355709532, 0, 276371631892909329522063341782530619, 0, 26816765516786570132093936052167570731, 0, 2602072115901631062752759912143202476421] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 5}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 2671339719446104 z + 1175459362555760 z + 192 z 24 22 4 6 - 419125077479566 z + 120613691389241 z - 15732 z + 738769 z 8 10 12 14 - 22499110 z + 476191652 z - 7322547876 z + 84378488890 z 18 16 50 + 5128768396652 z - 745169357101 z + 27861469238000 z 48 20 36 - 120613691389241 z - 27861469238000 z - 9094982852521184 z 34 66 64 30 + 9094982852521184 z + 15732 z - 738769 z + 4933500047045626 z 42 44 46 + 2671339719446104 z - 1175459362555760 z + 419125077479566 z 58 56 54 52 + 7322547876 z - 84378488890 z + 745169357101 z - 5128768396652 z 60 70 68 32 - 476191652 z + z - 192 z - 7418855022316640 z 38 40 62 / + 7418855022316640 z - 4933500047045626 z + 22499110 z ) / (1 / 28 26 2 + 12853003735829466 z - 5168059382027680 z - 236 z 24 22 4 6 + 1685904212359303 z - 444099019781276 z + 22719 z - 1222732 z 8 10 12 14 + 42033587 z - 994128032 z + 16961178266 z - 215706441108 z 18 16 50 - 15797742370648 z + 2094090606815 z - 444099019781276 z 48 20 36 + 1685904212359303 z + 93879744040735 z + 64211398407094236 z 34 66 64 - 58092326470722884 z - 1222732 z + 42033587 z 30 42 44 - 26023966585830844 z - 26023966585830844 z + 12853003735829466 z 46 58 56 - 5168059382027680 z - 215706441108 z + 2094090606815 z 54 52 60 70 - 15797742370648 z + 93879744040735 z + 16961178266 z - 236 z 68 32 38 + 22719 z + 43003867932491430 z - 58092326470722884 z 40 62 72 + 43003867932491430 z - 994128032 z + z ) And in Maple-input format, it is: -(-1-2671339719446104*z^28+1175459362555760*z^26+192*z^2-419125077479566*z^24+ 120613691389241*z^22-15732*z^4+738769*z^6-22499110*z^8+476191652*z^10-\ 7322547876*z^12+84378488890*z^14+5128768396652*z^18-745169357101*z^16+ 27861469238000*z^50-120613691389241*z^48-27861469238000*z^20-9094982852521184*z ^36+9094982852521184*z^34+15732*z^66-738769*z^64+4933500047045626*z^30+ 2671339719446104*z^42-1175459362555760*z^44+419125077479566*z^46+7322547876*z^ 58-84378488890*z^56+745169357101*z^54-5128768396652*z^52-476191652*z^60+z^70-\ 192*z^68-7418855022316640*z^32+7418855022316640*z^38-4933500047045626*z^40+ 22499110*z^62)/(1+12853003735829466*z^28-5168059382027680*z^26-236*z^2+ 1685904212359303*z^24-444099019781276*z^22+22719*z^4-1222732*z^6+42033587*z^8-\ 994128032*z^10+16961178266*z^12-215706441108*z^14-15797742370648*z^18+ 2094090606815*z^16-444099019781276*z^50+1685904212359303*z^48+93879744040735*z^ 20+64211398407094236*z^36-58092326470722884*z^34-1222732*z^66+42033587*z^64-\ 26023966585830844*z^30-26023966585830844*z^42+12853003735829466*z^44-\ 5168059382027680*z^46-215706441108*z^58+2094090606815*z^56-15797742370648*z^54+ 93879744040735*z^52+16961178266*z^60-236*z^70+22719*z^68+43003867932491430*z^32 -58092326470722884*z^38+43003867932491430*z^40-994128032*z^62+z^72) The first , 40, terms are: [0, 44, 0, 3397, 0, 286019, 0, 24589772, 0, 2127199687, 0, 184403587951, 0, 15997782677652, 0, 1388274949453867, 0, 120487044377095349, 0, 10457429377385827924, 0, 907648230464083921585, 0, 78779550605773201126081, 0, 6837712012551045561443572, 0, 593483547686484912895423749, 0, 51511809056557520831904891675, 0, 4471003612771204238470977530420, 0, 388063933552395196690731448669503, 0, 33682286063018183725276407994618327, 0, 2923478107178621309447954157274012236, 0, 253745374912778104299484201057746051187] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 36316716025325540764 z - 5552674095195937268 z - 260 z 24 22 4 6 + 731830147426001104 z - 82665334756978664 z + 30674 z - 2201708 z 102 8 10 12 - 3945897204 z + 108651071 z - 3945897204 z + 110156636814 z 14 18 16 - 2436568744080 z - 644646810193336 z + 43661522971117 z 50 48 - 4768038110989943616621820 z + 3091903222013735993128186 z 20 36 + 7946936314395296 z + 16377647912797844485348 z 34 66 - 4355741755631938498776 z - 1770316353002902967421320 z 80 100 90 + 1013463059036539406244 z + 110156636814 z - 82665334756978664 z 88 84 94 + 731830147426001104 z + 36316716025325540764 z - 644646810193336 z 86 96 98 - 5552674095195937268 z + 43661522971117 z - 2436568744080 z 92 82 + 7946936314395296 z - 205731269935184025128 z 64 112 110 106 + 3091903222013735993128186 z + z - 260 z - 2201708 z 108 30 42 + 30674 z - 205731269935184025128 z - 398488083694870849238936 z 44 46 + 894455856859384583828784 z - 1770316353002902967421320 z 58 56 - 7817623003781220166231840 z + 8315687735435857651579802 z 54 52 - 7817623003781220166231840 z + 6495012806521136096499264 z 60 70 + 6495012806521136096499264 z - 398488083694870849238936 z 68 78 + 894455856859384583828784 z - 4355741755631938498776 z 32 38 + 1013463059036539406244 z - 53995689626281575085884 z 40 62 + 156384429486411356234920 z - 4768038110989943616621820 z 76 74 + 16377647912797844485348 z - 53995689626281575085884 z 72 104 / 2 + 156384429486411356234920 z + 108651071 z ) / ((-1 + z ) (1 / 28 26 2 + 98759669843631715670 z - 14531086220992532035 z - 311 z 24 22 4 6 + 1839154677605369652 z - 199066269697866905 z + 41688 z - 3305143 z 102 8 10 12 - 6908053670 z + 177114431 z - 6908053670 z + 205542084342 z 14 18 16 - 4818755217820 z - 1415151195238523 z + 91130961317765 z 50 48 - 17012556872539745798420321 z + 10907637312137341555447498 z 20 36 + 18294973473636938 z + 50857477928023709230490 z 34 66 - 13126666833692582578667 z - 6156089882050411626094057 z 80 100 90 + 2957604432700929520928 z + 205542084342 z - 199066269697866905 z 88 84 + 1839154677605369652 z + 98759669843631715670 z 94 86 96 - 1415151195238523 z - 14531086220992532035 z + 91130961317765 z 98 92 82 - 4818755217820 z + 18294973473636938 z - 580167438613688225705 z 64 112 110 106 + 10907637312137341555447498 z + z - 311 z - 3305143 z 108 30 42 + 41688 z - 580167438613688225705 z - 1334784132025040546122707 z 44 46 + 3056964593590422595038118 z - 6156089882050411626094057 z 58 56 - 28262138245365248498315167 z + 30112485490136173950091970 z 54 52 - 28262138245365248498315167 z + 23364873022079621616496162 z 60 70 + 23364873022079621616496162 z - 1334784132025040546122707 z 68 78 + 3056964593590422595038118 z - 13126666833692582578667 z 32 38 + 2957604432700929520928 z - 172378685288508062370777 z 40 62 + 512036010667195554674748 z - 17012556872539745798420321 z 76 74 + 50857477928023709230490 z - 172378685288508062370777 z 72 104 + 512036010667195554674748 z + 177114431 z )) And in Maple-input format, it is: -(1+36316716025325540764*z^28-5552674095195937268*z^26-260*z^2+ 731830147426001104*z^24-82665334756978664*z^22+30674*z^4-2201708*z^6-3945897204 *z^102+108651071*z^8-3945897204*z^10+110156636814*z^12-2436568744080*z^14-\ 644646810193336*z^18+43661522971117*z^16-4768038110989943616621820*z^50+ 3091903222013735993128186*z^48+7946936314395296*z^20+16377647912797844485348*z^ 36-4355741755631938498776*z^34-1770316353002902967421320*z^66+ 1013463059036539406244*z^80+110156636814*z^100-82665334756978664*z^90+ 731830147426001104*z^88+36316716025325540764*z^84-644646810193336*z^94-\ 5552674095195937268*z^86+43661522971117*z^96-2436568744080*z^98+ 7946936314395296*z^92-205731269935184025128*z^82+3091903222013735993128186*z^64 +z^112-260*z^110-2201708*z^106+30674*z^108-205731269935184025128*z^30-\ 398488083694870849238936*z^42+894455856859384583828784*z^44-\ 1770316353002902967421320*z^46-7817623003781220166231840*z^58+ 8315687735435857651579802*z^56-7817623003781220166231840*z^54+ 6495012806521136096499264*z^52+6495012806521136096499264*z^60-\ 398488083694870849238936*z^70+894455856859384583828784*z^68-\ 4355741755631938498776*z^78+1013463059036539406244*z^32-53995689626281575085884 *z^38+156384429486411356234920*z^40-4768038110989943616621820*z^62+ 16377647912797844485348*z^76-53995689626281575085884*z^74+ 156384429486411356234920*z^72+108651071*z^104)/(-1+z^2)/(1+98759669843631715670 *z^28-14531086220992532035*z^26-311*z^2+1839154677605369652*z^24-\ 199066269697866905*z^22+41688*z^4-3305143*z^6-6908053670*z^102+177114431*z^8-\ 6908053670*z^10+205542084342*z^12-4818755217820*z^14-1415151195238523*z^18+ 91130961317765*z^16-17012556872539745798420321*z^50+10907637312137341555447498* z^48+18294973473636938*z^20+50857477928023709230490*z^36-\ 13126666833692582578667*z^34-6156089882050411626094057*z^66+ 2957604432700929520928*z^80+205542084342*z^100-199066269697866905*z^90+ 1839154677605369652*z^88+98759669843631715670*z^84-1415151195238523*z^94-\ 14531086220992532035*z^86+91130961317765*z^96-4818755217820*z^98+ 18294973473636938*z^92-580167438613688225705*z^82+10907637312137341555447498*z^ 64+z^112-311*z^110-3305143*z^106+41688*z^108-580167438613688225705*z^30-\ 1334784132025040546122707*z^42+3056964593590422595038118*z^44-\ 6156089882050411626094057*z^46-28262138245365248498315167*z^58+ 30112485490136173950091970*z^56-28262138245365248498315167*z^54+ 23364873022079621616496162*z^52+23364873022079621616496162*z^60-\ 1334784132025040546122707*z^70+3056964593590422595038118*z^68-\ 13126666833692582578667*z^78+2957604432700929520928*z^32-\ 172378685288508062370777*z^38+512036010667195554674748*z^40-\ 17012556872539745798420321*z^62+50857477928023709230490*z^76-\ 172378685288508062370777*z^74+512036010667195554674748*z^72+177114431*z^104) The first , 40, terms are: [0, 52, 0, 4899, 0, 489663, 0, 49288464, 0, 4966222549, 0, 500474290733, 0, 50437527175524, 0, 5083119352033235, 0, 512281011758500919, 0, 51628166104783989848, 0, 5203137599814847389881, 0, 524377423725112749421033, 0, 52847285188453017080302688, 0, 5326002755274175729868330287, 0, 536759938347219354581407826187, 0, 54095208987600846119860130066780, 0, 5451769830292846764043378987039277, 0, 549434873221391219564539573148084117, 0, 55372601805266957406865357200780768088, 0, 5580506772051853524221876923098962044263] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 92636 z + 992474 z + 117 z - 6342223 z + 25103201 z 4 6 8 10 12 - 4779 z + 92636 z - 992474 z + 6342223 z - 25103201 z 14 18 16 20 34 + 62568345 z + 98738390 z - 98738390 z - 62568345 z + z 30 32 / 24 20 8 + 4779 z - 117 z ) / (97962687 z + 583426401 z + 2624033 z / 28 26 36 10 34 4 + 2624033 z - 20349592 z + z - 20349592 z - 164 z + 8447 z 30 12 6 2 14 - 201252 z + 1 + 97962687 z - 201252 z - 164 z - 298984420 z 22 32 16 18 - 298984420 z + 8447 z + 583426401 z - 729057836 z ) And in Maple-input format, it is: -(-1-92636*z^28+992474*z^26+117*z^2-6342223*z^24+25103201*z^22-4779*z^4+92636*z ^6-992474*z^8+6342223*z^10-25103201*z^12+62568345*z^14+98738390*z^18-98738390*z ^16-62568345*z^20+z^34+4779*z^30-117*z^32)/(97962687*z^24+583426401*z^20+ 2624033*z^8+2624033*z^28-20349592*z^26+z^36-20349592*z^10-164*z^34+8447*z^4-\ 201252*z^30+1+97962687*z^12-201252*z^6-164*z^2-298984420*z^14-298984420*z^22+ 8447*z^32+583426401*z^16-729057836*z^18) The first , 40, terms are: [0, 47, 0, 4040, 0, 374167, 0, 35064793, 0, 3293773301, 0, 309570849995, 0, 29099994012424, 0, 2735553282646567, 0, 257159952730685393, 0, 24174822986360420753, 0, 2272604315753487876871, 0, 213640961950511732948296, 0, 20083771718246336733462179, 0, 1888017626845918195149765629, 0, 177487110091065600368750422945, 0, 16685053104344697679960087525327, 0, 1568513889682084925230199815468616, 0, 147451482930635960530539740362364351, 0, 13861490142931590233105417311427138449, 0, 1303078851229121372550053697048220517617] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 85789028531902 z - 50332449693130 z - 163 z 24 22 4 6 + 23778113310054 z - 9012590171736 z + 10965 z - 414328 z 8 10 12 14 + 10021048 z - 166959408 z + 2008323213 z - 18011543667 z 18 16 50 48 - 654079145992 z + 123232061113 z - 18011543667 z + 123232061113 z 20 36 34 64 + 2726873034192 z + 85789028531902 z - 118026602973752 z + z 30 42 44 - 118026602973752 z - 9012590171736 z + 2726873034192 z 46 58 56 54 - 654079145992 z - 414328 z + 10021048 z - 166959408 z 52 60 32 38 + 2008323213 z + 10965 z + 131249271266736 z - 50332449693130 z 40 62 / 28 + 23778113310054 z - 163 z ) / (-1 - 490502640798828 z / 26 2 24 22 + 259744594600678 z + 219 z - 110868760386626 z + 38005878730180 z 4 6 8 10 12 - 17688 z + 767678 z - 20890991 z + 387591243 z - 5164124538 z 14 18 16 + 51163781484 z + 2262132957709 z - 386284087183 z 50 48 20 + 386284087183 z - 2262132957709 z - 10409814767532 z 36 34 66 64 - 748448455315896 z + 924199805646614 z + z - 219 z 30 42 44 + 748448455315896 z + 110868760386626 z - 38005878730180 z 46 58 56 54 + 10409814767532 z + 20890991 z - 387591243 z + 5164124538 z 52 60 32 38 - 51163781484 z - 767678 z - 924199805646614 z + 490502640798828 z 40 62 - 259744594600678 z + 17688 z ) And in Maple-input format, it is: -(1+85789028531902*z^28-50332449693130*z^26-163*z^2+23778113310054*z^24-\ 9012590171736*z^22+10965*z^4-414328*z^6+10021048*z^8-166959408*z^10+2008323213* z^12-18011543667*z^14-654079145992*z^18+123232061113*z^16-18011543667*z^50+ 123232061113*z^48+2726873034192*z^20+85789028531902*z^36-118026602973752*z^34+z ^64-118026602973752*z^30-9012590171736*z^42+2726873034192*z^44-654079145992*z^ 46-414328*z^58+10021048*z^56-166959408*z^54+2008323213*z^52+10965*z^60+ 131249271266736*z^32-50332449693130*z^38+23778113310054*z^40-163*z^62)/(-1-\ 490502640798828*z^28+259744594600678*z^26+219*z^2-110868760386626*z^24+ 38005878730180*z^22-17688*z^4+767678*z^6-20890991*z^8+387591243*z^10-5164124538 *z^12+51163781484*z^14+2262132957709*z^18-386284087183*z^16+386284087183*z^50-\ 2262132957709*z^48-10409814767532*z^20-748448455315896*z^36+924199805646614*z^ 34+z^66-219*z^64+748448455315896*z^30+110868760386626*z^42-38005878730180*z^44+ 10409814767532*z^46+20890991*z^58-387591243*z^56+5164124538*z^54-51163781484*z^ 52-767678*z^60-924199805646614*z^32+490502640798828*z^38-259744594600678*z^40+ 17688*z^62) The first , 40, terms are: [0, 56, 0, 5541, 0, 576301, 0, 60320736, 0, 6321069233, 0, 662566909889, 0, 69454087980256, 0, 7280704525925149, 0, 763222196343300565, 0, 80007206968726378488, 0, 8387014024389531762513, 0, 879195926146899249070129, 0, 92164564475481518950210936, 0, 9661449452010840668504543989, 0, 1012792782175927388379423834557, 0, 106169289110700424928531629198816, 0, 11129540168570627630518242007474209, 0, 1166690154991969623352537426615124305, 0, 122302080512837516039616155297984191712, 0, 12820712366350288304606475276482622170509] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 8287110858792 z - 7311747429290 z - 174 z 24 22 4 6 + 5016859495784 z - 2668639265820 z + 12340 z - 475849 z 8 10 12 14 + 11261356 z - 175413837 z + 1886601300 z - 14501084166 z 18 16 50 48 - 343646911776 z + 81690819701 z - 475849 z + 11261356 z 20 36 34 + 1094706982538 z + 1094706982538 z - 2668639265820 z 30 42 44 46 - 7311747429290 z - 14501084166 z + 1886601300 z - 175413837 z 56 54 52 32 38 + z - 174 z + 12340 z + 5016859495784 z - 343646911776 z 40 / 28 26 + 81690819701 z ) / (-1 - 57740728339738 z + 45007400213086 z / 2 24 22 4 6 + 227 z - 27300655902470 z + 12843653276822 z - 19343 z + 864071 z 8 10 12 14 - 23285317 z + 409852305 z - 4966722891 z + 42996379099 z 18 16 50 48 + 1295737286733 z - 272996549495 z + 23285317 z - 409852305 z 20 36 34 - 4661750762146 z - 12843653276822 z + 27300655902470 z 30 42 44 46 + 57740728339738 z + 272996549495 z - 42996379099 z + 4966722891 z 58 56 54 52 32 + z - 227 z + 19343 z - 864071 z - 45007400213086 z 38 40 + 4661750762146 z - 1295737286733 z ) And in Maple-input format, it is: -(1+8287110858792*z^28-7311747429290*z^26-174*z^2+5016859495784*z^24-\ 2668639265820*z^22+12340*z^4-475849*z^6+11261356*z^8-175413837*z^10+1886601300* z^12-14501084166*z^14-343646911776*z^18+81690819701*z^16-475849*z^50+11261356*z ^48+1094706982538*z^20+1094706982538*z^36-2668639265820*z^34-7311747429290*z^30 -14501084166*z^42+1886601300*z^44-175413837*z^46+z^56-174*z^54+12340*z^52+ 5016859495784*z^32-343646911776*z^38+81690819701*z^40)/(-1-57740728339738*z^28+ 45007400213086*z^26+227*z^2-27300655902470*z^24+12843653276822*z^22-19343*z^4+ 864071*z^6-23285317*z^8+409852305*z^10-4966722891*z^12+42996379099*z^14+ 1295737286733*z^18-272996549495*z^16+23285317*z^50-409852305*z^48-4661750762146 *z^20-12843653276822*z^36+27300655902470*z^34+57740728339738*z^30+272996549495* z^42-42996379099*z^44+4966722891*z^46+z^58-227*z^56+19343*z^54-864071*z^52-\ 45007400213086*z^32+4661750762146*z^38-1295737286733*z^40) The first , 40, terms are: [0, 53, 0, 5028, 0, 504399, 0, 51013771, 0, 5168401815, 0, 523867864579, 0, 53106034835396, 0, 5383732011127953, 0, 545793860312927397, 0, 55331923277071609309, 0, 5609492209223006181145, 0, 568684730737402036355204, 0, 57652702591360607069017179, 0, 5844775095767612569261133823, 0, 592537648183407864556193990179, 0, 60070894448430873564372167850615, 0, 6089929278792858659616215067221220, 0, 617391151056735929985731251884470061, 0, 62590518893526697851452818348806456009, 0, 6345366384574447264000373628487267061561] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {4, 6}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 93544530304006 z - 54115739622610 z - 159 z 24 22 4 6 + 25103033294446 z - 9315560871728 z + 10469 z - 389416 z 8 10 12 14 + 9341572 z - 155584792 z + 1884426325 z - 17121156591 z 18 16 50 48 - 646208305488 z + 119231008209 z - 17121156591 z + 119231008209 z 20 36 34 64 + 2755493770800 z + 93544530304006 z - 129844460659136 z + z 30 42 44 - 129844460659136 z - 9315560871728 z + 2755493770800 z 46 58 56 54 - 646208305488 z - 389416 z + 9341572 z - 155584792 z 52 60 32 38 + 1884426325 z + 10469 z + 144830130248280 z - 54115739622610 z 40 62 / 28 + 25103033294446 z - 159 z ) / (-1 - 510818755627704 z / 26 2 24 22 + 267246236696234 z + 203 z - 112369372136526 z + 37865071282456 z 4 6 8 10 12 - 15936 z + 685502 z - 18680459 z + 349272667 z - 4710177046 z 14 18 16 + 47379470952 z + 2171122689105 z - 363944753771 z 50 48 20 + 363944753771 z - 2171122689105 z - 10181638990920 z 36 34 66 64 - 786111640757196 z + 974983661990138 z + z - 203 z 30 42 44 + 786111640757196 z + 112369372136526 z - 37865071282456 z 46 58 56 54 + 10181638990920 z + 18680459 z - 349272667 z + 4710177046 z 52 60 32 38 - 47379470952 z - 685502 z - 974983661990138 z + 510818755627704 z 40 62 - 267246236696234 z + 15936 z ) And in Maple-input format, it is: -(1+93544530304006*z^28-54115739622610*z^26-159*z^2+25103033294446*z^24-\ 9315560871728*z^22+10469*z^4-389416*z^6+9341572*z^8-155584792*z^10+1884426325*z ^12-17121156591*z^14-646208305488*z^18+119231008209*z^16-17121156591*z^50+ 119231008209*z^48+2755493770800*z^20+93544530304006*z^36-129844460659136*z^34+z ^64-129844460659136*z^30-9315560871728*z^42+2755493770800*z^44-646208305488*z^ 46-389416*z^58+9341572*z^56-155584792*z^54+1884426325*z^52+10469*z^60+ 144830130248280*z^32-54115739622610*z^38+25103033294446*z^40-159*z^62)/(-1-\ 510818755627704*z^28+267246236696234*z^26+203*z^2-112369372136526*z^24+ 37865071282456*z^22-15936*z^4+685502*z^6-18680459*z^8+349272667*z^10-4710177046 *z^12+47379470952*z^14+2171122689105*z^18-363944753771*z^16+363944753771*z^50-\ 2171122689105*z^48-10181638990920*z^20-786111640757196*z^36+974983661990138*z^ 34+z^66-203*z^64+786111640757196*z^30+112369372136526*z^42-37865071282456*z^44+ 10181638990920*z^46+18680459*z^58-349272667*z^56+4710177046*z^54-47379470952*z^ 52-685502*z^60-974983661990138*z^32+510818755627704*z^38-267246236696234*z^40+ 15936*z^62) The first , 40, terms are: [0, 44, 0, 3465, 0, 298297, 0, 26159252, 0, 2303679273, 0, 203070698833, 0, 17905053970244, 0, 1578807422160257, 0, 139215911854257441, 0, 12275809153937657340, 0, 1082459764589150201305, 0, 95449463753883820871497, 0, 8416572086001805696906140, 0, 742159084984457304907392913, 0, 65442332554318879230129672721, 0, 5770594176903796710418477178532, 0, 508841232621957358276053350294049, 0, 44868759107859150339004901417389977, 0, 3956451275655028377172095822510593716, 0, 348873180536900365898075144286174842857] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 147350 z + 1704705 z + 139 z - 11321267 z + 45281524 z 4 6 8 10 12 - 6722 z + 147350 z - 1704705 z + 11321267 z - 45281524 z 14 18 16 20 34 + 112248676 z + 175812720 z - 175812720 z - 112248676 z + z 30 32 / 10 2 8 + 6722 z - 139 z ) / (-37589911 z - 207 z + 4675685 z / 16 34 6 4 18 + 1108537962 z + 1 - 207 z - 336022 z + 12677 z - 1385700908 z 30 12 36 28 22 - 336022 z + 184255459 z + z + 4675685 z - 566761638 z 24 26 20 32 + 184255459 z - 37589911 z + 1108537962 z + 12677 z 14 - 566761638 z ) And in Maple-input format, it is: -(-1-147350*z^28+1704705*z^26+139*z^2-11321267*z^24+45281524*z^22-6722*z^4+ 147350*z^6-1704705*z^8+11321267*z^10-45281524*z^12+112248676*z^14+175812720*z^ 18-175812720*z^16-112248676*z^20+z^34+6722*z^30-139*z^32)/(-37589911*z^10-207*z ^2+4675685*z^8+1108537962*z^16+1-207*z^34-336022*z^6+12677*z^4-1385700908*z^18-\ 336022*z^30+184255459*z^12+z^36+4675685*z^28-566761638*z^22+184255459*z^24-\ 37589911*z^26+1108537962*z^20+12677*z^32-566761638*z^14) The first , 40, terms are: [0, 68, 0, 8121, 0, 1007683, 0, 125518980, 0, 15645188195, 0, 1950399406059, 0, 243157349524884, 0, 30315018279469851, 0, 3779465306074411665, 0, 471198133165112946196, 0, 58745820442786287476553, 0, 7324035733083474321451321, 0, 913111804868991574303953396, 0, 113840676381722016797448668129, 0, 14192894667637356004762865292395, 0, 1769475250708037769954548369471476, 0, 220606348307056700444979813434974811, 0, 27503725131048395158267720742086954227, 0, 3428980634195752501128706967470567482148, 0, 427502388635411027180291951239589271646451] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 9479952494620 z - 8346224233692 z - 176 z 24 22 4 6 + 5690261593548 z - 2995704186732 z + 12544 z - 483363 z 8 10 12 14 + 11444622 z - 179337329 z + 1951480004 z - 15238368884 z 18 16 50 48 - 374260465948 z + 87387296795 z - 483363 z + 11444622 z 20 36 34 + 1212019029524 z + 1212019029524 z - 2995704186732 z 30 42 44 46 - 8346224233692 z - 15238368884 z + 1951480004 z - 179337329 z 56 54 52 32 38 + z - 176 z + 12544 z + 5690261593548 z - 374260465948 z 40 / 2 28 + 87387296795 z ) / ((-1 + z ) (1 + 35409906568866 z / 26 2 24 22 - 30941090407632 z - 234 z + 20631322796014 z - 10479263156980 z 4 6 8 10 12 + 20059 z - 888172 z + 23615209 z - 409874916 z + 4890865635 z 14 18 16 50 - 41526782858 z - 1177085183480 z + 256889297897 z - 888172 z 48 20 36 + 23615209 z + 4041299947258 z + 4041299947258 z 34 30 42 - 10479263156980 z - 30941090407632 z - 41526782858 z 44 46 56 54 52 + 4890865635 z - 409874916 z + z - 234 z + 20059 z 32 38 40 + 20631322796014 z - 1177085183480 z + 256889297897 z )) And in Maple-input format, it is: -(1+9479952494620*z^28-8346224233692*z^26-176*z^2+5690261593548*z^24-\ 2995704186732*z^22+12544*z^4-483363*z^6+11444622*z^8-179337329*z^10+1951480004* z^12-15238368884*z^14-374260465948*z^18+87387296795*z^16-483363*z^50+11444622*z ^48+1212019029524*z^20+1212019029524*z^36-2995704186732*z^34-8346224233692*z^30 -15238368884*z^42+1951480004*z^44-179337329*z^46+z^56-176*z^54+12544*z^52+ 5690261593548*z^32-374260465948*z^38+87387296795*z^40)/(-1+z^2)/(1+ 35409906568866*z^28-30941090407632*z^26-234*z^2+20631322796014*z^24-\ 10479263156980*z^22+20059*z^4-888172*z^6+23615209*z^8-409874916*z^10+4890865635 *z^12-41526782858*z^14-1177085183480*z^18+256889297897*z^16-888172*z^50+ 23615209*z^48+4041299947258*z^20+4041299947258*z^36-10479263156980*z^34-\ 30941090407632*z^30-41526782858*z^42+4890865635*z^44-409874916*z^46+z^56-234*z^ 54+20059*z^52+20631322796014*z^32-1177085183480*z^38+256889297897*z^40) The first , 40, terms are: [0, 59, 0, 6116, 0, 664841, 0, 72652517, 0, 7944917195, 0, 868911198247, 0, 95032003981516, 0, 10393604065364293, 0, 1136744594546900735, 0, 124325369128608779535, 0, 13597424613990955179509, 0, 1487145896470315513761260, 0, 162648663859320670313204535, 0, 17788831642805193944946890683, 0, 1945558753912883565622304233301, 0, 212785130778556300486794741238681, 0, 23272240841450195363990697594670340, 0, 2545277444221830803306606920672465771, 0, 278376169808829172583272758317491106449, 0, 30445911542762072676841021780893866179953] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 7}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 18 16 14 12 10 8 6 f(z) = - (z - 83 z + 1275 z - 6443 z + 13910 z - 13910 z + 6443 z 4 2 / 20 18 16 14 - 1275 z + 83 z - 1) / (z - 151 z + 3192 z - 23881 z / 12 10 8 6 4 2 + 78223 z - 117360 z + 78223 z - 23881 z + 3192 z - 151 z + 1) And in Maple-input format, it is: -(z^18-83*z^16+1275*z^14-6443*z^12+13910*z^10-13910*z^8+6443*z^6-1275*z^4+83*z^ 2-1)/(z^20-151*z^18+3192*z^16-23881*z^14+78223*z^12-117360*z^10+78223*z^8-23881 *z^6+3192*z^4-151*z^2+1) The first , 40, terms are: [0, 68, 0, 8351, 0, 1061383, 0, 135172036, 0, 17217257417, 0, 2193038286905, 0, 279337289253700, 0, 35580468140000695, 0, 4532047004609738735, 0, 577267561233027898820, 0, 73529210297759472060817, 0, 9365751914952592037674609, 0, 1192958670129814657972600004, 0, 151952603652241160290350304079, 0, 19354898316957387339585286051927, 0, 2465321948132574858431680318596676, 0, 314019335488797017314831695247298393, 0, 39998079413327364282281414606228979881, 0, 5094738367828686066702555198489255616708, 0, 648940134559986960703696908844239714288487] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 7}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 76551457395598 z - 45038990688050 z - 163 z 24 22 4 6 + 21364553602902 z - 8142306608672 z + 10901 z - 408008 z 8 10 12 14 + 9753372 z - 160445800 z + 1905366461 z - 16879692363 z 18 16 50 48 - 600115978304 z + 114195286473 z - 16879692363 z + 114195286473 z 20 36 34 64 + 2480775125040 z + 76551457395598 z - 105146036282608 z + z 30 42 44 - 105146036282608 z - 8142306608672 z + 2480775125040 z 46 58 56 54 - 600115978304 z - 408008 z + 9753372 z - 160445800 z 52 60 32 38 + 1905366461 z + 10901 z + 116863187225768 z - 45038990688050 z 40 62 / 28 + 21364553602902 z - 163 z ) / (-1 - 438185795836032 z / 26 2 24 22 + 232780093475170 z + 219 z - 99805597973606 z + 34416439528272 z 4 6 8 10 12 - 17608 z + 758126 z - 20414867 z + 374189067 z - 4922304462 z 14 18 16 + 48158953176 z + 2082483548985 z - 359355996899 z 50 48 20 + 359355996899 z - 2082483548985 z - 9497155581840 z 36 34 66 64 - 667266920469732 z + 823147846310338 z + z - 219 z 30 42 44 + 667266920469732 z + 99805597973606 z - 34416439528272 z 46 58 56 54 + 9497155581840 z + 20414867 z - 374189067 z + 4922304462 z 52 60 32 38 - 48158953176 z - 758126 z - 823147846310338 z + 438185795836032 z 40 62 - 232780093475170 z + 17608 z ) And in Maple-input format, it is: -(1+76551457395598*z^28-45038990688050*z^26-163*z^2+21364553602902*z^24-\ 8142306608672*z^22+10901*z^4-408008*z^6+9753372*z^8-160445800*z^10+1905366461*z ^12-16879692363*z^14-600115978304*z^18+114195286473*z^16-16879692363*z^50+ 114195286473*z^48+2480775125040*z^20+76551457395598*z^36-105146036282608*z^34+z ^64-105146036282608*z^30-8142306608672*z^42+2480775125040*z^44-600115978304*z^ 46-408008*z^58+9753372*z^56-160445800*z^54+1905366461*z^52+10901*z^60+ 116863187225768*z^32-45038990688050*z^38+21364553602902*z^40-163*z^62)/(-1-\ 438185795836032*z^28+232780093475170*z^26+219*z^2-99805597973606*z^24+ 34416439528272*z^22-17608*z^4+758126*z^6-20414867*z^8+374189067*z^10-4922304462 *z^12+48158953176*z^14+2082483548985*z^18-359355996899*z^16+359355996899*z^50-\ 2082483548985*z^48-9497155581840*z^20-667266920469732*z^36+823147846310338*z^34 +z^66-219*z^64+667266920469732*z^30+99805597973606*z^42-34416439528272*z^44+ 9497155581840*z^46+20414867*z^58-374189067*z^56+4922304462*z^54-48158953176*z^ 52-758126*z^60-823147846310338*z^32+438185795836032*z^38-232780093475170*z^40+ 17608*z^62) The first , 40, terms are: [0, 56, 0, 5557, 0, 581053, 0, 61196512, 0, 6454271801, 0, 680940961633, 0, 71846798723440, 0, 7580795252780165, 0, 799879599205158093, 0, 84398582022340808744, 0, 8905244815683521805721, 0, 939629463166414292595785, 0, 99144220495029986035580232, 0, 10461119969353546079025573565, 0, 1103796375818859830273267750677, 0, 116466156932118085712940500780880, 0, 12288829724408827215262817079906545, 0, 1296645652097616618654074554181248873, 0, 136814488021223261182183423660933150464, 0, 14435866963571118585663262384235722931053] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 229838 z + 3020522 z + 163 z - 21849302 z + 89965010 z 4 6 8 10 12 - 9078 z + 229838 z - 3020522 z + 21849302 z - 89965010 z 14 18 16 20 34 + 223024490 z + 347431260 z - 347431260 z - 223024490 z + z 30 32 / 36 34 32 30 + 9078 z - 163 z ) / (z - 202 z + 14885 z - 504692 z / 28 26 24 22 + 8380084 z - 73536960 z + 362053740 z - 1075579660 z 20 18 16 14 + 2028901370 z - 2499472108 z + 2028901370 z - 1075579660 z 12 10 8 6 4 + 362053740 z - 73536960 z + 8380084 z - 504692 z + 14885 z 2 - 202 z + 1) And in Maple-input format, it is: -(-1-229838*z^28+3020522*z^26+163*z^2-21849302*z^24+89965010*z^22-9078*z^4+ 229838*z^6-3020522*z^8+21849302*z^10-89965010*z^12+223024490*z^14+347431260*z^ 18-347431260*z^16-223024490*z^20+z^34+9078*z^30-163*z^32)/(z^36-202*z^34+14885* z^32-504692*z^30+8380084*z^28-73536960*z^26+362053740*z^24-1075579660*z^22+ 2028901370*z^20-2499472108*z^18+2028901370*z^16-1075579660*z^14+362053740*z^12-\ 73536960*z^10+8380084*z^8-504692*z^6+14885*z^4-202*z^2+1) The first , 40, terms are: [0, 39, 0, 2071, 0, 112681, 0, 6258153, 0, 356971735, 0, 21065581063, 0, 1294914111393, 0, 83305078465249, 0, 5614565934538567, 0, 395249599421115735, 0, 28885886243965001449, 0, 2175150396015128399401, 0, 167521862521343167704471, 0, 13112621559033758226923239, 0, 1038021187139212064415514561, 0, 82804860459242622202989431361, 0, 6639534003467542600411548013927, 0, 534193447801853961350366672681623, 0, 43075751034996377943305491057103913, 0, 3478601537747005555389076899928016617] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 756058644251148 z - 372108151188506 z - 195 z 24 22 4 6 + 148650276356254 z - 47967337181184 z + 15450 z - 679923 z 8 10 12 14 + 19039793 z - 366325400 z + 5085048081 z - 52655316435 z 18 16 50 - 2562667235875 z + 416565341082 z - 2562667235875 z 48 20 36 + 12424678990689 z + 12424678990689 z + 1691469972776638 z 34 66 64 30 - 1869897840076496 z - 195 z + 15450 z - 1251385143867706 z 42 44 46 - 372108151188506 z + 148650276356254 z - 47967337181184 z 58 56 54 52 - 366325400 z + 5085048081 z - 52655316435 z + 416565341082 z 60 68 32 38 + 19039793 z + z + 1691469972776638 z - 1251385143867706 z 40 62 / 28 + 756058644251148 z - 679923 z ) / (-1 - 3923835137771362 z / 26 2 24 + 1776050736404102 z + 235 z - 653824759104738 z 22 4 6 8 + 194588807516957 z - 22465 z + 1174775 z - 38308485 z 10 12 14 18 + 841869255 z - 13133052075 z + 150896358689 z + 8813783540189 z 16 50 48 - 1312128339227 z + 46461325044623 z - 194588807516957 z 20 36 34 - 46461325044623 z - 12744461867515790 z + 12744461867515790 z 66 64 30 42 + 22465 z - 1174775 z + 7083106609331438 z + 3923835137771362 z 44 46 58 - 1776050736404102 z + 653824759104738 z + 13133052075 z 56 54 52 60 - 150896358689 z + 1312128339227 z - 8813783540189 z - 841869255 z 70 68 32 38 + z - 235 z - 10482019513267546 z + 10482019513267546 z 40 62 - 7083106609331438 z + 38308485 z ) And in Maple-input format, it is: -(1+756058644251148*z^28-372108151188506*z^26-195*z^2+148650276356254*z^24-\ 47967337181184*z^22+15450*z^4-679923*z^6+19039793*z^8-366325400*z^10+5085048081 *z^12-52655316435*z^14-2562667235875*z^18+416565341082*z^16-2562667235875*z^50+ 12424678990689*z^48+12424678990689*z^20+1691469972776638*z^36-1869897840076496* z^34-195*z^66+15450*z^64-1251385143867706*z^30-372108151188506*z^42+ 148650276356254*z^44-47967337181184*z^46-366325400*z^58+5085048081*z^56-\ 52655316435*z^54+416565341082*z^52+19039793*z^60+z^68+1691469972776638*z^32-\ 1251385143867706*z^38+756058644251148*z^40-679923*z^62)/(-1-3923835137771362*z^ 28+1776050736404102*z^26+235*z^2-653824759104738*z^24+194588807516957*z^22-\ 22465*z^4+1174775*z^6-38308485*z^8+841869255*z^10-13133052075*z^12+150896358689 *z^14+8813783540189*z^18-1312128339227*z^16+46461325044623*z^50-194588807516957 *z^48-46461325044623*z^20-12744461867515790*z^36+12744461867515790*z^34+22465*z ^66-1174775*z^64+7083106609331438*z^30+3923835137771362*z^42-1776050736404102*z ^44+653824759104738*z^46+13133052075*z^58-150896358689*z^56+1312128339227*z^54-\ 8813783540189*z^52-841869255*z^60+z^70-235*z^68-10482019513267546*z^32+ 10482019513267546*z^38-7083106609331438*z^40+38308485*z^62) The first , 40, terms are: [0, 40, 0, 2385, 0, 156727, 0, 10974128, 0, 803090855, 0, 60572556311, 0, 4662049100544, 0, 363629507286695, 0, 28608062945787329, 0, 2263186105813593464, 0, 179672175068648634801, 0, 14295822009757100683153, 0, 1139063536480854336190936, 0, 90838726155110625860104865, 0, 7248291157389580222349508743, 0, 578564610467223333576402190176, 0, 46191626439570227781746035164279, 0, 3688368707066619636079778987324615, 0, 294539015762437229306069020523970896, 0, 23522030069445299408679967497031402711] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 459514577934977600 z - 116544401992106368 z - 242 z 24 22 4 6 + 24799457030156949 z - 4401680610786714 z + 25260 z - 1533666 z 8 10 12 14 + 61481955 z - 1748430864 z + 36950415216 z - 599466568128 z 18 16 50 - 78062861325114 z + 7645872646767 z - 36106641421461893892 z 48 20 + 53410794551662314078 z + 646999910793500 z 36 34 + 20841467922754641848 z - 10257211745442464340 z 66 80 88 84 86 - 4401680610786714 z + 61481955 z + z + 25260 z - 242 z 82 64 30 - 1533666 z + 24799457030156949 z - 1527314273535886048 z 42 44 - 67528250065730286544 z + 73014615226848280480 z 46 58 - 67528250065730286544 z - 1527314273535886048 z 56 54 + 4295903447270175162 z - 10257211745442464340 z 52 60 70 + 20841467922754641848 z + 459514577934977600 z - 78062861325114 z 68 78 32 + 646999910793500 z - 1748430864 z + 4295903447270175162 z 38 40 - 36106641421461893892 z + 53410794551662314078 z 62 76 74 - 116544401992106368 z + 36950415216 z - 599466568128 z 72 / 2 28 + 7645872646767 z ) / ((-1 + z ) (1 + 1480028895860457104 z / 26 2 24 - 362739941813253368 z - 284 z + 74242040241484245 z 22 4 6 8 - 12610600101389860 z + 34198 z - 2358956 z + 105955375 z 10 12 14 - 3333443988 z + 77048758388 z - 1353328294652 z 18 16 50 - 201318405630868 z + 18521377985803 z - 129609134394700196288 z 48 20 + 193608074451401116118 z + 1764096028427230 z 36 34 + 73795763521426533004 z - 35684511768129371584 z 66 80 88 84 86 - 12610600101389860 z + 105955375 z + z + 34198 z - 284 z 82 64 30 - 2358956 z + 74242040241484245 z - 5068267321999328616 z 42 44 - 246222641494961062928 z + 266748397471866196472 z 46 58 - 246222641494961062928 z - 5068267321999328616 z 56 54 + 14626017857820371226 z - 35684511768129371584 z 52 60 70 + 73795763521426533004 z + 1480028895860457104 z - 201318405630868 z 68 78 32 + 1764096028427230 z - 3333443988 z + 14626017857820371226 z 38 40 - 129609134394700196288 z + 193608074451401116118 z 62 76 74 - 362739941813253368 z + 77048758388 z - 1353328294652 z 72 + 18521377985803 z )) And in Maple-input format, it is: -(1+459514577934977600*z^28-116544401992106368*z^26-242*z^2+24799457030156949*z ^24-4401680610786714*z^22+25260*z^4-1533666*z^6+61481955*z^8-1748430864*z^10+ 36950415216*z^12-599466568128*z^14-78062861325114*z^18+7645872646767*z^16-\ 36106641421461893892*z^50+53410794551662314078*z^48+646999910793500*z^20+ 20841467922754641848*z^36-10257211745442464340*z^34-4401680610786714*z^66+ 61481955*z^80+z^88+25260*z^84-242*z^86-1533666*z^82+24799457030156949*z^64-\ 1527314273535886048*z^30-67528250065730286544*z^42+73014615226848280480*z^44-\ 67528250065730286544*z^46-1527314273535886048*z^58+4295903447270175162*z^56-\ 10257211745442464340*z^54+20841467922754641848*z^52+459514577934977600*z^60-\ 78062861325114*z^70+646999910793500*z^68-1748430864*z^78+4295903447270175162*z^ 32-36106641421461893892*z^38+53410794551662314078*z^40-116544401992106368*z^62+ 36950415216*z^76-599466568128*z^74+7645872646767*z^72)/(-1+z^2)/(1+ 1480028895860457104*z^28-362739941813253368*z^26-284*z^2+74242040241484245*z^24 -12610600101389860*z^22+34198*z^4-2358956*z^6+105955375*z^8-3333443988*z^10+ 77048758388*z^12-1353328294652*z^14-201318405630868*z^18+18521377985803*z^16-\ 129609134394700196288*z^50+193608074451401116118*z^48+1764096028427230*z^20+ 73795763521426533004*z^36-35684511768129371584*z^34-12610600101389860*z^66+ 105955375*z^80+z^88+34198*z^84-284*z^86-2358956*z^82+74242040241484245*z^64-\ 5068267321999328616*z^30-246222641494961062928*z^42+266748397471866196472*z^44-\ 246222641494961062928*z^46-5068267321999328616*z^58+14626017857820371226*z^56-\ 35684511768129371584*z^54+73795763521426533004*z^52+1480028895860457104*z^60-\ 201318405630868*z^70+1764096028427230*z^68-3333443988*z^78+14626017857820371226 *z^32-129609134394700196288*z^38+193608074451401116118*z^40-362739941813253368* z^62+77048758388*z^76-1353328294652*z^74+18521377985803*z^72) The first , 40, terms are: [0, 43, 0, 3033, 0, 241167, 0, 20221935, 0, 1739219329, 0, 151479536339, 0, 13278443373473, 0, 1167881108438833, 0, 102901585637207651, 0, 9075231395117470449, 0, 800782550605055526975, 0, 70679036674191086658495, 0, 6239228623540606104312905, 0, 550815149683165480916312827, 0, 48629477455771800927691856753, 0, 4293420882963043909257106463313, 0, 379064230959267360197768763454811, 0, 33467641669227380153276186925975657, 0, 2954874347732108906777313226760415135, 0, 260887812863019651283650275293983812447] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 19352512575 z + 36456831684 z + 156 z - 49973497059 z 22 4 6 8 10 + 49973497059 z - 9077 z + 271123 z - 4766988 z + 53276663 z 12 14 18 16 - 397449381 z + 2045861012 z + 19352512575 z - 7433063097 z 20 36 34 30 - 36456831684 z - 53276663 z + 397449381 z + 7433063097 z 42 44 46 32 38 40 + 9077 z - 156 z + z - 2045861012 z + 4766988 z - 271123 z ) / 28 26 2 24 / (1 + 204765540080 z - 326543751060 z - 204 z + 381400541734 z / 22 4 6 8 10 - 326543751060 z + 14848 z - 536408 z + 11157992 z - 145394204 z 12 14 18 16 + 1254270572 z - 7441868292 z - 93770187816 z + 31170944424 z 48 20 36 34 + z + 204765540080 z + 1254270572 z - 7441868292 z 30 42 44 46 32 - 93770187816 z - 536408 z + 14848 z - 204 z + 31170944424 z 38 40 - 145394204 z + 11157992 z ) And in Maple-input format, it is: -(-1-19352512575*z^28+36456831684*z^26+156*z^2-49973497059*z^24+49973497059*z^ 22-9077*z^4+271123*z^6-4766988*z^8+53276663*z^10-397449381*z^12+2045861012*z^14 +19352512575*z^18-7433063097*z^16-36456831684*z^20-53276663*z^36+397449381*z^34 +7433063097*z^30+9077*z^42-156*z^44+z^46-2045861012*z^32+4766988*z^38-271123*z^ 40)/(1+204765540080*z^28-326543751060*z^26-204*z^2+381400541734*z^24-\ 326543751060*z^22+14848*z^4-536408*z^6+11157992*z^8-145394204*z^10+1254270572*z ^12-7441868292*z^14-93770187816*z^18+31170944424*z^16+z^48+204765540080*z^20+ 1254270572*z^36-7441868292*z^34-93770187816*z^30-536408*z^42+14848*z^44-204*z^ 46+31170944424*z^32-145394204*z^38+11157992*z^40) The first , 40, terms are: [0, 48, 0, 4021, 0, 372865, 0, 35717232, 0, 3463446301, 0, 337477168357, 0, 32948497077264, 0, 3219422582206921, 0, 314677175855689645, 0, 30761847000996510288, 0, 3007352716142572375609, 0, 294013041103312557019081, 0, 28744388500427935486938768, 0, 2810226439768587093034063165, 0, 274745309430297348010894584505, 0, 26860838162440222936195292681040, 0, 2626086088029900386914071437553493, 0, 256742880615934660956033966620795053, 0, 25100818404075445545083939422585729584, 0, 2454015846359218425515286361252721244817] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {3, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 43659080687 z + 84834224728 z + 168 z - 118049380455 z 22 4 6 8 10 + 118049380455 z - 10889 z + 365091 z - 7195456 z + 89571347 z 12 14 18 16 - 736436789 z + 4118104800 z + 43659080687 z - 15980888389 z 20 36 34 30 - 84834224728 z - 89571347 z + 736436789 z + 15980888389 z 42 44 46 32 38 40 + 10889 z - 168 z + z - 4118104800 z + 7195456 z - 365091 z ) / 28 26 2 24 / (1 + 472920624576 z - 768376715368 z - 216 z + 902770252406 z / 22 4 6 8 10 - 768376715368 z + 17296 z - 697444 z + 16270728 z - 237282744 z 12 14 18 16 + 2269897580 z - 14713450792 z - 209380444956 z + 66116700136 z 48 20 36 34 + z + 472920624576 z + 2269897580 z - 14713450792 z 30 42 44 46 32 - 209380444956 z - 697444 z + 17296 z - 216 z + 66116700136 z 38 40 - 237282744 z + 16270728 z ) And in Maple-input format, it is: -(-1-43659080687*z^28+84834224728*z^26+168*z^2-118049380455*z^24+118049380455*z ^22-10889*z^4+365091*z^6-7195456*z^8+89571347*z^10-736436789*z^12+4118104800*z^ 14+43659080687*z^18-15980888389*z^16-84834224728*z^20-89571347*z^36+736436789*z ^34+15980888389*z^30+10889*z^42-168*z^44+z^46-4118104800*z^32+7195456*z^38-\ 365091*z^40)/(1+472920624576*z^28-768376715368*z^26-216*z^2+902770252406*z^24-\ 768376715368*z^22+17296*z^4-697444*z^6+16270728*z^8-237282744*z^10+2269897580*z ^12-14713450792*z^14-209380444956*z^18+66116700136*z^16+z^48+472920624576*z^20+ 2269897580*z^36-14713450792*z^34-209380444956*z^30-697444*z^42+17296*z^44-216*z ^46+66116700136*z^32-237282744*z^38+16270728*z^40) The first , 40, terms are: [0, 48, 0, 3961, 0, 357721, 0, 33160320, 0, 3104778841, 0, 291989457373, 0, 27518070300960, 0, 2596066423131709, 0, 245038740344681101, 0, 23134686040206226128, 0, 2184475633543286689765, 0, 206280455364514227261949, 0, 19479713313020613147973008, 0, 1839559434330058360688815429, 0, 173719459735305194722292663125, 0, 16405323670375902745963933783008, 0, 1549251687715408893848271147784693, 0, 146305136029443261767070397907101057, 0, 13816478829202245416692349085964011456, 0, 1304773969672542321598829516120773893505] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 139900 z + 1525072 z + 141 z - 9569848 z + 36676660 z 4 6 8 10 12 - 6692 z + 139900 z - 1525072 z + 9569848 z - 36676660 z 14 18 16 20 34 + 88737580 z + 137673706 z - 137673706 z - 88737580 z + z 30 32 / 24 12 10 + 6692 z - 141 z ) / (154119860 z + 154119860 z - 33534448 z / 4 16 18 30 + 11897 z + 842447662 z + 1 - 1038883496 z - 325488 z 26 28 34 32 36 20 - 33534448 z + 4413348 z - 188 z + 11897 z + z + 842447662 z 8 22 6 2 14 + 4413348 z - 447727760 z - 325488 z - 188 z - 447727760 z ) And in Maple-input format, it is: -(-1-139900*z^28+1525072*z^26+141*z^2-9569848*z^24+36676660*z^22-6692*z^4+ 139900*z^6-1525072*z^8+9569848*z^10-36676660*z^12+88737580*z^14+137673706*z^18-\ 137673706*z^16-88737580*z^20+z^34+6692*z^30-141*z^32)/(154119860*z^24+154119860 *z^12-33534448*z^10+11897*z^4+842447662*z^16+1-1038883496*z^18-325488*z^30-\ 33534448*z^26+4413348*z^28-188*z^34+11897*z^32+z^36+842447662*z^20+4413348*z^8-\ 447727760*z^22-325488*z^6-188*z^2-447727760*z^14) The first , 40, terms are: [0, 47, 0, 3631, 0, 309057, 0, 27314369, 0, 2456634415, 0, 222916376111, 0, 20323101255297, 0, 1857588901190017, 0, 170027328105878063, 0, 15574828120018059311, 0, 1427291276320272467393, 0, 130828943865093460998721, 0, 11993645808215933528531503, 0, 1099586908396747777445803055, 0, 100814956257430804582725602561, 0, 9243358072993542462630651015937, 0, 847500128467805614211205468974127, 0, 77705655002067754848301015569619503, 0, 7124707729259019103231027588844729409, 0, 653254408360330148027551337278895573953] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4296757606191670 z - 1778151490042062 z - 207 z 24 22 4 6 + 601385830321543 z - 165517707823419 z + 17847 z - 865050 z 8 10 12 14 + 26840295 z - 574955043 z + 8934652182 z - 104245627755 z 18 16 50 - 6589465940006 z + 936131937391 z - 165517707823419 z 48 20 36 + 601385830321543 z + 36857900938043 z + 20431623413573700 z 34 66 64 30 - 18540377146048730 z - 865050 z + 26840295 z - 8511352614247296 z 42 44 46 - 8511352614247296 z + 4296757606191670 z - 1778151490042062 z 58 56 54 - 104245627755 z + 936131937391 z - 6589465940006 z 52 60 70 68 + 36857900938043 z + 8934652182 z - 207 z + 17847 z 32 38 40 + 13850335120422082 z - 18540377146048730 z + 13850335120422082 z 62 72 / 28 - 574955043 z + z ) / (-1 - 21416449890238048 z / 26 2 24 + 8113696813123183 z + 255 z - 2513352846509089 z 22 4 6 8 + 633501573494538 z - 26054 z + 1459056 z - 51432172 z 10 12 14 18 + 1237067918 z - 21400640128 z + 276200947036 z + 21087327320140 z 16 50 48 - 2730608039354 z + 2513352846509089 z - 8113696813123183 z 20 36 34 - 129092478215360 z - 147043962868148948 z + 121374650773695496 z 66 64 30 + 51432172 z - 1237067918 z + 46386572471278064 z 42 44 46 + 82654428723836136 z - 46386572471278064 z + 21416449890238048 z 58 56 54 + 2730608039354 z - 21087327320140 z + 129092478215360 z 52 60 70 68 - 633501573494538 z - 276200947036 z + 26054 z - 1459056 z 32 38 40 - 82654428723836136 z + 147043962868148948 z - 121374650773695496 z 62 74 72 + 21400640128 z + z - 255 z ) And in Maple-input format, it is: -(1+4296757606191670*z^28-1778151490042062*z^26-207*z^2+601385830321543*z^24-\ 165517707823419*z^22+17847*z^4-865050*z^6+26840295*z^8-574955043*z^10+ 8934652182*z^12-104245627755*z^14-6589465940006*z^18+936131937391*z^16-\ 165517707823419*z^50+601385830321543*z^48+36857900938043*z^20+20431623413573700 *z^36-18540377146048730*z^34-865050*z^66+26840295*z^64-8511352614247296*z^30-\ 8511352614247296*z^42+4296757606191670*z^44-1778151490042062*z^46-104245627755* z^58+936131937391*z^56-6589465940006*z^54+36857900938043*z^52+8934652182*z^60-\ 207*z^70+17847*z^68+13850335120422082*z^32-18540377146048730*z^38+ 13850335120422082*z^40-574955043*z^62+z^72)/(-1-21416449890238048*z^28+ 8113696813123183*z^26+255*z^2-2513352846509089*z^24+633501573494538*z^22-26054* z^4+1459056*z^6-51432172*z^8+1237067918*z^10-21400640128*z^12+276200947036*z^14 +21087327320140*z^18-2730608039354*z^16+2513352846509089*z^50-8113696813123183* z^48-129092478215360*z^20-147043962868148948*z^36+121374650773695496*z^34+ 51432172*z^66-1237067918*z^64+46386572471278064*z^30+82654428723836136*z^42-\ 46386572471278064*z^44+21416449890238048*z^46+2730608039354*z^58-21087327320140 *z^56+129092478215360*z^54-633501573494538*z^52-276200947036*z^60+26054*z^70-\ 1459056*z^68-82654428723836136*z^32+147043962868148948*z^38-121374650773695496* z^40+21400640128*z^62+z^74-255*z^72) The first , 40, terms are: [0, 48, 0, 4033, 0, 371829, 0, 35183424, 0, 3361881821, 0, 322617591325, 0, 31021449135128, 0, 2985736847181701, 0, 287502562084177281, 0, 27690408843261884040, 0, 2667254397699618876785, 0, 256934614636720189662001, 0, 24750960659791186162849496, 0, 2384333485146647002251940065, 0, 229691337836040198040729110277, 0, 22127051418842914750063457037224, 0, 2131587237925295956057580046989885, 0, 205344459486256830528782861864337149, 0, 19781673046886012785916436446160266160, 0, 1905649932878314438121097779252775101237] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {3, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 534627303570 z - 612133582552 z - 154 z + 534627303570 z 22 4 6 8 10 - 355888405079 z + 9170 z - 292364 z + 5717422 z - 73901855 z 12 14 18 16 + 661707005 z - 4236609990 z - 68964183464 z + 19832952756 z 50 48 20 36 - 154 z + 9170 z + 180102475058 z + 19832952756 z 34 30 42 44 - 68964183464 z - 355888405079 z - 73901855 z + 5717422 z 46 52 32 38 40 - 292364 z + z + 180102475058 z - 4236609990 z + 661707005 z ) / 2 28 26 2 / ((-1 + z ) (1 + 2009096556163 z - 2316977963279 z - 201 z / 24 22 4 6 + 2009096556163 z - 1308858395090 z + 14445 z - 538738 z 8 10 12 14 + 12093594 z - 176661754 z + 1761694705 z - 12382660633 z 18 16 50 48 - 232687406722 z + 62730442117 z - 201 z + 14445 z 20 36 34 + 638993459190 z + 62730442117 z - 232687406722 z 30 42 44 46 52 - 1308858395090 z - 176661754 z + 12093594 z - 538738 z + z 32 38 40 + 638993459190 z - 12382660633 z + 1761694705 z )) And in Maple-input format, it is: -(1+534627303570*z^28-612133582552*z^26-154*z^2+534627303570*z^24-355888405079* z^22+9170*z^4-292364*z^6+5717422*z^8-73901855*z^10+661707005*z^12-4236609990*z^ 14-68964183464*z^18+19832952756*z^16-154*z^50+9170*z^48+180102475058*z^20+ 19832952756*z^36-68964183464*z^34-355888405079*z^30-73901855*z^42+5717422*z^44-\ 292364*z^46+z^52+180102475058*z^32-4236609990*z^38+661707005*z^40)/(-1+z^2)/(1+ 2009096556163*z^28-2316977963279*z^26-201*z^2+2009096556163*z^24-1308858395090* z^22+14445*z^4-538738*z^6+12093594*z^8-176661754*z^10+1761694705*z^12-\ 12382660633*z^14-232687406722*z^18+62730442117*z^16-201*z^50+14445*z^48+ 638993459190*z^20+62730442117*z^36-232687406722*z^34-1308858395090*z^30-\ 176661754*z^42+12093594*z^44-538738*z^46+z^52+638993459190*z^32-12382660633*z^ 38+1761694705*z^40) The first , 40, terms are: [0, 48, 0, 4220, 0, 410251, 0, 40702456, 0, 4056293783, 0, 404662218733, 0, 40380183524240, 0, 4029700897608457, 0, 402147127585207671, 0, 40132774516658425320, 0, 4005105484750060651695, 0, 399695157176790839097172, 0, 39888146482812848337456960, 0, 3980694397200577172551793897, 0, 397259070666374114419897825873, 0, 39645035214200677806001371990848, 0, 3956432801908206164243155230705492, 0, 394837851295709533716493839809862639, 0, 39403406206607718429473297369921303880, 0, 3932319091512997664147021066992152612063] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 12164424 z + 59557560 z + 131 z - 193022220 z 22 4 6 8 10 + 420371402 z - 5931 z + 129525 z - 1600692 z + 12164424 z 12 14 18 16 - 59557560 z + 193022220 z + 619602642 z - 420371402 z 20 36 34 30 32 38 - 619602642 z - 131 z + 5931 z + 1600692 z - 129525 z + z ) / 40 36 38 28 32 / (1 + z + 10490 z - 182 z + 223230176 z + 4175905 z / 34 30 22 24 - 280218 z - 38074756 z - 4014362480 z + 2262163294 z 26 12 14 16 - 866107068 z + 223230176 z - 866107068 z + 2262163294 z 18 20 10 8 6 - 4014362480 z + 4858710860 z - 38074756 z + 4175905 z - 280218 z 4 2 + 10490 z - 182 z ) And in Maple-input format, it is: -(-1-12164424*z^28+59557560*z^26+131*z^2-193022220*z^24+420371402*z^22-5931*z^4 +129525*z^6-1600692*z^8+12164424*z^10-59557560*z^12+193022220*z^14+619602642*z^ 18-420371402*z^16-619602642*z^20-131*z^36+5931*z^34+1600692*z^30-129525*z^32+z^ 38)/(1+z^40+10490*z^36-182*z^38+223230176*z^28+4175905*z^32-280218*z^34-\ 38074756*z^30-4014362480*z^22+2262163294*z^24-866107068*z^26+223230176*z^12-\ 866107068*z^14+2262163294*z^16-4014362480*z^18+4858710860*z^20-38074756*z^10+ 4175905*z^8-280218*z^6+10490*z^4-182*z^2) The first , 40, terms are: [0, 51, 0, 4723, 0, 475289, 0, 48674233, 0, 5009337587, 0, 516346610291, 0, 53250881715041, 0, 5492716643434785, 0, 566594915826588275, 0, 58447583187792074099, 0, 6029249558857608933113, 0, 621957783882595273452121, 0, 64159189832851089478759667, 0, 6618459890509906299901528371, 0, 682739526884186299974430410561, 0, 70429266337804310170011072584385, 0, 7265262090796453696940921062013235, 0, 749461639394946115886813231768131059, 0, 77312111041031850428769300472593806553, 0, 7975274785938305054779215023894934669945] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 171765165005 z + 352215289406 z + 202 z - 503530718891 z 22 4 6 8 10 + 503530718891 z - 15815 z + 636877 z - 14980614 z + 220372067 z 12 14 18 16 - 2108538785 z + 13439965106 z + 171765165005 z - 58014843695 z 20 36 34 30 - 352215289406 z - 220372067 z + 2108538785 z + 58014843695 z 42 44 46 32 38 40 + 15815 z - 202 z + z - 13439965106 z + 14980614 z - 636877 z ) / 28 26 2 / (1 + 1922448312096 z - 3226857030800 z - 256 z / 24 22 4 6 + 3832653606022 z - 3226857030800 z + 24736 z - 1211752 z 8 10 12 14 + 33959736 z - 584776152 z + 6473194140 z - 47471011256 z 18 16 48 20 - 806088291704 z + 235732488232 z + z + 1922448312096 z 36 34 30 42 + 6473194140 z - 47471011256 z - 806088291704 z - 1211752 z 44 46 32 38 40 + 24736 z - 256 z + 235732488232 z - 584776152 z + 33959736 z ) And in Maple-input format, it is: -(-1-171765165005*z^28+352215289406*z^26+202*z^2-503530718891*z^24+503530718891 *z^22-15815*z^4+636877*z^6-14980614*z^8+220372067*z^10-2108538785*z^12+ 13439965106*z^14+171765165005*z^18-58014843695*z^16-352215289406*z^20-220372067 *z^36+2108538785*z^34+58014843695*z^30+15815*z^42-202*z^44+z^46-13439965106*z^ 32+14980614*z^38-636877*z^40)/(1+1922448312096*z^28-3226857030800*z^26-256*z^2+ 3832653606022*z^24-3226857030800*z^22+24736*z^4-1211752*z^6+33959736*z^8-\ 584776152*z^10+6473194140*z^12-47471011256*z^14-806088291704*z^18+235732488232* z^16+z^48+1922448312096*z^20+6473194140*z^36-47471011256*z^34-806088291704*z^30 -1211752*z^42+24736*z^44-256*z^46+235732488232*z^32-584776152*z^38+33959736*z^ 40) The first , 40, terms are: [0, 54, 0, 4903, 0, 494299, 0, 51715422, 0, 5483966365, 0, 584339183941, 0, 62371077614382, 0, 6661448861042419, 0, 711622964611059727, 0, 76026633351498689190, 0, 8122582288529086768537, 0, 867814874595552627779689, 0, 92717512927699227667618374, 0, 9905972187503243669226107839, 0, 1058358211957719539031925109827, 0, 113075459488198548109556457534990, 0, 12081033197676371572633839539054581, 0, 1290743094993827230700460978234165997, 0, 137903582921717936794405670011388326782, 0, 14733682004328677771604933701528497414795] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 10742978042036 z - 9439502507205 z - 181 z 24 22 4 6 + 6398283024309 z - 3337229014043 z + 12855 z - 492628 z 8 10 12 14 + 11664478 z - 183933055 z + 2023791090 z - 16027160634 z 18 16 50 48 - 406000296676 z + 93342727625 z - 492628 z + 11664478 z 20 36 34 + 1333793198182 z + 1333793198182 z - 3337229014043 z 30 42 44 46 - 9439502507205 z - 16027160634 z + 2023791090 z - 183933055 z 56 54 52 32 38 + z - 181 z + 12855 z + 6398283024309 z - 406000296676 z 40 / 28 26 + 93342727625 z ) / (-1 - 72839565208373 z + 56728788769963 z / 2 24 22 4 6 + 223 z - 34342095012925 z + 16096125942381 z - 19537 z + 906865 z 8 10 12 14 - 25386695 z + 462694745 z - 5779176443 z + 51287166317 z 18 16 50 48 + 1598083761589 z - 331975135747 z + 25386695 z - 462694745 z 20 36 34 - 5805754929531 z - 16096125942381 z + 34342095012925 z 30 42 44 46 + 72839565208373 z + 331975135747 z - 51287166317 z + 5779176443 z 58 56 54 52 32 + z - 223 z + 19537 z - 906865 z - 56728788769963 z 38 40 + 5805754929531 z - 1598083761589 z ) And in Maple-input format, it is: -(1+10742978042036*z^28-9439502507205*z^26-181*z^2+6398283024309*z^24-\ 3337229014043*z^22+12855*z^4-492628*z^6+11664478*z^8-183933055*z^10+2023791090* z^12-16027160634*z^14-406000296676*z^18+93342727625*z^16-492628*z^50+11664478*z ^48+1333793198182*z^20+1333793198182*z^36-3337229014043*z^34-9439502507205*z^30 -16027160634*z^42+2023791090*z^44-183933055*z^46+z^56-181*z^54+12855*z^52+ 6398283024309*z^32-406000296676*z^38+93342727625*z^40)/(-1-72839565208373*z^28+ 56728788769963*z^26+223*z^2-34342095012925*z^24+16096125942381*z^22-19537*z^4+ 906865*z^6-25386695*z^8+462694745*z^10-5779176443*z^12+51287166317*z^14+ 1598083761589*z^18-331975135747*z^16+25386695*z^50-462694745*z^48-5805754929531 *z^20-16096125942381*z^36+34342095012925*z^34+72839565208373*z^30+331975135747* z^42-51287166317*z^44+5779176443*z^46+z^58-223*z^56+19537*z^54-906865*z^52-\ 56728788769963*z^32+5805754929531*z^38-1598083761589*z^40) The first , 40, terms are: [0, 42, 0, 2684, 0, 192215, 0, 14792750, 0, 1190024955, 0, 98222568747, 0, 8223846224728, 0, 693937320273385, 0, 58801135131494221, 0, 4993729789569130914, 0, 424603786141509876717, 0, 36125961022982105749404, 0, 3074696578166722709088910, 0, 261736033074365445694979383, 0, 22282631809881834345491680087, 0, 1897106233166252478814436494990, 0, 161520864338596115780870664819948, 0, 13752190970748421868006711900415549, 0, 1170896489704107730877177201215366594, 0, 99693511486616285728725866680728744509] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 479470604525840259 z - 121350651173822556 z - 244 z 24 22 4 6 + 25762671536017577 z - 4561321610428832 z + 25579 z - 1557296 z 8 10 12 14 + 62574072 z - 1783629400 z + 37786793321 z - 614619326052 z 18 16 50 - 80467292420992 z + 7860088875579 z - 37931682763922558540 z 48 20 + 56144168764253532891 z + 668720151005616 z 36 34 + 21876342070256756329 z - 10754658271637419904 z 66 80 88 84 86 - 4561321610428832 z + 62574072 z + z + 25579 z - 244 z 82 64 30 - 1557296 z + 25762671536017577 z - 1596621404375258168 z 42 44 - 71009463617670471840 z + 76787763741017685280 z 46 58 - 71009463617670471840 z - 1596621404375258168 z 56 54 + 4498116514927080392 z - 10754658271637419904 z 52 60 70 + 21876342070256756329 z + 479470604525840259 z - 80467292420992 z 68 78 32 + 668720151005616 z - 1783629400 z + 4498116514927080392 z 38 40 - 37931682763922558540 z + 56144168764253532891 z 62 76 74 - 121350651173822556 z + 37786793321 z - 614619326052 z 72 / 2 28 + 7860088875579 z ) / ((-1 + z ) (1 + 1553288443315918219 z / 26 2 24 - 380404371595949795 z - 287 z + 77776935851999425 z 22 4 6 8 - 13193474194352048 z + 34803 z - 2412952 z + 108830710 z 10 12 14 - 3436384384 z + 79689104945 z - 1403846302001 z 18 16 50 - 209864454137056 z + 19263310751555 z - 136172204266103869837 z 48 20 + 203405032880921453691 z + 1842604321846036 z 36 34 + 77532950688672533713 z - 37489318578247142504 z 66 80 88 84 86 - 13193474194352048 z + 108830710 z + z + 34803 z - 287 z 82 64 30 - 2412952 z + 77776935851999425 z - 5321940429109569280 z 42 44 - 258673487840698498800 z + 280233493529120400920 z 46 58 - 258673487840698498800 z - 5321940429109569280 z 56 54 + 15363102571108751914 z - 37489318578247142504 z 52 60 70 + 77532950688672533713 z + 1553288443315918219 z - 209864454137056 z 68 78 32 + 1842604321846036 z - 3436384384 z + 15363102571108751914 z 38 40 - 136172204266103869837 z + 203405032880921453691 z 62 76 74 - 380404371595949795 z + 79689104945 z - 1403846302001 z 72 + 19263310751555 z )) And in Maple-input format, it is: -(1+479470604525840259*z^28-121350651173822556*z^26-244*z^2+25762671536017577*z ^24-4561321610428832*z^22+25579*z^4-1557296*z^6+62574072*z^8-1783629400*z^10+ 37786793321*z^12-614619326052*z^14-80467292420992*z^18+7860088875579*z^16-\ 37931682763922558540*z^50+56144168764253532891*z^48+668720151005616*z^20+ 21876342070256756329*z^36-10754658271637419904*z^34-4561321610428832*z^66+ 62574072*z^80+z^88+25579*z^84-244*z^86-1557296*z^82+25762671536017577*z^64-\ 1596621404375258168*z^30-71009463617670471840*z^42+76787763741017685280*z^44-\ 71009463617670471840*z^46-1596621404375258168*z^58+4498116514927080392*z^56-\ 10754658271637419904*z^54+21876342070256756329*z^52+479470604525840259*z^60-\ 80467292420992*z^70+668720151005616*z^68-1783629400*z^78+4498116514927080392*z^ 32-37931682763922558540*z^38+56144168764253532891*z^40-121350651173822556*z^62+ 37786793321*z^76-614619326052*z^74+7860088875579*z^72)/(-1+z^2)/(1+ 1553288443315918219*z^28-380404371595949795*z^26-287*z^2+77776935851999425*z^24 -13193474194352048*z^22+34803*z^4-2412952*z^6+108830710*z^8-3436384384*z^10+ 79689104945*z^12-1403846302001*z^14-209864454137056*z^18+19263310751555*z^16-\ 136172204266103869837*z^50+203405032880921453691*z^48+1842604321846036*z^20+ 77532950688672533713*z^36-37489318578247142504*z^34-13193474194352048*z^66+ 108830710*z^80+z^88+34803*z^84-287*z^86-2412952*z^82+77776935851999425*z^64-\ 5321940429109569280*z^30-258673487840698498800*z^42+280233493529120400920*z^44-\ 258673487840698498800*z^46-5321940429109569280*z^58+15363102571108751914*z^56-\ 37489318578247142504*z^54+77532950688672533713*z^52+1553288443315918219*z^60-\ 209864454137056*z^70+1842604321846036*z^68-3436384384*z^78+15363102571108751914 *z^32-136172204266103869837*z^38+203405032880921453691*z^40-380404371595949795* z^62+79689104945*z^76-1403846302001*z^74+19263310751555*z^72) The first , 40, terms are: [0, 44, 0, 3161, 0, 256867, 0, 22089836, 0, 1952627859, 0, 174981514283, 0, 15790464794972, 0, 1430110463657755, 0, 129768592740450769, 0, 11787080873523834812, 0, 1071211023362856179577, 0, 97379496035268608582665, 0, 8853725863725042832685276, 0, 805044571487669751534364321, 0, 73203637644510197322091770987, 0, 6656646597018345147432943550524, 0, 605318216912523873481869495741435, 0, 55044620272697044449924751048789699, 0, 5005501134902744489142267926466476044, 0, 455177804664520849478954567806003322515] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 2645920 z - 21606112 z - 148 z + 107760648 z 22 4 6 8 10 - 335572392 z + 7629 z - 191064 z + 2645920 z - 21606112 z 12 14 18 16 + 107760648 z - 335572392 z - 828025112 z + 660885130 z 20 36 34 30 32 / 38 + 660885130 z + z - 148 z - 191064 z + 7629 z ) / (z / 36 34 32 30 28 - 199 z + 13331 z - 417789 z + 7074144 z - 69580600 z 26 24 22 20 + 415419552 z - 1550511720 z + 3690778682 z - 5676226382 z 18 16 14 12 + 5676226382 z - 3690778682 z + 1550511720 z - 415419552 z 10 8 6 4 2 + 69580600 z - 7074144 z + 417789 z - 13331 z + 199 z - 1) And in Maple-input format, it is: -(1+2645920*z^28-21606112*z^26-148*z^2+107760648*z^24-335572392*z^22+7629*z^4-\ 191064*z^6+2645920*z^8-21606112*z^10+107760648*z^12-335572392*z^14-828025112*z^ 18+660885130*z^16+660885130*z^20+z^36-148*z^34-191064*z^30+7629*z^32)/(z^38-199 *z^36+13331*z^34-417789*z^32+7074144*z^30-69580600*z^28+415419552*z^26-\ 1550511720*z^24+3690778682*z^22-5676226382*z^20+5676226382*z^18-3690778682*z^16 +1550511720*z^14-415419552*z^12+69580600*z^10-7074144*z^8+417789*z^6-13331*z^4+ 199*z^2-1) The first , 40, terms are: [0, 51, 0, 4447, 0, 431797, 0, 43523661, 0, 4450023559, 0, 457523033611, 0, 47142385742697, 0, 4861646818892953, 0, 501536293127740507, 0, 51746303071116889143, 0, 5339236540672000613149, 0, 550919335123384177265125, 0, 56846067270092446340706735, 0, 5865624230189992633285627363, 0, 605241366446725201603548552913, 0, 62451545015543788985810548941105, 0, 6444034501954002799713323635096195, 0, 664924845267789258747440001134524687, 0, 68609977699592475383342570094208521093, 0, 7079490457647738736920911481806542582141] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 60442063653785573637 z - 9295839557431932346 z - 293 z 24 22 4 6 + 1229429237049123055 z - 138936718077943280 z + 38158 z - 2966360 z 102 8 10 12 - 5947385756 z + 155923559 z - 5947385756 z + 172352450669 z 14 18 16 - 3919405389596 z - 1071659393147446 z + 71633474201499 z 50 48 - 7171657668156387619851262 z + 4678199197125966269649506 z 20 36 + 13313419850560483 z + 26247023107558213516380 z 34 66 - 7055719465835449227182 z - 2698327922591870331875898 z 80 100 90 + 1658400275131904856659 z + 172352450669 z - 138936718077943280 z 88 84 + 1229429237049123055 z + 60442063653785573637 z 94 86 96 - 1071659393147446 z - 9295839557431932346 z + 71633474201499 z 98 92 82 - 3919405389596 z + 13313419850560483 z - 339752290222372078878 z 64 112 110 106 + 4678199197125966269649506 z + z - 293 z - 2966360 z 108 30 42 + 38158 z - 339752290222372078878 z - 618506234162361054371470 z 44 46 + 1375079400279876166020858 z - 2698327922591870331875898 z 58 56 - 11676547166838422184186478 z + 12409386188783909581310958 z 54 52 - 11676547166838422184186478 z + 9726852859346094759299050 z 60 70 + 9726852859346094759299050 z - 618506234162361054371470 z 68 78 + 1375079400279876166020858 z - 7055719465835449227182 z 32 38 + 1658400275131904856659 z - 85595059577417855545637 z 40 62 + 245247299529678666006345 z - 7171657668156387619851262 z 76 74 + 26247023107558213516380 z - 85595059577417855545637 z 72 104 / 2 + 245247299529678666006345 z + 155923559 z ) / ((-1 + z ) (1 / 28 26 2 + 178554118604621943698 z - 26514157853140233442 z - 339 z 24 22 4 6 + 3372957697798487312 z - 365104583188480900 z + 50204 z - 4378152 z 102 8 10 12 - 10668759310 z + 255090994 z - 10668759310 z + 335708364880 z 14 18 16 - 8217394715328 z - 2545949761144542 z + 160415903721079 z 50 48 - 25996056939291616745489674 z + 16836128693969803427794239 z 20 36 + 33353242836568599 z + 86455303846778255553054 z 34 66 - 22712849877209522134066 z - 9621157429633502194811138 z 80 100 90 + 5203233741537500118157 z + 335708364880 z - 365104583188480900 z 88 84 + 3372957697798487312 z + 178554118604621943698 z 94 86 96 - 2545949761144542 z - 26514157853140233442 z + 160415903721079 z 98 92 82 - 8217394715328 z + 33353242836568599 z - 1035972964763549967516 z 64 112 110 106 + 16836128693969803427794239 z + z - 339 z - 4378152 z 108 30 42 + 50204 z - 1035972964763549967516 z - 2150990912460373956288146 z 44 46 + 4847413844710647450136290 z - 9621157429633502194811138 z 58 56 - 42673944040520759337263434 z + 45398654843997837774129651 z 54 52 - 42673944040520759337263434 z + 35439431740593933342304986 z 60 70 + 35439431740593933342304986 z - 2150990912460373956288146 z 68 78 + 4847413844710647450136290 z - 22712849877209522134066 z 32 38 + 5203233741537500118157 z - 287770533774912894512067 z 40 62 + 839553360887845104270694 z - 25996056939291616745489674 z 76 74 + 86455303846778255553054 z - 287770533774912894512067 z 72 104 + 839553360887845104270694 z + 255090994 z )) And in Maple-input format, it is: -(1+60442063653785573637*z^28-9295839557431932346*z^26-293*z^2+ 1229429237049123055*z^24-138936718077943280*z^22+38158*z^4-2966360*z^6-\ 5947385756*z^102+155923559*z^8-5947385756*z^10+172352450669*z^12-3919405389596* z^14-1071659393147446*z^18+71633474201499*z^16-7171657668156387619851262*z^50+ 4678199197125966269649506*z^48+13313419850560483*z^20+26247023107558213516380*z ^36-7055719465835449227182*z^34-2698327922591870331875898*z^66+ 1658400275131904856659*z^80+172352450669*z^100-138936718077943280*z^90+ 1229429237049123055*z^88+60442063653785573637*z^84-1071659393147446*z^94-\ 9295839557431932346*z^86+71633474201499*z^96-3919405389596*z^98+ 13313419850560483*z^92-339752290222372078878*z^82+4678199197125966269649506*z^ 64+z^112-293*z^110-2966360*z^106+38158*z^108-339752290222372078878*z^30-\ 618506234162361054371470*z^42+1375079400279876166020858*z^44-\ 2698327922591870331875898*z^46-11676547166838422184186478*z^58+ 12409386188783909581310958*z^56-11676547166838422184186478*z^54+ 9726852859346094759299050*z^52+9726852859346094759299050*z^60-\ 618506234162361054371470*z^70+1375079400279876166020858*z^68-\ 7055719465835449227182*z^78+1658400275131904856659*z^32-85595059577417855545637 *z^38+245247299529678666006345*z^40-7171657668156387619851262*z^62+ 26247023107558213516380*z^76-85595059577417855545637*z^74+ 245247299529678666006345*z^72+155923559*z^104)/(-1+z^2)/(1+ 178554118604621943698*z^28-26514157853140233442*z^26-339*z^2+ 3372957697798487312*z^24-365104583188480900*z^22+50204*z^4-4378152*z^6-\ 10668759310*z^102+255090994*z^8-10668759310*z^10+335708364880*z^12-\ 8217394715328*z^14-2545949761144542*z^18+160415903721079*z^16-\ 25996056939291616745489674*z^50+16836128693969803427794239*z^48+ 33353242836568599*z^20+86455303846778255553054*z^36-22712849877209522134066*z^ 34-9621157429633502194811138*z^66+5203233741537500118157*z^80+335708364880*z^ 100-365104583188480900*z^90+3372957697798487312*z^88+178554118604621943698*z^84 -2545949761144542*z^94-26514157853140233442*z^86+160415903721079*z^96-\ 8217394715328*z^98+33353242836568599*z^92-1035972964763549967516*z^82+ 16836128693969803427794239*z^64+z^112-339*z^110-4378152*z^106+50204*z^108-\ 1035972964763549967516*z^30-2150990912460373956288146*z^42+ 4847413844710647450136290*z^44-9621157429633502194811138*z^46-\ 42673944040520759337263434*z^58+45398654843997837774129651*z^56-\ 42673944040520759337263434*z^54+35439431740593933342304986*z^52+ 35439431740593933342304986*z^60-2150990912460373956288146*z^70+ 4847413844710647450136290*z^68-22712849877209522134066*z^78+ 5203233741537500118157*z^32-287770533774912894512067*z^38+ 839553360887845104270694*z^40-25996056939291616745489674*z^62+ 86455303846778255553054*z^76-287770533774912894512067*z^74+ 839553360887845104270694*z^72+255090994*z^104) The first , 40, terms are: [0, 47, 0, 3595, 0, 308775, 0, 27868560, 0, 2570250081, 0, 239294734257, 0, 22371593831169, 0, 2095401189982403, 0, 196427344201962715, 0, 18420572685031503701, 0, 1727750128270675851769, 0, 162066875583813895995669, 0, 15202819757418044640424784, 0, 1426138795052046532167897915, 0, 133783666625421031102960286275, 0, 12550069232458580157077609231875, 0, 1177307655289001811388406021670505, 0, 110441985209032785593550812573157625, 0, 10360450201940749632685345038290422011, 0, 971903486623334339998341069298605985587] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 773344468561812245 z - 194834084140085517 z - 261 z 24 22 4 6 + 41090627879816136 z - 7208765517423724 z + 29372 z - 1909852 z 8 10 12 14 + 81267013 z - 2431165552 z + 53605616373 z - 900794464514 z 18 16 50 - 123601226819688 z + 11825552112153 z - 61569496533844599210 z 48 20 + 91126284944485080948 z + 1043909710639625 z 36 34 + 35505540458640097184 z - 17447480228260173944 z 66 80 88 84 86 - 7208765517423724 z + 81267013 z + z + 29372 z - 261 z 82 64 30 - 1909852 z + 41090627879816136 z - 2582958343813573886 z 42 44 - 115244118794329435494 z + 124617425757509685676 z 46 58 - 115244118794329435494 z - 2582958343813573886 z 56 54 + 7290192087095605274 z - 17447480228260173944 z 52 60 70 + 35505540458640097184 z + 773344468561812245 z - 123601226819688 z 68 78 32 + 1043909710639625 z - 2431165552 z + 7290192087095605274 z 38 40 - 61569496533844599210 z + 91126284944485080948 z 62 76 74 - 194834084140085517 z + 53605616373 z - 900794464514 z 72 / 2 28 + 11825552112153 z ) / ((-1 + z ) (1 + 2559771177929609936 z / 26 2 24 - 622736584918758164 z - 314 z + 126278112053883653 z 22 4 6 8 - 21202907754057091 z + 41164 z - 3050719 z + 145444627 z 10 12 14 - 4806007570 z + 115644402806 z - 2099118075813 z 18 16 50 - 327816651400848 z + 29508556919719 z - 228444395972269316486 z 48 20 + 341717077435596124653 z + 2923876451578078 z 36 34 + 129807748855594958213 z - 62597225875016625942 z 66 80 88 84 86 - 21202907754057091 z + 145444627 z + z + 41164 z - 314 z 82 64 30 - 3050719 z + 126278112053883653 z - 8817596229733136150 z 42 44 - 434930280524576129402 z + 471311350826062628293 z 46 58 - 434930280524576129402 z - 8817596229733136150 z 56 54 + 25564263520197786977 z - 62597225875016625942 z 52 60 + 129807748855594958213 z + 2559771177929609936 z 70 68 78 - 327816651400848 z + 2923876451578078 z - 4806007570 z 32 38 + 25564263520197786977 z - 228444395972269316486 z 40 62 76 + 341717077435596124653 z - 622736584918758164 z + 115644402806 z 74 72 - 2099118075813 z + 29508556919719 z )) And in Maple-input format, it is: -(1+773344468561812245*z^28-194834084140085517*z^26-261*z^2+41090627879816136*z ^24-7208765517423724*z^22+29372*z^4-1909852*z^6+81267013*z^8-2431165552*z^10+ 53605616373*z^12-900794464514*z^14-123601226819688*z^18+11825552112153*z^16-\ 61569496533844599210*z^50+91126284944485080948*z^48+1043909710639625*z^20+ 35505540458640097184*z^36-17447480228260173944*z^34-7208765517423724*z^66+ 81267013*z^80+z^88+29372*z^84-261*z^86-1909852*z^82+41090627879816136*z^64-\ 2582958343813573886*z^30-115244118794329435494*z^42+124617425757509685676*z^44-\ 115244118794329435494*z^46-2582958343813573886*z^58+7290192087095605274*z^56-\ 17447480228260173944*z^54+35505540458640097184*z^52+773344468561812245*z^60-\ 123601226819688*z^70+1043909710639625*z^68-2431165552*z^78+7290192087095605274* z^32-61569496533844599210*z^38+91126284944485080948*z^40-194834084140085517*z^ 62+53605616373*z^76-900794464514*z^74+11825552112153*z^72)/(-1+z^2)/(1+ 2559771177929609936*z^28-622736584918758164*z^26-314*z^2+126278112053883653*z^ 24-21202907754057091*z^22+41164*z^4-3050719*z^6+145444627*z^8-4806007570*z^10+ 115644402806*z^12-2099118075813*z^14-327816651400848*z^18+29508556919719*z^16-\ 228444395972269316486*z^50+341717077435596124653*z^48+2923876451578078*z^20+ 129807748855594958213*z^36-62597225875016625942*z^34-21202907754057091*z^66+ 145444627*z^80+z^88+41164*z^84-314*z^86-3050719*z^82+126278112053883653*z^64-\ 8817596229733136150*z^30-434930280524576129402*z^42+471311350826062628293*z^44-\ 434930280524576129402*z^46-8817596229733136150*z^58+25564263520197786977*z^56-\ 62597225875016625942*z^54+129807748855594958213*z^52+2559771177929609936*z^60-\ 327816651400848*z^70+2923876451578078*z^68-4806007570*z^78+25564263520197786977 *z^32-228444395972269316486*z^38+341717077435596124653*z^40-622736584918758164* z^62+115644402806*z^76-2099118075813*z^74+29508556919719*z^72) The first , 40, terms are: [0, 54, 0, 4904, 0, 486979, 0, 49723622, 0, 5128158161, 0, 530927966707, 0, 55053639199128, 0, 5712384208617331, 0, 592881134765641813, 0, 61541590131152660514, 0, 6388394686154851905223, 0, 663169118254503532261552, 0, 68843193527381782542483930, 0, 7146601190848002848757664425, 0, 741888929998804721465551185849, 0, 77015580023252076567207741255874, 0, 7994999642819013210769907546738512, 0, 829962323051512401170212534588715839, 0, 86158541442328468958710494894772817514, 0, 8944134332162526305465475592131557306493] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 828476790501024431 z - 208982403419520103 z - 263 z 24 22 4 6 + 44116013602267441 z - 7742930484969408 z + 29887 z - 1964850 z 8 10 12 14 + 84527272 z - 2553442926 z + 56751169277 z - 959411884965 z 18 16 50 - 132545249556736 z + 12648115403171 z - 65564496235017621293 z 48 20 + 96975868070026400619 z + 1120923409826312 z 36 34 + 37843075885988785365 z - 18616670031322699894 z 66 80 88 84 86 - 7742930484969408 z + 84527272 z + z + 29887 z - 263 z 82 64 30 - 1964850 z + 44116013602267441 z - 2763319511566505834 z 42 44 - 122593281221007240000 z + 132546714723262087792 z 46 58 - 122593281221007240000 z - 2763319511566505834 z 56 54 + 7788598089936932952 z - 18616670031322699894 z 52 60 70 + 37843075885988785365 z + 828476790501024431 z - 132545249556736 z 68 78 32 + 1120923409826312 z - 2553442926 z + 7788598089936932952 z 38 40 - 65564496235017621293 z + 96975868070026400619 z 62 76 74 - 208982403419520103 z + 56751169277 z - 959411884965 z 72 / 2 28 + 12648115403171 z ) / ((-1 + z ) (1 + 2775035437748896800 z / 26 2 24 - 674617806241904732 z - 324 z + 136659043428213939 z 22 4 6 8 - 22914011802923944 z + 43038 z - 3210604 z + 153764331 z 10 12 14 - 5101599356 z + 123238131644 z - 2245219826052 z 18 16 50 - 352799700958408 z + 31667271754757 z - 247961780911032515676 z 48 20 + 370927639290487373031 z + 3154046161336282 z 36 34 + 140887347128784076070 z - 67931413197423511060 z 66 80 88 84 86 - 22914011802923944 z + 153764331 z + z + 43038 z - 324 z 82 64 30 - 3210604 z + 136659043428213939 z - 9563847689417266308 z 42 44 - 472118674533169592080 z + 511613660018972346604 z 46 58 - 472118674533169592080 z - 9563847689417266308 z 56 54 + 27736906453050702501 z - 67931413197423511060 z 52 60 + 140887347128784076070 z + 2775035437748896800 z 70 68 78 - 352799700958408 z + 3154046161336282 z - 5101599356 z 32 38 + 27736906453050702501 z - 247961780911032515676 z 40 62 76 + 370927639290487373031 z - 674617806241904732 z + 123238131644 z 74 72 - 2245219826052 z + 31667271754757 z )) And in Maple-input format, it is: -(1+828476790501024431*z^28-208982403419520103*z^26-263*z^2+44116013602267441*z ^24-7742930484969408*z^22+29887*z^4-1964850*z^6+84527272*z^8-2553442926*z^10+ 56751169277*z^12-959411884965*z^14-132545249556736*z^18+12648115403171*z^16-\ 65564496235017621293*z^50+96975868070026400619*z^48+1120923409826312*z^20+ 37843075885988785365*z^36-18616670031322699894*z^34-7742930484969408*z^66+ 84527272*z^80+z^88+29887*z^84-263*z^86-1964850*z^82+44116013602267441*z^64-\ 2763319511566505834*z^30-122593281221007240000*z^42+132546714723262087792*z^44-\ 122593281221007240000*z^46-2763319511566505834*z^58+7788598089936932952*z^56-\ 18616670031322699894*z^54+37843075885988785365*z^52+828476790501024431*z^60-\ 132545249556736*z^70+1120923409826312*z^68-2553442926*z^78+7788598089936932952* z^32-65564496235017621293*z^38+96975868070026400619*z^40-208982403419520103*z^ 62+56751169277*z^76-959411884965*z^74+12648115403171*z^72)/(-1+z^2)/(1+ 2775035437748896800*z^28-674617806241904732*z^26-324*z^2+136659043428213939*z^ 24-22914011802923944*z^22+43038*z^4-3210604*z^6+153764331*z^8-5101599356*z^10+ 123238131644*z^12-2245219826052*z^14-352799700958408*z^18+31667271754757*z^16-\ 247961780911032515676*z^50+370927639290487373031*z^48+3154046161336282*z^20+ 140887347128784076070*z^36-67931413197423511060*z^34-22914011802923944*z^66+ 153764331*z^80+z^88+43038*z^84-324*z^86-3210604*z^82+136659043428213939*z^64-\ 9563847689417266308*z^30-472118674533169592080*z^42+511613660018972346604*z^44-\ 472118674533169592080*z^46-9563847689417266308*z^58+27736906453050702501*z^56-\ 67931413197423511060*z^54+140887347128784076070*z^52+2775035437748896800*z^60-\ 352799700958408*z^70+3154046161336282*z^68-5101599356*z^78+27736906453050702501 *z^32-247961780911032515676*z^38+370927639290487373031*z^40-674617806241904732* z^62+123238131644*z^76-2245219826052*z^74+31667271754757*z^72) The first , 40, terms are: [0, 62, 0, 6675, 0, 769723, 0, 89996766, 0, 10559755365, 0, 1240320103245, 0, 145733496607718, 0, 17125139768850995, 0, 2012453711630287787, 0, 236495939269218301206, 0, 27792244886717428608473, 0, 3266061412067937781046809, 0, 383818010931060839079985798, 0, 45105184107050416924900264299, 0, 5300631445134701866849518674099, 0, 622914976072545722443221381743990, 0, 73203178995820173777434294028701805, 0, 8602627401345670708330812149779091685, 0, 1010956072734793757472630933888603166990, 0, 118804655146123504363060424269097644769243] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 540533942539584512 z - 136537587828928408 z - 248 z 24 22 4 6 + 28921588118347317 z - 5107074098623392 z + 26476 z - 1639136 z 8 10 12 14 + 66816515 z - 1927386892 z + 41228697720 z - 675804658228 z 18 16 50 - 89458103276072 z + 8695777408527 z - 43045828255662267960 z 48 20 + 63756154712485483678 z + 746351199311404 z 36 34 + 24803389186499683560 z - 12179928438904985896 z 66 80 88 84 86 - 5107074098623392 z + 66816515 z + z + 26476 z - 248 z 82 64 30 - 1639136 z + 28921588118347317 z - 1803068978097123144 z 42 44 - 80669680553522798304 z + 87246021296805042704 z 46 58 - 80669680553522798304 z - 1803068978097123144 z 56 54 + 5087475857648192794 z - 12179928438904985896 z 52 60 70 + 24803389186499683560 z + 540533942539584512 z - 89458103276072 z 68 78 32 + 746351199311404 z - 1927386892 z + 5087475857648192794 z 38 40 - 43045828255662267960 z + 63756154712485483678 z 62 76 74 - 136537587828928408 z + 41228697720 z - 675804658228 z 72 / 28 + 8695777408527 z ) / (-1 - 2206369931028148400 z / 26 2 24 + 520848442379521013 z + 297 z - 103014805336732797 z 22 4 6 8 + 16959191774984634 z - 36946 z + 2617454 z - 120382075 z 10 12 14 + 3872337479 z - 91474153092 z + 1642608278388 z 18 16 50 + 256246911735895 z - 23004451689587 z + 393955231101907046782 z 48 20 - 536716720051629496182 z - 2305428074750678 z 36 34 - 132908346222573737204 z + 60907372292340766986 z 66 80 90 88 84 + 103014805336732797 z - 3872337479 z + z - 297 z - 2617454 z 86 82 64 + 36946 z + 120382075 z - 520848442379521013 z 30 42 + 7872671097330621264 z + 536716720051629496182 z 44 46 - 626336839540907122440 z + 626336839540907122440 z 58 56 + 23765070466605312730 z - 60907372292340766986 z 54 52 + 132908346222573737204 z - 247492779906120417468 z 60 70 68 - 7872671097330621264 z + 2305428074750678 z - 16959191774984634 z 78 32 38 + 91474153092 z - 23765070466605312730 z + 247492779906120417468 z 40 62 76 - 393955231101907046782 z + 2206369931028148400 z - 1642608278388 z 74 72 + 23004451689587 z - 256246911735895 z ) And in Maple-input format, it is: -(1+540533942539584512*z^28-136537587828928408*z^26-248*z^2+28921588118347317*z ^24-5107074098623392*z^22+26476*z^4-1639136*z^6+66816515*z^8-1927386892*z^10+ 41228697720*z^12-675804658228*z^14-89458103276072*z^18+8695777408527*z^16-\ 43045828255662267960*z^50+63756154712485483678*z^48+746351199311404*z^20+ 24803389186499683560*z^36-12179928438904985896*z^34-5107074098623392*z^66+ 66816515*z^80+z^88+26476*z^84-248*z^86-1639136*z^82+28921588118347317*z^64-\ 1803068978097123144*z^30-80669680553522798304*z^42+87246021296805042704*z^44-\ 80669680553522798304*z^46-1803068978097123144*z^58+5087475857648192794*z^56-\ 12179928438904985896*z^54+24803389186499683560*z^52+540533942539584512*z^60-\ 89458103276072*z^70+746351199311404*z^68-1927386892*z^78+5087475857648192794*z^ 32-43045828255662267960*z^38+63756154712485483678*z^40-136537587828928408*z^62+ 41228697720*z^76-675804658228*z^74+8695777408527*z^72)/(-1-2206369931028148400* z^28+520848442379521013*z^26+297*z^2-103014805336732797*z^24+16959191774984634* z^22-36946*z^4+2617454*z^6-120382075*z^8+3872337479*z^10-91474153092*z^12+ 1642608278388*z^14+256246911735895*z^18-23004451689587*z^16+ 393955231101907046782*z^50-536716720051629496182*z^48-2305428074750678*z^20-\ 132908346222573737204*z^36+60907372292340766986*z^34+103014805336732797*z^66-\ 3872337479*z^80+z^90-297*z^88-2617454*z^84+36946*z^86+120382075*z^82-\ 520848442379521013*z^64+7872671097330621264*z^30+536716720051629496182*z^42-\ 626336839540907122440*z^44+626336839540907122440*z^46+23765070466605312730*z^58 -60907372292340766986*z^56+132908346222573737204*z^54-247492779906120417468*z^ 52-7872671097330621264*z^60+2305428074750678*z^70-16959191774984634*z^68+ 91474153092*z^78-23765070466605312730*z^32+247492779906120417468*z^38-\ 393955231101907046782*z^40+2206369931028148400*z^62-1642608278388*z^76+ 23004451689587*z^74-256246911735895*z^72) The first , 40, terms are: [0, 49, 0, 4083, 0, 380615, 0, 36881823, 0, 3624993235, 0, 358208481321, 0, 35471494599953, 0, 3515565899556241, 0, 348552018449615049, 0, 34562711754163040483, 0, 3427505052174155030799, 0, 339908462698092329838327, 0, 33709480523151753740101027, 0, 3343066437969021040749654673, 0, 331542559466412487159877791009, 0, 32880179883926677778111100005729, 0, 3260839133133638252244796069422993, 0, 323388595313268287094584883196308355, 0, 32071560327114798689248150202867599255, 0, 3180647270462610184905875152639521075119] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 73660107594100416997 z - 11181099485792632002 z - 293 z 24 22 4 6 + 1457666389403645007 z - 162179007166872344 z + 38350 z - 3010748 z 102 8 10 12 - 6222500580 z + 160454183 z - 6222500580 z + 183645976141 z 14 18 16 - 4256330387292 z - 1208432586007766 z + 79287504336683 z 50 48 - 9418534006408892265917670 z + 6128509440460229726807698 z 20 36 + 15282150722685803 z + 33311930269293960212540 z 34 66 - 8879357792265589478826 z - 3523399191272660067955458 z 80 100 90 + 2067204592460926692227 z + 183645976141 z - 162179007166872344 z 88 84 + 1457666389403645007 z + 73660107594100416997 z 94 86 96 - 1208432586007766 z - 11181099485792632002 z + 79287504336683 z 98 92 82 - 4256330387292 z + 15282150722685803 z - 419001526938262070654 z 64 112 110 106 + 6128509440460229726807698 z + z - 293 z - 3010748 z 108 30 42 + 38350 z - 419001526938262070654 z - 800544879125857658137534 z 44 46 + 1788358527332745635069306 z - 3523399191272660067955458 z 58 56 - 15378477042280085195051510 z + 16349446363790304636527086 z 54 52 - 15378477042280085195051510 z + 12797023679778611660224778 z 60 70 + 12797023679778611660224778 z - 800544879125857658137534 z 68 78 + 1788358527332745635069306 z - 8879357792265589478826 z 32 38 + 2067204592460926692227 z - 109448625192223130613517 z 40 62 + 315644350626022875885497 z - 9418534006408892265917670 z 76 74 + 33311930269293960212540 z - 109448625192223130613517 z 72 104 / 2 + 315644350626022875885497 z + 160454183 z ) / ((-1 + z ) (1 / 28 26 2 + 216782313793757882798 z - 31673232761343673262 z - 347 z 24 22 4 6 + 3961894308186028252 z - 421494111737285080 z + 51844 z - 4543784 z 102 8 10 12 - 11233224230 z + 266310050 z - 11233224230 z + 357498029436 z 14 18 16 - 8871742151964 z - 2838642227160802 z + 175898176884555 z 50 48 - 35302246751023648066508802 z + 22761612245539009717917615 z 20 36 + 37837048938104667 z + 111004498655091491343942 z 34 66 - 28800006814636686793474 z - 12933623197303411170026978 z 80 100 90 + 6509074624627972870085 z + 357498029436 z - 421494111737285080 z 88 84 + 3961894308186028252 z + 216782313793757882798 z 94 86 96 - 2838642227160802 z - 31673232761343673262 z + 175898176884555 z 98 92 82 - 8871742151964 z + 37837048938104667 z - 1277305244378964334020 z 64 112 110 106 + 22761612245539009717917615 z + z - 347 z - 4543784 z 108 30 42 + 51844 z - 1277305244378964334020 z - 2848657370178324965842402 z 44 46 + 6471615310983233423188082 z - 12933623197303411170026978 z 58 56 - 58248554819288925701730706 z + 62007633925957004507896803 z 54 52 - 58248554819288925701730706 z + 48280478546205419994853490 z 60 70 + 48280478546205419994853490 z - 2848657370178324965842402 z 68 78 + 6471615310983233423188082 z - 28800006814636686793474 z 32 38 + 6509074624627972870085 z - 373731560475610259691051 z 40 62 + 1101665686696256085684110 z - 35302246751023648066508802 z 76 74 + 111004498655091491343942 z - 373731560475610259691051 z 72 104 + 1101665686696256085684110 z + 266310050 z )) And in Maple-input format, it is: -(1+73660107594100416997*z^28-11181099485792632002*z^26-293*z^2+ 1457666389403645007*z^24-162179007166872344*z^22+38350*z^4-3010748*z^6-\ 6222500580*z^102+160454183*z^8-6222500580*z^10+183645976141*z^12-4256330387292* z^14-1208432586007766*z^18+79287504336683*z^16-9418534006408892265917670*z^50+ 6128509440460229726807698*z^48+15282150722685803*z^20+33311930269293960212540*z ^36-8879357792265589478826*z^34-3523399191272660067955458*z^66+ 2067204592460926692227*z^80+183645976141*z^100-162179007166872344*z^90+ 1457666389403645007*z^88+73660107594100416997*z^84-1208432586007766*z^94-\ 11181099485792632002*z^86+79287504336683*z^96-4256330387292*z^98+ 15282150722685803*z^92-419001526938262070654*z^82+6128509440460229726807698*z^ 64+z^112-293*z^110-3010748*z^106+38350*z^108-419001526938262070654*z^30-\ 800544879125857658137534*z^42+1788358527332745635069306*z^44-\ 3523399191272660067955458*z^46-15378477042280085195051510*z^58+ 16349446363790304636527086*z^56-15378477042280085195051510*z^54+ 12797023679778611660224778*z^52+12797023679778611660224778*z^60-\ 800544879125857658137534*z^70+1788358527332745635069306*z^68-\ 8879357792265589478826*z^78+2067204592460926692227*z^32-\ 109448625192223130613517*z^38+315644350626022875885497*z^40-\ 9418534006408892265917670*z^62+33311930269293960212540*z^76-\ 109448625192223130613517*z^74+315644350626022875885497*z^72+160454183*z^104)/(-\ 1+z^2)/(1+216782313793757882798*z^28-31673232761343673262*z^26-347*z^2+ 3961894308186028252*z^24-421494111737285080*z^22+51844*z^4-4543784*z^6-\ 11233224230*z^102+266310050*z^8-11233224230*z^10+357498029436*z^12-\ 8871742151964*z^14-2838642227160802*z^18+175898176884555*z^16-\ 35302246751023648066508802*z^50+22761612245539009717917615*z^48+ 37837048938104667*z^20+111004498655091491343942*z^36-28800006814636686793474*z^ 34-12933623197303411170026978*z^66+6509074624627972870085*z^80+357498029436*z^ 100-421494111737285080*z^90+3961894308186028252*z^88+216782313793757882798*z^84 -2838642227160802*z^94-31673232761343673262*z^86+175898176884555*z^96-\ 8871742151964*z^98+37837048938104667*z^92-1277305244378964334020*z^82+ 22761612245539009717917615*z^64+z^112-347*z^110-4543784*z^106+51844*z^108-\ 1277305244378964334020*z^30-2848657370178324965842402*z^42+ 6471615310983233423188082*z^44-12933623197303411170026978*z^46-\ 58248554819288925701730706*z^58+62007633925957004507896803*z^56-\ 58248554819288925701730706*z^54+48280478546205419994853490*z^52+ 48280478546205419994853490*z^60-2848657370178324965842402*z^70+ 6471615310983233423188082*z^68-28800006814636686793474*z^78+ 6509074624627972870085*z^32-373731560475610259691051*z^38+ 1101665686696256085684110*z^40-35302246751023648066508802*z^62+ 111004498655091491343942*z^76-373731560475610259691051*z^74+ 1101665686696256085684110*z^72+266310050*z^104) The first , 40, terms are: [0, 55, 0, 5299, 0, 558427, 0, 60132376, 0, 6513508893, 0, 706789657613, 0, 76738918099969, 0, 8333533049577939, 0, 905056448223384439, 0, 98295852402438687925, 0, 10675790448533700557637, 0, 1159490285427710783178653, 0, 125931719560744106457315224, 0, 13677400309989253520477159179, 0, 1485498293252866567940144692331, 0, 161339546479901550914555992569963, 0, 17523043505459440377660228165563637, 0, 1903172969255254419607284791123340733, 0, 206703099332040298836918532950960410691, 0, 22449967620935002549025172459836185226827] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 452688752831032942 z + 118406271579730293 z + 249 z 24 22 4 6 - 25864915541824597 z + 4691974136609036 z - 26666 z + 1653334 z 8 10 12 14 - 67339134 z + 1935502098 z - 41128916215 z + 667586908903 z 18 16 50 + 85803775422724 z - 8478442077084 z + 17239509238281649371 z 48 20 - 28179145429880231043 z - 701838431000420 z 36 34 66 - 17239509238281649371 z + 8937795173427632086 z + 701838431000420 z 80 84 86 82 64 - 1653334 z - 249 z + z + 26666 z - 4691974136609036 z 30 42 + 1451914322954076434 z + 46010810345933118744 z 44 46 - 46010810345933118744 z + 39078146982817192408 z 58 56 + 452688752831032942 z - 1451914322954076434 z 54 52 + 3920549096606921626 z - 8937795173427632086 z 60 70 68 - 118406271579730293 z + 8478442077084 z - 85803775422724 z 78 32 38 + 67339134 z - 3920549096606921626 z + 28179145429880231043 z 40 62 76 - 39078146982817192408 z + 25864915541824597 z - 1935502098 z 74 72 / 28 + 41128916215 z - 667586908903 z ) / (1 + 1891271612693639237 z / 26 2 24 - 460263999294933321 z - 305 z + 93520639965771981 z 22 4 6 8 - 15768423186589640 z + 38289 z - 2712310 z + 124072306 z 10 12 14 - 3957000586 z + 92472289815 z - 1639624024219 z 18 16 50 - 247950784965064 z + 22630993942655 z - 170284372888416615659 z 48 20 + 255124519353657959219 z + 2189396931419876 z 36 34 + 96566249166015840147 z - 46465262700375851586 z 66 80 88 84 86 - 15768423186589640 z + 124072306 z + z + 38289 z - 305 z 82 64 30 - 2712310 z + 93520639965771981 z - 6520534399343948926 z 42 44 - 325048492580441253680 z + 352361788125577307000 z 46 58 - 325048492580441253680 z - 6520534399343948926 z 56 54 + 18936277629930771950 z - 46465262700375851586 z 52 60 70 + 96566249166015840147 z + 1891271612693639237 z - 247950784965064 z 68 78 32 + 2189396931419876 z - 3957000586 z + 18936277629930771950 z 38 40 - 170284372888416615659 z + 255124519353657959219 z 62 76 74 - 460263999294933321 z + 92472289815 z - 1639624024219 z 72 + 22630993942655 z ) And in Maple-input format, it is: -(-1-452688752831032942*z^28+118406271579730293*z^26+249*z^2-25864915541824597* z^24+4691974136609036*z^22-26666*z^4+1653334*z^6-67339134*z^8+1935502098*z^10-\ 41128916215*z^12+667586908903*z^14+85803775422724*z^18-8478442077084*z^16+ 17239509238281649371*z^50-28179145429880231043*z^48-701838431000420*z^20-\ 17239509238281649371*z^36+8937795173427632086*z^34+701838431000420*z^66-1653334 *z^80-249*z^84+z^86+26666*z^82-4691974136609036*z^64+1451914322954076434*z^30+ 46010810345933118744*z^42-46010810345933118744*z^44+39078146982817192408*z^46+ 452688752831032942*z^58-1451914322954076434*z^56+3920549096606921626*z^54-\ 8937795173427632086*z^52-118406271579730293*z^60+8478442077084*z^70-\ 85803775422724*z^68+67339134*z^78-3920549096606921626*z^32+28179145429880231043 *z^38-39078146982817192408*z^40+25864915541824597*z^62-1935502098*z^76+ 41128916215*z^74-667586908903*z^72)/(1+1891271612693639237*z^28-\ 460263999294933321*z^26-305*z^2+93520639965771981*z^24-15768423186589640*z^22+ 38289*z^4-2712310*z^6+124072306*z^8-3957000586*z^10+92472289815*z^12-\ 1639624024219*z^14-247950784965064*z^18+22630993942655*z^16-\ 170284372888416615659*z^50+255124519353657959219*z^48+2189396931419876*z^20+ 96566249166015840147*z^36-46465262700375851586*z^34-15768423186589640*z^66+ 124072306*z^80+z^88+38289*z^84-305*z^86-2712310*z^82+93520639965771981*z^64-\ 6520534399343948926*z^30-325048492580441253680*z^42+352361788125577307000*z^44-\ 325048492580441253680*z^46-6520534399343948926*z^58+18936277629930771950*z^56-\ 46465262700375851586*z^54+96566249166015840147*z^52+1891271612693639237*z^60-\ 247950784965064*z^70+2189396931419876*z^68-3957000586*z^78+18936277629930771950 *z^32-170284372888416615659*z^38+255124519353657959219*z^40-460263999294933321* z^62+92472289815*z^76-1639624024219*z^74+22630993942655*z^72) The first , 40, terms are: [0, 56, 0, 5457, 0, 579177, 0, 62862100, 0, 6871357369, 0, 752930704889, 0, 82575080251820, 0, 9059090152187929, 0, 993971357498868129, 0, 109064623985217916320, 0, 11967462680182915390369, 0, 1313177677537616101746529, 0, 144094100671244707035551440, 0, 15811367635014576462447160193, 0, 1734973667346579194150251595449, 0, 190377853958876239693159940872700, 0, 20890075164082805555068027900523513, 0, 2292258508244452301595322039734035897, 0, 251528493943601997106604514666787938756, 0, 27600108546910014101178909546122862305225] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1215813681294855165 z - 297693866062926030 z - 270 z 24 22 4 6 + 60924650568666485 z - 10360316398624244 z + 31433 z - 2114142 z 8 10 12 14 + 93015952 z - 2875959414 z + 65521851311 z - 1137528220650 z 18 16 50 - 166584218607204 z + 15427905926231 z - 107443938282533630394 z 48 20 + 160701978964125646643 z + 1453170671680352 z 36 34 + 61074249025049727755 z - 29476282363657299522 z 66 80 88 84 86 - 10360316398624244 z + 93015952 z + z + 31433 z - 270 z 82 64 30 - 2114142 z + 60924650568666485 z - 4170060112813216106 z 42 44 - 204536754950531049640 z + 221647496547315191616 z 46 58 - 204536754950531049640 z - 4170060112813216106 z 56 54 + 12057074357739710448 z - 29476282363657299522 z 52 60 70 + 61074249025049727755 z + 1215813681294855165 z - 166584218607204 z 68 78 32 + 1453170671680352 z - 2875959414 z + 12057074357739710448 z 38 40 - 107443938282533630394 z + 160701978964125646643 z 62 76 74 - 297693866062926030 z + 65521851311 z - 1137528220650 z 72 / 2 28 + 15427905926231 z ) / ((-1 + z ) (1 + 4071023044363053703 z / 26 2 24 - 956980957733570145 z - 333 z + 187293854068096737 z 22 4 6 8 - 30339668805079288 z + 45327 z - 3452804 z + 168574170 z 10 12 14 - 5703991404 z + 140799582629 z - 2628381853195 z 18 16 50 - 437343896632072 z + 38096606604323 z - 413628004674450889223 z 48 20 + 627042712428934936187 z + 4037732125517404 z 36 34 + 230787569998947266373 z - 108799147835989893532 z 66 80 88 84 86 - 30339668805079288 z + 168574170 z + z + 45327 z - 333 z 82 64 30 - 3452804 z + 187293854068096737 z - 14485585790304078324 z 42 44 - 804619060258157177984 z + 874316612382820544648 z 46 58 - 804619060258157177984 z - 14485585790304078324 z 56 54 + 43268577682715561158 z - 108799147835989893532 z 52 60 + 230787569998947266373 z + 4071023044363053703 z 70 68 78 - 437343896632072 z + 4037732125517404 z - 5703991404 z 32 38 + 43268577682715561158 z - 413628004674450889223 z 40 62 76 + 627042712428934936187 z - 956980957733570145 z + 140799582629 z 74 72 - 2628381853195 z + 38096606604323 z )) And in Maple-input format, it is: -(1+1215813681294855165*z^28-297693866062926030*z^26-270*z^2+60924650568666485* z^24-10360316398624244*z^22+31433*z^4-2114142*z^6+93015952*z^8-2875959414*z^10+ 65521851311*z^12-1137528220650*z^14-166584218607204*z^18+15427905926231*z^16-\ 107443938282533630394*z^50+160701978964125646643*z^48+1453170671680352*z^20+ 61074249025049727755*z^36-29476282363657299522*z^34-10360316398624244*z^66+ 93015952*z^80+z^88+31433*z^84-270*z^86-2114142*z^82+60924650568666485*z^64-\ 4170060112813216106*z^30-204536754950531049640*z^42+221647496547315191616*z^44-\ 204536754950531049640*z^46-4170060112813216106*z^58+12057074357739710448*z^56-\ 29476282363657299522*z^54+61074249025049727755*z^52+1215813681294855165*z^60-\ 166584218607204*z^70+1453170671680352*z^68-2875959414*z^78+12057074357739710448 *z^32-107443938282533630394*z^38+160701978964125646643*z^40-297693866062926030* z^62+65521851311*z^76-1137528220650*z^74+15427905926231*z^72)/(-1+z^2)/(1+ 4071023044363053703*z^28-956980957733570145*z^26-333*z^2+187293854068096737*z^ 24-30339668805079288*z^22+45327*z^4-3452804*z^6+168574170*z^8-5703991404*z^10+ 140799582629*z^12-2628381853195*z^14-437343896632072*z^18+38096606604323*z^16-\ 413628004674450889223*z^50+627042712428934936187*z^48+4037732125517404*z^20+ 230787569998947266373*z^36-108799147835989893532*z^34-30339668805079288*z^66+ 168574170*z^80+z^88+45327*z^84-333*z^86-3452804*z^82+187293854068096737*z^64-\ 14485585790304078324*z^30-804619060258157177984*z^42+874316612382820544648*z^44 -804619060258157177984*z^46-14485585790304078324*z^58+43268577682715561158*z^56 -108799147835989893532*z^54+230787569998947266373*z^52+4071023044363053703*z^60 -437343896632072*z^70+4037732125517404*z^68-5703991404*z^78+ 43268577682715561158*z^32-413628004674450889223*z^38+627042712428934936187*z^40 -956980957733570145*z^62+140799582629*z^76-2628381853195*z^74+38096606604323*z^ 72) The first , 40, terms are: [0, 64, 0, 7149, 0, 849515, 0, 102184032, 0, 12335630131, 0, 1491133955987, 0, 180345900510368, 0, 21817003711998627, 0, 2639530995224869429, 0, 319357481392023567808, 0, 38639865223192471875089, 0, 4675173225956345954958225, 0, 565667718710023335782835456, 0, 68442487013622874987198105925, 0, 8281146690030699953858415115283, 0, 1001971363415868921061465401711712, 0, 121232818352873126239696060219076915, 0, 14668480242469902277355598784130307411, 0, 1774802577292837443463201453981546986016, 0, 214741007530002414614587282795700954590907] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 596014244365810496 z - 150270596145137392 z - 252 z 24 22 4 6 + 31756207155228293 z - 5591647560589068 z + 27280 z - 1708784 z 8 10 12 14 + 70341943 z - 2046048872 z + 44081835808 z - 727079509704 z 18 16 50 - 97220868920736 z + 9406240068803 z - 47662504445548813808 z 48 20 + 70607858073410813462 z + 814383120656240 z 36 34 + 27454361268152732672 z - 13474616434241354104 z 66 80 88 84 86 - 5591647560589068 z + 70341943 z + z + 27280 z - 252 z 82 64 30 - 1708784 z + 31756207155228293 z - 1991021367812348272 z 42 44 - 89347904706285520496 z + 96634601835942465344 z 46 58 - 89347904706285520496 z - 1991021367812348272 z 56 54 + 5623892603028010346 z - 13474616434241354104 z 52 60 70 + 27454361268152732672 z + 596014244365810496 z - 97220868920736 z 68 78 32 + 814383120656240 z - 2046048872 z + 5623892603028010346 z 38 40 - 47662504445548813808 z + 70607858073410813462 z 62 76 74 - 150270596145137392 z + 44081835808 z - 727079509704 z 72 / 2 28 + 9406240068803 z ) / ((-1 + z ) (1 + 2008844202784985520 z / 26 2 24 - 486066177259600968 z - 318 z + 98188522453081557 z 22 4 6 8 - 16462755419196030 z + 40314 z - 2848142 z + 129466131 z 10 12 14 - 4104305792 z + 95525093548 z - 1690936966184 z 18 16 50 - 256531435699790 z + 23354120192383 z - 184779172980435889892 z 48 20 + 277421240146313992286 z + 2274320952204178 z 36 34 + 104482629339313649524 z - 50090423472860658308 z 66 80 88 84 86 - 16462755419196030 z + 129466131 z + z + 40314 z - 318 z 82 64 30 - 2848142 z + 98188522453081557 z - 6963823607842283064 z 42 44 - 353903309672604203480 z + 383803362495294002248 z 46 58 - 353903309672604203480 z - 6963823607842283064 z 56 54 + 20324635664242225210 z - 50090423472860658308 z 52 60 + 104482629339313649524 z + 2008844202784985520 z 70 68 78 - 256531435699790 z + 2274320952204178 z - 4104305792 z 32 38 + 20324635664242225210 z - 184779172980435889892 z 40 62 76 + 277421240146313992286 z - 486066177259600968 z + 95525093548 z 74 72 - 1690936966184 z + 23354120192383 z )) And in Maple-input format, it is: -(1+596014244365810496*z^28-150270596145137392*z^26-252*z^2+31756207155228293*z ^24-5591647560589068*z^22+27280*z^4-1708784*z^6+70341943*z^8-2046048872*z^10+ 44081835808*z^12-727079509704*z^14-97220868920736*z^18+9406240068803*z^16-\ 47662504445548813808*z^50+70607858073410813462*z^48+814383120656240*z^20+ 27454361268152732672*z^36-13474616434241354104*z^34-5591647560589068*z^66+ 70341943*z^80+z^88+27280*z^84-252*z^86-1708784*z^82+31756207155228293*z^64-\ 1991021367812348272*z^30-89347904706285520496*z^42+96634601835942465344*z^44-\ 89347904706285520496*z^46-1991021367812348272*z^58+5623892603028010346*z^56-\ 13474616434241354104*z^54+27454361268152732672*z^52+596014244365810496*z^60-\ 97220868920736*z^70+814383120656240*z^68-2046048872*z^78+5623892603028010346*z^ 32-47662504445548813808*z^38+70607858073410813462*z^40-150270596145137392*z^62+ 44081835808*z^76-727079509704*z^74+9406240068803*z^72)/(-1+z^2)/(1+ 2008844202784985520*z^28-486066177259600968*z^26-318*z^2+98188522453081557*z^24 -16462755419196030*z^22+40314*z^4-2848142*z^6+129466131*z^8-4104305792*z^10+ 95525093548*z^12-1690936966184*z^14-256531435699790*z^18+23354120192383*z^16-\ 184779172980435889892*z^50+277421240146313992286*z^48+2274320952204178*z^20+ 104482629339313649524*z^36-50090423472860658308*z^34-16462755419196030*z^66+ 129466131*z^80+z^88+40314*z^84-318*z^86-2848142*z^82+98188522453081557*z^64-\ 6963823607842283064*z^30-353903309672604203480*z^42+383803362495294002248*z^44-\ 353903309672604203480*z^46-6963823607842283064*z^58+20324635664242225210*z^56-\ 50090423472860658308*z^54+104482629339313649524*z^52+2008844202784985520*z^60-\ 256531435699790*z^70+2274320952204178*z^68-4104305792*z^78+20324635664242225210 *z^32-184779172980435889892*z^38+277421240146313992286*z^40-486066177259600968* z^62+95525093548*z^76-1690936966184*z^74+23354120192383*z^72) The first , 40, terms are: [0, 67, 0, 8021, 0, 1016027, 0, 129757563, 0, 16600425869, 0, 2124798208283, 0, 272009559461057, 0, 34823624072809953, 0, 4458328137703399083, 0, 570785688522124286301, 0, 73076074120593826302043, 0, 9355731260973426531874427, 0, 1197789263470142022215772741, 0, 153349776571753902119236464627, 0, 19632965309590642807732416933217, 0, 2513556537427632867169571992344353, 0, 321803986664969261430896153715867027, 0, 41199712253604420690338618027613478501, 0, 5274690064713149375625545623785496571579, 0, 675304602042552892578708845609427255010075] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 1297893000433900 z + 664082728927704 z + 208 z 24 22 4 6 - 270812622376354 z + 87719609830904 z - 17746 z + 842349 z 8 10 12 14 - 25404730 z + 524475546 z - 7767751225 z + 85182029778 z 18 16 50 + 4517024131389 z - 707193309000 z + 707193309000 z 48 20 36 - 4517024131389 z - 22461230926520 z - 2026429211931770 z 34 66 64 30 + 2531207794304380 z + z - 208 z + 2026429211931770 z 42 44 46 + 270812622376354 z - 87719609830904 z + 22461230926520 z 58 56 54 52 + 25404730 z - 524475546 z + 7767751225 z - 85182029778 z 60 32 38 - 842349 z - 2531207794304380 z + 1297893000433900 z 40 62 / 28 - 664082728927704 z + 17746 z ) / (1 + 7055631046656346 z / 26 2 24 - 3285815578777880 z - 256 z + 1223240884437666 z 22 4 6 8 - 362360334460472 z + 26301 z - 1470002 z + 51185401 z 10 12 14 18 - 1200575874 z + 19947499061 z - 243018631762 z - 15600468659288 z 16 50 48 + 2225372321769 z - 15600468659288 z + 84889502394653 z 20 36 34 + 84889502394653 z + 16823732460936978 z - 18748052551305356 z 66 64 30 42 - 256 z + 26301 z - 12151692390834444 z - 3285815578777880 z 44 46 58 + 1223240884437666 z - 362360334460472 z - 1200575874 z 56 54 52 60 + 19947499061 z - 243018631762 z + 2225372321769 z + 51185401 z 68 32 38 + z + 16823732460936978 z - 12151692390834444 z 40 62 + 7055631046656346 z - 1470002 z ) And in Maple-input format, it is: -(-1-1297893000433900*z^28+664082728927704*z^26+208*z^2-270812622376354*z^24+ 87719609830904*z^22-17746*z^4+842349*z^6-25404730*z^8+524475546*z^10-7767751225 *z^12+85182029778*z^14+4517024131389*z^18-707193309000*z^16+707193309000*z^50-\ 4517024131389*z^48-22461230926520*z^20-2026429211931770*z^36+2531207794304380*z ^34+z^66-208*z^64+2026429211931770*z^30+270812622376354*z^42-87719609830904*z^ 44+22461230926520*z^46+25404730*z^58-524475546*z^56+7767751225*z^54-85182029778 *z^52-842349*z^60-2531207794304380*z^32+1297893000433900*z^38-664082728927704*z ^40+17746*z^62)/(1+7055631046656346*z^28-3285815578777880*z^26-256*z^2+ 1223240884437666*z^24-362360334460472*z^22+26301*z^4-1470002*z^6+51185401*z^8-\ 1200575874*z^10+19947499061*z^12-243018631762*z^14-15600468659288*z^18+ 2225372321769*z^16-15600468659288*z^50+84889502394653*z^48+84889502394653*z^20+ 16823732460936978*z^36-18748052551305356*z^34-256*z^66+26301*z^64-\ 12151692390834444*z^30-3285815578777880*z^42+1223240884437666*z^44-\ 362360334460472*z^46-1200575874*z^58+19947499061*z^56-243018631762*z^54+ 2225372321769*z^52+51185401*z^60+z^68+16823732460936978*z^32-12151692390834444* z^38+7055631046656346*z^40-1470002*z^62) The first , 40, terms are: [0, 48, 0, 3733, 0, 320853, 0, 28736160, 0, 2624420753, 0, 242089312497, 0, 22451303364544, 0, 2088265608508629, 0, 194555190518160661, 0, 18142631364840545040, 0, 1692713017757964468129, 0, 157976947942026283380321, 0, 14746060753970383482901840, 0, 1376572057566504924712230933, 0, 128512342606095076785200043477, 0, 11997857665597342409112129445248, 0, 1120133801792120550111378602320945, 0, 104577978856571769216139747524076625, 0, 9763666281760129119315511994278221024, 0, 911563512584830901819602894027599819861] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1701243782711924 z - 810455443045798 z - 203 z 24 22 4 6 + 310805218405618 z - 95560310487008 z + 17018 z - 798421 z 8 10 12 14 + 23936235 z - 494128820 z + 7364179647 z - 81809009409 z 18 16 50 - 4548843832687 z + 692916685650 z - 4548843832687 z 48 20 36 + 23429631244653 z + 23429631244653 z + 3953955717378486 z 34 66 64 30 - 4392217008794200 z - 203 z + 17018 z - 2883379249009290 z 42 44 46 - 810455443045798 z + 310805218405618 z - 95560310487008 z 58 56 54 52 - 494128820 z + 7364179647 z - 81809009409 z + 692916685650 z 60 68 32 38 + 23936235 z + z + 3953955717378486 z - 2883379249009290 z 40 62 / 28 + 1701243782711924 z - 798421 z ) / (-1 - 9141276500525054 z / 26 2 24 + 3965566129652002 z + 255 z - 1386682237290590 z 22 4 6 8 + 388963522921313 z - 25525 z + 1389989 z - 47540811 z 10 12 14 18 + 1106572693 z - 18434520581 z + 227405640187 z + 15377913965837 z 16 50 48 - 2128052169341 z + 86971746870751 z - 388963522921313 z 20 36 34 - 86971746870751 z - 31712845246175398 z + 31712845246175398 z 66 64 30 42 + 25525 z - 1389989 z + 17048562394926110 z + 9141276500525054 z 44 46 58 - 3965566129652002 z + 1386682237290590 z + 18434520581 z 56 54 52 - 227405640187 z + 2128052169341 z - 15377913965837 z 60 70 68 32 - 1106572693 z + z - 255 z - 25793543781298362 z 38 40 62 + 25793543781298362 z - 17048562394926110 z + 47540811 z ) And in Maple-input format, it is: -(1+1701243782711924*z^28-810455443045798*z^26-203*z^2+310805218405618*z^24-\ 95560310487008*z^22+17018*z^4-798421*z^6+23936235*z^8-494128820*z^10+7364179647 *z^12-81809009409*z^14-4548843832687*z^18+692916685650*z^16-4548843832687*z^50+ 23429631244653*z^48+23429631244653*z^20+3953955717378486*z^36-4392217008794200* z^34-203*z^66+17018*z^64-2883379249009290*z^30-810455443045798*z^42+ 310805218405618*z^44-95560310487008*z^46-494128820*z^58+7364179647*z^56-\ 81809009409*z^54+692916685650*z^52+23936235*z^60+z^68+3953955717378486*z^32-\ 2883379249009290*z^38+1701243782711924*z^40-798421*z^62)/(-1-9141276500525054*z ^28+3965566129652002*z^26+255*z^2-1386682237290590*z^24+388963522921313*z^22-\ 25525*z^4+1389989*z^6-47540811*z^8+1106572693*z^10-18434520581*z^12+ 227405640187*z^14+15377913965837*z^18-2128052169341*z^16+86971746870751*z^50-\ 388963522921313*z^48-86971746870751*z^20-31712845246175398*z^36+ 31712845246175398*z^34+25525*z^66-1389989*z^64+17048562394926110*z^30+ 9141276500525054*z^42-3965566129652002*z^44+1386682237290590*z^46+18434520581*z ^58-227405640187*z^56+2128052169341*z^54-15377913965837*z^52-1106572693*z^60+z^ 70-255*z^68-25793543781298362*z^32+25793543781298362*z^38-17048562394926110*z^ 40+47540811*z^62) The first , 40, terms are: [0, 52, 0, 4753, 0, 476283, 0, 48806692, 0, 5035522303, 0, 520805469271, 0, 53918114371300, 0, 5584382895481339, 0, 578487961555746593, 0, 59930502709326848660, 0, 6208928779068123882121, 0, 643268255294072528518713, 0, 66645450877670691712701972, 0, 6904785671739402991913269825, 0, 715369580396187737810412526459, 0, 74115832069387548028693521707556, 0, 7678769309551739907742983731113255, 0, 795558832308468059161940084289368239, 0, 82423870535842303248096811808886379876, 0, 8539525059362781564297836682599177623163] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 872599136997268137 z - 213637340037723495 z - 255 z 24 22 4 6 + 43783898261200005 z - 7470672895095356 z + 28117 z - 1801984 z 8 10 12 14 + 76097932 z - 2274673820 z + 50428848979 z - 856835790869 z 18 16 50 - 121892716491724 z + 11429573079015 z - 77866229056414729925 z 48 20 + 116655930386976207699 z + 1053982350534696 z 36 34 + 44169137572703891183 z - 21268334705576802904 z 66 80 88 84 86 - 7470672895095356 z + 76097932 z + z + 28117 z - 255 z 82 64 30 - 1801984 z + 43783898261200005 z - 2996429109856626380 z 42 44 - 148633591547017180088 z + 161126390094960548592 z 46 58 - 148633591547017180088 z - 2996429109856626380 z 56 54 + 8680103678913505812 z - 21268334705576802904 z 52 60 70 + 44169137572703891183 z + 872599136997268137 z - 121892716491724 z 68 78 32 + 1053982350534696 z - 2274673820 z + 8680103678913505812 z 38 40 - 77866229056414729925 z + 116655930386976207699 z 62 76 74 - 213637340037723495 z + 50428848979 z - 856835790869 z 72 / 2 28 + 11429573079015 z ) / ((-1 + z ) (1 + 2903256537557664572 z / 26 2 24 - 683020632589497990 z - 314 z + 133974145554648755 z 22 4 6 8 - 21790539053011260 z + 40458 z - 2935164 z + 137498015 z 10 12 14 - 4497264624 z + 108028911984 z - 1973793188506 z 18 16 50 - 319001411388124 z + 28137896362789 z - 296699243755101517606 z 48 20 + 450344819015963357447 z + 2918393346266034 z 36 34 + 165293906702793411626 z - 77799250977210560060 z 66 80 88 84 86 - 21790539053011260 z + 137498015 z + z + 40458 z - 314 z 82 64 30 - 2935164 z + 133974145554648755 z - 10333107084423087320 z 42 44 - 578351453807700247656 z + 628626878168518605692 z 46 58 - 578351453807700247656 z - 10333107084423087320 z 56 54 + 30895845658363625681 z - 77799250977210560060 z 52 60 + 165293906702793411626 z + 2903256537557664572 z 70 68 78 - 319001411388124 z + 2918393346266034 z - 4497264624 z 32 38 + 30895845658363625681 z - 296699243755101517606 z 40 62 76 + 450344819015963357447 z - 683020632589497990 z + 108028911984 z 74 72 - 1973793188506 z + 28137896362789 z )) And in Maple-input format, it is: -(1+872599136997268137*z^28-213637340037723495*z^26-255*z^2+43783898261200005*z ^24-7470672895095356*z^22+28117*z^4-1801984*z^6+76097932*z^8-2274673820*z^10+ 50428848979*z^12-856835790869*z^14-121892716491724*z^18+11429573079015*z^16-\ 77866229056414729925*z^50+116655930386976207699*z^48+1053982350534696*z^20+ 44169137572703891183*z^36-21268334705576802904*z^34-7470672895095356*z^66+ 76097932*z^80+z^88+28117*z^84-255*z^86-1801984*z^82+43783898261200005*z^64-\ 2996429109856626380*z^30-148633591547017180088*z^42+161126390094960548592*z^44-\ 148633591547017180088*z^46-2996429109856626380*z^58+8680103678913505812*z^56-\ 21268334705576802904*z^54+44169137572703891183*z^52+872599136997268137*z^60-\ 121892716491724*z^70+1053982350534696*z^68-2274673820*z^78+8680103678913505812* z^32-77866229056414729925*z^38+116655930386976207699*z^40-213637340037723495*z^ 62+50428848979*z^76-856835790869*z^74+11429573079015*z^72)/(-1+z^2)/(1+ 2903256537557664572*z^28-683020632589497990*z^26-314*z^2+133974145554648755*z^ 24-21790539053011260*z^22+40458*z^4-2935164*z^6+137498015*z^8-4497264624*z^10+ 108028911984*z^12-1973793188506*z^14-319001411388124*z^18+28137896362789*z^16-\ 296699243755101517606*z^50+450344819015963357447*z^48+2918393346266034*z^20+ 165293906702793411626*z^36-77799250977210560060*z^34-21790539053011260*z^66+ 137498015*z^80+z^88+40458*z^84-314*z^86-2935164*z^82+133974145554648755*z^64-\ 10333107084423087320*z^30-578351453807700247656*z^42+628626878168518605692*z^44 -578351453807700247656*z^46-10333107084423087320*z^58+30895845658363625681*z^56 -77799250977210560060*z^54+165293906702793411626*z^52+2903256537557664572*z^60-\ 319001411388124*z^70+2918393346266034*z^68-4497264624*z^78+30895845658363625681 *z^32-296699243755101517606*z^38+450344819015963357447*z^40-683020632589497990* z^62+108028911984*z^76-1973793188506*z^74+28137896362789*z^72) The first , 40, terms are: [0, 60, 0, 6245, 0, 694493, 0, 78346228, 0, 8880050693, 0, 1008415437781, 0, 114609844310284, 0, 13030562517126949, 0, 1481751086683418909, 0, 168507419839659876372, 0, 19163595134316114416017, 0, 2179421837387866952011665, 0, 247861147234267158216965668, 0, 28188819475099436567469140877, 0, 3205869891806883440314258991925, 0, 364598731610764814295433043450076, 0, 41465272992513137530496032693071381, 0, 4715784557272389309592527869185779205, 0, 536319275557101521734088475870563478948, 0, 60994807691280243287540154982020638410893] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 5952461012417672 z - 2432947188214073 z - 213 z 24 22 4 6 + 810604208176584 z - 219240701964824 z + 18824 z - 935781 z 8 10 12 14 + 29836236 z - 657444344 z + 10508282408 z - 125992900836 z 18 16 50 - 8371008155642 z + 1161002355657 z - 219240701964824 z 48 20 36 + 810604208176584 z + 47864177134882 z + 28946648862705354 z 34 66 64 - 26230060653690134 z - 935781 z + 29836236 z 30 42 44 - 11906679207781047 z - 11906679207781047 z + 5952461012417672 z 46 58 56 - 2432947188214073 z - 125992900836 z + 1161002355657 z 54 52 60 70 - 8371008155642 z + 47864177134882 z + 10508282408 z - 213 z 68 32 38 + 18824 z + 19512000062948369 z - 26230060653690134 z 40 62 72 / 2 + 19512000062948369 z - 657444344 z + z ) / ((-1 + z ) (1 / 28 26 2 + 20890779097657933 z - 8307888002635088 z - 256 z 24 22 4 6 + 2675927352702981 z - 694950295570474 z + 26592 z - 1525808 z 8 10 12 14 + 55233967 z - 1361005514 z + 23991468566 z - 313325072906 z 18 16 50 - 23921671640558 z + 3110284603583 z - 694950295570474 z 48 20 36 + 2675927352702981 z + 144629114221325 z + 106621820529933411 z 34 66 64 - 96324859774033704 z - 1525808 z + 55233967 z 30 42 44 - 42682482772844196 z - 42682482772844196 z + 20890779097657933 z 46 58 56 - 8307888002635088 z - 313325072906 z + 3110284603583 z 54 52 60 70 - 23921671640558 z + 144629114221325 z + 23991468566 z - 256 z 68 32 38 + 26592 z + 71009037277747653 z - 96324859774033704 z 40 62 72 + 71009037277747653 z - 1361005514 z + z )) And in Maple-input format, it is: -(1+5952461012417672*z^28-2432947188214073*z^26-213*z^2+810604208176584*z^24-\ 219240701964824*z^22+18824*z^4-935781*z^6+29836236*z^8-657444344*z^10+ 10508282408*z^12-125992900836*z^14-8371008155642*z^18+1161002355657*z^16-\ 219240701964824*z^50+810604208176584*z^48+47864177134882*z^20+28946648862705354 *z^36-26230060653690134*z^34-935781*z^66+29836236*z^64-11906679207781047*z^30-\ 11906679207781047*z^42+5952461012417672*z^44-2432947188214073*z^46-125992900836 *z^58+1161002355657*z^56-8371008155642*z^54+47864177134882*z^52+10508282408*z^ 60-213*z^70+18824*z^68+19512000062948369*z^32-26230060653690134*z^38+ 19512000062948369*z^40-657444344*z^62+z^72)/(-1+z^2)/(1+20890779097657933*z^28-\ 8307888002635088*z^26-256*z^2+2675927352702981*z^24-694950295570474*z^22+26592* z^4-1525808*z^6+55233967*z^8-1361005514*z^10+23991468566*z^12-313325072906*z^14 -23921671640558*z^18+3110284603583*z^16-694950295570474*z^50+2675927352702981*z ^48+144629114221325*z^20+106621820529933411*z^36-96324859774033704*z^34-1525808 *z^66+55233967*z^64-42682482772844196*z^30-42682482772844196*z^42+ 20890779097657933*z^44-8307888002635088*z^46-313325072906*z^58+3110284603583*z^ 56-23921671640558*z^54+144629114221325*z^52+23991468566*z^60-256*z^70+26592*z^ 68+71009037277747653*z^32-96324859774033704*z^38+71009037277747653*z^40-\ 1361005514*z^62+z^72) The first , 40, terms are: [0, 44, 0, 3284, 0, 279295, 0, 24992044, 0, 2283889785, 0, 210622079825, 0, 19503674551040, 0, 1809500869950691, 0, 168033385449969887, 0, 15610704361133795248, 0, 1450580717844736159665, 0, 134805163784888441203228, 0, 12528334739239813931630768, 0, 1164370194353288684683770659, 0, 108216674016325484682668557699, 0, 10057729007368955593232292855048, 0, 934774769309461329288497311724220, 0, 86878972565590260896522724857257113, 0, 8074631054441377021342062252596354552, 0, 750465713335256275029339264616365134503] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1097206548848486976 z - 269197226589454872 z - 266 z 24 22 4 6 + 55234409998383829 z - 9422490546265674 z + 30540 z - 2029038 z 8 10 12 14 + 88338003 z - 2706774484 z + 61185862984 z - 1054988714684 z 18 16 50 - 152770109937694 z + 14222739161695 z - 96551447570468612564 z 48 20 + 144386607404703798718 z + 1326679042166412 z 36 34 + 54899231366769019432 z - 26509393964482091980 z 66 80 88 84 86 - 9422490546265674 z + 88338003 z + z + 30540 z - 266 z 82 64 30 - 2029038 z + 55234409998383829 z - 3757462397597008584 z 42 44 - 183756999148253523856 z + 199125075747992334576 z 46 58 - 183756999148253523856 z - 3757462397597008584 z 56 54 + 10851918058741892282 z - 26509393964482091980 z 52 60 70 + 54899231366769019432 z + 1097206548848486976 z - 152770109937694 z 68 78 32 + 1326679042166412 z - 2706774484 z + 10851918058741892282 z 38 40 - 96551447570468612564 z + 144386607404703798718 z 62 76 74 - 269197226589454872 z + 61185862984 z - 1054988714684 z 72 / 2 28 + 14222739161695 z ) / ((-1 + z ) (1 + 3588885721769477648 z / 26 2 24 - 850807008859217120 z - 316 z + 167929784635532389 z 22 4 6 8 - 27424979089910876 z + 42046 z - 3179876 z + 155231927 z 10 12 14 - 5262646284 z + 130066596332 z - 2426539918636 z 18 16 50 - 400771161127668 z + 35074841678083 z - 352832845693930143752 z 48 20 + 533061030437633129814 z + 3676974011337366 z 36 34 + 197784792688920810284 z - 93781974694079375624 z 66 80 88 84 86 - 27424979089910876 z + 155231927 z + z + 42046 z - 316 z 82 64 30 - 3179876 z + 167929784635532389 z - 12666493697943928480 z 42 44 - 682603466966678970440 z + 741214695184188153864 z 46 58 - 682603466966678970440 z - 12666493697943928480 z 56 54 + 37549717847827880330 z - 93781974694079375624 z 52 60 + 197784792688920810284 z + 3588885721769477648 z 70 68 78 - 400771161127668 z + 3676974011337366 z - 5262646284 z 32 38 + 37549717847827880330 z - 352832845693930143752 z 40 62 76 + 533061030437633129814 z - 850807008859217120 z + 130066596332 z 74 72 - 2426539918636 z + 35074841678083 z )) And in Maple-input format, it is: -(1+1097206548848486976*z^28-269197226589454872*z^26-266*z^2+55234409998383829* z^24-9422490546265674*z^22+30540*z^4-2029038*z^6+88338003*z^8-2706774484*z^10+ 61185862984*z^12-1054988714684*z^14-152770109937694*z^18+14222739161695*z^16-\ 96551447570468612564*z^50+144386607404703798718*z^48+1326679042166412*z^20+ 54899231366769019432*z^36-26509393964482091980*z^34-9422490546265674*z^66+ 88338003*z^80+z^88+30540*z^84-266*z^86-2029038*z^82+55234409998383829*z^64-\ 3757462397597008584*z^30-183756999148253523856*z^42+199125075747992334576*z^44-\ 183756999148253523856*z^46-3757462397597008584*z^58+10851918058741892282*z^56-\ 26509393964482091980*z^54+54899231366769019432*z^52+1097206548848486976*z^60-\ 152770109937694*z^70+1326679042166412*z^68-2706774484*z^78+10851918058741892282 *z^32-96551447570468612564*z^38+144386607404703798718*z^40-269197226589454872*z ^62+61185862984*z^76-1054988714684*z^74+14222739161695*z^72)/(-1+z^2)/(1+ 3588885721769477648*z^28-850807008859217120*z^26-316*z^2+167929784635532389*z^ 24-27424979089910876*z^22+42046*z^4-3179876*z^6+155231927*z^8-5262646284*z^10+ 130066596332*z^12-2426539918636*z^14-400771161127668*z^18+35074841678083*z^16-\ 352832845693930143752*z^50+533061030437633129814*z^48+3676974011337366*z^20+ 197784792688920810284*z^36-93781974694079375624*z^34-27424979089910876*z^66+ 155231927*z^80+z^88+42046*z^84-316*z^86-3179876*z^82+167929784635532389*z^64-\ 12666493697943928480*z^30-682603466966678970440*z^42+741214695184188153864*z^44 -682603466966678970440*z^46-12666493697943928480*z^58+37549717847827880330*z^56 -93781974694079375624*z^54+197784792688920810284*z^52+3588885721769477648*z^60-\ 400771161127668*z^70+3676974011337366*z^68-5262646284*z^78+37549717847827880330 *z^32-352832845693930143752*z^38+533061030437633129814*z^40-850807008859217120* z^62+130066596332*z^76-2426539918636*z^74+35074841678083*z^72) The first , 40, terms are: [0, 51, 0, 4345, 0, 409787, 0, 40083811, 0, 3978524057, 0, 397332600195, 0, 39788520401625, 0, 3989181562123737, 0, 400170733302647075, 0, 40152593471620672537, 0, 4029307516595193171875, 0, 404361056164645852247355, 0, 40580584465231529994974873, 0, 4072600983162222394561603827, 0, 408721532177325742915260844161, 0, 41018911117446191249957261705281, 0, 4116623719919328789723754100596595, 0, 413141112928350942588116775194557145, 0, 41462525202590647601044708978851123195, 0, 4161147621594903489457614748326755499619] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4782226005919442 z - 1975160875948946 z - 211 z 24 22 4 6 + 666191889976351 z - 182690646904763 z + 18439 z - 902328 z 8 10 12 14 + 28231205 z - 609702509 z + 9551797782 z - 112325556085 z 18 16 50 - 7199304163208 z + 1016101779461 z - 182690646904763 z 48 20 36 + 666191889976351 z + 40495220411719 z + 22804112287087764 z 34 66 64 30 - 20690205151356030 z - 902328 z + 28231205 z - 9485759810831168 z 42 44 46 - 9485759810831168 z + 4782226005919442 z - 1975160875948946 z 58 56 54 - 112325556085 z + 1016101779461 z - 7199304163208 z 52 60 70 68 + 40495220411719 z + 9551797782 z - 211 z + 18439 z 32 38 40 + 15449133796555574 z - 20690205151356030 z + 15449133796555574 z 62 72 / 28 - 609702509 z + z ) / (-1 - 24065970472430716 z / 26 2 24 + 9112094528097763 z + 263 z - 2819400464451797 z 22 4 6 8 + 709292803412298 z - 27322 z + 1547604 z - 55099884 z 10 12 14 18 + 1337705506 z - 23341679680 z + 303562899952 z + 23449077962252 z 16 50 48 - 3020615921618 z + 2819400464451797 z - 9112094528097763 z 20 36 34 - 144124180251572 z - 165262824455913232 z + 136419121773118348 z 66 64 30 + 55099884 z - 1337705506 z + 52137217042985784 z 42 44 46 + 92903994746660552 z - 52137217042985784 z + 24065970472430716 z 58 56 54 + 3020615921618 z - 23449077962252 z + 144124180251572 z 52 60 70 68 - 709292803412298 z - 303562899952 z + 27322 z - 1547604 z 32 38 40 - 92903994746660552 z + 165262824455913232 z - 136419121773118348 z 62 74 72 + 23341679680 z + z - 263 z ) And in Maple-input format, it is: -(1+4782226005919442*z^28-1975160875948946*z^26-211*z^2+666191889976351*z^24-\ 182690646904763*z^22+18439*z^4-902328*z^6+28231205*z^8-609702509*z^10+ 9551797782*z^12-112325556085*z^14-7199304163208*z^18+1016101779461*z^16-\ 182690646904763*z^50+666191889976351*z^48+40495220411719*z^20+22804112287087764 *z^36-20690205151356030*z^34-902328*z^66+28231205*z^64-9485759810831168*z^30-\ 9485759810831168*z^42+4782226005919442*z^44-1975160875948946*z^46-112325556085* z^58+1016101779461*z^56-7199304163208*z^54+40495220411719*z^52+9551797782*z^60-\ 211*z^70+18439*z^68+15449133796555574*z^32-20690205151356030*z^38+ 15449133796555574*z^40-609702509*z^62+z^72)/(-1-24065970472430716*z^28+ 9112094528097763*z^26+263*z^2-2819400464451797*z^24+709292803412298*z^22-27322* z^4+1547604*z^6-55099884*z^8+1337705506*z^10-23341679680*z^12+303562899952*z^14 +23449077962252*z^18-3020615921618*z^16+2819400464451797*z^50-9112094528097763* z^48-144124180251572*z^20-165262824455913232*z^36+136419121773118348*z^34+ 55099884*z^66-1337705506*z^64+52137217042985784*z^30+92903994746660552*z^42-\ 52137217042985784*z^44+24065970472430716*z^46+3020615921618*z^58-23449077962252 *z^56+144124180251572*z^54-709292803412298*z^52-303562899952*z^60+27322*z^70-\ 1547604*z^68-92903994746660552*z^32+165262824455913232*z^38-136419121773118348* z^40+23341679680*z^62+z^74-263*z^72) The first , 40, terms are: [0, 52, 0, 4793, 0, 485091, 0, 50231316, 0, 5237654807, 0, 547489030855, 0, 57285230706196, 0, 5996409207434475, 0, 627796613984678705, 0, 65732737130451513300, 0, 6882720030457861351617, 0, 720685025365608164397729, 0, 75462995053614349782379604, 0, 7901762845728549334547838625, 0, 827398215433142019736090792187, 0, 86637408741849982987786208125524, 0, 9071862466588368786117860728569863, 0, 949921064898962165143513128563358231, 0, 99466905160765323551200524856969666132, 0, 10415250123916110872936118748483646174483] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 73246883694100337435 z - 10939322358886354758 z - 287 z 24 22 4 6 + 1404882635346777077 z - 154223273810560940 z + 36852 z - 2847540 z 102 8 10 12 - 5768574148 z + 149943749 z - 5768574148 z + 169559169123 z 14 18 16 - 3928563348656 z - 1125897483893050 z + 73411293402449 z 50 48 - 10979923127524553344336654 z + 7089416831580983814687622 z 20 36 + 14367510089995773 z + 35508946029155234636146 z 34 66 - 9302365708754954212498 z - 4036351072099918801022706 z 80 100 90 + 2127915708386097481809 z + 169559169123 z - 154223273810560940 z 88 84 + 1404882635346777077 z + 73246883694100337435 z 94 86 96 - 1125897483893050 z - 10939322358886354758 z + 73411293402449 z 98 92 82 - 3928563348656 z + 14367510089995773 z - 423820634712986118222 z 64 112 110 106 + 7089416831580983814687622 z + z - 287 z - 2847540 z 108 30 42 + 36852 z - 423820634712986118222 z - 894619156750388591722174 z 44 46 + 2025130891171271692977902 z - 4036351072099918801022706 z 58 56 - 18090333031150189366931646 z + 19254476655112195355384106 z 54 52 - 18090333031150189366931646 z + 15002499280334251618252926 z 60 70 + 15002499280334251618252926 z - 894619156750388591722174 z 68 78 + 2025130891171271692977902 z - 9302365708754954212498 z 32 38 + 2127915708386097481809 z - 118636153424110229191827 z 40 62 + 347602963380553214050877 z - 10979923127524553344336654 z 76 74 + 35508946029155234636146 z - 118636153424110229191827 z 72 104 / 2 + 347602963380553214050877 z + 149943749 z ) / ((-1 + z ) (1 / 28 26 2 + 212537496449289966798 z - 30524497451014019542 z - 337 z 24 22 4 6 + 3757941303639818900 z - 394151799627809164 z + 49216 z - 4236348 z 102 8 10 12 - 10238045734 z + 244945082 z - 10238045734 z + 324262229012 z 14 18 16 - 8041290294784 z - 2597757856664378 z + 159926714672639 z 50 48 - 40887669084205413808884846 z + 26151201898061726199021995 z 20 36 + 34957387427968107 z + 117088082423387402555314 z 34 66 - 29830945403838431146410 z - 14709596902892392921563470 z 80 100 90 + 6618522620280956176361 z + 324262229012 z - 394151799627809164 z 88 84 + 3757941303639818900 z + 212537496449289966798 z 94 86 96 - 2597757856664378 z - 30524497451014019542 z + 159926714672639 z 98 92 82 - 8041290294784 z + 34957387427968107 z - 1275048299280382372240 z 64 112 110 106 + 26151201898061726199021995 z + z - 337 z - 4236348 z 108 30 42 + 49216 z - 1275048299280382372240 z - 3157019193597979839355614 z 44 46 + 7271901979698322113602474 z - 14709596902892392921563470 z 58 56 - 68101265955347612141268670 z + 72582361798960874826682487 z 54 52 - 68101265955347612141268670 z + 56247310079860760151028922 z 60 70 + 56247310079860760151028922 z - 3157019193597979839355614 z 68 78 + 7271901979698322113602474 z - 29830945403838431146410 z 32 38 + 6618522620280956176361 z - 401187268811106778382801 z 40 62 + 1202357527112836857203610 z - 40887669084205413808884846 z 76 74 + 117088082423387402555314 z - 401187268811106778382801 z 72 104 + 1202357527112836857203610 z + 244945082 z )) And in Maple-input format, it is: -(1+73246883694100337435*z^28-10939322358886354758*z^26-287*z^2+ 1404882635346777077*z^24-154223273810560940*z^22+36852*z^4-2847540*z^6-\ 5768574148*z^102+149943749*z^8-5768574148*z^10+169559169123*z^12-3928563348656* z^14-1125897483893050*z^18+73411293402449*z^16-10979923127524553344336654*z^50+ 7089416831580983814687622*z^48+14367510089995773*z^20+35508946029155234636146*z ^36-9302365708754954212498*z^34-4036351072099918801022706*z^66+ 2127915708386097481809*z^80+169559169123*z^100-154223273810560940*z^90+ 1404882635346777077*z^88+73246883694100337435*z^84-1125897483893050*z^94-\ 10939322358886354758*z^86+73411293402449*z^96-3928563348656*z^98+ 14367510089995773*z^92-423820634712986118222*z^82+7089416831580983814687622*z^ 64+z^112-287*z^110-2847540*z^106+36852*z^108-423820634712986118222*z^30-\ 894619156750388591722174*z^42+2025130891171271692977902*z^44-\ 4036351072099918801022706*z^46-18090333031150189366931646*z^58+ 19254476655112195355384106*z^56-18090333031150189366931646*z^54+ 15002499280334251618252926*z^52+15002499280334251618252926*z^60-\ 894619156750388591722174*z^70+2025130891171271692977902*z^68-\ 9302365708754954212498*z^78+2127915708386097481809*z^32-\ 118636153424110229191827*z^38+347602963380553214050877*z^40-\ 10979923127524553344336654*z^62+35508946029155234636146*z^76-\ 118636153424110229191827*z^74+347602963380553214050877*z^72+149943749*z^104)/(-\ 1+z^2)/(1+212537496449289966798*z^28-30524497451014019542*z^26-337*z^2+ 3757941303639818900*z^24-394151799627809164*z^22+49216*z^4-4236348*z^6-\ 10238045734*z^102+244945082*z^8-10238045734*z^10+324262229012*z^12-\ 8041290294784*z^14-2597757856664378*z^18+159926714672639*z^16-\ 40887669084205413808884846*z^50+26151201898061726199021995*z^48+ 34957387427968107*z^20+117088082423387402555314*z^36-29830945403838431146410*z^ 34-14709596902892392921563470*z^66+6618522620280956176361*z^80+324262229012*z^ 100-394151799627809164*z^90+3757941303639818900*z^88+212537496449289966798*z^84 -2597757856664378*z^94-30524497451014019542*z^86+159926714672639*z^96-\ 8041290294784*z^98+34957387427968107*z^92-1275048299280382372240*z^82+ 26151201898061726199021995*z^64+z^112-337*z^110-4236348*z^106+49216*z^108-\ 1275048299280382372240*z^30-3157019193597979839355614*z^42+ 7271901979698322113602474*z^44-14709596902892392921563470*z^46-\ 68101265955347612141268670*z^58+72582361798960874826682487*z^56-\ 68101265955347612141268670*z^54+56247310079860760151028922*z^52+ 56247310079860760151028922*z^60-3157019193597979839355614*z^70+ 7271901979698322113602474*z^68-29830945403838431146410*z^78+ 6618522620280956176361*z^32-401187268811106778382801*z^38+ 1202357527112836857203610*z^40-40887669084205413808884846*z^62+ 117088082423387402555314*z^76-401187268811106778382801*z^74+ 1202357527112836857203610*z^72+244945082*z^104) The first , 40, terms are: [0, 51, 0, 4537, 0, 444327, 0, 44686648, 0, 4536118799, 0, 462197759229, 0, 47172804969703, 0, 4818201671782825, 0, 492301816885996349, 0, 50309462807686005631, 0, 5141639976961182257313, 0, 525496142825283146852599, 0, 53708730269018712670519624, 0, 5489385927663920806536740111, 0, 561053538937024927893588721453, 0, 57343689355183541361209599643651, 0, 5860940670697095623204728230055005, 0, 599030855856634934698958429803900949, 0, 61225331259638533388496645816268575939, 0, 6257676854191231533574671920068883676149] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 72427817038777715481 z - 10808436494393422242 z - 287 z 24 22 4 6 + 1387609471152319023 z - 152345316394718452 z + 36826 z - 2842440 z 102 8 10 12 - 5741451420 z + 149471467 z - 5741451420 z + 168483062481 z 14 18 16 - 3897256987576 z - 1113845532764326 z + 72717165356355 z 50 48 - 11119484402207088465592482 z + 7166851294023208270854290 z 20 36 + 14200380246441179 z + 35364408248810079206328 z 34 66 - 9243282521525445112746 z - 4071735736354401220214574 z 80 100 90 + 2110177085630824530015 z + 168483062481 z - 152345316394718452 z 88 84 + 1387609471152319023 z + 72427817038777715481 z 94 86 96 - 1113845532764326 z - 10808436494393422242 z + 72717165356355 z 98 92 82 - 3897256987576 z + 14200380246441179 z - 419598422723454664690 z 64 112 110 106 + 7166851294023208270854290 z + z - 287 z - 2842440 z 108 30 42 + 36826 z - 419598422723454664690 z - 897959728360664034027818 z 44 46 + 2037962795497846366097714 z - 4071735736354401220214574 z 58 56 - 18359519362270325147327682 z + 19546396428583829497182438 z 54 52 - 18359519362270325147327682 z + 15213289325198529519784426 z 60 70 + 15213289325198529519784426 z - 897959728360664034027818 z 68 78 + 2037962795497846366097714 z - 9243282521525445112746 z 32 38 + 2110177085630824530015 z - 118449984760869764163759 z 40 62 + 347973012751595218408329 z - 11119484402207088465592482 z 76 74 + 35364408248810079206328 z - 118449984760869764163759 z 72 104 / 2 + 347973012751595218408329 z + 149471467 z ) / ((-1 + z ) (1 / 28 26 2 + 208329338616655884042 z - 29891204955904465622 z - 339 z 24 22 4 6 + 3679901862731633512 z - 386328472081363484 z + 49584 z - 4263760 z 102 8 10 12 - 10240136658 z + 245884218 z - 10240136658 z + 322995341744 z 14 18 16 - 7976312345400 z - 2557861630130722 z + 158010377373523 z 50 48 - 41782695839017562940275814 z + 26639310348890025315011395 z 20 36 + 34327405684750271 z + 116148625453240404323090 z 34 66 - 29473960315423877194038 z - 14927072414047866668732078 z 80 100 90 + 6517106312428857954905 z + 322995341744 z - 386328472081363484 z 88 84 + 3679901862731633512 z + 208329338616655884042 z 94 86 96 - 2557861630130722 z - 29891204955904465622 z + 158010377373523 z 98 92 82 - 7976312345400 z + 34327405684750271 z - 1252139092838323812212 z 64 112 110 106 + 26639310348890025315011395 z + z - 339 z - 4263760 z 108 30 42 + 49584 z - 1252139092838323812212 z - 3175234840309656416191854 z 44 46 + 7347699209674993033503978 z - 14927072414047866668732078 z 58 56 - 69859034007408560995467510 z + 74492702633769820657487999 z 54 52 - 69859034007408560995467510 z + 57614699743948274598565954 z 60 70 + 57614699743948274598565954 z - 3175234840309656416191854 z 68 78 + 7347699209674993033503978 z - 29473960315423877194038 z 32 38 + 6517106312428857954905 z - 399727078282558650903463 z 40 62 + 1203582163688008845186938 z - 41782695839017562940275814 z 76 74 + 116148625453240404323090 z - 399727078282558650903463 z 72 104 + 1203582163688008845186938 z + 245884218 z )) And in Maple-input format, it is: -(1+72427817038777715481*z^28-10808436494393422242*z^26-287*z^2+ 1387609471152319023*z^24-152345316394718452*z^22+36826*z^4-2842440*z^6-\ 5741451420*z^102+149471467*z^8-5741451420*z^10+168483062481*z^12-3897256987576* z^14-1113845532764326*z^18+72717165356355*z^16-11119484402207088465592482*z^50+ 7166851294023208270854290*z^48+14200380246441179*z^20+35364408248810079206328*z ^36-9243282521525445112746*z^34-4071735736354401220214574*z^66+ 2110177085630824530015*z^80+168483062481*z^100-152345316394718452*z^90+ 1387609471152319023*z^88+72427817038777715481*z^84-1113845532764326*z^94-\ 10808436494393422242*z^86+72717165356355*z^96-3897256987576*z^98+ 14200380246441179*z^92-419598422723454664690*z^82+7166851294023208270854290*z^ 64+z^112-287*z^110-2842440*z^106+36826*z^108-419598422723454664690*z^30-\ 897959728360664034027818*z^42+2037962795497846366097714*z^44-\ 4071735736354401220214574*z^46-18359519362270325147327682*z^58+ 19546396428583829497182438*z^56-18359519362270325147327682*z^54+ 15213289325198529519784426*z^52+15213289325198529519784426*z^60-\ 897959728360664034027818*z^70+2037962795497846366097714*z^68-\ 9243282521525445112746*z^78+2110177085630824530015*z^32-\ 118449984760869764163759*z^38+347973012751595218408329*z^40-\ 11119484402207088465592482*z^62+35364408248810079206328*z^76-\ 118449984760869764163759*z^74+347973012751595218408329*z^72+149471467*z^104)/(-\ 1+z^2)/(1+208329338616655884042*z^28-29891204955904465622*z^26-339*z^2+ 3679901862731633512*z^24-386328472081363484*z^22+49584*z^4-4263760*z^6-\ 10240136658*z^102+245884218*z^8-10240136658*z^10+322995341744*z^12-\ 7976312345400*z^14-2557861630130722*z^18+158010377373523*z^16-\ 41782695839017562940275814*z^50+26639310348890025315011395*z^48+ 34327405684750271*z^20+116148625453240404323090*z^36-29473960315423877194038*z^ 34-14927072414047866668732078*z^66+6517106312428857954905*z^80+322995341744*z^ 100-386328472081363484*z^90+3679901862731633512*z^88+208329338616655884042*z^84 -2557861630130722*z^94-29891204955904465622*z^86+158010377373523*z^96-\ 7976312345400*z^98+34327405684750271*z^92-1252139092838323812212*z^82+ 26639310348890025315011395*z^64+z^112-339*z^110-4263760*z^106+49584*z^108-\ 1252139092838323812212*z^30-3175234840309656416191854*z^42+ 7347699209674993033503978*z^44-14927072414047866668732078*z^46-\ 69859034007408560995467510*z^58+74492702633769820657487999*z^56-\ 69859034007408560995467510*z^54+57614699743948274598565954*z^52+ 57614699743948274598565954*z^60-3175234840309656416191854*z^70+ 7347699209674993033503978*z^68-29473960315423877194038*z^78+ 6517106312428857954905*z^32-399727078282558650903463*z^38+ 1203582163688008845186938*z^40-41782695839017562940275814*z^62+ 116148625453240404323090*z^76-399727078282558650903463*z^74+ 1203582163688008845186938*z^72+245884218*z^104) The first , 40, terms are: [0, 53, 0, 4923, 0, 498805, 0, 51753492, 0, 5415664399, 0, 568682063277, 0, 59804490870671, 0, 6293359580187747, 0, 662457424511395007, 0, 69741317935074142719, 0, 7342565675447817213625, 0, 773066793039504195811215, 0, 81393809903804914278868100, 0, 8569748435353674094794063749, 0, 902289334892203949371503822727, 0, 95000097441144017920988102678877, 0, 10002360254028356733740306240588469, 0, 1053127688346938582551796843464687181, 0, 110881633106458432310698230656469695581, 0, 11674498027784282797258462599637709192223] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 68928759341223824089 z - 10465139498393684114 z - 293 z 24 22 4 6 + 1366042876644820155 z - 152350094554016296 z + 38206 z - 2981416 z 102 8 10 12 - 6067244500 z + 157719135 z - 6067244500 z + 177610314081 z 14 18 16 - 4084688406508 z - 1144787819673062 z + 75561108622935 z 50 48 - 9038460606930731315803942 z + 5870352758084675718049226 z 20 36 + 14407770983596551 z + 31385414467042021732572 z 34 66 - 8344576827924966770262 z - 3367335458793252533631906 z 80 100 90 + 1938644878485756906027 z + 177610314081 z - 152350094554016296 z 88 84 + 1366042876644820155 z + 68928759341223824089 z 94 86 96 - 1144787819673062 z - 10465139498393684114 z + 75561108622935 z 98 92 82 - 4084688406508 z + 14407770983596551 z - 392364355663029760686 z 64 112 110 106 + 5870352758084675718049226 z + z - 293 z - 2981416 z 108 30 42 + 38206 z - 392364355663029760686 z - 760918445628203779028646 z 44 46 + 1704692464300847097949242 z - 3367335458793252533631906 z 58 56 - 14790373558457995411891958 z + 15728635929267263444769718 z 54 52 - 14790373558457995411891958 z + 12297363674615284497017010 z 60 70 + 12297363674615284497017010 z - 760918445628203779028646 z 68 78 + 1704692464300847097949242 z - 8344576827924966770262 z 32 38 + 1938644878485756906027 z - 103412238481014466351245 z 40 62 + 299126637847569587410633 z - 9038460606930731315803942 z 76 74 + 31385414467042021732572 z - 103412238481014466351245 z 72 104 / 2 + 299126637847569587410633 z + 157719135 z ) / ((-1 + z ) (1 / 28 26 2 + 202404885999876590678 z - 29583866619759030202 z - 347 z 24 22 4 6 + 3706684232177936620 z - 395544656767481048 z + 51772 z - 4519040 z 102 8 10 12 - 11009817906 z + 263143858 z - 11009817906 z + 347294682836 z 14 18 16 - 8542727296748 z - 2691637975941966 z + 167990016209299 z 50 48 - 33979477167904176763899726 z + 21858975238666121935418919 z 20 36 + 35666944159185511 z + 104473718123167670589470 z 34 66 - 27022999268877238766518 z - 12386144495547238408814126 z 80 100 90 + 6092339308512176543069 z + 347294682836 z - 395544656767481048 z 88 84 + 3706684232177936620 z + 202404885999876590678 z 94 86 96 - 2691637975941966 z - 29583866619759030202 z + 167990016209299 z 98 92 82 - 8542727296748 z + 35666944159185511 z - 1193486818952824446292 z 64 112 110 106 + 21858975238666121935418919 z + z - 347 z - 4519040 z 108 30 42 + 51772 z - 1193486818952824446292 z - 2709880098215392392884606 z 44 46 + 6177866738422940092293890 z - 12386144495547238408814126 z 58 56 - 56218238275286064450076494 z + 59867087732647533654859011 z 54 52 - 56218238275286064450076494 z + 46549590131928984142123218 z 60 70 + 46549590131928984142123218 z - 2709880098215392392884606 z 68 78 + 6177866738422940092293890 z - 27022999268877238766518 z 32 38 + 6092339308512176543069 z - 352952733145492188785343 z 40 62 + 1044191058118783382196022 z - 33979477167904176763899726 z 76 74 + 104473718123167670589470 z - 352952733145492188785343 z 72 104 + 1044191058118783382196022 z + 263143858 z )) And in Maple-input format, it is: -(1+68928759341223824089*z^28-10465139498393684114*z^26-293*z^2+ 1366042876644820155*z^24-152350094554016296*z^22+38206*z^4-2981416*z^6-\ 6067244500*z^102+157719135*z^8-6067244500*z^10+177610314081*z^12-4084688406508* z^14-1144787819673062*z^18+75561108622935*z^16-9038460606930731315803942*z^50+ 5870352758084675718049226*z^48+14407770983596551*z^20+31385414467042021732572*z ^36-8344576827924966770262*z^34-3367335458793252533631906*z^66+ 1938644878485756906027*z^80+177610314081*z^100-152350094554016296*z^90+ 1366042876644820155*z^88+68928759341223824089*z^84-1144787819673062*z^94-\ 10465139498393684114*z^86+75561108622935*z^96-4084688406508*z^98+ 14407770983596551*z^92-392364355663029760686*z^82+5870352758084675718049226*z^ 64+z^112-293*z^110-2981416*z^106+38206*z^108-392364355663029760686*z^30-\ 760918445628203779028646*z^42+1704692464300847097949242*z^44-\ 3367335458793252533631906*z^46-14790373558457995411891958*z^58+ 15728635929267263444769718*z^56-14790373558457995411891958*z^54+ 12297363674615284497017010*z^52+12297363674615284497017010*z^60-\ 760918445628203779028646*z^70+1704692464300847097949242*z^68-\ 8344576827924966770262*z^78+1938644878485756906027*z^32-\ 103412238481014466351245*z^38+299126637847569587410633*z^40-\ 9038460606930731315803942*z^62+31385414467042021732572*z^76-\ 103412238481014466351245*z^74+299126637847569587410633*z^72+157719135*z^104)/(-\ 1+z^2)/(1+202404885999876590678*z^28-29583866619759030202*z^26-347*z^2+ 3706684232177936620*z^24-395544656767481048*z^22+51772*z^4-4519040*z^6-\ 11009817906*z^102+263143858*z^8-11009817906*z^10+347294682836*z^12-\ 8542727296748*z^14-2691637975941966*z^18+167990016209299*z^16-\ 33979477167904176763899726*z^50+21858975238666121935418919*z^48+ 35666944159185511*z^20+104473718123167670589470*z^36-27022999268877238766518*z^ 34-12386144495547238408814126*z^66+6092339308512176543069*z^80+347294682836*z^ 100-395544656767481048*z^90+3706684232177936620*z^88+202404885999876590678*z^84 -2691637975941966*z^94-29583866619759030202*z^86+167990016209299*z^96-\ 8542727296748*z^98+35666944159185511*z^92-1193486818952824446292*z^82+ 21858975238666121935418919*z^64+z^112-347*z^110-4519040*z^106+51772*z^108-\ 1193486818952824446292*z^30-2709880098215392392884606*z^42+ 6177866738422940092293890*z^44-12386144495547238408814126*z^46-\ 56218238275286064450076494*z^58+59867087732647533654859011*z^56-\ 56218238275286064450076494*z^54+46549590131928984142123218*z^52+ 46549590131928984142123218*z^60-2709880098215392392884606*z^70+ 6177866738422940092293890*z^68-27022999268877238766518*z^78+ 6092339308512176543069*z^32-352952733145492188785343*z^38+ 1044191058118783382196022*z^40-33979477167904176763899726*z^62+ 104473718123167670589470*z^76-352952733145492188785343*z^74+ 1044191058118783382196022*z^72+263143858*z^104) The first , 40, terms are: [0, 55, 0, 5227, 0, 541847, 0, 57587640, 0, 6175867125, 0, 664717062617, 0, 71653813946609, 0, 7729134082412971, 0, 833970494822883863, 0, 89997037438137953465, 0, 9712519235371682203465, 0, 1048208058528285668161073, 0, 113127594903197437980022536, 0, 12209338047670864389711977459, 0, 1317700874421272044471904138331, 0, 142213910856957430502008493804751, 0, 15348557626147055704883685483642685, 0, 1656506601764640927297738194990033509, 0, 178779955486195714724153682148317851167, 0, 19294987540860450324045833904402553330219] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 81316875977442910732 z - 12177262418364551020 z - 290 z 24 22 4 6 + 1566099005419292524 z - 171904611864189004 z + 37714 z - 2955878 z 102 8 10 12 - 6160504582 z + 157929718 z - 6160504582 z + 183334507338 z 14 18 16 - 4292273446042 z - 1247811113959756 z + 80872316319699 z 50 48 - 11613303747200187197167720 z + 7518895017014009608275051 z 20 36 + 15984822516456172 z + 38707080422756880013286 z 34 66 - 10193265354491743867894 z - 4295529108180917008106254 z 80 100 90 + 2343251901464614797249 z + 183334507338 z - 171904611864189004 z 88 84 + 1566099005419292524 z + 81316875977442910732 z 94 86 96 - 1247811113959756 z - 12177262418364551020 z + 80872316319699 z 98 92 82 - 4292273446042 z + 15984822516456172 z - 468783119001090835852 z 64 112 110 106 + 7518895017014009608275051 z + z - 290 z - 2955878 z 108 30 42 + 37714 z - 468783119001090835852 z - 960254524736870661346586 z 44 46 + 2163861610362789380285886 z - 4295529108180917008106254 z 58 56 - 19072700545985155672900744 z + 20291817957367536526762728 z 54 52 - 19072700545985155672900744 z + 15836348509314535107374792 z 60 70 + 15836348509314535107374792 z - 960254524736870661346586 z 68 78 + 2163861610362789380285886 z - 10193265354491743867894 z 32 38 + 2343251901464614797249 z - 128637546370280982585658 z 40 62 + 374952524625891774657066 z - 11613303747200187197167720 z 76 74 + 38707080422756880013286 z - 128637546370280982585658 z 72 104 / 2 + 374952524625891774657066 z + 157929718 z ) / ((-1 + z ) (1 / 28 26 2 + 237869936198500303040 z - 34242381068564094028 z - 346 z 24 22 4 6 + 4220621278858453436 z - 442588015615743020 z + 51308 z - 4473450 z 102 8 10 12 - 11071360994 z + 261832518 z - 11071360994 z + 354362562772 z 14 18 16 - 8867572266218 z - 2901377113321812 z + 177651585652011 z 50 48 - 43856799006523399659060824 z + 28116321682957592265784475 z 20 36 + 39182530922182304 z + 128943662383071190872156 z 34 66 - 33003574231719657501334 z - 15861740438734771411383142 z 80 100 90 + 7354523774741955002937 z + 354362562772 z - 442588015615743020 z 88 84 + 4220621278858453436 z + 237869936198500303040 z 94 86 96 - 2901377113321812 z - 34242381068564094028 z + 177651585652011 z 98 92 82 - 8867572266218 z + 39182530922182304 z - 1422402710832278511636 z 64 112 110 106 + 28116321682957592265784475 z + z - 346 z - 4473450 z 108 30 42 + 51308 z - 1422402710832278511636 z - 3429776799691079427452262 z 44 46 + 7868945730140953261843684 z - 15861740438734771411383142 z 58 56 - 72846175023031697117381320 z + 77612514056606426850760648 z 54 52 - 72846175023031697117381320 z + 60228730519840216606225952 z 60 70 + 60228730519840216606225952 z - 3429776799691079427452262 z 68 78 + 7868945730140953261843684 z - 33003574231719657501334 z 32 38 + 7354523774741955002937 z - 439746022797073190682894 z 40 62 + 1311902678282812916879130 z - 43856799006523399659060824 z 76 74 + 128943662383071190872156 z - 439746022797073190682894 z 72 104 + 1311902678282812916879130 z + 261832518 z )) And in Maple-input format, it is: -(1+81316875977442910732*z^28-12177262418364551020*z^26-290*z^2+ 1566099005419292524*z^24-171904611864189004*z^22+37714*z^4-2955878*z^6-\ 6160504582*z^102+157929718*z^8-6160504582*z^10+183334507338*z^12-4292273446042* z^14-1247811113959756*z^18+80872316319699*z^16-11613303747200187197167720*z^50+ 7518895017014009608275051*z^48+15984822516456172*z^20+38707080422756880013286*z ^36-10193265354491743867894*z^34-4295529108180917008106254*z^66+ 2343251901464614797249*z^80+183334507338*z^100-171904611864189004*z^90+ 1566099005419292524*z^88+81316875977442910732*z^84-1247811113959756*z^94-\ 12177262418364551020*z^86+80872316319699*z^96-4292273446042*z^98+ 15984822516456172*z^92-468783119001090835852*z^82+7518895017014009608275051*z^ 64+z^112-290*z^110-2955878*z^106+37714*z^108-468783119001090835852*z^30-\ 960254524736870661346586*z^42+2163861610362789380285886*z^44-\ 4295529108180917008106254*z^46-19072700545985155672900744*z^58+ 20291817957367536526762728*z^56-19072700545985155672900744*z^54+ 15836348509314535107374792*z^52+15836348509314535107374792*z^60-\ 960254524736870661346586*z^70+2163861610362789380285886*z^68-\ 10193265354491743867894*z^78+2343251901464614797249*z^32-\ 128637546370280982585658*z^38+374952524625891774657066*z^40-\ 11613303747200187197167720*z^62+38707080422756880013286*z^76-\ 128637546370280982585658*z^74+374952524625891774657066*z^72+157929718*z^104)/(-\ 1+z^2)/(1+237869936198500303040*z^28-34242381068564094028*z^26-346*z^2+ 4220621278858453436*z^24-442588015615743020*z^22+51308*z^4-4473450*z^6-\ 11071360994*z^102+261832518*z^8-11071360994*z^10+354362562772*z^12-\ 8867572266218*z^14-2901377113321812*z^18+177651585652011*z^16-\ 43856799006523399659060824*z^50+28116321682957592265784475*z^48+ 39182530922182304*z^20+128943662383071190872156*z^36-33003574231719657501334*z^ 34-15861740438734771411383142*z^66+7354523774741955002937*z^80+354362562772*z^ 100-442588015615743020*z^90+4220621278858453436*z^88+237869936198500303040*z^84 -2901377113321812*z^94-34242381068564094028*z^86+177651585652011*z^96-\ 8867572266218*z^98+39182530922182304*z^92-1422402710832278511636*z^82+ 28116321682957592265784475*z^64+z^112-346*z^110-4473450*z^106+51308*z^108-\ 1422402710832278511636*z^30-3429776799691079427452262*z^42+ 7868945730140953261843684*z^44-15861740438734771411383142*z^46-\ 72846175023031697117381320*z^58+77612514056606426850760648*z^56-\ 72846175023031697117381320*z^54+60228730519840216606225952*z^52+ 60228730519840216606225952*z^60-3429776799691079427452262*z^70+ 7868945730140953261843684*z^68-33003574231719657501334*z^78+ 7354523774741955002937*z^32-439746022797073190682894*z^38+ 1311902678282812916879130*z^40-43856799006523399659060824*z^62+ 128943662383071190872156*z^76-439746022797073190682894*z^74+ 1311902678282812916879130*z^72+261832518*z^104) The first , 40, terms are: [0, 57, 0, 5839, 0, 650735, 0, 73732295, 0, 8385351391, 0, 954499430481, 0, 108674983844177, 0, 12373981851081209, 0, 1408952127689157529, 0, 160429721589887997143, 0, 18267281089131036748447, 0, 2079998991226241631338087, 0, 236838538403512240002465063, 0, 26967558508423551102538661441, 0, 3070654045852585894348097835625, 0, 349639226007129684908589466420409, 0, 39811579744729607599972347535341841, 0, 4533135197826086407153791160538650455, 0, 516164263115734534049497286700822360215, 0, 58772909894202663526647555878208147649295] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 850497340391256 z + 441283825086724 z + 205 z 24 22 4 6 - 183261032330148 z + 60693857015048 z - 16988 z + 777000 z 8 10 12 14 - 22496668 z + 445429436 z - 6332221304 z + 66785305620 z 18 16 50 + 3304257894528 z - 534720892856 z + 534720892856 z 48 20 36 - 3304257894528 z - 15950737892604 z - 1315379251145620 z 34 66 64 30 + 1635180643284986 z + z - 205 z + 1315379251145620 z 42 44 46 + 183261032330148 z - 60693857015048 z + 15950737892604 z 58 56 54 52 + 22496668 z - 445429436 z + 6332221304 z - 66785305620 z 60 32 38 - 777000 z - 1635180643284986 z + 850497340391256 z 40 62 / 28 - 441283825086724 z + 16988 z ) / (1 + 4647226659554120 z / 26 2 24 - 2196784454964376 z - 254 z + 833558634050496 z 22 4 6 8 - 252709813786632 z + 25565 z - 1383496 z + 46312744 z 10 12 14 18 - 1040797032 z + 16560433104 z - 193512180120 z - 11535550547288 z 16 50 48 + 1704531109880 z - 11535550547288 z + 60836523454464 z 20 36 34 + 60836523454464 z + 10891804199718954 z - 12111347202586452 z 66 64 30 42 - 254 z + 25565 z - 7918249182770712 z - 2196784454964376 z 44 46 58 + 833558634050496 z - 252709813786632 z - 1040797032 z 56 54 52 60 + 16560433104 z - 193512180120 z + 1704531109880 z + 46312744 z 68 32 38 + z + 10891804199718954 z - 7918249182770712 z 40 62 + 4647226659554120 z - 1383496 z ) And in Maple-input format, it is: -(-1-850497340391256*z^28+441283825086724*z^26+205*z^2-183261032330148*z^24+ 60693857015048*z^22-16988*z^4+777000*z^6-22496668*z^8+445429436*z^10-6332221304 *z^12+66785305620*z^14+3304257894528*z^18-534720892856*z^16+534720892856*z^50-\ 3304257894528*z^48-15950737892604*z^20-1315379251145620*z^36+1635180643284986*z ^34+z^66-205*z^64+1315379251145620*z^30+183261032330148*z^42-60693857015048*z^ 44+15950737892604*z^46+22496668*z^58-445429436*z^56+6332221304*z^54-66785305620 *z^52-777000*z^60-1635180643284986*z^32+850497340391256*z^38-441283825086724*z^ 40+16988*z^62)/(1+4647226659554120*z^28-2196784454964376*z^26-254*z^2+ 833558634050496*z^24-252709813786632*z^22+25565*z^4-1383496*z^6+46312744*z^8-\ 1040797032*z^10+16560433104*z^12-193512180120*z^14-11535550547288*z^18+ 1704531109880*z^16-11535550547288*z^50+60836523454464*z^48+60836523454464*z^20+ 10891804199718954*z^36-12111347202586452*z^34-254*z^66+25565*z^64-\ 7918249182770712*z^30-2196784454964376*z^42+833558634050496*z^44-\ 252709813786632*z^46-1040797032*z^58+16560433104*z^56-193512180120*z^54+ 1704531109880*z^52+46312744*z^60+z^68+10891804199718954*z^32-7918249182770712*z ^38+4647226659554120*z^40-1383496*z^62) The first , 40, terms are: [0, 49, 0, 3869, 0, 336537, 0, 30544641, 0, 2833559573, 0, 266034813961, 0, 25147438348969, 0, 2386388199215129, 0, 226967504185877209, 0, 21614687325755400485, 0, 2059960225160228695793, 0, 196406514075303141166889, 0, 18730994142635493369002253, 0, 1786602494756202275223637857, 0, 170424041209957714034542765841, 0, 16257524359675302827178100390897, 0, 1550921621173596046176138556361473, 0, 147955846685974991284110686247821613, 0, 14114917839398548640565585553726128393, 0, 1346563529052532701259628480582638427665] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 519706015705244421 z - 128314723455104641 z - 233 z 24 22 4 6 + 26554082658538244 z - 4582142661877892 z + 23596 z - 1406812 z 8 10 12 14 + 56038057 z - 1599296076 z + 34184750661 z - 564250729610 z 18 16 50 - 76954137730040 z + 7353756488621 z - 45154641878597496182 z 48 20 + 67482201969606384988 z + 655051228062277 z 36 34 + 25701579704266755136 z - 12430044087640554912 z 66 80 88 84 86 - 4582142661877892 z + 56038057 z + z + 23596 z - 233 z 82 64 30 - 1406812 z + 26554082658538244 z - 1771649892564054798 z 42 44 - 85852495562512047090 z + 93022157079690156644 z 46 58 - 85852495562512047090 z - 1771649892564054798 z 56 54 + 5100023056126682506 z - 12430044087640554912 z 52 60 70 + 25701579704266755136 z + 519706015705244421 z - 76954137730040 z 68 78 32 + 655051228062277 z - 1599296076 z + 5100023056126682506 z 38 40 - 45154641878597496182 z + 67482201969606384988 z 62 76 74 - 128314723455104641 z + 34184750661 z - 564250729610 z 72 / 2 28 + 7353756488621 z ) / ((-1 + z ) (1 + 1691994427003339160 z / 26 2 24 - 402439402511749296 z - 282 z + 79854568092127673 z 22 4 6 8 - 13147548605859091 z + 32944 z - 2211707 z + 97739067 z 10 12 14 - 3061613886 z + 71206113230 z - 1269197064057 z 18 16 50 - 197833864666744 z + 17739927154679 z - 165816625234630065342 z 48 20 + 250554501972147830505 z + 1784020421658806 z 36 34 + 92940036141197524161 z - 44070595527300985198 z 66 80 88 84 86 - 13147548605859091 z + 97739067 z + z + 32944 z - 282 z 82 64 30 - 2211707 z + 79854568092127673 z - 5960342480732536230 z 42 44 - 320881676893871872882 z + 348449060146017898637 z 46 58 - 320881676893871872882 z - 5960342480732536230 z 56 54 + 17652307180919336861 z - 44070595527300985198 z 52 60 70 + 92940036141197524161 z + 1691994427003339160 z - 197833864666744 z 68 78 32 + 1784020421658806 z - 3061613886 z + 17652307180919336861 z 38 40 - 165816625234630065342 z + 250554501972147830505 z 62 76 74 - 402439402511749296 z + 71206113230 z - 1269197064057 z 72 + 17739927154679 z )) And in Maple-input format, it is: -(1+519706015705244421*z^28-128314723455104641*z^26-233*z^2+26554082658538244*z ^24-4582142661877892*z^22+23596*z^4-1406812*z^6+56038057*z^8-1599296076*z^10+ 34184750661*z^12-564250729610*z^14-76954137730040*z^18+7353756488621*z^16-\ 45154641878597496182*z^50+67482201969606384988*z^48+655051228062277*z^20+ 25701579704266755136*z^36-12430044087640554912*z^34-4582142661877892*z^66+ 56038057*z^80+z^88+23596*z^84-233*z^86-1406812*z^82+26554082658538244*z^64-\ 1771649892564054798*z^30-85852495562512047090*z^42+93022157079690156644*z^44-\ 85852495562512047090*z^46-1771649892564054798*z^58+5100023056126682506*z^56-\ 12430044087640554912*z^54+25701579704266755136*z^52+519706015705244421*z^60-\ 76954137730040*z^70+655051228062277*z^68-1599296076*z^78+5100023056126682506*z^ 32-45154641878597496182*z^38+67482201969606384988*z^40-128314723455104641*z^62+ 34184750661*z^76-564250729610*z^74+7353756488621*z^72)/(-1+z^2)/(1+ 1691994427003339160*z^28-402439402511749296*z^26-282*z^2+79854568092127673*z^24 -13147548605859091*z^22+32944*z^4-2211707*z^6+97739067*z^8-3061613886*z^10+ 71206113230*z^12-1269197064057*z^14-197833864666744*z^18+17739927154679*z^16-\ 165816625234630065342*z^50+250554501972147830505*z^48+1784020421658806*z^20+ 92940036141197524161*z^36-44070595527300985198*z^34-13147548605859091*z^66+ 97739067*z^80+z^88+32944*z^84-282*z^86-2211707*z^82+79854568092127673*z^64-\ 5960342480732536230*z^30-320881676893871872882*z^42+348449060146017898637*z^44-\ 320881676893871872882*z^46-5960342480732536230*z^58+17652307180919336861*z^56-\ 44070595527300985198*z^54+92940036141197524161*z^52+1691994427003339160*z^60-\ 197833864666744*z^70+1784020421658806*z^68-3061613886*z^78+17652307180919336861 *z^32-165816625234630065342*z^38+250554501972147830505*z^40-402439402511749296* z^62+71206113230*z^76-1269197064057*z^74+17739927154679*z^72) The first , 40, terms are: [0, 50, 0, 4520, 0, 455699, 0, 47101130, 0, 4896905513, 0, 509834503563, 0, 53099754167432, 0, 5530907316784423, 0, 576117521691465157, 0, 60010705235772117882, 0, 6250966734966075466471, 0, 651127248514438755819712, 0, 67824189953403904237654842, 0, 7064857046978444774459540133, 0, 735905665826322896865931568333, 0, 76655075558913279752353553700570, 0, 7984719899831778518853364523989440, 0, 831722510615603381447781083689682367, 0, 86635767239631233025400479050239118394, 0, 9024351354636491547164956985326151888125] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 69316831737212016000 z - 10514963503259030632 z - 293 z 24 22 4 6 + 1371164263191169592 z - 152748847041417068 z + 38155 z - 2973718 z 102 8 10 12 - 6046825202 z + 157201864 z - 6046825202 z + 177089670751 z 14 18 16 - 4076056151513 z - 1144968493989044 z + 75482135591029 z 50 48 - 9105643967154318842612378 z + 5914581520629823961691714 z 20 36 + 14427979165794480 z + 31623723544274988692504 z 34 66 - 8405631787223069198692 z - 3393065098283562189101084 z 80 100 90 + 1952031557465296177536 z + 177089670751 z - 152748847041417068 z 88 84 + 1371164263191169592 z + 69316831737212016000 z 94 86 96 - 1144968493989044 z - 10514963503259030632 z + 75482135591029 z 98 92 82 - 4076056151513 z + 14427979165794480 z - 394853992186548913052 z 64 112 110 106 + 5914581520629823961691714 z + z - 293 z - 2973718 z 108 30 42 + 38155 z - 394853992186548913052 z - 766854427830325675583556 z 44 46 + 1717880597054808068416720 z - 3393065098283562189101084 z 58 56 - 14898306203077664530538360 z + 15843125272142175818250352 z 54 52 - 14898306203077664530538360 z + 12387759588709857173863166 z 60 70 + 12387759588709857173863166 z - 766854427830325675583556 z 68 78 + 1717880597054808068416720 z - 8405631787223069198692 z 32 38 + 1952031557465296177536 z - 104213860762162512013056 z 40 62 + 301463911681891198679824 z - 9105643967154318842612378 z 76 74 + 31623723544274988692504 z - 104213860762162512013056 z 72 104 / 2 + 301463911681891198679824 z + 157201864 z ) / ((-1 + z ) (1 / 28 26 2 + 203846173380909606168 z - 29792951669070763916 z - 344 z 24 22 4 6 + 3730752455686092780 z - 397667352976720580 z + 51134 z - 4459284 z 102 8 10 12 - 10898839200 z + 259918635 z - 10898839200 z + 344780156110 z 14 18 16 - 8507089078172 z - 2695592839418956 z + 167789260469249 z 50 48 - 33734690068409994742610300 z + 21725127233276086543951910 z 20 36 + 35798271416455328 z + 104819001056727284103416 z 34 66 - 27150532744245927013692 z - 12326643845006494006821444 z 80 100 90 + 6128152593286170237316 z + 344780156110 z - 397667352976720580 z 88 84 + 3730752455686092780 z + 203846173380909606168 z 94 86 96 - 2695592839418956 z - 29792951669070763916 z + 167789260469249 z 98 92 82 - 8507089078172 z + 35798271416455328 z - 1201486998048055485780 z 64 112 110 106 + 21725127233276086543951910 z + z - 344 z - 4459284 z 108 30 42 + 51134 z - 1201486998048055485780 z - 2705379322719279070344108 z 44 46 + 6157502231257849426541216 z - 12326643845006494006821444 z 58 56 - 55740478076617390671558332 z + 59348363847386498536070946 z 54 52 - 55740478076617390671558332 z + 46176938516131270014556516 z 60 70 + 46176938516131270014556516 z - 2705379322719279070344108 z 68 78 + 6157502231257849426541216 z - 27150532744245927013692 z 32 38 + 6128152593286170237316 z - 353558517195694259576324 z 40 62 + 1044227372113634971313900 z - 33734690068409994742610300 z 76 74 + 104819001056727284103416 z - 353558517195694259576324 z 72 104 + 1044227372113634971313900 z + 259918635 z )) And in Maple-input format, it is: -(1+69316831737212016000*z^28-10514963503259030632*z^26-293*z^2+ 1371164263191169592*z^24-152748847041417068*z^22+38155*z^4-2973718*z^6-\ 6046825202*z^102+157201864*z^8-6046825202*z^10+177089670751*z^12-4076056151513* z^14-1144968493989044*z^18+75482135591029*z^16-9105643967154318842612378*z^50+ 5914581520629823961691714*z^48+14427979165794480*z^20+31623723544274988692504*z ^36-8405631787223069198692*z^34-3393065098283562189101084*z^66+ 1952031557465296177536*z^80+177089670751*z^100-152748847041417068*z^90+ 1371164263191169592*z^88+69316831737212016000*z^84-1144968493989044*z^94-\ 10514963503259030632*z^86+75482135591029*z^96-4076056151513*z^98+ 14427979165794480*z^92-394853992186548913052*z^82+5914581520629823961691714*z^ 64+z^112-293*z^110-2973718*z^106+38155*z^108-394853992186548913052*z^30-\ 766854427830325675583556*z^42+1717880597054808068416720*z^44-\ 3393065098283562189101084*z^46-14898306203077664530538360*z^58+ 15843125272142175818250352*z^56-14898306203077664530538360*z^54+ 12387759588709857173863166*z^52+12387759588709857173863166*z^60-\ 766854427830325675583556*z^70+1717880597054808068416720*z^68-\ 8405631787223069198692*z^78+1952031557465296177536*z^32-\ 104213860762162512013056*z^38+301463911681891198679824*z^40-\ 9105643967154318842612378*z^62+31623723544274988692504*z^76-\ 104213860762162512013056*z^74+301463911681891198679824*z^72+157201864*z^104)/(-\ 1+z^2)/(1+203846173380909606168*z^28-29792951669070763916*z^26-344*z^2+ 3730752455686092780*z^24-397667352976720580*z^22+51134*z^4-4459284*z^6-\ 10898839200*z^102+259918635*z^8-10898839200*z^10+344780156110*z^12-\ 8507089078172*z^14-2695592839418956*z^18+167789260469249*z^16-\ 33734690068409994742610300*z^50+21725127233276086543951910*z^48+ 35798271416455328*z^20+104819001056727284103416*z^36-27150532744245927013692*z^ 34-12326643845006494006821444*z^66+6128152593286170237316*z^80+344780156110*z^ 100-397667352976720580*z^90+3730752455686092780*z^88+203846173380909606168*z^84 -2695592839418956*z^94-29792951669070763916*z^86+167789260469249*z^96-\ 8507089078172*z^98+35798271416455328*z^92-1201486998048055485780*z^82+ 21725127233276086543951910*z^64+z^112-344*z^110-4459284*z^106+51134*z^108-\ 1201486998048055485780*z^30-2705379322719279070344108*z^42+ 6157502231257849426541216*z^44-12326643845006494006821444*z^46-\ 55740478076617390671558332*z^58+59348363847386498536070946*z^56-\ 55740478076617390671558332*z^54+46176938516131270014556516*z^52+ 46176938516131270014556516*z^60-2705379322719279070344108*z^70+ 6157502231257849426541216*z^68-27150532744245927013692*z^78+ 6128152593286170237316*z^32-353558517195694259576324*z^38+ 1044227372113634971313900*z^40-33734690068409994742610300*z^62+ 104819001056727284103416*z^76-353558517195694259576324*z^74+ 1044227372113634971313900*z^72+259918635*z^104) The first , 40, terms are: [0, 52, 0, 4617, 0, 452709, 0, 45876360, 0, 4711671049, 0, 486843305025, 0, 50449324809956, 0, 5235189890356529, 0, 543639381810770029, 0, 56472733606140420128, 0, 5867336189908794388129, 0, 609649307111263336872625, 0, 63348682868676471404452616, 0, 6582702897590782976168502917, 0, 684030521930410014671634435017, 0, 71080255366480374474764493076300, 0, 7386243315042751325920247788612593, 0, 767536087892348490924346809758371001, 0, 79758004444805819250137152329573285024, 0, 8288002899609343217527612002790237727757] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 701449576634446608 z - 169651852690269216 z - 240 z 24 22 4 6 + 34374272846247397 z - 5806871179956400 z + 24946 z - 1521868 z 8 10 12 14 + 61889359 z - 1800748556 z + 39219280412 z - 659636610812 z 18 16 50 - 93554172470748 z + 8764013522539 z - 66015655217477744120 z 48 20 + 99493796508708181430 z + 812869559742378 z 36 34 + 37147914080814338276 z - 17713895169598955968 z 66 80 88 84 86 - 5806871179956400 z + 61889359 z + z + 24946 z - 240 z 82 64 30 - 1521868 z + 34374272846247397 z - 2438708291034090560 z 42 44 - 127234118739222454936 z + 138099776290577752488 z 46 58 - 127234118739222454936 z - 2438708291034090560 z 56 54 + 7149686100869515946 z - 17713895169598955968 z 52 60 70 + 37147914080814338276 z + 701449576634446608 z - 93554172470748 z 68 78 32 + 812869559742378 z - 1800748556 z + 7149686100869515946 z 38 40 - 66015655217477744120 z + 99493796508708181430 z 62 76 74 - 169651852690269216 z + 39219280412 z - 659636610812 z 72 / 28 + 8764013522539 z ) / (-1 - 2809496792881434960 z / 26 2 24 + 631759188200778181 z + 293 z - 118987083405086521 z 22 4 6 8 + 18673454935843850 z - 35410 z + 2446874 z - 110920315 z 10 12 14 + 3559679623 z - 84881916716 z + 1554753754860 z 18 16 50 + 258703081225447 z - 22407264732835 z + 624542157837915955278 z 48 20 - 864205694670311504006 z - 2425877550248178 z 36 34 - 199971468464495320932 z + 88459104748079350770 z 66 80 90 88 84 + 118987083405086521 z - 3559679623 z + z - 293 z - 2446874 z 86 82 64 + 35410 z + 110920315 z - 631759188200778181 z 30 42 + 10507345149833330128 z + 864205694670311504006 z 44 46 - 1016494824821567102664 z + 1016494824821567102664 z 58 56 + 33152349266196331658 z - 88459104748079350770 z 54 52 + 199971468464495320932 z - 383504783358472151860 z 60 70 68 - 10507345149833330128 z + 2425877550248178 z - 18673454935843850 z 78 32 38 + 84881916716 z - 33152349266196331658 z + 383504783358472151860 z 40 62 76 - 624542157837915955278 z + 2809496792881434960 z - 1554753754860 z 74 72 + 22407264732835 z - 258703081225447 z ) And in Maple-input format, it is: -(1+701449576634446608*z^28-169651852690269216*z^26-240*z^2+34374272846247397*z ^24-5806871179956400*z^22+24946*z^4-1521868*z^6+61889359*z^8-1800748556*z^10+ 39219280412*z^12-659636610812*z^14-93554172470748*z^18+8764013522539*z^16-\ 66015655217477744120*z^50+99493796508708181430*z^48+812869559742378*z^20+ 37147914080814338276*z^36-17713895169598955968*z^34-5806871179956400*z^66+ 61889359*z^80+z^88+24946*z^84-240*z^86-1521868*z^82+34374272846247397*z^64-\ 2438708291034090560*z^30-127234118739222454936*z^42+138099776290577752488*z^44-\ 127234118739222454936*z^46-2438708291034090560*z^58+7149686100869515946*z^56-\ 17713895169598955968*z^54+37147914080814338276*z^52+701449576634446608*z^60-\ 93554172470748*z^70+812869559742378*z^68-1800748556*z^78+7149686100869515946*z^ 32-66015655217477744120*z^38+99493796508708181430*z^40-169651852690269216*z^62+ 39219280412*z^76-659636610812*z^74+8764013522539*z^72)/(-1-2809496792881434960* z^28+631759188200778181*z^26+293*z^2-118987083405086521*z^24+18673454935843850* z^22-35410*z^4+2446874*z^6-110920315*z^8+3559679623*z^10-84881916716*z^12+ 1554753754860*z^14+258703081225447*z^18-22407264732835*z^16+ 624542157837915955278*z^50-864205694670311504006*z^48-2425877550248178*z^20-\ 199971468464495320932*z^36+88459104748079350770*z^34+118987083405086521*z^66-\ 3559679623*z^80+z^90-293*z^88-2446874*z^84+35410*z^86+110920315*z^82-\ 631759188200778181*z^64+10507345149833330128*z^30+864205694670311504006*z^42-\ 1016494824821567102664*z^44+1016494824821567102664*z^46+33152349266196331658*z^ 58-88459104748079350770*z^56+199971468464495320932*z^54-383504783358472151860*z ^52-10507345149833330128*z^60+2425877550248178*z^70-18673454935843850*z^68+ 84881916716*z^78-33152349266196331658*z^32+383504783358472151860*z^38-\ 624542157837915955278*z^40+2809496792881434960*z^62-1554753754860*z^76+ 22407264732835*z^74-258703081225447*z^72) The first , 40, terms are: [0, 53, 0, 5065, 0, 532321, 0, 57271769, 0, 6204712889, 0, 673698938981, 0, 73202648069529, 0, 7955968448752265, 0, 864757810770712709, 0, 93995639595129757337, 0, 10217034610295006004041, 0, 1110563365277216471431473, 0, 120715285121516008437365865, 0, 13121435029716613757273244693, 0, 1426265752511806892909105943505, 0, 155031371977087487140222446187185, 0, 16851506505345611325730649817670741, 0, 1831714891671312104577665274708728873, 0, 199102640951835973538710361750565763345, 0, 21641938837480963129259138968342278503913] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 620842455761240 z - 349365365223344 z - 208 z 24 22 4 6 + 155759284183916 z - 54840150685936 z + 17544 z - 816848 z 8 10 12 14 + 23951764 z - 476482192 z + 6743808728 z - 70141486416 z 18 16 50 - 3279825575088 z + 548549921956 z - 70141486416 z 48 20 36 + 548549921956 z + 15176754760008 z + 620842455761240 z 34 64 30 42 - 875924187305776 z + z - 875924187305776 z - 54840150685936 z 44 46 58 56 + 15176754760008 z - 3279825575088 z - 816848 z + 23951764 z 54 52 60 32 - 476482192 z + 6743808728 z + 17544 z + 982292679631030 z 38 40 62 / 2 - 349365365223344 z + 155759284183916 z - 208 z ) / ((-1 + z ) (1 / 28 26 2 + 2340625202858024 z - 1287079541876308 z - 264 z 24 22 4 6 + 555868604950860 z - 188003024080764 z + 26712 z - 1445404 z 8 10 12 14 + 48137780 z - 1068754708 z + 16647586744 z - 188361102424 z 18 16 50 - 10123392775464 z + 1586468122564 z - 188361102424 z 48 20 36 + 1586468122564 z + 49571505580744 z + 2340625202858024 z 34 64 30 42 - 3348959879762040 z + z - 3348959879762040 z - 188003024080764 z 44 46 58 56 + 49571505580744 z - 10123392775464 z - 1445404 z + 48137780 z 54 52 60 32 - 1068754708 z + 16647586744 z + 26712 z + 3773373845290870 z 38 40 62 - 1287079541876308 z + 555868604950860 z - 264 z )) And in Maple-input format, it is: -(1+620842455761240*z^28-349365365223344*z^26-208*z^2+155759284183916*z^24-\ 54840150685936*z^22+17544*z^4-816848*z^6+23951764*z^8-476482192*z^10+6743808728 *z^12-70141486416*z^14-3279825575088*z^18+548549921956*z^16-70141486416*z^50+ 548549921956*z^48+15176754760008*z^20+620842455761240*z^36-875924187305776*z^34 +z^64-875924187305776*z^30-54840150685936*z^42+15176754760008*z^44-\ 3279825575088*z^46-816848*z^58+23951764*z^56-476482192*z^54+6743808728*z^52+ 17544*z^60+982292679631030*z^32-349365365223344*z^38+155759284183916*z^40-208*z ^62)/(-1+z^2)/(1+2340625202858024*z^28-1287079541876308*z^26-264*z^2+ 555868604950860*z^24-188003024080764*z^22+26712*z^4-1445404*z^6+48137780*z^8-\ 1068754708*z^10+16647586744*z^12-188361102424*z^14-10123392775464*z^18+ 1586468122564*z^16-188361102424*z^50+1586468122564*z^48+49571505580744*z^20+ 2340625202858024*z^36-3348959879762040*z^34+z^64-3348959879762040*z^30-\ 188003024080764*z^42+49571505580744*z^44-10123392775464*z^46-1445404*z^58+ 48137780*z^56-1068754708*z^54+16647586744*z^52+26712*z^60+3773373845290870*z^32 -1287079541876308*z^38+555868604950860*z^40-264*z^62) The first , 40, terms are: [0, 57, 0, 5673, 0, 620981, 0, 69804309, 0, 7912041305, 0, 899210908297, 0, 102286656243777, 0, 11638696568005633, 0, 1324440993310570793, 0, 150721551294625817273, 0, 17152325245244441883157, 0, 1951966329009849206559285, 0, 222137669711655437856959049, 0, 25279722208982959852314192921, 0, 2876884658561292930914768184001, 0, 327395440218817406607127537976897, 0, 37258280736826989360049831779154265, 0, 4240069719393055920242503747133477129, 0, 482528740178351098617578262316039309365, 0, 54912772828295201956635936505054315077397] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 4}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 739149493518767 z + 454719263615305 z + 219 z 24 22 4 6 - 218593217996425 z + 81738615397484 z - 19953 z + 1011913 z 8 10 12 14 - 32218079 z + 688219915 z - 10290062879 z + 110905385209 z 18 16 50 + 5235981530043 z - 880523606143 z + 10290062879 z 48 20 36 - 110905385209 z - 23626245410784 z - 454719263615305 z 34 30 42 + 739149493518767 z + 941682336217347 z + 23626245410784 z 44 46 58 56 - 5235981530043 z + 880523606143 z + 19953 z - 1011913 z 54 52 60 32 + 32218079 z - 688219915 z - 219 z - 941682336217347 z 38 40 62 / + 218593217996425 z - 81738615397484 z + z ) / (1 / 28 26 2 + 4640841353479004 z - 2539325424850064 z - 279 z 24 22 4 6 + 1086761722572314 z - 362100252977018 z + 30413 z - 1784216 z 8 10 12 14 + 64518708 z - 1550274509 z + 25954727979 z - 312736128421 z 18 16 50 - 18469697591824 z + 2775941105348 z - 312736128421 z 48 20 36 + 2775941105348 z + 93347652938978 z + 4640841353479004 z 34 64 30 42 - 6655365769228385 z + z - 6655365769228385 z - 362100252977018 z 44 46 58 56 + 93347652938978 z - 18469697591824 z - 1784216 z + 64518708 z 54 52 60 32 - 1550274509 z + 25954727979 z + 30413 z + 7503645502820111 z 38 40 62 - 2539325424850064 z + 1086761722572314 z - 279 z ) And in Maple-input format, it is: -(-1-739149493518767*z^28+454719263615305*z^26+219*z^2-218593217996425*z^24+ 81738615397484*z^22-19953*z^4+1011913*z^6-32218079*z^8+688219915*z^10-\ 10290062879*z^12+110905385209*z^14+5235981530043*z^18-880523606143*z^16+ 10290062879*z^50-110905385209*z^48-23626245410784*z^20-454719263615305*z^36+ 739149493518767*z^34+941682336217347*z^30+23626245410784*z^42-5235981530043*z^ 44+880523606143*z^46+19953*z^58-1011913*z^56+32218079*z^54-688219915*z^52-219*z ^60-941682336217347*z^32+218593217996425*z^38-81738615397484*z^40+z^62)/(1+ 4640841353479004*z^28-2539325424850064*z^26-279*z^2+1086761722572314*z^24-\ 362100252977018*z^22+30413*z^4-1784216*z^6+64518708*z^8-1550274509*z^10+ 25954727979*z^12-312736128421*z^14-18469697591824*z^18+2775941105348*z^16-\ 312736128421*z^50+2775941105348*z^48+93347652938978*z^20+4640841353479004*z^36-\ 6655365769228385*z^34+z^64-6655365769228385*z^30-362100252977018*z^42+ 93347652938978*z^44-18469697591824*z^46-1784216*z^58+64518708*z^56-1550274509*z ^54+25954727979*z^52+30413*z^60+7503645502820111*z^32-2539325424850064*z^38+ 1086761722572314*z^40-279*z^62) The first , 40, terms are: [0, 60, 0, 6280, 0, 699643, 0, 78959088, 0, 8947151587, 0, 1015361103517, 0, 115296303380568, 0, 13095315842046661, 0, 1487511388986202987, 0, 168975148079456273172, 0, 19195215371584148548327, 0, 2180551439031595065095056, 0, 247708564978799209521326304, 0, 28139493356880891416916558901, 0, 3196625510604080727634203373621, 0, 363134363065017475223598371372424, 0, 41251806917059477033174265672392480, 0, 4686176291750394607905188219194963567, 0, 532346344392215131041298968282449648076, 0, 60474172284048295775896703243505821217427] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 4}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 58722114291 z + 113749232973 z + 183 z - 158022339035 z 22 4 6 8 10 + 158022339035 z - 12849 z + 458587 z - 9415377 z + 119929071 z 12 14 18 16 - 995742837 z + 5575675179 z + 58722114291 z - 21576894017 z 20 36 34 30 - 113749232973 z - 119929071 z + 995742837 z + 21576894017 z 42 44 46 32 38 40 + 12849 z - 183 z + z - 5575675179 z + 9415377 z - 458587 z ) / 28 26 2 24 / (1 + 676493791120 z - 1101680923893 z - 255 z + 1295495658754 z / 22 4 6 8 10 - 1101680923893 z + 22148 z - 934775 z + 22370710 z - 330983143 z 12 14 18 16 + 3193909002 z - 20820956429 z - 298575280765 z + 93951162734 z 48 20 36 34 + z + 676493791120 z + 3193909002 z - 20820956429 z 30 42 44 46 32 - 298575280765 z - 934775 z + 22148 z - 255 z + 93951162734 z 38 40 - 330983143 z + 22370710 z ) And in Maple-input format, it is: -(-1-58722114291*z^28+113749232973*z^26+183*z^2-158022339035*z^24+158022339035* z^22-12849*z^4+458587*z^6-9415377*z^8+119929071*z^10-995742837*z^12+5575675179* z^14+58722114291*z^18-21576894017*z^16-113749232973*z^20-119929071*z^36+ 995742837*z^34+21576894017*z^30+12849*z^42-183*z^44+z^46-5575675179*z^32+ 9415377*z^38-458587*z^40)/(1+676493791120*z^28-1101680923893*z^26-255*z^2+ 1295495658754*z^24-1101680923893*z^22+22148*z^4-934775*z^6+22370710*z^8-\ 330983143*z^10+3193909002*z^12-20820956429*z^14-298575280765*z^18+93951162734*z ^16+z^48+676493791120*z^20+3193909002*z^36-20820956429*z^34-298575280765*z^30-\ 934775*z^42+22148*z^44-255*z^46+93951162734*z^32-330983143*z^38+22370710*z^40) The first , 40, terms are: [0, 72, 0, 9061, 0, 1192087, 0, 157647624, 0, 20868160471, 0, 2763066085999, 0, 365875381062696, 0, 48449227350984103, 0, 6415707991867053925, 0, 849578946979224932328, 0, 112502815652808532040041, 0, 14897836932758681360642041, 0, 1972800194241969950475238056, 0, 261242005581648596611988545093, 0, 34594170695377105732103435215399, 0, 4581026894006013247166235483084264, 0, 606628429785076967353268524384772575, 0, 80330908465635167588544366341497636519, 0, 10637574072995587860561103744206575228232, 0, 1408648107314845889406389559732601353992023] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 5}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 62410261082 z + 118641477998 z + 195 z - 163242858394 z 22 4 6 8 10 + 163242858394 z - 14633 z + 549091 z - 11443331 z + 143963561 z 12 14 18 16 - 1161883811 z + 6288599329 z + 62410261082 z - 23556363742 z 20 36 34 30 - 118641477998 z - 143963561 z + 1161883811 z + 23556363742 z 42 44 46 32 38 40 + 14633 z - 195 z + z - 6288599329 z + 11443331 z - 549091 z ) / 28 26 2 24 / (1 + 693028438736 z - 1107999174476 z - 262 z + 1294713424804 z / 22 4 6 8 10 - 1107999174476 z + 24328 z - 1093730 z + 27013486 z - 399597202 z 12 14 18 16 + 3767029800 z - 23688898454 z - 314837628196 z + 102747191999 z 48 20 36 34 + z + 693028438736 z + 3767029800 z - 23688898454 z 30 42 44 46 32 - 314837628196 z - 1093730 z + 24328 z - 262 z + 102747191999 z 38 40 - 399597202 z + 27013486 z ) And in Maple-input format, it is: -(-1-62410261082*z^28+118641477998*z^26+195*z^2-163242858394*z^24+163242858394* z^22-14633*z^4+549091*z^6-11443331*z^8+143963561*z^10-1161883811*z^12+ 6288599329*z^14+62410261082*z^18-23556363742*z^16-118641477998*z^20-143963561*z ^36+1161883811*z^34+23556363742*z^30+14633*z^42-195*z^44+z^46-6288599329*z^32+ 11443331*z^38-549091*z^40)/(1+693028438736*z^28-1107999174476*z^26-262*z^2+ 1294713424804*z^24-1107999174476*z^22+24328*z^4-1093730*z^6+27013486*z^8-\ 399597202*z^10+3767029800*z^12-23688898454*z^14-314837628196*z^18+102747191999* z^16+z^48+693028438736*z^20+3767029800*z^36-23688898454*z^34-314837628196*z^30-\ 1093730*z^42+24328*z^44-262*z^46+102747191999*z^32-399597202*z^38+27013486*z^40 ) The first , 40, terms are: [0, 67, 0, 7859, 0, 973721, 0, 121630905, 0, 15219966771, 0, 1905451386563, 0, 238590126511905, 0, 29876650386174049, 0, 3741281661841947459, 0, 468502887822945978547, 0, 58668556783039388655353, 0, 7346813540299591671708249, 0, 920010563491317143554544179, 0, 115209072553195578833537841859, 0, 14427150900011174213843550958017, 0, 1806651885329482532713588452576321, 0, 226239476394514582737757646181469123, 0, 28331025629468572597325456626752709939, 0, 3547776131540144746540403471250424692441, 0, 444273202501339447206859516955258350564473] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 80555870865722461228 z - 11997405485121266120 z - 290 z 24 22 4 6 + 1536031740194290388 z - 168038023448071768 z + 37652 z - 2940658 z 102 8 10 12 - 6070692318 z + 156389634 z - 6070692318 z + 179864939760 z 14 18 16 - 4196526691694 z - 1216384690825224 z + 78895919632115 z 50 48 - 12345845727636444938791648 z + 7961679585022185869829835 z 20 36 + 15592864099354508 z + 39427941221155612194728 z 34 66 - 10305864273400118247866 z - 4526258301910702956198326 z 80 100 90 + 2351994876769045048129 z + 179864939760 z - 168038023448071768 z 88 84 + 1536031740194290388 z + 80555870865722461228 z 94 86 96 - 1216384690825224 z - 11997405485121266120 z + 78895919632115 z 98 92 82 - 4196526691694 z + 15592864099354508 z - 467314788015735952664 z 64 112 110 106 + 7961679585022185869829835 z + z - 290 z - 2940658 z 108 30 42 + 37652 z - 467314788015735952664 z - 999619095048704218035014 z 44 46 + 2267065277047647417352916 z - 4526258301910702956198326 z 58 56 - 20370513466833050958851808 z + 21685470212732610239794072 z 54 52 - 20370513466833050958851808 z + 16884074495684721385649864 z 60 70 + 16884074495684721385649864 z - 999619095048704218035014 z 68 78 + 2267065277047647417352916 z - 10305864273400118247866 z 32 38 + 2351994876769045048129 z - 132016765982740366607594 z 40 62 + 387622195344067937064542 z - 12345845727636444938791648 z 76 74 + 39427941221155612194728 z - 132016765982740366607594 z 72 104 / 2 + 387622195344067937064542 z + 156389634 z ) / ((-1 + z ) (1 / 28 26 2 + 233164985951147158360 z - 33357024982637380056 z - 348 z 24 22 4 6 + 4092633325947719636 z - 427989335172155336 z + 51708 z - 4495528 z 102 8 10 12 - 10972431232 z + 261508402 z - 10972431232 z + 348437073980 z 14 18 16 - 8658386840244 z - 2807608945105216 z + 172516380376475 z 50 48 - 47242300961709656934261008 z + 30119284608794355642427547 z 20 36 + 37862866382756328 z + 130962554615238042120404 z 34 66 - 33190575826500294680652 z - 16875567615207618157893988 z 80 100 90 + 7326896700182954700233 z + 348437073980 z - 427989335172155336 z 88 84 + 4092633325947719636 z + 233164985951147158360 z 94 86 96 - 2807608945105216 z - 33357024982637380056 z + 172516380376475 z 98 92 82 - 8658386840244 z + 37862866382756328 z - 1404866720628746937840 z 64 112 110 106 + 30119284608794355642427547 z + z - 348 z - 4495528 z 108 30 42 + 51708 z - 1404866720628746937840 z - 3587954292504865919910584 z 44 46 + 8305337282792366438075956 z - 16875567615207618157893988 z 58 56 - 78986631013834787881197472 z + 84225363958728733593213848 z 54 52 - 78986631013834787881197472 z + 65143034046371677922205312 z 60 70 + 65143034046371677922205312 z - 3587954292504865919910584 z 68 78 + 8305337282792366438075956 z - 33190575826500294680652 z 32 38 + 7326896700182954700233 z - 451147015870886148652000 z 40 62 + 1359365465217272999738926 z - 47242300961709656934261008 z 76 74 + 130962554615238042120404 z - 451147015870886148652000 z 72 104 + 1359365465217272999738926 z + 261508402 z )) And in Maple-input format, it is: -(1+80555870865722461228*z^28-11997405485121266120*z^26-290*z^2+ 1536031740194290388*z^24-168038023448071768*z^22+37652*z^4-2940658*z^6-\ 6070692318*z^102+156389634*z^8-6070692318*z^10+179864939760*z^12-4196526691694* z^14-1216384690825224*z^18+78895919632115*z^16-12345845727636444938791648*z^50+ 7961679585022185869829835*z^48+15592864099354508*z^20+39427941221155612194728*z ^36-10305864273400118247866*z^34-4526258301910702956198326*z^66+ 2351994876769045048129*z^80+179864939760*z^100-168038023448071768*z^90+ 1536031740194290388*z^88+80555870865722461228*z^84-1216384690825224*z^94-\ 11997405485121266120*z^86+78895919632115*z^96-4196526691694*z^98+ 15592864099354508*z^92-467314788015735952664*z^82+7961679585022185869829835*z^ 64+z^112-290*z^110-2940658*z^106+37652*z^108-467314788015735952664*z^30-\ 999619095048704218035014*z^42+2267065277047647417352916*z^44-\ 4526258301910702956198326*z^46-20370513466833050958851808*z^58+ 21685470212732610239794072*z^56-20370513466833050958851808*z^54+ 16884074495684721385649864*z^52+16884074495684721385649864*z^60-\ 999619095048704218035014*z^70+2267065277047647417352916*z^68-\ 10305864273400118247866*z^78+2351994876769045048129*z^32-\ 132016765982740366607594*z^38+387622195344067937064542*z^40-\ 12345845727636444938791648*z^62+39427941221155612194728*z^76-\ 132016765982740366607594*z^74+387622195344067937064542*z^72+156389634*z^104)/(-\ 1+z^2)/(1+233164985951147158360*z^28-33357024982637380056*z^26-348*z^2+ 4092633325947719636*z^24-427989335172155336*z^22+51708*z^4-4495528*z^6-\ 10972431232*z^102+261508402*z^8-10972431232*z^10+348437073980*z^12-\ 8658386840244*z^14-2807608945105216*z^18+172516380376475*z^16-\ 47242300961709656934261008*z^50+30119284608794355642427547*z^48+ 37862866382756328*z^20+130962554615238042120404*z^36-33190575826500294680652*z^ 34-16875567615207618157893988*z^66+7326896700182954700233*z^80+348437073980*z^ 100-427989335172155336*z^90+4092633325947719636*z^88+233164985951147158360*z^84 -2807608945105216*z^94-33357024982637380056*z^86+172516380376475*z^96-\ 8658386840244*z^98+37862866382756328*z^92-1404866720628746937840*z^82+ 30119284608794355642427547*z^64+z^112-348*z^110-4495528*z^106+51708*z^108-\ 1404866720628746937840*z^30-3587954292504865919910584*z^42+ 8305337282792366438075956*z^44-16875567615207618157893988*z^46-\ 78986631013834787881197472*z^58+84225363958728733593213848*z^56-\ 78986631013834787881197472*z^54+65143034046371677922205312*z^52+ 65143034046371677922205312*z^60-3587954292504865919910584*z^70+ 8305337282792366438075956*z^68-33190575826500294680652*z^78+ 7326896700182954700233*z^32-451147015870886148652000*z^38+ 1359365465217272999738926*z^40-47242300961709656934261008*z^62+ 130962554615238042120404*z^76-451147015870886148652000*z^74+ 1359365465217272999738926*z^72+261508402*z^104) The first , 40, terms are: [0, 59, 0, 6187, 0, 694537, 0, 78995569, 0, 9017400087, 0, 1030594498275, 0, 117839145566557, 0, 13476192982135781, 0, 1541257234490601203, 0, 176276854934992178911, 0, 20161389785836079217153, 0, 2305938361812308949884105, 0, 263739858567732716037923843, 0, 30165059939049643410075747403, 0, 3450108557545523644292091604873, 0, 394603913072584511614817921327865, 0, 45132567405253647214870518437980523, 0, 5162008334873519517058498459541236883, 0, 590401385052074540002437890100423253385, 0, 67526779162680170709648892941048275570209] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 5}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6394479267429106 z - 2619529123107025 z - 219 z 24 22 4 6 + 875058880457428 z - 237357096629528 z + 19870 z - 1006733 z 8 10 12 14 + 32453184 z - 718818168 z + 11506409286 z - 137862268582 z 18 16 50 - 9117158769576 z + 1267862914747 z - 237357096629528 z 48 20 36 + 875058880457428 z + 51975939176712 z + 30963195323139718 z 34 66 64 - 28065172876838038 z - 1006733 z + 32453184 z 30 42 44 - 12767440313026921 z - 12767440313026921 z + 6394479267429106 z 46 58 56 - 2619529123107025 z - 137862268582 z + 1267862914747 z 54 52 60 70 - 9117158769576 z + 51975939176712 z + 11506409286 z - 219 z 68 32 38 + 19870 z + 20894364008816325 z - 28065172876838038 z 40 62 72 / 2 + 20894364008816325 z - 718818168 z + z ) / ((-1 + z ) (1 / 28 26 2 + 23566766469197453 z - 9307585096219866 z - 278 z 24 22 4 6 + 2977031060605927 z - 768258798418634 z + 29976 z - 1736472 z 8 10 12 14 + 62540121 z - 1525309236 z + 26619533578 z - 345123349418 z 18 16 50 - 26244760130412 z + 3413192107261 z - 768258798418634 z 48 20 36 + 2977031060605927 z + 159102184269157 z + 122113384620357891 z 34 66 64 - 110203489299997110 z - 1736472 z + 62540121 z 30 42 44 - 48447007421426560 z - 48447007421426560 z + 23566766469197453 z 46 58 56 - 9307585096219866 z - 345123349418 z + 3413192107261 z 54 52 60 70 - 26244760130412 z + 159102184269157 z + 26619533578 z - 278 z 68 32 38 + 29976 z + 80989900554245821 z - 110203489299997110 z 40 62 72 + 80989900554245821 z - 1525309236 z + z )) And in Maple-input format, it is: -(1+6394479267429106*z^28-2619529123107025*z^26-219*z^2+875058880457428*z^24-\ 237357096629528*z^22+19870*z^4-1006733*z^6+32453184*z^8-718818168*z^10+ 11506409286*z^12-137862268582*z^14-9117158769576*z^18+1267862914747*z^16-\ 237357096629528*z^50+875058880457428*z^48+51975939176712*z^20+30963195323139718 *z^36-28065172876838038*z^34-1006733*z^66+32453184*z^64-12767440313026921*z^30-\ 12767440313026921*z^42+6394479267429106*z^44-2619529123107025*z^46-137862268582 *z^58+1267862914747*z^56-9117158769576*z^54+51975939176712*z^52+11506409286*z^ 60-219*z^70+19870*z^68+20894364008816325*z^32-28065172876838038*z^38+ 20894364008816325*z^40-718818168*z^62+z^72)/(-1+z^2)/(1+23566766469197453*z^28-\ 9307585096219866*z^26-278*z^2+2977031060605927*z^24-768258798418634*z^22+29976* z^4-1736472*z^6+62540121*z^8-1525309236*z^10+26619533578*z^12-345123349418*z^14 -26244760130412*z^18+3413192107261*z^16-768258798418634*z^50+2977031060605927*z ^48+159102184269157*z^20+122113384620357891*z^36-110203489299997110*z^34-\ 1736472*z^66+62540121*z^64-48447007421426560*z^30-48447007421426560*z^42+ 23566766469197453*z^44-9307585096219866*z^46-345123349418*z^58+3413192107261*z^ 56-26244760130412*z^54+159102184269157*z^52+26619533578*z^60-278*z^70+29976*z^ 68+80989900554245821*z^32-110203489299997110*z^38+80989900554245821*z^40-\ 1525309236*z^62+z^72) The first , 40, terms are: [0, 60, 0, 6356, 0, 717799, 0, 82134968, 0, 9439344223, 0, 1086710827081, 0, 125201436966480, 0, 14429288546082513, 0, 1663189966369400379, 0, 191719239143325531708, 0, 22100462243644947883163, 0, 2547664386332180621343980, 0, 293687432414948266902045672, 0, 33855522096728758669295446497, 0, 3902780370740239721378447500969, 0, 449903015057889424242157983334128, 0, 51863734001652522826015636733948252, 0, 5978726634346407371280256244385586771, 0, 689213266804774996498267844027852979588, 0, 79450853704913204833976636358575049937179] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 88751622704984628472 z - 13264214181783392008 z - 293 z 24 22 4 6 + 1701388007152194456 z - 186138247992728972 z + 38495 z - 3045758 z 102 8 10 12 - 6452752098 z + 164136688 z - 6452752098 z + 193398728843 z 14 18 16 - 4557216672553 z - 1339591321998772 z + 86367440591621 z 50 48 - 12618145903873894984979562 z + 8176850077722933005686882 z 20 36 + 17239849377171680 z + 42338673854018193154864 z 34 66 - 11151865676078378883804 z - 4676354558159243602297452 z 80 100 90 + 2562970745284520737360 z + 193398728843 z - 186138247992728972 z 88 84 + 1701388007152194456 z + 88751622704984628472 z 94 86 96 - 1339591321998772 z - 13264214181783392008 z + 86367440591621 z 98 92 82 - 4557216672553 z + 17239849377171680 z - 512340788316990381460 z 64 112 110 106 + 8176850077722933005686882 z + z - 293 z - 3045758 z 108 30 42 + 38495 z - 512340788316990381460 z - 1047762487003580267224660 z 44 46 + 2358380015634147992331168 z - 4676354558159243602297452 z 58 56 - 20699718716311001697679408 z + 22019592058872963660626560 z 54 52 - 20699718716311001697679408 z + 17194719667261662967667822 z 60 70 + 17194719667261662967667822 z - 1047762487003580267224660 z 68 78 + 2358380015634147992331168 z - 11151865676078378883804 z 32 38 + 2562970745284520737360 z - 140626187698755335785280 z 40 62 + 409547606215830122998400 z - 12618145903873894984979562 z 76 74 + 42338673854018193154864 z - 140626187698755335785280 z 72 104 / + 409547606215830122998400 z + 164136688 z ) / (-1 / 28 26 2 - 298193271879985865060 z + 42019090274984078480 z + 357 z 24 22 4 6 - 5077473988114524144 z + 522769237100176748 z - 53970 z + 4770410 z 102 8 10 12 + 390209154910 z - 282347443 z + 12063082975 z - 390209154910 z 14 18 16 + 9876577118134 z + 3318335448360381 z - 200370276001617 z 50 48 + 79831628744719992727511618 z - 48784224525658234038083610 z 20 36 - 45507829813795508 z - 179067265806573546718364 z 34 66 + 44550595078245334719680 z + 48784224525658234038083610 z 80 100 - 44550595078245334719680 z - 9876577118134 z 90 88 + 5077473988114524144 z - 42019090274984078480 z 84 94 - 1824509121218440010244 z + 45507829813795508 z 86 96 98 + 298193271879985865060 z - 3318335448360381 z + 200370276001617 z 92 82 - 522769237100176748 z + 9668781380640666290200 z 64 112 114 110 - 79831628744719992727511618 z - 357 z + z + 53970 z 106 108 30 + 282347443 z - 4770410 z + 1824509121218440010244 z 42 44 + 5257486408854348012916792 z - 12531974297066162704744452 z 46 58 + 26324328710240346325460124 z + 166840114698078289772544726 z 56 54 - 166840114698078289772544726 z + 147572388031679531666460056 z 52 60 - 115436749257817287902490424 z - 147572388031679531666460056 z 70 68 + 12531974297066162704744452 z - 26324328710240346325460124 z 78 32 + 179067265806573546718364 z - 9668781380640666290200 z 38 40 + 629606086448545094435356 z - 1941043693984932012214584 z 62 76 + 115436749257817287902490424 z - 629606086448545094435356 z 74 72 + 1941043693984932012214584 z - 5257486408854348012916792 z 104 - 12063082975 z ) And in Maple-input format, it is: -(1+88751622704984628472*z^28-13264214181783392008*z^26-293*z^2+ 1701388007152194456*z^24-186138247992728972*z^22+38495*z^4-3045758*z^6-\ 6452752098*z^102+164136688*z^8-6452752098*z^10+193398728843*z^12-4557216672553* z^14-1339591321998772*z^18+86367440591621*z^16-12618145903873894984979562*z^50+ 8176850077722933005686882*z^48+17239849377171680*z^20+42338673854018193154864*z ^36-11151865676078378883804*z^34-4676354558159243602297452*z^66+ 2562970745284520737360*z^80+193398728843*z^100-186138247992728972*z^90+ 1701388007152194456*z^88+88751622704984628472*z^84-1339591321998772*z^94-\ 13264214181783392008*z^86+86367440591621*z^96-4557216672553*z^98+ 17239849377171680*z^92-512340788316990381460*z^82+8176850077722933005686882*z^ 64+z^112-293*z^110-3045758*z^106+38495*z^108-512340788316990381460*z^30-\ 1047762487003580267224660*z^42+2358380015634147992331168*z^44-\ 4676354558159243602297452*z^46-20699718716311001697679408*z^58+ 22019592058872963660626560*z^56-20699718716311001697679408*z^54+ 17194719667261662967667822*z^52+17194719667261662967667822*z^60-\ 1047762487003580267224660*z^70+2358380015634147992331168*z^68-\ 11151865676078378883804*z^78+2562970745284520737360*z^32-\ 140626187698755335785280*z^38+409547606215830122998400*z^40-\ 12618145903873894984979562*z^62+42338673854018193154864*z^76-\ 140626187698755335785280*z^74+409547606215830122998400*z^72+164136688*z^104)/(-\ 1-298193271879985865060*z^28+42019090274984078480*z^26+357*z^2-\ 5077473988114524144*z^24+522769237100176748*z^22-53970*z^4+4770410*z^6+ 390209154910*z^102-282347443*z^8+12063082975*z^10-390209154910*z^12+ 9876577118134*z^14+3318335448360381*z^18-200370276001617*z^16+ 79831628744719992727511618*z^50-48784224525658234038083610*z^48-\ 45507829813795508*z^20-179067265806573546718364*z^36+44550595078245334719680*z^ 34+48784224525658234038083610*z^66-44550595078245334719680*z^80-9876577118134*z ^100+5077473988114524144*z^90-42019090274984078480*z^88-1824509121218440010244* z^84+45507829813795508*z^94+298193271879985865060*z^86-3318335448360381*z^96+ 200370276001617*z^98-522769237100176748*z^92+9668781380640666290200*z^82-\ 79831628744719992727511618*z^64-357*z^112+z^114+53970*z^110+282347443*z^106-\ 4770410*z^108+1824509121218440010244*z^30+5257486408854348012916792*z^42-\ 12531974297066162704744452*z^44+26324328710240346325460124*z^46+ 166840114698078289772544726*z^58-166840114698078289772544726*z^56+ 147572388031679531666460056*z^54-115436749257817287902490424*z^52-\ 147572388031679531666460056*z^60+12531974297066162704744452*z^70-\ 26324328710240346325460124*z^68+179067265806573546718364*z^78-\ 9668781380640666290200*z^32+629606086448545094435356*z^38-\ 1941043693984932012214584*z^40+115436749257817287902490424*z^62-\ 629606086448545094435356*z^76+1941043693984932012214584*z^74-\ 5257486408854348012916792*z^72-12063082975*z^104) The first , 40, terms are: [0, 64, 0, 7373, 0, 902733, 0, 111450356, 0, 13779604537, 0, 1704228824013, 0, 210791057121008, 0, 26072647085798533, 0, 3224931035132161133, 0, 398892983573516711244, 0, 49339250832575584904909, 0, 6102794680507442791443493, 0, 754857515231670597228413732, 0, 93368678514762283336616944861, 0, 11548815479165984732078452873013, 0, 1428478385127916417568899404185480, 0, 176689159255360811318573644483724997, 0, 21854764710743057076234843040705308257, 0, 2703226064287568444638868773416674027900, 0, 334363295667031112424547140992374167060477] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 7}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2748136956922612 z - 1300139815538542 z - 213 z 24 22 4 6 + 493945345056062 z - 150017925990112 z + 18886 z - 940877 z 8 10 12 14 + 29893181 z - 650415896 z + 10145141053 z - 117094912261 z 18 16 50 - 6889820058653 z + 1023357947590 z - 6889820058653 z 48 20 36 + 36206166593569 z + 36206166593569 z + 6435868087804326 z 34 66 64 30 - 7155886274233104 z - 213 z + 18886 z - 4680081647615438 z 42 44 46 - 1300139815538542 z + 493945345056062 z - 150017925990112 z 58 56 54 52 - 650415896 z + 10145141053 z - 117094912261 z + 1023357947590 z 60 68 32 38 + 29893181 z + z + 6435868087804326 z - 4680081647615438 z 40 62 / 2 + 2748136956922612 z - 940877 z ) / ((-1 + z ) (1 / 28 26 2 + 10316187595804974 z - 4730134486832844 z - 274 z 24 22 4 6 + 1727197065926826 z - 500164141136360 z + 28715 z - 1631100 z 8 10 12 14 + 57993161 z - 1395483906 z + 23859454189 z - 299560199988 z 18 16 50 - 20413776178386 z + 2827460040279 z - 20413776178386 z 48 20 36 + 114216366456677 z + 114216366456677 z + 25053741247721050 z 34 66 64 30 - 27984773386072012 z - 274 z + 28715 z - 17970793662041344 z 42 44 46 - 4730134486832844 z + 1727197065926826 z - 500164141136360 z 58 56 54 52 - 1395483906 z + 23859454189 z - 299560199988 z + 2827460040279 z 60 68 32 38 + 57993161 z + z + 25053741247721050 z - 17970793662041344 z 40 62 + 10316187595804974 z - 1631100 z )) And in Maple-input format, it is: -(1+2748136956922612*z^28-1300139815538542*z^26-213*z^2+493945345056062*z^24-\ 150017925990112*z^22+18886*z^4-940877*z^6+29893181*z^8-650415896*z^10+ 10145141053*z^12-117094912261*z^14-6889820058653*z^18+1023357947590*z^16-\ 6889820058653*z^50+36206166593569*z^48+36206166593569*z^20+6435868087804326*z^ 36-7155886274233104*z^34-213*z^66+18886*z^64-4680081647615438*z^30-\ 1300139815538542*z^42+493945345056062*z^44-150017925990112*z^46-650415896*z^58+ 10145141053*z^56-117094912261*z^54+1023357947590*z^52+29893181*z^60+z^68+ 6435868087804326*z^32-4680081647615438*z^38+2748136956922612*z^40-940877*z^62)/ (-1+z^2)/(1+10316187595804974*z^28-4730134486832844*z^26-274*z^2+ 1727197065926826*z^24-500164141136360*z^22+28715*z^4-1631100*z^6+57993161*z^8-\ 1395483906*z^10+23859454189*z^12-299560199988*z^14-20413776178386*z^18+ 2827460040279*z^16-20413776178386*z^50+114216366456677*z^48+114216366456677*z^ 20+25053741247721050*z^36-27984773386072012*z^34-274*z^66+28715*z^64-\ 17970793662041344*z^30-4730134486832844*z^42+1727197065926826*z^44-\ 500164141136360*z^46-1395483906*z^58+23859454189*z^56-299560199988*z^54+ 2827460040279*z^52+57993161*z^60+z^68+25053741247721050*z^32-17970793662041344* z^38+10316187595804974*z^40-1631100*z^62) The first , 40, terms are: [0, 62, 0, 6947, 0, 832045, 0, 100603242, 0, 12182830839, 0, 1475727910007, 0, 178767617701114, 0, 21655930522726397, 0, 2623409992141477171, 0, 317801420634904001550, 0, 38498658490837866004289, 0, 4663751337654923871565569, 0, 564969736747006197515669998, 0, 68440785573165628366846094291, 0, 8290959379532964635918834106781, 0, 1004371982204835512090836478535770, 0, 121670247389475021628879752226906423, 0, 14739209538750075674745453040413104567, 0, 1785517022375837894362212818737006172426, 0, 216298644023463885118501754191219743796493] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1229920292464016096 z - 300937169540168672 z - 272 z 24 22 4 6 + 61545031225939461 z - 10458832872254000 z + 31760 z - 2137356 z 8 10 12 14 + 93985499 z - 2903469272 z + 66100836296 z - 1147092349064 z 18 16 50 - 168000976140588 z + 15556107755447 z - 108975523428400907976 z 48 20 + 163035476851708682590 z + 1466144600580400 z 36 34 + 61922546057601507008 z - 29872203850477864128 z 66 80 88 84 86 - 10458832872254000 z + 93985499 z + z + 31760 z - 272 z 82 64 30 - 2137356 z + 61545031225939461 z - 4221247821070827392 z 42 44 - 207539834411565687360 z + 224913829973896313520 z 46 58 - 207539834411565687360 z - 4221247821070827392 z 56 54 + 12212489678184453642 z - 29872203850477864128 z 52 60 70 + 61922546057601507008 z + 1229920292464016096 z - 168000976140588 z 68 78 32 + 1466144600580400 z - 2903469272 z + 12212489678184453642 z 38 40 - 108975523428400907976 z + 163035476851708682590 z 62 76 74 - 300937169540168672 z + 66100836296 z - 1147092349064 z 72 / 28 + 15556107755447 z ) / (-1 - 5094613159810790168 z / 26 2 24 + 1159426954723733005 z + 337 z - 220542762530049149 z 22 4 6 8 + 34847071304739170 z - 46338 z + 3560318 z - 175248395 z 10 12 14 + 5981427167 z - 149106118028 z + 2815592729204 z 18 16 50 + 482503516728911 z - 41366258644691 z + 1055456377079089746518 z 48 20 - 1452149689931617076638 z - 4538418322365774 z 36 34 - 344302416498469939644 z + 154150226817694308482 z 66 80 90 88 84 + 220542762530049149 z - 5981427167 z + z - 337 z - 3560318 z 86 82 64 + 46338 z + 175248395 z - 1159426954723733005 z 30 42 + 18804294451385828168 z + 1452149689931617076638 z 44 46 - 1703051524392468028064 z + 1703051524392468028064 z 58 56 + 58534225484840489634 z - 154150226817694308482 z 54 52 + 344302416498469939644 z - 653477840695467094036 z 60 70 68 - 18804294451385828168 z + 4538418322365774 z - 34847071304739170 z 78 32 38 + 149106118028 z - 58534225484840489634 z + 653477840695467094036 z 40 62 76 - 1055456377079089746518 z + 5094613159810790168 z - 2815592729204 z 74 72 + 41366258644691 z - 482503516728911 z ) And in Maple-input format, it is: -(1+1229920292464016096*z^28-300937169540168672*z^26-272*z^2+61545031225939461* z^24-10458832872254000*z^22+31760*z^4-2137356*z^6+93985499*z^8-2903469272*z^10+ 66100836296*z^12-1147092349064*z^14-168000976140588*z^18+15556107755447*z^16-\ 108975523428400907976*z^50+163035476851708682590*z^48+1466144600580400*z^20+ 61922546057601507008*z^36-29872203850477864128*z^34-10458832872254000*z^66+ 93985499*z^80+z^88+31760*z^84-272*z^86-2137356*z^82+61545031225939461*z^64-\ 4221247821070827392*z^30-207539834411565687360*z^42+224913829973896313520*z^44-\ 207539834411565687360*z^46-4221247821070827392*z^58+12212489678184453642*z^56-\ 29872203850477864128*z^54+61922546057601507008*z^52+1229920292464016096*z^60-\ 168000976140588*z^70+1466144600580400*z^68-2903469272*z^78+12212489678184453642 *z^32-108975523428400907976*z^38+163035476851708682590*z^40-300937169540168672* z^62+66100836296*z^76-1147092349064*z^74+15556107755447*z^72)/(-1-\ 5094613159810790168*z^28+1159426954723733005*z^26+337*z^2-220542762530049149*z^ 24+34847071304739170*z^22-46338*z^4+3560318*z^6-175248395*z^8+5981427167*z^10-\ 149106118028*z^12+2815592729204*z^14+482503516728911*z^18-41366258644691*z^16+ 1055456377079089746518*z^50-1452149689931617076638*z^48-4538418322365774*z^20-\ 344302416498469939644*z^36+154150226817694308482*z^34+220542762530049149*z^66-\ 5981427167*z^80+z^90-337*z^88-3560318*z^84+46338*z^86+175248395*z^82-\ 1159426954723733005*z^64+18804294451385828168*z^30+1452149689931617076638*z^42-\ 1703051524392468028064*z^44+1703051524392468028064*z^46+58534225484840489634*z^ 58-154150226817694308482*z^56+344302416498469939644*z^54-653477840695467094036* z^52-18804294451385828168*z^60+4538418322365774*z^70-34847071304739170*z^68+ 149106118028*z^78-58534225484840489634*z^32+653477840695467094036*z^38-\ 1055456377079089746518*z^40+5094613159810790168*z^62-2815592729204*z^76+ 41366258644691*z^74-482503516728911*z^72) The first , 40, terms are: [0, 65, 0, 7327, 0, 880191, 0, 107263615, 0, 13134809903, 0, 1611551900137, 0, 197895207808009, 0, 24310418045598713, 0, 2986928807134359593, 0, 367021549928283101023, 0, 45099721610656144427327, 0, 5541958508219833914432543, 0, 681013815283237110189425007, 0, 83685468885097660931544219585, 0, 10283592528916148605017698658225, 0, 1263688327916471240765047322629073, 0, 155287046027622236238544047901638657, 0, 19082292435201083527301496599498186127, 0, 2344908467225825936905850151411401489247, 0, 288151746290553139080985419176929914992191] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 88650357874744210391 z - 13006741758488457410 z - 287 z 24 22 4 6 + 1640518065449402941 z - 176841451352003016 z + 36988 z - 2879260 z 102 8 10 12 - 5971803224 z + 153220265 z - 5971803224 z + 178160817567 z 14 18 16 - 4195273894600 z - 1244987738505130 z + 79749532929937 z 50 48 - 15313155101285726103622450 z + 9823613821460021415010022 z 20 36 + 16177078710589317 z + 45902071131310320747498 z 34 66 - 11840680038617708454734 z - 5547775616653434353756438 z 80 100 90 + 2665008367733237503725 z + 178160817567 z - 176841451352003016 z 88 84 + 1640518065449402941 z + 88650357874744210391 z 94 86 96 - 1244987738505130 z - 13006741758488457410 z + 79749532929937 z 98 92 82 - 4195273894600 z + 16177078710589317 z - 521929760753995802734 z 64 112 110 106 + 9823613821460021415010022 z + z - 287 z - 2879260 z 108 30 42 + 36988 z - 521929760753995802734 z - 1204301490968922565480850 z 44 46 + 2756617307988217592025894 z - 5547775616653434353756438 z 58 56 - 25420430915244873225590442 z + 27082080633714424572229714 z 54 52 - 25420430915244873225590442 z + 21021559638422502823401774 z 60 70 + 21021559638422502823401774 z - 1204301490968922565480850 z 68 78 + 2756617307988217592025894 z - 11840680038617708454734 z 32 38 + 2665008367733237503725 z - 155610803754373355782127 z 40 62 + 462161251179413269665653 z - 15313155101285726103622450 z 76 74 + 45902071131310320747498 z - 155610803754373355782127 z 72 104 / 2 + 462161251179413269665653 z + 153220265 z ) / ((-1 + z ) (1 / 28 26 2 + 255307275828426122254 z - 35942916135140886638 z - 345 z 24 22 4 6 + 4339499841429563156 z - 446664675511481664 z + 50704 z - 4377528 z 102 8 10 12 - 10686004046 z + 254100338 z - 10686004046 z + 341381146180 z 14 18 16 - 8559588273476 z - 2844249694263714 z + 172490662899055 z 50 48 - 58592665238314957703734762 z + 37159360459104899211731867 z 20 36 + 38914256940853675 z + 152210377613158981509378 z 34 66 - 38044610707341910542762 z - 20680807840815814112107594 z 80 100 90 + 8276164714442636132817 z + 341381146180 z - 446664675511481664 z 88 84 + 4339499841429563156 z + 255307275828426122254 z 94 86 96 - 2844249694263714 z - 35942916135140886638 z + 172490662899055 z 98 92 82 - 8559588273476 z + 38914256940853675 z - 1562759638599068645124 z 64 112 110 106 + 37159360459104899211731867 z + z - 345 z - 4377528 z 108 30 42 + 50704 z - 1562759638599068645124 z - 4320571500014503539333370 z 44 46 + 10095960783310675488128298 z - 20680807840815814112107594 z 58 56 - 98560860991605465946567290 z + 105178106666944846035631447 z 54 52 - 98560860991605465946567290 z + 81101054788356243283760170 z 60 70 + 81101054788356243283760170 z - 4320571500014503539333370 z 68 78 + 10095960783310675488128298 z - 38044610707341910542762 z 32 38 + 8276164714442636132817 z - 531158483519584667350613 z 40 62 + 1619532224620074868129690 z - 58592665238314957703734762 z 76 74 + 152210377613158981509378 z - 531158483519584667350613 z 72 104 + 1619532224620074868129690 z + 254100338 z )) And in Maple-input format, it is: -(1+88650357874744210391*z^28-13006741758488457410*z^26-287*z^2+ 1640518065449402941*z^24-176841451352003016*z^22+36988*z^4-2879260*z^6-\ 5971803224*z^102+153220265*z^8-5971803224*z^10+178160817567*z^12-4195273894600* z^14-1244987738505130*z^18+79749532929937*z^16-15313155101285726103622450*z^50+ 9823613821460021415010022*z^48+16177078710589317*z^20+45902071131310320747498*z ^36-11840680038617708454734*z^34-5547775616653434353756438*z^66+ 2665008367733237503725*z^80+178160817567*z^100-176841451352003016*z^90+ 1640518065449402941*z^88+88650357874744210391*z^84-1244987738505130*z^94-\ 13006741758488457410*z^86+79749532929937*z^96-4195273894600*z^98+ 16177078710589317*z^92-521929760753995802734*z^82+9823613821460021415010022*z^ 64+z^112-287*z^110-2879260*z^106+36988*z^108-521929760753995802734*z^30-\ 1204301490968922565480850*z^42+2756617307988217592025894*z^44-\ 5547775616653434353756438*z^46-25420430915244873225590442*z^58+ 27082080633714424572229714*z^56-25420430915244873225590442*z^54+ 21021559638422502823401774*z^52+21021559638422502823401774*z^60-\ 1204301490968922565480850*z^70+2756617307988217592025894*z^68-\ 11840680038617708454734*z^78+2665008367733237503725*z^32-\ 155610803754373355782127*z^38+462161251179413269665653*z^40-\ 15313155101285726103622450*z^62+45902071131310320747498*z^76-\ 155610803754373355782127*z^74+462161251179413269665653*z^72+153220265*z^104)/(-\ 1+z^2)/(1+255307275828426122254*z^28-35942916135140886638*z^26-345*z^2+ 4339499841429563156*z^24-446664675511481664*z^22+50704*z^4-4377528*z^6-\ 10686004046*z^102+254100338*z^8-10686004046*z^10+341381146180*z^12-\ 8559588273476*z^14-2844249694263714*z^18+172490662899055*z^16-\ 58592665238314957703734762*z^50+37159360459104899211731867*z^48+ 38914256940853675*z^20+152210377613158981509378*z^36-38044610707341910542762*z^ 34-20680807840815814112107594*z^66+8276164714442636132817*z^80+341381146180*z^ 100-446664675511481664*z^90+4339499841429563156*z^88+255307275828426122254*z^84 -2844249694263714*z^94-35942916135140886638*z^86+172490662899055*z^96-\ 8559588273476*z^98+38914256940853675*z^92-1562759638599068645124*z^82+ 37159360459104899211731867*z^64+z^112-345*z^110-4377528*z^106+50704*z^108-\ 1562759638599068645124*z^30-4320571500014503539333370*z^42+ 10095960783310675488128298*z^44-20680807840815814112107594*z^46-\ 98560860991605465946567290*z^58+105178106666944846035631447*z^56-\ 98560860991605465946567290*z^54+81101054788356243283760170*z^52+ 81101054788356243283760170*z^60-4320571500014503539333370*z^70+ 10095960783310675488128298*z^68-38044610707341910542762*z^78+ 8276164714442636132817*z^32-531158483519584667350613*z^38+ 1619532224620074868129690*z^40-58592665238314957703734762*z^62+ 152210377613158981509378*z^76-531158483519584667350613*z^74+ 1619532224620074868129690*z^72+254100338*z^104) The first , 40, terms are: [0, 59, 0, 6353, 0, 735219, 0, 86079564, 0, 10101999375, 0, 1186186367221, 0, 139302175465559, 0, 16359806215474589, 0, 1921331987611087885, 0, 225646047110926174707, 0, 26500454882507855193449, 0, 3112282445740533053997399, 0, 365514574025341828152042556, 0, 42926986178883943435071754915, 0, 5041457385820941058332524934101, 0, 592081924680189224729312140739647, 0, 69535647888737630957200076718796433, 0, 8166448131782072089483268594281864129, 0, 959089001330545710903299825866126812871, 0, 112637917688939882508088857193198046059293] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 647634711475914507 z - 162920440939359256 z - 256 z 24 22 4 6 + 34345191158252753 z - 6031388880505112 z + 28043 z - 1773664 z 8 10 12 14 + 73597960 z - 2154929392 z + 46683754001 z - 773591460280 z 18 16 50 - 104227763672024 z + 10048362941443 z - 52192455202076897584 z 48 20 + 77376986211805349995 z + 875873669565808 z 36 34 + 30032455112684529945 z - 14720671969265821472 z 66 80 88 84 86 - 6031388880505112 z + 73597960 z + z + 28043 z - 256 z 82 64 30 - 1773664 z + 34345191158252753 z - 2167843290582483152 z 42 44 - 97958522848984915152 z + 105963740979008238240 z 46 58 - 97958522848984915152 z - 2167843290582483152 z 56 54 + 6134372458830467384 z - 14720671969265821472 z 52 60 70 + 30032455112684529945 z + 647634711475914507 z - 104227763672024 z 68 78 32 + 875873669565808 z - 2154929392 z + 6134372458830467384 z 38 40 - 52192455202076897584 z + 77376986211805349995 z 62 76 74 - 162920440939359256 z + 46683754001 z - 773591460280 z 72 / 2 28 + 10048362941443 z ) / ((-1 + z ) (1 + 2185921094296754199 z / 26 2 24 - 528119523239811895 z - 323 z + 106510765636775769 z 22 4 6 8 - 17826528447760984 z + 41527 z - 2968892 z + 136260862 z 10 12 14 - 4352558148 z + 101897098181 z - 1811845378957 z 18 16 50 - 276598471937480 z + 25111289023899 z - 202172924234581271225 z 48 20 + 303707613609693269899 z + 2457858037742756 z 36 34 + 114229834907563820141 z - 54711130853405307652 z 66 80 88 84 86 - 17826528447760984 z + 136260862 z + z + 41527 z - 323 z 82 64 30 - 2968892 z + 106510765636775769 z - 7588168473036924412 z 42 44 - 387570917150681327200 z + 420364509679475305720 z 46 58 - 387570917150681327200 z - 7588168473036924412 z 56 54 + 22174746890235565090 z - 54711130853405307652 z 52 60 + 114229834907563820141 z + 2185921094296754199 z 70 68 78 - 276598471937480 z + 2457858037742756 z - 4352558148 z 32 38 + 22174746890235565090 z - 202172924234581271225 z 40 62 76 + 303707613609693269899 z - 528119523239811895 z + 101897098181 z 74 72 - 1811845378957 z + 25111289023899 z )) And in Maple-input format, it is: -(1+647634711475914507*z^28-162920440939359256*z^26-256*z^2+34345191158252753*z ^24-6031388880505112*z^22+28043*z^4-1773664*z^6+73597960*z^8-2154929392*z^10+ 46683754001*z^12-773591460280*z^14-104227763672024*z^18+10048362941443*z^16-\ 52192455202076897584*z^50+77376986211805349995*z^48+875873669565808*z^20+ 30032455112684529945*z^36-14720671969265821472*z^34-6031388880505112*z^66+ 73597960*z^80+z^88+28043*z^84-256*z^86-1773664*z^82+34345191158252753*z^64-\ 2167843290582483152*z^30-97958522848984915152*z^42+105963740979008238240*z^44-\ 97958522848984915152*z^46-2167843290582483152*z^58+6134372458830467384*z^56-\ 14720671969265821472*z^54+30032455112684529945*z^52+647634711475914507*z^60-\ 104227763672024*z^70+875873669565808*z^68-2154929392*z^78+6134372458830467384*z ^32-52192455202076897584*z^38+77376986211805349995*z^40-162920440939359256*z^62 +46683754001*z^76-773591460280*z^74+10048362941443*z^72)/(-1+z^2)/(1+ 2185921094296754199*z^28-528119523239811895*z^26-323*z^2+106510765636775769*z^ 24-17826528447760984*z^22+41527*z^4-2968892*z^6+136260862*z^8-4352558148*z^10+ 101897098181*z^12-1811845378957*z^14-276598471937480*z^18+25111289023899*z^16-\ 202172924234581271225*z^50+303707613609693269899*z^48+2457858037742756*z^20+ 114229834907563820141*z^36-54711130853405307652*z^34-17826528447760984*z^66+ 136260862*z^80+z^88+41527*z^84-323*z^86-2968892*z^82+106510765636775769*z^64-\ 7588168473036924412*z^30-387570917150681327200*z^42+420364509679475305720*z^44-\ 387570917150681327200*z^46-7588168473036924412*z^58+22174746890235565090*z^56-\ 54711130853405307652*z^54+114229834907563820141*z^52+2185921094296754199*z^60-\ 276598471937480*z^70+2457858037742756*z^68-4352558148*z^78+22174746890235565090 *z^32-202172924234581271225*z^38+303707613609693269899*z^40-528119523239811895* z^62+101897098181*z^76-1811845378957*z^74+25111289023899*z^72) The first , 40, terms are: [0, 68, 0, 8225, 0, 1055855, 0, 136957468, 0, 17813650503, 0, 2319027744119, 0, 301992358237164, 0, 39331108622430367, 0, 5122654302259400689, 0, 667207387414749418068, 0, 86901891762957788083761, 0, 11318752945018984821865297, 0, 1474240341045620685195626228, 0, 192016317263489398896283871953, 0, 25009674022350560861783837197247, 0, 3257451401160961732045649568351628, 0, 424275414086871560969529977672923799, 0, 55260879108757472083429761768890407719, 0, 7197600109137929409568345931668130915324, 0, 937470561635073862883264402105962447782991] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2900885503520844 z - 1366142152437758 z - 211 z 24 22 4 6 + 516158034537658 z - 155784648884912 z + 18622 z - 928281 z 8 10 12 14 + 29611951 z - 647999644 z + 10173380267 z - 118231042125 z 18 16 50 - 7056348561695 z + 1040626720630 z - 7056348561695 z 48 20 36 + 37344038152629 z + 37344038152629 z + 6831348701647542 z 34 66 64 30 - 7601065936074856 z - 211 z + 18622 z - 4957152116513834 z 42 44 46 - 1366142152437758 z + 516158034537658 z - 155784648884912 z 58 56 54 52 - 647999644 z + 10173380267 z - 118231042125 z + 1040626720630 z 60 68 32 38 + 29611951 z + z + 6831348701647542 z - 4957152116513834 z 40 62 / 28 + 2900885503520844 z - 928281 z ) / (-1 - 15866555094993986 z / 26 2 24 + 6762154126860226 z + 269 z - 2314051533529834 z 22 4 6 8 + 632882197466661 z - 28229 z + 1620251 z - 58482343 z 10 12 14 18 + 1433294615 z - 25045709843 z + 322739679363 z + 23537062054857 z 16 50 48 - 3142311296023 z + 137490291840673 z - 632882197466661 z 20 36 34 - 137490291840673 z - 56577959590360006 z + 56577959590360006 z 66 64 30 42 + 28229 z - 1620251 z + 29996224693789822 z + 15866555094993986 z 44 46 58 - 6762154126860226 z + 2314051533529834 z + 25045709843 z 56 54 52 - 322739679363 z + 3142311296023 z - 23537062054857 z 60 70 68 32 - 1433294615 z + z - 269 z - 45803264378202470 z 38 40 62 + 45803264378202470 z - 29996224693789822 z + 58482343 z ) And in Maple-input format, it is: -(1+2900885503520844*z^28-1366142152437758*z^26-211*z^2+516158034537658*z^24-\ 155784648884912*z^22+18622*z^4-928281*z^6+29611951*z^8-647999644*z^10+ 10173380267*z^12-118231042125*z^14-7056348561695*z^18+1040626720630*z^16-\ 7056348561695*z^50+37344038152629*z^48+37344038152629*z^20+6831348701647542*z^ 36-7601065936074856*z^34-211*z^66+18622*z^64-4957152116513834*z^30-\ 1366142152437758*z^42+516158034537658*z^44-155784648884912*z^46-647999644*z^58+ 10173380267*z^56-118231042125*z^54+1040626720630*z^52+29611951*z^60+z^68+ 6831348701647542*z^32-4957152116513834*z^38+2900885503520844*z^40-928281*z^62)/ (-1-15866555094993986*z^28+6762154126860226*z^26+269*z^2-2314051533529834*z^24+ 632882197466661*z^22-28229*z^4+1620251*z^6-58482343*z^8+1433294615*z^10-\ 25045709843*z^12+322739679363*z^14+23537062054857*z^18-3142311296023*z^16+ 137490291840673*z^50-632882197466661*z^48-137490291840673*z^20-\ 56577959590360006*z^36+56577959590360006*z^34+28229*z^66-1620251*z^64+ 29996224693789822*z^30+15866555094993986*z^42-6762154126860226*z^44+ 2314051533529834*z^46+25045709843*z^58-322739679363*z^56+3142311296023*z^54-\ 23537062054857*z^52-1433294615*z^60+z^70-269*z^68-45803264378202470*z^32+ 45803264378202470*z^38-29996224693789822*z^40+58482343*z^62) The first , 40, terms are: [0, 58, 0, 5995, 0, 667343, 0, 75386578, 0, 8547287757, 0, 970052971173, 0, 110124718002882, 0, 12502858957484215, 0, 1419528425514372963, 0, 161169130902413306442, 0, 18298712145256857230409, 0, 2077588114572405520265337, 0, 235883986225032849256164010, 0, 26781659766104076046477240371, 0, 3040720666169556133272867686055, 0, 345235593942023917610118636850274, 0, 39197160341649654526048358600385461, 0, 4450344653100549779965599732746070845, 0, 505280672414533743041327697607656270386, 0, 57368266466454658278322102993197979239007] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {3, 7}, {4, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 991183321763694597 z - 240116207939425547 z - 255 z 24 22 4 6 + 48626343055155746 z - 8188516590494196 z + 28062 z - 1798864 z 8 10 12 14 + 76270897 z - 2297624136 z + 51492305175 z - 886345469490 z 18 16 50 - 129811343408616 z + 11993238237907 z - 91305368317657564474 z 48 20 + 137142157609654861760 z + 1139074463525113 z 36 34 + 51600271064964357260 z - 24723757907549600440 z 66 80 88 84 86 - 8188516590494196 z + 76270897 z + z + 28062 z - 255 z 82 64 30 - 1798864 z + 48626343055155746 z - 3434893861730704622 z 42 44 - 174999070701622359266 z + 189802064117649567636 z 46 58 - 174999070701622359266 z - 3434893861730704622 z 56 54 + 10027104695608223042 z - 24723757907549600440 z 52 60 70 + 51600271064964357260 z + 991183321763694597 z - 129811343408616 z 68 78 32 + 1139074463525113 z - 2297624136 z + 10027104695608223042 z 38 40 - 91305368317657564474 z + 137142157609654861760 z 62 76 74 - 240116207939425547 z + 51492305175 z - 886345469490 z 72 / 2 28 + 11993238237907 z ) / ((-1 + z ) (1 + 3239848324662970556 z / 26 2 24 - 755682484430320328 z - 306 z + 146710109558857513 z 22 4 6 8 - 23578665366447795 z + 39056 z - 2836635 z + 133983155 z 10 12 14 - 4438299302 z + 108243306626 z - 2010167665273 z 18 16 50 - 335534757928116 z + 29130521638107 z - 338231841144798515922 z 48 20 + 514012344561700674613 z + 3115543122582282 z 36 34 + 188063282169557316833 z - 88259403423670708302 z 66 80 88 84 86 - 23578665366447795 z + 133983155 z + z + 39056 z - 306 z 82 64 30 - 2836635 z + 146710109558857513 z - 11611594936435822930 z 42 44 - 660548985659799160490 z + 718117094530062696653 z 46 58 - 660548985659799160490 z - 11611594936435822930 z 56 54 + 34907546654302605437 z - 88259403423670708302 z 52 60 + 188063282169557316833 z + 3239848324662970556 z 70 68 78 - 335534757928116 z + 3115543122582282 z - 4438299302 z 32 38 + 34907546654302605437 z - 338231841144798515922 z 40 62 76 + 514012344561700674613 z - 755682484430320328 z + 108243306626 z 74 72 - 2010167665273 z + 29130521638107 z )) And in Maple-input format, it is: -(1+991183321763694597*z^28-240116207939425547*z^26-255*z^2+48626343055155746*z ^24-8188516590494196*z^22+28062*z^4-1798864*z^6+76270897*z^8-2297624136*z^10+ 51492305175*z^12-886345469490*z^14-129811343408616*z^18+11993238237907*z^16-\ 91305368317657564474*z^50+137142157609654861760*z^48+1139074463525113*z^20+ 51600271064964357260*z^36-24723757907549600440*z^34-8188516590494196*z^66+ 76270897*z^80+z^88+28062*z^84-255*z^86-1798864*z^82+48626343055155746*z^64-\ 3434893861730704622*z^30-174999070701622359266*z^42+189802064117649567636*z^44-\ 174999070701622359266*z^46-3434893861730704622*z^58+10027104695608223042*z^56-\ 24723757907549600440*z^54+51600271064964357260*z^52+991183321763694597*z^60-\ 129811343408616*z^70+1139074463525113*z^68-2297624136*z^78+10027104695608223042 *z^32-91305368317657564474*z^38+137142157609654861760*z^40-240116207939425547*z ^62+51492305175*z^76-886345469490*z^74+11993238237907*z^72)/(-1+z^2)/(1+ 3239848324662970556*z^28-755682484430320328*z^26-306*z^2+146710109558857513*z^ 24-23578665366447795*z^22+39056*z^4-2836635*z^6+133983155*z^8-4438299302*z^10+ 108243306626*z^12-2010167665273*z^14-335534757928116*z^18+29130521638107*z^16-\ 338231841144798515922*z^50+514012344561700674613*z^48+3115543122582282*z^20+ 188063282169557316833*z^36-88259403423670708302*z^34-23578665366447795*z^66+ 133983155*z^80+z^88+39056*z^84-306*z^86-2836635*z^82+146710109558857513*z^64-\ 11611594936435822930*z^30-660548985659799160490*z^42+718117094530062696653*z^44 -660548985659799160490*z^46-11611594936435822930*z^58+34907546654302605437*z^56 -88259403423670708302*z^54+188063282169557316833*z^52+3239848324662970556*z^60-\ 335534757928116*z^70+3115543122582282*z^68-4438299302*z^78+34907546654302605437 *z^32-338231841144798515922*z^38+514012344561700674613*z^40-755682484430320328* z^62+108243306626*z^76-2010167665273*z^74+29130521638107*z^72) The first , 40, terms are: [0, 52, 0, 4664, 0, 461851, 0, 47190928, 0, 4880487899, 0, 507363127549, 0, 52868632950664, 0, 5515141291641189, 0, 575629443225414955, 0, 60095003234070309580, 0, 6274600349734119856159, 0, 655177469629462449762928, 0, 68413839103883293834034432, 0, 7143890822409425350307646309, 0, 745982207275654325491663042821, 0, 77897493767654579915296697698360, 0, 8134280690932309752298749497724960, 0, 849405637257321135395173254377307319, 0, 88697478750372819119450533989699909684, 0, 9262057745608556120380306759341342321763] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {3, 7}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1769707529507148 z - 851730673458066 z - 207 z 24 22 4 6 + 330479220125646 z - 102850377379312 z + 17646 z - 843731 z 8 10 12 14 + 25758653 z - 539587340 z + 8118661485 z - 90545499011 z 18 16 50 - 4997988260159 z + 765844599462 z - 4997988260159 z 48 20 36 + 25505212336721 z + 25505212336721 z + 4060437817989942 z 34 66 64 30 - 4502840506936776 z - 207 z + 17646 z - 2975813390659722 z 42 44 46 - 851730673458066 z + 330479220125646 z - 102850377379312 z 58 56 54 52 - 539587340 z + 8118661485 z - 90545499011 z + 765844599462 z 60 68 32 38 + 25758653 z + z + 4060437817989942 z - 2975813390659722 z 40 62 / 2 + 1769707529507148 z - 843731 z ) / ((-1 + z ) (1 / 28 26 2 + 6499270768576046 z - 3050033774081852 z - 262 z 24 22 4 6 + 1145518802749962 z - 342504284263136 z + 26531 z - 1462070 z 8 10 12 14 + 50472081 z - 1177996968 z + 19500308629 z - 236599403982 z 18 16 50 - 15003046726686 z + 2154972959135 z - 15003046726686 z 48 20 36 + 80969544492677 z + 80969544492677 z + 15347419658842474 z 34 66 64 30 - 17080881150655600 z - 262 z + 26531 z - 11127133841223516 z 42 44 46 - 3050033774081852 z + 1145518802749962 z - 342504284263136 z 58 56 54 52 - 1177996968 z + 19500308629 z - 236599403982 z + 2154972959135 z 60 68 32 38 + 50472081 z + z + 15347419658842474 z - 11127133841223516 z 40 62 + 6499270768576046 z - 1462070 z )) And in Maple-input format, it is: -(1+1769707529507148*z^28-851730673458066*z^26-207*z^2+330479220125646*z^24-\ 102850377379312*z^22+17646*z^4-843731*z^6+25758653*z^8-539587340*z^10+ 8118661485*z^12-90545499011*z^14-4997988260159*z^18+765844599462*z^16-\ 4997988260159*z^50+25505212336721*z^48+25505212336721*z^20+4060437817989942*z^ 36-4502840506936776*z^34-207*z^66+17646*z^64-2975813390659722*z^30-\ 851730673458066*z^42+330479220125646*z^44-102850377379312*z^46-539587340*z^58+ 8118661485*z^56-90545499011*z^54+765844599462*z^52+25758653*z^60+z^68+ 4060437817989942*z^32-2975813390659722*z^38+1769707529507148*z^40-843731*z^62)/ (-1+z^2)/(1+6499270768576046*z^28-3050033774081852*z^26-262*z^2+ 1145518802749962*z^24-342504284263136*z^22+26531*z^4-1462070*z^6+50472081*z^8-\ 1177996968*z^10+19500308629*z^12-236599403982*z^14-15003046726686*z^18+ 2154972959135*z^16-15003046726686*z^50+80969544492677*z^48+80969544492677*z^20+ 15347419658842474*z^36-17080881150655600*z^34-262*z^66+26531*z^64-\ 11127133841223516*z^30-3050033774081852*z^42+1145518802749962*z^44-\ 342504284263136*z^46-1177996968*z^58+19500308629*z^56-236599403982*z^54+ 2154972959135*z^52+50472081*z^60+z^68+15347419658842474*z^32-11127133841223516* z^38+6499270768576046*z^40-1462070*z^62) The first , 40, terms are: [0, 56, 0, 5581, 0, 612265, 0, 68680120, 0, 7746906849, 0, 875098463293, 0, 98891507853544, 0, 11176621832914541, 0, 1263213113014474561, 0, 142773312389885071464, 0, 16136850014892929155149, 0, 1823857454845857855473877, 0, 206140415503875683867051016, 0, 23298901211066513670922377657, 0, 2633344903331722050415298667845, 0, 297632295025362766968798952856328, 0, 33639719265363574058809435857968293, 0, 3802109959505727461981598656010750393, 0, 429731295729709878362650984327021656728, 0, 48570133034324216751216154388545672383377] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1226270420872012 z - 613362413887266 z - 227 z 24 22 4 6 + 249645261499662 z - 82144209759488 z + 20530 z - 1005359 z 8 10 12 14 + 30488969 z - 619704340 z + 8896603625 z - 93630387559 z 18 16 50 - 4534037582203 z + 742614183746 z - 4534037582203 z 48 20 36 + 21668685799665 z + 21668685799665 z + 2688689577998670 z 34 66 64 30 - 2964464725353304 z - 227 z + 20530 z - 2004658212465066 z 42 44 46 - 613362413887266 z + 249645261499662 z - 82144209759488 z 58 56 54 52 - 619704340 z + 8896603625 z - 93630387559 z + 742614183746 z 60 68 32 38 + 30488969 z + z + 2688689577998670 z - 2004658212465066 z 40 62 / 28 + 1226270420872012 z - 1005359 z ) / (-1 - 6539839592268946 z / 26 2 24 + 3013339370103858 z + 277 z - 1131951773037770 z 22 4 6 8 + 344053143773845 z - 30509 z + 1791023 z - 63680995 z 10 12 14 18 + 1483194515 z - 23937296879 z + 279161481215 z + 16157867822497 z 16 50 48 - 2428783785739 z + 83799861846137 z - 344053143773845 z 20 36 34 - 83799861846137 z - 20627705385777246 z + 20627705385777246 z 66 64 30 42 + 30509 z - 1791023 z + 11638034514262950 z + 6539839592268946 z 44 46 58 - 3013339370103858 z + 1131951773037770 z + 23937296879 z 56 54 52 - 279161481215 z + 2428783785739 z - 16157867822497 z 60 70 68 32 - 1483194515 z + z - 277 z - 17052387561932254 z 38 40 62 + 17052387561932254 z - 11638034514262950 z + 63680995 z ) And in Maple-input format, it is: -(1+1226270420872012*z^28-613362413887266*z^26-227*z^2+249645261499662*z^24-\ 82144209759488*z^22+20530*z^4-1005359*z^6+30488969*z^8-619704340*z^10+ 8896603625*z^12-93630387559*z^14-4534037582203*z^18+742614183746*z^16-\ 4534037582203*z^50+21668685799665*z^48+21668685799665*z^20+2688689577998670*z^ 36-2964464725353304*z^34-227*z^66+20530*z^64-2004658212465066*z^30-\ 613362413887266*z^42+249645261499662*z^44-82144209759488*z^46-619704340*z^58+ 8896603625*z^56-93630387559*z^54+742614183746*z^52+30488969*z^60+z^68+ 2688689577998670*z^32-2004658212465066*z^38+1226270420872012*z^40-1005359*z^62) /(-1-6539839592268946*z^28+3013339370103858*z^26+277*z^2-1131951773037770*z^24+ 344053143773845*z^22-30509*z^4+1791023*z^6-63680995*z^8+1483194515*z^10-\ 23937296879*z^12+279161481215*z^14+16157867822497*z^18-2428783785739*z^16+ 83799861846137*z^50-344053143773845*z^48-83799861846137*z^20-20627705385777246* z^36+20627705385777246*z^34+30509*z^66-1791023*z^64+11638034514262950*z^30+ 6539839592268946*z^42-3013339370103858*z^44+1131951773037770*z^46+23937296879*z ^58-279161481215*z^56+2428783785739*z^54-16157867822497*z^52-1483194515*z^60+z^ 70-277*z^68-17052387561932254*z^32+17052387561932254*z^38-11638034514262950*z^ 40+63680995*z^62) The first , 40, terms are: [0, 50, 0, 3871, 0, 332481, 0, 30356022, 0, 2877445723, 0, 279011608987, 0, 27423949035574, 0, 2717283113994217, 0, 270523771372389399, 0, 27008011173500728338, 0, 2700817089691910112849, 0, 270344800114612081985489, 0, 27076196011792784507522322, 0, 2712702362847705177877512327, 0, 271832683059992572235395468601, 0, 27242759888200587954055730561142, 0, 2730422299779356232953212494340987, 0, 273669079175613914834865379610606907, 0, 27430373904028958511696170433458776694, 0, 2749436188121452370100770197254315630513] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {4, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 5592391036216386 z - 2292116664183258 z - 211 z 24 22 4 6 + 765701521025879 z - 207555136710715 z + 18379 z - 902318 z 8 10 12 14 + 28525323 z - 625600283 z + 9979417698 z - 119593511775 z 18 16 50 - 7943943707442 z + 1102025165471 z - 207555136710715 z 48 20 36 + 765701521025879 z + 45384148119051 z + 27019352513246604 z 34 66 64 - 24495185036857374 z - 902318 z + 28525323 z 30 42 44 - 11157605431149216 z - 11157605431149216 z + 5592391036216386 z 46 58 56 - 2292116664183258 z - 119593511775 z + 1102025165471 z 54 52 60 70 - 7943943707442 z + 45384148119051 z + 9979417698 z - 211 z 68 32 38 + 18379 z + 18246261446965694 z - 24495185036857374 z 40 62 72 / + 18246261446965694 z - 625600283 z + z ) / ((1 / 28 26 2 + 20171329640371008 z - 8027064032543448 z - 266 z 24 22 4 6 + 2586785368639279 z - 672028670566398 z + 27428 z - 1547000 z 8 10 12 14 + 55086612 z - 1340353854 z + 23428310174 z - 304424886170 z 18 16 50 - 23151232551416 z + 3013837966652 z - 672028670566398 z 48 20 36 + 2586785368639279 z + 139898931411340 z + 102746511748303972 z 34 66 64 - 92838770790387272 z - 1547000 z + 55086612 z 30 42 44 - 41182963802757904 z - 41182963802757904 z + 20171329640371008 z 46 58 56 - 8027064032543448 z - 304424886170 z + 3013837966652 z 54 52 60 70 - 23151232551416 z + 139898931411340 z + 23428310174 z - 266 z 68 32 38 + 27428 z + 68469977262227392 z - 92838770790387272 z 40 62 72 2 + 68469977262227392 z - 1340353854 z + z ) (-1 + z )) And in Maple-input format, it is: -(1+5592391036216386*z^28-2292116664183258*z^26-211*z^2+765701521025879*z^24-\ 207555136710715*z^22+18379*z^4-902318*z^6+28525323*z^8-625600283*z^10+ 9979417698*z^12-119593511775*z^14-7943943707442*z^18+1102025165471*z^16-\ 207555136710715*z^50+765701521025879*z^48+45384148119051*z^20+27019352513246604 *z^36-24495185036857374*z^34-902318*z^66+28525323*z^64-11157605431149216*z^30-\ 11157605431149216*z^42+5592391036216386*z^44-2292116664183258*z^46-119593511775 *z^58+1102025165471*z^56-7943943707442*z^54+45384148119051*z^52+9979417698*z^60 -211*z^70+18379*z^68+18246261446965694*z^32-24495185036857374*z^38+ 18246261446965694*z^40-625600283*z^62+z^72)/(1+20171329640371008*z^28-\ 8027064032543448*z^26-266*z^2+2586785368639279*z^24-672028670566398*z^22+27428* z^4-1547000*z^6+55086612*z^8-1340353854*z^10+23428310174*z^12-304424886170*z^14 -23151232551416*z^18+3013837966652*z^16-672028670566398*z^50+2586785368639279*z ^48+139898931411340*z^20+102746511748303972*z^36-92838770790387272*z^34-1547000 *z^66+55086612*z^64-41182963802757904*z^30-41182963802757904*z^42+ 20171329640371008*z^44-8027064032543448*z^46-304424886170*z^58+3013837966652*z^ 56-23151232551416*z^54+139898931411340*z^52+23428310174*z^60-266*z^70+27428*z^ 68+68469977262227392*z^32-92838770790387272*z^38+68469977262227392*z^40-\ 1340353854*z^62+z^72)/(-1+z^2) The first , 40, terms are: [0, 56, 0, 5637, 0, 626325, 0, 71177376, 0, 8132323389, 0, 930359459941, 0, 106469330980152, 0, 12185184073278437, 0, 1394594695837454709, 0, 159612165035320184224, 0, 18267725605254002114585, 0, 2090754775560000085973033, 0, 239288455664120580012557648, 0, 27386744115213554986676632789, 0, 3134433520877374160123193990597, 0, 358738281061143124555484228236744, 0, 41057866908728492053493930770798421, 0, 4699103843128111143939391961710250573, 0, 537815980015972391868368635643997848336, 0, 61553444660470757452495673126349212726581] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 49252185627968 z + 38617358757699 z + 217 z 24 22 4 6 - 23687969820587 z + 11316195057170 z - 18310 z + 816954 z 8 10 12 14 - 22048983 z + 388637551 z - 4701073648 z + 40412957328 z 18 16 50 48 + 1183506535205 z - 253452562637 z + 22048983 z - 388637551 z 20 36 34 - 4181145699438 z - 11316195057170 z + 23687969820587 z 30 42 44 46 + 49252185627968 z + 253452562637 z - 40412957328 z + 4701073648 z 58 56 54 52 32 + z - 217 z + 18310 z - 816954 z - 38617358757699 z 38 40 / 28 + 4181145699438 z - 1183506535205 z ) / (1 + 329842532498115 z / 26 2 24 22 - 230965647589654 z - 274 z + 127089195183037 z - 54658742246636 z 4 6 8 10 12 + 28575 z - 1537812 z + 48778897 z - 987975230 z + 13502520735 z 14 18 16 50 - 129675595480 z - 4664476381082 z + 902645159901 z - 987975230 z 48 20 36 + 13502520735 z + 18229894911699 z + 127089195183037 z 34 30 42 - 230965647589654 z - 371316575634000 z - 4664476381082 z 44 46 58 56 54 + 902645159901 z - 129675595480 z - 274 z + 28575 z - 1537812 z 52 60 32 38 + 48778897 z + z + 329842532498115 z - 54658742246636 z 40 + 18229894911699 z ) And in Maple-input format, it is: -(-1-49252185627968*z^28+38617358757699*z^26+217*z^2-23687969820587*z^24+ 11316195057170*z^22-18310*z^4+816954*z^6-22048983*z^8+388637551*z^10-4701073648 *z^12+40412957328*z^14+1183506535205*z^18-253452562637*z^16+22048983*z^50-\ 388637551*z^48-4181145699438*z^20-11316195057170*z^36+23687969820587*z^34+ 49252185627968*z^30+253452562637*z^42-40412957328*z^44+4701073648*z^46+z^58-217 *z^56+18310*z^54-816954*z^52-38617358757699*z^32+4181145699438*z^38-\ 1183506535205*z^40)/(1+329842532498115*z^28-230965647589654*z^26-274*z^2+ 127089195183037*z^24-54658742246636*z^22+28575*z^4-1537812*z^6+48778897*z^8-\ 987975230*z^10+13502520735*z^12-129675595480*z^14-4664476381082*z^18+ 902645159901*z^16-987975230*z^50+13502520735*z^48+18229894911699*z^20+ 127089195183037*z^36-230965647589654*z^34-371316575634000*z^30-4664476381082*z^ 42+902645159901*z^44-129675595480*z^46-274*z^58+28575*z^56-1537812*z^54+ 48778897*z^52+z^60+329842532498115*z^32-54658742246636*z^38+18229894911699*z^40 ) The first , 40, terms are: [0, 57, 0, 5353, 0, 558805, 0, 61075965, 0, 6817809721, 0, 768570903721, 0, 87043225554921, 0, 9879770446231417, 0, 1122591700609642633, 0, 127620792274614275097, 0, 14512107284825434756429, 0, 1650414331907860328228293, 0, 187707513484888663697318793, 0, 21349275748641368939849013337, 0, 2428236159194335662629497429297, 0, 276186076449596398340738870804049, 0, 31413342657607876973751200325518617, 0, 3572952686102986610646505290495429193, 0, 406387876198765953866802807315763360869, 0, 46222603105552827331024670146667114700589] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 7}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 4 -18 z + z + 1 f(z) = - ----------------------- 4 6 2 -69 z + z + 69 z - 1 And in Maple-input format, it is: -(-18*z^2+z^4+1)/(-69*z^4+z^6+69*z^2-1) The first , 40, terms are: [0, 51, 0, 3451, 0, 234601, 0, 15949401, 0, 1084324651, 0, 73718126851, 0, 5011748301201, 0, 340725166354801, 0, 23164299563825251, 0, 1574831645173762251, 0, 107065387572252007801, 0, 7278871523267962768201, 0, 494856198194649216229851, 0, 33642942605712878740861651, 0, 2287225240990281105162362401, 0, 155497673444733402272299781601, 0, 10571554569000881073411222786451, 0, 718710213018615179589690849697051, 0, 48861722930696831331025566556613001, 0, 3321878449074365915330148834999987001] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 15533568332312 z - 13612564003627 z - 187 z 24 22 4 6 + 9154720531533 z - 4714816317487 z + 13881 z - 554824 z 8 10 12 14 + 13658900 z - 223252107 z + 2538598142 z - 20716439082 z 18 16 50 48 - 552352454560 z + 123968573421 z - 554824 z + 13658900 z 20 36 34 + 1852647536032 z + 1852647536032 z - 4714816317487 z 30 42 44 46 - 13612564003627 z - 20716439082 z + 2538598142 z - 223252107 z 56 54 52 32 38 + z - 187 z + 13881 z + 9154720531533 z - 552352454560 z 40 / 2 28 + 123968573421 z ) / ((-1 + z ) (1 + 58710171576395 z / 26 2 24 22 - 51151213879034 z - 238 z + 33811564733353 z - 16928343628756 z 4 6 8 10 12 + 21523 z - 1017892 z + 28909613 z - 533197622 z + 6712758771 z 14 18 16 50 - 59681976036 z - 1816014894958 z + 383809036053 z - 1017892 z 48 20 36 + 28909613 z + 6398606737495 z + 6398606737495 z 34 30 42 - 16928343628756 z - 51151213879034 z - 59681976036 z 44 46 56 54 52 + 6712758771 z - 533197622 z + z - 238 z + 21523 z 32 38 40 + 33811564733353 z - 1816014894958 z + 383809036053 z )) And in Maple-input format, it is: -(1+15533568332312*z^28-13612564003627*z^26-187*z^2+9154720531533*z^24-\ 4714816317487*z^22+13881*z^4-554824*z^6+13658900*z^8-223252107*z^10+2538598142* z^12-20716439082*z^14-552352454560*z^18+123968573421*z^16-554824*z^50+13658900* z^48+1852647536032*z^20+1852647536032*z^36-4714816317487*z^34-13612564003627*z^ 30-20716439082*z^42+2538598142*z^44-223252107*z^46+z^56-187*z^54+13881*z^52+ 9154720531533*z^32-552352454560*z^38+123968573421*z^40)/(-1+z^2)/(1+ 58710171576395*z^28-51151213879034*z^26-238*z^2+33811564733353*z^24-\ 16928343628756*z^22+21523*z^4-1017892*z^6+28909613*z^8-533197622*z^10+ 6712758771*z^12-59681976036*z^14-1816014894958*z^18+383809036053*z^16-1017892*z ^50+28909613*z^48+6398606737495*z^20+6398606737495*z^36-16928343628756*z^34-\ 51151213879034*z^30-59681976036*z^42+6712758771*z^44-533197622*z^46+z^56-238*z^ 54+21523*z^52+33811564733353*z^32-1816014894958*z^38+383809036053*z^40) The first , 40, terms are: [0, 52, 0, 4548, 0, 439991, 0, 43969796, 0, 4444021381, 0, 451039549797, 0, 45852488054496, 0, 4664449714515979, 0, 474634424814207531, 0, 48302530246269362068, 0, 4915898681194500531937, 0, 500317516125874910103260, 0, 50920510292336988026297028, 0, 5182527905353426989409344343, 0, 527462231387232290904645319111, 0, 53683576829601387839642727381844, 0, 5463760801750494992165179933640028, 0, 556086024901193836455069399091911217, 0, 56596856978340804749317882937427657188, 0, 5760267610577597493350995272438828867883] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1936879440450567221 z - 459509137058464749 z - 273 z 24 22 4 6 + 90856893974652769 z - 14889772648271740 z + 32365 z - 2231856 z 8 10 12 14 + 101275164 z - 3244574756 z + 76842562271 z - 1389523849135 z 18 16 50 - 221098825742732 z + 19645320946355 z - 190369661704960866067 z 48 20 + 287576108111018744363 z + 2008554111950216 z 36 34 + 106723742293749276095 z - 50605489758278014808 z 66 80 88 84 86 - 14889772648271740 z + 101275164 z + z + 32365 z - 273 z 82 64 30 - 2231856 z + 90856893974652769 z - 6834597471391030164 z 42 44 - 368214058681422079544 z + 399814865160211140144 z 46 58 - 368214058681422079544 z - 6834597471391030164 z 56 54 + 20261381422095685124 z - 50605489758278014808 z 52 60 + 106723742293749276095 z + 1936879440450567221 z 70 68 78 - 221098825742732 z + 2008554111950216 z - 3244574756 z 32 38 + 20261381422095685124 z - 190369661704960866067 z 40 62 76 + 287576108111018744363 z - 459509137058464749 z + 76842562271 z 74 72 / - 1389523849135 z + 19645320946355 z ) / (-1 / 28 26 2 - 7855953799723744870 z + 1738110412067965929 z + 331 z 24 22 4 6 - 320426977505805679 z + 48921522323745654 z - 45848 z + 3608886 z 8 10 12 14 - 183908903 z + 6537590459 z - 170221795920 z + 3359835554086 z 18 16 50 + 627219341330033 z - 51561946040767 z + 1804156458773936568941 z 48 20 - 2497667330103214953855 z - 6138987886223814 z 36 34 - 575882285457972994886 z + 253847174527553074385 z 66 80 90 88 84 + 320426977505805679 z - 6537590459 z + z - 331 z - 3608886 z 86 82 64 + 45848 z + 183908903 z - 1738110412067965929 z 30 42 + 29732627058896466580 z + 2497667330103214953855 z 44 46 - 2938286735860436309924 z + 2938286735860436309924 z 58 56 + 94601950881438922541 z - 253847174527553074385 z 54 52 + 575882285457972994886 z - 1106684781267628972732 z 60 70 68 - 29732627058896466580 z + 6138987886223814 z - 48921522323745654 z 78 32 38 + 170221795920 z - 94601950881438922541 z + 1106684781267628972732 z 40 62 76 - 1804156458773936568941 z + 7855953799723744870 z - 3359835554086 z 74 72 + 51561946040767 z - 627219341330033 z ) And in Maple-input format, it is: -(1+1936879440450567221*z^28-459509137058464749*z^26-273*z^2+90856893974652769* z^24-14889772648271740*z^22+32365*z^4-2231856*z^6+101275164*z^8-3244574756*z^10 +76842562271*z^12-1389523849135*z^14-221098825742732*z^18+19645320946355*z^16-\ 190369661704960866067*z^50+287576108111018744363*z^48+2008554111950216*z^20+ 106723742293749276095*z^36-50605489758278014808*z^34-14889772648271740*z^66+ 101275164*z^80+z^88+32365*z^84-273*z^86-2231856*z^82+90856893974652769*z^64-\ 6834597471391030164*z^30-368214058681422079544*z^42+399814865160211140144*z^44-\ 368214058681422079544*z^46-6834597471391030164*z^58+20261381422095685124*z^56-\ 50605489758278014808*z^54+106723742293749276095*z^52+1936879440450567221*z^60-\ 221098825742732*z^70+2008554111950216*z^68-3244574756*z^78+20261381422095685124 *z^32-190369661704960866067*z^38+287576108111018744363*z^40-459509137058464749* z^62+76842562271*z^76-1389523849135*z^74+19645320946355*z^72)/(-1-\ 7855953799723744870*z^28+1738110412067965929*z^26+331*z^2-320426977505805679*z^ 24+48921522323745654*z^22-45848*z^4+3608886*z^6-183908903*z^8+6537590459*z^10-\ 170221795920*z^12+3359835554086*z^14+627219341330033*z^18-51561946040767*z^16+ 1804156458773936568941*z^50-2497667330103214953855*z^48-6138987886223814*z^20-\ 575882285457972994886*z^36+253847174527553074385*z^34+320426977505805679*z^66-\ 6537590459*z^80+z^90-331*z^88-3608886*z^84+45848*z^86+183908903*z^82-\ 1738110412067965929*z^64+29732627058896466580*z^30+2497667330103214953855*z^42-\ 2938286735860436309924*z^44+2938286735860436309924*z^46+94601950881438922541*z^ 58-253847174527553074385*z^56+575882285457972994886*z^54-1106684781267628972732 *z^52-29732627058896466580*z^60+6138987886223814*z^70-48921522323745654*z^68+ 170221795920*z^78-94601950881438922541*z^32+1106684781267628972732*z^38-\ 1804156458773936568941*z^40+7855953799723744870*z^62-3359835554086*z^76+ 51561946040767*z^74-627219341330033*z^72) The first , 40, terms are: [0, 58, 0, 5715, 0, 609511, 0, 66408470, 0, 7287426061, 0, 801859840705, 0, 88327572531390, 0, 9734007122832211, 0, 1072928343401293583, 0, 118272965911092107426, 0, 13038137075501369593653, 0, 1437315633817106216900941, 0, 158449752956236670691983570, 0, 17467557799321174493562081239, 0, 1925632153040407488688501468219, 0, 212282743070222395243895001416878, 0, 23402170980940093749336978248652697, 0, 2579868953388095948285397863661685429, 0, 284406266881989109370605747650564834726, 0, 31353114249661647883900765253514909872127] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2282871365962822033 z - 540821122437271285 z - 277 z 24 22 4 6 + 106676053072274368 z - 17417426255066986 z + 33428 z - 2350986 z 8 10 12 14 + 108844035 z - 3554020634 z + 85616390017 z - 1570890692966 z 18 16 50 - 255391365606510 z + 22477382015517 z - 224442332625820201206 z 48 20 + 338960134625170179580 z + 2337024676707243 z 36 34 + 125861683287480826796 z - 59694744236640735680 z 66 80 88 84 86 - 17417426255066986 z + 108844035 z + z + 33428 z - 277 z 82 64 30 - 2350986 z + 106676053072274368 z - 8061160881201819326 z 42 44 - 433931578361677984562 z + 471143851553371629476 z 46 58 - 433931578361677984562 z - 8061160881201819326 z 56 54 + 23902660923927146086 z - 59694744236640735680 z 52 60 + 125861683287480826796 z + 2282871365962822033 z 70 68 78 - 255391365606510 z + 2337024676707243 z - 3554020634 z 32 38 + 23902660923927146086 z - 224442332625820201206 z 40 62 76 + 338960134625170179580 z - 540821122437271285 z + 85616390017 z 74 72 / - 1570890692966 z + 22477382015517 z ) / (-1 / 28 26 2 - 9362916429974714608 z + 2063953141428691361 z + 343 z 24 22 4 6 - 378742217490091142 z + 57493252781537063 z - 48506 z + 3878201 z 8 10 12 14 - 200452564 z + 7223595189 z - 190562564560 z + 3807670708285 z 18 16 50 + 725838506790559 z - 59087738448062 z + 2171289688419257303427 z 48 20 - 3007254605524104831575 z - 7164127416264582 z 36 34 - 691925779556465859823 z + 304603160095007935651 z 66 80 90 88 84 + 378742217490091142 z - 7223595189 z + z - 343 z - 3878201 z 86 82 64 + 48506 z + 200452564 z - 2063953141428691361 z 30 42 + 35537707856768395134 z + 3007254605524104831575 z 44 46 - 3538538499809079487155 z + 3538538499809079487155 z 58 56 + 113323671096698707327 z - 304603160095007935651 z 54 52 + 691925779556465859823 z - 1330972757876874036907 z 60 70 68 - 35537707856768395134 z + 7164127416264582 z - 57493252781537063 z 78 32 + 190562564560 z - 113323671096698707327 z 38 40 + 1330972757876874036907 z - 2171289688419257303427 z 62 76 74 + 9362916429974714608 z - 3807670708285 z + 59087738448062 z 72 - 725838506790559 z ) And in Maple-input format, it is: -(1+2282871365962822033*z^28-540821122437271285*z^26-277*z^2+106676053072274368 *z^24-17417426255066986*z^22+33428*z^4-2350986*z^6+108844035*z^8-3554020634*z^ 10+85616390017*z^12-1570890692966*z^14-255391365606510*z^18+22477382015517*z^16 -224442332625820201206*z^50+338960134625170179580*z^48+2337024676707243*z^20+ 125861683287480826796*z^36-59694744236640735680*z^34-17417426255066986*z^66+ 108844035*z^80+z^88+33428*z^84-277*z^86-2350986*z^82+106676053072274368*z^64-\ 8061160881201819326*z^30-433931578361677984562*z^42+471143851553371629476*z^44-\ 433931578361677984562*z^46-8061160881201819326*z^58+23902660923927146086*z^56-\ 59694744236640735680*z^54+125861683287480826796*z^52+2282871365962822033*z^60-\ 255391365606510*z^70+2337024676707243*z^68-3554020634*z^78+23902660923927146086 *z^32-224442332625820201206*z^38+338960134625170179580*z^40-540821122437271285* z^62+85616390017*z^76-1570890692966*z^74+22477382015517*z^72)/(-1-\ 9362916429974714608*z^28+2063953141428691361*z^26+343*z^2-378742217490091142*z^ 24+57493252781537063*z^22-48506*z^4+3878201*z^6-200452564*z^8+7223595189*z^10-\ 190562564560*z^12+3807670708285*z^14+725838506790559*z^18-59087738448062*z^16+ 2171289688419257303427*z^50-3007254605524104831575*z^48-7164127416264582*z^20-\ 691925779556465859823*z^36+304603160095007935651*z^34+378742217490091142*z^66-\ 7223595189*z^80+z^90-343*z^88-3878201*z^84+48506*z^86+200452564*z^82-\ 2063953141428691361*z^64+35537707856768395134*z^30+3007254605524104831575*z^42-\ 3538538499809079487155*z^44+3538538499809079487155*z^46+113323671096698707327*z ^58-304603160095007935651*z^56+691925779556465859823*z^54-\ 1330972757876874036907*z^52-35537707856768395134*z^60+7164127416264582*z^70-\ 57493252781537063*z^68+190562564560*z^78-113323671096698707327*z^32+ 1330972757876874036907*z^38-2171289688419257303427*z^40+9362916429974714608*z^ 62-3807670708285*z^76+59087738448062*z^74-725838506790559*z^72) The first , 40, terms are: [0, 66, 0, 7560, 0, 918899, 0, 112829734, 0, 13887388759, 0, 1710520011723, 0, 210737441881632, 0, 25965331691386097, 0, 3199341370857988569, 0, 394214737801353856806, 0, 48574382408263235899357, 0, 5985253660092787692417656, 0, 737493446266828179909952938, 0, 90872798038838203516056866947, 0, 11197206499225795090261197704699, 0, 1379702639850796292507235704702146, 0, 170004849926992719278991704373997176, 0, 20947737855224439454436411993182403357, 0, 2581148260013632103406161602393111702654, 0, 318045146178701217199623636073734490730345] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1883681112418 z - 2168307456904 z - 188 z 24 22 4 6 + 1883681112418 z - 1233642154928 z + 13693 z - 520176 z 8 10 12 14 + 11813424 z - 173367856 z + 1727091820 z - 12076947968 z 18 16 50 48 - 223089138864 z + 60701060660 z - 188 z + 13693 z 20 36 34 + 607014683472 z + 60701060660 z - 223089138864 z 30 42 44 46 52 - 1233642154928 z - 173367856 z + 11813424 z - 520176 z + z 32 38 40 / 2 + 607014683472 z - 12076947968 z + 1727091820 z ) / ((-1 + z ) (1 / 28 26 2 24 + 7358912737342 z - 8545686604442 z - 247 z + 7358912737342 z 22 4 6 8 10 - 4696505910974 z + 22329 z - 1004618 z + 26153966 z - 430422538 z 12 14 18 16 + 4735324148 z - 36143075524 z - 771181410814 z + 196264551044 z 50 48 20 36 - 247 z + 22329 z + 2217157028362 z + 196264551044 z 34 30 42 44 - 771181410814 z - 4696505910974 z - 430422538 z + 26153966 z 46 52 32 38 40 - 1004618 z + z + 2217157028362 z - 36143075524 z + 4735324148 z )) And in Maple-input format, it is: -(1+1883681112418*z^28-2168307456904*z^26-188*z^2+1883681112418*z^24-\ 1233642154928*z^22+13693*z^4-520176*z^6+11813424*z^8-173367856*z^10+1727091820* z^12-12076947968*z^14-223089138864*z^18+60701060660*z^16-188*z^50+13693*z^48+ 607014683472*z^20+60701060660*z^36-223089138864*z^34-1233642154928*z^30-\ 173367856*z^42+11813424*z^44-520176*z^46+z^52+607014683472*z^32-12076947968*z^ 38+1727091820*z^40)/(-1+z^2)/(1+7358912737342*z^28-8545686604442*z^26-247*z^2+ 7358912737342*z^24-4696505910974*z^22+22329*z^4-1004618*z^6+26153966*z^8-\ 430422538*z^10+4735324148*z^12-36143075524*z^14-771181410814*z^18+196264551044* z^16-247*z^50+22329*z^48+2217157028362*z^20+196264551044*z^36-771181410814*z^34 -4696505910974*z^30-430422538*z^42+26153966*z^44-1004618*z^46+z^52+ 2217157028362*z^32-36143075524*z^38+4735324148*z^40) The first , 40, terms are: [0, 60, 0, 5997, 0, 639467, 0, 69471204, 0, 7604546367, 0, 835330221887, 0, 91914828491076, 0, 10122456546824187, 0, 1115258752260987613, 0, 122902851792606192604, 0, 13545581321482320320801, 0, 1492995793016841856808097, 0, 164563091238146834766917148, 0, 18138981524531720234947233085, 0, 1999386436995820506984747661723, 0, 220385143778684151957677895112452, 0, 24292307737517864265390891533368959, 0, 2677661573132652056893865439280369791, 0, 295150028752876534293168828559476835108, 0, 32533447893115455131241283046899603703243] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 9568446233584300 z - 4433357019219978 z - 247 z 24 22 4 6 + 1637729778836854 z - 479776707420592 z + 25614 z - 1484423 z 8 10 12 14 + 54164713 z - 1332709620 z + 23141985489 z - 292980674607 z 18 16 50 - 19932777981151 z + 2770486346278 z - 19932777981151 z 48 20 36 + 110687694492649 z + 110687694492649 z + 22924899814098278 z 34 66 64 30 - 25560285926272632 z - 247 z + 25614 z - 16531053423766250 z 42 44 46 - 4433357019219978 z + 1637729778836854 z - 479776707420592 z 58 56 54 52 - 1332709620 z + 23141985489 z - 292980674607 z + 2770486346278 z 60 68 32 38 + 54164713 z + z + 22924899814098278 z - 16531053423766250 z 40 62 / 2 + 9568446233584300 z - 1484423 z ) / ((-1 + z ) (1 / 28 26 2 + 36823365735661902 z - 16464703234205196 z - 318 z 24 22 4 6 + 5820127051943994 z - 1619347845132992 z + 39363 z - 2614702 z 8 10 12 14 + 106832449 z - 2901522480 z + 55076976789 z - 756623931654 z 18 16 50 - 59527395715990 z + 7715810911839 z - 59527395715990 z 48 20 36 + 352472329957269 z + 352472329957269 z + 92037496772934138 z 34 66 64 30 - 103175726342647072 z - 318 z + 39363 z - 65309031967622188 z 42 44 46 - 16464703234205196 z + 5820127051943994 z - 1619347845132992 z 58 56 54 52 - 2901522480 z + 55076976789 z - 756623931654 z + 7715810911839 z 60 68 32 38 + 106832449 z + z + 92037496772934138 z - 65309031967622188 z 40 62 + 36823365735661902 z - 2614702 z )) And in Maple-input format, it is: -(1+9568446233584300*z^28-4433357019219978*z^26-247*z^2+1637729778836854*z^24-\ 479776707420592*z^22+25614*z^4-1484423*z^6+54164713*z^8-1332709620*z^10+ 23141985489*z^12-292980674607*z^14-19932777981151*z^18+2770486346278*z^16-\ 19932777981151*z^50+110687694492649*z^48+110687694492649*z^20+22924899814098278 *z^36-25560285926272632*z^34-247*z^66+25614*z^64-16531053423766250*z^30-\ 4433357019219978*z^42+1637729778836854*z^44-479776707420592*z^46-1332709620*z^ 58+23141985489*z^56-292980674607*z^54+2770486346278*z^52+54164713*z^60+z^68+ 22924899814098278*z^32-16531053423766250*z^38+9568446233584300*z^40-1484423*z^ 62)/(-1+z^2)/(1+36823365735661902*z^28-16464703234205196*z^26-318*z^2+ 5820127051943994*z^24-1619347845132992*z^22+39363*z^4-2614702*z^6+106832449*z^8 -2901522480*z^10+55076976789*z^12-756623931654*z^14-59527395715990*z^18+ 7715810911839*z^16-59527395715990*z^50+352472329957269*z^48+352472329957269*z^ 20+92037496772934138*z^36-103175726342647072*z^34-318*z^66+39363*z^64-\ 65309031967622188*z^30-16464703234205196*z^42+5820127051943994*z^44-\ 1619347845132992*z^46-2901522480*z^58+55076976789*z^56-756623931654*z^54+ 7715810911839*z^52+106832449*z^60+z^68+92037496772934138*z^32-65309031967622188 *z^38+36823365735661902*z^40-2614702*z^62) The first , 40, terms are: [0, 72, 0, 8901, 0, 1152029, 0, 150106912, 0, 19589725181, 0, 2557865755609, 0, 334048136119408, 0, 43628700487891729, 0, 5698337933973992729, 0, 744267786536070948216, 0, 97210303484810785038973, 0, 12696856373371667855139717, 0, 1658366252986920374249955032, 0, 216603182146931970883299504113, 0, 28291063219308787753107787582697, 0, 3695164085938787300845417939160720, 0, 482634314722793226770826948351407969, 0, 63038035206277313444861323766915118069, 0, 8233550278710386328261959339344490681984, 0, 1075403921873215949332813848149592998925509] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2040888085162797665 z - 483547963134406709 z - 277 z 24 22 4 6 + 95501545319298248 z - 15636192540930658 z + 33196 z - 2306770 z 8 10 12 14 + 105233947 z - 3383532066 z + 80320519481 z - 1454565082870 z 18 16 50 - 231842063358534 z + 20584846386949 z - 202025347870166147334 z 48 20 + 305469733361927357356 z + 2107601038314067 z 36 34 + 113120398899806406476 z - 53562980333266114576 z 66 80 88 84 86 - 15636192540930658 z + 105233947 z + z + 33196 z - 277 z 82 64 30 - 2306770 z + 95501545319298248 z - 7212080708569153678 z 42 44 - 391356708277238222162 z + 425029640176858114468 z 46 58 - 391356708277238222162 z - 7212080708569153678 z 56 54 + 21413064398609644486 z - 53562980333266114576 z 52 60 + 113120398899806406476 z + 2040888085162797665 z 70 68 78 - 231842063358534 z + 2107601038314067 z - 3383532066 z 32 38 + 21413064398609644486 z - 202025347870166147334 z 40 62 76 + 305469733361927357356 z - 483547963134406709 z + 80320519481 z 74 72 / - 1454565082870 z + 20584846386949 z ) / (-1 / 28 26 2 - 8389923017378755216 z + 1847840614946414361 z + 343 z 24 22 4 6 - 339274182802339262 z + 51624192224680951 z - 48258 z + 3817777 z 8 10 12 14 - 194560020 z + 6902658605 z - 179312551600 z + 3533000289501 z 18 16 50 + 659241718257631 z - 54176112311486 z + 1975956365684458822755 z 48 20 - 2741096008658716314391 z - 6461920285534638 z 36 34 - 626554127725224759487 z + 275005414303842304163 z 66 80 90 88 84 + 339274182802339262 z - 6902658605 z + z - 343 z - 3817777 z 86 82 64 + 48258 z + 194560020 z - 1847840614946414361 z 30 42 + 31906313188542686542 z + 2741096008658716314391 z 44 46 - 3228035642103474102899 z + 3228035642103474102899 z 58 56 + 102010487628743324223 z - 275005414303842304163 z 54 52 + 626554127725224759487 z - 1208514430680290993467 z 60 70 68 - 31906313188542686542 z + 6461920285534638 z - 51624192224680951 z 78 32 + 179312551600 z - 102010487628743324223 z 38 40 + 1208514430680290993467 z - 1975956365684458822755 z 62 76 74 + 8389923017378755216 z - 3533000289501 z + 54176112311486 z 72 - 659241718257631 z ) And in Maple-input format, it is: -(1+2040888085162797665*z^28-483547963134406709*z^26-277*z^2+95501545319298248* z^24-15636192540930658*z^22+33196*z^4-2306770*z^6+105233947*z^8-3383532066*z^10 +80320519481*z^12-1454565082870*z^14-231842063358534*z^18+20584846386949*z^16-\ 202025347870166147334*z^50+305469733361927357356*z^48+2107601038314067*z^20+ 113120398899806406476*z^36-53562980333266114576*z^34-15636192540930658*z^66+ 105233947*z^80+z^88+33196*z^84-277*z^86-2306770*z^82+95501545319298248*z^64-\ 7212080708569153678*z^30-391356708277238222162*z^42+425029640176858114468*z^44-\ 391356708277238222162*z^46-7212080708569153678*z^58+21413064398609644486*z^56-\ 53562980333266114576*z^54+113120398899806406476*z^52+2040888085162797665*z^60-\ 231842063358534*z^70+2107601038314067*z^68-3383532066*z^78+21413064398609644486 *z^32-202025347870166147334*z^38+305469733361927357356*z^40-483547963134406709* z^62+80320519481*z^76-1454565082870*z^74+20584846386949*z^72)/(-1-\ 8389923017378755216*z^28+1847840614946414361*z^26+343*z^2-339274182802339262*z^ 24+51624192224680951*z^22-48258*z^4+3817777*z^6-194560020*z^8+6902658605*z^10-\ 179312551600*z^12+3533000289501*z^14+659241718257631*z^18-54176112311486*z^16+ 1975956365684458822755*z^50-2741096008658716314391*z^48-6461920285534638*z^20-\ 626554127725224759487*z^36+275005414303842304163*z^34+339274182802339262*z^66-\ 6902658605*z^80+z^90-343*z^88-3817777*z^84+48258*z^86+194560020*z^82-\ 1847840614946414361*z^64+31906313188542686542*z^30+2741096008658716314391*z^42-\ 3228035642103474102899*z^44+3228035642103474102899*z^46+102010487628743324223*z ^58-275005414303842304163*z^56+626554127725224759487*z^54-\ 1208514430680290993467*z^52-31906313188542686542*z^60+6461920285534638*z^70-\ 51624192224680951*z^68+179312551600*z^78-102010487628743324223*z^32+ 1208514430680290993467*z^38-1975956365684458822755*z^40+8389923017378755216*z^ 62-3533000289501*z^76+54176112311486*z^74-659241718257631*z^72) The first , 40, terms are: [0, 66, 0, 7576, 0, 924547, 0, 114164222, 0, 14143182791, 0, 1754085668347, 0, 217641675671008, 0, 27009100390275249, 0, 3352050997460000849, 0, 416030393069884004782, 0, 51635163399153475493853, 0, 6408681637470779011763976, 0, 795413578276317722131700826, 0, 98722879353835738233403251307, 0, 12253011792734715103517674834707, 0, 1520785589597952205891477807679602, 0, 188752698670361082703060222106717768, 0, 23427091291231505201673675530772075661, 0, 2907659710468793756423336948216186969862, 0, 360884966908436940121647245916284090496561] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2560460111954 z - 2952381888456 z - 188 z 24 22 4 6 + 2560460111954 z - 1667950271172 z + 13989 z - 553740 z 8 10 12 14 + 13189352 z - 202647556 z + 2102208904 z - 15204940640 z 18 16 50 48 - 294448891948 z + 78503311744 z - 188 z + 13989 z 20 36 34 + 812919347992 z + 78503311744 z - 294448891948 z 30 42 44 46 52 - 1667950271172 z - 202647556 z + 13189352 z - 553740 z + z 32 38 40 / 2 + 812919347992 z - 15204940640 z + 2102208904 z ) / ((-1 + z ) (1 / 28 26 2 24 + 10051778794690 z - 11692563274506 z - 247 z + 10051778794690 z 22 4 6 8 10 - 6380233696956 z + 22293 z - 1032980 z + 28199248 z - 489267500 z 12 14 18 16 + 5659731496 z - 45093084548 z - 1019862297860 z + 253258409472 z 50 48 20 36 - 247 z + 22293 z + 2981226353328 z + 253258409472 z 34 30 42 44 - 1019862297860 z - 6380233696956 z - 489267500 z + 28199248 z 46 52 32 38 40 - 1032980 z + z + 2981226353328 z - 45093084548 z + 5659731496 z )) And in Maple-input format, it is: -(1+2560460111954*z^28-2952381888456*z^26-188*z^2+2560460111954*z^24-\ 1667950271172*z^22+13989*z^4-553740*z^6+13189352*z^8-202647556*z^10+2102208904* z^12-15204940640*z^14-294448891948*z^18+78503311744*z^16-188*z^50+13989*z^48+ 812919347992*z^20+78503311744*z^36-294448891948*z^34-1667950271172*z^30-\ 202647556*z^42+13189352*z^44-553740*z^46+z^52+812919347992*z^32-15204940640*z^ 38+2102208904*z^40)/(-1+z^2)/(1+10051778794690*z^28-11692563274506*z^26-247*z^2 +10051778794690*z^24-6380233696956*z^22+22293*z^4-1032980*z^6+28199248*z^8-\ 489267500*z^10+5659731496*z^12-45093084548*z^14-1019862297860*z^18+253258409472 *z^16-247*z^50+22293*z^48+2981226353328*z^20+253258409472*z^36-1019862297860*z^ 34-6380233696956*z^30-489267500*z^42+28199248*z^44-1032980*z^46+z^52+ 2981226353328*z^32-45093084548*z^38+5659731496*z^40) The first , 40, terms are: [0, 60, 0, 6329, 0, 718725, 0, 82861644, 0, 9589334541, 0, 1110895041109, 0, 128731747204140, 0, 14918859033842701, 0, 1729006385763234929, 0, 200383008832885692508, 0, 23223420889696568757369, 0, 2691483988680791751637257, 0, 311930257155748625275824156, 0, 36151243108214875153072449921, 0, 4189758372250120075904771639165, 0, 485573214881857482297832157997164, 0, 56275643265657321000205587784142661, 0, 6522081387779214212364801424578414621, 0, 755878443496984996338692527344629486092, 0, 87602743267789145000717805079896995835157] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 13183792802199628 z - 6022154157701902 z - 253 z 24 22 4 6 + 2185207188438462 z - 626784996596320 z + 26858 z - 1592441 z 8 10 12 14 + 59430689 z - 1496384224 z + 26620201217 z - 345692875393 z 18 16 50 - 24789637152725 z + 3355940197194 z - 24789637152725 z 48 20 36 + 141214060773921 z + 141214060773921 z + 32113476723123070 z 34 66 64 30 - 35879848528404608 z - 253 z + 26858 z - 23013073156581286 z 42 44 46 - 6022154157701902 z + 2185207188438462 z - 626784996596320 z 58 56 54 52 - 1496384224 z + 26620201217 z - 345692875393 z + 3355940197194 z 60 68 32 38 + 59430689 z + z + 32113476723123070 z - 23013073156581286 z 40 62 / 28 + 13183792802199628 z - 1592441 z ) / (-1 - 73746802226580770 z / 26 2 24 + 30300274314494830 z + 327 z - 9901214361764466 z 22 4 6 8 + 2560783139244973 z - 41529 z + 2831355 z - 118986153 z 10 12 14 18 + 3337231987 z - 65780283543 z + 944281895397 z + 82668729408197 z 16 50 48 - 10126892437599 z + 520918731974923 z - 2560783139244973 z 20 36 34 - 520918731974923 z - 277633601094252822 z + 277633601094252822 z 66 64 30 42 + 41529 z - 2831355 z + 143270428033725894 z + 73746802226580770 z 44 46 58 - 30300274314494830 z + 9901214361764466 z + 65780283543 z 56 54 52 - 944281895397 z + 10126892437599 z - 82668729408197 z 60 70 68 32 - 3337231987 z + z - 327 z - 222751663410763826 z 38 40 62 + 222751663410763826 z - 143270428033725894 z + 118986153 z ) And in Maple-input format, it is: -(1+13183792802199628*z^28-6022154157701902*z^26-253*z^2+2185207188438462*z^24-\ 626784996596320*z^22+26858*z^4-1592441*z^6+59430689*z^8-1496384224*z^10+ 26620201217*z^12-345692875393*z^14-24789637152725*z^18+3355940197194*z^16-\ 24789637152725*z^50+141214060773921*z^48+141214060773921*z^20+32113476723123070 *z^36-35879848528404608*z^34-253*z^66+26858*z^64-23013073156581286*z^30-\ 6022154157701902*z^42+2185207188438462*z^44-626784996596320*z^46-1496384224*z^ 58+26620201217*z^56-345692875393*z^54+3355940197194*z^52+59430689*z^60+z^68+ 32113476723123070*z^32-23013073156581286*z^38+13183792802199628*z^40-1592441*z^ 62)/(-1-73746802226580770*z^28+30300274314494830*z^26+327*z^2-9901214361764466* z^24+2560783139244973*z^22-41529*z^4+2831355*z^6-118986153*z^8+3337231987*z^10-\ 65780283543*z^12+944281895397*z^14+82668729408197*z^18-10126892437599*z^16+ 520918731974923*z^50-2560783139244973*z^48-520918731974923*z^20-\ 277633601094252822*z^36+277633601094252822*z^34+41529*z^66-2831355*z^64+ 143270428033725894*z^30+73746802226580770*z^42-30300274314494830*z^44+ 9901214361764466*z^46+65780283543*z^58-944281895397*z^56+10126892437599*z^54-\ 82668729408197*z^52-3337231987*z^60+z^70-327*z^68-222751663410763826*z^32+ 222751663410763826*z^38-143270428033725894*z^40+118986153*z^62) The first , 40, terms are: [0, 74, 0, 9527, 0, 1281097, 0, 173236742, 0, 23455928847, 0, 3177194475967, 0, 430427893727686, 0, 58315234821838489, 0, 7900841450568703751, 0, 1070455078041601928970, 0, 145032397531733221787697, 0, 19649983315712427357850065, 0, 2662315822203062937920102538, 0, 360709059270396656403698724455, 0, 48871375396284861275870143496953, 0, 6621434498993410332827577281651526, 0, 897118098642673130361034404275739743, 0, 121547813695299857482563953357388866415, 0, 16468145125068961626665226489285806156422, 0, 2231219104831973947883917190631028223779305] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 13175235542586140 z - 6011039876970226 z - 253 z 24 22 4 6 + 2178518833245626 z - 624192203805808 z + 26798 z - 1585155 z 8 10 12 14 + 59073007 z - 1486607556 z + 26447071451 z - 343527131807 z 18 16 50 - 24651867818273 z + 3335947416630 z - 24651867818273 z 48 20 36 + 140511784521109 z + 140511784521109 z + 32146405450410470 z 34 66 64 30 - 35925035861134744 z - 253 z + 26798 z - 23021287197815886 z 42 44 46 - 6011039876970226 z + 2178518833245626 z - 624192203805808 z 58 56 54 52 - 1486607556 z + 26447071451 z - 343527131807 z + 3335947416630 z 60 68 32 38 + 59073007 z + z + 32146405450410470 z - 23021287197815886 z 40 62 / 2 + 13175235542586140 z - 1585155 z ) / ((-1 + z ) (1 / 28 26 2 + 50892358977561050 z - 22364706555544704 z - 324 z 24 22 4 6 + 7739971631851218 z - 2101755306144248 z + 40869 z - 2766614 z 8 10 12 14 + 115325897 z - 3202101130 z + 62287689493 z - 878726541222 z 18 16 50 - 73150664407916 z + 9215302774401 z - 73150664407916 z 48 20 36 + 445452494428013 z + 445452494428013 z + 129864717590486754 z 34 66 64 30 - 145968654272429308 z - 324 z + 40869 z - 91426958164387860 z 42 44 46 - 22364706555544704 z + 7739971631851218 z - 2101755306144248 z 58 56 54 52 - 3202101130 z + 62287689493 z - 878726541222 z + 9215302774401 z 60 68 32 38 + 115325897 z + z + 129864717590486754 z - 91426958164387860 z 40 62 + 50892358977561050 z - 2766614 z )) And in Maple-input format, it is: -(1+13175235542586140*z^28-6011039876970226*z^26-253*z^2+2178518833245626*z^24-\ 624192203805808*z^22+26798*z^4-1585155*z^6+59073007*z^8-1486607556*z^10+ 26447071451*z^12-343527131807*z^14-24651867818273*z^18+3335947416630*z^16-\ 24651867818273*z^50+140511784521109*z^48+140511784521109*z^20+32146405450410470 *z^36-35925035861134744*z^34-253*z^66+26798*z^64-23021287197815886*z^30-\ 6011039876970226*z^42+2178518833245626*z^44-624192203805808*z^46-1486607556*z^ 58+26447071451*z^56-343527131807*z^54+3335947416630*z^52+59073007*z^60+z^68+ 32146405450410470*z^32-23021287197815886*z^38+13175235542586140*z^40-1585155*z^ 62)/(-1+z^2)/(1+50892358977561050*z^28-22364706555544704*z^26-324*z^2+ 7739971631851218*z^24-2101755306144248*z^22+40869*z^4-2766614*z^6+115325897*z^8 -3202101130*z^10+62287689493*z^12-878726541222*z^14-73150664407916*z^18+ 9215302774401*z^16-73150664407916*z^50+445452494428013*z^48+445452494428013*z^ 20+129864717590486754*z^36-145968654272429308*z^34-324*z^66+40869*z^64-\ 91426958164387860*z^30-22364706555544704*z^42+7739971631851218*z^44-\ 2101755306144248*z^46-3202101130*z^58+62287689493*z^56-878726541222*z^54+ 9215302774401*z^52+115325897*z^60+z^68+129864717590486754*z^32-\ 91426958164387860*z^38+50892358977561050*z^40-2766614*z^62) The first , 40, terms are: [0, 72, 0, 9005, 0, 1183057, 0, 156669832, 0, 20793571493, 0, 2762011726397, 0, 366997591286792, 0, 48770697384223657, 0, 6481548126293208213, 0, 861407212655054496712, 0, 114483367590017680580009, 0, 15215211941454277182860249, 0, 2022154480655680146079416520, 0, 268751532924119851924153128517, 0, 35718046073772788575951936068985, 0, 4747057445628128378629389005633288, 0, 630901126176629041680762713832427149, 0, 83849046202454730038581675497279620949, 0, 11143842222764818869861838794846486228104, 0, 1481057036956199947706386806860155879494785] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 21864300290344 z - 19107800668223 z - 191 z 24 22 4 6 + 12747405788261 z - 6481419735419 z + 14593 z - 605152 z 8 10 12 14 + 15493332 z - 262974127 z + 3095436454 z - 26054049474 z 18 16 50 48 - 731284739944 z + 160236177837 z - 605152 z + 15493332 z 20 36 34 + 2503921843088 z + 2503921843088 z - 6481419735419 z 30 42 44 46 - 19107800668223 z - 26054049474 z + 3095436454 z - 262974127 z 56 54 52 32 38 + z - 191 z + 14593 z + 12747405788261 z - 731284739944 z 40 / 2 28 + 160236177837 z ) / ((-1 + z ) (1 + 88515015875571 z / 26 2 24 22 - 76681079164498 z - 258 z + 49860387360125 z - 24333878609240 z 4 6 8 10 12 + 23703 z - 1131632 z + 32676205 z - 617981386 z + 8032513683 z 14 18 16 50 - 74067317228 z - 2435799011030 z + 495147800601 z - 1131632 z 48 20 36 + 32676205 z + 8904485692387 z + 8904485692387 z 34 30 42 - 24333878609240 z - 76681079164498 z - 74067317228 z 44 46 56 54 52 + 8032513683 z - 617981386 z + z - 258 z + 23703 z 32 38 40 + 49860387360125 z - 2435799011030 z + 495147800601 z )) And in Maple-input format, it is: -(1+21864300290344*z^28-19107800668223*z^26-191*z^2+12747405788261*z^24-\ 6481419735419*z^22+14593*z^4-605152*z^6+15493332*z^8-262974127*z^10+3095436454* z^12-26054049474*z^14-731284739944*z^18+160236177837*z^16-605152*z^50+15493332* z^48+2503921843088*z^20+2503921843088*z^36-6481419735419*z^34-19107800668223*z^ 30-26054049474*z^42+3095436454*z^44-262974127*z^46+z^56-191*z^54+14593*z^52+ 12747405788261*z^32-731284739944*z^38+160236177837*z^40)/(-1+z^2)/(1+ 88515015875571*z^28-76681079164498*z^26-258*z^2+49860387360125*z^24-\ 24333878609240*z^22+23703*z^4-1131632*z^6+32676205*z^8-617981386*z^10+ 8032513683*z^12-74067317228*z^14-2435799011030*z^18+495147800601*z^16-1131632*z ^50+32676205*z^48+8904485692387*z^20+8904485692387*z^36-24333878609240*z^34-\ 76681079164498*z^30-74067317228*z^42+8032513683*z^44-617981386*z^46+z^56-258*z^ 54+23703*z^52+49860387360125*z^32-2435799011030*z^38+495147800601*z^40) The first , 40, terms are: [0, 68, 0, 8244, 0, 1056031, 0, 136225820, 0, 17592260877, 0, 2272336118853, 0, 293523059040832, 0, 37915454665010651, 0, 4897691399199398843, 0, 632654872589286487756, 0, 81722637869401509137129, 0, 10556450545496083648000780, 0, 1363620320830701934534540164, 0, 176144470018339683178495992911, 0, 22753308877742525448389617586159, 0, 2939138906774713428347242651546836, 0, 379660714864560891888683104914345292, 0, 49042343008251612469456315907368090873, 0, 6335002052064568732404923363205894586268, 0, 818318386480561098240810113905059160016555] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 4}, {3, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6267534484742184 z - 2567666174513160 z - 214 z 24 22 4 6 + 856616021806972 z - 231646641227556 z + 19002 z - 948122 z 8 10 12 14 + 30363313 z - 672944512 z + 10829843992 z - 130777593960 z 18 16 50 - 8794152874124 z + 1213089946204 z - 231646641227556 z 48 20 36 + 856616021806972 z + 50474316662808 z + 30248148003916252 z 34 66 64 - 27426594553298592 z - 948122 z + 30363313 z 30 42 44 - 12502880267093840 z - 12502880267093840 z + 6267534484742184 z 46 58 56 - 2567666174513160 z - 130777593960 z + 1213089946204 z 54 52 60 70 - 8794152874124 z + 50474316662808 z + 10829843992 z - 214 z 68 32 38 + 19002 z + 20437790465299958 z - 27426594553298592 z 40 62 72 / + 20437790465299958 z - 672944512 z + z ) / (-1 / 28 26 2 - 32647108767611384 z + 12198310133888016 z + 279 z 24 22 4 6 - 3714864758085868 z + 917676141807508 z - 29788 z + 1713336 z 8 10 12 14 - 61853559 z + 1526008093 z - 27132836144 z + 360326380152 z 18 16 50 + 29096107959692 z - 3665487028648 z + 3714864758085868 z 48 20 36 - 12198310133888016 z - 182737041077100 z - 230099952529059500 z 34 66 64 + 189457898216388026 z + 61853559 z - 1526008093 z 30 42 44 + 71476812702307488 z + 128363546739672938 z - 71476812702307488 z 46 58 56 + 32647108767611384 z + 3665487028648 z - 29096107959692 z 54 52 60 70 + 182737041077100 z - 917676141807508 z - 360326380152 z + 29788 z 68 32 38 - 1713336 z - 128363546739672938 z + 230099952529059500 z 40 62 74 72 - 189457898216388026 z + 27132836144 z + z - 279 z ) And in Maple-input format, it is: -(1+6267534484742184*z^28-2567666174513160*z^26-214*z^2+856616021806972*z^24-\ 231646641227556*z^22+19002*z^4-948122*z^6+30363313*z^8-672944512*z^10+ 10829843992*z^12-130777593960*z^14-8794152874124*z^18+1213089946204*z^16-\ 231646641227556*z^50+856616021806972*z^48+50474316662808*z^20+30248148003916252 *z^36-27426594553298592*z^34-948122*z^66+30363313*z^64-12502880267093840*z^30-\ 12502880267093840*z^42+6267534484742184*z^44-2567666174513160*z^46-130777593960 *z^58+1213089946204*z^56-8794152874124*z^54+50474316662808*z^52+10829843992*z^ 60-214*z^70+19002*z^68+20437790465299958*z^32-27426594553298592*z^38+ 20437790465299958*z^40-672944512*z^62+z^72)/(-1-32647108767611384*z^28+ 12198310133888016*z^26+279*z^2-3714864758085868*z^24+917676141807508*z^22-29788 *z^4+1713336*z^6-61853559*z^8+1526008093*z^10-27132836144*z^12+360326380152*z^ 14+29096107959692*z^18-3665487028648*z^16+3714864758085868*z^50-\ 12198310133888016*z^48-182737041077100*z^20-230099952529059500*z^36+ 189457898216388026*z^34+61853559*z^66-1526008093*z^64+71476812702307488*z^30+ 128363546739672938*z^42-71476812702307488*z^44+32647108767611384*z^46+ 3665487028648*z^58-29096107959692*z^56+182737041077100*z^54-917676141807508*z^ 52-360326380152*z^60+29788*z^70-1713336*z^68-128363546739672938*z^32+ 230099952529059500*z^38-189457898216388026*z^40+27132836144*z^62+z^74-279*z^72) The first , 40, terms are: [0, 65, 0, 7349, 0, 879365, 0, 106307417, 0, 12889133233, 0, 1564356274853, 0, 189942917554749, 0, 23066440881308117, 0, 2801342933507532509, 0, 340222752550500731289, 0, 41320449051360481493761, 0, 5018438099845713468887037, 0, 609498811299607443615616717, 0, 74024836753198219525936436489, 0, 8990465454677724211508720092745, 0, 1091910254568712753851506280472441, 0, 132614720029728664064845350139663865, 0, 16106327625411447293078543465291095965, 0, 1956146282112953122123220328854865580237, 0, 237577949345280876915429591842964877321841] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 4}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1031374221940363513 z - 252102517542904245 z - 261 z 24 22 4 6 + 51536157956878709 z - 8760971154494068 z + 29413 z - 1923600 z 8 10 12 14 + 82763788 z - 2515944748 z + 56610603747 z - 974163944543 z 18 16 50 - 141265131655924 z + 13132969212535 z - 92124629760404821031 z 48 20 + 137985364895605800147 z + 1229739568936064 z 36 34 + 52269364351852352255 z - 25172475285977334712 z 66 80 88 84 86 - 8760971154494068 z + 82763788 z + z + 29413 z - 261 z 82 64 30 - 1923600 z + 51536157956878709 z - 3544870433501247308 z 42 44 - 175780947247422924760 z + 190544253061540355008 z 46 58 - 175780947247422924760 z - 3544870433501247308 z 56 54 + 10272919370462822100 z - 25172475285977334712 z 52 60 70 + 52269364351852352255 z + 1031374221940363513 z - 141265131655924 z 68 78 32 + 1229739568936064 z - 2515944748 z + 10272919370462822100 z 38 40 - 92124629760404821031 z + 137985364895605800147 z 62 76 74 - 252102517542904245 z + 56610603747 z - 974163944543 z 72 / 28 + 13132969212535 z ) / (-1 - 4306818662417440546 z / 26 2 24 + 976376035667837389 z + 331 z - 185212417502633547 z 22 4 6 8 + 29226658860469890 z - 44168 z + 3289714 z - 157569627 z 10 12 14 + 5260116891 z - 128897966168 z + 2403441921474 z 18 16 50 + 406031107155021 z - 35002582575747 z + 916809569617649346629 z 48 20 - 1264448808700543988039 z - 3808473079463378 z 36 34 - 296780607411458070546 z + 132236690363428473637 z 66 80 90 88 84 + 185212417502633547 z - 5260116891 z + z - 331 z - 3289714 z 86 82 64 + 44168 z + 157569627 z - 976376035667837389 z 30 42 + 15969722288342621444 z + 1264448808700543988039 z 44 46 - 1484772268615579982732 z + 1484772268615579982732 z 58 56 + 49959662854524604357 z - 132236690363428473637 z 54 52 + 296780607411458070546 z - 565684651414570553460 z 60 70 68 - 15969722288342621444 z + 3808473079463378 z - 29226658860469890 z 78 32 38 + 128897966168 z - 49959662854524604357 z + 565684651414570553460 z 40 62 76 - 916809569617649346629 z + 4306818662417440546 z - 2403441921474 z 74 72 + 35002582575747 z - 406031107155021 z ) And in Maple-input format, it is: -(1+1031374221940363513*z^28-252102517542904245*z^26-261*z^2+51536157956878709* z^24-8760971154494068*z^22+29413*z^4-1923600*z^6+82763788*z^8-2515944748*z^10+ 56610603747*z^12-974163944543*z^14-141265131655924*z^18+13132969212535*z^16-\ 92124629760404821031*z^50+137985364895605800147*z^48+1229739568936064*z^20+ 52269364351852352255*z^36-25172475285977334712*z^34-8760971154494068*z^66+ 82763788*z^80+z^88+29413*z^84-261*z^86-1923600*z^82+51536157956878709*z^64-\ 3544870433501247308*z^30-175780947247422924760*z^42+190544253061540355008*z^44-\ 175780947247422924760*z^46-3544870433501247308*z^58+10272919370462822100*z^56-\ 25172475285977334712*z^54+52269364351852352255*z^52+1031374221940363513*z^60-\ 141265131655924*z^70+1229739568936064*z^68-2515944748*z^78+10272919370462822100 *z^32-92124629760404821031*z^38+137985364895605800147*z^40-252102517542904245*z ^62+56610603747*z^76-974163944543*z^74+13132969212535*z^72)/(-1-\ 4306818662417440546*z^28+976376035667837389*z^26+331*z^2-185212417502633547*z^ 24+29226658860469890*z^22-44168*z^4+3289714*z^6-157569627*z^8+5260116891*z^10-\ 128897966168*z^12+2403441921474*z^14+406031107155021*z^18-35002582575747*z^16+ 916809569617649346629*z^50-1264448808700543988039*z^48-3808473079463378*z^20-\ 296780607411458070546*z^36+132236690363428473637*z^34+185212417502633547*z^66-\ 5260116891*z^80+z^90-331*z^88-3289714*z^84+44168*z^86+157569627*z^82-\ 976376035667837389*z^64+15969722288342621444*z^30+1264448808700543988039*z^42-\ 1484772268615579982732*z^44+1484772268615579982732*z^46+49959662854524604357*z^ 58-132236690363428473637*z^56+296780607411458070546*z^54-565684651414570553460* z^52-15969722288342621444*z^60+3808473079463378*z^70-29226658860469890*z^68+ 128897966168*z^78-49959662854524604357*z^32+565684651414570553460*z^38-\ 916809569617649346629*z^40+4306818662417440546*z^62-2403441921474*z^76+ 35002582575747*z^74-406031107155021*z^72) The first , 40, terms are: [0, 70, 0, 8415, 0, 1059719, 0, 134567410, 0, 17133385481, 0, 2183722068441, 0, 278443703778026, 0, 35510371863260759, 0, 4529035710916325183, 0, 577657009279635408766, 0, 73678395183155192833873, 0, 9397508284283719341693537, 0, 1198633219138213127585606926, 0, 152883401839904612826129218991, 0, 19499997226255325978926300776007, 0, 2487189290125733446658691532746810, 0, 317236508235826166635665897259378697, 0, 40462945765068411978187648009815958201, 0, 5160975989255165508629630861363306377250, 0, 658273212900363805864600763237886118696439] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 4}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 14924729277504 z - 13120694650591 z - 193 z 24 22 4 6 + 8906887596279 z - 4656285114609 z + 14807 z - 606964 z 8 10 12 14 + 15133626 z - 247408313 z + 2784944218 z - 22331344870 z 18 16 50 48 - 568932974786 z + 130728948403 z - 606964 z + 15133626 z 20 36 34 + 1865710910768 z + 1865710910768 z - 4656285114609 z 30 42 44 46 - 13120694650591 z - 22331344870 z + 2784944218 z - 247408313 z 56 54 52 32 38 + z - 193 z + 14807 z + 8906887596279 z - 568932974786 z 40 / 2 28 + 130728948403 z ) / ((-1 + z ) (1 + 60947402012039 z / 26 2 24 22 - 53128536089450 z - 262 z + 35179742384005 z - 17671778527140 z 4 6 8 10 12 + 24307 z - 1149838 z + 32371057 z - 589989600 z + 7334337117 z 14 18 16 50 - 64411927502 z - 1919927378570 z + 409629178279 z - 1149838 z 48 20 36 + 32371057 z + 6714834860165 z + 6714834860165 z 34 30 42 - 17671778527140 z - 53128536089450 z - 64411927502 z 44 46 56 54 52 + 7334337117 z - 589989600 z + z - 262 z + 24307 z 32 38 40 + 35179742384005 z - 1919927378570 z + 409629178279 z )) And in Maple-input format, it is: -(1+14924729277504*z^28-13120694650591*z^26-193*z^2+8906887596279*z^24-\ 4656285114609*z^22+14807*z^4-606964*z^6+15133626*z^8-247408313*z^10+2784944218* z^12-22331344870*z^14-568932974786*z^18+130728948403*z^16-606964*z^50+15133626* z^48+1865710910768*z^20+1865710910768*z^36-4656285114609*z^34-13120694650591*z^ 30-22331344870*z^42+2784944218*z^44-247408313*z^46+z^56-193*z^54+14807*z^52+ 8906887596279*z^32-568932974786*z^38+130728948403*z^40)/(-1+z^2)/(1+ 60947402012039*z^28-53128536089450*z^26-262*z^2+35179742384005*z^24-\ 17671778527140*z^22+24307*z^4-1149838*z^6+32371057*z^8-589989600*z^10+ 7334337117*z^12-64411927502*z^14-1919927378570*z^18+409629178279*z^16-1149838*z ^50+32371057*z^48+6714834860165*z^20+6714834860165*z^36-17671778527140*z^34-\ 53128536089450*z^30-64411927502*z^42+7334337117*z^44-589989600*z^46+z^56-262*z^ 54+24307*z^52+35179742384005*z^32-1919927378570*z^38+409629178279*z^40) The first , 40, terms are: [0, 70, 0, 8648, 0, 1121775, 0, 146356994, 0, 19113495101, 0, 2496667896883, 0, 326142513486560, 0, 42605142740344379, 0, 5565692355063358565, 0, 727071643159801844618, 0, 94980727706910630084927, 0, 12407773859825512611718072, 0, 1620885273799943079165098222, 0, 211743795525764419473659520641, 0, 27661078799824944999909200975937, 0, 3613495641714570769329486954027038, 0, 472047777241041162421632176378970712, 0, 61665801249947356718237022194858807711, 0, 8055691028500584304402880996807118511482, 0, 1052352464951973310942478195079796551867669] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6511935119036790 z - 2664887409232139 z - 217 z 24 22 4 6 + 888794698335934 z - 240527972573242 z + 19530 z - 985451 z 8 10 12 14 + 31793638 z - 707122858 z + 11383893038 z - 137224596770 z 18 16 50 - 9172506501912 z + 1269230963177 z - 240527972573242 z 48 20 36 + 888794698335934 z + 52504716556860 z + 31569924854445658 z 34 66 64 - 28613655516532842 z - 985451 z + 31793638 z 30 42 44 - 13010129814530471 z - 13010129814530471 z + 6511935119036790 z 46 58 56 - 2664887409232139 z - 137224596770 z + 1269230963177 z 54 52 60 70 - 9172506501912 z + 52504716556860 z + 11383893038 z - 217 z 68 32 38 + 19530 z + 21299033557122745 z - 28613655516532842 z 40 62 72 / 2 + 21299033557122745 z - 707122858 z + z ) / ((-1 + z ) (1 / 28 26 2 + 23489833524267783 z - 9315613817763224 z - 270 z 24 22 4 6 + 2991050157504547 z - 774145741792620 z + 28714 z - 1663916 z 8 10 12 14 + 60461917 z - 1492290432 z + 26342901324 z - 344691077040 z 18 16 50 - 26464754562216 z + 3430374933079 z - 774145741792620 z 48 20 36 + 2991050157504547 z + 160548329099171 z + 120532748409901211 z 34 66 64 - 108853875430645912 z - 1663916 z + 60461917 z 30 42 44 - 48102171176351438 z - 48102171176351438 z + 23489833524267783 z 46 58 56 - 9315613817763224 z - 344691077040 z + 3430374933079 z 54 52 60 70 - 26464754562216 z + 160548329099171 z + 26342901324 z - 270 z 68 32 38 + 28714 z + 80161354358634393 z - 108853875430645912 z 40 62 72 + 80161354358634393 z - 1492290432 z + z )) And in Maple-input format, it is: -(1+6511935119036790*z^28-2664887409232139*z^26-217*z^2+888794698335934*z^24-\ 240527972573242*z^22+19530*z^4-985451*z^6+31793638*z^8-707122858*z^10+ 11383893038*z^12-137224596770*z^14-9172506501912*z^18+1269230963177*z^16-\ 240527972573242*z^50+888794698335934*z^48+52504716556860*z^20+31569924854445658 *z^36-28613655516532842*z^34-985451*z^66+31793638*z^64-13010129814530471*z^30-\ 13010129814530471*z^42+6511935119036790*z^44-2664887409232139*z^46-137224596770 *z^58+1269230963177*z^56-9172506501912*z^54+52504716556860*z^52+11383893038*z^ 60-217*z^70+19530*z^68+21299033557122745*z^32-28613655516532842*z^38+ 21299033557122745*z^40-707122858*z^62+z^72)/(-1+z^2)/(1+23489833524267783*z^28-\ 9315613817763224*z^26-270*z^2+2991050157504547*z^24-774145741792620*z^22+28714* z^4-1663916*z^6+60461917*z^8-1492290432*z^10+26342901324*z^12-344691077040*z^14 -26464754562216*z^18+3430374933079*z^16-774145741792620*z^50+2991050157504547*z ^48+160548329099171*z^20+120532748409901211*z^36-108853875430645912*z^34-\ 1663916*z^66+60461917*z^64-48102171176351438*z^30-48102171176351438*z^42+ 23489833524267783*z^44-9315613817763224*z^46-344691077040*z^58+3430374933079*z^ 56-26464754562216*z^54+160548329099171*z^52+26342901324*z^60-270*z^70+28714*z^ 68+80161354358634393*z^32-108853875430645912*z^38+80161354358634393*z^40-\ 1492290432*z^62+z^72) The first , 40, terms are: [0, 54, 0, 5180, 0, 545823, 0, 58850738, 0, 6387074075, 0, 694629181811, 0, 75598335590952, 0, 8229690462067325, 0, 895978209985186613, 0, 97550207340469419322, 0, 10621007365793039376589, 0, 1156394417949235787587452, 0, 125906282933524854606651342, 0, 13708479638519394714459749883, 0, 1492558550220072112142535328539, 0, 162507550472995809504707415441910, 0, 17693581294485451309023356461706396, 0, 1926450976850982003417341419592961253, 0, 209749137591778302506062594313267560594, 0, 22837176498745392078474020507566786783053] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 116061685914415 z - 68139381009368 z - 184 z 24 22 4 6 + 32220849392076 z - 12226264365976 z + 13364 z - 528784 z 8 10 12 14 + 13151600 z - 222857615 z + 2707384508 z - 24409700430 z 18 16 50 48 - 888807825473 z + 167387642520 z - 24409700430 z + 167387642520 z 20 36 34 64 + 3703152939704 z + 116061685914415 z - 159612440464676 z + z 30 42 44 - 159612440464676 z - 12226264365976 z + 3703152939704 z 46 58 56 54 - 888807825473 z - 528784 z + 13151600 z - 222857615 z 52 60 32 38 + 2707384508 z + 13364 z + 177471461345884 z - 68139381009368 z 40 62 / 2 28 + 32220849392076 z - 184 z ) / ((-1 + z ) (1 + 430504217016388 z / 26 2 24 22 - 248169072050664 z - 236 z + 114410209915691 z - 42028832868542 z 4 6 8 10 12 + 20986 z - 979606 z + 27865705 z - 527035986 z + 7012927246 z 14 18 16 50 - 68248974594 z - 2802073188034 z + 499344755615 z - 68248974594 z 48 20 36 + 499344755615 z + 12236543840700 z + 430504217016388 z 34 64 30 42 - 598623261765708 z + z - 598623261765708 z - 42028832868542 z 44 46 58 56 + 12236543840700 z - 2802073188034 z - 979606 z + 27865705 z 54 52 60 32 - 527035986 z + 7012927246 z + 20986 z + 668069332812501 z 38 40 62 - 248169072050664 z + 114410209915691 z - 236 z )) And in Maple-input format, it is: -(1+116061685914415*z^28-68139381009368*z^26-184*z^2+32220849392076*z^24-\ 12226264365976*z^22+13364*z^4-528784*z^6+13151600*z^8-222857615*z^10+2707384508 *z^12-24409700430*z^14-888807825473*z^18+167387642520*z^16-24409700430*z^50+ 167387642520*z^48+3703152939704*z^20+116061685914415*z^36-159612440464676*z^34+ z^64-159612440464676*z^30-12226264365976*z^42+3703152939704*z^44-888807825473*z ^46-528784*z^58+13151600*z^56-222857615*z^54+2707384508*z^52+13364*z^60+ 177471461345884*z^32-68139381009368*z^38+32220849392076*z^40-184*z^62)/(-1+z^2) /(1+430504217016388*z^28-248169072050664*z^26-236*z^2+114410209915691*z^24-\ 42028832868542*z^22+20986*z^4-979606*z^6+27865705*z^8-527035986*z^10+7012927246 *z^12-68248974594*z^14-2802073188034*z^18+499344755615*z^16-68248974594*z^50+ 499344755615*z^48+12236543840700*z^20+430504217016388*z^36-598623261765708*z^34 +z^64-598623261765708*z^30-42028832868542*z^42+12236543840700*z^44-\ 2802073188034*z^46-979606*z^58+27865705*z^56-527035986*z^54+7012927246*z^52+ 20986*z^60+668069332812501*z^32-248169072050664*z^38+114410209915691*z^40-236*z ^62) The first , 40, terms are: [0, 53, 0, 4703, 0, 461653, 0, 46942360, 0, 4837166123, 0, 501041619073, 0, 52007086578919, 0, 5402837289292411, 0, 561480555883733235, 0, 58359499361171933231, 0, 6066178672501603102521, 0, 630565331589427007304131, 0, 65546530837086710814477560, 0, 6813516650267992098982122957, 0, 708261719493046351034887473575, 0, 73623518674928211308101378137277, 0, 7653137540097910965018738825651945, 0, 795540931553494723497311183520000505, 0, 82696202541971813399563815433158448461, 0, 8596241648870044829017416278999113547927] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 78752174281081566684 z - 11734922212695329836 z - 290 z 24 22 4 6 + 1503275329802802764 z - 164565194344240300 z + 37576 z - 2926414 z 102 8 10 12 - 6005277022 z + 155158054 z - 6005277022 z + 177461499388 z 14 18 16 - 4131251800650 z - 1193583119794444 z + 77527977118499 z 50 48 - 11957572157129595044939112 z + 7717415349715412037138347 z 20 36 + 15283633517651868 z + 38444710420432734423468 z 34 66 - 10056783246048067930470 z - 4391491068180886663288942 z 80 100 90 + 2296722669745682514481 z + 177461499388 z - 164565194344240300 z 88 84 + 1503275329802802764 z + 78752174281081566684 z 94 86 96 - 1193583119794444 z - 11734922212695329836 z + 77527977118499 z 98 92 82 - 4131251800650 z + 15283633517651868 z - 456605324161559952972 z 64 112 110 106 + 7717415349715412037138347 z + z - 290 z - 2926414 z 108 30 42 + 37576 z - 456605324161559952972 z - 971866746365764505108658 z 44 46 + 2201808058311324281300368 z - 4391491068180886663288942 z 58 56 - 19710390861769771614306536 z + 20980012536587896272673192 z 54 52 - 19710390861769771614306536 z + 16343153990043457280700296 z 60 70 + 16343153990043457280700296 z - 971866746365764505108658 z 68 78 + 2201808058311324281300368 z - 10056783246048067930470 z 32 38 + 2296722669745682514481 z - 128610070022421732134114 z 40 62 + 377250822458519461556922 z - 11957572157129595044939112 z 76 74 + 38444710420432734423468 z - 128610070022421732134114 z 72 104 / + 377250822458519461556922 z + 155158054 z ) / (-1 / 28 26 2 - 262673493310163299104 z + 36960689349690887164 z + 345 z 24 22 4 6 - 4466034987168045452 z + 460493545257822784 z - 51244 z + 4471724 z 102 8 10 12 + 354352923572 z - 261692410 z + 11062945762 z - 354352923572 z 14 18 16 + 8889830106948 z + 2945698578471651 z - 178978600545835 z 50 48 + 74105670104597611666787835 z - 45112915981368027941923107 z 20 36 - 40209491392040368 z - 160312424810812121178796 z 34 66 + 39677348156067541062937 z + 45112915981368027941923107 z 80 100 - 39677348156067541062937 z - 8889830106948 z 90 88 + 4466034987168045452 z - 36960689349690887164 z 84 94 - 1611488290934036650672 z + 40209491392040368 z 86 96 98 + 262673493310163299104 z - 2945698578471651 z + 178978600545835 z 92 82 - 460493545257822784 z + 8571815006381019655521 z 64 112 114 110 - 74105670104597611666787835 z - 345 z + z + 51244 z 106 108 30 + 261692410 z - 4471724 z + 1611488290934036650672 z 42 44 + 4788457126228633724008134 z - 11477413942448536202731556 z 46 58 + 24233009830771598431498980 z + 155808522950482725132949752 z 56 54 - 155808522950482725132949752 z + 137673079310962988334792096 z 52 60 - 107475073002451503471080864 z - 137673079310962988334792096 z 70 68 + 11477413942448536202731556 z - 24233009830771598431498980 z 78 32 + 160312424810812121178796 z - 8571815006381019655521 z 38 40 + 566837889644221774848636 z - 1757701709232282838833886 z 62 76 + 107475073002451503471080864 z - 566837889644221774848636 z 74 72 + 1757701709232282838833886 z - 4788457126228633724008134 z 104 - 11062945762 z ) And in Maple-input format, it is: -(1+78752174281081566684*z^28-11734922212695329836*z^26-290*z^2+ 1503275329802802764*z^24-164565194344240300*z^22+37576*z^4-2926414*z^6-\ 6005277022*z^102+155158054*z^8-6005277022*z^10+177461499388*z^12-4131251800650* z^14-1193583119794444*z^18+77527977118499*z^16-11957572157129595044939112*z^50+ 7717415349715412037138347*z^48+15283633517651868*z^20+38444710420432734423468*z ^36-10056783246048067930470*z^34-4391491068180886663288942*z^66+ 2296722669745682514481*z^80+177461499388*z^100-164565194344240300*z^90+ 1503275329802802764*z^88+78752174281081566684*z^84-1193583119794444*z^94-\ 11734922212695329836*z^86+77527977118499*z^96-4131251800650*z^98+ 15283633517651868*z^92-456605324161559952972*z^82+7717415349715412037138347*z^ 64+z^112-290*z^110-2926414*z^106+37576*z^108-456605324161559952972*z^30-\ 971866746365764505108658*z^42+2201808058311324281300368*z^44-\ 4391491068180886663288942*z^46-19710390861769771614306536*z^58+ 20980012536587896272673192*z^56-19710390861769771614306536*z^54+ 16343153990043457280700296*z^52+16343153990043457280700296*z^60-\ 971866746365764505108658*z^70+2201808058311324281300368*z^68-\ 10056783246048067930470*z^78+2296722669745682514481*z^32-\ 128610070022421732134114*z^38+377250822458519461556922*z^40-\ 11957572157129595044939112*z^62+38444710420432734423468*z^76-\ 128610070022421732134114*z^74+377250822458519461556922*z^72+155158054*z^104)/(-\ 1-262673493310163299104*z^28+36960689349690887164*z^26+345*z^2-\ 4466034987168045452*z^24+460493545257822784*z^22-51244*z^4+4471724*z^6+ 354352923572*z^102-261692410*z^8+11062945762*z^10-354352923572*z^12+ 8889830106948*z^14+2945698578471651*z^18-178978600545835*z^16+ 74105670104597611666787835*z^50-45112915981368027941923107*z^48-\ 40209491392040368*z^20-160312424810812121178796*z^36+39677348156067541062937*z^ 34+45112915981368027941923107*z^66-39677348156067541062937*z^80-8889830106948*z ^100+4466034987168045452*z^90-36960689349690887164*z^88-1611488290934036650672* z^84+40209491392040368*z^94+262673493310163299104*z^86-2945698578471651*z^96+ 178978600545835*z^98-460493545257822784*z^92+8571815006381019655521*z^82-\ 74105670104597611666787835*z^64-345*z^112+z^114+51244*z^110+261692410*z^106-\ 4471724*z^108+1611488290934036650672*z^30+4788457126228633724008134*z^42-\ 11477413942448536202731556*z^44+24233009830771598431498980*z^46+ 155808522950482725132949752*z^58-155808522950482725132949752*z^56+ 137673079310962988334792096*z^54-107475073002451503471080864*z^52-\ 137673079310962988334792096*z^60+11477413942448536202731556*z^70-\ 24233009830771598431498980*z^68+160312424810812121178796*z^78-\ 8571815006381019655521*z^32+566837889644221774848636*z^38-\ 1757701709232282838833886*z^40+107475073002451503471080864*z^62-\ 566837889644221774848636*z^76+1757701709232282838833886*z^74-\ 4788457126228633724008134*z^72-11062945762*z^104) The first , 40, terms are: [0, 55, 0, 5307, 0, 557805, 0, 59901281, 0, 6477807983, 0, 702381489247, 0, 76241703296529, 0, 8279714968155329, 0, 899346211369770207, 0, 97696163004123030279, 0, 10613176443415382599305, 0, 1152977726751223959461701, 0, 125256377153107702918959859, 0, 13607560730842192486470815015, 0, 1478295942087365552603984999249, 0, 160598980857194225573692603955249, 0, 17447142847636077161279938669996487, 0, 1895421956519038239304230063108664131, 0, 205914781533140813664337155156469167893, 0, 22370163135521092517760776238978975196409] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 75796812173227576443 z - 11477532548928266072 z - 296 z 24 22 4 6 + 1493298531163978919 z - 165887372283133248 z + 38989 z - 3071660 z 102 8 10 12 - 6360607508 z + 163956382 z - 6360607508 z + 187674363868 z 14 18 16 - 4347850806824 z - 1234144046438580 z + 80969471265465 z 50 48 - 9985216008124838199409438 z + 6487059663534366786762589 z 20 36 + 15615460199240873 z + 34697468333868708535313 z 34 66 - 9218913613883264545814 z - 3722207794032087570514814 z 80 100 90 + 2139480522212911513573 z + 187674363868 z - 165887372283133248 z 88 84 + 1493298531163978919 z + 75796812173227576443 z 94 86 96 - 1234144046438580 z - 11477532548928266072 z + 80969471265465 z 98 92 82 - 4347850806824 z + 15615460199240873 z - 432348435243595054438 z 64 112 110 106 + 6487059663534366786762589 z + z - 296 z - 3071660 z 108 30 42 + 38989 z - 432348435243595054438 z - 841525301184331909266916 z 44 46 + 1884875105881632484221306 z - 3722207794032087570514814 z 58 56 - 16333658869573829968571482 z + 17368967563120272010194096 z 54 52 - 16333658869573829968571482 z + 13582460502782789125302509 z 60 70 + 13582460502782789125302509 z - 841525301184331909266916 z 68 78 + 1884875105881632484221306 z - 9218913613883264545814 z 32 38 + 2139480522212911513573 z - 114365691660924764952856 z 40 62 + 330843262861241851871884 z - 9985216008124838199409438 z 76 74 + 34697468333868708535313 z - 114365691660924764952856 z 72 104 / 2 + 330843262861241851871884 z + 163956382 z ) / ((-1 + z ) (1 / 28 26 2 + 224430560209090706858 z - 32714867035916272938 z - 353 z 24 22 4 6 + 4085200378799050609 z - 434160785551026306 z + 53280 z - 4694057 z 102 8 10 12 - 11626669365 z + 275651619 z - 11626669365 z + 369615801843 z 14 18 16 - 9159154875982 z - 2923945551310460 z + 181352852790620 z 50 48 - 37867262122637815236633711 z + 24365073488438892416100531 z 20 36 + 38960862784672689 z + 116466642396573296066107 z 34 66 - 30104564845643476353952 z - 13809322278493891814170071 z 80 100 90 + 6779815637339277533830 z + 369615801843 z - 434160785551026306 z 88 84 + 4085200378799050609 z + 224430560209090706858 z 94 86 96 - 2923945551310460 z - 32714867035916272938 z + 181352852790620 z 98 92 82 - 9159154875982 z + 38960862784672689 z - 1326120912246325940634 z 64 112 110 106 + 24365073488438892416100531 z + z - 353 z - 4694057 z 108 30 42 + 53280 z - 1326120912246325940634 z - 3022339752512537960972915 z 44 46 + 6889156051710343317518264 z - 13809322278493891814170071 z 58 56 - 62633401154719556196593268 z + 66696187582996498610316987 z 54 52 - 62633401154719556196593268 z + 51866967397067637519566802 z 60 70 + 51866967397067637519566802 z - 3022339752512537960972915 z 68 78 + 6889156051710343317518264 z - 30104564845643476353952 z 32 38 + 6779815637339277533830 z - 393614499145748193479702 z 40 62 + 1164637903576500428914476 z - 37867262122637815236633711 z 76 74 + 116466642396573296066107 z - 393614499145748193479702 z 72 104 + 1164637903576500428914476 z + 275651619 z )) And in Maple-input format, it is: -(1+75796812173227576443*z^28-11477532548928266072*z^26-296*z^2+ 1493298531163978919*z^24-165887372283133248*z^22+38989*z^4-3071660*z^6-\ 6360607508*z^102+163956382*z^8-6360607508*z^10+187674363868*z^12-4347850806824* z^14-1234144046438580*z^18+80969471265465*z^16-9985216008124838199409438*z^50+ 6487059663534366786762589*z^48+15615460199240873*z^20+34697468333868708535313*z ^36-9218913613883264545814*z^34-3722207794032087570514814*z^66+ 2139480522212911513573*z^80+187674363868*z^100-165887372283133248*z^90+ 1493298531163978919*z^88+75796812173227576443*z^84-1234144046438580*z^94-\ 11477532548928266072*z^86+80969471265465*z^96-4347850806824*z^98+ 15615460199240873*z^92-432348435243595054438*z^82+6487059663534366786762589*z^ 64+z^112-296*z^110-3071660*z^106+38989*z^108-432348435243595054438*z^30-\ 841525301184331909266916*z^42+1884875105881632484221306*z^44-\ 3722207794032087570514814*z^46-16333658869573829968571482*z^58+ 17368967563120272010194096*z^56-16333658869573829968571482*z^54+ 13582460502782789125302509*z^52+13582460502782789125302509*z^60-\ 841525301184331909266916*z^70+1884875105881632484221306*z^68-\ 9218913613883264545814*z^78+2139480522212911513573*z^32-\ 114365691660924764952856*z^38+330843262861241851871884*z^40-\ 9985216008124838199409438*z^62+34697468333868708535313*z^76-\ 114365691660924764952856*z^74+330843262861241851871884*z^72+163956382*z^104)/(-\ 1+z^2)/(1+224430560209090706858*z^28-32714867035916272938*z^26-353*z^2+ 4085200378799050609*z^24-434160785551026306*z^22+53280*z^4-4694057*z^6-\ 11626669365*z^102+275651619*z^8-11626669365*z^10+369615801843*z^12-\ 9159154875982*z^14-2923945551310460*z^18+181352852790620*z^16-\ 37867262122637815236633711*z^50+24365073488438892416100531*z^48+ 38960862784672689*z^20+116466642396573296066107*z^36-30104564845643476353952*z^ 34-13809322278493891814170071*z^66+6779815637339277533830*z^80+369615801843*z^ 100-434160785551026306*z^90+4085200378799050609*z^88+224430560209090706858*z^84 -2923945551310460*z^94-32714867035916272938*z^86+181352852790620*z^96-\ 9159154875982*z^98+38960862784672689*z^92-1326120912246325940634*z^82+ 24365073488438892416100531*z^64+z^112-353*z^110-4694057*z^106+53280*z^108-\ 1326120912246325940634*z^30-3022339752512537960972915*z^42+ 6889156051710343317518264*z^44-13809322278493891814170071*z^46-\ 62633401154719556196593268*z^58+66696187582996498610316987*z^56-\ 62633401154719556196593268*z^54+51866967397067637519566802*z^52+ 51866967397067637519566802*z^60-3022339752512537960972915*z^70+ 6889156051710343317518264*z^68-30104564845643476353952*z^78+ 6779815637339277533830*z^32-393614499145748193479702*z^38+ 1164637903576500428914476*z^40-37867262122637815236633711*z^62+ 116466642396573296066107*z^76-393614499145748193479702*z^74+ 1164637903576500428914476*z^72+275651619*z^104) The first , 40, terms are: [0, 58, 0, 5888, 0, 649315, 0, 73022658, 0, 8259294061, 0, 935974174679, 0, 106144375507288, 0, 12040797930525863, 0, 1366048148600493921, 0, 154988445081399344630, 0, 17585005176865300575983, 0, 1995216502334059871218152, 0, 226380743490546526103388694, 0, 25685604499133069072192463569, 0, 2914341769375860897474177272129, 0, 330667369028835332087696821913742, 0, 37518224212830272477177640116588088, 0, 4256897985326186751798218930875827815, 0, 482996769239996878376141069081308493550, 0, 54801849488880045265242377543259144081337] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 157540965555839 z - 90572012124416 z - 180 z 24 22 4 6 + 41661229036024 z - 15298227845178 z + 12920 z - 513318 z 8 10 12 14 + 12968434 z - 225066119 z + 2817245146 z - 26285642134 z 18 16 50 - 1032225427625 z + 187065914334 z - 26285642134 z 48 20 36 + 187065914334 z + 4468214177266 z + 157540965555839 z 34 64 30 42 - 219515868776280 z + z - 219515868776280 z - 15298227845178 z 44 46 58 56 + 4468214177266 z - 1032225427625 z - 513318 z + 12968434 z 54 52 60 32 - 225066119 z + 2817245146 z + 12920 z + 245169081999308 z 38 40 62 / - 90572012124416 z + 41661229036024 z - 180 z ) / (-1 / 28 26 2 24 - 954009478126942 z + 494319461594363 z + 243 z - 205303985579387 z 22 4 6 8 10 + 68172899446542 z - 21126 z + 974022 z - 27955891 z + 544579513 z 12 14 18 16 - 7595184436 z + 78596763328 z + 3770986401277 z - 618691867537 z 50 48 20 + 618691867537 z - 3770986401277 z - 18024542842680 z 36 34 66 64 - 1477962741226832 z + 1839316979119345 z + z - 243 z 30 42 44 + 1477962741226832 z + 205303985579387 z - 68172899446542 z 46 58 56 54 + 18024542842680 z + 27955891 z - 544579513 z + 7595184436 z 52 60 32 - 78596763328 z - 974022 z - 1839316979119345 z 38 40 62 + 954009478126942 z - 494319461594363 z + 21126 z ) And in Maple-input format, it is: -(1+157540965555839*z^28-90572012124416*z^26-180*z^2+41661229036024*z^24-\ 15298227845178*z^22+12920*z^4-513318*z^6+12968434*z^8-225066119*z^10+2817245146 *z^12-26285642134*z^14-1032225427625*z^18+187065914334*z^16-26285642134*z^50+ 187065914334*z^48+4468214177266*z^20+157540965555839*z^36-219515868776280*z^34+ z^64-219515868776280*z^30-15298227845178*z^42+4468214177266*z^44-1032225427625* z^46-513318*z^58+12968434*z^56-225066119*z^54+2817245146*z^52+12920*z^60+ 245169081999308*z^32-90572012124416*z^38+41661229036024*z^40-180*z^62)/(-1-\ 954009478126942*z^28+494319461594363*z^26+243*z^2-205303985579387*z^24+ 68172899446542*z^22-21126*z^4+974022*z^6-27955891*z^8+544579513*z^10-7595184436 *z^12+78596763328*z^14+3770986401277*z^18-618691867537*z^16+618691867537*z^50-\ 3770986401277*z^48-18024542842680*z^20-1477962741226832*z^36+1839316979119345*z ^34+z^66-243*z^64+1477962741226832*z^30+205303985579387*z^42-68172899446542*z^ 44+18024542842680*z^46+27955891*z^58-544579513*z^56+7595184436*z^54-78596763328 *z^52-974022*z^60-1839316979119345*z^32+954009478126942*z^38-494319461594363*z^ 40+21126*z^62) The first , 40, terms are: [0, 63, 0, 7103, 0, 855795, 0, 104276136, 0, 12736346405, 0, 1556517561025, 0, 190250414760257, 0, 23254829898117423, 0, 2842528599484847855, 0, 347454216459196212801, 0, 42470816147541610712225, 0, 5191390512973047119200373, 0, 634566016442305252956900232, 0, 77565737487579477484256384195, 0, 9481194223412068608747104349919, 0, 1158927212988793373182453722956991, 0, 141660665694704336100129405703239169, 0, 17315793417548566093797978987052361089, 0, 2116584022902813041407352077699450694079, 0, 258719183002190149837969223896711157861663] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 5}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 12865467164980 z - 11297827890164 z - 188 z 24 22 4 6 + 7644475396554 z - 3975585693240 z + 13782 z - 540688 z 8 10 12 14 + 13038117 z - 208688584 z + 2325229196 z - 18610695988 z 18 16 50 48 - 478997993228 z + 109347072144 z - 540688 z + 13038117 z 20 36 34 + 1582337510276 z + 1582337510276 z - 3975585693240 z 30 42 44 46 - 11297827890164 z - 18610695988 z + 2325229196 z - 208688584 z 56 54 52 32 38 + z - 188 z + 13782 z + 7644475396554 z - 478997993228 z 40 / 2 28 + 109347072144 z ) / ((-1 + z ) (1 + 48276286779260 z / 26 2 24 22 - 42163624884572 z - 239 z + 28069892667548 z - 14212617276892 z 4 6 8 10 12 + 21546 z - 1005093 z + 27977907 z - 504236500 z + 6200063950 z 14 18 16 50 - 53884521514 z - 1574196115014 z + 339239365480 z - 1005093 z 48 20 36 + 27977907 z + 5451329974530 z + 5451329974530 z 34 30 42 - 14212617276892 z - 42163624884572 z - 53884521514 z 44 46 56 54 52 + 6200063950 z - 504236500 z + z - 239 z + 21546 z 32 38 40 + 28069892667548 z - 1574196115014 z + 339239365480 z )) And in Maple-input format, it is: -(1+12865467164980*z^28-11297827890164*z^26-188*z^2+7644475396554*z^24-\ 3975585693240*z^22+13782*z^4-540688*z^6+13038117*z^8-208688584*z^10+2325229196* z^12-18610695988*z^14-478997993228*z^18+109347072144*z^16-540688*z^50+13038117* z^48+1582337510276*z^20+1582337510276*z^36-3975585693240*z^34-11297827890164*z^ 30-18610695988*z^42+2325229196*z^44-208688584*z^46+z^56-188*z^54+13782*z^52+ 7644475396554*z^32-478997993228*z^38+109347072144*z^40)/(-1+z^2)/(1+ 48276286779260*z^28-42163624884572*z^26-239*z^2+28069892667548*z^24-\ 14212617276892*z^22+21546*z^4-1005093*z^6+27977907*z^8-504236500*z^10+ 6200063950*z^12-53884521514*z^14-1574196115014*z^18+339239365480*z^16-1005093*z ^50+27977907*z^48+5451329974530*z^20+5451329974530*z^36-14212617276892*z^34-\ 42163624884572*z^30-53884521514*z^42+6200063950*z^44-504236500*z^46+z^56-239*z^ 54+21546*z^52+28069892667548*z^32-1574196115014*z^38+339239365480*z^40) The first , 40, terms are: [0, 52, 0, 4477, 0, 427611, 0, 42535540, 0, 4305696591, 0, 439271759279, 0, 44976390795300, 0, 4612793204952939, 0, 473462396352860877, 0, 48614788212726748388, 0, 4992609453512357889057, 0, 512770328600231716461921, 0, 52666597864956025311286148, 0, 5409482653596449647967214765, 0, 555622764051896271068893333707, 0, 57069772677536821288803646851268, 0, 5861828575158840036224439192722287, 0, 602088720812372322203012529114621583, 0, 61842645835776544072766916792755229140, 0, 6352076554690718094803003465032960099323] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1417936373273632271 z - 346710956931839314 z - 274 z 24 22 4 6 + 70785657884006853 z - 11992425643120792 z + 32435 z - 2220600 z 8 10 12 14 + 99464812 z - 3128353824 z + 72388231765 z - 1273887547478 z 18 16 50 - 190435424844760 z + 17475125734375 z - 125225324100685813062 z 48 20 + 187218272639610448467 z + 1673173688332712 z 36 34 + 71216919389233421545 z - 34387454717193775928 z 66 80 88 84 86 - 11992425643120792 z + 99464812 z + z + 32435 z - 274 z 82 64 30 - 2220600 z + 70785657884006853 z - 4866136877623855856 z 42 44 - 238219238718351139600 z + 258122517372014007472 z 46 58 - 238219238718351139600 z - 4866136877623855856 z 56 54 + 14070186041754091924 z - 34387454717193775928 z 52 60 70 + 71216919389233421545 z + 1417936373273632271 z - 190435424844760 z 68 78 32 + 1673173688332712 z - 3128353824 z + 14070186041754091924 z 38 40 - 125225324100685813062 z + 187218272639610448467 z 62 76 74 - 346710956931839314 z + 72388231765 z - 1273887547478 z 72 / 2 28 + 17475125734375 z ) / ((-1 + z ) (1 + 4770782794966461001 z / 26 2 24 - 1122494397135246573 z - 341 z + 219598363987629209 z 22 4 6 8 - 35498921069106420 z + 47297 z - 3668922 z + 182408074 z 10 12 14 - 6281085354 z + 157548408267 z - 2982211156295 z 18 16 50 - 506394047733636 z + 43722232334075 z - 478937526341853801591 z 48 20 + 724719170248051854251 z + 4705108973472788 z 36 34 + 267856344350015312875 z - 126610196731054240438 z 66 80 88 84 86 - 35498921069106420 z + 182408074 z + z + 47297 z - 341 z 82 64 30 - 3668922 z + 219598363987629209 z - 16943998273273746278 z 42 44 - 928876280323198761992 z + 1008935372154837326872 z 46 58 - 928876280323198761992 z - 16943998273273746278 z 56 54 + 50488173554778805846 z - 126610196731054240438 z 52 60 + 267856344350015312875 z + 4770782794966461001 z 70 68 78 - 506394047733636 z + 4705108973472788 z - 6281085354 z 32 38 + 50488173554778805846 z - 478937526341853801591 z 40 62 76 + 724719170248051854251 z - 1122494397135246573 z + 157548408267 z 74 72 - 2982211156295 z + 43722232334075 z )) And in Maple-input format, it is: -(1+1417936373273632271*z^28-346710956931839314*z^26-274*z^2+70785657884006853* z^24-11992425643120792*z^22+32435*z^4-2220600*z^6+99464812*z^8-3128353824*z^10+ 72388231765*z^12-1273887547478*z^14-190435424844760*z^18+17475125734375*z^16-\ 125225324100685813062*z^50+187218272639610448467*z^48+1673173688332712*z^20+ 71216919389233421545*z^36-34387454717193775928*z^34-11992425643120792*z^66+ 99464812*z^80+z^88+32435*z^84-274*z^86-2220600*z^82+70785657884006853*z^64-\ 4866136877623855856*z^30-238219238718351139600*z^42+258122517372014007472*z^44-\ 238219238718351139600*z^46-4866136877623855856*z^58+14070186041754091924*z^56-\ 34387454717193775928*z^54+71216919389233421545*z^52+1417936373273632271*z^60-\ 190435424844760*z^70+1673173688332712*z^68-3128353824*z^78+14070186041754091924 *z^32-125225324100685813062*z^38+187218272639610448467*z^40-346710956931839314* z^62+72388231765*z^76-1273887547478*z^74+17475125734375*z^72)/(-1+z^2)/(1+ 4770782794966461001*z^28-1122494397135246573*z^26-341*z^2+219598363987629209*z^ 24-35498921069106420*z^22+47297*z^4-3668922*z^6+182408074*z^8-6281085354*z^10+ 157548408267*z^12-2982211156295*z^14-506394047733636*z^18+43722232334075*z^16-\ 478937526341853801591*z^50+724719170248051854251*z^48+4705108973472788*z^20+ 267856344350015312875*z^36-126610196731054240438*z^34-35498921069106420*z^66+ 182408074*z^80+z^88+47297*z^84-341*z^86-3668922*z^82+219598363987629209*z^64-\ 16943998273273746278*z^30-928876280323198761992*z^42+1008935372154837326872*z^ 44-928876280323198761992*z^46-16943998273273746278*z^58+50488173554778805846*z^ 56-126610196731054240438*z^54+267856344350015312875*z^52+4770782794966461001*z^ 60-506394047733636*z^70+4705108973472788*z^68-6281085354*z^78+ 50488173554778805846*z^32-478937526341853801591*z^38+724719170248051854251*z^40 -1122494397135246573*z^62+157548408267*z^76-2982211156295*z^74+43722232334075*z ^72) The first , 40, terms are: [0, 68, 0, 8053, 0, 1010361, 0, 128005356, 0, 16254869917, 0, 2065567350005, 0, 262541515935612, 0, 33372922446046177, 0, 4242333116470898605, 0, 539288090167375428340, 0, 68554989196600685946953, 0, 8714814912676997618005241, 0, 1107841460822943803317995284, 0, 140830655307851888530700916221, 0, 17902630240402525492247477229393, 0, 2275812646141194139313074219385308, 0, 289305160247818932787932411223576197, 0, 36776962555704030228457075040303341197, 0, 4675149860580958739075594508479687840140, 0, 594312981047858643420264496455876376847561] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 86635251427697134548 z - 12884170088937902436 z - 293 z 24 22 4 6 + 1646258069409866412 z - 179630575621513088 z + 38363 z - 3018430 z 102 8 10 12 - 6313307766 z + 161616868 z - 6313307766 z + 188155321339 z 14 18 16 - 4413783710589 z - 1291270605716048 z + 83388694586545 z 50 48 - 13202390227066450810336902 z + 8521552184073352045402918 z 20 36 + 16615544804283128 z + 42446938253240283228772 z 34 66 - 11098908841506831510048 z - 4849531663484787870114336 z 80 100 90 + 2532947234317555588752 z + 188155321339 z - 179630575621513088 z 88 84 + 1646258069409866412 z + 86635251427697134548 z 94 86 96 - 1291270605716048 z - 12884170088937902436 z + 83388694586545 z 98 92 82 - 4413783710589 z + 16615544804283128 z - 503045696808732813824 z 64 112 110 106 + 8521552184073352045402918 z + z - 293 z - 3018430 z 108 30 42 + 38363 z - 503045696808732813824 z - 1073392005199217097342224 z 44 46 + 2431673751928379613824392 z - 4849531663484787870114336 z 58 56 - 21759910238633495586319020 z + 23161213041349791620917272 z 54 52 - 21759910238633495586319020 z + 18043309933390061856872306 z 60 70 + 18043309933390061856872306 z - 1073392005199217097342224 z 68 78 + 2431673751928379613824392 z - 11098908841506831510048 z 32 38 + 2532947234317555588752 z - 142032255501232314730100 z 40 62 + 416660835757004604702460 z - 13202390227066450810336902 z 76 74 + 42446938253240283228772 z - 142032255501232314730100 z 72 104 / 2 + 416660835757004604702460 z + 161616868 z ) / ((-1 + z ) (1 / 28 26 2 + 254206547241486374472 z - 36335216239015063216 z - 352 z 24 22 4 6 + 4450292060954860460 z - 464163661399865368 z + 52822 z - 4634044 z 102 8 10 12 - 11499668724 z + 271870651 z - 11499668724 z + 367954835750 z 14 18 16 - 9207193312896 z - 3020441048179272 z + 184596024750513 z 50 48 - 50537117033866365694079168 z + 32276344390761233136431574 z 20 36 + 40916709684996976 z + 142290664444131041478376 z 34 66 - 36123236139906855203336 z - 18122101860108006173388440 z 80 100 90 + 7983979388919422625396 z + 367954835750 z - 464163661399865368 z 88 84 + 4450292060954860460 z + 254206547241486374472 z 94 86 96 - 3020441048179272 z - 36335216239015063216 z + 184596024750513 z 98 92 82 - 9207193312896 z + 40916709684996976 z - 1531789996420665643192 z 64 112 110 106 + 32276344390761233136431574 z + z - 352 z - 4634044 z 108 30 42 + 52822 z - 1531789996420665643192 z - 3871530329264534690024840 z 44 46 + 8939700436435645119718672 z - 18122101860108006173388440 z 58 56 - 84315149431881457062906896 z + 89882358953751959838016674 z 54 52 - 84315149431881457062906896 z + 69594556749411390280313556 z 60 70 + 69594556749411390280313556 z - 3871530329264534690024840 z 68 78 + 8939700436435645119718672 z - 36123236139906855203336 z 32 38 + 7983979388919422625396 z - 489148768714682151422128 z 40 62 + 1470424857123417357944140 z - 50537117033866365694079168 z 76 74 + 142290664444131041478376 z - 489148768714682151422128 z 72 104 + 1470424857123417357944140 z + 271870651 z )) And in Maple-input format, it is: -(1+86635251427697134548*z^28-12884170088937902436*z^26-293*z^2+ 1646258069409866412*z^24-179630575621513088*z^22+38363*z^4-3018430*z^6-\ 6313307766*z^102+161616868*z^8-6313307766*z^10+188155321339*z^12-4413783710589* z^14-1291270605716048*z^18+83388694586545*z^16-13202390227066450810336902*z^50+ 8521552184073352045402918*z^48+16615544804283128*z^20+42446938253240283228772*z ^36-11098908841506831510048*z^34-4849531663484787870114336*z^66+ 2532947234317555588752*z^80+188155321339*z^100-179630575621513088*z^90+ 1646258069409866412*z^88+86635251427697134548*z^84-1291270605716048*z^94-\ 12884170088937902436*z^86+83388694586545*z^96-4413783710589*z^98+ 16615544804283128*z^92-503045696808732813824*z^82+8521552184073352045402918*z^ 64+z^112-293*z^110-3018430*z^106+38363*z^108-503045696808732813824*z^30-\ 1073392005199217097342224*z^42+2431673751928379613824392*z^44-\ 4849531663484787870114336*z^46-21759910238633495586319020*z^58+ 23161213041349791620917272*z^56-21759910238633495586319020*z^54+ 18043309933390061856872306*z^52+18043309933390061856872306*z^60-\ 1073392005199217097342224*z^70+2431673751928379613824392*z^68-\ 11098908841506831510048*z^78+2532947234317555588752*z^32-\ 142032255501232314730100*z^38+416660835757004604702460*z^40-\ 13202390227066450810336902*z^62+42446938253240283228772*z^76-\ 142032255501232314730100*z^74+416660835757004604702460*z^72+161616868*z^104)/(-\ 1+z^2)/(1+254206547241486374472*z^28-36335216239015063216*z^26-352*z^2+ 4450292060954860460*z^24-464163661399865368*z^22+52822*z^4-4634044*z^6-\ 11499668724*z^102+271870651*z^8-11499668724*z^10+367954835750*z^12-\ 9207193312896*z^14-3020441048179272*z^18+184596024750513*z^16-\ 50537117033866365694079168*z^50+32276344390761233136431574*z^48+ 40916709684996976*z^20+142290664444131041478376*z^36-36123236139906855203336*z^ 34-18122101860108006173388440*z^66+7983979388919422625396*z^80+367954835750*z^ 100-464163661399865368*z^90+4450292060954860460*z^88+254206547241486374472*z^84 -3020441048179272*z^94-36335216239015063216*z^86+184596024750513*z^96-\ 9207193312896*z^98+40916709684996976*z^92-1531789996420665643192*z^82+ 32276344390761233136431574*z^64+z^112-352*z^110-4634044*z^106+52822*z^108-\ 1531789996420665643192*z^30-3871530329264534690024840*z^42+ 8939700436435645119718672*z^44-18122101860108006173388440*z^46-\ 84315149431881457062906896*z^58+89882358953751959838016674*z^56-\ 84315149431881457062906896*z^54+69594556749411390280313556*z^52+ 69594556749411390280313556*z^60-3871530329264534690024840*z^70+ 8939700436435645119718672*z^68-36123236139906855203336*z^78+ 7983979388919422625396*z^32-489148768714682151422128*z^38+ 1470424857123417357944140*z^40-50537117033866365694079168*z^62+ 142290664444131041478376*z^76-489148768714682151422128*z^74+ 1470424857123417357944140*z^72+271870651*z^104) The first , 40, terms are: [0, 60, 0, 6369, 0, 726253, 0, 84026236, 0, 9762083749, 0, 1135789760341, 0, 132219116938580, 0, 15395257630267885, 0, 1792746906012612385, 0, 208769558533036426708, 0, 24312075953498193342113, 0, 2831259523413162206390545, 0, 329714815741114580842329188, 0, 38397037713533009950848356785, 0, 4471540568351902011719336587965, 0, 520734931963149306592628969030756, 0, 60642386314450686860642086524949733, 0, 7062132596650360263378531913342140789, 0, 822423399132996380237069654354090181548, 0, 95775637452975640805109856570154417157565] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 82662150011748421353 z - 12496377295788644558 z - 299 z 24 22 4 6 + 1622332159540223875 z - 179725693333488556 z + 39786 z - 3165388 z 102 8 10 12 - 6671139164 z + 170520603 z - 6671139164 z + 198327643921 z 14 18 16 - 4625332636400 z - 1327037937924914 z + 86635826387135 z 50 48 - 10908855932811379505406038 z + 7088707966753471983345306 z 20 36 + 16860478557859723 z + 37949127704615199211768 z 34 66 - 10080025336044812278306 z - 4068457949816316514098186 z 80 100 90 + 2338077163123410872423 z + 198327643921 z - 179725693333488556 z 88 84 + 1622332159540223875 z + 82662150011748421353 z 94 86 96 - 1327037937924914 z - 12496377295788644558 z + 86635826387135 z 98 92 82 - 4625332636400 z + 16860478557859723 z - 472082817363776260270 z 64 112 110 106 + 7088707966753471983345306 z + z - 299 z - 3165388 z 108 30 42 + 39786 z - 472082817363776260270 z - 920253583017556081485702 z 44 46 + 2060740363295895790802450 z - 4068457949816316514098186 z 58 56 - 17839428615422109879645110 z + 18969459177064379240489054 z 54 52 - 17839428615422109879645110 z + 14836249984735277080517618 z 60 70 + 14836249984735277080517618 z - 920253583017556081485702 z 68 78 + 2060740363295895790802450 z - 10080025336044812278306 z 32 38 + 2338077163123410872423 z - 125093924807063803469327 z 40 62 + 361855381848441235962161 z - 10908855932811379505406038 z 76 74 + 37949127704615199211768 z - 125093924807063803469327 z 72 104 / 2 + 361855381848441235962161 z + 170520603 z ) / ((-1 + z ) (1 / 28 26 2 + 245663973588883153438 z - 35721535527320456974 z - 363 z 24 22 4 6 + 4449482107676565096 z - 471681457087398332 z + 55680 z - 4955240 z 102 8 10 12 - 12414848290 z + 292942486 z - 12414848290 z + 396123521860 z 14 18 16 - 9846354694484 z - 3160417219254318 z + 195503074807835 z 50 48 - 42262079329799103017809178 z + 27170103396909761807922707 z 20 36 + 42220308872915427 z + 128662053350919513161490 z 34 66 - 33185526204454556075370 z - 15382742610963345889718866 z 80 100 90 + 7456793706344414969869 z + 396123521860 z - 471681457087398332 z 88 84 + 4449482107676565096 z + 245663973588883153438 z 94 86 96 - 3160417219254318 z - 35721535527320456974 z + 195503074807835 z 98 92 82 - 9846354694484 z + 42220308872915427 z - 1455108176980572103496 z 64 112 110 106 + 27170103396909761807922707 z + z - 363 z - 4955240 z 108 30 42 + 55680 z - 1455108176980572103496 z - 3357386873427907335207850 z 44 46 + 7664265854086605330485522 z - 15382742610963345889718866 z 58 56 - 69970070540113147280163346 z + 74517802914598589480100335 z 54 52 - 69970070540113147280163346 z + 57921448647590478710234802 z 60 70 + 57921448647590478710234802 z - 3357386873427907335207850 z 68 78 + 7664265854086605330485522 z - 33185526204454556075370 z 32 38 + 7456793706344414969869 z - 435706107155140171701139 z 40 62 + 1291573933412274848769314 z - 42262079329799103017809178 z 76 74 + 128662053350919513161490 z - 435706107155140171701139 z 72 104 + 1291573933412274848769314 z + 292942486 z )) And in Maple-input format, it is: -(1+82662150011748421353*z^28-12496377295788644558*z^26-299*z^2+ 1622332159540223875*z^24-179725693333488556*z^22+39786*z^4-3165388*z^6-\ 6671139164*z^102+170520603*z^8-6671139164*z^10+198327643921*z^12-4625332636400* z^14-1327037937924914*z^18+86635826387135*z^16-10908855932811379505406038*z^50+ 7088707966753471983345306*z^48+16860478557859723*z^20+37949127704615199211768*z ^36-10080025336044812278306*z^34-4068457949816316514098186*z^66+ 2338077163123410872423*z^80+198327643921*z^100-179725693333488556*z^90+ 1622332159540223875*z^88+82662150011748421353*z^84-1327037937924914*z^94-\ 12496377295788644558*z^86+86635826387135*z^96-4625332636400*z^98+ 16860478557859723*z^92-472082817363776260270*z^82+7088707966753471983345306*z^ 64+z^112-299*z^110-3165388*z^106+39786*z^108-472082817363776260270*z^30-\ 920253583017556081485702*z^42+2060740363295895790802450*z^44-\ 4068457949816316514098186*z^46-17839428615422109879645110*z^58+ 18969459177064379240489054*z^56-17839428615422109879645110*z^54+ 14836249984735277080517618*z^52+14836249984735277080517618*z^60-\ 920253583017556081485702*z^70+2060740363295895790802450*z^68-\ 10080025336044812278306*z^78+2338077163123410872423*z^32-\ 125093924807063803469327*z^38+361855381848441235962161*z^40-\ 10908855932811379505406038*z^62+37949127704615199211768*z^76-\ 125093924807063803469327*z^74+361855381848441235962161*z^72+170520603*z^104)/(-\ 1+z^2)/(1+245663973588883153438*z^28-35721535527320456974*z^26-363*z^2+ 4449482107676565096*z^24-471681457087398332*z^22+55680*z^4-4955240*z^6-\ 12414848290*z^102+292942486*z^8-12414848290*z^10+396123521860*z^12-\ 9846354694484*z^14-3160417219254318*z^18+195503074807835*z^16-\ 42262079329799103017809178*z^50+27170103396909761807922707*z^48+ 42220308872915427*z^20+128662053350919513161490*z^36-33185526204454556075370*z^ 34-15382742610963345889718866*z^66+7456793706344414969869*z^80+396123521860*z^ 100-471681457087398332*z^90+4449482107676565096*z^88+245663973588883153438*z^84 -3160417219254318*z^94-35721535527320456974*z^86+195503074807835*z^96-\ 9846354694484*z^98+42220308872915427*z^92-1455108176980572103496*z^82+ 27170103396909761807922707*z^64+z^112-363*z^110-4955240*z^106+55680*z^108-\ 1455108176980572103496*z^30-3357386873427907335207850*z^42+ 7664265854086605330485522*z^44-15382742610963345889718866*z^46-\ 69970070540113147280163346*z^58+74517802914598589480100335*z^56-\ 69970070540113147280163346*z^54+57921448647590478710234802*z^52+ 57921448647590478710234802*z^60-3357386873427907335207850*z^70+ 7664265854086605330485522*z^68-33185526204454556075370*z^78+ 7456793706344414969869*z^32-435706107155140171701139*z^38+ 1291573933412274848769314*z^40-42262079329799103017809178*z^62+ 128662053350919513161490*z^76-435706107155140171701139*z^74+ 1291573933412274848769314*z^72+292942486*z^104) The first , 40, terms are: [0, 65, 0, 7403, 0, 897429, 0, 110110504, 0, 13554750191, 0, 1670409827165, 0, 205931107125995, 0, 25391223021867531, 0, 3130902197798967523, 0, 386068975839394901707, 0, 47606263328753349962361, 0, 5870360952317673454848139, 0, 723879253468388259818431528, 0, 89262223781399007797431454257, 0, 11007010358646663950213008190047, 0, 1357285123692763538329686774985841, 0, 167368151527347222111694301263932897, 0, 20638330217763441251837229546260001297, 0, 2544932670849134218671986807152608741457, 0, 313818135845523136568103808493600731233319] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 82786733539780697576 z - 12285600628331804216 z - 293 z 24 22 4 6 + 1568113364088645336 z - 171128028299070376 z + 38255 z - 2994842 z 102 8 10 12 - 6179566282 z + 159319612 z - 6179566282 z + 182884069547 z 14 18 16 - 4262905615177 z - 1235363079963160 z + 80107961833381 z 50 48 - 13219909988006683800690666 z + 8507335647636007211021298 z 20 36 + 15851877193346320 z + 41218276566285727839920 z 34 66 - 10725998491280873415912 z - 4823641646721116713288328 z 80 100 90 + 2437105414892236432592 z + 182884069547 z - 171128028299070376 z 88 84 + 1568113364088645336 z + 82786733539780697576 z 94 86 96 - 1235363079963160 z - 12285600628331804216 z + 80107961833381 z 98 92 82 - 4262905615177 z + 15851877193346320 z - 482169914424280687944 z 64 112 110 106 + 8507335647636007211021298 z + z - 293 z - 2994842 z 108 30 42 + 38255 z - 482169914424280687944 z - 1058161866516839681674296 z 44 46 + 2408428355989559310416112 z - 4823641646721116713288328 z 58 56 - 21866677698245427929892332 z + 23285498330286872941307368 z 54 52 - 21866677698245427929892332 z + 18107261544799635686973134 z 60 70 + 18107261544799635686973134 z - 1058161866516839681674296 z 68 78 + 2408428355989559310416112 z - 10725998491280873415912 z 32 38 + 2437105414892236432592 z - 138616770232323790681456 z 40 62 + 408714129767663599857168 z - 13219909988006683800690666 z 76 74 + 41218276566285727839920 z - 138616770232323790681456 z 72 104 / 2 + 408714129767663599857168 z + 159319612 z ) / ((-1 + z ) (1 / 28 26 2 + 241320465601587318456 z - 34402657765919889236 z - 356 z 24 22 4 6 + 4210008118244703292 z - 439582390047799676 z + 53534 z - 4676716 z 102 8 10 12 - 11394650080 z + 272141995 z - 11394650080 z + 360716773886 z 14 18 16 - 8934929986160 z - 2884443805018596 z + 177552706860161 z 50 48 - 51597882543182905607896412 z + 32792854692117555966227782 z 20 36 + 38871666468199792 z + 138314780771290545506232 z 34 66 - 34856323446300573357492 z - 18302036941029844564452204 z 80 100 90 + 7653381651250345989252 z + 360716773886 z - 439582390047799676 z 88 84 + 4210008118244703292 z + 241320465601587318456 z 94 86 96 - 2884443805018596 z - 34402657765919889236 z + 177552706860161 z 98 92 82 - 8934929986160 z + 38871666468199792 z - 1460265701261417471260 z 64 112 110 106 + 32792854692117555966227782 z + z - 356 z - 4676716 z 108 30 42 + 53534 z - 1460265701261417471260 z - 3853979834517216964499252 z 44 46 + 8966502080149373882408080 z - 18302036941029844564452204 z 58 56 - 86590315646742666094300388 z + 92377146554013860304964002 z 54 52 - 86590315646742666094300388 z + 71313441071404738063541108 z 60 70 + 71313441071404738063541108 z - 3853979834517216964499252 z 68 78 + 8966502080149373882408080 z - 34856323446300573357492 z 32 38 + 7653381651250345989252 z - 479224991776772636454268 z 40 62 + 1452206538911702173697788 z - 51597882543182905607896412 z 76 74 + 138314780771290545506232 z - 479224991776772636454268 z 72 104 + 1452206538911702173697788 z + 272141995 z )) And in Maple-input format, it is: -(1+82786733539780697576*z^28-12285600628331804216*z^26-293*z^2+ 1568113364088645336*z^24-171128028299070376*z^22+38255*z^4-2994842*z^6-\ 6179566282*z^102+159319612*z^8-6179566282*z^10+182884069547*z^12-4262905615177* z^14-1235363079963160*z^18+80107961833381*z^16-13219909988006683800690666*z^50+ 8507335647636007211021298*z^48+15851877193346320*z^20+41218276566285727839920*z ^36-10725998491280873415912*z^34-4823641646721116713288328*z^66+ 2437105414892236432592*z^80+182884069547*z^100-171128028299070376*z^90+ 1568113364088645336*z^88+82786733539780697576*z^84-1235363079963160*z^94-\ 12285600628331804216*z^86+80107961833381*z^96-4262905615177*z^98+ 15851877193346320*z^92-482169914424280687944*z^82+8507335647636007211021298*z^ 64+z^112-293*z^110-2994842*z^106+38255*z^108-482169914424280687944*z^30-\ 1058161866516839681674296*z^42+2408428355989559310416112*z^44-\ 4823641646721116713288328*z^46-21866677698245427929892332*z^58+ 23285498330286872941307368*z^56-21866677698245427929892332*z^54+ 18107261544799635686973134*z^52+18107261544799635686973134*z^60-\ 1058161866516839681674296*z^70+2408428355989559310416112*z^68-\ 10725998491280873415912*z^78+2437105414892236432592*z^32-\ 138616770232323790681456*z^38+408714129767663599857168*z^40-\ 13219909988006683800690666*z^62+41218276566285727839920*z^76-\ 138616770232323790681456*z^74+408714129767663599857168*z^72+159319612*z^104)/(-\ 1+z^2)/(1+241320465601587318456*z^28-34402657765919889236*z^26-356*z^2+ 4210008118244703292*z^24-439582390047799676*z^22+53534*z^4-4676716*z^6-\ 11394650080*z^102+272141995*z^8-11394650080*z^10+360716773886*z^12-\ 8934929986160*z^14-2884443805018596*z^18+177552706860161*z^16-\ 51597882543182905607896412*z^50+32792854692117555966227782*z^48+ 38871666468199792*z^20+138314780771290545506232*z^36-34856323446300573357492*z^ 34-18302036941029844564452204*z^66+7653381651250345989252*z^80+360716773886*z^ 100-439582390047799676*z^90+4210008118244703292*z^88+241320465601587318456*z^84 -2884443805018596*z^94-34402657765919889236*z^86+177552706860161*z^96-\ 8934929986160*z^98+38871666468199792*z^92-1460265701261417471260*z^82+ 32792854692117555966227782*z^64+z^112-356*z^110-4676716*z^106+53534*z^108-\ 1460265701261417471260*z^30-3853979834517216964499252*z^42+ 8966502080149373882408080*z^44-18302036941029844564452204*z^46-\ 86590315646742666094300388*z^58+92377146554013860304964002*z^56-\ 86590315646742666094300388*z^54+71313441071404738063541108*z^52+ 71313441071404738063541108*z^60-3853979834517216964499252*z^70+ 8966502080149373882408080*z^68-34856323446300573357492*z^78+ 7653381651250345989252*z^32-479224991776772636454268*z^38+ 1452206538911702173697788*z^40-51597882543182905607896412*z^62+ 138314780771290545506232*z^76-479224991776772636454268*z^74+ 1452206538911702173697788*z^72+272141995*z^104) The first , 40, terms are: [0, 64, 0, 7213, 0, 861489, 0, 104079904, 0, 12621005057, 0, 1532645096977, 0, 186226209722400, 0, 22633071953859697, 0, 2750989989546886829, 0, 334389213193110057488, 0, 40646473451224779187585, 0, 4940791653701546727939713, 0, 600580849073999982791175632, 0, 73004048676630220425443786733, 0, 8874065533574732063441478645105, 0, 1078694360601801119658227834184064, 0, 131121594785732422345902346885783985, 0, 15938595648358668138401500152269093665, 0, 1937429417941104691651346013308576717824, 0, 235505866124902898972086181547314533603633] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 95318321780786444977 z - 14211413179472644976 z - 296 z 24 22 4 6 + 1817788681245989817 z - 198243786742489744 z + 39223 z - 3125660 z 102 8 10 12 - 6699546192 z + 169483370 z - 6699546192 z + 201803945462 z 14 18 16 - 4777505429304 z - 1416299918001900 z + 90939322519703 z 50 48 - 13607195090400522237427250 z + 8821918262180395192220371 z 20 36 + 18296752894496379 z + 45721941052493603233887 z 34 66 - 12033754308224648177082 z - 5047779308576046830206350 z 80 100 90 + 2762439801475083879623 z + 201803945462 z - 198243786742489744 z 88 84 + 1817788681245989817 z + 95318321780786444977 z 94 86 96 - 1416299918001900 z - 14211413179472644976 z + 90939322519703 z 98 92 82 - 4777505429304 z + 18296752894496379 z - 551346294355314141070 z 64 112 110 106 + 8821918262180395192220371 z + z - 296 z - 3125660 z 108 30 42 + 39223 z - 551346294355314141070 z - 1131885672911286911559084 z 44 46 + 2546858628005475054650838 z - 5047779308576046830206350 z 58 56 - 22308139995596838567264242 z + 23728522795796828998892232 z 54 52 - 22308139995596838567264242 z + 18535406910406272053799261 z 60 70 + 18535406910406272053799261 z - 1131885672911286911559084 z 68 78 + 2546858628005475054650838 z - 12033754308224648177082 z 32 38 + 2762439801475083879623 z - 151925766264026069664484 z 40 62 + 442498611586471039985438 z - 13607195090400522237427250 z 76 74 + 45721941052493603233887 z - 151925766264026069664484 z 72 104 / 2 + 442498611586471039985438 z + 169483370 z ) / ((-1 + z ) (1 / 28 26 2 + 281605665064042618658 z - 40322238282683437726 z - 363 z 24 22 4 6 + 4941769572720657041 z - 515138043905070522 z + 55336 z - 4908999 z 102 8 10 12 - 12401232155 z + 290725707 z - 12401232155 z + 399803129895 z 14 18 16 - 10069981905054 z - 3335912020404560 z + 203002556778880 z 50 48 - 53232360526994777047207133 z + 34118950276399480481641187 z 20 36 + 45328592420551885 z + 155157884015106274286967 z 34 66 - 39586018553222141161012 z - 19239746760061556126622417 z 80 100 90 + 8787869097618079517914 z + 399803129895 z - 515138043905070522 z 88 84 + 4941769572720657041 z + 281605665064042618658 z 94 86 96 - 3335912020404560 z - 40322238282683437726 z + 203002556778880 z 98 92 82 - 10069981905054 z + 45328592420551885 z - 1692198503125236909530 z 64 112 110 106 + 34118950276399480481641187 z + z - 363 z - 4908999 z 108 30 42 + 55336 z - 1692198503125236909530 z - 4152704603235238802660021 z 44 46 + 9537912960618634308506472 z - 19239746760061556126622417 z 58 56 - 88432013836605705708339084 z + 94218978293098548086824259 z 54 52 - 88432013836605705708339084 z + 73112100503656678973142330 z 60 70 + 73112100503656678973142330 z - 4152704603235238802660021 z 68 78 + 9537912960618634308506472 z - 39586018553222141161012 z 32 38 + 8787869097618079517914 z - 530531465359906529478590 z 40 62 + 1585991016732484741490344 z - 53232360526994777047207133 z 76 74 + 155157884015106274286967 z - 530531465359906529478590 z 72 104 + 1585991016732484741490344 z + 290725707 z )) And in Maple-input format, it is: -(1+95318321780786444977*z^28-14211413179472644976*z^26-296*z^2+ 1817788681245989817*z^24-198243786742489744*z^22+39223*z^4-3125660*z^6-\ 6699546192*z^102+169483370*z^8-6699546192*z^10+201803945462*z^12-4777505429304* z^14-1416299918001900*z^18+90939322519703*z^16-13607195090400522237427250*z^50+ 8821918262180395192220371*z^48+18296752894496379*z^20+45721941052493603233887*z ^36-12033754308224648177082*z^34-5047779308576046830206350*z^66+ 2762439801475083879623*z^80+201803945462*z^100-198243786742489744*z^90+ 1817788681245989817*z^88+95318321780786444977*z^84-1416299918001900*z^94-\ 14211413179472644976*z^86+90939322519703*z^96-4777505429304*z^98+ 18296752894496379*z^92-551346294355314141070*z^82+8821918262180395192220371*z^ 64+z^112-296*z^110-3125660*z^106+39223*z^108-551346294355314141070*z^30-\ 1131885672911286911559084*z^42+2546858628005475054650838*z^44-\ 5047779308576046830206350*z^46-22308139995596838567264242*z^58+ 23728522795796828998892232*z^56-22308139995596838567264242*z^54+ 18535406910406272053799261*z^52+18535406910406272053799261*z^60-\ 1131885672911286911559084*z^70+2546858628005475054650838*z^68-\ 12033754308224648177082*z^78+2762439801475083879623*z^32-\ 151925766264026069664484*z^38+442498611586471039985438*z^40-\ 13607195090400522237427250*z^62+45721941052493603233887*z^76-\ 151925766264026069664484*z^74+442498611586471039985438*z^72+169483370*z^104)/(-\ 1+z^2)/(1+281605665064042618658*z^28-40322238282683437726*z^26-363*z^2+ 4941769572720657041*z^24-515138043905070522*z^22+55336*z^4-4908999*z^6-\ 12401232155*z^102+290725707*z^8-12401232155*z^10+399803129895*z^12-\ 10069981905054*z^14-3335912020404560*z^18+203002556778880*z^16-\ 53232360526994777047207133*z^50+34118950276399480481641187*z^48+ 45328592420551885*z^20+155157884015106274286967*z^36-39586018553222141161012*z^ 34-19239746760061556126622417*z^66+8787869097618079517914*z^80+399803129895*z^ 100-515138043905070522*z^90+4941769572720657041*z^88+281605665064042618658*z^84 -3335912020404560*z^94-40322238282683437726*z^86+203002556778880*z^96-\ 10069981905054*z^98+45328592420551885*z^92-1692198503125236909530*z^82+ 34118950276399480481641187*z^64+z^112-363*z^110-4908999*z^106+55336*z^108-\ 1692198503125236909530*z^30-4152704603235238802660021*z^42+ 9537912960618634308506472*z^44-19239746760061556126622417*z^46-\ 88432013836605705708339084*z^58+94218978293098548086824259*z^56-\ 88432013836605705708339084*z^54+73112100503656678973142330*z^52+ 73112100503656678973142330*z^60-4152704603235238802660021*z^70+ 9537912960618634308506472*z^68-39586018553222141161012*z^78+ 8787869097618079517914*z^32-530531465359906529478590*z^38+ 1585991016732484741490344*z^40-53232360526994777047207133*z^62+ 155157884015106274286967*z^76-530531465359906529478590*z^74+ 1585991016732484741490344*z^72+290725707*z^104) The first , 40, terms are: [0, 68, 0, 8276, 0, 1063607, 0, 137611468, 0, 17822816181, 0, 2308765281269, 0, 299088834417792, 0, 38745786699785219, 0, 5019375772566745455, 0, 650242242816380311680, 0, 84236577841790246433553, 0, 10912550550278111441394884, 0, 1413682321921304418647531848, 0, 183137544725396739011472268343, 0, 23724821202233609136323453414583, 0, 3073466678332543518725527554293872, 0, 398156738180567982206597972711431724, 0, 51579797263250890130371016693013807813, 0, 6681980312333527053364263393718231546392, 0, 865626917195551264407730363953900313288071] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 74872753654759547161 z - 11336252318211518524 z - 296 z 24 22 4 6 + 1474789627328440453 z - 163825697072050224 z + 38891 z - 3055412 z 102 8 10 12 - 6301094780 z + 162707954 z - 6301094780 z + 185688380286 z 14 18 16 - 4298269292268 z - 1219047017368572 z + 80003812625239 z 50 48 - 9872354366538426609717474 z + 6413697287506794395686391 z 20 36 + 15422060223929299 z + 34293025809397481247167 z 34 66 - 9110317100289991177322 z - 3680056144049238835360062 z 80 100 90 + 2113987698444044234123 z + 185688380286 z - 163825697072050224 z 88 84 + 1474789627328440453 z + 74872753654759547161 z 94 86 96 - 1219047017368572 z - 11336252318211518524 z + 80003812625239 z 98 92 82 - 4298269292268 z + 15422060223929299 z - 427136102679889587378 z 64 112 110 106 + 6413697287506794395686391 z + z - 296 z - 3055412 z 108 30 42 + 38891 z - 427136102679889587378 z - 831933057431805256997364 z 44 46 + 1863476100893345237123114 z - 3680056144049238835360062 z 58 56 - 16149043355268593260757834 z + 17172640714504552880966000 z 54 52 - 16149043355268593260757834 z + 13428953019344303219116565 z 60 70 + 13428953019344303219116565 z - 831933057431805256997364 z 68 78 + 1863476100893345237123114 z - 9110317100289991177322 z 32 38 + 2113987698444044234123 z - 113044764059165816011712 z 40 62 + 327050500301477785694378 z - 9872354366538426609717474 z 76 74 + 34293025809397481247167 z - 113044764059165816011712 z 72 104 / + 327050500301477785694378 z + 162707954 z ) / (-1 / 28 26 2 - 253530857094181812804 z + 36310966622904359515 z + 352 z 24 22 4 6 - 4462860137190909839 z + 467579207216922619 z - 53223 z + 4707695 z 102 8 10 12 + 377838808802 z - 277934382 z + 11798292814 z - 377838808802 z 14 18 16 + 9440753635917 z + 3073399254178932 z - 188666176632854 z 50 48 + 61044593993541535914086440 z - 37453584148086977417979800 z 20 36 - 41425766365762521 z - 144133108491805777423347 z 34 66 + 36291491771003189766310 z + 37453584148086977417979800 z 80 100 - 36291491771003189766310 z - 9440753635917 z 90 88 + 4462860137190909839 z - 36310966622904359515 z 84 94 - 1527631252787869816072 z + 41425766365762521 z 86 96 98 + 253530857094181812804 z - 3073399254178932 z + 188666176632854 z 92 82 - 467579207216922619 z + 7980648311748518418140 z 64 112 114 110 - 61044593993541535914086440 z - 352 z + z + 53223 z 106 108 30 + 277934382 z - 4707695 z + 1527631252787869816072 z 42 44 + 4111682047225248143346541 z - 9729809441542361414870393 z 46 58 + 20312858700860113657672217 z + 126821126572588576795290639 z 56 54 - 126821126572588576795290639 z + 112285510083481165989752630 z 52 60 - 88007752872593945123832935 z - 112285510083481165989752630 z 70 68 + 9729809441542361414870393 z - 20312858700860113657672217 z 78 32 + 144133108491805777423347 z - 7980648311748518418140 z 38 40 + 501339671559499083166817 z - 1530848656382977738156394 z 62 76 + 88007752872593945123832935 z - 501339671559499083166817 z 74 72 + 1530848656382977738156394 z - 4111682047225248143346541 z 104 - 11798292814 z ) And in Maple-input format, it is: -(1+74872753654759547161*z^28-11336252318211518524*z^26-296*z^2+ 1474789627328440453*z^24-163825697072050224*z^22+38891*z^4-3055412*z^6-\ 6301094780*z^102+162707954*z^8-6301094780*z^10+185688380286*z^12-4298269292268* z^14-1219047017368572*z^18+80003812625239*z^16-9872354366538426609717474*z^50+ 6413697287506794395686391*z^48+15422060223929299*z^20+34293025809397481247167*z ^36-9110317100289991177322*z^34-3680056144049238835360062*z^66+ 2113987698444044234123*z^80+185688380286*z^100-163825697072050224*z^90+ 1474789627328440453*z^88+74872753654759547161*z^84-1219047017368572*z^94-\ 11336252318211518524*z^86+80003812625239*z^96-4298269292268*z^98+ 15422060223929299*z^92-427136102679889587378*z^82+6413697287506794395686391*z^ 64+z^112-296*z^110-3055412*z^106+38891*z^108-427136102679889587378*z^30-\ 831933057431805256997364*z^42+1863476100893345237123114*z^44-\ 3680056144049238835360062*z^46-16149043355268593260757834*z^58+ 17172640714504552880966000*z^56-16149043355268593260757834*z^54+ 13428953019344303219116565*z^52+13428953019344303219116565*z^60-\ 831933057431805256997364*z^70+1863476100893345237123114*z^68-\ 9110317100289991177322*z^78+2113987698444044234123*z^32-\ 113044764059165816011712*z^38+327050500301477785694378*z^40-\ 9872354366538426609717474*z^62+34293025809397481247167*z^76-\ 113044764059165816011712*z^74+327050500301477785694378*z^72+162707954*z^104)/(-\ 1-253530857094181812804*z^28+36310966622904359515*z^26+352*z^2-\ 4462860137190909839*z^24+467579207216922619*z^22-53223*z^4+4707695*z^6+ 377838808802*z^102-277934382*z^8+11798292814*z^10-377838808802*z^12+ 9440753635917*z^14+3073399254178932*z^18-188666176632854*z^16+ 61044593993541535914086440*z^50-37453584148086977417979800*z^48-\ 41425766365762521*z^20-144133108491805777423347*z^36+36291491771003189766310*z^ 34+37453584148086977417979800*z^66-36291491771003189766310*z^80-9440753635917*z ^100+4462860137190909839*z^90-36310966622904359515*z^88-1527631252787869816072* z^84+41425766365762521*z^94+253530857094181812804*z^86-3073399254178932*z^96+ 188666176632854*z^98-467579207216922619*z^92+7980648311748518418140*z^82-\ 61044593993541535914086440*z^64-352*z^112+z^114+53223*z^110+277934382*z^106-\ 4707695*z^108+1527631252787869816072*z^30+4111682047225248143346541*z^42-\ 9729809441542361414870393*z^44+20312858700860113657672217*z^46+ 126821126572588576795290639*z^58-126821126572588576795290639*z^56+ 112285510083481165989752630*z^54-88007752872593945123832935*z^52-\ 112285510083481165989752630*z^60+9729809441542361414870393*z^70-\ 20312858700860113657672217*z^68+144133108491805777423347*z^78-\ 7980648311748518418140*z^32+501339671559499083166817*z^38-\ 1530848656382977738156394*z^40+88007752872593945123832935*z^62-\ 501339671559499083166817*z^76+1530848656382977738156394*z^74-\ 4111682047225248143346541*z^72-11798292814*z^104) The first , 40, terms are: [0, 56, 0, 5380, 0, 565555, 0, 61140112, 0, 6681057401, 0, 733399463809, 0, 80670839747232, 0, 8881687746659915, 0, 978276029331554471, 0, 107774190055593508924, 0, 11874332083837253355553, 0, 1308347010046031058247076, 0, 144160345931507739268348292, 0, 15884479369640561939586833927, 0, 1750258451677410252911510453203, 0, 192855638157781016873465378046540, 0, 21250196435361646612405644425105172, 0, 2341497894324657696116129708150113357, 0, 258002963845114305359576435974739004468, 0, 28428612648314965784829935920949608652051] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 82605597373360993508 z - 12472706555333723912 z - 299 z 24 22 4 6 + 1617478082848180660 z - 179015802266735688 z + 39743 z - 3157334 z 102 8 10 12 - 6638504966 z + 169857752 z - 6638504966 z + 197235378015 z 14 18 16 - 4598712533543 z - 1319968409983096 z + 86143745499193 z 50 48 - 11042228557237721954368406 z + 7170604981680872234999214 z 20 36 + 16780580900422216 z + 38128496289930195280260 z 34 66 - 10113732331075708903816 z - 4112044656540051481688904 z 80 100 90 + 2342666021242133135632 z + 197235378015 z - 179015802266735688 z 88 84 + 1617478082848180660 z + 82605597373360993508 z 94 86 96 - 1319968409983096 z - 12472706555333723912 z + 86143745499193 z 98 92 82 - 4598712533543 z + 16780580900422216 z - 472371111871757329160 z 64 112 110 106 + 7170604981680872234999214 z + z - 299 z - 3157334 z 108 30 42 + 39743 z - 472371111871757329160 z - 928174511413252206362744 z 44 46 + 2080779943909305352070584 z - 4112044656540051481688904 z 58 56 - 18071641557232526313975956 z + 19218287998304803660533808 z 54 52 - 18071641557232526313975956 z + 15024930831849681513876474 z 60 70 + 15024930831849681513876474 z - 928174511413252206362744 z 68 78 + 2080779943909305352070584 z - 10113732331075708903816 z 32 38 + 2342666021242133135632 z - 125854833539575079733008 z 40 62 + 364527713607199020960612 z - 11042228557237721954368406 z 76 74 + 38128496289930195280260 z - 125854833539575079733008 z 72 104 / + 364527713607199020960612 z + 169857752 z ) / (-1 / 28 26 2 - 281793144440395059612 z + 40210742862286299616 z + 361 z 24 22 4 6 - 4922706309537003320 z + 513589425797409548 z - 55398 z + 4948470 z 102 8 10 12 + 404932699946 z - 294326675 z + 12572641623 z - 404932699946 z 14 18 16 + 10172132302162 z + 3344986529945205 z - 204328816202517 z 50 48 + 69475673829095615926918682 z - 42585401962631486069566594 z 20 36 - 45299197866996340 z - 162138564376299238118412 z 34 66 + 40718332616214670530360 z + 42585401962631486069566594 z 80 100 - 40718332616214670530360 z - 10172132302162 z 90 88 + 4922706309537003320 z - 40210742862286299616 z 84 94 - 1703708521752324628788 z + 45299197866996340 z 86 96 98 + 281793144440395059612 z - 3344986529945205 z + 204328816202517 z 92 82 - 513589425797409548 z + 8928444969182849759232 z 64 112 114 110 - 69475673829095615926918682 z - 361 z + z + 55398 z 106 108 30 + 294326675 z - 4948470 z + 1703708521752324628788 z 42 44 + 4655029780564012894334032 z - 11033919210003555753799780 z 46 58 + 23068433592186448584649244 z + 144546802741113386962817502 z 56 54 - 144546802741113386962817502 z + 127948441186963038576718880 z 52 60 - 100235483066921373985052584 z - 127948441186963038576718880 z 70 68 + 11033919210003555753799780 z - 23068433592186448584649244 z 78 32 + 162138564376299238118412 z - 8928444969182849759232 z 38 40 + 565307945407787115223940 z - 1729861846416755506439176 z 62 76 + 100235483066921373985052584 z - 565307945407787115223940 z 74 72 + 1729861846416755506439176 z - 4655029780564012894334032 z 104 - 12572641623 z ) And in Maple-input format, it is: -(1+82605597373360993508*z^28-12472706555333723912*z^26-299*z^2+ 1617478082848180660*z^24-179015802266735688*z^22+39743*z^4-3157334*z^6-\ 6638504966*z^102+169857752*z^8-6638504966*z^10+197235378015*z^12-4598712533543* z^14-1319968409983096*z^18+86143745499193*z^16-11042228557237721954368406*z^50+ 7170604981680872234999214*z^48+16780580900422216*z^20+38128496289930195280260*z ^36-10113732331075708903816*z^34-4112044656540051481688904*z^66+ 2342666021242133135632*z^80+197235378015*z^100-179015802266735688*z^90+ 1617478082848180660*z^88+82605597373360993508*z^84-1319968409983096*z^94-\ 12472706555333723912*z^86+86143745499193*z^96-4598712533543*z^98+ 16780580900422216*z^92-472371111871757329160*z^82+7170604981680872234999214*z^ 64+z^112-299*z^110-3157334*z^106+39743*z^108-472371111871757329160*z^30-\ 928174511413252206362744*z^42+2080779943909305352070584*z^44-\ 4112044656540051481688904*z^46-18071641557232526313975956*z^58+ 19218287998304803660533808*z^56-18071641557232526313975956*z^54+ 15024930831849681513876474*z^52+15024930831849681513876474*z^60-\ 928174511413252206362744*z^70+2080779943909305352070584*z^68-\ 10113732331075708903816*z^78+2342666021242133135632*z^32-\ 125854833539575079733008*z^38+364527713607199020960612*z^40-\ 11042228557237721954368406*z^62+38128496289930195280260*z^76-\ 125854833539575079733008*z^74+364527713607199020960612*z^72+169857752*z^104)/(-\ 1-281793144440395059612*z^28+40210742862286299616*z^26+361*z^2-\ 4922706309537003320*z^24+513589425797409548*z^22-55398*z^4+4948470*z^6+ 404932699946*z^102-294326675*z^8+12572641623*z^10-404932699946*z^12+ 10172132302162*z^14+3344986529945205*z^18-204328816202517*z^16+ 69475673829095615926918682*z^50-42585401962631486069566594*z^48-\ 45299197866996340*z^20-162138564376299238118412*z^36+40718332616214670530360*z^ 34+42585401962631486069566594*z^66-40718332616214670530360*z^80-10172132302162* z^100+4922706309537003320*z^90-40210742862286299616*z^88-1703708521752324628788 *z^84+45299197866996340*z^94+281793144440395059612*z^86-3344986529945205*z^96+ 204328816202517*z^98-513589425797409548*z^92+8928444969182849759232*z^82-\ 69475673829095615926918682*z^64-361*z^112+z^114+55398*z^110+294326675*z^106-\ 4948470*z^108+1703708521752324628788*z^30+4655029780564012894334032*z^42-\ 11033919210003555753799780*z^44+23068433592186448584649244*z^46+ 144546802741113386962817502*z^58-144546802741113386962817502*z^56+ 127948441186963038576718880*z^54-100235483066921373985052584*z^52-\ 127948441186963038576718880*z^60+11033919210003555753799780*z^70-\ 23068433592186448584649244*z^68+162138564376299238118412*z^78-\ 8928444969182849759232*z^32+565307945407787115223940*z^38-\ 1729861846416755506439176*z^40+100235483066921373985052584*z^62-\ 565307945407787115223940*z^76+1729861846416755506439176*z^74-\ 4655029780564012894334032*z^72-12572641623*z^104) The first , 40, terms are: [0, 62, 0, 6727, 0, 784907, 0, 93025298, 0, 11074095089, 0, 1320292526785, 0, 157500463898686, 0, 18792962020307935, 0, 2242598752534640843, 0, 267624823037528402938, 0, 31938112163262259485537, 0, 3811497163195291544785809, 0, 454866024261216114525848834, 0, 54284028838666704024327724707, 0, 6478298321786985118237677048999, 0, 773125404633775253493891905095718, 0, 92265428452159957975691070053666721, 0, 11011033567604444711397680311516202449, 0, 1314066008920174392285379525794367057322, 0, 156821744718320712267066791079399751700963] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 96109920598761637908 z - 14046907367896123344 z - 290 z 24 22 4 6 + 1764590919382941284 z - 189419313093907584 z + 37696 z - 2956106 z 102 8 10 12 - 6209302798 z + 158351626 z - 6209302798 z + 186295128300 z 14 18 16 - 4410179528118 z - 1321666333956528 z + 84257340343235 z 50 48 - 17058202614612502481646704 z + 10931654642158854501902507 z 20 36 + 17252038924974324 z + 50441512765707266413260 z 34 66 - 12971953835819428718114 z - 6165196438630841523253766 z 80 100 90 + 2910055614768443574161 z + 186295128300 z - 189419313093907584 z 88 84 + 1764590919382941284 z + 96109920598761637908 z 94 86 96 - 1321666333956528 z - 14046907367896123344 z + 84257340343235 z 98 92 82 - 4410179528118 z + 17252038924974324 z - 567934877170456545504 z 64 112 110 106 + 10931654642158854501902507 z + z - 290 z - 2956106 z 108 30 42 + 37696 z - 567934877170456545504 z - 1333498822365549100438110 z 44 46 + 3058334751673073856904328 z - 6165196438630841523253766 z 58 56 - 28350897027740215125611440 z + 30208559329293706291029368 z 54 52 - 28350897027740215125611440 z + 23434522116262099042920568 z 60 70 + 23434522116262099042920568 z - 1333498822365549100438110 z 68 78 + 3058334751673073856904328 z - 12971953835819428718114 z 32 38 + 2910055614768443574161 z - 171480485122894494816586 z 40 62 + 510588879868465246815958 z - 17058202614612502481646704 z 76 74 + 50441512765707266413260 z - 171480485122894494816586 z 72 104 / + 510588879868465246815958 z + 158351626 z ) / (-1 / 28 26 2 - 318084632496549085712 z + 43856976654330066292 z + 353 z 24 22 4 6 - 5194330630440501220 z + 525263098097198288 z - 52720 z + 4612956 z 102 8 10 12 + 373358085636 z - 271115118 z + 11542097126 z - 373358085636 z 14 18 16 + 9483165739288 z + 3240314815509635 z - 193710919295123 z 50 48 + 107560527585436780272580187 z - 64874178614569087240867899 z 20 36 - 45016592882621576 z - 210307063844282596090584 z 34 66 + 51052049889841552727825 z + 64874178614569087240867899 z 80 100 - 51052049889841552727825 z - 9483165739288 z 90 88 + 5194330630440501220 z - 43856976654330066292 z 84 94 - 1991689434411834798760 z + 45016592882621576 z 86 96 98 + 318084632496549085712 z - 3240314815509635 z + 193710919295123 z 92 82 - 525263098097198288 z + 10811362325363708896737 z 64 112 114 110 - 107560527585436780272580187 z - 353 z + z + 52720 z 106 108 30 + 271115118 z - 4612956 z + 1991689434411834798760 z 42 44 + 6620202508465575248576338 z - 16105440353620220260426516 z 46 58 + 34455803203922327152273568 z + 229407553090191154341067176 z 56 54 - 229407553090191154341067176 z + 202215879864262028971634336 z 52 60 - 157105321193451884914045776 z - 202215879864262028971634336 z 70 68 + 16105440353620220260426516 z - 34455803203922327152273568 z 78 32 + 210307063844282596090584 z - 10811362325363708896737 z 38 40 + 757560253720415524680972 z - 2390740005635685561908058 z 62 76 + 157105321193451884914045776 z - 757560253720415524680972 z 74 72 + 2390740005635685561908058 z - 6620202508465575248576338 z 104 - 11542097126 z ) And in Maple-input format, it is: -(1+96109920598761637908*z^28-14046907367896123344*z^26-290*z^2+ 1764590919382941284*z^24-189419313093907584*z^22+37696*z^4-2956106*z^6-\ 6209302798*z^102+158351626*z^8-6209302798*z^10+186295128300*z^12-4410179528118* z^14-1321666333956528*z^18+84257340343235*z^16-17058202614612502481646704*z^50+ 10931654642158854501902507*z^48+17252038924974324*z^20+50441512765707266413260* z^36-12971953835819428718114*z^34-6165196438630841523253766*z^66+ 2910055614768443574161*z^80+186295128300*z^100-189419313093907584*z^90+ 1764590919382941284*z^88+96109920598761637908*z^84-1321666333956528*z^94-\ 14046907367896123344*z^86+84257340343235*z^96-4410179528118*z^98+ 17252038924974324*z^92-567934877170456545504*z^82+10931654642158854501902507*z^ 64+z^112-290*z^110-2956106*z^106+37696*z^108-567934877170456545504*z^30-\ 1333498822365549100438110*z^42+3058334751673073856904328*z^44-\ 6165196438630841523253766*z^46-28350897027740215125611440*z^58+ 30208559329293706291029368*z^56-28350897027740215125611440*z^54+ 23434522116262099042920568*z^52+23434522116262099042920568*z^60-\ 1333498822365549100438110*z^70+3058334751673073856904328*z^68-\ 12971953835819428718114*z^78+2910055614768443574161*z^32-\ 171480485122894494816586*z^38+510588879868465246815958*z^40-\ 17058202614612502481646704*z^62+50441512765707266413260*z^76-\ 171480485122894494816586*z^74+510588879868465246815958*z^72+158351626*z^104)/(-\ 1-318084632496549085712*z^28+43856976654330066292*z^26+353*z^2-\ 5194330630440501220*z^24+525263098097198288*z^22-52720*z^4+4612956*z^6+ 373358085636*z^102-271115118*z^8+11542097126*z^10-373358085636*z^12+ 9483165739288*z^14+3240314815509635*z^18-193710919295123*z^16+ 107560527585436780272580187*z^50-64874178614569087240867899*z^48-\ 45016592882621576*z^20-210307063844282596090584*z^36+51052049889841552727825*z^ 34+64874178614569087240867899*z^66-51052049889841552727825*z^80-9483165739288*z ^100+5194330630440501220*z^90-43856976654330066292*z^88-1991689434411834798760* z^84+45016592882621576*z^94+318084632496549085712*z^86-3240314815509635*z^96+ 193710919295123*z^98-525263098097198288*z^92+10811362325363708896737*z^82-\ 107560527585436780272580187*z^64-353*z^112+z^114+52720*z^110+271115118*z^106-\ 4612956*z^108+1991689434411834798760*z^30+6620202508465575248576338*z^42-\ 16105440353620220260426516*z^44+34455803203922327152273568*z^46+ 229407553090191154341067176*z^58-229407553090191154341067176*z^56+ 202215879864262028971634336*z^54-157105321193451884914045776*z^52-\ 202215879864262028971634336*z^60+16105440353620220260426516*z^70-\ 34455803203922327152273568*z^68+210307063844282596090584*z^78-\ 10811362325363708896737*z^32+757560253720415524680972*z^38-\ 2390740005635685561908058*z^40+157105321193451884914045776*z^62-\ 757560253720415524680972*z^76+2390740005635685561908058*z^74-\ 6620202508465575248576338*z^72-11542097126*z^104) The first , 40, terms are: [0, 63, 0, 7215, 0, 882385, 0, 108959841, 0, 13478506107, 0, 1667946603543, 0, 206425005573397, 0, 25547758143253261, 0, 3161885244205707159, 0, 391327352815860169459, 0, 48432237691927570641377, 0, 5994168291116884840945697, 0, 741862380467529546326943543, 0, 91815873754794781355096885375, 0, 11363502086154732375178619110169, 0, 1406392756963184285512025294552553, 0, 174060828483242458387142546201607679, 0, 21542468749046170631149173476166940839, 0, 2666182643526973524300407622969541655745, 0, 329977495686773266305574990658185306659617] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 7}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 8703873931932 z + 8703873931932 z + 193 z 24 22 4 6 - 6580113657420 z + 3754798934511 z - 14761 z + 603985 z 8 10 12 14 - 15033802 z + 244877906 z - 2734782834 z + 21607824586 z 18 16 50 48 + 517700509719 z - 123435544031 z + 14761 z - 603985 z 20 36 34 - 1611826890527 z - 517700509719 z + 1611826890527 z 30 42 44 46 54 + 6580113657420 z + 2734782834 z - 244877906 z + 15033802 z + z 52 32 38 40 / - 193 z - 3754798934511 z + 123435544031 z - 21607824586 z ) / (1 / 28 26 2 24 + 69865259096200 z - 60903693341728 z - 264 z + 40315326230843 z 22 4 6 8 10 - 20218072499128 z + 24518 z - 1170600 z + 33489947 z - 622645936 z 12 14 18 16 + 7903420764 z - 70770864688 z - 2170150421912 z + 457438620329 z 50 48 20 36 - 1170600 z + 33489947 z + 7649980101274 z + 7649980101274 z 34 30 42 - 20218072499128 z - 60903693341728 z - 70770864688 z 44 46 56 54 52 + 7903420764 z - 622645936 z + z - 264 z + 24518 z 32 38 40 + 40315326230843 z - 2170150421912 z + 457438620329 z ) And in Maple-input format, it is: -(-1-8703873931932*z^28+8703873931932*z^26+193*z^2-6580113657420*z^24+ 3754798934511*z^22-14761*z^4+603985*z^6-15033802*z^8+244877906*z^10-2734782834* z^12+21607824586*z^14+517700509719*z^18-123435544031*z^16+14761*z^50-603985*z^ 48-1611826890527*z^20-517700509719*z^36+1611826890527*z^34+6580113657420*z^30+ 2734782834*z^42-244877906*z^44+15033802*z^46+z^54-193*z^52-3754798934511*z^32+ 123435544031*z^38-21607824586*z^40)/(1+69865259096200*z^28-60903693341728*z^26-\ 264*z^2+40315326230843*z^24-20218072499128*z^22+24518*z^4-1170600*z^6+33489947* z^8-622645936*z^10+7903420764*z^12-70770864688*z^14-2170150421912*z^18+ 457438620329*z^16-1170600*z^50+33489947*z^48+7649980101274*z^20+7649980101274*z ^36-20218072499128*z^34-60903693341728*z^30-70770864688*z^42+7903420764*z^44-\ 622645936*z^46+z^56-264*z^54+24518*z^52+40315326230843*z^32-2170150421912*z^38+ 457438620329*z^40) The first , 40, terms are: [0, 71, 0, 8987, 0, 1198405, 0, 160692109, 0, 21560386979, 0, 2893010996831, 0, 388192737265817, 0, 52088902490604969, 0, 6989451162624836591, 0, 937866348137304639315, 0, 125845831190777349976637, 0, 16886386078888698938935061, 0, 2265867944804894472901605611, 0, 304041227038367483610260962359, 0, 40797200010184213931838491958641, 0, 5474295525292225200358220888329105, 0, 734558045424741388861968760142289751, 0, 98565289288006233001748005464379226187, 0, 13225797897036300872402329221764598798453, 0, 1774678807081242224111375781721018502001117] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 1011437263820919642 z + 255969891173431953 z + 273 z 24 22 4 6 - 53927211679949409 z + 9406878794409804 z - 31942 z + 2147550 z 8 10 12 14 - 94021110 z + 2882829494 z - 64960108483 z + 1112987855731 z 18 16 50 + 157757180842228 z - 14866518362844 z + 42353206660306484763 z 48 20 - 70164930431548632235 z - 1349303867191812 z 36 34 - 42353206660306484763 z + 21572211524833732522 z 66 80 84 86 82 + 1349303867191812 z - 2147550 z - 273 z + z + 31942 z 64 30 - 9406878794409804 z + 3341193769126867850 z 42 44 + 116126642781140209928 z - 116126642781140209928 z 46 58 + 98183992082044873048 z + 1011437263820919642 z 56 54 - 3341193769126867850 z + 9257971846811177378 z 52 60 70 - 21572211524833732522 z - 255969891173431953 z + 14866518362844 z 68 78 32 - 157757180842228 z + 94021110 z - 9257971846811177378 z 38 40 + 70164930431548632235 z - 98183992082044873048 z 62 76 74 + 53927211679949409 z - 2882829494 z + 64960108483 z 72 / 28 - 1112987855731 z ) / (1 + 4291780937821153469 z / 26 2 24 - 1005971469154272353 z - 345 z + 196430967587215265 z 22 4 6 8 - 31772752730592684 z + 47773 z - 3663782 z + 178989866 z 10 12 14 - 6042189142 z + 148636910775 z - 2765278213251 z 18 16 50 - 458066606787644 z + 39971394756163 z - 442655794989681486275 z 48 20 + 672249373909676359771 z + 4226410396289012 z 36 34 + 246393740882042402223 z - 115822778741093938634 z 66 80 88 84 86 - 31772752730592684 z + 178989866 z + z + 47773 z - 345 z 82 64 30 - 3663782 z + 196430967587215265 z - 15320431528516668346 z 42 44 - 863581476783976829848 z + 938736767086321951320 z 46 58 - 863581476783976829848 z - 15320431528516668346 z 56 54 + 45914710332843681718 z - 115822778741093938634 z 52 60 + 246393740882042402223 z + 4291780937821153469 z 70 68 78 - 458066606787644 z + 4226410396289012 z - 6042189142 z 32 38 + 45914710332843681718 z - 442655794989681486275 z 40 62 76 + 672249373909676359771 z - 1005971469154272353 z + 148636910775 z 74 72 - 2765278213251 z + 39971394756163 z ) And in Maple-input format, it is: -(-1-1011437263820919642*z^28+255969891173431953*z^26+273*z^2-53927211679949409 *z^24+9406878794409804*z^22-31942*z^4+2147550*z^6-94021110*z^8+2882829494*z^10-\ 64960108483*z^12+1112987855731*z^14+157757180842228*z^18-14866518362844*z^16+ 42353206660306484763*z^50-70164930431548632235*z^48-1349303867191812*z^20-\ 42353206660306484763*z^36+21572211524833732522*z^34+1349303867191812*z^66-\ 2147550*z^80-273*z^84+z^86+31942*z^82-9406878794409804*z^64+3341193769126867850 *z^30+116126642781140209928*z^42-116126642781140209928*z^44+ 98183992082044873048*z^46+1011437263820919642*z^58-3341193769126867850*z^56+ 9257971846811177378*z^54-21572211524833732522*z^52-255969891173431953*z^60+ 14866518362844*z^70-157757180842228*z^68+94021110*z^78-9257971846811177378*z^32 +70164930431548632235*z^38-98183992082044873048*z^40+53927211679949409*z^62-\ 2882829494*z^76+64960108483*z^74-1112987855731*z^72)/(1+4291780937821153469*z^ 28-1005971469154272353*z^26-345*z^2+196430967587215265*z^24-31772752730592684*z ^22+47773*z^4-3663782*z^6+178989866*z^8-6042189142*z^10+148636910775*z^12-\ 2765278213251*z^14-458066606787644*z^18+39971394756163*z^16-\ 442655794989681486275*z^50+672249373909676359771*z^48+4226410396289012*z^20+ 246393740882042402223*z^36-115822778741093938634*z^34-31772752730592684*z^66+ 178989866*z^80+z^88+47773*z^84-345*z^86-3663782*z^82+196430967587215265*z^64-\ 15320431528516668346*z^30-863581476783976829848*z^42+938736767086321951320*z^44 -863581476783976829848*z^46-15320431528516668346*z^58+45914710332843681718*z^56 -115822778741093938634*z^54+246393740882042402223*z^52+4291780937821153469*z^60 -458066606787644*z^70+4226410396289012*z^68-6042189142*z^78+ 45914710332843681718*z^32-442655794989681486275*z^38+672249373909676359771*z^40 -1005971469154272353*z^62+148636910775*z^76-2765278213251*z^74+39971394756163*z ^72) The first , 40, terms are: [0, 72, 0, 9009, 0, 1184681, 0, 157151536, 0, 20900615841, 0, 2782366172497, 0, 370538799127776, 0, 49353666177942505, 0, 6574036192964922001, 0, 875700755635612239064, 0, 116649752278628768823345, 0, 15538666092491949836488593, 0, 2069876279964748188784143544, 0, 275724490214067151321765140369, 0, 36728772980651519573131729923369, 0, 4892575624497958229682622481316992, 0, 651731474282477870170615321997892529, 0, 86816015791544099573153468080555652033, 0, 11564610508962521536362172980962287119248, 0, 1540501662372664923182603260728908922221033] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 91926716167929002892 z - 13708747392371923388 z - 296 z 24 22 4 6 + 1755054914469942492 z - 191701401702216684 z + 39138 z - 3108828 z 102 8 10 12 - 6611088600 z + 167930806 z - 6611088600 z + 198314123322 z 14 18 16 - 4676011277724 z - 1376445003520044 z + 88674786643499 z 50 48 - 13415801369213574529584632 z + 8681471659973482252701787 z 20 36 + 17732053980639212 z + 44318111421408956351670 z 34 66 - 11640769483776795119004 z - 4956197921040340357805664 z 80 100 90 + 2668041908726497456505 z + 198314123322 z - 191701401702216684 z 88 84 + 1755054914469942492 z + 91926716167929002892 z 94 86 96 - 1376445003520044 z - 13708747392371923388 z + 88674786643499 z 98 92 82 - 4676011277724 z + 17732053980639212 z - 531956614722493675292 z 64 112 110 106 + 8681471659973482252701787 z + z - 296 z - 3108828 z 108 30 42 + 39138 z - 531956614722493675292 z - 1105572339956835524924476 z 44 46 + 2494329011074367265859438 z - 4956197921040340357805664 z 58 56 - 22044797245572446433883800 z + 23455378238945295155905672 z 54 52 - 22044797245572446433883800 z + 18300550448328323056389960 z 60 70 + 18300550448328323056389960 z - 1105572339956835524924476 z 68 78 + 2494329011074367265859438 z - 11640769483776795119004 z 32 38 + 2668041908726497456505 z - 147609682782430572880592 z 40 62 + 431045094723086611229994 z - 13415801369213574529584632 z 76 74 + 44318111421408956351670 z - 147609682782430572880592 z 72 104 / 2 + 431045094723086611229994 z + 167930806 z ) / ((-1 + z ) (1 / 28 26 2 + 271866328908676048784 z - 38953906630331587484 z - 362 z 24 22 4 6 + 4780951542469623420 z - 499453492131995836 z + 55100 z - 4880850 z 102 8 10 12 - 12272214410 z + 288477558 z - 12272214410 z + 394348038724 z 14 18 16 - 9896382718530 z - 3254172229178668 z + 198752012573371 z 50 48 - 52648674820373671550493000 z + 33668885910760402076801531 z 20 36 + 44071753810307056 z + 150476733939811974167404 z 34 66 - 38310967935601814750702 z - 18935024196442856321188390 z 80 100 90 + 8491910692871263013705 z + 394348038724 z - 499453492131995836 z 88 84 + 4780951542469623420 z + 271866328908676048784 z 94 86 96 - 3254172229178668 z - 38953906630331587484 z + 198752012573371 z 98 92 82 - 9896382718530 z + 44071753810307056 z - 1633835271989068692940 z 64 112 110 106 + 33668885910760402076801531 z + z - 362 z - 4880850 z 108 30 42 + 55100 z - 1633835271989068692940 z - 4062105086582385847248190 z 44 46 + 9358936143073328040354580 z - 18935024196442856321188390 z 58 56 - 87704126191430619580477672 z + 93477047066384628448491592 z 54 52 - 87704126191430619580477672 z + 72433575477960767751237760 z 60 70 + 72433575477960767751237760 z - 4062105086582385847248190 z 68 78 + 9358936143073328040354580 z - 38310967935601814750702 z 32 38 + 8491910692871263013705 z - 515851936183899277586982 z 40 62 + 1546600089674863855432682 z - 52648674820373671550493000 z 76 74 + 150476733939811974167404 z - 515851936183899277586982 z 72 104 + 1546600089674863855432682 z + 288477558 z )) And in Maple-input format, it is: -(1+91926716167929002892*z^28-13708747392371923388*z^26-296*z^2+ 1755054914469942492*z^24-191701401702216684*z^22+39138*z^4-3108828*z^6-\ 6611088600*z^102+167930806*z^8-6611088600*z^10+198314123322*z^12-4676011277724* z^14-1376445003520044*z^18+88674786643499*z^16-13415801369213574529584632*z^50+ 8681471659973482252701787*z^48+17732053980639212*z^20+44318111421408956351670*z ^36-11640769483776795119004*z^34-4956197921040340357805664*z^66+ 2668041908726497456505*z^80+198314123322*z^100-191701401702216684*z^90+ 1755054914469942492*z^88+91926716167929002892*z^84-1376445003520044*z^94-\ 13708747392371923388*z^86+88674786643499*z^96-4676011277724*z^98+ 17732053980639212*z^92-531956614722493675292*z^82+8681471659973482252701787*z^ 64+z^112-296*z^110-3108828*z^106+39138*z^108-531956614722493675292*z^30-\ 1105572339956835524924476*z^42+2494329011074367265859438*z^44-\ 4956197921040340357805664*z^46-22044797245572446433883800*z^58+ 23455378238945295155905672*z^56-22044797245572446433883800*z^54+ 18300550448328323056389960*z^52+18300550448328323056389960*z^60-\ 1105572339956835524924476*z^70+2494329011074367265859438*z^68-\ 11640769483776795119004*z^78+2668041908726497456505*z^32-\ 147609682782430572880592*z^38+431045094723086611229994*z^40-\ 13415801369213574529584632*z^62+44318111421408956351670*z^76-\ 147609682782430572880592*z^74+431045094723086611229994*z^72+167930806*z^104)/(-\ 1+z^2)/(1+271866328908676048784*z^28-38953906630331587484*z^26-362*z^2+ 4780951542469623420*z^24-499453492131995836*z^22+55100*z^4-4880850*z^6-\ 12272214410*z^102+288477558*z^8-12272214410*z^10+394348038724*z^12-\ 9896382718530*z^14-3254172229178668*z^18+198752012573371*z^16-\ 52648674820373671550493000*z^50+33668885910760402076801531*z^48+ 44071753810307056*z^20+150476733939811974167404*z^36-38310967935601814750702*z^ 34-18935024196442856321188390*z^66+8491910692871263013705*z^80+394348038724*z^ 100-499453492131995836*z^90+4780951542469623420*z^88+271866328908676048784*z^84 -3254172229178668*z^94-38953906630331587484*z^86+198752012573371*z^96-\ 9896382718530*z^98+44071753810307056*z^92-1633835271989068692940*z^82+ 33668885910760402076801531*z^64+z^112-362*z^110-4880850*z^106+55100*z^108-\ 1633835271989068692940*z^30-4062105086582385847248190*z^42+ 9358936143073328040354580*z^44-18935024196442856321188390*z^46-\ 87704126191430619580477672*z^58+93477047066384628448491592*z^56-\ 87704126191430619580477672*z^54+72433575477960767751237760*z^52+ 72433575477960767751237760*z^60-4062105086582385847248190*z^70+ 9358936143073328040354580*z^68-38310967935601814750702*z^78+ 8491910692871263013705*z^32-515851936183899277586982*z^38+ 1546600089674863855432682*z^40-52648674820373671550493000*z^62+ 150476733939811974167404*z^76-515851936183899277586982*z^74+ 1546600089674863855432682*z^72+288477558*z^104) The first , 40, terms are: [0, 67, 0, 7997, 0, 1014079, 0, 129862111, 0, 16664478977, 0, 2139509751687, 0, 274720008791873, 0, 35276063345421857, 0, 4529742928297318527, 0, 581658338029559230153, 0, 74690027239689185755727, 0, 9590855487353968632669615, 0, 1231550133277839455826312341, 0, 158141862427141752358786257099, 0, 20306805269108300784304649075617, 0, 2607572304357802517704288592508513, 0, 334835205939769163277777225022201275, 0, 42995783839960803545183830405504197413, 0, 5521036603258458832970078424214515648271, 0, 708949633019203527977592072942382281838767] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 7883749452526780 z - 3222970306522329 z - 225 z 24 22 4 6 + 1072939668291844 z - 289515623847076 z + 20884 z - 1081701 z 8 10 12 14 + 35658288 z - 807176300 z + 13183384124 z - 160793449688 z 18 16 50 - 10931382070894 z + 1501356579601 z - 289515623847076 z 48 20 36 + 1072939668291844 z + 62932377896230 z + 38240359035682426 z 34 66 64 - 34660258889521422 z - 1081701 z + 35658288 z 30 42 44 - 15757961020362059 z - 15757961020362059 z + 7883749452526780 z 46 58 56 - 3222970306522329 z - 160793449688 z + 1501356579601 z 54 52 60 70 - 10931382070894 z + 62932377896230 z + 13183384124 z - 225 z 68 32 38 + 20884 z + 25800483235726809 z - 34660258889521422 z 40 62 72 / 2 + 25800483235726809 z - 807176300 z + z ) / ((-1 + z ) (1 / 28 26 2 + 29905452402594117 z - 11728500155408312 z - 296 z 24 22 4 6 + 3719834659880365 z - 950596816375882 z + 32712 z - 1924424 z 8 10 12 14 + 70327071 z - 1741166034 z + 30850222574 z - 405943086346 z 18 16 50 - 31727121771334 z + 4071869116871 z - 950596816375882 z 48 20 36 + 3719834659880365 z + 194700166010917 z + 156937907958099307 z 34 66 64 - 141517958603507568 z - 1924424 z + 70327071 z 30 42 44 - 61817867593867636 z - 61817867593867636 z + 29905452402594117 z 46 58 56 - 11728500155408312 z - 405943086346 z + 4071869116871 z 54 52 60 70 - 31727121771334 z + 194700166010917 z + 30850222574 z - 296 z 68 32 38 + 32712 z + 103754014419271181 z - 141517958603507568 z 40 62 72 + 103754014419271181 z - 1741166034 z + z )) And in Maple-input format, it is: -(1+7883749452526780*z^28-3222970306522329*z^26-225*z^2+1072939668291844*z^24-\ 289515623847076*z^22+20884*z^4-1081701*z^6+35658288*z^8-807176300*z^10+ 13183384124*z^12-160793449688*z^14-10931382070894*z^18+1501356579601*z^16-\ 289515623847076*z^50+1072939668291844*z^48+62932377896230*z^20+ 38240359035682426*z^36-34660258889521422*z^34-1081701*z^66+35658288*z^64-\ 15757961020362059*z^30-15757961020362059*z^42+7883749452526780*z^44-\ 3222970306522329*z^46-160793449688*z^58+1501356579601*z^56-10931382070894*z^54+ 62932377896230*z^52+13183384124*z^60-225*z^70+20884*z^68+25800483235726809*z^32 -34660258889521422*z^38+25800483235726809*z^40-807176300*z^62+z^72)/(-1+z^2)/(1 +29905452402594117*z^28-11728500155408312*z^26-296*z^2+3719834659880365*z^24-\ 950596816375882*z^22+32712*z^4-1924424*z^6+70327071*z^8-1741166034*z^10+ 30850222574*z^12-405943086346*z^14-31727121771334*z^18+4071869116871*z^16-\ 950596816375882*z^50+3719834659880365*z^48+194700166010917*z^20+ 156937907958099307*z^36-141517958603507568*z^34-1924424*z^66+70327071*z^64-\ 61817867593867636*z^30-61817867593867636*z^42+29905452402594117*z^44-\ 11728500155408312*z^46-405943086346*z^58+4071869116871*z^56-31727121771334*z^54 +194700166010917*z^52+30850222574*z^60-296*z^70+32712*z^68+103754014419271181*z ^32-141517958603507568*z^38+103754014419271181*z^40-1741166034*z^62+z^72) The first , 40, terms are: [0, 72, 0, 9260, 0, 1249079, 0, 169642968, 0, 23079650389, 0, 3141669210909, 0, 427733036328160, 0, 58238925465476831, 0, 7929828487129157091, 0, 1079736301908130928484, 0, 147018785340557038514137, 0, 20018354956057018674145860, 0, 2725737891679206065497773860, 0, 371141783391480786798199517619, 0, 50535390690441808918306461048851, 0, 6880997680533163058709368394237068, 0, 936930113393129336290952704602559540, 0, 127574238471064188321470013600723742617, 0, 17370758074120427523811501648392710488412, 0, 2365236428257366853356756799210983448381995] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 14822841762808 z - 13043940947072 z - 192 z 24 22 4 6 + 8879059588347 z - 4659865114368 z + 14650 z - 601952 z 8 10 12 14 + 15104635 z - 248471424 z + 2809207140 z - 22573830912 z 18 16 50 48 - 573811601888 z + 132130701449 z - 601952 z + 15104635 z 20 36 34 + 1875005086566 z + 1875005086566 z - 4659865114368 z 30 42 44 46 - 13043940947072 z - 22573830912 z + 2809207140 z - 248471424 z 56 54 52 32 38 + z - 192 z + 14650 z + 8879059588347 z - 573811601888 z 40 / 2 28 + 132130701449 z ) / ((-1 + z ) (1 + 58599279552408 z / 26 2 24 22 - 51190824953144 z - 250 z + 34098455170107 z - 17274270916134 z 4 6 8 10 12 + 23026 z - 1103418 z + 31606811 z - 584147460 z + 7324361748 z 14 18 16 50 - 64532269188 z - 1909840689894 z + 409740823465 z - 1103418 z 48 20 36 + 31606811 z + 6624873066030 z + 6624873066030 z 34 30 42 - 17274270916134 z - 51190824953144 z - 64532269188 z 44 46 56 54 52 + 7324361748 z - 584147460 z + z - 250 z + 23026 z 32 38 40 + 34098455170107 z - 1909840689894 z + 409740823465 z )) And in Maple-input format, it is: -(1+14822841762808*z^28-13043940947072*z^26-192*z^2+8879059588347*z^24-\ 4659865114368*z^22+14650*z^4-601952*z^6+15104635*z^8-248471424*z^10+2809207140* z^12-22573830912*z^14-573811601888*z^18+132130701449*z^16-601952*z^50+15104635* z^48+1875005086566*z^20+1875005086566*z^36-4659865114368*z^34-13043940947072*z^ 30-22573830912*z^42+2809207140*z^44-248471424*z^46+z^56-192*z^54+14650*z^52+ 8879059588347*z^32-573811601888*z^38+132130701449*z^40)/(-1+z^2)/(1+ 58599279552408*z^28-51190824953144*z^26-250*z^2+34098455170107*z^24-\ 17274270916134*z^22+23026*z^4-1103418*z^6+31606811*z^8-584147460*z^10+ 7324361748*z^12-64532269188*z^14-1909840689894*z^18+409740823465*z^16-1103418*z ^50+31606811*z^48+6624873066030*z^20+6624873066030*z^36-17274270916134*z^34-\ 51190824953144*z^30-64532269188*z^42+7324361748*z^44-584147460*z^46+z^56-250*z^ 54+23026*z^52+34098455170107*z^32-1909840689894*z^38+409740823465*z^40) The first , 40, terms are: [0, 59, 0, 6183, 0, 703141, 0, 81427485, 0, 9474171407, 0, 1103742696915, 0, 128632091454729, 0, 14992512957152249, 0, 1747478791519478691, 0, 203682130935718176575, 0, 23740778800175910436397, 0, 2767179210087289455141525, 0, 322537113950637582532238583, 0, 37594310989779821477786039403, 0, 4381921276994492162285469807377, 0, 510748399571482435586002943791345, 0, 59531860922809335612523465281040395, 0, 6938920355684351882999383696530691927, 0, 808787344493074571864265909031216973493, 0, 94270712892708378328955489158667960102413] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 4}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 59429383139 z + 114684505594 z + 182 z - 159006750247 z 22 4 6 8 10 + 159006750247 z - 12813 z + 462291 z - 9598266 z + 123038235 z 12 14 18 16 - 1021913485 z + 5701456278 z + 59429383139 z - 21947871469 z 20 36 34 30 - 114684505594 z - 123038235 z + 1021913485 z + 21947871469 z 42 44 46 32 38 40 + 12813 z - 182 z + z - 5701456278 z + 9598266 z - 462291 z ) / 28 26 2 24 / (1 + 694582776112 z - 1134998997970 z - 254 z + 1336511222222 z / 22 4 6 8 10 - 1134998997970 z + 22352 z - 956684 z + 23054292 z - 341108446 z 12 14 18 16 + 3278697112 z - 21290206034 z - 305452618228 z + 95944284204 z 48 20 36 34 + z + 694582776112 z + 3278697112 z - 21290206034 z 30 42 44 46 32 - 305452618228 z - 956684 z + 22352 z - 254 z + 95944284204 z 38 40 - 341108446 z + 23054292 z ) And in Maple-input format, it is: -(-1-59429383139*z^28+114684505594*z^26+182*z^2-159006750247*z^24+159006750247* z^22-12813*z^4+462291*z^6-9598266*z^8+123038235*z^10-1021913485*z^12+5701456278 *z^14+59429383139*z^18-21947871469*z^16-114684505594*z^20-123038235*z^36+ 1021913485*z^34+21947871469*z^30+12813*z^42-182*z^44+z^46-5701456278*z^32+ 9598266*z^38-462291*z^40)/(1+694582776112*z^28-1134998997970*z^26-254*z^2+ 1336511222222*z^24-1134998997970*z^22+22352*z^4-956684*z^6+23054292*z^8-\ 341108446*z^10+3278697112*z^12-21290206034*z^14-305452618228*z^18+95944284204*z ^16+z^48+694582776112*z^20+3278697112*z^36-21290206034*z^34-305452618228*z^30-\ 956684*z^42+22352*z^44-254*z^46+95944284204*z^32-341108446*z^38+23054292*z^40) The first , 40, terms are: [0, 72, 0, 8749, 0, 1107295, 0, 141120504, 0, 18022539679, 0, 2303332006615, 0, 294450229115544, 0, 37645204481453359, 0, 4813080919120634701, 0, 615378782873729091720, 0, 78679945810333489807465, 0, 10059731678730939799839577, 0, 1286201595914682405704252040, 0, 164449214658313597095755245453, 0, 21025900527766668096588662428495, 0, 2688298085444805412029785579613144, 0, 343716393198808524134545306889636455, 0, 43946376402191996289444272124018738319, 0, 5618830060759951400452108926012304091064, 0, 718403969981161672932540559318453997438527] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 82303260568871564813 z - 12218645396540180754 z - 293 z 24 22 4 6 + 1560332693962910415 z - 170372833377046988 z + 38230 z - 2991072 z 102 8 10 12 - 6167706840 z + 159055623 z - 6167706840 z + 182489742725 z 14 18 16 - 4252468718016 z - 1231278434916502 z + 79881103996515 z 50 48 - 13186114362836634962337594 z + 8481605307569785997043418 z 20 36 + 15790902564451503 z + 40971103141457951513412 z 34 66 - 10659669173869641422138 z - 4806441580409799656314334 z 80 100 90 + 2421945519309894626843 z + 182489742725 z - 170372833377046988 z 88 84 + 1560332693962910415 z + 82303260568871564813 z 94 86 96 - 1231278434916502 z - 12218645396540180754 z + 79881103996515 z 98 92 82 - 4252468718016 z + 15790902564451503 z - 479227172126186555670 z 64 112 110 106 + 8481605307569785997043418 z + z - 293 z - 2991072 z 108 30 42 + 38230 z - 479227172126186555670 z - 1053177184401108867269490 z 44 46 + 2398453837796691836606282 z - 4806441580409799656314334 z 58 56 - 21823651972258360258678058 z + 23241492850316952184276782 z 54 52 - 21823651972258360258678058 z + 18067509141832592912186362 z 60 70 + 18067509141832592912186362 z - 1053177184401108867269490 z 68 78 + 2398453837796691836606282 z - 10659669173869641422138 z 32 38 + 2421945519309894626843 z - 137830521820678094046705 z 40 62 + 406575267844011458616977 z - 13186114362836634962337594 z 76 74 + 40971103141457951513412 z - 137830521820678094046705 z 72 104 / 2 + 406575267844011458616977 z + 159055623 z ) / ((-1 + z ) (1 / 28 26 2 + 240112461848831728130 z - 34256695315793001014 z - 355 z 24 22 4 6 + 4195280903119990144 z - 438341362711172520 z + 53352 z - 4661224 z 102 8 10 12 - 11366975462 z + 271348278 z - 11366975462 z + 359993198112 z 14 18 16 - 8919311464484 z - 2879008807182530 z + 177250112152023 z 50 48 - 51215021111005016636072434 z + 32543836433933482609190303 z 20 36 + 38783447580241135 z + 137290275663814328457402 z 34 66 - 34613467038993670767250 z - 18160019501184534220826722 z 80 100 90 + 7604366293079461202097 z + 359993198112 z - 438341362711172520 z 88 84 + 4195280903119990144 z + 240112461848831728130 z 94 86 96 - 2879008807182530 z - 34256695315793001014 z + 177250112152023 z 98 92 82 - 8919311464484 z + 38783447580241135 z - 1451883901969180895748 z 64 112 110 106 + 32543836433933482609190303 z + z - 355 z - 4661224 z 108 30 42 + 53352 z - 1451883901969180895748 z - 3823396162238481656095634 z 44 46 + 8895806295783580679568346 z - 18160019501184534220826722 z 58 56 - 85969541534969592565151650 z + 91718173905350232133112803 z 54 52 - 85969541534969592565151650 z + 70794983007996591661250346 z 60 70 + 70794983007996591661250346 z - 3823396162238481656095634 z 68 78 + 8895806295783580679568346 z - 34613467038993670767250 z 32 38 + 7604366293079461202097 z - 475528805478594128807619 z 40 62 + 1440755484702203015338718 z - 51215021111005016636072434 z 76 74 + 137290275663814328457402 z - 475528805478594128807619 z 72 104 + 1440755484702203015338718 z + 271348278 z )) And in Maple-input format, it is: -(1+82303260568871564813*z^28-12218645396540180754*z^26-293*z^2+ 1560332693962910415*z^24-170372833377046988*z^22+38230*z^4-2991072*z^6-\ 6167706840*z^102+159055623*z^8-6167706840*z^10+182489742725*z^12-4252468718016* z^14-1231278434916502*z^18+79881103996515*z^16-13186114362836634962337594*z^50+ 8481605307569785997043418*z^48+15790902564451503*z^20+40971103141457951513412*z ^36-10659669173869641422138*z^34-4806441580409799656314334*z^66+ 2421945519309894626843*z^80+182489742725*z^100-170372833377046988*z^90+ 1560332693962910415*z^88+82303260568871564813*z^84-1231278434916502*z^94-\ 12218645396540180754*z^86+79881103996515*z^96-4252468718016*z^98+ 15790902564451503*z^92-479227172126186555670*z^82+8481605307569785997043418*z^ 64+z^112-293*z^110-2991072*z^106+38230*z^108-479227172126186555670*z^30-\ 1053177184401108867269490*z^42+2398453837796691836606282*z^44-\ 4806441580409799656314334*z^46-21823651972258360258678058*z^58+ 23241492850316952184276782*z^56-21823651972258360258678058*z^54+ 18067509141832592912186362*z^52+18067509141832592912186362*z^60-\ 1053177184401108867269490*z^70+2398453837796691836606282*z^68-\ 10659669173869641422138*z^78+2421945519309894626843*z^32-\ 137830521820678094046705*z^38+406575267844011458616977*z^40-\ 13186114362836634962337594*z^62+40971103141457951513412*z^76-\ 137830521820678094046705*z^74+406575267844011458616977*z^72+159055623*z^104)/(-\ 1+z^2)/(1+240112461848831728130*z^28-34256695315793001014*z^26-355*z^2+ 4195280903119990144*z^24-438341362711172520*z^22+53352*z^4-4661224*z^6-\ 11366975462*z^102+271348278*z^8-11366975462*z^10+359993198112*z^12-\ 8919311464484*z^14-2879008807182530*z^18+177250112152023*z^16-\ 51215021111005016636072434*z^50+32543836433933482609190303*z^48+ 38783447580241135*z^20+137290275663814328457402*z^36-34613467038993670767250*z^ 34-18160019501184534220826722*z^66+7604366293079461202097*z^80+359993198112*z^ 100-438341362711172520*z^90+4195280903119990144*z^88+240112461848831728130*z^84 -2879008807182530*z^94-34256695315793001014*z^86+177250112152023*z^96-\ 8919311464484*z^98+38783447580241135*z^92-1451883901969180895748*z^82+ 32543836433933482609190303*z^64+z^112-355*z^110-4661224*z^106+53352*z^108-\ 1451883901969180895748*z^30-3823396162238481656095634*z^42+ 8895806295783580679568346*z^44-18160019501184534220826722*z^46-\ 85969541534969592565151650*z^58+91718173905350232133112803*z^56-\ 85969541534969592565151650*z^54+70794983007996591661250346*z^52+ 70794983007996591661250346*z^60-3823396162238481656095634*z^70+ 8895806295783580679568346*z^68-34613467038993670767250*z^78+ 7604366293079461202097*z^32-475528805478594128807619*z^38+ 1440755484702203015338718*z^40-51215021111005016636072434*z^62+ 137290275663814328457402*z^76-475528805478594128807619*z^74+ 1440755484702203015338718*z^72+271348278*z^104) The first , 40, terms are: [0, 63, 0, 6951, 0, 814519, 0, 96715816, 0, 11538494613, 0, 1379301397629, 0, 165021582076169, 0, 19750787747542659, 0, 2364280257297939539, 0, 283037825121883919637, 0, 33884693073201494671921, 0, 4056659568057796478538421, 0, 485664282436702844551397400, 0, 58143999425119640866255338343, 0, 6961040077135159578978035498987, 0, 833380979468790826502833009321195, 0, 99773022597068187068766516375393793, 0, 11944905435078918614873601069197020001, 0, 1430053616849621770091493463758444483235, 0, 171207164062832878396030790318953865833283] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 603863654981299969 z - 149170516129402547 z - 239 z 24 22 4 6 + 30863389489436044 z - 5319351520677146 z + 24796 z - 1510698 z 8 10 12 14 + 61313635 z - 1777683250 z + 38492808493 z - 641944217998 z 18 16 50 - 88761175710510 z + 8432878872385 z - 52132277374764476014 z 48 20 + 77836693418233641312 z + 758575772847307 z 36 34 + 29709344150231088708 z - 14388336677163363744 z 66 80 88 84 86 - 5319351520677146 z + 61313635 z + z + 24796 z - 239 z 82 64 30 - 1510698 z + 30863389489436044 z - 2056430442741549626 z 42 44 - 98966923032477007870 z + 107210044925148751196 z 46 58 - 98966923032477007870 z - 2056430442741549626 z 56 54 + 5911992677074435650 z - 14388336677163363744 z 52 60 70 + 29709344150231088708 z + 603863654981299969 z - 88761175710510 z 68 78 32 + 758575772847307 z - 1777683250 z + 5911992677074435650 z 38 40 - 52132277374764476014 z + 77836693418233641312 z 62 76 74 - 149170516129402547 z + 38492808493 z - 641944217998 z 72 / 28 + 8432878872385 z ) / (-1 - 2476593211884251638 z / 26 2 24 + 569573080494610225 z + 297 z - 109669304631511302 z 22 4 6 8 + 17575735585722369 z - 36002 z + 2488685 z - 112707654 z 10 12 14 + 3607523029 z - 85595000122 z + 1555537983197 z 18 16 50 + 252433556345997 z - 22173749599262 z + 494853662767599603311 z 48 20 - 679410910115235426135 z - 2327526233571914 z 36 34 - 162632890911855101499 z + 73206408944119398635 z 66 80 90 88 84 + 109669304631511302 z - 3607523029 z + z - 297 z - 2488685 z 86 82 64 + 36002 z + 112707654 z - 569573080494610225 z 30 42 + 9058718054662553694 z + 679410910115235426135 z 44 46 - 795962376387856713955 z + 795962376387856713955 z 58 56 + 27980359527758709663 z - 73206408944119398635 z 54 52 + 162632890911855101499 z - 307359766036429145639 z 60 70 68 - 9058718054662553694 z + 2327526233571914 z - 17575735585722369 z 78 32 38 + 85595000122 z - 27980359527758709663 z + 307359766036429145639 z 40 62 76 - 494853662767599603311 z + 2476593211884251638 z - 1555537983197 z 74 72 + 22173749599262 z - 252433556345997 z ) And in Maple-input format, it is: -(1+603863654981299969*z^28-149170516129402547*z^26-239*z^2+30863389489436044*z ^24-5319351520677146*z^22+24796*z^4-1510698*z^6+61313635*z^8-1777683250*z^10+ 38492808493*z^12-641944217998*z^14-88761175710510*z^18+8432878872385*z^16-\ 52132277374764476014*z^50+77836693418233641312*z^48+758575772847307*z^20+ 29709344150231088708*z^36-14388336677163363744*z^34-5319351520677146*z^66+ 61313635*z^80+z^88+24796*z^84-239*z^86-1510698*z^82+30863389489436044*z^64-\ 2056430442741549626*z^30-98966923032477007870*z^42+107210044925148751196*z^44-\ 98966923032477007870*z^46-2056430442741549626*z^58+5911992677074435650*z^56-\ 14388336677163363744*z^54+29709344150231088708*z^52+603863654981299969*z^60-\ 88761175710510*z^70+758575772847307*z^68-1777683250*z^78+5911992677074435650*z^ 32-52132277374764476014*z^38+77836693418233641312*z^40-149170516129402547*z^62+ 38492808493*z^76-641944217998*z^74+8432878872385*z^72)/(-1-2476593211884251638* z^28+569573080494610225*z^26+297*z^2-109669304631511302*z^24+17575735585722369* z^22-36002*z^4+2488685*z^6-112707654*z^8+3607523029*z^10-85595000122*z^12+ 1555537983197*z^14+252433556345997*z^18-22173749599262*z^16+ 494853662767599603311*z^50-679410910115235426135*z^48-2327526233571914*z^20-\ 162632890911855101499*z^36+73206408944119398635*z^34+109669304631511302*z^66-\ 3607523029*z^80+z^90-297*z^88-2488685*z^84+36002*z^86+112707654*z^82-\ 569573080494610225*z^64+9058718054662553694*z^30+679410910115235426135*z^42-\ 795962376387856713955*z^44+795962376387856713955*z^46+27980359527758709663*z^58 -73206408944119398635*z^56+162632890911855101499*z^54-307359766036429145639*z^ 52-9058718054662553694*z^60+2327526233571914*z^70-17575735585722369*z^68+ 85595000122*z^78-27980359527758709663*z^32+307359766036429145639*z^38-\ 494853662767599603311*z^40+2476593211884251638*z^62-1555537983197*z^76+ 22173749599262*z^74-252433556345997*z^72) The first , 40, terms are: [0, 58, 0, 6020, 0, 677811, 0, 77527538, 0, 8897806711, 0, 1021994305599, 0, 117406977188320, 0, 13488341624318653, 0, 1549629527627828513, 0, 178032116957340143238, 0, 20453569550889866856969, 0, 2349848904524330219839404, 0, 269967062858234559340164870, 0, 31015703019686278239845212743, 0, 3563300745145842076614913304575, 0, 409376895324333285380289384998454, 0, 47032079085932576864230422474946460, 0, 5403373977626719270719649269213684289, 0, 620777369613156851931596297277697939910, 0, 71319243165571118237672118750471168408745] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6946463273272320 z - 2846282474690859 z - 223 z 24 22 4 6 + 950828747565008 z - 257823104909868 z + 20512 z - 1050791 z 8 10 12 14 + 34208036 z - 764178108 z + 12318265852 z - 148387641584 z 18 16 50 - 9878136315542 z + 1370026340357 z - 257823104909868 z 48 20 36 + 950828747565008 z + 56407563918542 z + 33608672296590234 z 34 66 64 - 30465036493551118 z - 1050791 z + 34208036 z 30 42 44 - 13865495324992557 z - 13865495324992557 z + 6946463273272320 z 46 58 56 - 2846282474690859 z - 148387641584 z + 1370026340357 z 54 52 60 70 - 9878136315542 z + 56407563918542 z + 12318265852 z - 223 z 68 32 38 + 20512 z + 22685246584670441 z - 30465036493551118 z 40 62 72 / + 22685246584670441 z - 764178108 z + z ) / (-1 / 28 26 2 - 36351081316960127 z + 13563060633115927 z + 291 z 24 22 4 6 - 4128482715727395 z + 1020578837692055 z - 32184 z + 1896398 z 8 10 12 14 - 69400141 z + 1721973601 z - 30632870588 z + 405817835612 z 18 16 50 + 32524416670539 z - 4112976989567 z + 4128482715727395 z 48 20 36 - 13563060633115927 z - 203633196894935 z - 257886111783998887 z 34 66 64 + 212140698921040381 z + 69400141 z - 1721973601 z 30 42 44 + 79741361890659573 z + 143495260363494923 z - 79741361890659573 z 46 58 56 + 36351081316960127 z + 4112976989567 z - 32524416670539 z 54 52 60 + 203633196894935 z - 1020578837692055 z - 405817835612 z 70 68 32 + 32184 z - 1896398 z - 143495260363494923 z 38 40 62 74 + 257886111783998887 z - 212140698921040381 z + 30632870588 z + z 72 - 291 z ) And in Maple-input format, it is: -(1+6946463273272320*z^28-2846282474690859*z^26-223*z^2+950828747565008*z^24-\ 257823104909868*z^22+20512*z^4-1050791*z^6+34208036*z^8-764178108*z^10+ 12318265852*z^12-148387641584*z^14-9878136315542*z^18+1370026340357*z^16-\ 257823104909868*z^50+950828747565008*z^48+56407563918542*z^20+33608672296590234 *z^36-30465036493551118*z^34-1050791*z^66+34208036*z^64-13865495324992557*z^30-\ 13865495324992557*z^42+6946463273272320*z^44-2846282474690859*z^46-148387641584 *z^58+1370026340357*z^56-9878136315542*z^54+56407563918542*z^52+12318265852*z^ 60-223*z^70+20512*z^68+22685246584670441*z^32-30465036493551118*z^38+ 22685246584670441*z^40-764178108*z^62+z^72)/(-1-36351081316960127*z^28+ 13563060633115927*z^26+291*z^2-4128482715727395*z^24+1020578837692055*z^22-\ 32184*z^4+1896398*z^6-69400141*z^8+1721973601*z^10-30632870588*z^12+ 405817835612*z^14+32524416670539*z^18-4112976989567*z^16+4128482715727395*z^50-\ 13563060633115927*z^48-203633196894935*z^20-257886111783998887*z^36+ 212140698921040381*z^34+69400141*z^66-1721973601*z^64+79741361890659573*z^30+ 143495260363494923*z^42-79741361890659573*z^44+36351081316960127*z^46+ 4112976989567*z^58-32524416670539*z^56+203633196894935*z^54-1020578837692055*z^ 52-405817835612*z^60+32184*z^70-1896398*z^68-143495260363494923*z^32+ 257886111783998887*z^38-212140698921040381*z^40+30632870588*z^62+z^74-291*z^72) The first , 40, terms are: [0, 68, 0, 8116, 0, 1018851, 0, 129043256, 0, 16390638985, 0, 2084222848005, 0, 265151425949872, 0, 33738662449429415, 0, 4293355994671690323, 0, 546362190214016080740, 0, 69529726492244868924237, 0, 8848363909574390459137228, 0, 1126046968499288269073417848, 0, 143301417234821926392992797503, 0, 18236632797160291847614040840607, 0, 2320806328732037622904293694812144, 0, 295347418767617810868953089217871020, 0, 37586118004629928556942447883189843485, 0, 4783235591772161847491865279342512305660, 0, 608717900860331096458805855540575965526443] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 183683653841474 z - 105371034165992 z - 180 z 24 22 4 6 + 48286304298882 z - 17627092389408 z + 12969 z - 520866 z 8 10 12 14 + 13377360 z - 236744810 z + 3024234997 z - 28767640164 z 18 16 50 - 1165967677168 z + 208280039837 z - 28767640164 z 48 20 36 + 208280039837 z + 5104990208608 z + 183683653841474 z 34 64 30 42 - 256221579330020 z + z - 256221579330020 z - 17627092389408 z 44 46 58 56 + 5104990208608 z - 1165967677168 z - 520866 z + 13377360 z 54 52 60 32 - 236744810 z + 3024234997 z + 12969 z + 286256829400224 z 38 40 62 / - 105371034165992 z + 48286304298882 z - 180 z ) / (-1 / 28 26 2 - 1123188168861240 z + 579853120981944 z + 248 z 24 22 4 6 - 239461551065850 z + 78864466923672 z - 21432 z + 989191 z 8 10 12 14 - 28685080 z + 568253696 z - 8086466643 z + 85439605904 z 18 16 50 + 4253030328981 z - 685867224016 z + 685867224016 z 48 20 36 - 4253030328981 z - 20620346244712 z - 1743738756058646 z 34 66 64 30 + 2172139799436776 z + z - 248 z + 1743738756058646 z 42 44 46 + 239461551065850 z - 78864466923672 z + 20620346244712 z 58 56 54 52 + 28685080 z - 568253696 z + 8086466643 z - 85439605904 z 60 32 38 - 989191 z - 2172139799436776 z + 1123188168861240 z 40 62 - 579853120981944 z + 21432 z ) And in Maple-input format, it is: -(1+183683653841474*z^28-105371034165992*z^26-180*z^2+48286304298882*z^24-\ 17627092389408*z^22+12969*z^4-520866*z^6+13377360*z^8-236744810*z^10+3024234997 *z^12-28767640164*z^14-1165967677168*z^18+208280039837*z^16-28767640164*z^50+ 208280039837*z^48+5104990208608*z^20+183683653841474*z^36-256221579330020*z^34+ z^64-256221579330020*z^30-17627092389408*z^42+5104990208608*z^44-1165967677168* z^46-520866*z^58+13377360*z^56-236744810*z^54+3024234997*z^52+12969*z^60+ 286256829400224*z^32-105371034165992*z^38+48286304298882*z^40-180*z^62)/(-1-\ 1123188168861240*z^28+579853120981944*z^26+248*z^2-239461551065850*z^24+ 78864466923672*z^22-21432*z^4+989191*z^6-28685080*z^8+568253696*z^10-8086466643 *z^12+85439605904*z^14+4253030328981*z^18-685867224016*z^16+685867224016*z^50-\ 4253030328981*z^48-20620346244712*z^20-1743738756058646*z^36+2172139799436776*z ^34+z^66-248*z^64+1743738756058646*z^30+239461551065850*z^42-78864466923672*z^ 44+20620346244712*z^46+28685080*z^58-568253696*z^56+8086466643*z^54-85439605904 *z^52-989191*z^60-2172139799436776*z^32+1123188168861240*z^38-579853120981944*z ^40+21432*z^62) The first , 40, terms are: [0, 68, 0, 8401, 0, 1094397, 0, 143317492, 0, 18778738549, 0, 2460709997037, 0, 322446353949332, 0, 42252737603432613, 0, 5536716348827452137, 0, 725520522416938334436, 0, 95070795750754167734857, 0, 12457891853965332904885689, 0, 1632457877548488347813093412, 0, 213914099850567854649651829689, 0, 28030886887943204522060318740565, 0, 3673112806839668975482748118399188, 0, 481317546094940738045713157524694013, 0, 63070913517132235791793516209923544933, 0, 8264689629870502499232031334144441126580, 0, 1082988827481228807979835050808499465621837] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 6072762306793192 z + 3031657139313188 z + 249 z 24 22 4 6 - 1194182718611252 z + 369429491241624 z - 25592 z + 1454456 z 8 10 12 14 - 51848060 z + 1245433644 z - 21108769032 z + 260558566856 z 18 16 50 + 16692420974076 z - 2396076121836 z + 2396076121836 z 48 20 36 - 16692420974076 z - 89217985049944 z - 9631442494723304 z 34 66 64 30 + 12122549870032414 z + z - 249 z + 9631442494723304 z 42 44 46 + 1194182718611252 z - 369429491241624 z + 89217985049944 z 58 56 54 52 + 51848060 z - 1245433644 z + 21108769032 z - 260558566856 z 60 32 38 - 1454456 z - 12122549870032414 z + 6072762306793192 z 40 62 / 28 - 3031657139313188 z + 25592 z ) / (1 + 33741710542405780 z / 26 2 24 - 15230281754629160 z - 310 z + 5440704465797460 z 22 4 6 8 - 1530065649514596 z + 38209 z - 2544708 z + 104324204 z 10 12 14 18 - 2838407640 z + 53850250668 z - 737627689964 z - 57322510089816 z 16 50 48 + 7483483706644 z - 57322510089816 z + 336419806236692 z 20 36 34 + 336419806236692 z + 83295651793568150 z - 93219108908727524 z 66 64 30 42 - 310 z + 38209 z - 59394059210382188 z - 15230281754629160 z 44 46 58 + 5440704465797460 z - 1530065649514596 z - 2838407640 z 56 54 52 60 + 53850250668 z - 737627689964 z + 7483483706644 z + 104324204 z 68 32 38 + z + 83295651793568150 z - 59394059210382188 z 40 62 + 33741710542405780 z - 2544708 z ) And in Maple-input format, it is: -(-1-6072762306793192*z^28+3031657139313188*z^26+249*z^2-1194182718611252*z^24+ 369429491241624*z^22-25592*z^4+1454456*z^6-51848060*z^8+1245433644*z^10-\ 21108769032*z^12+260558566856*z^14+16692420974076*z^18-2396076121836*z^16+ 2396076121836*z^50-16692420974076*z^48-89217985049944*z^20-9631442494723304*z^ 36+12122549870032414*z^34+z^66-249*z^64+9631442494723304*z^30+1194182718611252* z^42-369429491241624*z^44+89217985049944*z^46+51848060*z^58-1245433644*z^56+ 21108769032*z^54-260558566856*z^52-1454456*z^60-12122549870032414*z^32+ 6072762306793192*z^38-3031657139313188*z^40+25592*z^62)/(1+33741710542405780*z^ 28-15230281754629160*z^26-310*z^2+5440704465797460*z^24-1530065649514596*z^22+ 38209*z^4-2544708*z^6+104324204*z^8-2838407640*z^10+53850250668*z^12-\ 737627689964*z^14-57322510089816*z^18+7483483706644*z^16-57322510089816*z^50+ 336419806236692*z^48+336419806236692*z^20+83295651793568150*z^36-\ 93219108908727524*z^34-310*z^66+38209*z^64-59394059210382188*z^30-\ 15230281754629160*z^42+5440704465797460*z^44-1530065649514596*z^46-2838407640*z ^58+53850250668*z^56-737627689964*z^54+7483483706644*z^52+104324204*z^60+z^68+ 83295651793568150*z^32-59394059210382188*z^38+33741710542405780*z^40-2544708*z^ 62) The first , 40, terms are: [0, 61, 0, 6293, 0, 710333, 0, 82505037, 0, 9678492869, 0, 1139377067053, 0, 134301962759953, 0, 15838058889422449, 0, 1868092779581603405, 0, 220355751416606413413, 0, 25993326632813998045357, 0, 3066224470498707968509021, 0, 361699499748962272475539189, 0, 42667053800698957869115492893, 0, 5033125228415376691473564898209, 0, 593721752721672710612993652961889, 0, 70037115093258306514168156737461853, 0, 8261778829036171327238458201509986869, 0, 974583137834963361175089218106049780509, 0, 114964625004161689125940661366893594342381] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 76266246431092277196 z - 11659345368275673476 z - 305 z 24 22 4 6 + 1531772863990301860 z - 171825664774493084 z + 41079 z - 3282770 z 102 8 10 12 - 6861047430 z + 176513756 z - 6861047430 z + 202027415343 z 14 18 16 - 4657557390793 z - 1302289507060356 z + 86143654170121 z 50 48 - 9376107985087294398279278 z + 6108217292817355105633726 z 20 36 + 16329309861915592 z + 33741790811969934024092 z 34 66 - 9034710973315664693836 z - 3517262099936444117069900 z 80 100 90 + 2114235038412271103200 z + 202027415343 z - 171825664774493084 z 88 84 + 1531772863990301860 z + 76266246431092277196 z 94 86 96 - 1302289507060356 z - 11659345368275673476 z + 86143654170121 z 98 92 82 - 4657557390793 z + 16329309861915592 z - 431029061265455101364 z 64 112 110 106 + 6108217292817355105633726 z + z - 305 z - 3282770 z 108 30 42 + 41079 z - 431029061265455101364 z - 802657660845540878529428 z 44 46 + 1788770632816684913026360 z - 3517262099936444117069900 z 58 56 - 15288667888027997175675944 z + 16251272082952807920306824 z 54 52 - 15288667888027997175675944 z + 12728653385422351088519282 z 60 70 + 12728653385422351088519282 z - 802657660845540878529428 z 68 78 + 1788770632816684913026360 z - 9034710973315664693836 z 32 38 + 2114235038412271103200 z - 110425964287156608859668 z 40 62 + 317387977878876705638676 z - 9376107985087294398279278 z 76 74 + 33741790811969934024092 z - 110425964287156608859668 z 72 104 / 2 + 317387977878876705638676 z + 176513756 z ) / ((-1 + z ) (1 / 28 26 2 + 229973149526450954928 z - 33852551321954355720 z - 372 z 24 22 4 6 + 4271734982922868492 z - 459010205318330696 z + 58038 z - 5208732 z 102 8 10 12 - 13005602244 z + 308261935 z - 13005602244 z + 411485800318 z 14 18 16 - 10115470895484 z - 3162375806284344 z + 198306635141009 z 50 48 - 36625619408267727197542032 z + 23610484163132944019133302 z 20 36 + 41660895264629288 z + 115663241441529090349472 z 34 66 - 30095585453164295512120 z - 13414558492131230626940904 z 80 100 90 + 6828312568529029090500 z + 411485800318 z - 459010205318330696 z 88 84 + 4271734982922868492 z + 229973149526450954928 z 94 86 96 - 3162375806284344 z - 33852551321954355720 z + 198306635141009 z 98 92 82 - 10115470895484 z + 41660895264629288 z - 1346652039338430942536 z 64 112 110 106 + 23610484163132944019133302 z + z - 372 z - 5208732 z 108 30 42 + 58038 z - 1346652039338430942536 z - 2955871828546790481155736 z 44 46 + 6712746314346862168765208 z - 13414558492131230626940904 z 58 56 - 60451807793878374831940128 z + 64356247613005601920647322 z 54 52 - 60451807793878374831940128 z + 50099851874254078605374588 z 60 70 + 50099851874254078605374588 z - 2955871828546790481155736 z 68 78 + 6712746314346862168765208 z - 30095585453164295512120 z 32 38 + 6828312568529029090500 z - 388621899746428737110136 z 40 62 + 1144023975806334522859244 z - 36625619408267727197542032 z 76 74 + 115663241441529090349472 z - 388621899746428737110136 z 72 104 + 1144023975806334522859244 z + 308261935 z )) And in Maple-input format, it is: -(1+76266246431092277196*z^28-11659345368275673476*z^26-305*z^2+ 1531772863990301860*z^24-171825664774493084*z^22+41079*z^4-3282770*z^6-\ 6861047430*z^102+176513756*z^8-6861047430*z^10+202027415343*z^12-4657557390793* z^14-1302289507060356*z^18+86143654170121*z^16-9376107985087294398279278*z^50+ 6108217292817355105633726*z^48+16329309861915592*z^20+33741790811969934024092*z ^36-9034710973315664693836*z^34-3517262099936444117069900*z^66+ 2114235038412271103200*z^80+202027415343*z^100-171825664774493084*z^90+ 1531772863990301860*z^88+76266246431092277196*z^84-1302289507060356*z^94-\ 11659345368275673476*z^86+86143654170121*z^96-4657557390793*z^98+ 16329309861915592*z^92-431029061265455101364*z^82+6108217292817355105633726*z^ 64+z^112-305*z^110-3282770*z^106+41079*z^108-431029061265455101364*z^30-\ 802657660845540878529428*z^42+1788770632816684913026360*z^44-\ 3517262099936444117069900*z^46-15288667888027997175675944*z^58+ 16251272082952807920306824*z^56-15288667888027997175675944*z^54+ 12728653385422351088519282*z^52+12728653385422351088519282*z^60-\ 802657660845540878529428*z^70+1788770632816684913026360*z^68-\ 9034710973315664693836*z^78+2114235038412271103200*z^32-\ 110425964287156608859668*z^38+317387977878876705638676*z^40-\ 9376107985087294398279278*z^62+33741790811969934024092*z^76-\ 110425964287156608859668*z^74+317387977878876705638676*z^72+176513756*z^104)/(-\ 1+z^2)/(1+229973149526450954928*z^28-33852551321954355720*z^26-372*z^2+ 4271734982922868492*z^24-459010205318330696*z^22+58038*z^4-5208732*z^6-\ 13005602244*z^102+308261935*z^8-13005602244*z^10+411485800318*z^12-\ 10115470895484*z^14-3162375806284344*z^18+198306635141009*z^16-\ 36625619408267727197542032*z^50+23610484163132944019133302*z^48+ 41660895264629288*z^20+115663241441529090349472*z^36-30095585453164295512120*z^ 34-13414558492131230626940904*z^66+6828312568529029090500*z^80+411485800318*z^ 100-459010205318330696*z^90+4271734982922868492*z^88+229973149526450954928*z^84 -3162375806284344*z^94-33852551321954355720*z^86+198306635141009*z^96-\ 10115470895484*z^98+41660895264629288*z^92-1346652039338430942536*z^82+ 23610484163132944019133302*z^64+z^112-372*z^110-5208732*z^106+58038*z^108-\ 1346652039338430942536*z^30-2955871828546790481155736*z^42+ 6712746314346862168765208*z^44-13414558492131230626940904*z^46-\ 60451807793878374831940128*z^58+64356247613005601920647322*z^56-\ 60451807793878374831940128*z^54+50099851874254078605374588*z^52+ 50099851874254078605374588*z^60-2955871828546790481155736*z^70+ 6712746314346862168765208*z^68-30095585453164295512120*z^78+ 6828312568529029090500*z^32-388621899746428737110136*z^38+ 1144023975806334522859244*z^40-36625619408267727197542032*z^62+ 115663241441529090349472*z^76-388621899746428737110136*z^74+ 1144023975806334522859244*z^72+308261935*z^104) The first , 40, terms are: [0, 68, 0, 8033, 0, 1008429, 0, 128119936, 0, 16331173041, 0, 2083974595805, 0, 266038285865732, 0, 33967716837010653, 0, 4337283997474017337, 0, 553836821402663213736, 0, 70721452697135926234725, 0, 9030730124734604798095821, 0, 1153175964868999762230362528, 0, 147254559770467030877785530025, 0, 18803648336735869221217080563277, 0, 2401129465095906104758139790451804, 0, 306611946821136705796226086074736421, 0, 39152778314960682146696559730356951625, 0, 4999609742881842013557958980766690889448, 0, 638424624011728086877484613313488843485565] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 89301240197155410579 z - 13478707878117482408 z - 302 z 24 22 4 6 + 1746215522328161795 z - 192940767941385148 z + 40549 z - 3252830 z 102 8 10 12 - 6957104442 z + 176576990 z - 6957104442 z + 208169689604 z 14 18 16 - 4883176436036 z - 1414585836268416 z + 91937787366781 z 50 48 - 11775994921955740595984884 z + 7655593329291469517882297 z 20 36 + 18041993465649889 z + 41090225226590617265573 z 34 66 - 10913625777475800133406 z - 4396085623570493828384748 z 80 100 90 + 2530512254413960949997 z + 208169689604 z - 192940767941385148 z 88 84 + 1746215522328161795 z + 89301240197155410579 z 94 86 96 - 1414585836268416 z - 13478707878117482408 z + 91937787366781 z 98 92 82 - 4883176436036 z + 18041993465649889 z - 510571732749968685606 z 64 112 110 106 + 7655593329291469517882297 z + z - 302 z - 3252830 z 108 30 42 + 40549 z - 510571732749968685606 z - 995439368437474349578950 z 44 46 + 2227912420915492870335698 z - 4396085623570493828384748 z 58 56 - 19246770657173742431165922 z + 20464452934671506597907552 z 54 52 - 19246770657173742431165922 z + 16010106176195976830186253 z 60 70 + 16010106176195976830186253 z - 995439368437474349578950 z 68 78 + 2227912420915492870335698 z - 10913625777475800133406 z 32 38 + 2530512254413960949997 z - 135425066063208970880480 z 40 62 + 391603310969460590237988 z - 11775994921955740595984884 z 76 74 + 41090225226590617265573 z - 135425066063208970880480 z 72 104 / 2 + 391603310969460590237988 z + 176576990 z ) / ((-1 + z ) (1 / 28 26 2 + 267746308955655315366 z - 38844267450678898086 z - 373 z 24 22 4 6 + 4825205876801510813 z - 509874926772229062 z + 57852 z - 5178749 z 102 8 10 12 - 13086763273 z + 307502491 z - 13086763273 z + 419378138331 z 14 18 16 - 10470924574706 z - 3390330650495960 z + 208829234790640 z 50 48 - 46244106049889811375126823 z + 29738650678788433745217351 z 20 36 + 45473453254932673 z + 140897075971105823874531 z 34 66 - 36318770457178613073604 z - 16842165421421740072212487 z 80 100 90 + 8152917572878787640666 z + 419378138331 z - 509874926772229062 z 88 84 + 4825205876801510813 z + 267746308955655315366 z 94 86 96 - 3390330650495960 z - 38844267450678898086 z + 208829234790640 z 98 92 82 - 10470924574706 z + 45473453254932673 z - 1588772098212707902582 z 64 112 110 106 + 29738650678788433745217351 z + z - 373 z - 5178749 z 108 30 42 + 57852 z - 1588772098212707902582 z - 3677789242279443717853331 z 44 46 + 8393832632061803214989368 z - 16842165421421740072212487 z 58 56 - 76533861548453718535829340 z + 81504043643744857120856467 z 54 52 - 76533861548453718535829340 z + 63364365360520937853118034 z 60 70 + 63364365360520937853118034 z - 3677789242279443717853331 z 68 78 + 8393832632061803214989368 z - 36318770457178613073604 z 32 38 + 8152917572878787640666 z - 477293187994837236523466 z 40 62 + 1414969246153324513975632 z - 46244106049889811375126823 z 76 74 + 140897075971105823874531 z - 477293187994837236523466 z 72 104 + 1414969246153324513975632 z + 307502491 z )) And in Maple-input format, it is: -(1+89301240197155410579*z^28-13478707878117482408*z^26-302*z^2+ 1746215522328161795*z^24-192940767941385148*z^22+40549*z^4-3252830*z^6-\ 6957104442*z^102+176576990*z^8-6957104442*z^10+208169689604*z^12-4883176436036* z^14-1414585836268416*z^18+91937787366781*z^16-11775994921955740595984884*z^50+ 7655593329291469517882297*z^48+18041993465649889*z^20+41090225226590617265573*z ^36-10913625777475800133406*z^34-4396085623570493828384748*z^66+ 2530512254413960949997*z^80+208169689604*z^100-192940767941385148*z^90+ 1746215522328161795*z^88+89301240197155410579*z^84-1414585836268416*z^94-\ 13478707878117482408*z^86+91937787366781*z^96-4883176436036*z^98+ 18041993465649889*z^92-510571732749968685606*z^82+7655593329291469517882297*z^ 64+z^112-302*z^110-3252830*z^106+40549*z^108-510571732749968685606*z^30-\ 995439368437474349578950*z^42+2227912420915492870335698*z^44-\ 4396085623570493828384748*z^46-19246770657173742431165922*z^58+ 20464452934671506597907552*z^56-19246770657173742431165922*z^54+ 16010106176195976830186253*z^52+16010106176195976830186253*z^60-\ 995439368437474349578950*z^70+2227912420915492870335698*z^68-\ 10913625777475800133406*z^78+2530512254413960949997*z^32-\ 135425066063208970880480*z^38+391603310969460590237988*z^40-\ 11775994921955740595984884*z^62+41090225226590617265573*z^76-\ 135425066063208970880480*z^74+391603310969460590237988*z^72+176576990*z^104)/(-\ 1+z^2)/(1+267746308955655315366*z^28-38844267450678898086*z^26-373*z^2+ 4825205876801510813*z^24-509874926772229062*z^22+57852*z^4-5178749*z^6-\ 13086763273*z^102+307502491*z^8-13086763273*z^10+419378138331*z^12-\ 10470924574706*z^14-3390330650495960*z^18+208829234790640*z^16-\ 46244106049889811375126823*z^50+29738650678788433745217351*z^48+ 45473453254932673*z^20+140897075971105823874531*z^36-36318770457178613073604*z^ 34-16842165421421740072212487*z^66+8152917572878787640666*z^80+419378138331*z^ 100-509874926772229062*z^90+4825205876801510813*z^88+267746308955655315366*z^84 -3390330650495960*z^94-38844267450678898086*z^86+208829234790640*z^96-\ 10470924574706*z^98+45473453254932673*z^92-1588772098212707902582*z^82+ 29738650678788433745217351*z^64+z^112-373*z^110-5178749*z^106+57852*z^108-\ 1588772098212707902582*z^30-3677789242279443717853331*z^42+ 8393832632061803214989368*z^44-16842165421421740072212487*z^46-\ 76533861548453718535829340*z^58+81504043643744857120856467*z^56-\ 76533861548453718535829340*z^54+63364365360520937853118034*z^52+ 63364365360520937853118034*z^60-3677789242279443717853331*z^70+ 8393832632061803214989368*z^68-36318770457178613073604*z^78+ 8152917572878787640666*z^32-477293187994837236523466*z^38+ 1414969246153324513975632*z^40-46244106049889811375126823*z^62+ 140897075971105823874531*z^76-477293187994837236523466*z^74+ 1414969246153324513975632*z^72+307502491*z^104) The first , 40, terms are: [0, 72, 0, 9252, 0, 1251819, 0, 170413628, 0, 23220680091, 0, 3164642581481, 0, 431313485008328, 0, 58784991683780025, 0, 8012008373306770503, 0, 1091985339195916493964, 0, 148830650423909948026259, 0, 20284672758472212120394140, 0, 2764672232866882400947217112, 0, 376807294901714122344618712985, 0, 51356445278905296024536370741589, 0, 6999557896801264847992599635837048, 0, 953995365467599987515043928516783260, 0, 130023520194321609934328009426388083003, 0, 17721381483687633464662207436679740368300, 0, 2415311946871211471154332963868684361150811] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 82828192428016524011 z - 12306393182827000078 z - 293 z 24 22 4 6 + 1572182954947244853 z - 171670943947557140 z + 38240 z - 2993188 z 102 8 10 12 - 6182139576 z + 159278777 z - 6182139576 z + 183117759763 z 14 18 16 - 4272014607296 z - 1239418710541530 z + 80335501731825 z 50 48 - 12995257965709396841714534 z + 8371399316702915282263670 z 20 36 + 15905811071839045 z + 40964355274402979984750 z 34 66 - 10679667914061803208554 z - 4752626794320561866788610 z 80 100 90 + 2430869821219875556005 z + 183117759763 z - 171670943947557140 z 88 84 + 1572182954947244853 z + 82828192428016524011 z 94 86 96 - 1239418710541530 z - 12306393182827000078 z + 80335501731825 z 98 92 82 - 4272014607296 z + 15905811071839045 z - 481719785603763063930 z 64 112 110 106 + 8371399316702915282263670 z + z - 293 z - 2993188 z 108 30 42 + 38240 z - 481719785603763063930 z - 1045861822795694207513734 z 44 46 + 2376498142971420261790806 z - 4752626794320561866788610 z 58 56 - 21468819896349380703277286 z + 22858254732358336393144778 z 54 52 - 21468819896349380703277286 z + 17786057668867452121413038 z 60 70 + 17786057668867452121413038 z - 1045861822795694207513734 z 68 78 + 2376498142971420261790806 z - 10679667914061803208554 z 32 38 + 2430869821219875556005 z - 137504169242585015212261 z 40 62 + 404682379246830080391429 z - 12995257965709396841714534 z 76 74 + 40964355274402979984750 z - 137504169242585015212261 z 72 104 / 2 + 404682379246830080391429 z + 159278777 z ) / ((-1 + z ) (1 / 28 26 2 + 243003089537304333870 z - 34727847571260256246 z - 349 z 24 22 4 6 + 4255160734679483192 z - 444268647528557568 z + 52112 z - 4553076 z 102 8 10 12 - 11207643474 z + 266048718 z - 11207643474 z + 357172044772 z 14 18 16 - 8903540476512 z - 2902484600728542 z + 177901178164051 z 50 48 - 49214818105707875729657486 z + 31388367488412075874398843 z 20 36 + 39228379027921215 z + 136728576318603604286786 z 34 66 - 34648062197025593660482 z - 17593874016475099782154806 z 80 100 90 + 7646234648912325563277 z + 357172044772 z - 444268647528557568 z 88 84 + 4255160734679483192 z + 243003089537304333870 z 94 86 96 - 2902484600728542 z - 34727847571260256246 z + 177901178164051 z 98 92 82 - 8903540476512 z + 39228379027921215 z - 1465301481462365837404 z 64 112 110 106 + 31388367488412075874398843 z + z - 349 z - 4553076 z 108 30 42 + 52112 z - 1465301481462365837404 z - 3743703885200654803657582 z 44 46 + 8662493360112724031374386 z - 17593874016475099782154806 z 58 56 - 82245726080455403517139398 z + 87695070666262133229382863 z 54 52 - 82245726080455403517139398 z + 67843406061044299199018562 z 60 70 + 67843406061044299199018562 z - 3743703885200654803657582 z 68 78 + 8662493360112724031374386 z - 34648062197025593660482 z 32 38 + 7646234648912325563277 z - 470979309443904109159925 z 40 62 + 1418825584664867504513914 z - 49214818105707875729657486 z 76 74 + 136728576318603604286786 z - 470979309443904109159925 z 72 104 + 1418825584664867504513914 z + 266048718 z )) And in Maple-input format, it is: -(1+82828192428016524011*z^28-12306393182827000078*z^26-293*z^2+ 1572182954947244853*z^24-171670943947557140*z^22+38240*z^4-2993188*z^6-\ 6182139576*z^102+159278777*z^8-6182139576*z^10+183117759763*z^12-4272014607296* z^14-1239418710541530*z^18+80335501731825*z^16-12995257965709396841714534*z^50+ 8371399316702915282263670*z^48+15905811071839045*z^20+40964355274402979984750*z ^36-10679667914061803208554*z^34-4752626794320561866788610*z^66+ 2430869821219875556005*z^80+183117759763*z^100-171670943947557140*z^90+ 1572182954947244853*z^88+82828192428016524011*z^84-1239418710541530*z^94-\ 12306393182827000078*z^86+80335501731825*z^96-4272014607296*z^98+ 15905811071839045*z^92-481719785603763063930*z^82+8371399316702915282263670*z^ 64+z^112-293*z^110-2993188*z^106+38240*z^108-481719785603763063930*z^30-\ 1045861822795694207513734*z^42+2376498142971420261790806*z^44-\ 4752626794320561866788610*z^46-21468819896349380703277286*z^58+ 22858254732358336393144778*z^56-21468819896349380703277286*z^54+ 17786057668867452121413038*z^52+17786057668867452121413038*z^60-\ 1045861822795694207513734*z^70+2376498142971420261790806*z^68-\ 10679667914061803208554*z^78+2430869821219875556005*z^32-\ 137504169242585015212261*z^38+404682379246830080391429*z^40-\ 12995257965709396841714534*z^62+40964355274402979984750*z^76-\ 137504169242585015212261*z^74+404682379246830080391429*z^72+159278777*z^104)/(-\ 1+z^2)/(1+243003089537304333870*z^28-34727847571260256246*z^26-349*z^2+ 4255160734679483192*z^24-444268647528557568*z^22+52112*z^4-4553076*z^6-\ 11207643474*z^102+266048718*z^8-11207643474*z^10+357172044772*z^12-\ 8903540476512*z^14-2902484600728542*z^18+177901178164051*z^16-\ 49214818105707875729657486*z^50+31388367488412075874398843*z^48+ 39228379027921215*z^20+136728576318603604286786*z^36-34648062197025593660482*z^ 34-17593874016475099782154806*z^66+7646234648912325563277*z^80+357172044772*z^ 100-444268647528557568*z^90+4255160734679483192*z^88+243003089537304333870*z^84 -2902484600728542*z^94-34727847571260256246*z^86+177901178164051*z^96-\ 8903540476512*z^98+39228379027921215*z^92-1465301481462365837404*z^82+ 31388367488412075874398843*z^64+z^112-349*z^110-4553076*z^106+52112*z^108-\ 1465301481462365837404*z^30-3743703885200654803657582*z^42+ 8662493360112724031374386*z^44-17593874016475099782154806*z^46-\ 82245726080455403517139398*z^58+87695070666262133229382863*z^56-\ 82245726080455403517139398*z^54+67843406061044299199018562*z^52+ 67843406061044299199018562*z^60-3743703885200654803657582*z^70+ 8662493360112724031374386*z^68-34648062197025593660482*z^78+ 7646234648912325563277*z^32-470979309443904109159925*z^38+ 1418825584664867504513914*z^40-49214818105707875729657486*z^62+ 136728576318603604286786*z^76-470979309443904109159925*z^74+ 1418825584664867504513914*z^72+266048718*z^104) The first , 40, terms are: [0, 57, 0, 5729, 0, 626873, 0, 70029180, 0, 7874200957, 0, 887498388729, 0, 100124398877585, 0, 11300132660777409, 0, 1275558061877118469, 0, 143995360082188897813, 0, 16255880719127139663701, 0, 1835179334881611164738977, 0, 207180626747422016679565836, 0, 23389499323615412962908844237, 0, 2640542808228784374398510269501, 0, 298102570594407394586562381028381, 0, 33654126784392055742005958238219029, 0, 3799364622584909249760304083139725357, 0, 428927253066683090520102801365656736373, 0, 48423515553117418284759648107671759038469] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 728881932107337760 z - 183062840718118712 z - 260 z 24 22 4 6 + 38502826828103221 z - 6740958933749980 z + 28960 z - 1859852 z 8 10 12 14 + 78214459 z - 2316476188 z + 50671311120 z - 846527197636 z 18 16 50 - 115493730469092 z + 11071078960551 z - 58815649773473995408 z 48 20 + 87171907151623598990 z + 975151860275504 z 36 34 + 33852299567875447856 z - 16595082691861077920 z 66 80 88 84 86 - 6740958933749980 z + 78214459 z + z + 28960 z - 260 z 82 64 30 - 1859852 z + 38502826828103221 z - 2442288829879281768 z 42 44 - 110336118245797230976 z + 119343858915073776800 z 46 58 - 110336118245797230976 z - 2442288829879281768 z 56 54 + 6914552473581004122 z - 16595082691861077920 z 52 60 70 + 33852299567875447856 z + 728881932107337760 z - 115493730469092 z 68 78 32 + 975151860275504 z - 2316476188 z + 6914552473581004122 z 38 40 - 58815649773473995408 z + 87171907151623598990 z 62 76 74 - 183062840718118712 z + 50671311120 z - 846527197636 z 72 / 2 28 + 11071078960551 z ) / ((-1 + z ) (1 + 2485916783933082608 z / 26 2 24 - 598786984536390336 z - 332 z + 120344467653442805 z 22 4 6 8 - 20063575583007292 z + 43514 z - 3157776 z + 146679959 z 10 12 14 - 4731189552 z + 111650814948 z - 1998700068288 z 18 16 50 - 308514448480448 z + 27863385974099 z - 231834999748926676656 z 48 20 + 348457573225095906150 z + 2754427598661026 z 36 34 + 130878327117107165540 z - 62609493100677942776 z 66 80 88 84 86 - 20063575583007292 z + 146679959 z + z + 43514 z - 332 z 82 64 30 - 3157776 z + 120344467653442805 z - 8651600041017155744 z 42 44 - 444813016357706360240 z + 482496819500502126168 z 46 58 - 444813016357706360240 z - 8651600041017155744 z 56 54 + 25334916579688043450 z - 62609493100677942776 z 52 60 + 130878327117107165540 z + 2485916783933082608 z 70 68 78 - 308514448480448 z + 2754427598661026 z - 4731189552 z 32 38 + 25334916579688043450 z - 231834999748926676656 z 40 62 76 + 348457573225095906150 z - 598786984536390336 z + 111650814948 z 74 72 - 1998700068288 z + 27863385974099 z )) And in Maple-input format, it is: -(1+728881932107337760*z^28-183062840718118712*z^26-260*z^2+38502826828103221*z ^24-6740958933749980*z^22+28960*z^4-1859852*z^6+78214459*z^8-2316476188*z^10+ 50671311120*z^12-846527197636*z^14-115493730469092*z^18+11071078960551*z^16-\ 58815649773473995408*z^50+87171907151623598990*z^48+975151860275504*z^20+ 33852299567875447856*z^36-16595082691861077920*z^34-6740958933749980*z^66+ 78214459*z^80+z^88+28960*z^84-260*z^86-1859852*z^82+38502826828103221*z^64-\ 2442288829879281768*z^30-110336118245797230976*z^42+119343858915073776800*z^44-\ 110336118245797230976*z^46-2442288829879281768*z^58+6914552473581004122*z^56-\ 16595082691861077920*z^54+33852299567875447856*z^52+728881932107337760*z^60-\ 115493730469092*z^70+975151860275504*z^68-2316476188*z^78+6914552473581004122*z ^32-58815649773473995408*z^38+87171907151623598990*z^40-183062840718118712*z^62 +50671311120*z^76-846527197636*z^74+11071078960551*z^72)/(-1+z^2)/(1+ 2485916783933082608*z^28-598786984536390336*z^26-332*z^2+120344467653442805*z^ 24-20063575583007292*z^22+43514*z^4-3157776*z^6+146679959*z^8-4731189552*z^10+ 111650814948*z^12-1998700068288*z^14-308514448480448*z^18+27863385974099*z^16-\ 231834999748926676656*z^50+348457573225095906150*z^48+2754427598661026*z^20+ 130878327117107165540*z^36-62609493100677942776*z^34-20063575583007292*z^66+ 146679959*z^80+z^88+43514*z^84-332*z^86-3157776*z^82+120344467653442805*z^64-\ 8651600041017155744*z^30-444813016357706360240*z^42+482496819500502126168*z^44-\ 444813016357706360240*z^46-8651600041017155744*z^58+25334916579688043450*z^56-\ 62609493100677942776*z^54+130878327117107165540*z^52+2485916783933082608*z^60-\ 308514448480448*z^70+2754427598661026*z^68-4731189552*z^78+25334916579688043450 *z^32-231834999748926676656*z^38+348457573225095906150*z^40-598786984536390336* z^62+111650814948*z^76-1998700068288*z^74+27863385974099*z^72) The first , 40, terms are: [0, 73, 0, 9423, 0, 1278539, 0, 174663523, 0, 23893126503, 0, 3269541215369, 0, 447446503502097, 0, 61236165192192913, 0, 8380674986587676057, 0, 1146968344147526091495, 0, 156972784891124012506099, 0, 21483125820730300538738715, 0, 2940157829558279234601883919, 0, 402386905430682134787577294457, 0, 55070249329279407141360569174433, 0, 7536856545569205925060411852772449, 0, 1031486282007841649680574714912362841, 0, 141168130837667642909339476046204745167, 0, 19320122349972913062358341543268353232251, 0, 2644131685014150747068172565047193972893779] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 94919420037561696415 z - 13927068254223121752 z - 290 z 24 22 4 6 + 1755670264064490539 z - 189033126749760628 z + 37717 z - 2960810 z 102 8 10 12 - 6232416890 z + 158787678 z - 6232416890 z + 187090653448 z 14 18 16 - 4428968641228 z - 1324772584112952 z + 84562877787333 z 50 48 - 16144773007635768101014120 z + 10369153044762641468445329 z 20 36 + 17259779349712509 z + 48939645225883745237505 z 34 66 - 12643430429673394148790 z - 5864143941981045328263532 z 80 100 90 + 2849295329610778503345 z + 187090653448 z - 189033126749760628 z 88 84 + 1755670264064490539 z + 94919420037561696415 z 94 86 96 - 1324772584112952 z - 13927068254223121752 z + 84562877787333 z 98 92 82 - 4428968641228 z + 17259779349712509 z - 558544589159842933142 z 64 112 110 106 + 10369153044762641468445329 z + z - 290 z - 2960810 z 108 30 42 + 37717 z - 558544589159842933142 z - 1277244801647231599661070 z 44 46 + 2918505381307150932573166 z - 5864143941981045328263532 z 58 56 - 26763501846052038044365518 z + 28507807571882732103742800 z 54 52 - 26763501846052038044365518 z + 22144023910253300856709125 z 60 70 + 22144023910253300856709125 z - 1277244801647231599661070 z 68 78 + 2918505381307150932573166 z - 12643430429673394148790 z 32 38 + 2849295329610778503345 z - 165628097843069086827060 z 40 62 + 491036032353952376505696 z - 16144773007635768101014120 z 76 74 + 48939645225883745237505 z - 165628097843069086827060 z 72 104 / + 491036032353952376505696 z + 158787678 z ) / (-1 / 28 26 2 - 315758895809762413112 z + 43713291978745803199 z + 350 z 24 22 4 6 - 5194802357807489047 z + 526668457686433487 z - 52205 z + 4576677 z 102 8 10 12 + 373922446932 z - 269835324 z + 11525997228 z - 373922446932 z 14 18 16 + 9517925620277 z + 3256956349165444 z - 194659376867710 z 50 48 + 101309879481851297563458754 z - 61292624754047622039960926 z 20 36 - 45213836864805397 z - 204524701267571369521783 z 34 66 + 49922718203808741430762 z + 61292624754047622039960926 z 80 100 - 49922718203808741430762 z - 9517925620277 z 90 88 + 5194802357807489047 z - 43713291978745803199 z 84 94 - 1968010538076773423980 z + 45213836864805397 z 86 96 98 + 315758895809762413112 z - 3256956349165444 z + 194659376867710 z 92 82 - 526668457686433487 z + 10628951684552977150256 z 64 112 114 110 - 101309879481851297563458754 z - 350 z + z + 52205 z 106 108 30 + 269835324 z - 4576677 z + 1968010538076773423980 z 42 44 + 6335427893604390473981899 z - 15339083479625571546188255 z 46 58 + 32675290128725333933439207 z + 215046624633591016247674535 z 56 54 - 215046624633591016247674535 z + 189710535904344737996085434 z 52 60 - 147626021001302698211790945 z - 189710535904344737996085434 z 70 68 + 15339083479625571546188255 z - 32675290128725333933439207 z 78 32 + 204524701267571369521783 z - 10628951684552977150256 z 38 40 + 732670683358962002531197 z - 2299725827541330231094798 z 62 76 + 147626021001302698211790945 z - 732670683358962002531197 z 74 72 + 2299725827541330231094798 z - 6335427893604390473981899 z 104 - 11525997228 z ) And in Maple-input format, it is: -(1+94919420037561696415*z^28-13927068254223121752*z^26-290*z^2+ 1755670264064490539*z^24-189033126749760628*z^22+37717*z^4-2960810*z^6-\ 6232416890*z^102+158787678*z^8-6232416890*z^10+187090653448*z^12-4428968641228* z^14-1324772584112952*z^18+84562877787333*z^16-16144773007635768101014120*z^50+ 10369153044762641468445329*z^48+17259779349712509*z^20+48939645225883745237505* z^36-12643430429673394148790*z^34-5864143941981045328263532*z^66+ 2849295329610778503345*z^80+187090653448*z^100-189033126749760628*z^90+ 1755670264064490539*z^88+94919420037561696415*z^84-1324772584112952*z^94-\ 13927068254223121752*z^86+84562877787333*z^96-4428968641228*z^98+ 17259779349712509*z^92-558544589159842933142*z^82+10369153044762641468445329*z^ 64+z^112-290*z^110-2960810*z^106+37717*z^108-558544589159842933142*z^30-\ 1277244801647231599661070*z^42+2918505381307150932573166*z^44-\ 5864143941981045328263532*z^46-26763501846052038044365518*z^58+ 28507807571882732103742800*z^56-26763501846052038044365518*z^54+ 22144023910253300856709125*z^52+22144023910253300856709125*z^60-\ 1277244801647231599661070*z^70+2918505381307150932573166*z^68-\ 12643430429673394148790*z^78+2849295329610778503345*z^32-\ 165628097843069086827060*z^38+491036032353952376505696*z^40-\ 16144773007635768101014120*z^62+48939645225883745237505*z^76-\ 165628097843069086827060*z^74+491036032353952376505696*z^72+158787678*z^104)/(-\ 1-315758895809762413112*z^28+43713291978745803199*z^26+350*z^2-\ 5194802357807489047*z^24+526668457686433487*z^22-52205*z^4+4576677*z^6+ 373922446932*z^102-269835324*z^8+11525997228*z^10-373922446932*z^12+ 9517925620277*z^14+3256956349165444*z^18-194659376867710*z^16+ 101309879481851297563458754*z^50-61292624754047622039960926*z^48-\ 45213836864805397*z^20-204524701267571369521783*z^36+49922718203808741430762*z^ 34+61292624754047622039960926*z^66-49922718203808741430762*z^80-9517925620277*z ^100+5194802357807489047*z^90-43713291978745803199*z^88-1968010538076773423980* z^84+45213836864805397*z^94+315758895809762413112*z^86-3256956349165444*z^96+ 194659376867710*z^98-526668457686433487*z^92+10628951684552977150256*z^82-\ 101309879481851297563458754*z^64-350*z^112+z^114+52205*z^110+269835324*z^106-\ 4576677*z^108+1968010538076773423980*z^30+6335427893604390473981899*z^42-\ 15339083479625571546188255*z^44+32675290128725333933439207*z^46+ 215046624633591016247674535*z^58-215046624633591016247674535*z^56+ 189710535904344737996085434*z^54-147626021001302698211790945*z^52-\ 189710535904344737996085434*z^60+15339083479625571546188255*z^70-\ 32675290128725333933439207*z^68+204524701267571369521783*z^78-\ 10628951684552977150256*z^32+732670683358962002531197*z^38-\ 2299725827541330231094798*z^40+147626021001302698211790945*z^62-\ 732670683358962002531197*z^76+2299725827541330231094798*z^74-\ 6335427893604390473981899*z^72-11525997228*z^104) The first , 40, terms are: [0, 60, 0, 6512, 0, 762767, 0, 90562464, 0, 10783392687, 0, 1284872602897, 0, 153122965371600, 0, 18249063514029813, 0, 2174934811517116895, 0, 259210979080227809036, 0, 30893061758004468764683, 0, 3681871505540797787586824, 0, 438809817502516353663746128, 0, 52297876143792450123216938809, 0, 6232923139516359442621415145189, 0, 742847201318660708763123507672072, 0, 88533414004627206402706065190064640, 0, 10551517704397460001787948239727832359, 0, 1257542444536073152129960970871436085060, 0, 149875405996622942899018379447752079364179] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {4, 6}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1482225780605610208 z - 361683924017102600 z - 278 z 24 22 4 6 + 73683409286105205 z - 12456547306697238 z + 33208 z - 2284162 z 8 10 12 14 + 102510563 z - 3226366588 z + 74684602272 z - 1315052882676 z 18 16 50 - 197057044056978 z + 18057389403023 z - 131812979490409750284 z 48 20 + 197184808293497354878 z + 1734416538585128 z 36 34 + 74899265659764829728 z - 36124809245302821300 z 66 80 88 84 86 - 12456547306697238 z + 102510563 z + z + 33208 z - 278 z 82 64 30 - 2284162 z + 73683409286105205 z - 5096383526232528792 z 42 44 - 250988930487855579632 z + 271990610679573168896 z 46 58 - 250988930487855579632 z - 5096383526232528792 z 56 54 + 14760400564884461978 z - 36124809245302821300 z 52 60 70 + 74899265659764829728 z + 1482225780605610208 z - 197057044056978 z 68 78 32 + 1734416538585128 z - 3226366588 z + 14760400564884461978 z 38 40 - 131812979490409750284 z + 197184808293497354878 z 62 76 74 - 361683924017102600 z + 74684602272 z - 1315052882676 z 72 / 28 + 18057389403023 z ) / (-1 - 6247390894415011872 z / 26 2 24 + 1414905290628686869 z + 353 z - 267595610173518565 z 22 4 6 8 + 41998057987800694 z - 49918 z + 3911822 z - 195658055 z 10 12 14 + 6773185251 z - 171045087176 z + 3268687990808 z 18 16 50 + 571946548827523 z - 48551414395359 z + 1313327831977330047974 z 48 20 - 1808221449684380923470 z - 5427438412627478 z 36 34 - 427290648377907336428 z + 190914240540423933450 z 66 80 90 88 84 + 267595610173518565 z - 6773185251 z + z - 353 z - 3911822 z 86 82 64 + 49918 z + 195658055 z - 1414905290628686869 z 30 42 + 23152036584701715456 z + 1808221449684380923470 z 44 46 - 2121378258907600464384 z + 2121378258907600464384 z 58 56 + 72304489334113929178 z - 190914240540423933450 z 54 52 + 427290648377907336428 z - 812241089199298409212 z 60 70 68 - 23152036584701715456 z + 5427438412627478 z - 41998057987800694 z 78 32 38 + 171045087176 z - 72304489334113929178 z + 812241089199298409212 z 40 62 76 - 1313327831977330047974 z + 6247390894415011872 z - 3268687990808 z 74 72 + 48551414395359 z - 571946548827523 z ) And in Maple-input format, it is: -(1+1482225780605610208*z^28-361683924017102600*z^26-278*z^2+73683409286105205* z^24-12456547306697238*z^22+33208*z^4-2284162*z^6+102510563*z^8-3226366588*z^10 +74684602272*z^12-1315052882676*z^14-197057044056978*z^18+18057389403023*z^16-\ 131812979490409750284*z^50+197184808293497354878*z^48+1734416538585128*z^20+ 74899265659764829728*z^36-36124809245302821300*z^34-12456547306697238*z^66+ 102510563*z^80+z^88+33208*z^84-278*z^86-2284162*z^82+73683409286105205*z^64-\ 5096383526232528792*z^30-250988930487855579632*z^42+271990610679573168896*z^44-\ 250988930487855579632*z^46-5096383526232528792*z^58+14760400564884461978*z^56-\ 36124809245302821300*z^54+74899265659764829728*z^52+1482225780605610208*z^60-\ 197057044056978*z^70+1734416538585128*z^68-3226366588*z^78+14760400564884461978 *z^32-131812979490409750284*z^38+197184808293497354878*z^40-361683924017102600* z^62+74684602272*z^76-1315052882676*z^74+18057389403023*z^72)/(-1-\ 6247390894415011872*z^28+1414905290628686869*z^26+353*z^2-267595610173518565*z^ 24+41998057987800694*z^22-49918*z^4+3911822*z^6-195658055*z^8+6773185251*z^10-\ 171045087176*z^12+3268687990808*z^14+571946548827523*z^18-48551414395359*z^16+ 1313327831977330047974*z^50-1808221449684380923470*z^48-5427438412627478*z^20-\ 427290648377907336428*z^36+190914240540423933450*z^34+267595610173518565*z^66-\ 6773185251*z^80+z^90-353*z^88-3911822*z^84+49918*z^86+195658055*z^82-\ 1414905290628686869*z^64+23152036584701715456*z^30+1808221449684380923470*z^42-\ 2121378258907600464384*z^44+2121378258907600464384*z^46+72304489334113929178*z^ 58-190914240540423933450*z^56+427290648377907336428*z^54-812241089199298409212* z^52-23152036584701715456*z^60+5427438412627478*z^70-41998057987800694*z^68+ 171045087176*z^78-72304489334113929178*z^32+812241089199298409212*z^38-\ 1313327831977330047974*z^40+6247390894415011872*z^62-3268687990808*z^76+ 48551414395359*z^74-571946548827523*z^72) The first , 40, terms are: [0, 75, 0, 9765, 0, 1330855, 0, 182581703, 0, 25089127637, 0, 3449443975163, 0, 474354679200561, 0, 65237007724532577, 0, 8972223525158112347, 0, 1233992377804163987221, 0, 169717849856388647934487, 0, 23342301549983401663283159, 0, 3210408484864672843175582533, 0, 441547173226058794078503087435, 0, 60728702837965631511979460669009, 0, 8352393272762033215610480755839089, 0, 1148756229418277059328185519017609419, 0, 157995540897741730865101590244097363845, 0, 21730102914252870431610478884364289748151, 0, 2988675320401704133511600670887257761080119] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1278353600584152865 z - 312279503845777922 z - 274 z 24 22 4 6 + 63750647755211001 z - 10812719616817336 z + 32129 z - 2168442 z 8 10 12 14 + 95594284 z - 2960498566 z + 67565810771 z - 1175376312614 z 18 16 50 - 172948365313544 z + 15977585516171 z - 113878147794854682310 z 48 20 + 170453944143689523067 z + 1512623436708568 z 36 34 + 64663753353918229003 z - 31166959041812068702 z 66 80 88 84 86 - 10812719616817336 z + 95594284 z + z + 32129 z - 274 z 82 64 30 - 2168442 z + 63750647755211001 z - 4393861148193610402 z 42 44 - 217047257516337116480 z + 235240226191550779216 z 46 58 - 217047257516337116480 z - 4393861148193610402 z 56 54 + 12728048341053038420 z - 31166959041812068702 z 52 60 70 + 64663753353918229003 z + 1278353600584152865 z - 172948365313544 z 68 78 32 + 1512623436708568 z - 2960498566 z + 12728048341053038420 z 38 40 - 113878147794854682310 z + 170453944143689523067 z 62 76 74 - 312279503845777922 z + 67565810771 z - 1175376312614 z 72 / 2 28 + 15977585516171 z ) / ((-1 + z ) (1 + 4307986376294544151 z / 26 2 24 - 1012447120971210607 z - 339 z + 198090329026113777 z 22 4 6 8 - 32076261123139088 z + 46719 z - 3587324 z + 176089566 z 10 12 14 - 5981084876 z + 148043691621 z - 2768993307493 z 18 16 50 - 461821253099536 z + 40189823389507 z - 437957494257141905049 z 48 20 + 663950865711997044875 z + 4266674808956068 z 36 34 + 244348195898560419013 z - 115182551553965973124 z 66 80 88 84 86 - 32076261123139088 z + 176089566 z + z + 46719 z - 339 z 82 64 30 - 3587324 z + 198090329026113777 z - 15331581527492554228 z 42 44 - 851998206051348207648 z + 925806419880382217208 z 46 58 - 851998206051348207648 z - 15331581527492554228 z 56 54 + 45802166106092147906 z - 115182551553965973124 z 52 60 + 244348195898560419013 z + 4307986376294544151 z 70 68 78 - 461821253099536 z + 4266674808956068 z - 5981084876 z 32 38 + 45802166106092147906 z - 437957494257141905049 z 40 62 76 + 663950865711997044875 z - 1012447120971210607 z + 148043691621 z 74 72 - 2768993307493 z + 40189823389507 z )) And in Maple-input format, it is: -(1+1278353600584152865*z^28-312279503845777922*z^26-274*z^2+63750647755211001* z^24-10812719616817336*z^22+32129*z^4-2168442*z^6+95594284*z^8-2960498566*z^10+ 67565810771*z^12-1175376312614*z^14-172948365313544*z^18+15977585516171*z^16-\ 113878147794854682310*z^50+170453944143689523067*z^48+1512623436708568*z^20+ 64663753353918229003*z^36-31166959041812068702*z^34-10812719616817336*z^66+ 95594284*z^80+z^88+32129*z^84-274*z^86-2168442*z^82+63750647755211001*z^64-\ 4393861148193610402*z^30-217047257516337116480*z^42+235240226191550779216*z^44-\ 217047257516337116480*z^46-4393861148193610402*z^58+12728048341053038420*z^56-\ 31166959041812068702*z^54+64663753353918229003*z^52+1278353600584152865*z^60-\ 172948365313544*z^70+1512623436708568*z^68-2960498566*z^78+12728048341053038420 *z^32-113878147794854682310*z^38+170453944143689523067*z^40-312279503845777922* z^62+67565810771*z^76-1175376312614*z^74+15977585516171*z^72)/(-1+z^2)/(1+ 4307986376294544151*z^28-1012447120971210607*z^26-339*z^2+198090329026113777*z^ 24-32076261123139088*z^22+46719*z^4-3587324*z^6+176089566*z^8-5981084876*z^10+ 148043691621*z^12-2768993307493*z^14-461821253099536*z^18+40189823389507*z^16-\ 437957494257141905049*z^50+663950865711997044875*z^48+4266674808956068*z^20+ 244348195898560419013*z^36-115182551553965973124*z^34-32076261123139088*z^66+ 176089566*z^80+z^88+46719*z^84-339*z^86-3587324*z^82+198090329026113777*z^64-\ 15331581527492554228*z^30-851998206051348207648*z^42+925806419880382217208*z^44 -851998206051348207648*z^46-15331581527492554228*z^58+45802166106092147906*z^56 -115182551553965973124*z^54+244348195898560419013*z^52+4307986376294544151*z^60 -461821253099536*z^70+4266674808956068*z^68-5981084876*z^78+ 45802166106092147906*z^32-437957494257141905049*z^38+663950865711997044875*z^40 -1012447120971210607*z^62+148043691621*z^76-2768993307493*z^74+40189823389507*z ^72) The first , 40, terms are: [0, 66, 0, 7511, 0, 913513, 0, 112906014, 0, 14033248115, 0, 1748280102003, 0, 218029342576942, 0, 27203559068392065, 0, 3394942002895600223, 0, 423724510420455421042, 0, 52887811269785957233457, 0, 6601418900144042342334513, 0, 823992976608740688668243986, 0, 102851785683948605626750325359, 0, 12838111585753607932570141144625, 0, 1602473716623399026287854214208526, 0, 200023442815393320621837352508826387, 0, 24967265587569115002969738399403110419, 0, 3116456796549039390553984812434382760510, 0, 389001487451856755722994553241329456398457] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1877986578883507105 z - 444221969519710084 z - 268 z 24 22 4 6 + 87547949580571441 z - 14297006265227264 z + 31241 z - 2128702 z 8 10 12 14 + 95941188 z - 3065465298 z + 72612620275 z - 1315627341948 z 18 16 50 - 210722462925376 z + 18656896576371 z - 186370696533793246964 z 48 20 + 281783161649140920923 z + 1921465121644840 z 36 34 + 104353304527037608291 z - 49403297391317196122 z 66 80 88 84 86 - 14297006265227264 z + 95941188 z + z + 31241 z - 268 z 82 64 30 - 2128702 z + 87547949580571441 z - 6644195008913572246 z 42 44 - 360987544942047005120 z + 392037280830764540976 z 46 58 - 360987544942047005120 z - 6644195008913572246 z 56 54 + 19741883578505462396 z - 49403297391317196122 z 52 60 + 104353304527037608291 z + 1877986578883507105 z 70 68 78 - 210722462925376 z + 1921465121644840 z - 3065465298 z 32 38 + 19741883578505462396 z - 186370696533793246964 z 40 62 76 + 281783161649140920923 z - 444221969519710084 z + 72612620275 z 74 72 / 2 - 1315627341948 z + 18656896576371 z ) / ((-1 + z ) (1 / 28 26 2 + 6291142130078258253 z - 1426104285456242217 z - 337 z 24 22 4 6 + 268317709645708981 z - 41675161086330144 z + 45761 z - 3504414 z 8 10 12 14 + 174106286 z - 6059995770 z + 155107113647 z - 3018002645563 z 18 16 50 - 549395729843152 z + 45728389282551 z - 723598264371196472907 z 48 20 + 1109924871936948434835 z + 5307254047383980 z 36 34 + 397199026532027319555 z - 183402613723005020698 z 66 80 88 84 86 - 41675161086330144 z + 174106286 z + z + 45761 z - 337 z 82 64 30 - 3504414 z + 268317709645708981 z - 23132871927770645710 z 42 44 - 1434398014545243415696 z + 1562351264354399045608 z 46 58 - 1434398014545243415696 z - 23132871927770645710 z 56 54 + 71135752842776333234 z - 183402613723005020698 z 52 60 + 397199026532027319555 z + 6291142130078258253 z 70 68 78 - 549395729843152 z + 5307254047383980 z - 6059995770 z 32 38 + 71135752842776333234 z - 723598264371196472907 z 40 62 76 + 1109924871936948434835 z - 1426104285456242217 z + 155107113647 z 74 72 - 3018002645563 z + 45728389282551 z )) And in Maple-input format, it is: -(1+1877986578883507105*z^28-444221969519710084*z^26-268*z^2+87547949580571441* z^24-14297006265227264*z^22+31241*z^4-2128702*z^6+95941188*z^8-3065465298*z^10+ 72612620275*z^12-1315627341948*z^14-210722462925376*z^18+18656896576371*z^16-\ 186370696533793246964*z^50+281783161649140920923*z^48+1921465121644840*z^20+ 104353304527037608291*z^36-49403297391317196122*z^34-14297006265227264*z^66+ 95941188*z^80+z^88+31241*z^84-268*z^86-2128702*z^82+87547949580571441*z^64-\ 6644195008913572246*z^30-360987544942047005120*z^42+392037280830764540976*z^44-\ 360987544942047005120*z^46-6644195008913572246*z^58+19741883578505462396*z^56-\ 49403297391317196122*z^54+104353304527037608291*z^52+1877986578883507105*z^60-\ 210722462925376*z^70+1921465121644840*z^68-3065465298*z^78+19741883578505462396 *z^32-186370696533793246964*z^38+281783161649140920923*z^40-444221969519710084* z^62+72612620275*z^76-1315627341948*z^74+18656896576371*z^72)/(-1+z^2)/(1+ 6291142130078258253*z^28-1426104285456242217*z^26-337*z^2+268317709645708981*z^ 24-41675161086330144*z^22+45761*z^4-3504414*z^6+174106286*z^8-6059995770*z^10+ 155107113647*z^12-3018002645563*z^14-549395729843152*z^18+45728389282551*z^16-\ 723598264371196472907*z^50+1109924871936948434835*z^48+5307254047383980*z^20+ 397199026532027319555*z^36-183402613723005020698*z^34-41675161086330144*z^66+ 174106286*z^80+z^88+45761*z^84-337*z^86-3504414*z^82+268317709645708981*z^64-\ 23132871927770645710*z^30-1434398014545243415696*z^42+1562351264354399045608*z^ 44-1434398014545243415696*z^46-23132871927770645710*z^58+71135752842776333234*z ^56-183402613723005020698*z^54+397199026532027319555*z^52+6291142130078258253*z ^60-549395729843152*z^70+5307254047383980*z^68-6059995770*z^78+ 71135752842776333234*z^32-723598264371196472907*z^38+1109924871936948434835*z^ 40-1426104285456242217*z^62+155107113647*z^76-3018002645563*z^74+45728389282551 *z^72) The first , 40, terms are: [0, 70, 0, 8803, 0, 1170027, 0, 156511170, 0, 20952949097, 0, 2805371147529, 0, 375613714014186, 0, 50291355626796619, 0, 6733569547313219907, 0, 901565682477959191742, 0, 120711708394926310964961, 0, 16162235139486937219812321, 0, 2163981010686874695990046286, 0, 289738008035288459053383250627, 0, 38793368743095342486392639478091, 0, 5194090580812118674994579104280474, 0, 695443005745416647505963020693953545, 0, 93113696558721945484684474743216686569, 0, 12467104299275740701934211091986224386834, 0, 1669235519083917225294539368261826898697195] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6469256855774010 z - 2665502548978474 z - 223 z 24 22 4 6 + 895703331100543 z - 244312113659575 z + 20443 z - 1040360 z 8 10 12 14 + 33630289 z - 746569897 z + 11969967990 z - 143502285809 z 18 16 50 - 9462714417096 z + 1318780369585 z - 244312113659575 z 48 20 36 + 895703331100543 z + 53753550130667 z + 30933525972302100 z 34 66 64 - 28062979356249638 z - 1040360 z + 33630289 z 30 42 44 - 12850766555970768 z - 12850766555970768 z + 6469256855774010 z 46 58 56 - 2665502548978474 z - 143502285809 z + 1318780369585 z 54 52 60 70 - 9462714417096 z + 53753550130667 z + 11969967990 z - 223 z 68 32 38 + 20443 z + 20946360036662910 z - 28062979356249638 z 40 62 72 / + 20946360036662910 z - 746569897 z + z ) / (-1 / 28 26 2 - 34567955556679300 z + 12941673664673563 z + 299 z 24 22 4 6 - 3953246749804089 z + 980682210196910 z - 32846 z + 1910180 z 8 10 12 14 - 69077020 z + 1698259230 z - 30007224940 z + 395533352188 z 18 16 50 + 31464269290828 z - 3992792355102 z + 3953246749804089 z 48 20 36 - 12941673664673563 z - 196338389342196 z - 243191884290533144 z 34 66 64 + 200242066594627908 z + 69077020 z - 1698259230 z 30 42 44 + 75597037616933224 z + 135692605335455384 z - 75597037616933224 z 46 58 56 + 34567955556679300 z + 3992792355102 z - 31464269290828 z 54 52 60 70 + 196338389342196 z - 980682210196910 z - 395533352188 z + 32846 z 68 32 38 - 1910180 z - 135692605335455384 z + 243191884290533144 z 40 62 74 72 - 200242066594627908 z + 30007224940 z + z - 299 z ) And in Maple-input format, it is: -(1+6469256855774010*z^28-2665502548978474*z^26-223*z^2+895703331100543*z^24-\ 244312113659575*z^22+20443*z^4-1040360*z^6+33630289*z^8-746569897*z^10+ 11969967990*z^12-143502285809*z^14-9462714417096*z^18+1318780369585*z^16-\ 244312113659575*z^50+895703331100543*z^48+53753550130667*z^20+30933525972302100 *z^36-28062979356249638*z^34-1040360*z^66+33630289*z^64-12850766555970768*z^30-\ 12850766555970768*z^42+6469256855774010*z^44-2665502548978474*z^46-143502285809 *z^58+1318780369585*z^56-9462714417096*z^54+53753550130667*z^52+11969967990*z^ 60-223*z^70+20443*z^68+20946360036662910*z^32-28062979356249638*z^38+ 20946360036662910*z^40-746569897*z^62+z^72)/(-1-34567955556679300*z^28+ 12941673664673563*z^26+299*z^2-3953246749804089*z^24+980682210196910*z^22-32846 *z^4+1910180*z^6-69077020*z^8+1698259230*z^10-30007224940*z^12+395533352188*z^ 14+31464269290828*z^18-3992792355102*z^16+3953246749804089*z^50-\ 12941673664673563*z^48-196338389342196*z^20-243191884290533144*z^36+ 200242066594627908*z^34+69077020*z^66-1698259230*z^64+75597037616933224*z^30+ 135692605335455384*z^42-75597037616933224*z^44+34567955556679300*z^46+ 3992792355102*z^58-31464269290828*z^56+196338389342196*z^54-980682210196910*z^ 52-395533352188*z^60+32846*z^70-1910180*z^68-135692605335455384*z^32+ 243191884290533144*z^38-200242066594627908*z^40+30007224940*z^62+z^74-299*z^72) The first , 40, terms are: [0, 76, 0, 10321, 0, 1459503, 0, 207114780, 0, 29405287275, 0, 4175288792995, 0, 592872891293964, 0, 84186275043923087, 0, 11954256581140362977, 0, 1697478941638243504412, 0, 241038493708837833020937, 0, 34226971935775717648576473, 0, 4860160093413811889809912892, 0, 690132816808170359655929945121, 0, 97997452470832348971941859999727, 0, 13915438415924452638325900453257580, 0, 1975963880352799753518637171000761427, 0, 280582841930198480634122639647159248891, 0, 39842191436432347110605192911063091092028, 0, 5657509944603217850173108794818233086161935] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 101160611120045725247 z - 14769136331067553206 z - 293 z 24 22 4 6 + 1853052034994664701 z - 198641463842289692 z + 38392 z - 3029888 z 102 8 10 12 - 6422860704 z + 163130237 z - 6422860704 z + 193344448103 z 14 18 16 - 4589692516644 z - 1381440549605494 z + 87891488103089 z 50 48 - 18038704294523440041889234 z + 11559296009608582289802182 z 20 36 + 18063951755352321 z + 53253489887537910583702 z 34 66 - 13687198150179968409626 z - 6518533176007968890113358 z 80 100 90 + 3068380937560616398369 z + 193344448103 z - 198641463842289692 z 88 84 + 1853052034994664701 z + 101160611120045725247 z 94 86 96 - 1381440549605494 z - 14769136331067553206 z + 87891488103089 z 98 92 82 - 4589692516644 z + 18063951755352321 z - 598345565016450047614 z 64 112 110 106 + 11559296009608582289802182 z + z - 293 z - 3029888 z 108 30 42 + 38392 z - 598345565016450047614 z - 1409422086445946116765874 z 44 46 + 3233137587565665723617230 z - 6518533176007968890113358 z 58 56 - 29982085089058502109398138 z + 31946797458449769511844650 z 54 52 - 29982085089058502109398138 z + 24782406769799088459711966 z 60 70 + 24782406769799088459711966 z - 1409422086445946116765874 z 68 78 + 3233137587565665723617230 z - 13687198150179968409626 z 32 38 + 3068380937560616398369 z - 181125088633465850723889 z 40 62 + 539506248709922160261629 z - 18038704294523440041889234 z 76 74 + 53253489887537910583702 z - 181125088633465850723889 z 72 104 / 2 + 539506248709922160261629 z + 163130237 z ) / ((-1 + z ) (1 / 28 26 2 + 295485450601081896910 z - 41404948749484183806 z - 361 z 24 22 4 6 + 4976699276728829304 z - 510081408544848532 z + 54428 z - 4775108 z 102 8 10 12 - 11875504414 z + 280198058 z - 11875504414 z + 381645721684 z 14 18 16 - 9615980422172 z - 3222008875077742 z + 194613346838395 z 50 48 - 71190862185737036307797778 z + 45032837793386979761515379 z 20 36 + 44258626756324883 z + 179761310476847916233710 z 34 66 - 44700105228558936184382 z - 24982389586822671663601842 z 80 100 90 + 9674173473061131240385 z + 381645721684 z - 510081408544848532 z 88 84 + 4976699276728829304 z + 295485450601081896910 z 94 86 96 - 3222008875077742 z - 41404948749484183806 z + 194613346838395 z 98 92 82 - 9615980422172 z + 44258626756324883 z - 1817551859423556066972 z 64 112 110 106 + 45032837793386979761515379 z + z - 361 z - 4775108 z 108 30 42 + 54428 z - 1817551859423556066972 z - 5177599899567778147023850 z 44 46 + 12150022335952415416232426 z - 24982389586822671663601842 z 58 56 - 120116845212813599574406442 z + 128230846170912294154097391 z 54 52 - 120116845212813599574406442 z + 98724889615237020648689698 z 60 70 + 98724889615237020648689698 z - 5177599899567778147023850 z 68 78 + 12150022335952415416232426 z - 44700105228558936184382 z 32 38 + 9674173473061131240385 z - 630491087943703430017933 z 40 62 + 1931838727653085947016650 z - 71190862185737036307797778 z 76 74 + 179761310476847916233710 z - 630491087943703430017933 z 72 104 + 1931838727653085947016650 z + 280198058 z )) And in Maple-input format, it is: -(1+101160611120045725247*z^28-14769136331067553206*z^26-293*z^2+ 1853052034994664701*z^24-198641463842289692*z^22+38392*z^4-3029888*z^6-\ 6422860704*z^102+163130237*z^8-6422860704*z^10+193344448103*z^12-4589692516644* z^14-1381440549605494*z^18+87891488103089*z^16-18038704294523440041889234*z^50+ 11559296009608582289802182*z^48+18063951755352321*z^20+53253489887537910583702* z^36-13687198150179968409626*z^34-6518533176007968890113358*z^66+ 3068380937560616398369*z^80+193344448103*z^100-198641463842289692*z^90+ 1853052034994664701*z^88+101160611120045725247*z^84-1381440549605494*z^94-\ 14769136331067553206*z^86+87891488103089*z^96-4589692516644*z^98+ 18063951755352321*z^92-598345565016450047614*z^82+11559296009608582289802182*z^ 64+z^112-293*z^110-3029888*z^106+38392*z^108-598345565016450047614*z^30-\ 1409422086445946116765874*z^42+3233137587565665723617230*z^44-\ 6518533176007968890113358*z^46-29982085089058502109398138*z^58+ 31946797458449769511844650*z^56-29982085089058502109398138*z^54+ 24782406769799088459711966*z^52+24782406769799088459711966*z^60-\ 1409422086445946116765874*z^70+3233137587565665723617230*z^68-\ 13687198150179968409626*z^78+3068380937560616398369*z^32-\ 181125088633465850723889*z^38+539506248709922160261629*z^40-\ 18038704294523440041889234*z^62+53253489887537910583702*z^76-\ 181125088633465850723889*z^74+539506248709922160261629*z^72+163130237*z^104)/(-\ 1+z^2)/(1+295485450601081896910*z^28-41404948749484183806*z^26-361*z^2+ 4976699276728829304*z^24-510081408544848532*z^22+54428*z^4-4775108*z^6-\ 11875504414*z^102+280198058*z^8-11875504414*z^10+381645721684*z^12-\ 9615980422172*z^14-3222008875077742*z^18+194613346838395*z^16-\ 71190862185737036307797778*z^50+45032837793386979761515379*z^48+ 44258626756324883*z^20+179761310476847916233710*z^36-44700105228558936184382*z^ 34-24982389586822671663601842*z^66+9674173473061131240385*z^80+381645721684*z^ 100-510081408544848532*z^90+4976699276728829304*z^88+295485450601081896910*z^84 -3222008875077742*z^94-41404948749484183806*z^86+194613346838395*z^96-\ 9615980422172*z^98+44258626756324883*z^92-1817551859423556066972*z^82+ 45032837793386979761515379*z^64+z^112-361*z^110-4775108*z^106+54428*z^108-\ 1817551859423556066972*z^30-5177599899567778147023850*z^42+ 12150022335952415416232426*z^44-24982389586822671663601842*z^46-\ 120116845212813599574406442*z^58+128230846170912294154097391*z^56-\ 120116845212813599574406442*z^54+98724889615237020648689698*z^52+ 98724889615237020648689698*z^60-5177599899567778147023850*z^70+ 12150022335952415416232426*z^68-44700105228558936184382*z^78+ 9674173473061131240385*z^32-630491087943703430017933*z^38+ 1931838727653085947016650*z^40-71190862185737036307797778*z^62+ 179761310476847916233710*z^76-630491087943703430017933*z^74+ 1931838727653085947016650*z^72+280198058*z^104) The first , 40, terms are: [0, 69, 0, 8581, 0, 1125529, 0, 148692144, 0, 19671889477, 0, 2603524892729, 0, 344604410361597, 0, 45613386956583037, 0, 6037642508180274905, 0, 799177981431084370973, 0, 105783986361367525778769, 0, 14002205160293402404516473, 0, 1853416266417296852308503744, 0, 245329351856195695831325635309, 0, 32473272345600799363317658896493, 0, 4298358147632942055771487696671045, 0, 568956604580936016007571743245876813, 0, 75310527140662240439139262362370430373, 0, 9968555515044542333737727396452014622437, 0, 1319498121052953502108528218976332202681869] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 102272053583964646152 z - 15010064190810109776 z - 293 z 24 22 4 6 + 1891565372645253264 z - 203452891274592576 z + 38503 z - 3052418 z 102 8 10 12 - 6536095138 z + 165187968 z - 6536095138 z + 197580960387 z 14 18 16 - 4704833738113 z - 1419296685577216 z + 90259771263037 z 50 48 - 17095781837626217030942898 z + 10993249434057295707855082 z 20 36 + 18542100179425776 z + 52452241982694167443040 z 34 66 - 13574804590782033851224 z - 6226314896007492708365776 z 80 100 90 + 3063874085746486766096 z + 197580960387 z - 203452891274592576 z 88 84 + 1891565372645253264 z + 102272053583964646152 z 94 86 96 - 1419296685577216 z - 15010064190810109776 z + 90259771263037 z 98 92 82 - 4704833738113 z + 18542100179425776 z - 601334308051645188104 z 64 112 110 106 + 10993249434057295707855082 z + z - 293 z - 3052418 z 108 30 42 + 38503 z - 601334308051645188104 z - 1360965677195396608601200 z 44 46 + 3104029245200333162299536 z - 6226314896007492708365776 z 58 56 - 28299014025582228777527036 z + 30137795088987446159013344 z 54 52 - 28299014025582228777527036 z + 23427404783678322334016302 z 60 70 + 23427404783678322334016302 z - 1360965677195396608601200 z 68 78 + 3104029245200333162299536 z - 13574804590782033851224 z 32 38 + 3063874085746486766096 z - 177177706550007340469848 z 40 62 + 524244191051660788322248 z - 17095781837626217030942898 z 76 74 + 52452241982694167443040 z - 177177706550007340469848 z 72 104 / + 524244191051660788322248 z + 165187968 z ) / (-1 / 28 26 2 - 343178369516488341924 z + 47539232486736956080 z + 357 z 24 22 4 6 - 5650910574966493344 z + 572778390552771004 z - 54050 z + 4795634 z 102 8 10 12 + 401294375334 z - 285521915 z + 12292683847 z - 401294375334 z 14 18 16 + 10263882294102 z + 3533518220234605 z - 210663790595121 z 50 48 + 108761550959239953747094882 z - 65849652617006878157206186 z 20 36 - 49131054932008148 z - 221268822688368688488156 z 34 66 + 54080738698591128667360 z + 65849652617006878157206186 z 80 100 - 54080738698591128667360 z - 10263882294102 z 90 88 + 5650910574966493344 z - 47539232486736956080 z 84 94 - 2136953207276689737604 z + 49131054932008148 z 86 96 98 + 343178369516488341924 z - 3533518220234605 z + 210663790595121 z 92 82 - 572778390552771004 z + 11528546100412005367608 z 64 112 114 110 - 108761550959239953747094882 z - 357 z + z + 54050 z 106 108 30 + 285521915 z - 4795634 z + 2136953207276689737604 z 42 44 + 6827528015575769433045192 z - 16511455702259267412181140 z 46 58 + 35136201390598706911223260 z + 230599051845459732638735526 z 56 54 - 230599051845459732638735526 z + 203470202623387770511595400 z 52 60 - 158394318729657487516828136 z - 203470202623387770511595400 z 70 68 + 16511455702259267412181140 z - 35136201390598706911223260 z 78 32 + 221268822688368688488156 z - 11528546100412005367608 z 38 40 + 791598907190776525270140 z - 2481446199744279356470744 z 62 76 + 158394318729657487516828136 z - 791598907190776525270140 z 74 72 + 2481446199744279356470744 z - 6827528015575769433045192 z 104 - 12292683847 z ) And in Maple-input format, it is: -(1+102272053583964646152*z^28-15010064190810109776*z^26-293*z^2+ 1891565372645253264*z^24-203452891274592576*z^22+38503*z^4-3052418*z^6-\ 6536095138*z^102+165187968*z^8-6536095138*z^10+197580960387*z^12-4704833738113* z^14-1419296685577216*z^18+90259771263037*z^16-17095781837626217030942898*z^50+ 10993249434057295707855082*z^48+18542100179425776*z^20+52452241982694167443040* z^36-13574804590782033851224*z^34-6226314896007492708365776*z^66+ 3063874085746486766096*z^80+197580960387*z^100-203452891274592576*z^90+ 1891565372645253264*z^88+102272053583964646152*z^84-1419296685577216*z^94-\ 15010064190810109776*z^86+90259771263037*z^96-4704833738113*z^98+ 18542100179425776*z^92-601334308051645188104*z^82+10993249434057295707855082*z^ 64+z^112-293*z^110-3052418*z^106+38503*z^108-601334308051645188104*z^30-\ 1360965677195396608601200*z^42+3104029245200333162299536*z^44-\ 6226314896007492708365776*z^46-28299014025582228777527036*z^58+ 30137795088987446159013344*z^56-28299014025582228777527036*z^54+ 23427404783678322334016302*z^52+23427404783678322334016302*z^60-\ 1360965677195396608601200*z^70+3104029245200333162299536*z^68-\ 13574804590782033851224*z^78+3063874085746486766096*z^32-\ 177177706550007340469848*z^38+524244191051660788322248*z^40-\ 17095781837626217030942898*z^62+52452241982694167443040*z^76-\ 177177706550007340469848*z^74+524244191051660788322248*z^72+165187968*z^104)/(-\ 1-343178369516488341924*z^28+47539232486736956080*z^26+357*z^2-\ 5650910574966493344*z^24+572778390552771004*z^22-54050*z^4+4795634*z^6+ 401294375334*z^102-285521915*z^8+12292683847*z^10-401294375334*z^12+ 10263882294102*z^14+3533518220234605*z^18-210663790595121*z^16+ 108761550959239953747094882*z^50-65849652617006878157206186*z^48-\ 49131054932008148*z^20-221268822688368688488156*z^36+54080738698591128667360*z^ 34+65849652617006878157206186*z^66-54080738698591128667360*z^80-10263882294102* z^100+5650910574966493344*z^90-47539232486736956080*z^88-2136953207276689737604 *z^84+49131054932008148*z^94+343178369516488341924*z^86-3533518220234605*z^96+ 210663790595121*z^98-572778390552771004*z^92+11528546100412005367608*z^82-\ 108761550959239953747094882*z^64-357*z^112+z^114+54050*z^110+285521915*z^106-\ 4795634*z^108+2136953207276689737604*z^30+6827528015575769433045192*z^42-\ 16511455702259267412181140*z^44+35136201390598706911223260*z^46+ 230599051845459732638735526*z^58-230599051845459732638735526*z^56+ 203470202623387770511595400*z^54-158394318729657487516828136*z^52-\ 203470202623387770511595400*z^60+16511455702259267412181140*z^70-\ 35136201390598706911223260*z^68+221268822688368688488156*z^78-\ 11528546100412005367608*z^32+791598907190776525270140*z^38-\ 2481446199744279356470744*z^40+158394318729657487516828136*z^62-\ 791598907190776525270140*z^76+2481446199744279356470744*z^74-\ 6827528015575769433045192*z^72-12292683847*z^104) The first , 40, terms are: [0, 64, 0, 7301, 0, 890473, 0, 109866440, 0, 13588363413, 0, 1681570101169, 0, 208124255853408, 0, 25759973892654989, 0, 3188393807549338969, 0, 394638523458361501544, 0, 48845802883852968783373, 0, 6045818341213203255108181, 0, 748312418125470557840239592, 0, 92621287865348029189634291041, 0, 11464066052522016514978342502741, 0, 1418948209207681333503299917483584, 0, 175628264140288319217428616047806457, 0, 21738134603014885300129231315294132877, 0, 2690606197915335280588801769884707084872, 0, 333025894102306980694483505766751685403265] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}, {4, 5}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 11969550431948 z - 10523886343916 z - 190 z 24 22 4 6 + 7147397306878 z - 3741012836736 z + 14050 z - 554598 z 8 10 12 14 + 13366313 z - 212398720 z + 2337845748 z - 18433835264 z 18 16 50 48 - 460462352112 z + 106637575608 z - 554598 z + 13366313 z 20 36 34 + 1502847488484 z + 1502847488484 z - 3741012836736 z 30 42 44 46 - 10523886343916 z - 18433835264 z + 2337845748 z - 212398720 z 56 54 52 32 38 + z - 190 z + 14050 z + 7147397306878 z - 460462352112 z 40 / 28 26 + 106637575608 z ) / (-1 - 86769890151046 z + 67379783887716 z / 2 24 22 4 6 + 246 z - 40571925820798 z + 18886148090020 z - 22285 z + 1044917 z 8 10 12 14 - 29273942 z + 532533353 z - 6641133940 z + 58945275144 z 18 16 50 48 + 1849765712564 z - 382468037124 z + 29273942 z - 532533353 z 20 36 34 - 6763226749768 z - 18886148090020 z + 40571925820798 z 30 42 44 46 + 86769890151046 z + 382468037124 z - 58945275144 z + 6641133940 z 58 56 54 52 32 + z - 246 z + 22285 z - 1044917 z - 67379783887716 z 38 40 + 6763226749768 z - 1849765712564 z ) And in Maple-input format, it is: -(1+11969550431948*z^28-10523886343916*z^26-190*z^2+7147397306878*z^24-\ 3741012836736*z^22+14050*z^4-554598*z^6+13366313*z^8-212398720*z^10+2337845748* z^12-18433835264*z^14-460462352112*z^18+106637575608*z^16-554598*z^50+13366313* z^48+1502847488484*z^20+1502847488484*z^36-3741012836736*z^34-10523886343916*z^ 30-18433835264*z^42+2337845748*z^44-212398720*z^46+z^56-190*z^54+14050*z^52+ 7147397306878*z^32-460462352112*z^38+106637575608*z^40)/(-1-86769890151046*z^28 +67379783887716*z^26+246*z^2-40571925820798*z^24+18886148090020*z^22-22285*z^4+ 1044917*z^6-29273942*z^8+532533353*z^10-6641133940*z^12+58945275144*z^14+ 1849765712564*z^18-382468037124*z^16+29273942*z^50-532533353*z^48-6763226749768 *z^20-18886148090020*z^36+40571925820798*z^34+86769890151046*z^30+382468037124* z^42-58945275144*z^44+6641133940*z^46+z^58-246*z^56+22285*z^54-1044917*z^52-\ 67379783887716*z^32+6763226749768*z^38-1849765712564*z^40) The first , 40, terms are: [0, 56, 0, 5541, 0, 605445, 0, 68066008, 0, 7722575121, 0, 878853931505, 0, 100119422894104, 0, 11409654063399173, 0, 1300405922754817957, 0, 148218844730031670264, 0, 16894063998452418756769, 0, 1925604098130540635512673, 0, 219482862354201936580056696, 0, 25016957374107003509227053733, 0, 2851467670115268791402952946181, 0, 325014282865557106785742393171864, 0, 37045584568689800107744915834254897, 0, 4222507810806948248705036908280545489, 0, 481287378080883266149417882525306498904, 0, 54857812211110299280009112563199277238789] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}, {4, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1108093971864389697 z - 268015022880163723 z - 259 z 24 22 4 6 + 54174755335100913 z - 9101675674946272 z + 28945 z - 1882380 z 8 10 12 14 + 80858192 z - 2463762388 z + 55752817603 z - 967365129537 z 18 16 50 - 143302260974256 z + 13173468100115 z - 102670800149353890161 z 48 20 + 154321077791363558267 z + 1262332940740928 z 36 34 + 57970739417803783659 z - 27746598889898080148 z 66 80 88 84 86 - 9101675674946272 z + 80858192 z + z + 28945 z - 259 z 82 64 30 - 1882380 z + 54174755335100913 z - 3845404733149015948 z 42 44 - 197007290910386467056 z + 213704363387875735744 z 46 58 - 197007290910386467056 z - 3845404733149015948 z 56 54 + 11239812461385633104 z - 27746598889898080148 z 52 60 70 + 57970739417803783659 z + 1108093971864389697 z - 143302260974256 z 68 78 32 + 1262332940740928 z - 2463762388 z + 11239812461385633104 z 38 40 - 102670800149353890161 z + 154321077791363558267 z 62 76 74 - 268015022880163723 z + 55752817603 z - 967365129537 z 72 / 2 28 + 13173468100115 z ) / ((-1 + z ) (1 + 3658554189919728048 z / 26 2 24 - 851101189185971190 z - 314 z + 164763095040995019 z 22 4 6 8 - 26396052011621668 z + 40806 z - 3005528 z + 143604919 z 10 12 14 - 4803400460 z + 118114592764 z - 2208796076522 z 18 16 50 - 372700659310596 z + 32197360769757 z - 385921203672732620134 z 48 20 + 587196264012558745271 z + 3475219638816082 z 36 34 + 214238529470183714894 z - 100353646392920983584 z 66 80 88 84 86 - 26396052011621668 z + 143604919 z + z + 40806 z - 314 z 82 64 30 - 3005528 z + 164763095040995019 z - 13144501770997530876 z 42 44 - 755169617624211930456 z + 821196739910643292860 z 46 58 - 755169617624211930456 z - 13144501770997530876 z 56 54 + 39606853378539900537 z - 100353646392920983584 z 52 60 + 214238529470183714894 z + 3658554189919728048 z 70 68 78 - 372700659310596 z + 3475219638816082 z - 4803400460 z 32 38 + 39606853378539900537 z - 385921203672732620134 z 40 62 76 + 587196264012558745271 z - 851101189185971190 z + 118114592764 z 74 72 - 2208796076522 z + 32197360769757 z )) And in Maple-input format, it is: -(1+1108093971864389697*z^28-268015022880163723*z^26-259*z^2+54174755335100913* z^24-9101675674946272*z^22+28945*z^4-1882380*z^6+80858192*z^8-2463762388*z^10+ 55752817603*z^12-967365129537*z^14-143302260974256*z^18+13173468100115*z^16-\ 102670800149353890161*z^50+154321077791363558267*z^48+1262332940740928*z^20+ 57970739417803783659*z^36-27746598889898080148*z^34-9101675674946272*z^66+ 80858192*z^80+z^88+28945*z^84-259*z^86-1882380*z^82+54174755335100913*z^64-\ 3845404733149015948*z^30-197007290910386467056*z^42+213704363387875735744*z^44-\ 197007290910386467056*z^46-3845404733149015948*z^58+11239812461385633104*z^56-\ 27746598889898080148*z^54+57970739417803783659*z^52+1108093971864389697*z^60-\ 143302260974256*z^70+1262332940740928*z^68-2463762388*z^78+11239812461385633104 *z^32-102670800149353890161*z^38+154321077791363558267*z^40-268015022880163723* z^62+55752817603*z^76-967365129537*z^74+13173468100115*z^72)/(-1+z^2)/(1+ 3658554189919728048*z^28-851101189185971190*z^26-314*z^2+164763095040995019*z^ 24-26396052011621668*z^22+40806*z^4-3005528*z^6+143604919*z^8-4803400460*z^10+ 118114592764*z^12-2208796076522*z^14-372700659310596*z^18+32197360769757*z^16-\ 385921203672732620134*z^50+587196264012558745271*z^48+3475219638816082*z^20+ 214238529470183714894*z^36-100353646392920983584*z^34-26396052011621668*z^66+ 143604919*z^80+z^88+40806*z^84-314*z^86-3005528*z^82+164763095040995019*z^64-\ 13144501770997530876*z^30-755169617624211930456*z^42+821196739910643292860*z^44 -755169617624211930456*z^46-13144501770997530876*z^58+39606853378539900537*z^56 -100353646392920983584*z^54+214238529470183714894*z^52+3658554189919728048*z^60 -372700659310596*z^70+3475219638816082*z^68-4803400460*z^78+ 39606853378539900537*z^32-385921203672732620134*z^38+587196264012558745271*z^40 -851101189185971190*z^62+118114592764*z^76-2208796076522*z^74+32197360769757*z^ 72) The first , 40, terms are: [0, 56, 0, 5465, 0, 582709, 0, 63674984, 0, 7017899149, 0, 776090161409, 0, 85947951900128, 0, 9524216363770561, 0, 1055707372209915941, 0, 117034042426450333136, 0, 12974944103719774510981, 0, 1438500404417246126634429, 0, 159484905004435048900526976, 0, 17682004888896121783427553197, 0, 1960399079137115154989745338313, 0, 217349155754826598314064979204688, 0, 24097480991783200299894116154353945, 0, 2671686130980496854115966750989934837, 0, 296209696837133885206047184287810302776, 0, 32840754656096878028303111997223810320109] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}, {4, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 5449847773264104 z - 2234102094895647 z - 211 z 24 22 4 6 + 746834832776936 z - 202720864812194 z + 18396 z - 904007 z 8 10 12 14 + 28558392 z - 624695662 z + 9925784838 z - 118414044756 z 18 16 50 - 7800250514870 z + 1086319495191 z - 202720864812194 z 48 20 36 + 746834832776936 z + 44425357839608 z + 26347573198197118 z 34 66 64 - 23884267968479754 z - 904007 z + 28558392 z 30 42 44 - 10874517872158009 z - 10874517872158009 z + 5449847773264104 z 46 58 56 - 2234102094895647 z - 118414044756 z + 1086319495191 z 54 52 60 70 - 7800250514870 z + 44425357839608 z + 9925784838 z - 211 z 68 32 38 + 18396 z + 17787537258419933 z - 23884267968479754 z 40 62 72 / + 17787537258419933 z - 624695662 z + z ) / (-1 / 28 26 2 - 27931115293995819 z + 10468438008004307 z + 271 z 24 22 4 6 - 3201292613877679 z + 794971982694291 z - 28084 z + 1586918 z 8 10 12 14 - 56638393 z + 1384730293 z - 24413768020 z + 321519236488 z 18 16 50 + 25548176619911 z - 3243909213915 z + 3201292613877679 z 48 20 36 - 10468438008004307 z - 159303538403683 z - 195949257086604187 z 34 66 64 + 161395339579073129 z + 56638393 z - 1384730293 z 30 42 44 + 61020937479461165 z + 109435966851643351 z - 61020937479461165 z 46 58 56 + 27931115293995819 z + 3243909213915 z - 25548176619911 z 54 52 60 70 + 159303538403683 z - 794971982694291 z - 321519236488 z + 28084 z 68 32 38 - 1586918 z - 109435966851643351 z + 195949257086604187 z 40 62 74 72 - 161395339579073129 z + 24413768020 z + z - 271 z ) And in Maple-input format, it is: -(1+5449847773264104*z^28-2234102094895647*z^26-211*z^2+746834832776936*z^24-\ 202720864812194*z^22+18396*z^4-904007*z^6+28558392*z^8-624695662*z^10+ 9925784838*z^12-118414044756*z^14-7800250514870*z^18+1086319495191*z^16-\ 202720864812194*z^50+746834832776936*z^48+44425357839608*z^20+26347573198197118 *z^36-23884267968479754*z^34-904007*z^66+28558392*z^64-10874517872158009*z^30-\ 10874517872158009*z^42+5449847773264104*z^44-2234102094895647*z^46-118414044756 *z^58+1086319495191*z^56-7800250514870*z^54+44425357839608*z^52+9925784838*z^60 -211*z^70+18396*z^68+17787537258419933*z^32-23884267968479754*z^38+ 17787537258419933*z^40-624695662*z^62+z^72)/(-1-27931115293995819*z^28+ 10468438008004307*z^26+271*z^2-3201292613877679*z^24+794971982694291*z^22-28084 *z^4+1586918*z^6-56638393*z^8+1384730293*z^10-24413768020*z^12+321519236488*z^ 14+25548176619911*z^18-3243909213915*z^16+3201292613877679*z^50-\ 10468438008004307*z^48-159303538403683*z^20-195949257086604187*z^36+ 161395339579073129*z^34+56638393*z^66-1384730293*z^64+61020937479461165*z^30+ 109435966851643351*z^42-61020937479461165*z^44+27931115293995819*z^46+ 3243909213915*z^58-25548176619911*z^56+159303538403683*z^54-794971982694291*z^ 52-321519236488*z^60+28084*z^70-1586918*z^68-109435966851643351*z^32+ 195949257086604187*z^38-161395339579073129*z^40+24413768020*z^62+z^74-271*z^72) The first , 40, terms are: [0, 60, 0, 6572, 0, 778883, 0, 93644324, 0, 11294417779, 0, 1363271791089, 0, 164584134974224, 0, 19870855611000853, 0, 2399116696392363451, 0, 289659555124885086100, 0, 34972348877228880019111, 0, 4222424272292244081168468, 0, 509798989784112350123328604, 0, 61551137159471389154356860577, 0, 7431443732993943227203045916585, 0, 897243472556559337122716031320796, 0, 108329670293109820072942684354184964, 0, 13079301021583806796474103084016074335, 0, 1579143689406174700866124056758306143476, 0, 190659637521808354236867625406168469385907] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}, {4, 5}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 195487530818032 z - 111475785557378 z - 182 z 24 22 4 6 + 50702883992896 z - 18357900900806 z + 13216 z - 532278 z 8 10 12 14 + 13659584 z - 241229570 z + 3076987224 z - 29281200594 z 18 16 50 - 1196225227094 z + 212592122780 z - 29281200594 z 48 20 36 + 212592122780 z + 5274077782296 z + 195487530818032 z 34 64 30 42 - 273733045399202 z + z - 273733045399202 z - 18357900900806 z 44 46 58 56 + 5274077782296 z - 1196225227094 z - 532278 z + 13659584 z 54 52 60 32 - 241229570 z + 3076987224 z + 13216 z + 306224609931974 z 38 40 62 / 2 - 111475785557378 z + 50702883992896 z - 182 z ) / ((-1 + z ) (1 / 28 26 2 24 + 733806156867764 z - 408892995998734 z - 234 z + 180144020102112 z 22 4 6 8 10 - 62643443368274 z + 20236 z - 940042 z + 27285792 z - 536983574 z 12 14 18 16 + 7538093468 z - 78084170758 z - 3671899244546 z + 611024440636 z 50 48 20 - 78084170758 z + 611024440636 z + 17140940473508 z 36 34 64 30 + 733806156867764 z - 1042031930936718 z + z - 1042031930936718 z 42 44 46 58 - 62643443368274 z + 17140940473508 z - 3671899244546 z - 940042 z 56 54 52 60 + 27285792 z - 536983574 z + 7538093468 z + 20236 z 32 38 40 + 1171218440677254 z - 408892995998734 z + 180144020102112 z 62 - 234 z )) And in Maple-input format, it is: -(1+195487530818032*z^28-111475785557378*z^26-182*z^2+50702883992896*z^24-\ 18357900900806*z^22+13216*z^4-532278*z^6+13659584*z^8-241229570*z^10+3076987224 *z^12-29281200594*z^14-1196225227094*z^18+212592122780*z^16-29281200594*z^50+ 212592122780*z^48+5274077782296*z^20+195487530818032*z^36-273733045399202*z^34+ z^64-273733045399202*z^30-18357900900806*z^42+5274077782296*z^44-1196225227094* z^46-532278*z^58+13659584*z^56-241229570*z^54+3076987224*z^52+13216*z^60+ 306224609931974*z^32-111475785557378*z^38+50702883992896*z^40-182*z^62)/(-1+z^2 )/(1+733806156867764*z^28-408892995998734*z^26-234*z^2+180144020102112*z^24-\ 62643443368274*z^22+20236*z^4-940042*z^6+27285792*z^8-536983574*z^10+7538093468 *z^12-78084170758*z^14-3671899244546*z^18+611024440636*z^16-78084170758*z^50+ 611024440636*z^48+17140940473508*z^20+733806156867764*z^36-1042031930936718*z^ 34+z^64-1042031930936718*z^30-62643443368274*z^42+17140940473508*z^44-\ 3671899244546*z^46-940042*z^58+27285792*z^56-536983574*z^54+7538093468*z^52+ 20236*z^60+1171218440677254*z^32-408892995998734*z^38+180144020102112*z^40-234* z^62) The first , 40, terms are: [0, 53, 0, 5201, 0, 565325, 0, 62715389, 0, 6987390137, 0, 779227453661, 0, 86917141171009, 0, 9695443632718977, 0, 1081520655763882221, 0, 120643267090425690921, 0, 13457724329738639942397, 0, 1501205783793930909184525, 0, 167459135510207614167438561, 0, 18680025480014560200533959653, 0, 2083752263125847604997811801281, 0, 232442054251717355528672850533185, 0, 25928854186111869893964220829038341, 0, 2892357330028547649056303682622880705, 0, 322641751331290837632119178143341893901, 0, 35990608290866360332840670857029348711293] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 134590044332244 z - 79014256735306 z - 190 z 24 22 4 6 + 37350957347648 z - 14161201297054 z + 14188 z - 575278 z 8 10 12 14 + 14583872 z - 250597610 z + 3074779852 z - 27918931914 z 18 16 50 - 1025399045854 z + 192422191708 z - 27918931914 z 48 20 36 + 192422191708 z + 4282666082036 z + 134590044332244 z 34 64 30 42 - 185081332916634 z + z - 185081332916634 z - 14161201297054 z 44 46 58 56 + 4282666082036 z - 1025399045854 z - 575278 z + 14583872 z 54 52 60 32 - 250597610 z + 3074779852 z + 14188 z + 205782361679942 z 38 40 62 / 2 - 79014256735306 z + 37350957347648 z - 190 z ) / ((-1 + z ) (1 / 28 26 2 24 + 534708344292324 z - 306386475754738 z - 262 z + 140107189204144 z 22 4 6 8 10 - 50963277967678 z + 23772 z - 1112982 z + 31696720 z - 601483834 z 12 14 18 16 + 8053972716 z - 79081801754 z - 3320645160766 z + 584864402428 z 50 48 20 - 79081801754 z + 584864402428 z + 14673345225812 z 36 34 64 30 + 534708344292324 z - 746280008115106 z + z - 746280008115106 z 42 44 46 - 50963277967678 z + 14673345225812 z - 3320645160766 z 58 56 54 52 60 - 1112982 z + 31696720 z - 601483834 z + 8053972716 z + 23772 z 32 38 40 + 833896535742470 z - 306386475754738 z + 140107189204144 z 62 - 262 z )) And in Maple-input format, it is: -(1+134590044332244*z^28-79014256735306*z^26-190*z^2+37350957347648*z^24-\ 14161201297054*z^22+14188*z^4-575278*z^6+14583872*z^8-250597610*z^10+3074779852 *z^12-27918931914*z^14-1025399045854*z^18+192422191708*z^16-27918931914*z^50+ 192422191708*z^48+4282666082036*z^20+134590044332244*z^36-185081332916634*z^34+ z^64-185081332916634*z^30-14161201297054*z^42+4282666082036*z^44-1025399045854* z^46-575278*z^58+14583872*z^56-250597610*z^54+3074779852*z^52+14188*z^60+ 205782361679942*z^32-79014256735306*z^38+37350957347648*z^40-190*z^62)/(-1+z^2) /(1+534708344292324*z^28-306386475754738*z^26-262*z^2+140107189204144*z^24-\ 50963277967678*z^22+23772*z^4-1112982*z^6+31696720*z^8-601483834*z^10+ 8053972716*z^12-79081801754*z^14-3320645160766*z^18+584864402428*z^16-\ 79081801754*z^50+584864402428*z^48+14673345225812*z^20+534708344292324*z^36-\ 746280008115106*z^34+z^64-746280008115106*z^30-50963277967678*z^42+ 14673345225812*z^44-3320645160766*z^46-1112982*z^58+31696720*z^56-601483834*z^ 54+8053972716*z^52+23772*z^60+833896535742470*z^32-306386475754738*z^38+ 140107189204144*z^40-262*z^62) The first , 40, terms are: [0, 73, 0, 9353, 0, 1266833, 0, 173144289, 0, 23709418545, 0, 3248077076529, 0, 445019661936705, 0, 60973919398843649, 0, 8354339870686614913, 0, 1144671794865647012129, 0, 156837546497221919967585, 0, 21489145921183898424726609, 0, 2944342212365382967165480985, 0, 403419995221154669003564379065, 0, 55274720536614757284096135390273, 0, 7573483636524568818668831820843969, 0, 1037683299819507488146754867455218777, 0, 142178511561480460102211764406391730937, 0, 19480634557399030065521408088219672434769, 0, 2669145418617965990480116420344393601436641] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1169781387304741533 z - 283177763990742265 z - 261 z 24 22 4 6 + 57276608383793934 z - 9625645947228202 z + 29434 z - 1931358 z 8 10 12 14 + 83634503 z - 2565842146 z + 58380995371 z - 1017139834234 z 18 16 50 - 151380854313890 z + 13891124880295 z - 107928772540876642182 z 48 20 + 162147921730402437988 z + 1334712919425607 z 36 34 + 60978481507073824448 z - 29209116014912360272 z 66 80 88 84 86 - 9625645947228202 z + 83634503 z + z + 29434 z - 261 z 82 64 30 - 1931358 z + 57276608383793934 z - 4055612447568992142 z 42 44 - 206939668691215047090 z + 224456884004679123108 z 46 58 - 206939668691215047090 z - 4055612447568992142 z 56 54 + 11842873607477409630 z - 29209116014912360272 z 52 60 70 + 60978481507073824448 z + 1169781387304741533 z - 151380854313890 z 68 78 32 + 1334712919425607 z - 2565842146 z + 11842873607477409630 z 38 40 - 107928772540876642182 z + 162147921730402437988 z 62 76 74 - 283177763990742265 z + 58380995371 z - 1017139834234 z 72 / 28 + 13891124880295 z ) / (-1 - 4810256112827614046 z / 26 2 24 + 1081668761707720181 z + 325 z - 203193736309749042 z 22 4 6 8 + 31700748782050849 z - 42998 z + 3203681 z - 154543578 z 10 12 14 + 5219824129 z - 129772086602 z + 2458151350001 z 18 16 50 + 428523656381529 z - 36375833179786 z + 1047096069018332056607 z 48 20 - 1445383905114305448251 z - 4077620164267342 z 36 34 - 337763100824096737451 z + 150051797344476984211 z 66 80 90 88 84 + 203193736309749042 z - 5219824129 z + z - 325 z - 3203681 z 86 82 64 + 42998 z + 154543578 z - 1081668761707720181 z 30 42 + 17954787329577588902 z + 1445383905114305448251 z 44 46 - 1697923304589108403667 z + 1697923304589108403667 z 58 56 + 56463858913337425883 z - 150051797344476984211 z 54 52 + 337763100824096737451 z - 645158625529374038803 z 60 70 68 - 17954787329577588902 z + 4077620164267342 z - 31700748782050849 z 78 32 38 + 129772086602 z - 56463858913337425883 z + 645158625529374038803 z 40 62 76 - 1047096069018332056607 z + 4810256112827614046 z - 2458151350001 z 74 72 + 36375833179786 z - 428523656381529 z ) And in Maple-input format, it is: -(1+1169781387304741533*z^28-283177763990742265*z^26-261*z^2+57276608383793934* z^24-9625645947228202*z^22+29434*z^4-1931358*z^6+83634503*z^8-2565842146*z^10+ 58380995371*z^12-1017139834234*z^14-151380854313890*z^18+13891124880295*z^16-\ 107928772540876642182*z^50+162147921730402437988*z^48+1334712919425607*z^20+ 60978481507073824448*z^36-29209116014912360272*z^34-9625645947228202*z^66+ 83634503*z^80+z^88+29434*z^84-261*z^86-1931358*z^82+57276608383793934*z^64-\ 4055612447568992142*z^30-206939668691215047090*z^42+224456884004679123108*z^44-\ 206939668691215047090*z^46-4055612447568992142*z^58+11842873607477409630*z^56-\ 29209116014912360272*z^54+60978481507073824448*z^52+1169781387304741533*z^60-\ 151380854313890*z^70+1334712919425607*z^68-2565842146*z^78+11842873607477409630 *z^32-107928772540876642182*z^38+162147921730402437988*z^40-283177763990742265* z^62+58380995371*z^76-1017139834234*z^74+13891124880295*z^72)/(-1-\ 4810256112827614046*z^28+1081668761707720181*z^26+325*z^2-203193736309749042*z^ 24+31700748782050849*z^22-42998*z^4+3203681*z^6-154543578*z^8+5219824129*z^10-\ 129772086602*z^12+2458151350001*z^14+428523656381529*z^18-36375833179786*z^16+ 1047096069018332056607*z^50-1445383905114305448251*z^48-4077620164267342*z^20-\ 337763100824096737451*z^36+150051797344476984211*z^34+203193736309749042*z^66-\ 5219824129*z^80+z^90-325*z^88-3203681*z^84+42998*z^86+154543578*z^82-\ 1081668761707720181*z^64+17954787329577588902*z^30+1445383905114305448251*z^42-\ 1697923304589108403667*z^44+1697923304589108403667*z^46+56463858913337425883*z^ 58-150051797344476984211*z^56+337763100824096737451*z^54-645158625529374038803* z^52-17954787329577588902*z^60+4077620164267342*z^70-31700748782050849*z^68+ 129772086602*z^78-56463858913337425883*z^32+645158625529374038803*z^38-\ 1047096069018332056607*z^40+4810256112827614046*z^62-2458151350001*z^76+ 36375833179786*z^74-428523656381529*z^72) The first , 40, terms are: [0, 64, 0, 7236, 0, 872151, 0, 106442056, 0, 13037948209, 0, 1599031554485, 0, 196206859799584, 0, 24079842420739711, 0, 2955466302378717619, 0, 362753550091287321996, 0, 44524876330736062603297, 0, 5465072980172484060799052, 0, 670795443222043742401581924, 0, 82335022066444117483436671943, 0, 10105999153482418550543293049423, 0, 1240434879495049687213434617713148, 0, 152253998320282518704797220878508300, 0, 18688027020428526941771688029891854441, 0, 2293814028498102645912198579380542453428, 0, 281548331027363974432456759482468990612483] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 5}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1043341570582272441 z - 252383043924779329 z - 257 z 24 22 4 6 + 51017841165726865 z - 8572270697829356 z + 28433 z - 1830186 z 8 10 12 14 + 77898750 z - 2355867050 z + 53008267771 z - 916034853507 z 18 16 50 - 135125470363500 z + 12441424716723 z - 96452510848503454523 z 48 20 + 144906773651727236795 z + 1189325514660700 z 36 34 + 54490043672683234875 z - 26095449491181157526 z 66 80 88 84 86 - 8572270697829356 z + 77898750 z + z + 28433 z - 257 z 82 64 30 - 1830186 z + 51017841165726865 z - 3619807763244862486 z 42 44 - 184931323518049773864 z + 200583003283607139400 z 46 58 - 184931323518049773864 z - 3619807763244862486 z 56 54 + 10576302781521524354 z - 26095449491181157526 z 52 60 70 + 54490043672683234875 z + 1043341570582272441 z - 135125470363500 z 68 78 32 + 1189325514660700 z - 2355867050 z + 10576302781521524354 z 38 40 - 96452510848503454523 z + 144906773651727236795 z 62 76 74 - 252383043924779329 z + 53008267771 z - 916034853507 z 72 / 28 + 12441424716723 z ) / (-1 - 4228886974960030874 z / 26 2 24 + 954374325991129645 z + 311 z - 179886810766232807 z 22 4 6 8 + 28151591097798250 z - 40144 z + 2945954 z - 140729111 z 10 12 14 + 4722539755 z - 116883852848 z + 2206642119882 z 18 16 50 + 382629081592921 z - 32563766050891 z + 902312973426603998033 z 48 20 - 1243448324158211747263 z - 3631350934572826 z 36 34 - 292625756149093535058 z + 130443418453523627857 z 66 80 90 88 84 + 179886810766232807 z - 4722539755 z + z - 311 z - 2945954 z 86 82 64 + 40144 z + 140729111 z - 954374325991129645 z 30 42 + 15725552973619170780 z + 1243448324158211747263 z 44 46 - 1459448328457446559100 z + 1459448328457446559100 z 58 56 + 49266274938059919277 z - 130443418453523627857 z 54 52 + 292625756149093535058 z - 557281448613014716620 z 60 70 68 - 15725552973619170780 z + 3631350934572826 z - 28151591097798250 z 78 32 38 + 116883852848 z - 49266274938059919277 z + 557281448613014716620 z 40 62 76 - 902312973426603998033 z + 4228886974960030874 z - 2206642119882 z 74 72 + 32563766050891 z - 382629081592921 z ) And in Maple-input format, it is: -(1+1043341570582272441*z^28-252383043924779329*z^26-257*z^2+51017841165726865* z^24-8572270697829356*z^22+28433*z^4-1830186*z^6+77898750*z^8-2355867050*z^10+ 53008267771*z^12-916034853507*z^14-135125470363500*z^18+12441424716723*z^16-\ 96452510848503454523*z^50+144906773651727236795*z^48+1189325514660700*z^20+ 54490043672683234875*z^36-26095449491181157526*z^34-8572270697829356*z^66+ 77898750*z^80+z^88+28433*z^84-257*z^86-1830186*z^82+51017841165726865*z^64-\ 3619807763244862486*z^30-184931323518049773864*z^42+200583003283607139400*z^44-\ 184931323518049773864*z^46-3619807763244862486*z^58+10576302781521524354*z^56-\ 26095449491181157526*z^54+54490043672683234875*z^52+1043341570582272441*z^60-\ 135125470363500*z^70+1189325514660700*z^68-2355867050*z^78+10576302781521524354 *z^32-96452510848503454523*z^38+144906773651727236795*z^40-252383043924779329*z ^62+53008267771*z^76-916034853507*z^74+12441424716723*z^72)/(-1-\ 4228886974960030874*z^28+954374325991129645*z^26+311*z^2-179886810766232807*z^ 24+28151591097798250*z^22-40144*z^4+2945954*z^6-140729111*z^8+4722539755*z^10-\ 116883852848*z^12+2206642119882*z^14+382629081592921*z^18-32563766050891*z^16+ 902312973426603998033*z^50-1243448324158211747263*z^48-3631350934572826*z^20-\ 292625756149093535058*z^36+130443418453523627857*z^34+179886810766232807*z^66-\ 4722539755*z^80+z^90-311*z^88-2945954*z^84+40144*z^86+140729111*z^82-\ 954374325991129645*z^64+15725552973619170780*z^30+1243448324158211747263*z^42-\ 1459448328457446559100*z^44+1459448328457446559100*z^46+49266274938059919277*z^ 58-130443418453523627857*z^56+292625756149093535058*z^54-557281448613014716620* z^52-15725552973619170780*z^60+3631350934572826*z^70-28151591097798250*z^68+ 116883852848*z^78-49266274938059919277*z^32+557281448613014716620*z^38-\ 902312973426603998033*z^40+4228886974960030874*z^62-2206642119882*z^76+ 32563766050891*z^74-382629081592921*z^72) The first , 40, terms are: [0, 54, 0, 5083, 0, 528805, 0, 56657558, 0, 6133737511, 0, 666782053019, 0, 72612310712550, 0, 7913689026173721, 0, 862786741495345599, 0, 94080473257882264342, 0, 10259555482028219811541, 0, 1118852585405303210809485, 0, 122018102992906890046852598, 0, 13306962883945364171488813687, 0, 1451226304292918916918751002225, 0, 158267609237439472321969410208230, 0, 17260336925974700920313851537799059, 0, 1882377171764634591228737818081074479, 0, 205288251597969797222555745515721114838, 0, 22388323676453322320137426291102477024141] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 5}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 141633091 z - 530599372 z - 148 z + 1346362967 z 22 4 6 8 10 - 2343089604 z + 7781 z - 199182 z + 2861424 z - 25102638 z 12 14 18 16 + 141633091 z - 530599372 z - 2343089604 z + 1346362967 z 20 36 34 30 32 + 2816284128 z + 7781 z - 199182 z - 25102638 z + 2861424 z 38 40 / 28 8 14 - 148 z + z ) / (-2322601616 z - 7564908 z + 2322601616 z / 10 12 6 2 4 + 78482108 z - 1 - 523157813 z + 441255 z + 212 z - 14120 z 16 20 36 34 - 7013448244 z - 20991014808 z - 441255 z + 7564908 z 30 42 32 38 40 + 523157813 z + z - 78482108 z + 14120 z - 212 z 26 24 22 18 + 7013448244 z - 14578273635 z + 20991014808 z + 14578273635 z ) And in Maple-input format, it is: -(1+141633091*z^28-530599372*z^26-148*z^2+1346362967*z^24-2343089604*z^22+7781* z^4-199182*z^6+2861424*z^8-25102638*z^10+141633091*z^12-530599372*z^14-\ 2343089604*z^18+1346362967*z^16+2816284128*z^20+7781*z^36-199182*z^34-25102638* z^30+2861424*z^32-148*z^38+z^40)/(-2322601616*z^28-7564908*z^8+2322601616*z^14+ 78482108*z^10-1-523157813*z^12+441255*z^6+212*z^2-14120*z^4-7013448244*z^16-\ 20991014808*z^20-441255*z^36+7564908*z^34+523157813*z^30+z^42-78482108*z^32+ 14120*z^38-212*z^40+7013448244*z^26-14578273635*z^24+20991014808*z^22+ 14578273635*z^18) The first , 40, terms are: [0, 64, 0, 7229, 0, 870941, 0, 106102848, 0, 12955174609, 0, 1582586484561, 0, 193346749964608, 0, 23621981476143485, 0, 2886011347694706973, 0, 352598340208518698816, 0, 43078702508424474561985, 0, 5263140864761079718113985, 0, 643024292611506440419386688, 0, 78561500278789398311002704861, 0, 9598252201331920632417564581693, 0, 1172666573524215835208306357252928, 0, 143270552169375081026118041329581777, 0, 17504081366923431207085837791322177617, 0, 2138561343289054234884965838036096697920, 0, 261278756830803685329305859376034618651997] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1460725619287348347 z - 339393933578942979 z - 251 z 24 22 4 6 + 65805010716926417 z - 10599507349539840 z + 27451 z - 1768538 z 8 10 12 14 + 76136916 z - 2348674262 z + 54264443105 z - 967943049193 z 18 16 50 - 153816811073888 z + 13624550360483 z - 157247299121149580633 z 48 20 + 239983774292019567947 z + 1410181866953680 z 36 34 + 86952071821162410025 z - 40547854316885326126 z 66 80 88 84 86 - 10599507349539840 z + 76136916 z + z + 27451 z - 251 z 82 64 30 - 1768538 z + 65805010716926417 z - 5264455517730802850 z 42 44 - 309221781221213792032 z + 336476538033088535520 z 46 58 - 309221781221213792032 z - 5264455517730802850 z 56 54 + 15929468786555357868 z - 40547854316885326126 z 52 60 70 + 86952071821162410025 z + 1460725619287348347 z - 153816811073888 z 68 78 32 + 1410181866953680 z - 2348674262 z + 15929468786555357868 z 38 40 - 157247299121149580633 z + 239983774292019567947 z 62 76 74 - 339393933578942979 z + 54264443105 z - 967943049193 z 72 / 28 + 13624550360483 z ) / (-1 - 5850674775600440248 z / 26 2 24 + 1260111459783406707 z + 313 z - 226530654825507739 z 22 4 6 8 + 33830323635258738 z - 39806 z + 2887658 z - 137607895 z 10 12 14 + 4652613575 z - 117109607304 z + 2267049726292 z 18 16 50 + 421486779718701 z - 34544372354997 z + 1537623132579133880639 z 48 20 - 2150386418154447232215 z - 4171719404885618 z 36 34 - 474570922772161593690 z + 204575861376675598233 z 66 80 90 88 84 + 226530654825507739 z - 4652613575 z + z - 313 z - 2887658 z 86 82 64 + 39806 z + 137607895 z - 1260111459783406707 z 30 42 + 22756864097237410132 z + 2150386418154447232215 z 44 46 - 2542839578043402415548 z + 2542839578043402415548 z 58 56 + 74362365077619880937 z - 204575861376675598233 z 54 52 + 474570922772161593690 z - 929364264041218638486 z 60 70 68 - 22756864097237410132 z + 4171719404885618 z - 33830323635258738 z 78 32 38 + 117109607304 z - 74362365077619880937 z + 929364264041218638486 z 40 62 76 - 1537623132579133880639 z + 5850674775600440248 z - 2267049726292 z 74 72 + 34544372354997 z - 421486779718701 z ) And in Maple-input format, it is: -(1+1460725619287348347*z^28-339393933578942979*z^26-251*z^2+65805010716926417* z^24-10599507349539840*z^22+27451*z^4-1768538*z^6+76136916*z^8-2348674262*z^10+ 54264443105*z^12-967943049193*z^14-153816811073888*z^18+13624550360483*z^16-\ 157247299121149580633*z^50+239983774292019567947*z^48+1410181866953680*z^20+ 86952071821162410025*z^36-40547854316885326126*z^34-10599507349539840*z^66+ 76136916*z^80+z^88+27451*z^84-251*z^86-1768538*z^82+65805010716926417*z^64-\ 5264455517730802850*z^30-309221781221213792032*z^42+336476538033088535520*z^44-\ 309221781221213792032*z^46-5264455517730802850*z^58+15929468786555357868*z^56-\ 40547854316885326126*z^54+86952071821162410025*z^52+1460725619287348347*z^60-\ 153816811073888*z^70+1410181866953680*z^68-2348674262*z^78+15929468786555357868 *z^32-157247299121149580633*z^38+239983774292019567947*z^40-339393933578942979* z^62+54264443105*z^76-967943049193*z^74+13624550360483*z^72)/(-1-\ 5850674775600440248*z^28+1260111459783406707*z^26+313*z^2-226530654825507739*z^ 24+33830323635258738*z^22-39806*z^4+2887658*z^6-137607895*z^8+4652613575*z^10-\ 117109607304*z^12+2267049726292*z^14+421486779718701*z^18-34544372354997*z^16+ 1537623132579133880639*z^50-2150386418154447232215*z^48-4171719404885618*z^20-\ 474570922772161593690*z^36+204575861376675598233*z^34+226530654825507739*z^66-\ 4652613575*z^80+z^90-313*z^88-2887658*z^84+39806*z^86+137607895*z^82-\ 1260111459783406707*z^64+22756864097237410132*z^30+2150386418154447232215*z^42-\ 2542839578043402415548*z^44+2542839578043402415548*z^46+74362365077619880937*z^ 58-204575861376675598233*z^56+474570922772161593690*z^54-929364264041218638486* z^52-22756864097237410132*z^60+4171719404885618*z^70-33830323635258738*z^68+ 117109607304*z^78-74362365077619880937*z^32+929364264041218638486*z^38-\ 1537623132579133880639*z^40+5850674775600440248*z^62-2267049726292*z^76+ 34544372354997*z^74-421486779718701*z^72) The first , 40, terms are: [0, 62, 0, 7051, 0, 858111, 0, 105480454, 0, 12990542017, 0, 1600559403241, 0, 197226095477694, 0, 24303555927640743, 0, 2994875708589180755, 0, 369053027814945016262, 0, 45477755263918492753881, 0, 5604144778131878660103225, 0, 690589061174916968466636886, 0, 85100095964055134693698660307, 0, 10486737729562792202880421086119, 0, 1292262565566076448879959762686574, 0, 159243282515641119093605096947309545, 0, 19623274483494126381790476188293237793, 0, 2418142199672186910082128444991863305238, 0, 297983483989948255944241668538279131999327] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 12447672629404540 z - 5705750013938610 z - 255 z 24 22 4 6 + 2078612751356046 z - 598674767851264 z + 26962 z - 1582523 z 8 10 12 14 + 58488461 z - 1462056428 z + 25881018397 z - 334880325803 z 18 16 50 - 23859756793167 z + 3240700886834 z - 23859756793167 z 48 20 36 + 135420778222449 z + 135420778222449 z + 30185698826065750 z 34 66 64 30 - 33705945486458408 z - 255 z + 26962 z - 21669078000483738 z 42 44 46 - 5705750013938610 z + 2078612751356046 z - 598674767851264 z 58 56 54 52 - 1462056428 z + 25881018397 z - 334880325803 z + 3240700886834 z 60 68 32 38 + 58488461 z + z + 30185698826065750 z - 21669078000483738 z 40 62 / 28 + 12447672629404540 z - 1582523 z ) / (-1 - 67073229374018338 z / 26 2 24 + 27943596254454198 z + 319 z - 9274237620941194 z 22 4 6 8 + 2437565200165237 z - 40405 z + 2770235 z - 117411329 z 10 12 14 18 + 3316982679 z - 65608623843 z + 940563424885 z + 80998367505417 z 16 50 48 - 10025201995335 z + 503561149188875 z - 2437565200165237 z 20 36 34 - 503561149188875 z - 246696277949153414 z + 246696277949153414 z 66 64 30 42 + 40405 z - 2770235 z + 128844823171101054 z + 67073229374018338 z 44 46 58 - 27943596254454198 z + 9274237620941194 z + 65608623843 z 56 54 52 - 940563424885 z + 10025201995335 z - 80998367505417 z 60 70 68 32 - 3316982679 z + z - 319 z - 198741095448105354 z 38 40 62 + 198741095448105354 z - 128844823171101054 z + 117411329 z ) And in Maple-input format, it is: -(1+12447672629404540*z^28-5705750013938610*z^26-255*z^2+2078612751356046*z^24-\ 598674767851264*z^22+26962*z^4-1582523*z^6+58488461*z^8-1462056428*z^10+ 25881018397*z^12-334880325803*z^14-23859756793167*z^18+3240700886834*z^16-\ 23859756793167*z^50+135420778222449*z^48+135420778222449*z^20+30185698826065750 *z^36-33705945486458408*z^34-255*z^66+26962*z^64-21669078000483738*z^30-\ 5705750013938610*z^42+2078612751356046*z^44-598674767851264*z^46-1462056428*z^ 58+25881018397*z^56-334880325803*z^54+3240700886834*z^52+58488461*z^60+z^68+ 30185698826065750*z^32-21669078000483738*z^38+12447672629404540*z^40-1582523*z^ 62)/(-1-67073229374018338*z^28+27943596254454198*z^26+319*z^2-9274237620941194* z^24+2437565200165237*z^22-40405*z^4+2770235*z^6-117411329*z^8+3316982679*z^10-\ 65608623843*z^12+940563424885*z^14+80998367505417*z^18-10025201995335*z^16+ 503561149188875*z^50-2437565200165237*z^48-503561149188875*z^20-\ 246696277949153414*z^36+246696277949153414*z^34+40405*z^66-2770235*z^64+ 128844823171101054*z^30+67073229374018338*z^42-27943596254454198*z^44+ 9274237620941194*z^46+65608623843*z^58-940563424885*z^56+10025201995335*z^54-\ 80998367505417*z^52-3316982679*z^60+z^70-319*z^68-198741095448105354*z^32+ 198741095448105354*z^38-128844823171101054*z^40+117411329*z^62) The first , 40, terms are: [0, 64, 0, 6973, 0, 826179, 0, 100179208, 0, 12232854707, 0, 1497099823251, 0, 183359574466520, 0, 22463438262942379, 0, 2752298104748522933, 0, 337236034254011951184, 0, 41321951697692181309073, 0, 5063273916302447807591121, 0, 620417069548187553022887472, 0, 76021573287200389688156074661, 0, 9315160539676251359054338280699, 0, 1141416147751577498651380318042424, 0, 139861365669042739527936573864771059, 0, 17137661353878076723374964146058482067, 0, 2099932659024170590653447249280032794536, 0, 257311495013618789525226608796467498298931] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1168619034904923917 z - 285315123965142899 z - 267 z 24 22 4 6 + 58203831377957133 z - 9863553868875908 z + 30641 z - 2032400 z 8 10 12 14 + 88475336 z - 2717512948 z + 61721539503 z - 1071155778265 z 18 16 50 - 157521865963956 z + 14549639436367 z - 104178383323777168105 z 48 20 + 155929601457421064099 z + 1378647896681728 z 36 34 + 59157051415719527443 z - 28512033783877195640 z 66 80 88 84 86 - 9863553868875908 z + 88475336 z + z + 30641 z - 267 z 82 64 30 - 2032400 z + 58203831377957133 z - 4018219015602227972 z 42 44 - 198546595464292614344 z + 215186295443230295232 z 46 58 - 198546595464292614344 z - 4018219015602227972 z 56 54 + 11642552140811155992 z - 28512033783877195640 z 52 60 70 + 59157051415719527443 z + 1168619034904923917 z - 157521865963956 z 68 78 32 + 1378647896681728 z - 2717512948 z + 11642552140811155992 z 38 40 - 104178383323777168105 z + 155929601457421064099 z 62 76 74 - 285315123965142899 z + 61721539503 z - 1071155778265 z 72 / 28 + 14549639436367 z ) / (-1 - 4919719442315164182 z / 26 2 24 + 1114928413895870429 z + 343 z - 211159736173149495 z 22 4 6 8 + 33221565187235142 z - 46496 z + 3499958 z - 169287451 z 10 12 14 + 5707966263 z - 141281912600 z + 2659786990006 z 18 16 50 + 456684887968009 z - 39075370530427 z + 1037087947560935453489 z 48 20 - 1428520508194134191903 z - 4309566317686118 z 36 34 - 336982173138035563686 z + 150462145106725353533 z 66 80 90 88 84 + 211159736173149495 z - 5707966263 z + z - 343 z - 3499958 z 86 82 64 + 46496 z + 169287451 z - 1114928413895870429 z 30 42 + 18230461415105467116 z + 1428520508194134191903 z 44 46 - 1676311929487694982324 z + 1676311929487694982324 z 58 56 + 56952228223121805521 z - 150462145106725353533 z 54 52 + 336982173138035563686 z - 641012351915691369588 z 60 70 68 - 18230461415105467116 z + 4309566317686118 z - 33221565187235142 z 78 32 38 + 141281912600 z - 56952228223121805521 z + 641012351915691369588 z 40 62 76 - 1037087947560935453489 z + 4919719442315164182 z - 2659786990006 z 74 72 + 39075370530427 z - 456684887968009 z ) And in Maple-input format, it is: -(1+1168619034904923917*z^28-285315123965142899*z^26-267*z^2+58203831377957133* z^24-9863553868875908*z^22+30641*z^4-2032400*z^6+88475336*z^8-2717512948*z^10+ 61721539503*z^12-1071155778265*z^14-157521865963956*z^18+14549639436367*z^16-\ 104178383323777168105*z^50+155929601457421064099*z^48+1378647896681728*z^20+ 59157051415719527443*z^36-28512033783877195640*z^34-9863553868875908*z^66+ 88475336*z^80+z^88+30641*z^84-267*z^86-2032400*z^82+58203831377957133*z^64-\ 4018219015602227972*z^30-198546595464292614344*z^42+215186295443230295232*z^44-\ 198546595464292614344*z^46-4018219015602227972*z^58+11642552140811155992*z^56-\ 28512033783877195640*z^54+59157051415719527443*z^52+1168619034904923917*z^60-\ 157521865963956*z^70+1378647896681728*z^68-2717512948*z^78+11642552140811155992 *z^32-104178383323777168105*z^38+155929601457421064099*z^40-285315123965142899* z^62+61721539503*z^76-1071155778265*z^74+14549639436367*z^72)/(-1-\ 4919719442315164182*z^28+1114928413895870429*z^26+343*z^2-211159736173149495*z^ 24+33221565187235142*z^22-46496*z^4+3499958*z^6-169287451*z^8+5707966263*z^10-\ 141281912600*z^12+2659786990006*z^14+456684887968009*z^18-39075370530427*z^16+ 1037087947560935453489*z^50-1428520508194134191903*z^48-4309566317686118*z^20-\ 336982173138035563686*z^36+150462145106725353533*z^34+211159736173149495*z^66-\ 5707966263*z^80+z^90-343*z^88-3499958*z^84+46496*z^86+169287451*z^82-\ 1114928413895870429*z^64+18230461415105467116*z^30+1428520508194134191903*z^42-\ 1676311929487694982324*z^44+1676311929487694982324*z^46+56952228223121805521*z^ 58-150462145106725353533*z^56+336982173138035563686*z^54-641012351915691369588* z^52-18230461415105467116*z^60+4309566317686118*z^70-33221565187235142*z^68+ 141281912600*z^78-56952228223121805521*z^32+641012351915691369588*z^38-\ 1037087947560935453489*z^40+4919719442315164182*z^62-2659786990006*z^76+ 39075370530427*z^74-456684887968009*z^72) The first , 40, terms are: [0, 76, 0, 10213, 0, 1436921, 0, 203184948, 0, 28751036441, 0, 4068793632201, 0, 575820781636180, 0, 81491259947232793, 0, 11532811970199895989, 0, 1632147924007334621740, 0, 230985043175067128300657, 0, 32689495109469800574764305, 0, 4626286988787507971497486092, 0, 654721992343682849068648018293, 0, 92657651535390576431560112821049, 0, 13113108298897100545107294975278068, 0, 1855795030765085567349150686647183593, 0, 262636067504233502792885703127050823801, 0, 37168815958549374558996573772653430092244, 0, 5260210042350775072725570437071585953990681] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 15523792037824 z - 13617227127027 z - 189 z 24 22 4 6 + 9183620727415 z - 4750192589949 z + 14031 z - 563332 z 8 10 12 14 + 13952522 z - 229106461 z + 2609970538 z - 21276468902 z 18 16 50 48 - 562567736034 z + 126886251211 z - 563332 z + 13952522 z 20 36 34 + 1876467040368 z + 1876467040368 z - 4750192589949 z 30 42 44 46 - 13617227127027 z - 21276468902 z + 2609970538 z - 229106461 z 56 54 52 32 38 + z - 189 z + 14031 z + 9183620727415 z - 562567736034 z 40 / 2 28 + 126886251211 z ) / ((-1 + z ) (1 + 60724288528575 z / 26 2 24 22 - 52881149531026 z - 250 z + 34908561891585 z - 17443028780576 z 4 6 8 10 12 + 22487 z - 1052194 z + 29676185 z - 545501404 z + 6860142229 z 14 18 16 50 - 61012006590 z - 1862563215642 z + 392872789563 z - 1052194 z 48 20 36 + 29676185 z + 6577855367793 z + 6577855367793 z 34 30 42 - 17443028780576 z - 52881149531026 z - 61012006590 z 44 46 56 54 52 + 6860142229 z - 545501404 z + z - 250 z + 22487 z 32 38 40 + 34908561891585 z - 1862563215642 z + 392872789563 z )) And in Maple-input format, it is: -(1+15523792037824*z^28-13617227127027*z^26-189*z^2+9183620727415*z^24-\ 4750192589949*z^22+14031*z^4-563332*z^6+13952522*z^8-229106461*z^10+2609970538* z^12-21276468902*z^14-562567736034*z^18+126886251211*z^16-563332*z^50+13952522* z^48+1876467040368*z^20+1876467040368*z^36-4750192589949*z^34-13617227127027*z^ 30-21276468902*z^42+2609970538*z^44-229106461*z^46+z^56-189*z^54+14031*z^52+ 9183620727415*z^32-562567736034*z^38+126886251211*z^40)/(-1+z^2)/(1+ 60724288528575*z^28-52881149531026*z^26-250*z^2+34908561891585*z^24-\ 17443028780576*z^22+22487*z^4-1052194*z^6+29676185*z^8-545501404*z^10+ 6860142229*z^12-61012006590*z^14-1862563215642*z^18+392872789563*z^16-1052194*z ^50+29676185*z^48+6577855367793*z^20+6577855367793*z^36-17443028780576*z^34-\ 52881149531026*z^30-61012006590*z^42+6860142229*z^44-545501404*z^46+z^56-250*z^ 54+22487*z^52+34908561891585*z^32-1862563215642*z^38+392872789563*z^40) The first , 40, terms are: [0, 62, 0, 6856, 0, 822511, 0, 100419754, 0, 12312850213, 0, 1511409971755, 0, 185582262851184, 0, 22789067243452003, 0, 2798508471706103965, 0, 343660407485879809522, 0, 42202012524710476519263, 0, 5182472960439927513115400, 0, 636415854106010274959052118, 0, 78152873901072559051280336273, 0, 9597296688032862663141957443601, 0, 1178563236609598823347250365629094, 0, 144729432579029799238784464294590088, 0, 17773003610975741179671234342838448383, 0, 2182553000794444113137876438002171508258, 0, 268020966272266251206190434290692644289229] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3253112310663964 z - 1507308950440970 z - 215 z 24 22 4 6 + 558924188408766 z - 165388949858624 z + 19018 z - 941707 z 8 10 12 14 + 29774689 z - 647269220 z + 10142211825 z - 118295887539 z 18 16 50 - 7222952087775 z + 1050671090906 z - 7222952087775 z 48 20 36 + 38889612260865 z + 38889612260865 z + 7819453057535022 z 34 66 64 30 - 8724122310107960 z - 215 z + 19018 z - 5629339100329026 z 42 44 46 - 1507308950440970 z + 558924188408766 z - 165388949858624 z 58 56 54 52 - 647269220 z + 10142211825 z - 118295887539 z + 1050671090906 z 60 68 32 38 + 29774689 z + z + 7819453057535022 z - 5629339100329026 z 40 62 / 28 + 3253112310663964 z - 941707 z ) / (-1 - 18139570224656434 z / 26 2 24 + 7562658001204946 z + 285 z - 2527049898903754 z 22 4 6 8 + 674978054669397 z - 30205 z + 1713759 z - 60760979 z 10 12 14 18 + 1465413859 z - 25345900063 z + 325680788511 z + 24145613702257 z 16 50 48 - 3186072524891 z + 143520518917585 z - 674978054669397 z 20 36 34 - 143520518917585 z - 67211672444297886 z + 67211672444297886 z 66 64 30 42 + 30205 z - 1713759 z + 34928109059843686 z + 18139570224656434 z 44 46 58 - 7562658001204946 z + 2527049898903754 z + 25345900063 z 56 54 52 - 325680788511 z + 3186072524891 z - 24145613702257 z 60 70 68 32 - 1465413859 z + z - 285 z - 54039911079095310 z 38 40 62 + 54039911079095310 z - 34928109059843686 z + 60760979 z ) And in Maple-input format, it is: -(1+3253112310663964*z^28-1507308950440970*z^26-215*z^2+558924188408766*z^24-\ 165388949858624*z^22+19018*z^4-941707*z^6+29774689*z^8-647269220*z^10+ 10142211825*z^12-118295887539*z^14-7222952087775*z^18+1050671090906*z^16-\ 7222952087775*z^50+38889612260865*z^48+38889612260865*z^20+7819453057535022*z^ 36-8724122310107960*z^34-215*z^66+19018*z^64-5629339100329026*z^30-\ 1507308950440970*z^42+558924188408766*z^44-165388949858624*z^46-647269220*z^58+ 10142211825*z^56-118295887539*z^54+1050671090906*z^52+29774689*z^60+z^68+ 7819453057535022*z^32-5629339100329026*z^38+3253112310663964*z^40-941707*z^62)/ (-1-18139570224656434*z^28+7562658001204946*z^26+285*z^2-2527049898903754*z^24+ 674978054669397*z^22-30205*z^4+1713759*z^6-60760979*z^8+1465413859*z^10-\ 25345900063*z^12+325680788511*z^14+24145613702257*z^18-3186072524891*z^16+ 143520518917585*z^50-674978054669397*z^48-143520518917585*z^20-\ 67211672444297886*z^36+67211672444297886*z^34+30205*z^66-1713759*z^64+ 34928109059843686*z^30+18139570224656434*z^42-7562658001204946*z^44+ 2527049898903754*z^46+25345900063*z^58-325680788511*z^56+3186072524891*z^54-\ 24145613702257*z^52-1465413859*z^60+z^70-285*z^68-54039911079095310*z^32+ 54039911079095310*z^38-34928109059843686*z^40+60760979*z^62) The first , 40, terms are: [0, 70, 0, 8763, 0, 1155157, 0, 153510170, 0, 20441427491, 0, 2723619678163, 0, 362963849664106, 0, 48373356103879613, 0, 6446995275090604147, 0, 859233374417510043830, 0, 114515895555996692837889, 0, 15262324489149597344440673, 0, 2034115828391568792226501270, 0, 271100740505368629785461823267, 0, 36131478847732935097881355898573, 0, 4815493199987760555356218308017674, 0, 641794234506436858711748646542609555, 0, 85536376570755258179632252306117025187, 0, 11400027181749485947728645939047663631354, 0, 1519360825932321584799550149732520810750469] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 683470779 z + 1944179267 z + 149 z - 3880791683 z 22 4 6 8 10 + 5474387724 z - 7821 z + 202119 z - 2988110 z + 27525574 z 12 14 18 16 - 166491573 z + 683470779 z + 3880791683 z - 1944179267 z 20 36 34 30 42 - 5474387724 z - 202119 z + 2988110 z + 166491573 z + z 32 38 40 / 44 42 40 - 27525574 z + 7821 z - 149 z ) / (z - 217 z + 14104 z / 38 36 34 32 30 - 436923 z + 7676843 z - 83586848 z + 595217561 z - 2870821241 z 28 26 24 22 + 9593987560 z - 22533426931 z + 37505781179 z - 44428827584 z 20 18 16 14 + 37505781179 z - 22533426931 z + 9593987560 z - 2870821241 z 12 10 8 6 4 + 595217561 z - 83586848 z + 7676843 z - 436923 z + 14104 z 2 - 217 z + 1) And in Maple-input format, it is: -(-1-683470779*z^28+1944179267*z^26+149*z^2-3880791683*z^24+5474387724*z^22-\ 7821*z^4+202119*z^6-2988110*z^8+27525574*z^10-166491573*z^12+683470779*z^14+ 3880791683*z^18-1944179267*z^16-5474387724*z^20-202119*z^36+2988110*z^34+ 166491573*z^30+z^42-27525574*z^32+7821*z^38-149*z^40)/(z^44-217*z^42+14104*z^40 -436923*z^38+7676843*z^36-83586848*z^34+595217561*z^32-2870821241*z^30+ 9593987560*z^28-22533426931*z^26+37505781179*z^24-44428827584*z^22+37505781179* z^20-22533426931*z^18+9593987560*z^16-2870821241*z^14+595217561*z^12-83586848*z ^10+7676843*z^8-436923*z^6+14104*z^4-217*z^2+1) The first , 40, terms are: [0, 68, 0, 8473, 0, 1114373, 0, 147337780, 0, 19491265997, 0, 2578657155445, 0, 341154129315796, 0, 45134450163518189, 0, 5971256973984017841, 0, 789993252588366901156, 0, 104515572887291595607001, 0, 13827339608591741000203561, 0, 1829347679300764107841566436, 0, 242021460868496119968046628737, 0, 32019275605503795917756592291357, 0, 4236128509536413360424245233437332, 0, 560436937125340094986937052699836197, 0, 74145427785630957289397021773609089565, 0, 9809389955118015179758208076802876455796, 0, 1297775657452188784633862394018999587730389] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 5}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 100253509120114555665 z - 14643161549801073354 z - 293 z 24 22 4 6 + 1838120248219043083 z - 197142551764410212 z + 38378 z - 3026468 z 102 8 10 12 - 6403010536 z + 162789107 z - 6403010536 z + 192563481993 z 14 18 16 - 4567190886920 z - 1372646806990174 z + 87392661627775 z 50 48 - 17822273815681219944462618 z + 11422089565979198838426002 z 20 36 + 17937773082824139 z + 52697386884282900712408 z 34 66 - 13548806423428718042638 z - 6442219581011166270095166 z 80 100 90 + 3038448128866664301543 z + 192563481993 z - 197142551764410212 z 88 84 + 1838120248219043083 z + 100253509120114555665 z 94 86 96 - 1372646806990174 z - 14643161549801073354 z + 87392661627775 z 98 92 82 - 4567190886920 z + 17937773082824139 z - 592736384694396652646 z 64 112 110 106 + 11422089565979198838426002 z + z - 293 z - 3026468 z 108 30 42 + 38378 z - 592736384694396652646 z - 1393515369438593931081466 z 44 46 + 3195917903656621216940402 z - 6442219581011166270095166 z 58 56 - 29617899399890291022393002 z + 31558148371046224205468086 z 54 52 - 29617899399890291022393002 z + 24482768202059993019024386 z 60 70 + 24482768202059993019024386 z - 1393515369438593931081466 z 68 78 + 3195917903656621216940402 z - 13548806423428718042638 z 32 38 + 3038448128866664301543 z - 179178061282642718478257 z 40 62 + 533554809906625202686217 z - 17822273815681219944462618 z 76 74 + 52697386884282900712408 z - 179178061282642718478257 z 72 104 / 2 + 533554809906625202686217 z + 162789107 z ) / ((-1 + z ) (1 / 28 26 2 + 292266648339184860430 z - 40979176258496128466 z - 359 z 24 22 4 6 + 4927825848884744612 z - 505229227559397092 z + 53984 z - 4730096 z 102 8 10 12 - 11756398974 z + 277411830 z - 11756398974 z + 377870869404 z 14 18 16 - 9522976778080 z - 3191872453663070 z + 192770212596459 z 50 48 - 69688028605544795318290682 z + 44111857263092965823372151 z 20 36 + 43844372397192035 z + 177156813856363454196386 z 34 66 - 44098491218332985446106 z - 24491584138049085533455346 z 80 100 90 + 9553242252543056767329 z + 377870869404 z - 505229227559397092 z 88 84 + 4927825848884744612 z + 292266648339184860430 z 94 86 96 - 3191872453663070 z - 40979176258496128466 z + 192770212596459 z 98 92 82 - 9522976778080 z + 43844372397192035 z - 1796398973876439687292 z 64 112 110 106 + 44111857263092965823372151 z + z - 359 z - 4730096 z 108 30 42 + 53984 z - 1796398973876439687292 z - 5085896962994890239073122 z 44 46 + 11922545732678943187133578 z - 24491584138049085533455346 z 58 56 - 117486435404249967148696042 z + 125409748021545623358266075 z 54 52 - 117486435404249967148696042 z + 96592596870875508603753682 z 60 70 + 96592596870875508603753682 z - 5085896962994890239073122 z 68 78 + 11922545732678943187133578 z - 44098491218332985446106 z 32 38 + 9553242252543056767329 z - 620679696065457889897423 z 40 62 + 1899680012666785582071774 z - 69688028605544795318290682 z 76 74 + 177156813856363454196386 z - 620679696065457889897423 z 72 104 + 1899680012666785582071774 z + 277411830 z )) And in Maple-input format, it is: -(1+100253509120114555665*z^28-14643161549801073354*z^26-293*z^2+ 1838120248219043083*z^24-197142551764410212*z^22+38378*z^4-3026468*z^6-\ 6403010536*z^102+162789107*z^8-6403010536*z^10+192563481993*z^12-4567190886920* z^14-1372646806990174*z^18+87392661627775*z^16-17822273815681219944462618*z^50+ 11422089565979198838426002*z^48+17937773082824139*z^20+52697386884282900712408* z^36-13548806423428718042638*z^34-6442219581011166270095166*z^66+ 3038448128866664301543*z^80+192563481993*z^100-197142551764410212*z^90+ 1838120248219043083*z^88+100253509120114555665*z^84-1372646806990174*z^94-\ 14643161549801073354*z^86+87392661627775*z^96-4567190886920*z^98+ 17937773082824139*z^92-592736384694396652646*z^82+11422089565979198838426002*z^ 64+z^112-293*z^110-3026468*z^106+38378*z^108-592736384694396652646*z^30-\ 1393515369438593931081466*z^42+3195917903656621216940402*z^44-\ 6442219581011166270095166*z^46-29617899399890291022393002*z^58+ 31558148371046224205468086*z^56-29617899399890291022393002*z^54+ 24482768202059993019024386*z^52+24482768202059993019024386*z^60-\ 1393515369438593931081466*z^70+3195917903656621216940402*z^68-\ 13548806423428718042638*z^78+3038448128866664301543*z^32-\ 179178061282642718478257*z^38+533554809906625202686217*z^40-\ 17822273815681219944462618*z^62+52697386884282900712408*z^76-\ 179178061282642718478257*z^74+533554809906625202686217*z^72+162789107*z^104)/(-\ 1+z^2)/(1+292266648339184860430*z^28-40979176258496128466*z^26-359*z^2+ 4927825848884744612*z^24-505229227559397092*z^22+53984*z^4-4730096*z^6-\ 11756398974*z^102+277411830*z^8-11756398974*z^10+377870869404*z^12-\ 9522976778080*z^14-3191872453663070*z^18+192770212596459*z^16-\ 69688028605544795318290682*z^50+44111857263092965823372151*z^48+ 43844372397192035*z^20+177156813856363454196386*z^36-44098491218332985446106*z^ 34-24491584138049085533455346*z^66+9553242252543056767329*z^80+377870869404*z^ 100-505229227559397092*z^90+4927825848884744612*z^88+292266648339184860430*z^84 -3191872453663070*z^94-40979176258496128466*z^86+192770212596459*z^96-\ 9522976778080*z^98+43844372397192035*z^92-1796398973876439687292*z^82+ 44111857263092965823372151*z^64+z^112-359*z^110-4730096*z^106+53984*z^108-\ 1796398973876439687292*z^30-5085896962994890239073122*z^42+ 11922545732678943187133578*z^44-24491584138049085533455346*z^46-\ 117486435404249967148696042*z^58+125409748021545623358266075*z^56-\ 117486435404249967148696042*z^54+96592596870875508603753682*z^52+ 96592596870875508603753682*z^60-5085896962994890239073122*z^70+ 11922545732678943187133578*z^68-44098491218332985446106*z^78+ 9553242252543056767329*z^32-620679696065457889897423*z^38+ 1899680012666785582071774*z^40-69688028605544795318290682*z^62+ 177156813856363454196386*z^76-620679696065457889897423*z^74+ 1899680012666785582071774*z^72+277411830*z^104) The first , 40, terms are: [0, 67, 0, 8155, 0, 1052431, 0, 136888536, 0, 17829078753, 0, 2322782868665, 0, 302631360915753, 0, 39429845813377487, 0, 5137331587227508251, 0, 669345654037703616109, 0, 87209416238642499980845, 0, 11362563714336142357016133, 0, 1480434812040012165099576776, 0, 192886684275842016948902313811, 0, 25131314597825029053897371354063, 0, 3274373116706140242421901510139431, 0, 426619915419644595946190805061204477, 0, 55584548781998990402134000313697159381, 0, 7242142130836888987833984778071903663903, 0, 943582772418001463442327888118416414702183] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 94380843798410112652 z - 14221712461141985944 z - 305 z 24 22 4 6 + 1839130205556549420 z - 202806162032599188 z + 41275 z - 3331838 z 102 8 10 12 - 7192462082 z + 181783680 z - 7192462082 z + 215988320355 z 14 18 16 - 5082592576117 z - 1480277705808844 z + 95962222836177 z 50 48 - 12555227003479048433531186 z + 8160568272730877329765574 z 20 36 + 18923910826795152 z + 43651059406193795462540 z 34 66 - 11581576661047728251628 z - 4684751769669951229016628 z 80 100 90 + 2682148048335784244048 z + 215988320355 z - 202806162032599188 z 88 84 + 1839130205556549420 z + 94380843798410112652 z 94 86 96 - 1480277705808844 z - 14221712461141985944 z + 95962222836177 z 98 92 82 - 5082592576117 z + 18923910826795152 z - 540430857529680161188 z 64 112 110 106 + 8160568272730877329765574 z + z - 305 z - 3331838 z 108 30 42 + 41275 z - 540430857529680161188 z - 1059885464447436251908012 z 44 46 + 2373312802080931808295600 z - 4684751769669951229016628 z 58 56 - 20524478654143675407056688 z + 21823504561853356485415392 z 54 52 - 20524478654143675407056688 z + 17071712365191543619228042 z 60 70 + 17071712365191543619228042 z - 1059885464447436251908012 z 68 78 + 2373312802080931808295600 z - 11581576661047728251628 z 32 38 + 2682148048335784244048 z - 143994905125737914899600 z 40 62 + 416699063009764761517820 z - 12555227003479048433531186 z 76 74 + 43651059406193795462540 z - 143994905125737914899600 z 72 104 / 2 + 416699063009764761517820 z + 181783680 z ) / ((-1 + z ) (1 / 28 26 2 + 284369778159119988560 z - 41181778625082495948 z - 380 z 24 22 4 6 + 5106918319031875708 z - 538791693071624044 z + 59594 z - 5373008 z 102 8 10 12 - 13688771652 z + 320575475 z - 13688771652 z + 439733264130 z 14 18 16 - 10999669636656 z - 3572231145087812 z + 219717985548417 z 50 48 - 49984407184363029193787876 z + 32117514459909898634605382 z 20 36 + 47981349818740616 z + 150804372181895546116960 z 34 66 - 38797498494160258272020 z - 18170390564218780819401708 z 80 100 90 + 8692314123130546886756 z + 439733264130 z - 538791693071624044 z 88 84 + 5106918319031875708 z + 284369778159119988560 z 94 86 96 - 3572231145087812 z - 41181778625082495948 z + 219717985548417 z 98 92 82 - 10999669636656 z + 47981349818740616 z - 1690600021862245105724 z 64 112 110 106 + 32117514459909898634605382 z + z - 380 z - 5373008 z 108 30 42 + 59594 z - 1690600021862245105724 z - 3957104047754935318469476 z 44 46 + 9044420772933682507137784 z - 18170390564218780819401708 z 58 56 - 82802606831164368340560412 z + 88190397778165155037162066 z 54 52 - 82802606831164368340560412 z + 68529960251757462091884164 z 60 70 + 68529960251757462091884164 z - 3957104047754935318469476 z 68 78 + 9044420772933682507137784 z - 38797498494160258272020 z 32 38 + 8692314123130546886756 z - 511808702363788961718948 z 40 62 + 1519966567647976003855036 z - 49984407184363029193787876 z 76 74 + 150804372181895546116960 z - 511808702363788961718948 z 72 104 + 1519966567647976003855036 z + 320575475 z )) And in Maple-input format, it is: -(1+94380843798410112652*z^28-14221712461141985944*z^26-305*z^2+ 1839130205556549420*z^24-202806162032599188*z^22+41275*z^4-3331838*z^6-\ 7192462082*z^102+181783680*z^8-7192462082*z^10+215988320355*z^12-5082592576117* z^14-1480277705808844*z^18+95962222836177*z^16-12555227003479048433531186*z^50+ 8160568272730877329765574*z^48+18923910826795152*z^20+43651059406193795462540*z ^36-11581576661047728251628*z^34-4684751769669951229016628*z^66+ 2682148048335784244048*z^80+215988320355*z^100-202806162032599188*z^90+ 1839130205556549420*z^88+94380843798410112652*z^84-1480277705808844*z^94-\ 14221712461141985944*z^86+95962222836177*z^96-5082592576117*z^98+ 18923910826795152*z^92-540430857529680161188*z^82+8160568272730877329765574*z^ 64+z^112-305*z^110-3331838*z^106+41275*z^108-540430857529680161188*z^30-\ 1059885464447436251908012*z^42+2373312802080931808295600*z^44-\ 4684751769669951229016628*z^46-20524478654143675407056688*z^58+ 21823504561853356485415392*z^56-20524478654143675407056688*z^54+ 17071712365191543619228042*z^52+17071712365191543619228042*z^60-\ 1059885464447436251908012*z^70+2373312802080931808295600*z^68-\ 11581576661047728251628*z^78+2682148048335784244048*z^32-\ 143994905125737914899600*z^38+416699063009764761517820*z^40-\ 12555227003479048433531186*z^62+43651059406193795462540*z^76-\ 143994905125737914899600*z^74+416699063009764761517820*z^72+181783680*z^104)/(-\ 1+z^2)/(1+284369778159119988560*z^28-41181778625082495948*z^26-380*z^2+ 5106918319031875708*z^24-538791693071624044*z^22+59594*z^4-5373008*z^6-\ 13688771652*z^102+320575475*z^8-13688771652*z^10+439733264130*z^12-\ 10999669636656*z^14-3572231145087812*z^18+219717985548417*z^16-\ 49984407184363029193787876*z^50+32117514459909898634605382*z^48+ 47981349818740616*z^20+150804372181895546116960*z^36-38797498494160258272020*z^ 34-18170390564218780819401708*z^66+8692314123130546886756*z^80+439733264130*z^ 100-538791693071624044*z^90+5106918319031875708*z^88+284369778159119988560*z^84 -3572231145087812*z^94-41181778625082495948*z^86+219717985548417*z^96-\ 10999669636656*z^98+47981349818740616*z^92-1690600021862245105724*z^82+ 32117514459909898634605382*z^64+z^112-380*z^110-5373008*z^106+59594*z^108-\ 1690600021862245105724*z^30-3957104047754935318469476*z^42+ 9044420772933682507137784*z^44-18170390564218780819401708*z^46-\ 82802606831164368340560412*z^58+88190397778165155037162066*z^56-\ 82802606831164368340560412*z^54+68529960251757462091884164*z^52+ 68529960251757462091884164*z^60-3957104047754935318469476*z^70+ 9044420772933682507137784*z^68-38797498494160258272020*z^78+ 8692314123130546886756*z^32-511808702363788961718948*z^38+ 1519966567647976003855036*z^40-49984407184363029193787876*z^62+ 150804372181895546116960*z^76-511808702363788961718948*z^74+ 1519966567647976003855036*z^72+320575475*z^104) The first , 40, terms are: [0, 76, 0, 10257, 0, 1450657, 0, 206259948, 0, 29350336321, 0, 4177109212557, 0, 594500906076804, 0, 84612122787554469, 0, 12042414148945526325, 0, 1713936720544871852196, 0, 243936105444247761698813, 0, 34718216949911381059021893, 0, 4941271847237189856117803668, 0, 703266750500958195827186155405, 0, 100092473895157787208430759856125, 0, 14245666143501211478424213946944436, 0, 2027515117002489049691541295300549813, 0, 288566186277356962272563893751410238169, 0, 41070196304189730528501759121387035905404, 0, 5845317659151503790900126512024901991775689] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6455484991814200 z - 2661905745336300 z - 222 z 24 22 4 6 + 895238247109668 z - 244386558302656 z + 20270 z - 1031662 z 8 10 12 14 + 33434297 z - 744474196 z + 11965290504 z - 143655811956 z 18 16 50 - 9477366597760 z + 1320943438804 z - 244386558302656 z 48 20 36 + 895238247109668 z + 53809360400328 z + 30815770841526420 z 34 66 64 - 27959483321094324 z - 1031662 z + 33434297 z 30 42 44 - 12814660426537100 z - 12814660426537100 z + 6455484991814200 z 46 58 56 - 2661905745336300 z - 143655811956 z + 1320943438804 z 54 52 60 70 - 9477366597760 z + 53809360400328 z + 11965290504 z - 222 z 68 32 38 + 20270 z + 20876306857049038 z - 27959483321094324 z 40 62 72 / 2 + 20876306857049038 z - 744474196 z + z ) / ((-1 + z ) (1 / 28 26 2 + 24121542873227320 z - 9597991008024932 z - 286 z 24 22 4 6 + 3093808586650048 z - 804340957768160 z + 31214 z - 1818126 z 8 10 12 14 + 65808341 z - 1612718604 z + 28246681928 z - 366819066524 z 18 16 50 - 27793040260448 z + 3625286689896 z - 804340957768160 z 48 20 36 + 3093808586650048 z + 167655892708040 z + 122995809772086676 z 34 66 64 - 111124255331636180 z - 1818126 z + 65808341 z 30 42 44 - 49263297420945076 z - 49263297420945076 z + 24121542873227320 z 46 58 56 - 9597991008024932 z - 366819066524 z + 3625286689896 z 54 52 60 70 - 27793040260448 z + 167655892708040 z + 28246681928 z - 286 z 68 32 38 + 31214 z + 81933480394900994 z - 111124255331636180 z 40 62 72 + 81933480394900994 z - 1612718604 z + z )) And in Maple-input format, it is: -(1+6455484991814200*z^28-2661905745336300*z^26-222*z^2+895238247109668*z^24-\ 244386558302656*z^22+20270*z^4-1031662*z^6+33434297*z^8-744474196*z^10+ 11965290504*z^12-143655811956*z^14-9477366597760*z^18+1320943438804*z^16-\ 244386558302656*z^50+895238247109668*z^48+53809360400328*z^20+30815770841526420 *z^36-27959483321094324*z^34-1031662*z^66+33434297*z^64-12814660426537100*z^30-\ 12814660426537100*z^42+6455484991814200*z^44-2661905745336300*z^46-143655811956 *z^58+1320943438804*z^56-9477366597760*z^54+53809360400328*z^52+11965290504*z^ 60-222*z^70+20270*z^68+20876306857049038*z^32-27959483321094324*z^38+ 20876306857049038*z^40-744474196*z^62+z^72)/(-1+z^2)/(1+24121542873227320*z^28-\ 9597991008024932*z^26-286*z^2+3093808586650048*z^24-804340957768160*z^22+31214* z^4-1818126*z^6+65808341*z^8-1612718604*z^10+28246681928*z^12-366819066524*z^14 -27793040260448*z^18+3625286689896*z^16-804340957768160*z^50+3093808586650048*z ^48+167655892708040*z^20+122995809772086676*z^36-111124255331636180*z^34-\ 1818126*z^66+65808341*z^64-49263297420945076*z^30-49263297420945076*z^42+ 24121542873227320*z^44-9597991008024932*z^46-366819066524*z^58+3625286689896*z^ 56-27793040260448*z^54+167655892708040*z^52+28246681928*z^60-286*z^70+31214*z^ 68+81933480394900994*z^32-111124255331636180*z^38+81933480394900994*z^40-\ 1612718604*z^62+z^72) The first , 40, terms are: [0, 65, 0, 7425, 0, 901153, 0, 110758341, 0, 13671006261, 0, 1690312968349, 0, 209144772891693, 0, 25885746397478133, 0, 3204289189760950677, 0, 396668134625655410525, 0, 49105877765169852883613, 0, 6079168387324999953649161, 0, 752587173116137858160940985, 0, 93168753576876623360761020329, 0, 11534110747264099926098382056889, 0, 1427901015213958919081864840621385, 0, 176771461941695225612228151538640153, 0, 21883976173074407358801890330300792329, 0, 2709195371726501059285686293714374544953, 0, 335393330366180315597772718726636806233325] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 670739379699255744 z - 168459937464410752 z - 258 z 24 22 4 6 + 35447048446057381 z - 6211988098985098 z + 28352 z - 1796366 z 8 10 12 14 + 74654095 z - 2189584484 z + 47528223232 z - 789323888228 z 18 16 50 - 106854488501750 z + 10276864709563 z - 54303031317839656636 z 48 20 + 80533542957696108998 z + 900073328935040 z 36 34 + 31231141820638982976 z - 15297663530726909108 z 66 80 88 84 86 - 6211988098985098 z + 74654095 z + z + 28352 z - 258 z 82 64 30 - 1796366 z + 35447048446057381 z - 2248259219909768832 z 42 44 - 101974834361646930488 z + 110315381390552578944 z 46 58 - 101974834361646930488 z - 2248259219909768832 z 56 54 + 6369095702139856202 z - 15297663530726909108 z 52 60 70 + 31231141820638982976 z + 670739379699255744 z - 106854488501750 z 68 78 32 + 900073328935040 z - 2189584484 z + 6369095702139856202 z 38 40 - 54303031317839656636 z + 80533542957696108998 z 62 76 74 - 168459937464410752 z + 47528223232 z - 789323888228 z 72 / 2 28 + 10276864709563 z ) / ((-1 + z ) (1 + 2268098766956937904 z / 26 2 24 - 547629153329138872 z - 326 z + 110367231173159445 z 22 4 6 8 - 18457031247144358 z + 42154 z - 3026634 z + 139387099 z 10 12 14 - 4464977596 z + 104773404620 z - 1866599764164 z 18 16 50 - 285794224304218 z + 25911654188519 z - 210231579454591619452 z 48 20 + 315883766928800607342 z + 2542396184462242 z 36 34 + 118747148003979502132 z - 56853414026209256820 z 66 80 88 84 86 - 18457031247144358 z + 139387099 z + z + 42154 z - 326 z 82 64 30 - 3026634 z + 110367231173159445 z - 7877868120430430824 z 42 44 - 403164288122764427872 z + 437297388247394073096 z 46 58 - 403164288122764427872 z - 7877868120430430824 z 56 54 + 23032880106723371962 z - 56853414026209256820 z 52 60 + 118747148003979502132 z + 2268098766956937904 z 70 68 78 - 285794224304218 z + 2542396184462242 z - 4464977596 z 32 38 + 23032880106723371962 z - 210231579454591619452 z 40 62 76 + 315883766928800607342 z - 547629153329138872 z + 104773404620 z 74 72 - 1866599764164 z + 25911654188519 z )) And in Maple-input format, it is: -(1+670739379699255744*z^28-168459937464410752*z^26-258*z^2+35447048446057381*z ^24-6211988098985098*z^22+28352*z^4-1796366*z^6+74654095*z^8-2189584484*z^10+ 47528223232*z^12-789323888228*z^14-106854488501750*z^18+10276864709563*z^16-\ 54303031317839656636*z^50+80533542957696108998*z^48+900073328935040*z^20+ 31231141820638982976*z^36-15297663530726909108*z^34-6211988098985098*z^66+ 74654095*z^80+z^88+28352*z^84-258*z^86-1796366*z^82+35447048446057381*z^64-\ 2248259219909768832*z^30-101974834361646930488*z^42+110315381390552578944*z^44-\ 101974834361646930488*z^46-2248259219909768832*z^58+6369095702139856202*z^56-\ 15297663530726909108*z^54+31231141820638982976*z^52+670739379699255744*z^60-\ 106854488501750*z^70+900073328935040*z^68-2189584484*z^78+6369095702139856202*z ^32-54303031317839656636*z^38+80533542957696108998*z^40-168459937464410752*z^62 +47528223232*z^76-789323888228*z^74+10276864709563*z^72)/(-1+z^2)/(1+ 2268098766956937904*z^28-547629153329138872*z^26-326*z^2+110367231173159445*z^ 24-18457031247144358*z^22+42154*z^4-3026634*z^6+139387099*z^8-4464977596*z^10+ 104773404620*z^12-1866599764164*z^14-285794224304218*z^18+25911654188519*z^16-\ 210231579454591619452*z^50+315883766928800607342*z^48+2542396184462242*z^20+ 118747148003979502132*z^36-56853414026209256820*z^34-18457031247144358*z^66+ 139387099*z^80+z^88+42154*z^84-326*z^86-3026634*z^82+110367231173159445*z^64-\ 7877868120430430824*z^30-403164288122764427872*z^42+437297388247394073096*z^44-\ 403164288122764427872*z^46-7877868120430430824*z^58+23032880106723371962*z^56-\ 56853414026209256820*z^54+118747148003979502132*z^52+2268098766956937904*z^60-\ 285794224304218*z^70+2542396184462242*z^68-4464977596*z^78+23032880106723371962 *z^32-210231579454591619452*z^38+315883766928800607342*z^40-547629153329138872* z^62+104773404620*z^76-1866599764164*z^74+25911654188519*z^72) The first , 40, terms are: [0, 69, 0, 8435, 0, 1099547, 0, 145219803, 0, 19251578435, 0, 2555336722957, 0, 339328669527929, 0, 45067302568755785, 0, 5985872461779383869, 0, 795064588389847145987, 0, 105604071717536976430715, 0, 14026848946036243813035963, 0, 1863116518159315051206223923, 0, 247468583657504129329460522581, 0, 32870036604576219586436164537905, 0, 4365965746308270541440568913095505, 0, 579909826981661708900973547184053429, 0, 77026579954017668492003714995116244403, 0, 10231063102504456807620350054395827273083, 0, 1358941969919580233315799801058853993425467] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 146404053149727 z - 85626724162276 z - 188 z 24 22 4 6 + 40259204831070 z - 15156091025290 z + 13934 z - 564642 z 8 10 12 14 + 14397380 z - 249939655 z + 3105783240 z - 28582355370 z 18 16 50 - 1076442158393 z + 199590730216 z - 28582355370 z 48 20 36 + 199590730216 z + 4543316769404 z + 146404053149727 z 34 64 30 42 - 201775392454568 z + z - 201775392454568 z - 15156091025290 z 44 46 58 56 + 4543316769404 z - 1076442158393 z - 564642 z + 14397380 z 54 52 60 32 - 249939655 z + 3105783240 z + 13934 z + 224508394033540 z 38 40 62 / - 85626724162276 z + 40259204831070 z - 188 z ) / (-1 / 28 26 2 24 - 908993155073982 z + 479739982375399 z + 259 z - 203702299957357 z 22 4 6 8 10 + 69295408482942 z - 23314 z + 1094492 z - 31639003 z + 615763153 z 12 14 18 16 - 8524739044 z + 87092437384 z + 4021529029245 z - 673769224081 z 50 48 20 + 673769224081 z - 4021529029245 z - 18776960781904 z 36 34 66 64 - 1389734951242048 z + 1717605259559385 z + z - 259 z 30 42 44 + 1389734951242048 z + 203702299957357 z - 69295408482942 z 46 58 56 54 + 18776960781904 z + 31639003 z - 615763153 z + 8524739044 z 52 60 32 - 87092437384 z - 1094492 z - 1717605259559385 z 38 40 62 + 908993155073982 z - 479739982375399 z + 23314 z ) And in Maple-input format, it is: -(1+146404053149727*z^28-85626724162276*z^26-188*z^2+40259204831070*z^24-\ 15156091025290*z^22+13934*z^4-564642*z^6+14397380*z^8-249939655*z^10+3105783240 *z^12-28582355370*z^14-1076442158393*z^18+199590730216*z^16-28582355370*z^50+ 199590730216*z^48+4543316769404*z^20+146404053149727*z^36-201775392454568*z^34+ z^64-201775392454568*z^30-15156091025290*z^42+4543316769404*z^44-1076442158393* z^46-564642*z^58+14397380*z^56-249939655*z^54+3105783240*z^52+13934*z^60+ 224508394033540*z^32-85626724162276*z^38+40259204831070*z^40-188*z^62)/(-1-\ 908993155073982*z^28+479739982375399*z^26+259*z^2-203702299957357*z^24+ 69295408482942*z^22-23314*z^4+1094492*z^6-31639003*z^8+615763153*z^10-\ 8524739044*z^12+87092437384*z^14+4021529029245*z^18-673769224081*z^16+ 673769224081*z^50-4021529029245*z^48-18776960781904*z^20-1389734951242048*z^36+ 1717605259559385*z^34+z^66-259*z^64+1389734951242048*z^30+203702299957357*z^42-\ 69295408482942*z^44+18776960781904*z^46+31639003*z^58-615763153*z^56+8524739044 *z^54-87092437384*z^52-1094492*z^60-1717605259559385*z^32+908993155073982*z^38-\ 479739982375399*z^40+23314*z^62) The first , 40, terms are: [0, 71, 0, 9009, 0, 1207887, 0, 163274216, 0, 22107077139, 0, 2994445015613, 0, 405643509281999, 0, 54952045790308713, 0, 7444338366214159737, 0, 1008484214792571868207, 0, 136619378159322266026437, 0, 18507832312256131514907179, 0, 2507256820689682451007711368, 0, 339658189625244733140928641047, 0, 46013509696679037558064758810185, 0, 6233452160106995441054651221919751, 0, 844446035399522254173833853486474865, 0, 114397141169019292023019197693799952273, 0, 15497385693318405544079365154633751810567, 0, 2099431514405402211495644389467057388348761] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1680088401578621045 z - 409103473688327514 z - 282 z 24 22 4 6 + 83103429627273805 z - 13995890335557404 z + 34169 z - 2383270 z 8 10 12 14 + 108408944 z - 3455255606 z + 80899037127 z - 1438803295950 z 18 16 50 - 219036768531660 z + 19927054808239 z - 149841281284399571262 z 48 20 + 224128661200432197123 z + 1939393896099328 z 36 34 + 85146390443445828811 z - 41060940706598429322 z 66 80 88 84 86 - 13995890335557404 z + 108408944 z + z + 34169 z - 282 z 82 64 30 - 2383270 z + 83103429627273805 z - 5784997790730692762 z 42 44 - 285254584177405602200 z + 309110582991432603136 z 46 58 - 285254584177405602200 z - 5784997790730692762 z 56 54 + 16769629429433472528 z - 41060940706598429322 z 52 60 70 + 85146390443445828811 z + 1680088401578621045 z - 219036768531660 z 68 78 32 + 1939393896099328 z - 3455255606 z + 16769629429433472528 z 38 40 - 149841281284399571262 z + 224128661200432197123 z 62 76 74 - 409103473688327514 z + 80899037127 z - 1438803295950 z 72 / 2 28 + 19927054808239 z ) / ((-1 + z ) (1 + 5793074428552428755 z / 26 2 24 - 1350904358830265869 z - 361 z + 261880373955468089 z 22 4 6 8 - 41951496845344416 z + 51603 z - 4071696 z + 204579538 z 10 12 14 - 7098221320 z + 179213634809 z - 3414479803119 z 18 16 50 - 588328489531056 z + 50410396710459 z - 601450370978887260251 z 48 20 + 913174808236872949003 z + 5511775674041036 z 36 34 + 334816274950919512609 z - 157340989875920351664 z 66 80 88 84 86 - 41951496845344416 z + 204579538 z + z + 51603 z - 361 z 82 64 30 - 4071696 z + 261880373955468089 z - 20749912446516586168 z 42 44 - 1172811960273062283824 z + 1274766423730018170280 z 46 58 - 1172811960273062283824 z - 20749912446516586168 z 56 54 + 62311868960036176334 z - 157340989875920351664 z 52 60 + 334816274950919512609 z + 5793074428552428755 z 70 68 78 - 588328489531056 z + 5511775674041036 z - 7098221320 z 32 38 + 62311868960036176334 z - 601450370978887260251 z 40 62 76 + 913174808236872949003 z - 1350904358830265869 z + 179213634809 z 74 72 - 3414479803119 z + 50410396710459 z )) And in Maple-input format, it is: -(1+1680088401578621045*z^28-409103473688327514*z^26-282*z^2+83103429627273805* z^24-13995890335557404*z^22+34169*z^4-2383270*z^6+108408944*z^8-3455255606*z^10 +80899037127*z^12-1438803295950*z^14-219036768531660*z^18+19927054808239*z^16-\ 149841281284399571262*z^50+224128661200432197123*z^48+1939393896099328*z^20+ 85146390443445828811*z^36-41060940706598429322*z^34-13995890335557404*z^66+ 108408944*z^80+z^88+34169*z^84-282*z^86-2383270*z^82+83103429627273805*z^64-\ 5784997790730692762*z^30-285254584177405602200*z^42+309110582991432603136*z^44-\ 285254584177405602200*z^46-5784997790730692762*z^58+16769629429433472528*z^56-\ 41060940706598429322*z^54+85146390443445828811*z^52+1680088401578621045*z^60-\ 219036768531660*z^70+1939393896099328*z^68-3455255606*z^78+16769629429433472528 *z^32-149841281284399571262*z^38+224128661200432197123*z^40-409103473688327514* z^62+80899037127*z^76-1438803295950*z^74+19927054808239*z^72)/(-1+z^2)/(1+ 5793074428552428755*z^28-1350904358830265869*z^26-361*z^2+261880373955468089*z^ 24-41951496845344416*z^22+51603*z^4-4071696*z^6+204579538*z^8-7098221320*z^10+ 179213634809*z^12-3414479803119*z^14-588328489531056*z^18+50410396710459*z^16-\ 601450370978887260251*z^50+913174808236872949003*z^48+5511775674041036*z^20+ 334816274950919512609*z^36-157340989875920351664*z^34-41951496845344416*z^66+ 204579538*z^80+z^88+51603*z^84-361*z^86-4071696*z^82+261880373955468089*z^64-\ 20749912446516586168*z^30-1172811960273062283824*z^42+1274766423730018170280*z^ 44-1172811960273062283824*z^46-20749912446516586168*z^58+62311868960036176334*z ^56-157340989875920351664*z^54+334816274950919512609*z^52+5793074428552428755*z ^60-588328489531056*z^70+5511775674041036*z^68-7098221320*z^78+ 62311868960036176334*z^32-601450370978887260251*z^38+913174808236872949003*z^40 -1350904358830265869*z^62+179213634809*z^76-3414479803119*z^74+50410396710459*z ^72) The first , 40, terms are: [0, 80, 0, 11165, 0, 1624639, 0, 237562888, 0, 34767104327, 0, 5089066440431, 0, 744949841792008, 0, 109048811851550271, 0, 15963061909625591949, 0, 2336747792104140185904, 0, 342064178975657849048569, 0, 50072974209791910932392681, 0, 7329919251820083699065601840, 0, 1072988324493687606835034223053, 0, 157069117455497006647847578754847, 0, 22992522025472136677924048519323848, 0, 3365754375236754321756000804963563327, 0, 492695081619549829052107332264889060407, 0, 72123041791278633864793325162476830764488, 0, 10557712774857821534451936731699772148564511] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 6}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 227636888815903 z + 141129843753649 z + 197 z 24 22 4 6 - 68776768565477 z + 26284005815667 z - 15531 z + 674471 z 8 10 12 14 - 18474753 z + 344021405 z - 4563738611 z + 44501525783 z 18 16 50 48 + 1821390791209 z - 325971056085 z + 4563738611 z - 44501525783 z 20 36 34 - 7848554048263 z - 141129843753649 z + 227636888815903 z 30 42 44 + 289011763379475 z + 7848554048263 z - 1821390791209 z 46 58 56 54 52 + 325971056085 z + 15531 z - 674471 z + 18474753 z - 344021405 z 60 32 38 40 - 197 z - 289011763379475 z + 68776768565477 z - 26284005815667 z 62 / 28 26 2 + z ) / (1 + 1384509556130308 z - 771377365423462 z - 254 z / 24 22 4 6 + 339177760932776 z - 117319781051338 z + 24236 z - 1237506 z 8 10 12 14 + 39085656 z - 826794190 z + 12317560916 z - 133754893490 z 18 16 50 - 6689303628726 z + 1084826067020 z - 133754893490 z 48 20 36 + 1084826067020 z + 31769613852412 z + 1384509556130308 z 34 64 30 42 - 1965155503577754 z + z - 1965155503577754 z - 117319781051338 z 44 46 58 56 + 31769613852412 z - 6689303628726 z - 1237506 z + 39085656 z 54 52 60 32 - 826794190 z + 12317560916 z + 24236 z + 2208244905151334 z 38 40 62 - 771377365423462 z + 339177760932776 z - 254 z ) And in Maple-input format, it is: -(-1-227636888815903*z^28+141129843753649*z^26+197*z^2-68776768565477*z^24+ 26284005815667*z^22-15531*z^4+674471*z^6-18474753*z^8+344021405*z^10-4563738611 *z^12+44501525783*z^14+1821390791209*z^18-325971056085*z^16+4563738611*z^50-\ 44501525783*z^48-7848554048263*z^20-141129843753649*z^36+227636888815903*z^34+ 289011763379475*z^30+7848554048263*z^42-1821390791209*z^44+325971056085*z^46+ 15531*z^58-674471*z^56+18474753*z^54-344021405*z^52-197*z^60-289011763379475*z^ 32+68776768565477*z^38-26284005815667*z^40+z^62)/(1+1384509556130308*z^28-\ 771377365423462*z^26-254*z^2+339177760932776*z^24-117319781051338*z^22+24236*z^ 4-1237506*z^6+39085656*z^8-826794190*z^10+12317560916*z^12-133754893490*z^14-\ 6689303628726*z^18+1084826067020*z^16-133754893490*z^50+1084826067020*z^48+ 31769613852412*z^20+1384509556130308*z^36-1965155503577754*z^34+z^64-\ 1965155503577754*z^30-117319781051338*z^42+31769613852412*z^44-6689303628726*z^ 46-1237506*z^58+39085656*z^56-826794190*z^54+12317560916*z^52+24236*z^60+ 2208244905151334*z^32-771377365423462*z^38+339177760932776*z^40-254*z^62) The first , 40, terms are: [0, 57, 0, 5773, 0, 647925, 0, 74585461, 0, 8640609325, 0, 1002604565241, 0, 116383369061345, 0, 13511292230802465, 0, 1568607345442762873, 0, 182110300045703764973, 0, 21142460043253424504949, 0, 2454577319710637395284981, 0, 284969226848976553414358221, 0, 33084092288045555375431989433, 0, 3840966207592643962180571298753, 0, 445924926911290065225444520383553, 0, 51770577971484593766780964393591737, 0, 6010412475467843801540495215242073165, 0, 697791284970561198219831386727335731701, 0, 81011524478919730158233776487641992743541] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {4, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 100055697481570215908 z - 14647517489900492276 z - 293 z 24 22 4 6 + 1842334995441628324 z - 197920347478565084 z + 38391 z - 3030586 z 102 8 10 12 - 6429683750 z + 163238152 z - 6429683750 z + 193586505383 z 14 18 16 - 4594770032013 z - 1380854139228132 z + 87940360410105 z 50 48 - 17355260630826301980546806 z + 11136267820251558463013822 z 20 36 + 18030906457107464 z + 52044919187691683613140 z 34 66 - 13417087238111325647076 z - 6290629952905364178152060 z 80 100 90 + 3017013684772359684864 z + 193586505383 z - 197920347478565084 z 88 84 + 1842334995441628324 z + 100055697481570215908 z 94 86 96 - 1380854139228132 z - 14647517489900492276 z + 87940360410105 z 98 92 82 - 4594770032013 z + 18030906457107464 z - 590100036487193246252 z 64 112 110 106 + 11136267820251558463013822 z + z - 293 z - 3030586 z 108 30 42 + 38391 z - 590100036487193246252 z - 1366064256044126971984580 z 44 46 + 3126418295362265997272824 z - 6290629952905364178152060 z 58 56 - 28801282182643394421720208 z + 30682597276067399698088944 z 54 52 - 28801282182643394421720208 z + 23820348971182319145468330 z 60 70 + 23820348971182319145468330 z - 1366064256044126971984580 z 68 78 + 3126418295362265997272824 z - 13417087238111325647076 z 32 38 + 3017013684772359684864 z - 176495986564517923426404 z 40 62 + 524257469227839137579684 z - 17355260630826301980546806 z 76 74 + 52044919187691683613140 z - 176495986564517923426404 z 72 104 / + 524257469227839137579684 z + 163238152 z ) / (-1 / 28 26 2 - 334821465350205993328 z + 46268773989105243524 z + 357 z 24 22 4 6 - 5489179355118341372 z + 555629393669847920 z - 53818 z + 4748338 z 102 8 10 12 + 391496842818 z - 281115831 z + 12042158151 z - 391496842818 z 14 18 16 + 9982363924594 z + 3425958176045921 z - 204466553253117 z 50 48 + 109647493005706137302290374 z - 66254614786573090140729510 z 20 36 - 47628580655296768 z - 218625032733250384409080 z 34 66 + 53252846058590615352340 z + 66254614786573090140729510 z 80 100 - 53252846058590615352340 z - 9982363924594 z 90 88 + 5489179355118341372 z - 46268773989105243524 z 84 94 - 2090801078657765559768 z + 47628580655296768 z 86 96 98 + 334821465350205993328 z - 3425958176045921 z + 204466553253117 z 92 82 - 555629393669847920 z + 11314653445007131370100 z 64 112 114 110 - 109647493005706137302290374 z - 357 z + z + 53818 z 106 108 30 + 281115831 z - 4748338 z + 2090801078657765559768 z 42 44 + 6813891787886411964266876 z - 16528220650681439791726512 z 46 58 + 35268073623126164728293792 z + 233199461932096969581319098 z 56 54 - 233199461932096969581319098 z + 205656116631550209199352892 z 52 60 - 159929194431761604892606244 z - 205656116631550209199352892 z 70 68 + 16528220650681439791726512 z - 35268073623126164728293792 z 78 32 + 218625032733250384409080 z - 11314653445007131370100 z 38 40 + 784825869135874549012096 z - 2468510572582414323094676 z 62 76 + 159929194431761604892606244 z - 784825869135874549012096 z 74 72 + 2468510572582414323094676 z - 6813891787886411964266876 z 104 - 12042158151 z ) And in Maple-input format, it is: -(1+100055697481570215908*z^28-14647517489900492276*z^26-293*z^2+ 1842334995441628324*z^24-197920347478565084*z^22+38391*z^4-3030586*z^6-\ 6429683750*z^102+163238152*z^8-6429683750*z^10+193586505383*z^12-4594770032013* z^14-1380854139228132*z^18+87940360410105*z^16-17355260630826301980546806*z^50+ 11136267820251558463013822*z^48+18030906457107464*z^20+52044919187691683613140* z^36-13417087238111325647076*z^34-6290629952905364178152060*z^66+ 3017013684772359684864*z^80+193586505383*z^100-197920347478565084*z^90+ 1842334995441628324*z^88+100055697481570215908*z^84-1380854139228132*z^94-\ 14647517489900492276*z^86+87940360410105*z^96-4594770032013*z^98+ 18030906457107464*z^92-590100036487193246252*z^82+11136267820251558463013822*z^ 64+z^112-293*z^110-3030586*z^106+38391*z^108-590100036487193246252*z^30-\ 1366064256044126971984580*z^42+3126418295362265997272824*z^44-\ 6290629952905364178152060*z^46-28801282182643394421720208*z^58+ 30682597276067399698088944*z^56-28801282182643394421720208*z^54+ 23820348971182319145468330*z^52+23820348971182319145468330*z^60-\ 1366064256044126971984580*z^70+3126418295362265997272824*z^68-\ 13417087238111325647076*z^78+3017013684772359684864*z^32-\ 176495986564517923426404*z^38+524257469227839137579684*z^40-\ 17355260630826301980546806*z^62+52044919187691683613140*z^76-\ 176495986564517923426404*z^74+524257469227839137579684*z^72+163238152*z^104)/(-\ 1-334821465350205993328*z^28+46268773989105243524*z^26+357*z^2-\ 5489179355118341372*z^24+555629393669847920*z^22-53818*z^4+4748338*z^6+ 391496842818*z^102-281115831*z^8+12042158151*z^10-391496842818*z^12+ 9982363924594*z^14+3425958176045921*z^18-204466553253117*z^16+ 109647493005706137302290374*z^50-66254614786573090140729510*z^48-\ 47628580655296768*z^20-218625032733250384409080*z^36+53252846058590615352340*z^ 34+66254614786573090140729510*z^66-53252846058590615352340*z^80-9982363924594*z ^100+5489179355118341372*z^90-46268773989105243524*z^88-2090801078657765559768* z^84+47628580655296768*z^94+334821465350205993328*z^86-3425958176045921*z^96+ 204466553253117*z^98-555629393669847920*z^92+11314653445007131370100*z^82-\ 109647493005706137302290374*z^64-357*z^112+z^114+53818*z^110+281115831*z^106-\ 4748338*z^108+2090801078657765559768*z^30+6813891787886411964266876*z^42-\ 16528220650681439791726512*z^44+35268073623126164728293792*z^46+ 233199461932096969581319098*z^58-233199461932096969581319098*z^56+ 205656116631550209199352892*z^54-159929194431761604892606244*z^52-\ 205656116631550209199352892*z^60+16528220650681439791726512*z^70-\ 35268073623126164728293792*z^68+218625032733250384409080*z^78-\ 11314653445007131370100*z^32+784825869135874549012096*z^38-\ 2468510572582414323094676*z^40+159929194431761604892606244*z^62-\ 784825869135874549012096*z^76+2468510572582414323094676*z^74-\ 6813891787886411964266876*z^72-12042158151*z^104) The first , 40, terms are: [0, 64, 0, 7421, 0, 922697, 0, 116035404, 0, 14625409597, 0, 1844382283621, 0, 232621417394496, 0, 29340189631271029, 0, 3700666659313681649, 0, 466764744958215878644, 0, 58873031912465996442929, 0, 7425656124139442825586833, 0, 936598127862466996661205036, 0, 118133138635444039654838153849, 0, 14900134943244485152001558993725, 0, 1879354295534219300441359485989784, 0, 237042992061615329046979326151421637, 0, 29898236975900064358153905872177481245, 0, 3771065183198195142977427194001438675508, 0, 475644521363233916314673569946449835176737] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2531337115150 z - 2912633056576 z - 192 z 24 22 4 6 + 2531337115150 z - 1659142655584 z + 14513 z - 580896 z 8 10 12 14 + 13908464 z - 213690840 z + 2206608760 z - 15824742176 z 18 16 50 48 - 299221306024 z + 80780844488 z - 192 z + 14513 z 20 36 34 + 816381003408 z + 80780844488 z - 299221306024 z 30 42 44 46 52 - 1659142655584 z - 213690840 z + 13908464 z - 580896 z + z 32 38 40 / 2 + 816381003408 z - 15824742176 z + 2206608760 z ) / ((-1 + z ) (1 / 28 26 2 24 + 9921281812022 z - 11508824661674 z - 255 z + 9921281812022 z 22 4 6 8 10 - 6348536432758 z + 23425 z - 1100498 z + 30326510 z - 527416114 z 12 14 18 16 + 6067280044 z - 47756545428 z - 1044374551446 z + 263901519116 z 50 48 20 36 - 255 z + 23425 z + 3004326003178 z + 263901519116 z 34 30 42 44 - 1044374551446 z - 6348536432758 z - 527416114 z + 30326510 z 46 52 32 38 40 - 1100498 z + z + 3004326003178 z - 47756545428 z + 6067280044 z )) And in Maple-input format, it is: -(1+2531337115150*z^28-2912633056576*z^26-192*z^2+2531337115150*z^24-\ 1659142655584*z^22+14513*z^4-580896*z^6+13908464*z^8-213690840*z^10+2206608760* z^12-15824742176*z^14-299221306024*z^18+80780844488*z^16-192*z^50+14513*z^48+ 816381003408*z^20+80780844488*z^36-299221306024*z^34-1659142655584*z^30-\ 213690840*z^42+13908464*z^44-580896*z^46+z^52+816381003408*z^32-15824742176*z^ 38+2206608760*z^40)/(-1+z^2)/(1+9921281812022*z^28-11508824661674*z^26-255*z^2+ 9921281812022*z^24-6348536432758*z^22+23425*z^4-1100498*z^6+30326510*z^8-\ 527416114*z^10+6067280044*z^12-47756545428*z^14-1044374551446*z^18+263901519116 *z^16-255*z^50+23425*z^48+3004326003178*z^20+263901519116*z^36-1044374551446*z^ 34-6348536432758*z^30-527416114*z^42+30326510*z^44-1100498*z^46+z^52+ 3004326003178*z^32-47756545428*z^38+6067280044*z^40) The first , 40, terms are: [0, 64, 0, 7217, 0, 875059, 0, 107529072, 0, 13250120875, 0, 1633740179299, 0, 201469701623120, 0, 24845752396187131, 0, 3064068895822408665, 0, 377873052272574972704, 0, 46600822684941513921561, 0, 5747001571840171763748137, 0, 708743463549499594247235936, 0, 87405109692759646149544115657, 0, 10779151577646121753876980821547, 0, 1329328562536433128837772494049680, 0, 163938174059368692029657902903364531, 0, 20217518582576241910089555875363058139, 0, 2493306150249671850934936653644656600880, 0, 307484597256911027366741642970884502829955] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 12 10 8 6 18 4 f(z) = - (-56538 z + 188156 z - 188156 z + 56538 z + z - 5858 z 16 2 14 / 12 10 - 147 z + 147 z + 5858 z - 1) / (979178 z - 1951872 z / 8 20 6 18 4 16 2 + 979178 z + z - 186152 z - 248 z + 13585 z + 13585 z - 248 z 14 - 186152 z + 1) And in Maple-input format, it is: -(-56538*z^12+188156*z^10-188156*z^8+56538*z^6+z^18-5858*z^4-147*z^16+147*z^2+ 5858*z^14-1)/(979178*z^12-1951872*z^10+979178*z^8+z^20-186152*z^6-248*z^18+ 13585*z^4+13585*z^16-248*z^2-186152*z^14+1) The first , 40, terms are: [0, 101, 0, 17321, 0, 3053137, 0, 539882521, 0, 95541204593, 0, 16911498124397, 0, 2993668626580585, 0, 529949604993368473, 0, 93814121894714334749, 0, 16607440837448139043745, 0, 2939933241786012019745545, 0, 520441958327205121694468449, 0, 92131291643615670862045548281, 0, 16309551730662007956003367172405, 0, 2887200149537007965589992718984145, 0, 511106917839498320501706455973215281, 0, 90478757293070878277868561973840301717, 0, 16017011777254296072577091473516573053785, 0, 2835413239075595605821052928573104010171713, 0, 501939334767090971244144398772444672627182377] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 24 22 4 6 8 f(z) = - (1 - 132 z + z - 132 z + 4418 z - 56228 z + 319551 z 10 12 14 18 16 - 865112 z + 1195516 z - 865112 z - 56228 z + 319551 z 20 / 2 24 22 20 18 + 4418 z ) / ((-1 + z ) (z - 222 z + 9958 z - 158998 z / 16 14 12 10 8 + 1105039 z - 3590380 z + 5385044 z - 3590380 z + 1105039 z 6 4 2 - 158998 z + 9958 z - 222 z + 1)) And in Maple-input format, it is: -(1-132*z^2+z^24-132*z^22+4418*z^4-56228*z^6+319551*z^8-865112*z^10+1195516*z^ 12-865112*z^14-56228*z^18+319551*z^16+4418*z^20)/(-1+z^2)/(z^24-222*z^22+9958*z ^20-158998*z^18+1105039*z^16-3590380*z^14+5385044*z^12-3590380*z^10+1105039*z^8 -158998*z^6+9958*z^4-222*z^2+1) The first , 40, terms are: [0, 91, 0, 14531, 0, 2426761, 0, 407672633, 0, 68550472755, 0, 11528715633515, 0, 1938938415795153, 0, 326098907006140209, 0, 54844751076834614859, 0, 9224033408538049345107, 0, 1551338881704182845848281, 0, 260911060466462578874825193, 0, 43881180533591816498915571875, 0, 7380131765581018678665526187899, 0, 1241223326643375799144364627268257, 0, 208754449859128659856218445237262689, 0, 35109250205502389321083833795024416059, 0, 5904829577643000209667895179471822544099, 0, 993100454635861913807332521230850715771177, 0, 167024043628984209026389684163215461080088473] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 4 f(z) = - (1 + z - 126 z - 126 z + 3915 z - 49340 z + 3915 z 6 8 10 12 14 - 49340 z + 313600 z - 1113968 z + 2337472 z - 2983172 z 18 16 20 / 28 26 24 - 1113968 z + 2337472 z + 313600 z ) / ((z - 225 z + 8765 z / 22 20 18 16 14 - 134896 z + 1032048 z - 4314928 z + 10197678 z - 13630034 z 12 10 8 6 4 2 + 10197678 z - 4314928 z + 1032048 z - 134896 z + 8765 z - 225 z 2 + 1) (-1 + z )) And in Maple-input format, it is: -(1+z^28-126*z^26-126*z^2+3915*z^24-49340*z^22+3915*z^4-49340*z^6+313600*z^8-\ 1113968*z^10+2337472*z^12-2983172*z^14-1113968*z^18+2337472*z^16+313600*z^20)/( z^28-225*z^26+8765*z^24-134896*z^22+1032048*z^20-4314928*z^18+10197678*z^16-\ 13630034*z^14+10197678*z^12-4314928*z^10+1032048*z^8-134896*z^6+8765*z^4-225*z^ 2+1)/(-1+z^2) The first , 40, terms are: [0, 100, 0, 17525, 0, 3155971, 0, 569212452, 0, 102675032495, 0, 18520799179087, 0, 3340835136326100, 0, 602629537906438483, 0, 108704066388197785349, 0, 19608355253958258336660, 0, 3537012078767023593866081, 0, 638016513039401163313782817, 0, 115087271925202743081193988148, 0, 20759776414079546532313430203493, 0, 3744708772336516711083803800123443, 0, 675481445941953011729150784026683252, 0, 121845305349910602822154297430106505871, 0, 21978810113888394917948312697344316389743, 0, 3964601612143419570207336201589233433676420, 0, 715146355128568423560229044984921080456123427] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 19683728435172 z - 17251504818553 z - 193 z 24 22 4 6 + 11605148174821 z - 5977784059039 z + 14879 z - 621124 z 8 10 12 14 + 15913454 z - 268415123 z + 3121486482 z - 25851765186 z 18 16 50 48 - 698608048108 z + 156069173377 z - 621124 z + 15913454 z 20 36 34 + 2347799796326 z + 2347799796326 z - 5977784059039 z 30 42 44 46 - 17251504818553 z - 25851765186 z + 3121486482 z - 268415123 z 56 54 52 32 38 + z - 193 z + 14879 z + 11605148174821 z - 698608048108 z 40 / 28 26 + 156069173377 z ) / (-1 - 149452892002773 z + 114962331158903 z / 2 24 22 4 6 + 259 z - 67951767839429 z + 30791556119725 z - 24137 z + 1172289 z 8 10 12 14 - 34363403 z + 657978613 z - 8654788639 z + 80972218693 z 18 16 50 48 + 2797502149817 z - 552386452863 z + 34363403 z - 657978613 z 20 36 34 - 10654119403575 z - 30791556119725 z + 67951767839429 z 30 42 44 + 149452892002773 z + 552386452863 z - 80972218693 z 46 58 56 54 52 + 8654788639 z + z - 259 z + 24137 z - 1172289 z 32 38 40 - 114962331158903 z + 10654119403575 z - 2797502149817 z ) And in Maple-input format, it is: -(1+19683728435172*z^28-17251504818553*z^26-193*z^2+11605148174821*z^24-\ 5977784059039*z^22+14879*z^4-621124*z^6+15913454*z^8-268415123*z^10+3121486482* z^12-25851765186*z^14-698608048108*z^18+156069173377*z^16-621124*z^50+15913454* z^48+2347799796326*z^20+2347799796326*z^36-5977784059039*z^34-17251504818553*z^ 30-25851765186*z^42+3121486482*z^44-268415123*z^46+z^56-193*z^54+14879*z^52+ 11605148174821*z^32-698608048108*z^38+156069173377*z^40)/(-1-149452892002773*z^ 28+114962331158903*z^26+259*z^2-67951767839429*z^24+30791556119725*z^22-24137*z ^4+1172289*z^6-34363403*z^8+657978613*z^10-8654788639*z^12+80972218693*z^14+ 2797502149817*z^18-552386452863*z^16+34363403*z^50-657978613*z^48-\ 10654119403575*z^20-30791556119725*z^36+67951767839429*z^34+149452892002773*z^ 30+552386452863*z^42-80972218693*z^44+8654788639*z^46+z^58-259*z^56+24137*z^54-\ 1172289*z^52-114962331158903*z^32+10654119403575*z^38-2797502149817*z^40) The first , 40, terms are: [0, 66, 0, 7836, 0, 987647, 0, 125584166, 0, 15995098851, 0, 2037934962043, 0, 259673303510584, 0, 33088108089245065, 0, 4216172138174572725, 0, 537236044088848811482, 0, 68456083215343076211941, 0, 8722861436317076990512156, 0, 1111490885610947603151379398, 0, 141629212079875771888821204151, 0, 18046782018130134204133173666391, 0, 2299570381549070737519829302275014, 0, 293017554851629749510140249145376172, 0, 37337099199431811713356437210504826389, 0, 4757595418939807162580096563359691370778, 0, 606225835848339297956233919814067223850053] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 8625408350754814 z - 3541034339030273 z - 231 z 24 22 4 6 + 1184007465484696 z - 320824078254300 z + 21978 z - 1160925 z 8 10 12 14 + 38852596 z - 889236116 z + 14630212626 z - 179164417378 z 18 16 50 - 12186349644332 z + 1675113301699 z - 320824078254300 z 48 20 36 + 1184007465484696 z + 69985830982436 z + 41463640320632566 z 34 66 64 - 37605161336348950 z - 1160925 z + 38852596 z 30 42 44 - 17177019253925565 z - 17177019253925565 z + 8625408350754814 z 46 58 56 - 3541034339030273 z - 179164417378 z + 1675113301699 z 54 52 60 70 - 12186349644332 z + 69985830982436 z + 14630212626 z - 231 z 68 32 38 + 21978 z + 28043493823076509 z - 37605161336348950 z 40 62 72 / + 28043493823076509 z - 889236116 z + z ) / (-1 / 28 26 2 - 46420211420177671 z + 17277991407025417 z + 311 z 24 22 4 6 - 5240452563109377 z + 1289039521500447 z - 35494 z + 2142312 z 8 10 12 14 - 80120329 z + 2027504493 z - 36700394254 z + 493518031236 z 18 16 50 + 40471025551497 z - 5065098091831 z + 5240452563109377 z 48 20 36 - 17277991407025417 z - 255515066244617 z - 330283904414722753 z 34 66 64 + 271653999219279203 z + 80120329 z - 2027504493 z 30 42 44 + 101983578170015837 z + 183675633052781909 z - 101983578170015837 z 46 58 56 + 46420211420177671 z + 5065098091831 z - 40471025551497 z 54 52 60 + 255515066244617 z - 1289039521500447 z - 493518031236 z 70 68 32 + 35494 z - 2142312 z - 183675633052781909 z 38 40 62 74 + 330283904414722753 z - 271653999219279203 z + 36700394254 z + z 72 - 311 z ) And in Maple-input format, it is: -(1+8625408350754814*z^28-3541034339030273*z^26-231*z^2+1184007465484696*z^24-\ 320824078254300*z^22+21978*z^4-1160925*z^6+38852596*z^8-889236116*z^10+ 14630212626*z^12-179164417378*z^14-12186349644332*z^18+1675113301699*z^16-\ 320824078254300*z^50+1184007465484696*z^48+69985830982436*z^20+ 41463640320632566*z^36-37605161336348950*z^34-1160925*z^66+38852596*z^64-\ 17177019253925565*z^30-17177019253925565*z^42+8625408350754814*z^44-\ 3541034339030273*z^46-179164417378*z^58+1675113301699*z^56-12186349644332*z^54+ 69985830982436*z^52+14630212626*z^60-231*z^70+21978*z^68+28043493823076509*z^32 -37605161336348950*z^38+28043493823076509*z^40-889236116*z^62+z^72)/(-1-\ 46420211420177671*z^28+17277991407025417*z^26+311*z^2-5240452563109377*z^24+ 1289039521500447*z^22-35494*z^4+2142312*z^6-80120329*z^8+2027504493*z^10-\ 36700394254*z^12+493518031236*z^14+40471025551497*z^18-5065098091831*z^16+ 5240452563109377*z^50-17277991407025417*z^48-255515066244617*z^20-\ 330283904414722753*z^36+271653999219279203*z^34+80120329*z^66-2027504493*z^64+ 101983578170015837*z^30+183675633052781909*z^42-101983578170015837*z^44+ 46420211420177671*z^46+5065098091831*z^58-40471025551497*z^56+255515066244617*z ^54-1289039521500447*z^52-493518031236*z^60+35494*z^70-2142312*z^68-\ 183675633052781909*z^32+330283904414722753*z^38-271653999219279203*z^40+ 36700394254*z^62+z^74-311*z^72) The first , 40, terms are: [0, 80, 0, 11364, 0, 1676071, 0, 248021492, 0, 36718095563, 0, 5436362658253, 0, 804907644557696, 0, 119175344129555557, 0, 17645242789355611223, 0, 2612577264867037659552, 0, 386821621530483334669843, 0, 57273321476669533535867468, 0, 8479964016215509926988609044, 0, 1255554740609855897092439037633, 0, 185899103801150020337192714640273, 0, 27524468433926522921292287526477020, 0, 4075309387222473629552868432282115900, 0, 603395725629250572874537254049716993131, 0, 89339573299982425156362851652359015759160, 0, 13227735992310930710906843331323231278227967] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}, {3, 6}, {4, 5}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 4155850846 z - 9610335640 z - 156 z + 15846875394 z 22 4 6 8 10 - 18712989848 z + 8719 z - 242052 z + 3907183 z - 39957196 z 12 14 18 16 + 272113697 z - 1271738160 z - 9610335640 z + 4155850846 z 20 36 34 30 42 + 15846875394 z + 3907183 z - 39957196 z - 1271738160 z - 156 z 44 32 38 40 / 44 46 + z + 272113697 z - 242052 z + 8719 z ) / (-223 z + z / 26 6 42 4 38 + 100765357134 z + 509927 z + 15381 z - 15381 z + 9711759 z 24 32 10 2 - 140032476618 z - 5044881993 z + 116394301 z + 223 z 34 28 20 12 + 925519991 z - 52079113482 z - 100765357134 z - 925519991 z 30 16 18 22 + 19246591862 z - 19246591862 z + 52079113482 z + 140032476618 z 8 36 40 14 - 1 - 9711759 z - 116394301 z - 509927 z + 5044881993 z ) And in Maple-input format, it is: -(1+4155850846*z^28-9610335640*z^26-156*z^2+15846875394*z^24-18712989848*z^22+ 8719*z^4-242052*z^6+3907183*z^8-39957196*z^10+272113697*z^12-1271738160*z^14-\ 9610335640*z^18+4155850846*z^16+15846875394*z^20+3907183*z^36-39957196*z^34-\ 1271738160*z^30-156*z^42+z^44+272113697*z^32-242052*z^38+8719*z^40)/(-223*z^44+ z^46+100765357134*z^26+509927*z^6+15381*z^42-15381*z^4+9711759*z^38-\ 140032476618*z^24-5044881993*z^32+116394301*z^10+223*z^2+925519991*z^34-\ 52079113482*z^28-100765357134*z^20-925519991*z^12+19246591862*z^30-19246591862* z^16+52079113482*z^18+140032476618*z^22-1-9711759*z^8-116394301*z^36-509927*z^ 40+5044881993*z^14) The first , 40, terms are: [0, 67, 0, 8279, 0, 1083565, 0, 142656229, 0, 18793460687, 0, 2476026683819, 0, 326218064131753, 0, 42979493316514009, 0, 5662584293517630043, 0, 746050266158481623519, 0, 98292754127857794971509, 0, 12950153598842532891763741, 0, 1706193705012740257790547527, 0, 224792465762898324690725004435, 0, 29616597762223674789763101670097, 0, 3902011840305318525284652442323761, 0, 514093364948975819906175279473560051, 0, 67732236267138447518113018088346906215, 0, 8923779497141615503013201841487742564669, 0, 1175715507155699448933336935884719948943957] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 200746256 z - 749342144 z - 164 z + 1884765494 z 22 4 6 8 10 - 3253699720 z + 9490 z - 261076 z + 3926857 z - 35304256 z 12 14 18 16 + 200746256 z - 749342144 z - 3253699720 z + 1884765494 z 20 36 34 30 32 + 3898320572 z + 9490 z - 261076 z - 35304256 z + 3926857 z 38 40 / 28 36 16 - 164 z + z ) / (-3415536768 z - 596664 z - 10323574902 z / 40 18 20 6 - 251 z - 1 + 21418060498 z - 30781425328 z + 596664 z 8 42 30 14 22 - 10606803 z + z + 764160256 z + 3415536768 z + 30781425328 z 32 2 4 12 34 - 112899305 z + 251 z - 18200 z - 764160256 z + 10606803 z 10 24 38 26 + 112899305 z - 21418060498 z + 18200 z + 10323574902 z ) And in Maple-input format, it is: -(1+200746256*z^28-749342144*z^26-164*z^2+1884765494*z^24-3253699720*z^22+9490* z^4-261076*z^6+3926857*z^8-35304256*z^10+200746256*z^12-749342144*z^14-\ 3253699720*z^18+1884765494*z^16+3898320572*z^20+9490*z^36-261076*z^34-35304256* z^30+3926857*z^32-164*z^38+z^40)/(-3415536768*z^28-596664*z^36-10323574902*z^16 -251*z^40-1+21418060498*z^18-30781425328*z^20+596664*z^6-10606803*z^8+z^42+ 764160256*z^30+3415536768*z^14+30781425328*z^22-112899305*z^32+251*z^2-18200*z^ 4-764160256*z^12+10606803*z^34+112899305*z^10-21418060498*z^24+18200*z^38+ 10323574902*z^26) The first , 40, terms are: [0, 87, 0, 13127, 0, 2047065, 0, 320131737, 0, 50083694503, 0, 7836043020567, 0, 1226040039178977, 0, 191828999885133729, 0, 30014029470495061143, 0, 4696068828998648276199, 0, 734758517346783516078745, 0, 114962132511705788303425369, 0, 17987259282422550693014429703, 0, 2814331029947154746606222251223, 0, 440337186650309877410833401633345, 0, 68896244221137987806264849061953473, 0, 10779676601823757695765135426804506583, 0, 1686614835887853978007163179483206237831, 0, 263891924564627564553652567952834244762457, 0, 41289182549956529142166992340669801591733401] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 10 12 6 8 2 4 14 f(z) = - (3668 z - 1 - 141 z + 27714 z - 27714 z + 141 z - 3668 z + z / 12 10 8 6 4 16 ) / (10109 z - 113270 z + 225714 z - 113270 z + 10109 z + z / 2 14 - 193 z - 193 z + 1) And in Maple-input format, it is: -(3668*z^10-1-141*z^12+27714*z^6-27714*z^8+141*z^2-3668*z^4+z^14)/(10109*z^12-\ 113270*z^10+225714*z^8-113270*z^6+10109*z^4+z^16-193*z^2-193*z^14+1) The first , 40, terms are: [0, 52, 0, 3595, 0, 253723, 0, 18318724, 0, 1366206049, 0, 106427428993, 0, 8747616560740, 0, 763059186106267, 0, 70589502553557451, 0, 6876983612639782996, 0, 698137851638872188577, 0, 73045607606152043392225, 0, 7803435963899213545768468, 0, 845178924678825997175284939, 0, 92351673032952077348131040539, 0, 10147445929705798644562181364388, 0, 1118854340616146855974352263134145, 0, 123629145281066267276977288304402977, 0, 13678578299433938400388577864040820036, 0, 1514651244827061238111962452351688029851] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1962956952032892 z - 963537852726858 z - 245 z 24 22 4 6 + 383210939922866 z - 122784332796344 z + 23412 z - 1190663 z 8 10 12 14 + 37184639 z - 776350344 z + 11453374159 z - 124051937671 z 18 16 50 - 6384891208197 z + 1014000154872 z - 6384891208197 z 48 20 36 + 31460066663985 z + 31460066663985 z + 4401454505319502 z 34 66 64 30 - 4866843045621680 z - 245 z + 23412 z - 3253854525724990 z 42 44 46 - 963537852726858 z + 383210939922866 z - 122784332796344 z 58 56 54 52 - 776350344 z + 11453374159 z - 124051937671 z + 1014000154872 z 60 68 32 38 + 37184639 z + z + 4401454505319502 z - 3253854525724990 z 40 62 / 28 + 1962956952032892 z - 1190663 z ) / (-1 - 11097032635825086 z / 26 2 24 + 5013985307590002 z + 301 z - 1839651027741610 z 22 4 6 8 + 544322102730809 z - 35371 z + 2181377 z - 80605475 z 10 12 14 18 + 1941451791 z - 32355625915 z + 389656713975 z + 24056915698867 z 16 50 48 - 3501638497853 z + 128727189244277 z - 544322102730809 z 20 36 34 - 128727189244277 z - 36089234583044170 z + 36089234583044170 z 66 64 30 42 + 35371 z - 2181377 z + 20048604773374574 z + 11097032635825086 z 44 46 58 - 5013985307590002 z + 1839651027741610 z + 32355625915 z 56 54 52 - 389656713975 z + 3501638497853 z - 24056915698867 z 60 70 68 32 - 1941451791 z + z - 301 z - 29679312475446906 z 38 40 62 + 29679312475446906 z - 20048604773374574 z + 80605475 z ) And in Maple-input format, it is: -(1+1962956952032892*z^28-963537852726858*z^26-245*z^2+383210939922866*z^24-\ 122784332796344*z^22+23412*z^4-1190663*z^6+37184639*z^8-776350344*z^10+ 11453374159*z^12-124051937671*z^14-6384891208197*z^18+1014000154872*z^16-\ 6384891208197*z^50+31460066663985*z^48+31460066663985*z^20+4401454505319502*z^ 36-4866843045621680*z^34-245*z^66+23412*z^64-3253854525724990*z^30-\ 963537852726858*z^42+383210939922866*z^44-122784332796344*z^46-776350344*z^58+ 11453374159*z^56-124051937671*z^54+1014000154872*z^52+37184639*z^60+z^68+ 4401454505319502*z^32-3253854525724990*z^38+1962956952032892*z^40-1190663*z^62) /(-1-11097032635825086*z^28+5013985307590002*z^26+301*z^2-1839651027741610*z^24 +544322102730809*z^22-35371*z^4+2181377*z^6-80605475*z^8+1941451791*z^10-\ 32355625915*z^12+389656713975*z^14+24056915698867*z^18-3501638497853*z^16+ 128727189244277*z^50-544322102730809*z^48-128727189244277*z^20-\ 36089234583044170*z^36+36089234583044170*z^34+35371*z^66-2181377*z^64+ 20048604773374574*z^30+11097032635825086*z^42-5013985307590002*z^44+ 1839651027741610*z^46+32355625915*z^58-389656713975*z^56+3501638497853*z^54-\ 24056915698867*z^52-1941451791*z^60+z^70-301*z^68-29679312475446906*z^32+ 29679312475446906*z^38-20048604773374574*z^40+80605475*z^62) The first , 40, terms are: [0, 56, 0, 4897, 0, 483935, 0, 51188924, 0, 5623999255, 0, 630959060911, 0, 71607709755420, 0, 8178646892025123, 0, 937423402353468361, 0, 107657325832014188632, 0, 12377361818514032667445, 0, 1423899832540955028003989, 0, 163862812868450797863764856, 0, 18861027302047566751211224025, 0, 2171188069770914171225965789619, 0, 249951671267526242995351975856668, 0, 28775941464350543068289909217939455, 0, 3312923532611888616250948121403303447, 0, 381415194915720619309096230860996730076, 0, 43912404355657271686744217797882213034015] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 73264561998082 z + 57045240873186 z + 227 z 24 22 4 6 - 34515941221016 z + 16164453409330 z - 19704 z + 903035 z 8 10 12 14 - 25096193 z + 456939117 z - 5723667667 z + 51005298160 z 18 16 50 48 + 1599828323837 z - 331400836635 z + 25096193 z - 456939117 z 20 36 34 - 5823165030834 z - 16164453409330 z + 34515941221016 z 30 42 44 46 + 73264561998082 z + 331400836635 z - 51005298160 z + 5723667667 z 58 56 54 52 32 + z - 227 z + 19704 z - 903035 z - 57045240873186 z 38 40 / 28 + 5823165030834 z - 1599828323837 z ) / (1 + 520772260204408 z / 26 2 24 22 - 361107728682152 z - 288 z + 195528029774952 z - 82256658621712 z 4 6 8 10 12 + 31076 z - 1727156 z + 56723860 z - 1193927702 z + 16999092052 z 14 18 16 - 170160225956 z - 6609039776624 z + 1232688509428 z 50 48 20 - 1193927702 z + 16999092052 z + 26686516258079 z 36 34 30 + 195528029774952 z - 361107728682152 z - 588183538582292 z 42 44 46 58 - 6609039776624 z + 1232688509428 z - 170160225956 z - 288 z 56 54 52 60 32 + 31076 z - 1727156 z + 56723860 z + z + 520772260204408 z 38 40 - 82256658621712 z + 26686516258079 z ) And in Maple-input format, it is: -(-1-73264561998082*z^28+57045240873186*z^26+227*z^2-34515941221016*z^24+ 16164453409330*z^22-19704*z^4+903035*z^6-25096193*z^8+456939117*z^10-5723667667 *z^12+51005298160*z^14+1599828323837*z^18-331400836635*z^16+25096193*z^50-\ 456939117*z^48-5823165030834*z^20-16164453409330*z^36+34515941221016*z^34+ 73264561998082*z^30+331400836635*z^42-51005298160*z^44+5723667667*z^46+z^58-227 *z^56+19704*z^54-903035*z^52-57045240873186*z^32+5823165030834*z^38-\ 1599828323837*z^40)/(1+520772260204408*z^28-361107728682152*z^26-288*z^2+ 195528029774952*z^24-82256658621712*z^22+31076*z^4-1727156*z^6+56723860*z^8-\ 1193927702*z^10+16999092052*z^12-170160225956*z^14-6609039776624*z^18+ 1232688509428*z^16-1193927702*z^50+16999092052*z^48+26686516258079*z^20+ 195528029774952*z^36-361107728682152*z^34-588183538582292*z^30-6609039776624*z^ 42+1232688509428*z^44-170160225956*z^46-288*z^58+31076*z^56-1727156*z^54+ 56723860*z^52+z^60+520772260204408*z^32-82256658621712*z^38+26686516258079*z^40 ) The first , 40, terms are: [0, 61, 0, 6196, 0, 712933, 0, 86506657, 0, 10737103009, 0, 1345444431085, 0, 169271749395124, 0, 21332235639400933, 0, 2690285082349749553, 0, 339384312525005161633, 0, 42819457482623838640021, 0, 5402743973218651537538068, 0, 681707040717976048103221789, 0, 86017241457881799595617856177, 0, 10853632080627645866025515539057, 0, 1369510835045507501606922607833781, 0, 172804953706046314084474118929078804, 0, 21804545981700870486038611816194016525, 0, 2751300139244341606462913991054053546737, 0, 347159391604690582108775541710634613004881] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2339441415550 z - 2687533203988 z - 214 z 24 22 4 6 + 2339441415550 z - 1541080683224 z + 16897 z - 671656 z 8 10 12 14 + 15589864 z - 230110632 z + 2282322752 z - 15798959776 z 18 16 50 48 - 284500799832 z + 78413092032 z - 214 z + 16897 z 20 36 34 + 765225386136 z + 78413092032 z - 284500799832 z 30 42 44 46 52 - 1541080683224 z - 230110632 z + 15589864 z - 671656 z + z 32 38 40 / 2 + 765225386136 z - 15798959776 z + 2282322752 z ) / ((-1 + z ) (1 / 28 26 2 24 + 9207341528018 z - 10627688212862 z - 273 z + 9207341528018 z 22 4 6 8 10 - 5978724078304 z + 27437 z - 1348672 z + 37185668 z - 627631248 z 12 14 18 16 + 6903834508 z - 51772054128 z - 1039934241360 z + 273360128628 z 50 48 20 36 - 273 z + 27437 z + 2897260222524 z + 273360128628 z 34 30 42 44 - 1039934241360 z - 5978724078304 z - 627631248 z + 37185668 z 46 52 32 38 40 - 1348672 z + z + 2897260222524 z - 51772054128 z + 6903834508 z )) And in Maple-input format, it is: -(1+2339441415550*z^28-2687533203988*z^26-214*z^2+2339441415550*z^24-\ 1541080683224*z^22+16897*z^4-671656*z^6+15589864*z^8-230110632*z^10+2282322752* z^12-15798959776*z^14-284500799832*z^18+78413092032*z^16-214*z^50+16897*z^48+ 765225386136*z^20+78413092032*z^36-284500799832*z^34-1541080683224*z^30-\ 230110632*z^42+15589864*z^44-671656*z^46+z^52+765225386136*z^32-15798959776*z^ 38+2282322752*z^40)/(-1+z^2)/(1+9207341528018*z^28-10627688212862*z^26-273*z^2+ 9207341528018*z^24-5978724078304*z^22+27437*z^4-1348672*z^6+37185668*z^8-\ 627631248*z^10+6903834508*z^12-51772054128*z^14-1039934241360*z^18+273360128628 *z^16-273*z^50+27437*z^48+2897260222524*z^20+273360128628*z^36-1039934241360*z^ 34-5978724078304*z^30-627631248*z^42+37185668*z^44-1348672*z^46+z^52+ 2897260222524*z^32-51772054128*z^38+6903834508*z^40) The first , 40, terms are: [0, 60, 0, 5627, 0, 583651, 0, 63618268, 0, 7124447449, 0, 810210929481, 0, 92966334719708, 0, 10723274476813043, 0, 1240693186509961771, 0, 143808670550035142588, 0, 16686500354038203443153, 0, 1937379201536316009823537, 0, 225020235408262772525895100, 0, 26140910348278604249444122571, 0, 3037202596323641466410083466451, 0, 352905436793012904201796790720476, 0, 41007318573526873497477868711464297, 0, 4765134616193484059116287637014253753, 0, 553726460790597066743115111909321780700, 0, 64345631116686191700059179278202914265603] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 1801999924894427 z + 932896466588250 z + 243 z 24 22 4 6 - 386053901927360 z + 127159536851734 z - 23277 z + 1195568 z 8 10 12 14 - 37867677 z + 802381310 z - 11993308480 z + 131177867179 z 18 16 50 + 6781468089550 z - 1077810038837 z + 1077810038837 z 48 20 36 - 6781468089550 z - 33142722964795 z - 2790597725701459 z 34 66 64 30 + 3471140562317083 z + z - 243 z + 2790597725701459 z 42 44 46 + 386053901927360 z - 127159536851734 z + 33142722964795 z 58 56 54 52 + 37867677 z - 802381310 z + 11993308480 z - 131177867179 z 60 32 38 - 1195568 z - 3471140562317083 z + 1801999924894427 z 40 62 / 28 - 932896466588250 z + 23277 z ) / (1 + 10656707483202168 z / 26 2 24 - 5027331763464620 z - 304 z + 1900825271808192 z 22 4 6 8 - 572864375565638 z + 35585 z - 2183990 z + 80919638 z 10 12 14 18 - 1968569766 z + 33257556638 z - 406095878390 z - 25527766798270 z 16 50 48 + 3691008842939 z - 25527766798270 z + 136608182480835 z 20 36 34 + 136608182480835 z + 25015953766100697 z - 27821041000480364 z 66 64 30 42 - 304 z + 35585 z - 18177166579621156 z - 5027331763464620 z 44 46 58 + 1900825271808192 z - 572864375565638 z - 1968569766 z 56 54 52 60 + 33257556638 z - 406095878390 z + 3691008842939 z + 80919638 z 68 32 38 + z + 25015953766100697 z - 18177166579621156 z 40 62 + 10656707483202168 z - 2183990 z ) And in Maple-input format, it is: -(-1-1801999924894427*z^28+932896466588250*z^26+243*z^2-386053901927360*z^24+ 127159536851734*z^22-23277*z^4+1195568*z^6-37867677*z^8+802381310*z^10-\ 11993308480*z^12+131177867179*z^14+6781468089550*z^18-1077810038837*z^16+ 1077810038837*z^50-6781468089550*z^48-33142722964795*z^20-2790597725701459*z^36 +3471140562317083*z^34+z^66-243*z^64+2790597725701459*z^30+386053901927360*z^42 -127159536851734*z^44+33142722964795*z^46+37867677*z^58-802381310*z^56+ 11993308480*z^54-131177867179*z^52-1195568*z^60-3471140562317083*z^32+ 1801999924894427*z^38-932896466588250*z^40+23277*z^62)/(1+10656707483202168*z^ 28-5027331763464620*z^26-304*z^2+1900825271808192*z^24-572864375565638*z^22+ 35585*z^4-2183990*z^6+80919638*z^8-1968569766*z^10+33257556638*z^12-\ 406095878390*z^14-25527766798270*z^18+3691008842939*z^16-25527766798270*z^50+ 136608182480835*z^48+136608182480835*z^20+25015953766100697*z^36-\ 27821041000480364*z^34-304*z^66+35585*z^64-18177166579621156*z^30-\ 5027331763464620*z^42+1900825271808192*z^44-572864375565638*z^46-1968569766*z^ 58+33257556638*z^56-406095878390*z^54+3691008842939*z^52+80919638*z^60+z^68+ 25015953766100697*z^32-18177166579621156*z^38+10656707483202168*z^40-2183990*z^ 62) The first , 40, terms are: [0, 61, 0, 6236, 0, 713481, 0, 85161593, 0, 10349355065, 0, 1268167667045, 0, 156020822166916, 0, 19233527850090721, 0, 2373434858467706245, 0, 293036694558999683697, 0, 36189543622789143575245, 0, 4469965979630508267639652, 0, 552149017146229347476609697, 0, 68206283475809495542723932029, 0, 8425597954309729664762073388525, 0, 1040833706975287049149607135782037, 0, 128577261268346771911475983105419516, 0, 15883570457178780362820946638215067529, 0, 1962152174245268812294149390764203502477, 0, 242391587247283905825596072677258922749429] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1705818694844269599 z - 410274596420826223 z - 287 z 24 22 4 6 + 82371997334059637 z - 13728903526773924 z + 34723 z - 2396522 z 8 10 12 14 + 107561168 z - 3386015138 z + 78543454569 z - 1389465975305 z 18 16 50 - 211806138334540 z + 19219885868763 z - 160355183479882697865 z 48 20 + 241249688073187709183 z + 1886354057584100 z 36 34 + 90411022293479887205 z - 43185010895944280070 z 66 80 88 84 86 - 13728903526773924 z + 107561168 z + z + 34723 z - 287 z 82 64 30 - 2396522 z + 82371997334059637 z - 5946826397613489310 z 42 44 - 308141405640039557200 z + 334313270365290190632 z 46 58 - 308141405640039557200 z - 5946826397613489310 z 56 54 + 17445201218658232936 z - 43185010895944280070 z 52 60 70 + 90411022293479887205 z + 1705818694844269599 z - 211806138334540 z 68 78 32 + 1886354057584100 z - 3386015138 z + 17445201218658232936 z 38 40 - 160355183479882697865 z + 241249688073187709183 z 62 76 74 - 410274596420826223 z + 78543454569 z - 1389465975305 z 72 / 28 + 19219885868763 z ) / (-1 - 7462750212816543992 z / 26 2 24 + 1671209721134428123 z + 355 z - 312226523794400117 z 22 4 6 8 + 48369027462882756 z - 50880 z + 4041448 z - 204797969 z 10 12 14 + 7182626631 z - 183817039780 z + 3561137228404 z 18 16 50 + 640819593244263 z - 53638575492297 z + 1636438873682195683249 z 48 20 - 2258785756001143764699 z - 6166376086130832 z 36 34 - 527688788171131023660 z + 234264811478925022371 z 66 80 90 88 84 + 312226523794400117 z - 7182626631 z + z - 355 z - 4041448 z 86 82 64 + 50880 z + 204797969 z - 1671209721134428123 z 30 42 + 27938901367759406096 z + 2258785756001143764699 z 44 46 - 2653309296576687998260 z + 2653309296576687998260 z 58 56 + 88040876691977492941 z - 234264811478925022371 z 54 52 + 527688788171131023660 z - 1008225255316137133372 z 60 70 68 - 27938901367759406096 z + 6166376086130832 z - 48369027462882756 z 78 32 38 + 183817039780 z - 88040876691977492941 z + 1008225255316137133372 z 40 62 76 - 1636438873682195683249 z + 7462750212816543992 z - 3561137228404 z 74 72 + 53638575492297 z - 640819593244263 z ) And in Maple-input format, it is: -(1+1705818694844269599*z^28-410274596420826223*z^26-287*z^2+82371997334059637* z^24-13728903526773924*z^22+34723*z^4-2396522*z^6+107561168*z^8-3386015138*z^10 +78543454569*z^12-1389465975305*z^14-211806138334540*z^18+19219885868763*z^16-\ 160355183479882697865*z^50+241249688073187709183*z^48+1886354057584100*z^20+ 90411022293479887205*z^36-43185010895944280070*z^34-13728903526773924*z^66+ 107561168*z^80+z^88+34723*z^84-287*z^86-2396522*z^82+82371997334059637*z^64-\ 5946826397613489310*z^30-308141405640039557200*z^42+334313270365290190632*z^44-\ 308141405640039557200*z^46-5946826397613489310*z^58+17445201218658232936*z^56-\ 43185010895944280070*z^54+90411022293479887205*z^52+1705818694844269599*z^60-\ 211806138334540*z^70+1886354057584100*z^68-3386015138*z^78+17445201218658232936 *z^32-160355183479882697865*z^38+241249688073187709183*z^40-410274596420826223* z^62+78543454569*z^76-1389465975305*z^74+19219885868763*z^72)/(-1-\ 7462750212816543992*z^28+1671209721134428123*z^26+355*z^2-312226523794400117*z^ 24+48369027462882756*z^22-50880*z^4+4041448*z^6-204797969*z^8+7182626631*z^10-\ 183817039780*z^12+3561137228404*z^14+640819593244263*z^18-53638575492297*z^16+ 1636438873682195683249*z^50-2258785756001143764699*z^48-6166376086130832*z^20-\ 527688788171131023660*z^36+234264811478925022371*z^34+312226523794400117*z^66-\ 7182626631*z^80+z^90-355*z^88-4041448*z^84+50880*z^86+204797969*z^82-\ 1671209721134428123*z^64+27938901367759406096*z^30+2258785756001143764699*z^42-\ 2653309296576687998260*z^44+2653309296576687998260*z^46+88040876691977492941*z^ 58-234264811478925022371*z^56+527688788171131023660*z^54-1008225255316137133372 *z^52-27938901367759406096*z^60+6166376086130832*z^70-48369027462882756*z^68+ 183817039780*z^78-88040876691977492941*z^32+1008225255316137133372*z^38-\ 1636438873682195683249*z^40+7462750212816543992*z^62-3561137228404*z^76+ 53638575492297*z^74-640819593244263*z^72) The first , 40, terms are: [0, 68, 0, 7983, 0, 1019051, 0, 133169728, 0, 17559167545, 0, 2324513182853, 0, 308301711945272, 0, 40927239791025027, 0, 5435495970693380511, 0, 722035026058111807804, 0, 95922888528688022323161, 0, 12744067421677834336223141, 0, 1693185299228373205212116156, 0, 224960403091085325422498264435, 0, 29888917447880607493138951530711, 0, 3971142863277573962705433167642104, 0, 527620220668686043029365694127029929, 0, 70101553164057914756034932562993858285, 0, 9313951881676312660676598460118178136624, 0, 1237486323787277667188679579876169376251119] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2777746879345236 z - 1322951697705742 z - 227 z 24 22 4 6 + 506563465923802 z - 155176931956752 z + 20618 z - 1033849 z 8 10 12 14 + 32784779 z - 709119868 z + 10975059271 z - 125588807629 z 18 16 50 - 7257203511903 z + 1087799788322 z - 7257203511903 z 48 20 36 + 37789667430485 z + 37789667430485 z + 6451119838019582 z 34 66 64 30 - 7164938842455272 z - 227 z + 20618 z - 4706337105377130 z 42 44 46 - 1322951697705742 z + 506563465923802 z - 155176931956752 z 58 56 54 52 - 709119868 z + 10975059271 z - 125588807629 z + 1087799788322 z 60 68 32 38 + 32784779 z + z + 6451119838019582 z - 4706337105377130 z 40 62 / 28 + 2777746879345236 z - 1033849 z ) / (-1 - 16095834687712890 z / 26 2 24 + 6983049638713382 z + 295 z - 2438622880914858 z 22 4 6 8 + 681565205754085 z - 32361 z + 1898343 z - 69143889 z 10 12 14 18 + 1695641295 z - 29456991723 z + 375371570933 z + 26455307595565 z 16 50 48 - 3598686880163 z + 151342033502691 z - 681565205754085 z 20 36 34 - 151342033502691 z - 55726050282111158 z + 55726050282111158 z 66 64 30 42 + 32361 z - 1898343 z + 29997059384135398 z + 16095834687712890 z 44 46 58 - 6983049638713382 z + 2438622880914858 z + 29456991723 z 56 54 52 - 375371570933 z + 3598686880163 z - 26455307595565 z 60 70 68 32 - 1695641295 z + z - 295 z - 45347309218114698 z 38 40 62 + 45347309218114698 z - 29997059384135398 z + 69143889 z ) And in Maple-input format, it is: -(1+2777746879345236*z^28-1322951697705742*z^26-227*z^2+506563465923802*z^24-\ 155176931956752*z^22+20618*z^4-1033849*z^6+32784779*z^8-709119868*z^10+ 10975059271*z^12-125588807629*z^14-7257203511903*z^18+1087799788322*z^16-\ 7257203511903*z^50+37789667430485*z^48+37789667430485*z^20+6451119838019582*z^ 36-7164938842455272*z^34-227*z^66+20618*z^64-4706337105377130*z^30-\ 1322951697705742*z^42+506563465923802*z^44-155176931956752*z^46-709119868*z^58+ 10975059271*z^56-125588807629*z^54+1087799788322*z^52+32784779*z^60+z^68+ 6451119838019582*z^32-4706337105377130*z^38+2777746879345236*z^40-1033849*z^62) /(-1-16095834687712890*z^28+6983049638713382*z^26+295*z^2-2438622880914858*z^24 +681565205754085*z^22-32361*z^4+1898343*z^6-69143889*z^8+1695641295*z^10-\ 29456991723*z^12+375371570933*z^14+26455307595565*z^18-3598686880163*z^16+ 151342033502691*z^50-681565205754085*z^48-151342033502691*z^20-\ 55726050282111158*z^36+55726050282111158*z^34+32361*z^66-1898343*z^64+ 29997059384135398*z^30+16095834687712890*z^42-6983049638713382*z^44+ 2438622880914858*z^46+29456991723*z^58-375371570933*z^56+3598686880163*z^54-\ 26455307595565*z^52-1695641295*z^60+z^70-295*z^68-45347309218114698*z^32+ 45347309218114698*z^38-29997059384135398*z^40+69143889*z^62) The first , 40, terms are: [0, 68, 0, 8317, 0, 1117461, 0, 153232772, 0, 21114768025, 0, 2913167050601, 0, 402061388597892, 0, 55495906607154853, 0, 7660224183726430317, 0, 1057366338236867629124, 0, 145952149683567046559537, 0, 20146323787466050082159313, 0, 2780873354159168340357669444, 0, 383854501335358370167679352205, 0, 52984894314852458915222776750213, 0, 7313706183620466750127388247640708, 0, 1009538641404544247893115989447387081, 0, 139350452883611853322153883515165685305, 0, 19235072264706319731374801815729032862084, 0, 2655090079627394662920939989580197910160693] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 14572880 z + 70400288 z + 154 z - 225078469 z 22 4 6 8 10 + 484692256 z - 7184 z + 157613 z - 1938021 z + 14572880 z 12 14 18 16 - 70400288 z + 225078469 z + 709809390 z - 484692256 z 20 36 34 30 32 38 - 709809390 z - 154 z + 7184 z + 1938021 z - 157613 z + z ) / 18 16 12 24 / (-4876219316 z + 2809170759 z + 296863171 z + 2809170759 z / 22 26 14 8 - 4876219316 z - 1110667462 z - 1110667462 z + 5851108 z 6 20 2 40 32 - 390342 z + 5855446264 z - 218 z + z + 5851108 z + 1 10 38 34 36 28 - 52327398 z - 218 z - 390342 z + 13973 z + 296863171 z 30 4 - 52327398 z + 13973 z ) And in Maple-input format, it is: -(-1-14572880*z^28+70400288*z^26+154*z^2-225078469*z^24+484692256*z^22-7184*z^4 +157613*z^6-1938021*z^8+14572880*z^10-70400288*z^12+225078469*z^14+709809390*z^ 18-484692256*z^16-709809390*z^20-154*z^36+7184*z^34+1938021*z^30-157613*z^32+z^ 38)/(-4876219316*z^18+2809170759*z^16+296863171*z^12+2809170759*z^24-4876219316 *z^22-1110667462*z^26-1110667462*z^14+5851108*z^8-390342*z^6+5855446264*z^20-\ 218*z^2+z^40+5851108*z^32+1-52327398*z^10-218*z^38-390342*z^34+13973*z^36+ 296863171*z^28-52327398*z^30+13973*z^4) The first , 40, terms are: [0, 64, 0, 7163, 0, 899991, 0, 117178240, 0, 15428585429, 0, 2038615366909, 0, 269664877302720, 0, 35683209600808911, 0, 4722267480652923299, 0, 624959723486072241024, 0, 82710016205161129537913, 0, 10946256818966318747213769, 0, 1448683838839245024562830336, 0, 191726321438233523674412970515, 0, 25374056868547267680509277916959, 0, 3358134542406531697436066957359680, 0, 444432980393049269461287133769474029, 0, 58818570903468561355969940159759130469, 0, 7784355435315591027077314470396234577664, 0, 1030222064755727499517656651263805733573575] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2207141703653405070 z - 533809049158710619 z - 301 z 24 22 4 6 + 107686670480354510 z - 18008798946206245 z + 38245 z - 2764006 z 8 10 12 14 + 129090267 z - 4195300487 z + 99666411032 z - 1792753733989 z 18 16 50 - 277723030725457 z + 25061278272892 z - 202300812041635454659 z 48 20 + 303512201782094042889 z + 2477555561559290 z 36 34 + 114491146766699076945 z - 54940616107163660596 z 66 80 88 84 86 - 18008798946206245 z + 129090267 z + z + 38245 z - 301 z 82 64 30 - 2764006 z + 107686670480354510 z - 7649658353536330483 z 42 44 - 387010077508255323462 z + 419640919689245147156 z 46 58 - 387010077508255323462 z - 7649658353536330483 z 56 54 + 22312121894990563697 z - 54940616107163660596 z 52 60 + 114491146766699076945 z + 2207141703653405070 z 70 68 78 - 277723030725457 z + 2477555561559290 z - 4195300487 z 32 38 + 22312121894990563697 z - 202300812041635454659 z 40 62 76 + 303512201782094042889 z - 533809049158710619 z + 99666411032 z 74 72 / - 1792753733989 z + 25061278272892 z ) / (-1 / 28 26 2 - 9729304158608089122 z + 2189812087837026343 z + 373 z 24 22 4 6 - 410867537248181915 z + 63836366168534956 z - 56172 z + 4674700 z 8 10 12 14 - 246640110 z + 8934743998 z - 234281304762 z + 4616643843738 z 18 16 50 + 844769484107337 z - 70295310677389 z + 2077795024217532097599 z 48 20 - 2862517222227202614787 z - 8145324868964924 z 36 34 - 674354422663958131456 z + 300688761925868539690 z 66 80 90 88 84 + 410867537248181915 z - 8934743998 z + z - 373 z - 4674700 z 86 82 64 + 56172 z + 246640110 z - 2189812087837026343 z 30 42 + 36229751829128834258 z + 2862517222227202614787 z 44 46 - 3359246722569387590536 z + 3359246722569387590536 z 58 56 + 113564110270860881978 z - 300688761925868539690 z 54 52 + 674354422663958131456 z - 1283757886559946648816 z 60 70 68 - 36229751829128834258 z + 8145324868964924 z - 63836366168534956 z 78 32 + 234281304762 z - 113564110270860881978 z 38 40 + 1283757886559946648816 z - 2077795024217532097599 z 62 76 74 + 9729304158608089122 z - 4616643843738 z + 70295310677389 z 72 - 844769484107337 z ) And in Maple-input format, it is: -(1+2207141703653405070*z^28-533809049158710619*z^26-301*z^2+107686670480354510 *z^24-18008798946206245*z^22+38245*z^4-2764006*z^6+129090267*z^8-4195300487*z^ 10+99666411032*z^12-1792753733989*z^14-277723030725457*z^18+25061278272892*z^16 -202300812041635454659*z^50+303512201782094042889*z^48+2477555561559290*z^20+ 114491146766699076945*z^36-54940616107163660596*z^34-18008798946206245*z^66+ 129090267*z^80+z^88+38245*z^84-301*z^86-2764006*z^82+107686670480354510*z^64-\ 7649658353536330483*z^30-387010077508255323462*z^42+419640919689245147156*z^44-\ 387010077508255323462*z^46-7649658353536330483*z^58+22312121894990563697*z^56-\ 54940616107163660596*z^54+114491146766699076945*z^52+2207141703653405070*z^60-\ 277723030725457*z^70+2477555561559290*z^68-4195300487*z^78+22312121894990563697 *z^32-202300812041635454659*z^38+303512201782094042889*z^40-533809049158710619* z^62+99666411032*z^76-1792753733989*z^74+25061278272892*z^72)/(-1-\ 9729304158608089122*z^28+2189812087837026343*z^26+373*z^2-410867537248181915*z^ 24+63836366168534956*z^22-56172*z^4+4674700*z^6-246640110*z^8+8934743998*z^10-\ 234281304762*z^12+4616643843738*z^14+844769484107337*z^18-70295310677389*z^16+ 2077795024217532097599*z^50-2862517222227202614787*z^48-8145324868964924*z^20-\ 674354422663958131456*z^36+300688761925868539690*z^34+410867537248181915*z^66-\ 8934743998*z^80+z^90-373*z^88-4674700*z^84+56172*z^86+246640110*z^82-\ 2189812087837026343*z^64+36229751829128834258*z^30+2862517222227202614787*z^42-\ 3359246722569387590536*z^44+3359246722569387590536*z^46+113564110270860881978*z ^58-300688761925868539690*z^56+674354422663958131456*z^54-\ 1283757886559946648816*z^52-36229751829128834258*z^60+8145324868964924*z^70-\ 63836366168534956*z^68+234281304762*z^78-113564110270860881978*z^32+ 1283757886559946648816*z^38-2077795024217532097599*z^40+9729304158608089122*z^ 62-4616643843738*z^76+70295310677389*z^74-844769484107337*z^72) The first , 40, terms are: [0, 72, 0, 8929, 0, 1196827, 0, 163885240, 0, 22622780167, 0, 3133779609847, 0, 434795740754264, 0, 60370789291732375, 0, 8385358879706476817, 0, 1164901101913684152232, 0, 161841886272530447643197, 0, 22485844146735259016729901, 0, 3124174317253828464381015624, 0, 434075256492077887324473134865, 0, 60311005603513436513729848382887, 0, 8379709249372277154736148105372664, 0, 1164291511303868549055870097111483607, 0, 161768777205135024739033532836637459463, 0, 22476452415168881329925957315616769806424, 0, 3122920085149728260033816085696039670864075] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 8993887686490308 z - 3678454845067608 z - 236 z 24 22 4 6 + 1224719701326591 z - 330325772975260 z + 22426 z - 1178942 z 8 10 12 14 + 39301122 z - 897807192 z + 14775249150 z - 181321355480 z 18 16 50 - 12428754611206 z + 1701126106634 z - 330325772975260 z 48 20 36 + 1224719701326591 z + 71714475926562 z + 43556095733636100 z 34 66 64 - 39483392884299504 z - 1178942 z + 39301122 z 30 42 44 - 17966666946544828 z - 17966666946544828 z + 8993887686490308 z 46 58 56 - 3678454845067608 z - 181321355480 z + 1701126106634 z 54 52 60 70 - 12428754611206 z + 71714475926562 z + 14775249150 z - 236 z 68 32 38 + 22426 z + 29401365759984500 z - 39483392884299504 z 40 62 72 / + 29401365759984500 z - 897807192 z + z ) / (-1 / 28 26 2 - 48948962003708866 z + 18360465939969157 z + 304 z 24 22 4 6 - 5609028617270618 z + 1387663168162033 z - 34717 z + 2129119 z 8 10 12 14 - 81310177 z + 2100008659 z - 38672749755 z + 526782735327 z 18 16 50 + 43721756368517 z - 5451240548607 z + 5609028617270618 z 48 20 36 - 18360465939969157 z - 275995982075067 z - 341207936578473046 z 34 66 64 + 281304127972619230 z + 81310177 z - 2100008659 z 30 42 44 + 106749718082099670 z + 191062005222335530 z - 106749718082099670 z 46 58 56 + 48948962003708866 z + 5451240548607 z - 43721756368517 z 54 52 60 + 275995982075067 z - 1387663168162033 z - 526782735327 z 70 68 32 + 34717 z - 2129119 z - 191062005222335530 z 38 40 62 74 + 341207936578473046 z - 281304127972619230 z + 38672749755 z + z 72 - 304 z ) And in Maple-input format, it is: -(1+8993887686490308*z^28-3678454845067608*z^26-236*z^2+1224719701326591*z^24-\ 330325772975260*z^22+22426*z^4-1178942*z^6+39301122*z^8-897807192*z^10+ 14775249150*z^12-181321355480*z^14-12428754611206*z^18+1701126106634*z^16-\ 330325772975260*z^50+1224719701326591*z^48+71714475926562*z^20+ 43556095733636100*z^36-39483392884299504*z^34-1178942*z^66+39301122*z^64-\ 17966666946544828*z^30-17966666946544828*z^42+8993887686490308*z^44-\ 3678454845067608*z^46-181321355480*z^58+1701126106634*z^56-12428754611206*z^54+ 71714475926562*z^52+14775249150*z^60-236*z^70+22426*z^68+29401365759984500*z^32 -39483392884299504*z^38+29401365759984500*z^40-897807192*z^62+z^72)/(-1-\ 48948962003708866*z^28+18360465939969157*z^26+304*z^2-5609028617270618*z^24+ 1387663168162033*z^22-34717*z^4+2129119*z^6-81310177*z^8+2100008659*z^10-\ 38672749755*z^12+526782735327*z^14+43721756368517*z^18-5451240548607*z^16+ 5609028617270618*z^50-18360465939969157*z^48-275995982075067*z^20-\ 341207936578473046*z^36+281304127972619230*z^34+81310177*z^66-2100008659*z^64+ 106749718082099670*z^30+191062005222335530*z^42-106749718082099670*z^44+ 48948962003708866*z^46+5451240548607*z^58-43721756368517*z^56+275995982075067*z ^54-1387663168162033*z^52-526782735327*z^60+34717*z^70-2129119*z^68-\ 191062005222335530*z^32+341207936578473046*z^38-281304127972619230*z^40+ 38672749755*z^62+z^74-304*z^72) The first , 40, terms are: [0, 68, 0, 8381, 0, 1137245, 0, 157530340, 0, 21924744465, 0, 3054913935505, 0, 425779617497796, 0, 59347305945977117, 0, 8272270738734926333, 0, 1153055949448490066916, 0, 160722435172253537898113, 0, 22402823049112950522770817, 0, 3122691088661632249872899876, 0, 435266563861230453274798595069, 0, 60671061302971237671339990011933, 0, 8456835395697063410104709389051524, 0, 1178783812160061230608404611194899601, 0, 164308658132182129891647081257151036177, 0, 22902702649286881083072434787641391104548, 0, 3192368525230821520838928915236930214318685] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1263518619161063750 z - 305969712711169772 z - 275 z 24 22 4 6 + 61937868228967784 z - 10424242515873668 z + 31870 z - 2119951 z 8 10 12 14 + 92288127 z - 2832029998 z + 64275670340 z - 1115658768540 z 18 16 50 - 164788047262282 z + 15176190537252 z - 116741665430137623026 z 48 20 + 175437307021305577552 z + 1448629967481162 z 36 34 + 65933985886474995276 z - 31570351004923642420 z 66 80 88 84 86 - 10424242515873668 z + 92288127 z + z + 31870 z - 275 z 82 64 30 - 2119951 z + 61937868228967784 z - 4380728825680004214 z 42 44 - 223939771096353296974 z + 242910689564478558420 z 46 58 - 223939771096353296974 z - 4380728825680004214 z 56 54 + 12795518171072616108 z - 31570351004923642420 z 52 60 70 + 65933985886474995276 z + 1263518619161063750 z - 164788047262282 z 68 78 32 + 1448629967481162 z - 2832029998 z + 12795518171072616108 z 38 40 - 116741665430137623026 z + 175437307021305577552 z 62 76 74 - 305969712711169772 z + 64275670340 z - 1115658768540 z 72 / 28 + 15176190537252 z ) / (-1 - 5523456353183115890 z / 26 2 24 + 1241663976053408806 z + 348 z - 233290784760487618 z 22 4 6 8 + 36422137398101878 z - 47738 z + 3632420 z - 177288154 z 10 12 14 + 6022290425 z - 150014325154 z + 2840891683194 z 18 16 50 + 493736924184354 z - 41983201278098 z + 1209466693321355098030 z 48 20 - 1670577373826639558642 z - 4690642771168876 z 36 34 - 389376333836291053326 z + 172781894568008444022 z 66 80 90 88 84 + 233290784760487618 z - 6022290425 z + z - 348 z - 3632420 z 86 82 64 + 47738 z + 177288154 z - 1241663976053408806 z 30 42 + 20631281254104099324 z + 1670577373826639558642 z 44 46 - 1963112068253075496714 z + 1963112068253075496714 z 58 56 + 64944051354678312646 z - 172781894568008444022 z 54 52 + 389376333836291053326 z - 744539519365244695430 z 60 70 68 - 20631281254104099324 z + 4690642771168876 z - 36422137398101878 z 78 32 38 + 150014325154 z - 64944051354678312646 z + 744539519365244695430 z 40 62 76 - 1209466693321355098030 z + 5523456353183115890 z - 2840891683194 z 74 72 + 41983201278098 z - 493736924184354 z ) And in Maple-input format, it is: -(1+1263518619161063750*z^28-305969712711169772*z^26-275*z^2+61937868228967784* z^24-10424242515873668*z^22+31870*z^4-2119951*z^6+92288127*z^8-2832029998*z^10+ 64275670340*z^12-1115658768540*z^14-164788047262282*z^18+15176190537252*z^16-\ 116741665430137623026*z^50+175437307021305577552*z^48+1448629967481162*z^20+ 65933985886474995276*z^36-31570351004923642420*z^34-10424242515873668*z^66+ 92288127*z^80+z^88+31870*z^84-275*z^86-2119951*z^82+61937868228967784*z^64-\ 4380728825680004214*z^30-223939771096353296974*z^42+242910689564478558420*z^44-\ 223939771096353296974*z^46-4380728825680004214*z^58+12795518171072616108*z^56-\ 31570351004923642420*z^54+65933985886474995276*z^52+1263518619161063750*z^60-\ 164788047262282*z^70+1448629967481162*z^68-2832029998*z^78+12795518171072616108 *z^32-116741665430137623026*z^38+175437307021305577552*z^40-305969712711169772* z^62+64275670340*z^76-1115658768540*z^74+15176190537252*z^72)/(-1-\ 5523456353183115890*z^28+1241663976053408806*z^26+348*z^2-233290784760487618*z^ 24+36422137398101878*z^22-47738*z^4+3632420*z^6-177288154*z^8+6022290425*z^10-\ 150014325154*z^12+2840891683194*z^14+493736924184354*z^18-41983201278098*z^16+ 1209466693321355098030*z^50-1670577373826639558642*z^48-4690642771168876*z^20-\ 389376333836291053326*z^36+172781894568008444022*z^34+233290784760487618*z^66-\ 6022290425*z^80+z^90-348*z^88-3632420*z^84+47738*z^86+177288154*z^82-\ 1241663976053408806*z^64+20631281254104099324*z^30+1670577373826639558642*z^42-\ 1963112068253075496714*z^44+1963112068253075496714*z^46+64944051354678312646*z^ 58-172781894568008444022*z^56+389376333836291053326*z^54-744539519365244695430* z^52-20631281254104099324*z^60+4690642771168876*z^70-36422137398101878*z^68+ 150014325154*z^78-64944051354678312646*z^32+744539519365244695430*z^38-\ 1209466693321355098030*z^40+5523456353183115890*z^62-2840891683194*z^76+ 41983201278098*z^74-493736924184354*z^72) The first , 40, terms are: [0, 73, 0, 9536, 0, 1346123, 0, 193387869, 0, 27924740943, 0, 4038812575169, 0, 584452543079128, 0, 84590444604523103, 0, 12243872119090452423, 0, 1772248921591700632335, 0, 256527230244275218469095, 0, 37131557020674005296111048, 0, 5374686797321205117979922993, 0, 777970745067445895303240760655, 0, 112609078671978729618397719218189, 0, 16299848054846188908816718799325195, 0, 2359357274661043145904295853180463856, 0, 341510347037311846533160160953405082977, 0, 49432664777106221059105909939722468722329, 0, 7155239567296271005652968202312088354562057] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {3, 4}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 81772104468 z + 162417574181 z + 194 z - 228597436663 z 22 4 6 8 10 + 228597436663 z - 13895 z + 504807 z - 10668033 z + 141273295 z 12 14 18 16 - 1226448412 z + 7187615067 z + 81772104468 z - 29011869329 z 20 36 34 30 - 162417574181 z - 141273295 z + 1226448412 z + 29011869329 z 42 44 46 32 38 40 + 13895 z - 194 z + z - 7187615067 z + 10668033 z - 504807 z ) / 28 26 2 / (1 + 1015190179646 z - 1684125659988 z - 270 z / 24 22 4 6 + 1992809782217 z - 1684125659988 z + 24046 z - 1041932 z 8 10 12 14 + 25850493 z - 399970492 z + 4056154754 z - 27806064790 z 18 16 48 20 - 434829966134 z + 131488801478 z + z + 1015190179646 z 36 34 30 42 + 4056154754 z - 27806064790 z - 434829966134 z - 1041932 z 44 46 32 38 40 + 24046 z - 270 z + 131488801478 z - 399970492 z + 25850493 z ) And in Maple-input format, it is: -(-1-81772104468*z^28+162417574181*z^26+194*z^2-228597436663*z^24+228597436663* z^22-13895*z^4+504807*z^6-10668033*z^8+141273295*z^10-1226448412*z^12+ 7187615067*z^14+81772104468*z^18-29011869329*z^16-162417574181*z^20-141273295*z ^36+1226448412*z^34+29011869329*z^30+13895*z^42-194*z^44+z^46-7187615067*z^32+ 10668033*z^38-504807*z^40)/(1+1015190179646*z^28-1684125659988*z^26-270*z^2+ 1992809782217*z^24-1684125659988*z^22+24046*z^4-1041932*z^6+25850493*z^8-\ 399970492*z^10+4056154754*z^12-27806064790*z^14-434829966134*z^18+131488801478* z^16+z^48+1015190179646*z^20+4056154754*z^36-27806064790*z^34-434829966134*z^30 -1041932*z^42+24046*z^44-270*z^46+131488801478*z^32-399970492*z^38+25850493*z^ 40) The first , 40, terms are: [0, 76, 0, 10369, 0, 1509259, 0, 222171328, 0, 32792469283, 0, 4843704490843, 0, 715604467779568, 0, 105729384439515727, 0, 15621636844637219257, 0, 2308127709232557227596, 0, 341031013860237186188005, 0, 50388118572928821972980533, 0, 7444961860902232712547883084, 0, 1100010504698442096327547826809, 0, 162529123781992701661862766457423, 0, 24014058119900463110897327171816368, 0, 3548133249764946749885052379527505147, 0, 524244986010062971842701772957160456339, 0, 77458422794045853943667963472690920756320, 0, 11444663129015705120466975269913440451094747] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {3, 5}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 189793 z + 2274240 z + 160 z - 15321391 z + 61452128 z 4 6 8 10 12 - 8231 z + 189793 z - 2274240 z + 15321391 z - 61452128 z 14 18 16 20 34 + 152297792 z + 238454984 z - 238454984 z - 152297792 z + z 30 32 / 14 28 10 + 8231 z - 160 z ) / (-854617904 z + 6866016 z - 57488708 z / 2 34 32 26 30 - 236 z - 236 z + 15608 z - 57488708 z - 456344 z 16 36 22 20 18 + 1636862632 z + z - 854617904 z + 1636862632 z - 2028149600 z 12 4 6 8 24 + 282893591 z + 1 + 15608 z - 456344 z + 6866016 z + 282893591 z ) And in Maple-input format, it is: -(-1-189793*z^28+2274240*z^26+160*z^2-15321391*z^24+61452128*z^22-8231*z^4+ 189793*z^6-2274240*z^8+15321391*z^10-61452128*z^12+152297792*z^14+238454984*z^ 18-238454984*z^16-152297792*z^20+z^34+8231*z^30-160*z^32)/(-854617904*z^14+ 6866016*z^28-57488708*z^10-236*z^2-236*z^34+15608*z^32-57488708*z^26-456344*z^ 30+1636862632*z^16+z^36-854617904*z^22+1636862632*z^20-2028149600*z^18+ 282893591*z^12+1+15608*z^4-456344*z^6+6866016*z^8+282893591*z^24) The first , 40, terms are: [0, 76, 0, 10559, 0, 1572267, 0, 236340508, 0, 35575302949, 0, 5356112896349, 0, 806426903189820, 0, 121417861483751379, 0, 18281024616991347543, 0, 2752444394981473196716, 0, 414416068968083193202953, 0, 62395694328135121191935865, 0, 9394478077109978884397076716, 0, 1414460073089255436295082223527, 0, 212965242137899724101366716692483, 0, 32064669213473715456713415190307836, 0, 4827750300707655509405656676695505261, 0, 726880193611948081093785124650602226709, 0, 109441206142733974840540669151917080047708, 0, 16477787821483652473958150156469709729283707] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 4 f(z) = - (-1 - 165 z + 8176 z + 165 z - 142241 z + 1197844 z - 8176 z 6 8 10 12 14 + 142241 z - 1197844 z + 5583988 z - 15296729 z + 25232517 z 18 16 20 30 / 26 + 15296729 z - 25232517 z - 5583988 z + z ) / (-373374 z / 10 22 14 16 - 24109122 z - 24109122 z - 176222612 z + 225659905 z 18 24 28 30 32 6 - 176222612 z + 1 + 4051219 z + 16410 z - 242 z + z - 373374 z 4 12 2 8 20 + 16410 z + 83915880 z - 242 z + 4051219 z + 83915880 z ) And in Maple-input format, it is: -(-1-165*z^28+8176*z^26+165*z^2-142241*z^24+1197844*z^22-8176*z^4+142241*z^6-\ 1197844*z^8+5583988*z^10-15296729*z^12+25232517*z^14+15296729*z^18-25232517*z^ 16-5583988*z^20+z^30)/(-373374*z^26-24109122*z^10-24109122*z^22-176222612*z^14+ 225659905*z^16-176222612*z^18+1+4051219*z^24+16410*z^28-242*z^30+z^32-373374*z^ 6+16410*z^4+83915880*z^12-242*z^2+4051219*z^8+83915880*z^20) The first , 40, terms are: [0, 77, 0, 10400, 0, 1484363, 0, 214448269, 0, 31127755139, 0, 4527698305753, 0, 659196880898480, 0, 96014560857625527, 0, 13987569799519980291, 0, 2037909953092232520591, 0, 296923562130582651396667, 0, 43262536727978917151682640, 0, 6303514416469595241853656973, 0, 918449065673752444441518107719, 0, 133822181543036338415436176490609, 0, 19498511143415090341938145774579343, 0, 2841024242821563070606416293337800640, 0, 413950577456919391696152909238953059537, 0, 60314547647852709864185561610188768196141, 0, 8788113765486507232396050711411449104692421] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 271987477628078 z - 155589877001672 z - 203 z 24 22 4 6 + 71018025803326 z - 25792723884558 z + 15856 z - 671417 z 8 10 12 14 + 17865309 z - 324063660 z + 4215219046 z - 40660848590 z 18 16 50 - 1682205487990 z + 297718549812 z - 40660848590 z 48 20 36 + 297718549812 z + 7422385133096 z + 271987477628078 z 34 64 30 42 - 380034568225208 z + z - 380034568225208 z - 25792723884558 z 44 46 58 56 + 7422385133096 z - 1682205487990 z - 671417 z + 17865309 z 54 52 60 32 - 324063660 z + 4215219046 z + 15856 z + 424821047138572 z 38 40 62 / 2 - 155589877001672 z + 71018025803326 z - 203 z ) / ((-1 + z ) (1 / 28 26 2 + 1130512133154590 z - 630066166007559 z - 277 z 24 22 4 6 + 277448392695290 z - 96304428775521 z + 26116 z - 1275074 z 8 10 12 14 + 38248807 z - 771459663 z + 11044533388 z - 116243847221 z 18 16 50 - 5587652761139 z + 921146043006 z - 116243847221 z 48 20 36 + 921146043006 z + 26251100302936 z + 1130512133154590 z 34 64 30 42 - 1604818186401487 z + z - 1604818186401487 z - 96304428775521 z 44 46 58 56 + 26251100302936 z - 5587652761139 z - 1275074 z + 38248807 z 54 52 60 32 - 771459663 z + 11044533388 z + 26116 z + 1803499325647694 z 38 40 62 - 630066166007559 z + 277448392695290 z - 277 z )) And in Maple-input format, it is: -(1+271987477628078*z^28-155589877001672*z^26-203*z^2+71018025803326*z^24-\ 25792723884558*z^22+15856*z^4-671417*z^6+17865309*z^8-324063660*z^10+4215219046 *z^12-40660848590*z^14-1682205487990*z^18+297718549812*z^16-40660848590*z^50+ 297718549812*z^48+7422385133096*z^20+271987477628078*z^36-380034568225208*z^34+ z^64-380034568225208*z^30-25792723884558*z^42+7422385133096*z^44-1682205487990* z^46-671417*z^58+17865309*z^56-324063660*z^54+4215219046*z^52+15856*z^60+ 424821047138572*z^32-155589877001672*z^38+71018025803326*z^40-203*z^62)/(-1+z^2 )/(1+1130512133154590*z^28-630066166007559*z^26-277*z^2+277448392695290*z^24-\ 96304428775521*z^22+26116*z^4-1275074*z^6+38248807*z^8-771459663*z^10+ 11044533388*z^12-116243847221*z^14-5587652761139*z^18+921146043006*z^16-\ 116243847221*z^50+921146043006*z^48+26251100302936*z^20+1130512133154590*z^36-\ 1604818186401487*z^34+z^64-1604818186401487*z^30-96304428775521*z^42+ 26251100302936*z^44-5587652761139*z^46-1275074*z^58+38248807*z^56-771459663*z^ 54+11044533388*z^52+26116*z^60+1803499325647694*z^32-630066166007559*z^38+ 277448392695290*z^40-277*z^62) The first , 40, terms are: [0, 75, 0, 10313, 0, 1517312, 0, 225552405, 0, 33597679179, 0, 5006978964369, 0, 746264031961617, 0, 111229991834080651, 0, 16578854311341730093, 0, 2471086480149448852480, 0, 368316846246257557904785, 0, 54897842310059605529469531, 0, 8182555904651104657123736689, 0, 1219614812769021946453992281553, 0, 181784311883737538287808604837083, 0, 27095059619140801853767871975181729, 0, 4038534724232020003717745037211684288, 0, 601945998593190372088460801593962246941, 0, 89720408506435051480818351041132100747179, 0, 13372880160998915396354945409689489426165009] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 19921120374261380 z - 8906316557069666 z - 273 z 24 22 4 6 + 3150045281805562 z - 878173527855424 z + 30230 z - 1833051 z 8 10 12 14 + 69562331 z - 1782720032 z + 32384570983 z - 430937977087 z 18 16 50 - 32701955433293 z + 4299264486630 z - 32701955433293 z 48 20 36 + 192005409760677 z + 192005409760677 z + 49818726351050270 z 34 66 64 30 - 55852819087604160 z - 273 z + 30230 z - 35342194841944358 z 42 44 46 - 8906316557069666 z + 3150045281805562 z - 878173527855424 z 58 56 54 52 - 1782720032 z + 32384570983 z - 430937977087 z + 4299264486630 z 60 68 32 38 + 69562331 z + z + 49818726351050270 z - 35342194841944358 z 40 62 / 2 + 19921120374261380 z - 1833051 z ) / ((-1 + z ) (1 / 28 26 2 + 78471764330526826 z - 34133024806223576 z - 340 z 24 22 4 6 + 11660812850996006 z - 3117512072216368 z + 45279 z - 3227648 z 8 10 12 14 + 140867985 z - 4070802728 z + 81982221321 z - 1192157745344 z 18 16 50 - 104356607514772 z + 12840220520407 z - 104356607514772 z 48 20 36 + 648838108857913 z + 648838108857913 z + 202652317927661382 z 34 66 64 30 - 228129260648293392 z - 340 z + 45279 z - 142025943268056960 z 42 44 46 - 34133024806223576 z + 11660812850996006 z - 3117512072216368 z 58 56 54 - 4070802728 z + 81982221321 z - 1192157745344 z 52 60 68 32 + 12840220520407 z + 140867985 z + z + 202652317927661382 z 38 40 62 - 142025943268056960 z + 78471764330526826 z - 3227648 z )) And in Maple-input format, it is: -(1+19921120374261380*z^28-8906316557069666*z^26-273*z^2+3150045281805562*z^24-\ 878173527855424*z^22+30230*z^4-1833051*z^6+69562331*z^8-1782720032*z^10+ 32384570983*z^12-430937977087*z^14-32701955433293*z^18+4299264486630*z^16-\ 32701955433293*z^50+192005409760677*z^48+192005409760677*z^20+49818726351050270 *z^36-55852819087604160*z^34-273*z^66+30230*z^64-35342194841944358*z^30-\ 8906316557069666*z^42+3150045281805562*z^44-878173527855424*z^46-1782720032*z^ 58+32384570983*z^56-430937977087*z^54+4299264486630*z^52+69562331*z^60+z^68+ 49818726351050270*z^32-35342194841944358*z^38+19921120374261380*z^40-1833051*z^ 62)/(-1+z^2)/(1+78471764330526826*z^28-34133024806223576*z^26-340*z^2+ 11660812850996006*z^24-3117512072216368*z^22+45279*z^4-3227648*z^6+140867985*z^ 8-4070802728*z^10+81982221321*z^12-1192157745344*z^14-104356607514772*z^18+ 12840220520407*z^16-104356607514772*z^50+648838108857913*z^48+648838108857913*z ^20+202652317927661382*z^36-228129260648293392*z^34-340*z^66+45279*z^64-\ 142025943268056960*z^30-34133024806223576*z^42+11660812850996006*z^44-\ 3117512072216368*z^46-4070802728*z^58+81982221321*z^56-1192157745344*z^54+ 12840220520407*z^52+140867985*z^60+z^68+202652317927661382*z^32-\ 142025943268056960*z^38+78471764330526826*z^40-3227648*z^62) The first , 40, terms are: [0, 68, 0, 7799, 0, 997243, 0, 132303016, 0, 17778105349, 0, 2399629491017, 0, 324412436673712, 0, 43883937328528871, 0, 5937584760368857779, 0, 803436206800332507516, 0, 108719609349544604939941, 0, 14711957716080723998076637, 0, 1990836541669324273838024716, 0, 269402625935708939141759483515, 0, 36455957893435959923669639061199, 0, 4933275934358197582315679197180640, 0, 667578576625750881283545564535321857, 0, 90337780148689441831043508352100868173, 0, 12224650610453023432462172222445234774904, 0, 1654258993449704895613814408843762550811923] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 24647345010360056 z - 11150871321739006 z - 283 z 24 22 4 6 + 3993328682850842 z - 1125907069503584 z + 32764 z - 2081625 z 8 10 12 14 + 82435355 z - 2188170776 z + 40813121511 z - 552757492701 z 18 16 50 - 42439887862679 z + 5567036487180 z - 42439887862679 z 48 20 36 + 248238244152437 z + 248238244152437 z + 60654535013183086 z 34 66 64 30 - 67851027355970832 z - 283 z + 32764 z - 43303767985133338 z 42 44 46 - 11150871321739006 z + 3993328682850842 z - 1125907069503584 z 58 56 54 52 - 2188170776 z + 40813121511 z - 552757492701 z + 5567036487180 z 60 68 32 38 + 82435355 z + z + 60654535013183086 z - 43303767985133338 z 40 62 / 28 + 24647345010360056 z - 2081625 z ) / (-1 - 142481942202038250 z / 26 2 24 + 58462177748070486 z + 359 z - 19031353509535994 z 22 4 6 8 + 4884974285341397 z - 50137 z + 3741883 z - 170886853 z 10 12 14 + 5156648195 z - 108096509679 z + 1630750937465 z 18 16 50 + 152625549777165 z - 18173739833479 z + 980983970156787 z 48 20 36 - 4884974285341397 z - 980983970156787 z - 535979375806400574 z 34 66 64 + 535979375806400574 z + 50137 z - 3741883 z 30 42 44 + 276812815256848878 z + 142481942202038250 z - 58462177748070486 z 46 58 56 + 19031353509535994 z + 108096509679 z - 1630750937465 z 54 52 60 70 + 18173739833479 z - 152625549777165 z - 5156648195 z + z 68 32 38 - 359 z - 430185229692279058 z + 430185229692279058 z 40 62 - 276812815256848878 z + 170886853 z ) And in Maple-input format, it is: -(1+24647345010360056*z^28-11150871321739006*z^26-283*z^2+3993328682850842*z^24 -1125907069503584*z^22+32764*z^4-2081625*z^6+82435355*z^8-2188170776*z^10+ 40813121511*z^12-552757492701*z^14-42439887862679*z^18+5567036487180*z^16-\ 42439887862679*z^50+248238244152437*z^48+248238244152437*z^20+60654535013183086 *z^36-67851027355970832*z^34-283*z^66+32764*z^64-43303767985133338*z^30-\ 11150871321739006*z^42+3993328682850842*z^44-1125907069503584*z^46-2188170776*z ^58+40813121511*z^56-552757492701*z^54+5567036487180*z^52+82435355*z^60+z^68+ 60654535013183086*z^32-43303767985133338*z^38+24647345010360056*z^40-2081625*z^ 62)/(-1-142481942202038250*z^28+58462177748070486*z^26+359*z^2-\ 19031353509535994*z^24+4884974285341397*z^22-50137*z^4+3741883*z^6-170886853*z^ 8+5156648195*z^10-108096509679*z^12+1630750937465*z^14+152625549777165*z^18-\ 18173739833479*z^16+980983970156787*z^50-4884974285341397*z^48-980983970156787* z^20-535979375806400574*z^36+535979375806400574*z^34+50137*z^66-3741883*z^64+ 276812815256848878*z^30+142481942202038250*z^42-58462177748070486*z^44+ 19031353509535994*z^46+108096509679*z^58-1630750937465*z^56+18173739833479*z^54 -152625549777165*z^52-5156648195*z^60+z^70-359*z^68-430185229692279058*z^32+ 430185229692279058*z^38-276812815256848878*z^40+170886853*z^62) The first , 40, terms are: [0, 76, 0, 9911, 0, 1407895, 0, 204458108, 0, 29879708161, 0, 4375039709857, 0, 640998099373340, 0, 93934126179053687, 0, 13766488360498844503, 0, 2017601768746631388716, 0, 295700893808490685147361, 0, 43338293829262184664522913, 0, 6351726688732489085413170028, 0, 930919524419662210712231323287, 0, 136437144442416694655016040439351, 0, 19996462162958928559440524972848860, 0, 2930716009704390011461512305291671009, 0, 429530808278946073803247361819841464769, 0, 62952779027013511792208411870145760189628, 0, 9226468346783434157707924606089896632550103] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 10529433125156 z - 12219478996724 z - 246 z 24 22 4 6 + 10529433125156 z - 6725942247394 z + 23035 z - 1094910 z 8 10 12 14 + 30295488 z - 529026194 z + 6134112174 z - 48840054384 z 18 16 50 48 - 1093505523918 z + 273312441842 z - 246 z + 23035 z 20 36 34 + 3169647345312 z + 273312441842 z - 1093505523918 z 30 42 44 46 52 - 6725942247394 z - 529026194 z + 30295488 z - 1094910 z + z 32 38 40 / 2 + 3169647345312 z - 48840054384 z + 6134112174 z ) / ((-1 + z ) (1 / 28 26 2 24 + 46645515837470 z - 54738247255970 z - 339 z + 46645515837470 z 22 4 6 8 - 28840594542954 z + 39209 z - 2195102 z + 69654186 z 10 12 14 18 - 1372647238 z + 17762617068 z - 156378118500 z - 4159404409922 z 16 50 48 20 + 958802637404 z - 339 z + 39209 z + 12904677463966 z 36 34 30 + 958802637404 z - 4159404409922 z - 28840594542954 z 42 44 46 52 32 - 1372647238 z + 69654186 z - 2195102 z + z + 12904677463966 z 38 40 - 156378118500 z + 17762617068 z )) And in Maple-input format, it is: -(1+10529433125156*z^28-12219478996724*z^26-246*z^2+10529433125156*z^24-\ 6725942247394*z^22+23035*z^4-1094910*z^6+30295488*z^8-529026194*z^10+6134112174 *z^12-48840054384*z^14-1093505523918*z^18+273312441842*z^16-246*z^50+23035*z^48 +3169647345312*z^20+273312441842*z^36-1093505523918*z^34-6725942247394*z^30-\ 529026194*z^42+30295488*z^44-1094910*z^46+z^52+3169647345312*z^32-48840054384*z ^38+6134112174*z^40)/(-1+z^2)/(1+46645515837470*z^28-54738247255970*z^26-339*z^ 2+46645515837470*z^24-28840594542954*z^22+39209*z^4-2195102*z^6+69654186*z^8-\ 1372647238*z^10+17762617068*z^12-156378118500*z^14-4159404409922*z^18+ 958802637404*z^16-339*z^50+39209*z^48+12904677463966*z^20+958802637404*z^36-\ 4159404409922*z^34-28840594542954*z^30-1372647238*z^42+69654186*z^44-2195102*z^ 46+z^52+12904677463966*z^32-156378118500*z^38+17762617068*z^40) The first , 40, terms are: [0, 94, 0, 15447, 0, 2673869, 0, 466688938, 0, 81600911883, 0, 14274670069443, 0, 2497453420291130, 0, 436966528109477909, 0, 76454946709592464415, 0, 13377204161373107328750, 0, 2340593339873055012982457, 0, 409531093961794218746098377, 0, 71655232623394055439955648718, 0, 12537443247171097736703754986767, 0, 2193663782608117061179359787654917, 0, 383823141426695784205871503607722010, 0, 67157148581403101919733806022571539923, 0, 11750418686596875772805203992382485809147, 0, 2055958930873619273218754953269959717966282, 0, 359729064909558002324202334460089949555115229] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 18004039166884 z - 15830069666932 z - 216 z 24 22 4 6 + 10749772238808 z - 5621464580824 z + 17470 z - 730779 z 8 10 12 14 + 18325122 z - 299569441 z + 3367036850 z - 26967843928 z 18 16 50 48 - 686839856516 z + 157808398747 z - 730779 z + 18325122 z 20 36 34 + 2252656649860 z + 2252656649860 z - 5621464580824 z 30 42 44 46 - 15830069666932 z - 26967843928 z + 3367036850 z - 299569441 z 56 54 52 32 38 + z - 216 z + 17470 z + 10749772238808 z - 686839856516 z 40 / 28 26 + 157808398747 z ) / (-1 - 134423562427882 z + 104832595880294 z / 2 24 22 4 6 + 279 z - 63623425173606 z + 29920636047414 z - 28667 z + 1480531 z 8 10 12 14 - 44305233 z + 839324585 z - 10700273667 z + 95841671835 z 18 16 50 48 + 2990507819729 z - 621966105391 z + 44305233 z - 839324585 z 20 36 34 - 10831372138658 z - 29920636047414 z + 63623425173606 z 30 42 44 + 134423562427882 z + 621966105391 z - 95841671835 z 46 58 56 54 52 + 10700273667 z + z - 279 z + 28667 z - 1480531 z 32 38 40 - 104832595880294 z + 10831372138658 z - 2990507819729 z ) And in Maple-input format, it is: -(1+18004039166884*z^28-15830069666932*z^26-216*z^2+10749772238808*z^24-\ 5621464580824*z^22+17470*z^4-730779*z^6+18325122*z^8-299569441*z^10+3367036850* z^12-26967843928*z^14-686839856516*z^18+157808398747*z^16-730779*z^50+18325122* z^48+2252656649860*z^20+2252656649860*z^36-5621464580824*z^34-15830069666932*z^ 30-26967843928*z^42+3367036850*z^44-299569441*z^46+z^56-216*z^54+17470*z^52+ 10749772238808*z^32-686839856516*z^38+157808398747*z^40)/(-1-134423562427882*z^ 28+104832595880294*z^26+279*z^2-63623425173606*z^24+29920636047414*z^22-28667*z ^4+1480531*z^6-44305233*z^8+839324585*z^10-10700273667*z^12+95841671835*z^14+ 2990507819729*z^18-621966105391*z^16+44305233*z^50-839324585*z^48-\ 10831372138658*z^20-29920636047414*z^36+63623425173606*z^34+134423562427882*z^ 30+621966105391*z^42-95841671835*z^44+10700273667*z^46+z^58-279*z^56+28667*z^54 -1480531*z^52-104832595880294*z^32+10831372138658*z^38-2990507819729*z^40) The first , 40, terms are: [0, 63, 0, 6380, 0, 723751, 0, 86324411, 0, 10531053997, 0, 1297914792305, 0, 160714159413540, 0, 19943624017001705, 0, 2477385033469782567, 0, 307885287395308660375, 0, 38272002142401564193097, 0, 4757940517772944040505732, 0, 591532093666615013085587601, 0, 73544083882928402765990958205, 0, 9143699569257974066083412505611, 0, 1136837540895100527619727798269095, 0, 141343528060178747862370397381189324, 0, 17573325357924226564367624585776733503, 0, 2184903239292015275748228749531700400945, 0, 271650543896856327218686936449276994802385] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 170991378640855098962 z - 25190562801774431371 z - 335 z 24 22 4 6 + 3181597934017917344 z - 342265912022305621 z + 48910 z - 4204693 z 102 8 10 12 - 10021233743 z + 241944944 z - 10021233743 z + 313241776258 z 14 18 16 - 7639074165005 z - 2367344895153827 z + 148970935322756 z 50 48 - 26926860582483909078876699 z + 17372921944839816061706468 z 20 36 + 31114550364400918 z + 85689857941640682302390 z 34 66 - 22319404552634690845659 z - 9880599579111229655916313 z 80 100 90 + 5068710738005770465572 z + 313241776258 z - 342265912022305621 z 88 84 + 3181597934017917344 z + 170991378640855098962 z 94 86 96 - 2367344895153827 z - 25190562801774431371 z + 148970935322756 z 98 92 82 - 7639074165005 z + 31114550364400918 z - 1000477010913886087741 z 64 112 110 106 + 17372921944839816061706468 z + z - 335 z - 4204693 z 108 30 42 + 48910 z - 1000477010913886087741 z - 2182225693414036102514211 z 44 46 + 4949901924164984455902330 z - 9880599579111229655916313 z 58 56 - 44397728172279951933879881 z + 47258908611554403137114106 z 54 52 - 44397728172279951933879881 z + 36809524216745213267018318 z 60 70 + 36809524216745213267018318 z - 2182225693414036102514211 z 68 78 + 4949901924164984455902330 z - 22319404552634690845659 z 32 38 + 5068710738005770465572 z - 287592479120104965188821 z 40 62 + 845616758729255360353312 z - 26926860582483909078876699 z 76 74 + 85689857941640682302390 z - 287592479120104965188821 z 72 104 / 2 + 845616758729255360353312 z + 241944944 z ) / ((-1 + z ) (1 / 28 26 2 + 545272863312330820568 z - 77409338901973858816 z - 399 z 24 22 4 6 + 9387212812598636170 z - 965678802799296354 z + 67352 z - 6555304 z 102 8 10 12 - 19139827394 z + 420227294 z - 19139827394 z + 650779028332 z 14 18 16 - 17105810318051 z - 6021379875618693 z + 356731754585468 z 50 48 - 107620283117639135576776049 z + 68857120745369615214502536 z 20 36 + 83566991824865640 z + 307326895725730592660176 z 34 66 - 78055947846190901892839 z - 38743583200361061583109431 z 80 100 90 + 17239196776763522357132 z + 650779028332 z - 965678802799296354 z 88 84 + 9387212812598636170 z + 545272863312330820568 z 94 86 96 - 6021379875618693 z - 77409338901973858816 z + 356731754585468 z 98 92 82 - 17105810318051 z + 83566991824865640 z - 3299863644344161631529 z 64 112 110 106 + 68857120745369615214502536 z + z - 399 z - 6555304 z 108 30 42 + 67352 z - 3299863644344161631529 z - 8316243843826725535546062 z 44 46 + 19157119825469112961563516 z - 38743583200361061583109431 z 58 56 - 179156123093806260947347370 z + 190930742044951194848790764 z 54 52 - 179156123093806260947347370 z + 148002358602659442071119728 z 60 70 + 148002358602659442071119728 z - 8316243843826725535546062 z 68 78 + 19157119825469112961563516 z - 78055947846190901892839 z 32 38 + 17239196776763522357132 z - 1055097875019187743133668 z 40 62 + 3165674290723208917604154 z - 107620283117639135576776049 z 76 74 + 307326895725730592660176 z - 1055097875019187743133668 z 72 104 + 3165674290723208917604154 z + 420227294 z )) And in Maple-input format, it is: -(1+170991378640855098962*z^28-25190562801774431371*z^26-335*z^2+ 3181597934017917344*z^24-342265912022305621*z^22+48910*z^4-4204693*z^6-\ 10021233743*z^102+241944944*z^8-10021233743*z^10+313241776258*z^12-\ 7639074165005*z^14-2367344895153827*z^18+148970935322756*z^16-\ 26926860582483909078876699*z^50+17372921944839816061706468*z^48+ 31114550364400918*z^20+85689857941640682302390*z^36-22319404552634690845659*z^ 34-9880599579111229655916313*z^66+5068710738005770465572*z^80+313241776258*z^ 100-342265912022305621*z^90+3181597934017917344*z^88+170991378640855098962*z^84 -2367344895153827*z^94-25190562801774431371*z^86+148970935322756*z^96-\ 7639074165005*z^98+31114550364400918*z^92-1000477010913886087741*z^82+ 17372921944839816061706468*z^64+z^112-335*z^110-4204693*z^106+48910*z^108-\ 1000477010913886087741*z^30-2182225693414036102514211*z^42+ 4949901924164984455902330*z^44-9880599579111229655916313*z^46-\ 44397728172279951933879881*z^58+47258908611554403137114106*z^56-\ 44397728172279951933879881*z^54+36809524216745213267018318*z^52+ 36809524216745213267018318*z^60-2182225693414036102514211*z^70+ 4949901924164984455902330*z^68-22319404552634690845659*z^78+ 5068710738005770465572*z^32-287592479120104965188821*z^38+ 845616758729255360353312*z^40-26926860582483909078876699*z^62+ 85689857941640682302390*z^76-287592479120104965188821*z^74+ 845616758729255360353312*z^72+241944944*z^104)/(-1+z^2)/(1+ 545272863312330820568*z^28-77409338901973858816*z^26-399*z^2+ 9387212812598636170*z^24-965678802799296354*z^22+67352*z^4-6555304*z^6-\ 19139827394*z^102+420227294*z^8-19139827394*z^10+650779028332*z^12-\ 17105810318051*z^14-6021379875618693*z^18+356731754585468*z^16-\ 107620283117639135576776049*z^50+68857120745369615214502536*z^48+ 83566991824865640*z^20+307326895725730592660176*z^36-78055947846190901892839*z^ 34-38743583200361061583109431*z^66+17239196776763522357132*z^80+650779028332*z^ 100-965678802799296354*z^90+9387212812598636170*z^88+545272863312330820568*z^84 -6021379875618693*z^94-77409338901973858816*z^86+356731754585468*z^96-\ 17105810318051*z^98+83566991824865640*z^92-3299863644344161631529*z^82+ 68857120745369615214502536*z^64+z^112-399*z^110-6555304*z^106+67352*z^108-\ 3299863644344161631529*z^30-8316243843826725535546062*z^42+ 19157119825469112961563516*z^44-38743583200361061583109431*z^46-\ 179156123093806260947347370*z^58+190930742044951194848790764*z^56-\ 179156123093806260947347370*z^54+148002358602659442071119728*z^52+ 148002358602659442071119728*z^60-8316243843826725535546062*z^70+ 19157119825469112961563516*z^68-78055947846190901892839*z^78+ 17239196776763522357132*z^32-1055097875019187743133668*z^38+ 3165674290723208917604154*z^40-107620283117639135576776049*z^62+ 307326895725730592660176*z^76-1055097875019187743133668*z^74+ 3165674290723208917604154*z^72+420227294*z^104) The first , 40, terms are: [0, 65, 0, 7159, 0, 877748, 0, 111704777, 0, 14423152431, 0, 1873563540731, 0, 244020376808707, 0, 31820430796497231, 0, 4151726487385527265, 0, 541833410253760298612, 0, 70722428273081642165263, 0, 9231547104625679020746721, 0, 1205047699802129484532477433, 0, 157304064282029226363755191801, 0, 20534234120605316566607198520849, 0, 2680516276652996939334211653745535, 0, 349912170454803715445663435145920084, 0, 45677251627534521521008002550563946321, 0, 5962673564471744593244236375443988437023, 0, 778362987448707723837105145743849197372323] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1787229898339752171 z - 429289364717729433 z - 289 z 24 22 4 6 + 86043459371293693 z - 14310574375422664 z + 35119 z - 2431870 z 8 10 12 14 + 109509352 z - 3459699178 z + 80555100969 z - 1430435997275 z 18 16 50 - 219569944480424 z + 19858219635967 z - 168455461565529656491 z 48 20 + 253462122795079446147 z + 1961287973540992 z 36 34 + 94960542086191852989 z - 45344135616688164122 z 66 80 88 84 86 - 14310574375422664 z + 109509352 z + z + 35119 z - 289 z 82 64 30 - 2431870 z + 86043459371293693 z - 6236772706371073694 z 42 44 - 323755769482410726704 z + 351258800448598377696 z 46 58 - 323755769482410726704 z - 6236772706371073694 z 56 54 + 18308640623877987880 z - 45344135616688164122 z 52 60 70 + 94960542086191852989 z + 1787229898339752171 z - 219569944480424 z 68 78 32 + 1961287973540992 z - 3459699178 z + 18308640623877987880 z 38 40 - 168455461565529656491 z + 253462122795079446147 z 62 76 74 - 429289364717729433 z + 80555100969 z - 1430435997275 z 72 / 28 + 19858219635967 z ) / (-1 - 7862099173797302004 z / 26 2 24 + 1759574702192624377 z + 359 z - 328391809497137239 z 22 4 6 8 + 50792774564053682 z - 51706 z + 4121016 z - 209602637 z 10 12 14 + 7382260017 z - 189775871742 z + 3692811089448 z 18 16 50 + 669539125068041 z - 55847085362743 z + 1723340446465262520473 z 48 20 - 2378416572206851332367 z - 6461146260734578 z 36 34 - 555926958574360757500 z + 246847301763608076187 z 66 80 90 88 84 + 328391809497137239 z - 7382260017 z + z - 359 z - 4121016 z 86 82 64 + 51706 z + 209602637 z - 1759574702192624377 z 30 42 + 29441893262128105694 z + 2378416572206851332367 z 44 46 - 2793637456331423813548 z + 2793637456331423813548 z 58 56 + 92780161991104486743 z - 246847301763608076187 z 54 52 + 555926958574360757500 z - 1061960387103402777978 z 60 70 68 - 29441893262128105694 z + 6461146260734578 z - 50792774564053682 z 78 32 38 + 189775871742 z - 92780161991104486743 z + 1061960387103402777978 z 40 62 76 - 1723340446465262520473 z + 7862099173797302004 z - 3692811089448 z 74 72 + 55847085362743 z - 669539125068041 z ) And in Maple-input format, it is: -(1+1787229898339752171*z^28-429289364717729433*z^26-289*z^2+86043459371293693* z^24-14310574375422664*z^22+35119*z^4-2431870*z^6+109509352*z^8-3459699178*z^10 +80555100969*z^12-1430435997275*z^14-219569944480424*z^18+19858219635967*z^16-\ 168455461565529656491*z^50+253462122795079446147*z^48+1961287973540992*z^20+ 94960542086191852989*z^36-45344135616688164122*z^34-14310574375422664*z^66+ 109509352*z^80+z^88+35119*z^84-289*z^86-2431870*z^82+86043459371293693*z^64-\ 6236772706371073694*z^30-323755769482410726704*z^42+351258800448598377696*z^44-\ 323755769482410726704*z^46-6236772706371073694*z^58+18308640623877987880*z^56-\ 45344135616688164122*z^54+94960542086191852989*z^52+1787229898339752171*z^60-\ 219569944480424*z^70+1961287973540992*z^68-3459699178*z^78+18308640623877987880 *z^32-168455461565529656491*z^38+253462122795079446147*z^40-429289364717729433* z^62+80555100969*z^76-1430435997275*z^74+19858219635967*z^72)/(-1-\ 7862099173797302004*z^28+1759574702192624377*z^26+359*z^2-328391809497137239*z^ 24+50792774564053682*z^22-51706*z^4+4121016*z^6-209602637*z^8+7382260017*z^10-\ 189775871742*z^12+3692811089448*z^14+669539125068041*z^18-55847085362743*z^16+ 1723340446465262520473*z^50-2378416572206851332367*z^48-6461146260734578*z^20-\ 555926958574360757500*z^36+246847301763608076187*z^34+328391809497137239*z^66-\ 7382260017*z^80+z^90-359*z^88-4121016*z^84+51706*z^86+209602637*z^82-\ 1759574702192624377*z^64+29441893262128105694*z^30+2378416572206851332367*z^42-\ 2793637456331423813548*z^44+2793637456331423813548*z^46+92780161991104486743*z^ 58-246847301763608076187*z^56+555926958574360757500*z^54-1061960387103402777978 *z^52-29441893262128105694*z^60+6461146260734578*z^70-50792774564053682*z^68+ 189775871742*z^78-92780161991104486743*z^32+1061960387103402777978*z^38-\ 1723340446465262520473*z^40+7862099173797302004*z^62-3692811089448*z^76+ 55847085362743*z^74-669539125068041*z^72) The first , 40, terms are: [0, 70, 0, 8543, 0, 1136663, 0, 154715494, 0, 21226781205, 0, 2921803631965, 0, 402757730326742, 0, 55554770496010639, 0, 7665308756396212727, 0, 1057787283418675084822, 0, 145980539004244938739225, 0, 20146727559597496890418937, 0, 2780481767747299013874832470, 0, 383741141273281647184840073655, 0, 52961217285579463013152255981455, 0, 7309339754874090893895156233484278, 0, 1008784999689754043556151729769907677, 0, 139225635823888167744077396881179221045, 0, 19214976746178543194354978073437513348998, 0, 2651920764950684426085127577013186009657271] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 24009961538332 z - 21050877215002 z - 213 z 24 22 4 6 + 14175239630590 z - 7311723244004 z + 17242 z - 738193 z 8 10 12 14 + 19173729 z - 326065100 z + 3811201160 z - 31655717676 z 18 16 50 48 - 856470999348 z + 191344306632 z - 738193 z + 19173729 z 20 36 34 + 2875659299192 z + 2875659299192 z - 7311723244004 z 30 42 44 46 - 21050877215002 z - 31655717676 z + 3811201160 z - 326065100 z 56 54 52 32 38 + z - 213 z + 17242 z + 14175239630590 z - 856470999348 z 40 / 28 26 + 191344306632 z ) / (-1 - 190734541410388 z + 147335555159408 z / 2 24 22 4 6 + 291 z - 87774177679072 z + 40192620026010 z - 29177 z + 1486427 z 8 10 12 14 - 44981301 z + 878847879 z - 11680849422 z + 109523891010 z 18 16 50 48 + 3735647923430 z - 744032699778 z + 44981301 z - 878847879 z 20 36 34 - 14069930897734 z - 40192620026010 z + 87774177679072 z 30 42 44 + 190734541410388 z + 744032699778 z - 109523891010 z 46 58 56 54 52 + 11680849422 z + z - 291 z + 29177 z - 1486427 z 32 38 40 - 147335555159408 z + 14069930897734 z - 3735647923430 z ) And in Maple-input format, it is: -(1+24009961538332*z^28-21050877215002*z^26-213*z^2+14175239630590*z^24-\ 7311723244004*z^22+17242*z^4-738193*z^6+19173729*z^8-326065100*z^10+3811201160* z^12-31655717676*z^14-856470999348*z^18+191344306632*z^16-738193*z^50+19173729* z^48+2875659299192*z^20+2875659299192*z^36-7311723244004*z^34-21050877215002*z^ 30-31655717676*z^42+3811201160*z^44-326065100*z^46+z^56-213*z^54+17242*z^52+ 14175239630590*z^32-856470999348*z^38+191344306632*z^40)/(-1-190734541410388*z^ 28+147335555159408*z^26+291*z^2-87774177679072*z^24+40192620026010*z^22-29177*z ^4+1486427*z^6-44981301*z^8+878847879*z^10-11680849422*z^12+109523891010*z^14+ 3735647923430*z^18-744032699778*z^16+44981301*z^50-878847879*z^48-\ 14069930897734*z^20-40192620026010*z^36+87774177679072*z^34+190734541410388*z^ 30+744032699778*z^42-109523891010*z^44+11680849422*z^46+z^58-291*z^56+29177*z^ 54-1486427*z^52-147335555159408*z^32+14069930897734*z^38-3735647923430*z^40) The first , 40, terms are: [0, 78, 0, 10763, 0, 1604461, 0, 242999834, 0, 36942248199, 0, 5621648963775, 0, 855692445465754, 0, 130257502641627045, 0, 19828797277409483235, 0, 3018508294610500517070, 0, 459503722082707124982809, 0, 69949703291049564506504169, 0, 10648361108072330426278113102, 0, 1620987547338660750885837386163, 0, 246761039158467906421553892775989, 0, 37564144585407603348994818415864346, 0, 5718345830562861960015245922295301583, 0, 870497103159162372952688497461750436343, 0, 132514756732487053813976587475759350447002, 0, 20172566558124463668670194089516346354294237] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 20121594812696 z - 17623360913670 z - 210 z 24 22 4 6 + 11833463105068 z - 6080350321400 z + 16532 z - 683459 z 8 10 12 14 + 17134612 z - 282662999 z + 3229613584 z - 26434735398 z 18 16 50 48 - 708458926976 z + 158589835381 z - 683459 z + 17134612 z 20 36 34 + 2382792437162 z + 2382792437162 z - 6080350321400 z 30 42 44 46 - 17623360913670 z - 26434735398 z + 3229613584 z - 282662999 z 56 54 52 32 38 + z - 210 z + 16532 z + 11833463105068 z - 708458926976 z 40 / 2 28 + 158589835381 z ) / ((-1 + z ) (1 + 82661020545618 z / 26 2 24 22 - 71941907067968 z - 272 z + 47407718517210 z - 23620526746812 z 4 6 8 10 12 + 26415 z - 1304246 z + 37946227 z - 709727522 z + 9022225807 z 14 18 16 50 - 80883838028 z - 2500088121916 z + 524316518497 z - 1304246 z 48 20 36 + 37946227 z + 8872550793270 z + 8872550793270 z 34 30 42 - 23620526746812 z - 71941907067968 z - 80883838028 z 44 46 56 54 52 + 9022225807 z - 709727522 z + z - 272 z + 26415 z 32 38 40 + 47407718517210 z - 2500088121916 z + 524316518497 z )) And in Maple-input format, it is: -(1+20121594812696*z^28-17623360913670*z^26-210*z^2+11833463105068*z^24-\ 6080350321400*z^22+16532*z^4-683459*z^6+17134612*z^8-282662999*z^10+3229613584* z^12-26434735398*z^14-708458926976*z^18+158589835381*z^16-683459*z^50+17134612* z^48+2382792437162*z^20+2382792437162*z^36-6080350321400*z^34-17623360913670*z^ 30-26434735398*z^42+3229613584*z^44-282662999*z^46+z^56-210*z^54+16532*z^52+ 11833463105068*z^32-708458926976*z^38+158589835381*z^40)/(-1+z^2)/(1+ 82661020545618*z^28-71941907067968*z^26-272*z^2+47407718517210*z^24-\ 23620526746812*z^22+26415*z^4-1304246*z^6+37946227*z^8-709727522*z^10+ 9022225807*z^12-80883838028*z^14-2500088121916*z^18+524316518497*z^16-1304246*z ^50+37946227*z^48+8872550793270*z^20+8872550793270*z^36-23620526746812*z^34-\ 71941907067968*z^30-80883838028*z^42+9022225807*z^44-709727522*z^46+z^56-272*z^ 54+26415*z^52+47407718517210*z^32-2500088121916*z^38+524316518497*z^40) The first , 40, terms are: [0, 63, 0, 7044, 0, 888933, 0, 116411263, 0, 15422726863, 0, 2050726307261, 0, 272991829165572, 0, 36353557876266695, 0, 4841644627304194225, 0, 644843285064221171409, 0, 85885579588975122813399, 0, 11438993825364069780853380, 0, 1523547058907105143607532589, 0, 202919633466758557113113508847, 0, 27026655873282466080134389954527, 0, 3599652433096353423737571398008629, 0, 479434001644947061972572756841952004, 0, 63855321304139125203801138068803043951, 0, 8504824541293641466528131973475724357089, 0, 1132748830085049169919070763817524028935713] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2307438960124784274 z - 556237069195963268 z - 303 z 24 22 4 6 + 111811884601719256 z - 18628568186801484 z + 38602 z - 2793555 z 8 10 12 14 + 130646195 z - 4253689198 z + 101298638596 z - 1827484922644 z 18 16 50 - 285092286060866 z + 25632669511468 z - 213750104253924497858 z 48 20 + 320980883186513390364 z + 2552995119205886 z 36 34 + 120813759227141577404 z - 57875693941470182204 z 66 80 88 84 86 - 18628568186801484 z + 130646195 z + z + 38602 z - 303 z 82 64 30 - 2793555 z + 111811884601719256 z - 8020783397718862958 z 42 44 - 409503669552879853966 z + 444109555015063103580 z 46 58 - 409503669552879853966 z - 8020783397718862958 z 56 54 + 23454154212368572020 z - 57875693941470182204 z 52 60 + 120813759227141577404 z + 2307438960124784274 z 70 68 78 - 285092286060866 z + 2552995119205886 z - 4253689198 z 32 38 + 23454154212368572020 z - 213750104253924497858 z 40 62 76 + 320980883186513390364 z - 556237069195963268 z + 101298638596 z 74 72 / 2 - 1827484922644 z + 25632669511468 z ) / ((-1 + z ) (1 / 28 26 2 + 8279375134988783016 z - 1920735893443588070 z - 375 z 24 22 4 6 + 369963529626654444 z - 58794680184334922 z + 56515 z - 4693109 z 8 10 12 14 + 246639989 z - 8885970932 z + 231339426882 z - 4517440613836 z 18 16 50 - 806279821288892 z + 68015299338330 z - 871553334626459694492 z 48 20 + 1324723825351638766594 z + 7648240459496592 z 36 34 + 484416536144605107766 z - 227169240102039163332 z 66 80 88 84 86 - 58794680184334922 z + 246639989 z + z + 56515 z - 375 z 82 64 30 - 4693109 z + 369963529626654444 z - 29779246965047656916 z 42 44 - 1702483520940502938884 z + 1850883329108507190354 z 46 58 - 1702483520940502938884 z - 29779246965047656916 z 56 54 + 89726602815441052638 z - 227169240102039163332 z 52 60 + 484416536144605107766 z + 8279375134988783016 z 70 68 78 - 806279821288892 z + 7648240459496592 z - 8885970932 z 32 38 + 89726602815441052638 z - 871553334626459694492 z 40 62 76 + 1324723825351638766594 z - 1920735893443588070 z + 231339426882 z 74 72 - 4517440613836 z + 68015299338330 z )) And in Maple-input format, it is: -(1+2307438960124784274*z^28-556237069195963268*z^26-303*z^2+111811884601719256 *z^24-18628568186801484*z^22+38602*z^4-2793555*z^6+130646195*z^8-4253689198*z^ 10+101298638596*z^12-1827484922644*z^14-285092286060866*z^18+25632669511468*z^ 16-213750104253924497858*z^50+320980883186513390364*z^48+2552995119205886*z^20+ 120813759227141577404*z^36-57875693941470182204*z^34-18628568186801484*z^66+ 130646195*z^80+z^88+38602*z^84-303*z^86-2793555*z^82+111811884601719256*z^64-\ 8020783397718862958*z^30-409503669552879853966*z^42+444109555015063103580*z^44-\ 409503669552879853966*z^46-8020783397718862958*z^58+23454154212368572020*z^56-\ 57875693941470182204*z^54+120813759227141577404*z^52+2307438960124784274*z^60-\ 285092286060866*z^70+2552995119205886*z^68-4253689198*z^78+23454154212368572020 *z^32-213750104253924497858*z^38+320980883186513390364*z^40-556237069195963268* z^62+101298638596*z^76-1827484922644*z^74+25632669511468*z^72)/(-1+z^2)/(1+ 8279375134988783016*z^28-1920735893443588070*z^26-375*z^2+369963529626654444*z^ 24-58794680184334922*z^22+56515*z^4-4693109*z^6+246639989*z^8-8885970932*z^10+ 231339426882*z^12-4517440613836*z^14-806279821288892*z^18+68015299338330*z^16-\ 871553334626459694492*z^50+1324723825351638766594*z^48+7648240459496592*z^20+ 484416536144605107766*z^36-227169240102039163332*z^34-58794680184334922*z^66+ 246639989*z^80+z^88+56515*z^84-375*z^86-4693109*z^82+369963529626654444*z^64-\ 29779246965047656916*z^30-1702483520940502938884*z^42+1850883329108507190354*z^ 44-1702483520940502938884*z^46-29779246965047656916*z^58+89726602815441052638*z ^56-227169240102039163332*z^54+484416536144605107766*z^52+8279375134988783016*z ^60-806279821288892*z^70+7648240459496592*z^68-8885970932*z^78+ 89726602815441052638*z^32-871553334626459694492*z^38+1324723825351638766594*z^ 40-1920735893443588070*z^62+231339426882*z^76-4517440613836*z^74+68015299338330 *z^72) The first , 40, terms are: [0, 73, 0, 9160, 0, 1247259, 0, 173892633, 0, 24465226907, 0, 3455727366613, 0, 489013442862880, 0, 69258406734766259, 0, 9812934141671694095, 0, 1390618250039391611351, 0, 197086160517722216797795, 0, 27933342354394568119958544, 0, 3959118389956694233345959437, 0, 561149077082591941242782598843, 0, 79535314116486368012510541480009, 0, 11273082607611426996365378192222243, 0, 1597812538907185732314812309462858936, 0, 226469213831998178363622472896715437193, 0, 32099082617898452805875317447498682192569, 0, 4549630392352415476770004259799343091516521] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 615941586772676 z - 349452471705580 z - 223 z 24 22 4 6 + 157479871263970 z - 56157588613220 z + 19408 z - 915197 z 8 10 12 14 + 26868583 z - 531157824 z + 7435090058 z - 76271180376 z 18 16 50 - 3458749273694 z + 587517601544 z - 76271180376 z 48 20 36 + 587517601544 z + 15763970366900 z + 615941586772676 z 34 64 30 42 - 864562754197168 z + z - 864562754197168 z - 56157588613220 z 44 46 58 56 + 15763970366900 z - 3458749273694 z - 915197 z + 26868583 z 54 52 60 32 - 531157824 z + 7435090058 z + 19408 z + 967855920018588 z 38 40 62 / - 349452471705580 z + 157479871263970 z - 223 z ) / (-1 / 28 26 2 - 3886809701007049 z + 1993893444481417 z + 292 z 24 22 4 6 - 814572886234855 z + 263756906813473 z - 30953 z + 1724982 z 8 10 12 14 - 58726015 z + 1326651458 z - 20968133395 z + 240571761765 z 18 16 50 + 13389609408457 z - 2057960556171 z + 2057960556171 z 48 20 36 - 13389609408457 z - 67254600362611 z - 6054421754649993 z 34 66 64 30 + 7552349566861981 z + z - 292 z + 6054421754649993 z 42 44 46 + 814572886234855 z - 263756906813473 z + 67254600362611 z 58 56 54 52 + 58726015 z - 1326651458 z + 20968133395 z - 240571761765 z 60 32 38 - 1724982 z - 7552349566861981 z + 3886809701007049 z 40 62 - 1993893444481417 z + 30953 z ) And in Maple-input format, it is: -(1+615941586772676*z^28-349452471705580*z^26-223*z^2+157479871263970*z^24-\ 56157588613220*z^22+19408*z^4-915197*z^6+26868583*z^8-531157824*z^10+7435090058 *z^12-76271180376*z^14-3458749273694*z^18+587517601544*z^16-76271180376*z^50+ 587517601544*z^48+15763970366900*z^20+615941586772676*z^36-864562754197168*z^34 +z^64-864562754197168*z^30-56157588613220*z^42+15763970366900*z^44-\ 3458749273694*z^46-915197*z^58+26868583*z^56-531157824*z^54+7435090058*z^52+ 19408*z^60+967855920018588*z^32-349452471705580*z^38+157479871263970*z^40-223*z ^62)/(-1-3886809701007049*z^28+1993893444481417*z^26+292*z^2-814572886234855*z^ 24+263756906813473*z^22-30953*z^4+1724982*z^6-58726015*z^8+1326651458*z^10-\ 20968133395*z^12+240571761765*z^14+13389609408457*z^18-2057960556171*z^16+ 2057960556171*z^50-13389609408457*z^48-67254600362611*z^20-6054421754649993*z^ 36+7552349566861981*z^34+z^66-292*z^64+6054421754649993*z^30+814572886234855*z^ 42-263756906813473*z^44+67254600362611*z^46+58726015*z^58-1326651458*z^56+ 20968133395*z^54-240571761765*z^52-1724982*z^60-7552349566861981*z^32+ 3886809701007049*z^38-1993893444481417*z^40+30953*z^62) The first , 40, terms are: [0, 69, 0, 8603, 0, 1186104, 0, 167220035, 0, 23698191853, 0, 3362704328069, 0, 477306602359357, 0, 67754815675423213, 0, 9618148868198806147, 0, 1365353192313200589592, 0, 193820220974443291120907, 0, 27513972683115864103882485, 0, 3905778009033313930006856361, 0, 554449274294665702242379491961, 0, 78707494069693710181386154255093, 0, 11173014230254051162517191812123627, 0, 1586078282777867571865796468789603288, 0, 225153594857951213559024602907688998851, 0, 31961941493788243529922855679913745533261, 0, 4537194730128078369058614073576899535186413] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1226089063498501974 z - 301204467273414411 z - 281 z 24 22 4 6 + 61833423999251038 z - 10544589673298169 z + 33009 z - 2215194 z 8 10 12 14 + 96949867 z - 2981813771 z + 67633673916 z - 1170043597861 z 18 16 50 - 170411803386561 z + 15822911969336 z - 106664041925574161143 z 48 20 + 159231503649347637941 z + 1482867664532750 z 36 34 + 60783742030320319345 z - 29423872924022241820 z 66 80 88 84 86 - 10544589673298169 z + 96949867 z + z + 33009 z - 281 z 82 64 30 - 2215194 z + 61833423999251038 z - 4190845251596770075 z 42 44 - 202424163210076768938 z + 219269480679871536828 z 46 58 - 202424163210076768938 z - 4190845251596770075 z 56 54 + 12075436687652044649 z - 29423872924022241820 z 52 60 70 + 60783742030320319345 z + 1226089063498501974 z - 170411803386561 z 68 78 32 + 1482867664532750 z - 2981813771 z + 12075436687652044649 z 38 40 - 106664041925574161143 z + 159231503649347637941 z 62 76 74 - 301204467273414411 z + 67633673916 z - 1170043597861 z 72 / 2 28 + 15822911969336 z ) / ((-1 + z ) (1 + 4436637439269285374 z / 26 2 24 - 1048531438163692304 z - 356 z + 206167514092960779 z 22 4 6 8 - 33522250903002328 z + 49576 z - 3807012 z + 186461802 z 10 12 14 - 6320261768 z + 156205783110 z - 2918324037880 z 18 16 50 - 485345422707640 z + 42306419220777 z - 438575555486399428004 z 48 20 + 662580553465936724883 z + 4473438833396660 z 36 34 + 245814149995952464092 z - 116500939601733666020 z 66 80 88 84 86 - 33522250903002328 z + 186461802 z + z + 49576 z - 356 z 82 64 30 - 3807012 z + 206167514092960779 z - 15694944089711080128 z 42 44 - 848393547839219725904 z + 921207867679606769640 z 46 58 - 848393547839219725904 z - 15694944089711080128 z 56 54 + 46601570820255135774 z - 116500939601733666020 z 52 60 + 245814149995952464092 z + 4436637439269285374 z 70 68 78 - 485345422707640 z + 4473438833396660 z - 6320261768 z 32 38 + 46601570820255135774 z - 438575555486399428004 z 40 62 76 + 662580553465936724883 z - 1048531438163692304 z + 156205783110 z 74 72 - 2918324037880 z + 42306419220777 z )) And in Maple-input format, it is: -(1+1226089063498501974*z^28-301204467273414411*z^26-281*z^2+61833423999251038* z^24-10544589673298169*z^22+33009*z^4-2215194*z^6+96949867*z^8-2981813771*z^10+ 67633673916*z^12-1170043597861*z^14-170411803386561*z^18+15822911969336*z^16-\ 106664041925574161143*z^50+159231503649347637941*z^48+1482867664532750*z^20+ 60783742030320319345*z^36-29423872924022241820*z^34-10544589673298169*z^66+ 96949867*z^80+z^88+33009*z^84-281*z^86-2215194*z^82+61833423999251038*z^64-\ 4190845251596770075*z^30-202424163210076768938*z^42+219269480679871536828*z^44-\ 202424163210076768938*z^46-4190845251596770075*z^58+12075436687652044649*z^56-\ 29423872924022241820*z^54+60783742030320319345*z^52+1226089063498501974*z^60-\ 170411803386561*z^70+1482867664532750*z^68-2981813771*z^78+12075436687652044649 *z^32-106664041925574161143*z^38+159231503649347637941*z^40-301204467273414411* z^62+67633673916*z^76-1170043597861*z^74+15822911969336*z^72)/(-1+z^2)/(1+ 4436637439269285374*z^28-1048531438163692304*z^26-356*z^2+206167514092960779*z^ 24-33522250903002328*z^22+49576*z^4-3807012*z^6+186461802*z^8-6320261768*z^10+ 156205783110*z^12-2918324037880*z^14-485345422707640*z^18+42306419220777*z^16-\ 438575555486399428004*z^50+662580553465936724883*z^48+4473438833396660*z^20+ 245814149995952464092*z^36-116500939601733666020*z^34-33522250903002328*z^66+ 186461802*z^80+z^88+49576*z^84-356*z^86-3807012*z^82+206167514092960779*z^64-\ 15694944089711080128*z^30-848393547839219725904*z^42+921207867679606769640*z^44 -848393547839219725904*z^46-15694944089711080128*z^58+46601570820255135774*z^56 -116500939601733666020*z^54+245814149995952464092*z^52+4436637439269285374*z^60 -485345422707640*z^70+4473438833396660*z^68-6320261768*z^78+ 46601570820255135774*z^32-438575555486399428004*z^38+662580553465936724883*z^40 -1048531438163692304*z^62+156205783110*z^76-2918324037880*z^74+42306419220777*z ^72) The first , 40, terms are: [0, 76, 0, 10209, 0, 1491175, 0, 222375428, 0, 33367064523, 0, 5016404072947, 0, 754637944421388, 0, 113546267236812211, 0, 17085816218397909969, 0, 2571034616223247839972, 0, 386886124411922316262645, 0, 58218280410997050260809941, 0, 8760641635912305204758678100, 0, 1318294870568356115623515215585, 0, 198376052236943778536228021555539, 0, 29851484664858965066072431750261292, 0, 4492029829779103994056139278214016659, 0, 675957402899008248135057637156170131531, 0, 101717581640211489339279203648801680963780, 0, 15306388203403434042542222634538316918201415] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3140170704245767705 z - 752407347726617929 z - 317 z 24 22 4 6 + 150147195560217381 z - 24799241804681768 z + 42309 z - 3187984 z 8 10 12 14 + 154065432 z - 5152764644 z + 125499215455 z - 2307783815551 z 18 16 50 - 371155400940408 z + 32904360527363 z - 295751461390776491515 z 48 20 + 444687487426837733943 z + 3363971335491544 z 36 34 + 166852278941847509523 z - 79731456592290792408 z 66 80 88 84 86 - 24799241804681768 z + 154065432 z + z + 42309 z - 317 z 82 64 30 - 3187984 z + 150147195560217381 z - 10969876631934599252 z 42 44 - 567752170175960868768 z + 615883033204484391536 z 46 58 - 567752170175960868768 z - 10969876631934599252 z 56 54 + 32207546976766323592 z - 79731456592290792408 z 52 60 + 166852278941847509523 z + 3140170704245767705 z 70 68 78 - 371155400940408 z + 3363971335491544 z - 5152764644 z 32 38 + 32207546976766323592 z - 295751461390776491515 z 40 62 76 + 444687487426837733943 z - 752407347726617929 z + 125499215455 z 74 72 / 2 - 2307783815551 z + 32904360527363 z ) / ((-1 + z ) (1 / 28 26 2 + 11263314460307516168 z - 2610710623914343008 z - 386 z 24 22 4 6 + 501785391141013659 z - 79437723575172362 z + 60922 z - 5316926 z 8 10 12 14 + 292124519 z - 10916428988 z + 292519859732 z - 5840525463668 z 18 16 50 - 1073261870234130 z + 89423707552089 z - 1181104168684041604786 z 48 20 + 1793698821355810789107 z + 10271267507896374 z 36 34 + 657158862348874505602 z - 308520704498303264574 z 66 80 88 84 86 - 79437723575172362 z + 292124519 z + z + 60922 z - 386 z 82 64 30 - 5316926 z + 501785391141013659 z - 40510023876923068176 z 42 44 - 2303921379584276782124 z + 2504269525193935209420 z 46 58 - 2303921379584276782124 z - 40510023876923068176 z 56 54 + 121979596968095810229 z - 308520704498303264574 z 52 60 + 657158862348874505602 z + 11263314460307516168 z 70 68 78 - 1073261870234130 z + 10271267507896374 z - 10916428988 z 32 38 + 121979596968095810229 z - 1181104168684041604786 z 40 62 76 + 1793698821355810789107 z - 2610710623914343008 z + 292519859732 z 74 72 - 5840525463668 z + 89423707552089 z )) And in Maple-input format, it is: -(1+3140170704245767705*z^28-752407347726617929*z^26-317*z^2+150147195560217381 *z^24-24799241804681768*z^22+42309*z^4-3187984*z^6+154065432*z^8-5152764644*z^ 10+125499215455*z^12-2307783815551*z^14-371155400940408*z^18+32904360527363*z^ 16-295751461390776491515*z^50+444687487426837733943*z^48+3363971335491544*z^20+ 166852278941847509523*z^36-79731456592290792408*z^34-24799241804681768*z^66+ 154065432*z^80+z^88+42309*z^84-317*z^86-3187984*z^82+150147195560217381*z^64-\ 10969876631934599252*z^30-567752170175960868768*z^42+615883033204484391536*z^44 -567752170175960868768*z^46-10969876631934599252*z^58+32207546976766323592*z^56 -79731456592290792408*z^54+166852278941847509523*z^52+3140170704245767705*z^60-\ 371155400940408*z^70+3363971335491544*z^68-5152764644*z^78+32207546976766323592 *z^32-295751461390776491515*z^38+444687487426837733943*z^40-752407347726617929* z^62+125499215455*z^76-2307783815551*z^74+32904360527363*z^72)/(-1+z^2)/(1+ 11263314460307516168*z^28-2610710623914343008*z^26-386*z^2+501785391141013659*z ^24-79437723575172362*z^22+60922*z^4-5316926*z^6+292124519*z^8-10916428988*z^10 +292519859732*z^12-5840525463668*z^14-1073261870234130*z^18+89423707552089*z^16 -1181104168684041604786*z^50+1793698821355810789107*z^48+10271267507896374*z^20 +657158862348874505602*z^36-308520704498303264574*z^34-79437723575172362*z^66+ 292124519*z^80+z^88+60922*z^84-386*z^86-5316926*z^82+501785391141013659*z^64-\ 40510023876923068176*z^30-2303921379584276782124*z^42+2504269525193935209420*z^ 44-2303921379584276782124*z^46-40510023876923068176*z^58+121979596968095810229* z^56-308520704498303264574*z^54+657158862348874505602*z^52+11263314460307516168 *z^60-1073261870234130*z^70+10271267507896374*z^68-10916428988*z^78+ 121979596968095810229*z^32-1181104168684041604786*z^38+1793698821355810789107*z ^40-2610710623914343008*z^62+292519859732*z^76-5840525463668*z^74+ 89423707552089*z^72) The first , 40, terms are: [0, 70, 0, 8091, 0, 1029521, 0, 135454946, 0, 18050246515, 0, 2417643963475, 0, 324510724784154, 0, 43597593131501281, 0, 5859600856689234663, 0, 787678061352508845118, 0, 105891835339632285771865, 0, 14236090287663272798670661, 0, 1913927143392281665617902982, 0, 257313695081336013497933230067, 0, 34594069528194756524758157611613, 0, 4650942404907574535075044516585522, 0, 625288606115015603217588548701641015, 0, 84065960258721909739042240735205603543, 0, 11302119367766376756603193927917593045418, 0, 1519496216797065375333829775721067223407661] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 14824582777893200 z - 6952497942346062 z - 291 z 24 22 4 6 + 2605612308470834 z - 775459243501496 z + 34482 z - 2207439 z 8 10 12 14 + 86079117 z - 2202904004 z + 38991627497 z - 496030597419 z 18 16 50 - 33213943077723 z + 4667762847006 z - 33213943077723 z 48 20 36 + 181778302965057 z + 181778302965057 z + 34994857481080038 z 34 66 64 30 - 38942609004374440 z - 291 z + 34482 z - 25378563844257818 z 42 44 46 - 6952497942346062 z + 2605612308470834 z - 775459243501496 z 58 56 54 52 - 2202904004 z + 38991627497 z - 496030597419 z + 4667762847006 z 60 68 32 38 + 86079117 z + z + 34994857481080038 z - 25378563844257818 z 40 62 / 28 + 14824582777893200 z - 2207439 z ) / (-1 - 88724178342352846 z / 26 2 24 + 37786040425703738 z + 381 z - 12878017891957970 z 22 4 6 8 + 3488189807525425 z - 54927 z + 4126285 z - 185039855 z 10 12 14 + 5373054559 z - 106785568863 z + 1512891439887 z 18 16 50 + 123632258292847 z - 15759629437893 z + 744204638694333 z 48 20 36 - 3488189807525425 z - 744204638694333 z - 315407402576328526 z 34 66 64 + 315407402576328526 z + 54927 z - 4126285 z 30 42 44 + 167582060272268642 z + 88724178342352846 z - 37786040425703738 z 46 58 56 + 12878017891957970 z + 106785568863 z - 1512891439887 z 54 52 60 70 + 15759629437893 z - 123632258292847 z - 5373054559 z + z 68 32 38 - 381 z - 255559926646701486 z + 255559926646701486 z 40 62 - 167582060272268642 z + 185039855 z ) And in Maple-input format, it is: -(1+14824582777893200*z^28-6952497942346062*z^26-291*z^2+2605612308470834*z^24-\ 775459243501496*z^22+34482*z^4-2207439*z^6+86079117*z^8-2202904004*z^10+ 38991627497*z^12-496030597419*z^14-33213943077723*z^18+4667762847006*z^16-\ 33213943077723*z^50+181778302965057*z^48+181778302965057*z^20+34994857481080038 *z^36-38942609004374440*z^34-291*z^66+34482*z^64-25378563844257818*z^30-\ 6952497942346062*z^42+2605612308470834*z^44-775459243501496*z^46-2202904004*z^ 58+38991627497*z^56-496030597419*z^54+4667762847006*z^52+86079117*z^60+z^68+ 34994857481080038*z^32-25378563844257818*z^38+14824582777893200*z^40-2207439*z^ 62)/(-1-88724178342352846*z^28+37786040425703738*z^26+381*z^2-12878017891957970 *z^24+3488189807525425*z^22-54927*z^4+4126285*z^6-185039855*z^8+5373054559*z^10 -106785568863*z^12+1512891439887*z^14+123632258292847*z^18-15759629437893*z^16+ 744204638694333*z^50-3488189807525425*z^48-744204638694333*z^20-\ 315407402576328526*z^36+315407402576328526*z^34+54927*z^66-4126285*z^64+ 167582060272268642*z^30+88724178342352846*z^42-37786040425703738*z^44+ 12878017891957970*z^46+106785568863*z^58-1512891439887*z^56+15759629437893*z^54 -123632258292847*z^52-5373054559*z^60+z^70-381*z^68-255559926646701486*z^32+ 255559926646701486*z^38-167582060272268642*z^40+185039855*z^62) The first , 40, terms are: [0, 90, 0, 13845, 0, 2250361, 0, 369328138, 0, 60753421361, 0, 10000501917969, 0, 1646530838434370, 0, 271114121345382109, 0, 44642329836866126917, 0, 7350995857155175030570, 0, 1210451064933462620395661, 0, 199319116378045133952702653, 0, 32820932886118944522387529386, 0, 5404468449557750089840021039541, 0, 889928444395248348917812550337085, 0, 146540343454712149622657140949823362, 0, 24130111445861580036436793639527995617, 0, 3973392361874575401143421426640497377041, 0, 654279899567931918261488669167867752311914, 0, 107737204970570429226989678641524178544537193] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 9428182427983241 z - 3895462673723607 z - 252 z 24 22 4 6 + 1312530053226535 z - 358782122430898 z + 25271 z - 1368037 z 8 10 12 14 + 46129233 z - 1053664690 z + 17213850349 z - 208808538536 z 18 16 50 - 13904324110208 z + 1931829254042 z - 358782122430898 z 48 20 36 + 1312530053226535 z + 79028480330927 z + 44800227769856132 z 34 66 64 - 40660908490939728 z - 1368037 z + 46129233 z 30 42 44 - 18681170999656264 z - 18681170999656264 z + 9428182427983241 z 46 58 56 - 3895462673723607 z - 208808538536 z + 1931829254042 z 54 52 60 70 - 13904324110208 z + 79028480330927 z + 17213850349 z - 252 z 68 32 38 + 25271 z + 30388634542253931 z - 40660908490939728 z 40 62 72 / 2 + 30388634542253931 z - 1053664690 z + z ) / ((-1 + z ) (1 / 28 26 2 + 37110629581995412 z - 14908037493921376 z - 322 z 24 22 4 6 + 4853827025459711 z - 1273991470768970 z + 39118 z - 2486808 z 8 10 12 14 + 96020195 z - 2459974686 z + 44319862340 z - 584720372180 z 18 16 50 - 44585584889644 z + 5816739961149 z - 1273991470768970 z 48 20 36 + 4853827025459711 z + 267649460452158 z + 185471615474933146 z 34 66 64 - 167799658468110848 z - 2486808 z + 96020195 z 30 42 44 - 75165811645752422 z - 75165811645752422 z + 37110629581995412 z 46 58 56 - 14908037493921376 z - 584720372180 z + 5816739961149 z 54 52 60 70 - 44585584889644 z + 267649460452158 z + 44319862340 z - 322 z 68 32 38 + 39118 z + 124218897025259953 z - 167799658468110848 z 40 62 72 + 124218897025259953 z - 2459974686 z + z )) And in Maple-input format, it is: -(1+9428182427983241*z^28-3895462673723607*z^26-252*z^2+1312530053226535*z^24-\ 358782122430898*z^22+25271*z^4-1368037*z^6+46129233*z^8-1053664690*z^10+ 17213850349*z^12-208808538536*z^14-13904324110208*z^18+1931829254042*z^16-\ 358782122430898*z^50+1312530053226535*z^48+79028480330927*z^20+ 44800227769856132*z^36-40660908490939728*z^34-1368037*z^66+46129233*z^64-\ 18681170999656264*z^30-18681170999656264*z^42+9428182427983241*z^44-\ 3895462673723607*z^46-208808538536*z^58+1931829254042*z^56-13904324110208*z^54+ 79028480330927*z^52+17213850349*z^60-252*z^70+25271*z^68+30388634542253931*z^32 -40660908490939728*z^38+30388634542253931*z^40-1053664690*z^62+z^72)/(-1+z^2)/( 1+37110629581995412*z^28-14908037493921376*z^26-322*z^2+4853827025459711*z^24-\ 1273991470768970*z^22+39118*z^4-2486808*z^6+96020195*z^8-2459974686*z^10+ 44319862340*z^12-584720372180*z^14-44585584889644*z^18+5816739961149*z^16-\ 1273991470768970*z^50+4853827025459711*z^48+267649460452158*z^20+ 185471615474933146*z^36-167799658468110848*z^34-2486808*z^66+96020195*z^64-\ 75165811645752422*z^30-75165811645752422*z^42+37110629581995412*z^44-\ 14908037493921376*z^46-584720372180*z^58+5816739961149*z^56-44585584889644*z^54 +267649460452158*z^52+44319862340*z^60-322*z^70+39118*z^68+124218897025259953*z ^32-167799658468110848*z^38+124218897025259953*z^40-2459974686*z^62+z^72) The first , 40, terms are: [0, 71, 0, 8764, 0, 1188421, 0, 165170799, 0, 23124392279, 0, 3245300171985, 0, 455846951094228, 0, 64051761445815871, 0, 9001265177431738765, 0, 1265032025716858026045, 0, 177791265824600063770439, 0, 24987572421333383608468308, 0, 3511880886534505976402352417, 0, 493578681318009793783922466047, 0, 69370273311587664687336103852263, 0, 9749685160786634205833127118065109, 0, 1370275377135099274279265270895765724, 0, 192586193595864979714320819349973045919, 0, 27067145653517878533028229207869456496081, 0, 3804168777776130609041709022452124042559921] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 204464343236274612953 z - 29992700261291386236 z - 341 z 24 22 4 6 + 3767642188831286519 z - 402621309612823545 z + 50775 z - 4450268 z 102 8 10 12 - 10987394897 z + 260804575 z - 10987394897 z + 348817658110 z 14 18 16 - 8626700834565 z - 2736684777389751 z + 170340943895386 z 50 48 - 32446128595137466069766746 z + 20941737826133308911944257 z 20 36 + 36309671175853536 z + 103334965586237515961983 z 34 66 - 26886830336058659370517 z - 11915133880345161099196413 z 80 100 90 + 6095721076014747636244 z + 348817658110 z - 402621309612823545 z 88 84 + 3767642188831286519 z + 204464343236274612953 z 94 86 96 - 2736684777389751 z - 29992700261291386236 z + 170340943895386 z 98 92 82 - 8626700834565 z + 36309671175853536 z - 1200278477078919250291 z 64 112 110 106 + 20941737826133308911944257 z + z - 341 z - 4450268 z 108 30 42 + 50775 z - 1200278477078919250291 z - 2633296164742206636701211 z 44 46 + 5971396193959915069279080 z - 11915133880345161099196413 z 58 56 - 53471735893102075918729571 z + 56913913605096582903833834 z 54 52 - 53471735893102075918729571 z + 44341185746608886722144235 z 60 70 + 44341185746608886722144235 z - 2633296164742206636701211 z 68 78 + 5971396193959915069279080 z - 26886830336058659370517 z 32 38 + 6095721076014747636244 z - 347008728444113499975088 z 40 62 + 1020505244272702401607653 z - 32446128595137466069766746 z 76 74 + 103334965586237515961983 z - 347008728444113499975088 z 72 104 / + 1020505244272702401607653 z + 260804575 z ) / (-1 / 28 26 2 - 751272969152094399076 z + 104071064164374686464 z + 421 z 24 22 4 - 12330695474456429352 z + 1240835505133947956 z - 73332 z 6 102 8 10 + 7292936 z + 760972965460 z - 475739514 z + 22020872958 z 12 14 18 - 760972965460 z + 20342861196816 z + 7427953419779269 z 16 50 - 431860571719817 z + 221581782911465560658632866 z 48 20 - 134883128668041555527567062 z - 105155145893683208 z 36 34 - 474511083772565437165048 z + 116818120092202618332132 z 66 80 + 134883128668041555527567062 z - 116818120092202618332132 z 100 90 - 20342861196816 z + 12330695474456429352 z 88 84 - 104071064164374686464 z - 4666003731843976906428 z 94 86 + 105155145893683208 z + 751272969152094399076 z 96 98 92 - 7427953419779269 z + 431860571719817 z - 1240835505133947956 z 82 64 112 + 25056197079619227793808 z - 221581782911465560658632866 z - 421 z 114 110 106 108 + z + 73332 z + 475739514 z - 7292936 z 30 42 + 4666003731843976906428 z + 14290004286383901482657260 z 44 46 - 34287824936243704873604880 z + 72435851128017250569239492 z 58 56 + 465812688452396843239608980 z - 465812688452396843239608980 z 54 52 + 411611825621708953453994384 z - 321346888807270873052322732 z 60 70 - 411611825621708953453994384 z + 34287824936243704873604880 z 68 78 - 72435851128017250569239492 z + 474511083772565437165048 z 32 38 - 25056197079619227793808 z + 1684214146160794312006748 z 40 62 - 5236363024472323697141920 z + 321346888807270873052322732 z 76 74 - 1684214146160794312006748 z + 5236363024472323697141920 z 72 104 - 14290004286383901482657260 z - 22020872958 z ) And in Maple-input format, it is: -(1+204464343236274612953*z^28-29992700261291386236*z^26-341*z^2+ 3767642188831286519*z^24-402621309612823545*z^22+50775*z^4-4450268*z^6-\ 10987394897*z^102+260804575*z^8-10987394897*z^10+348817658110*z^12-\ 8626700834565*z^14-2736684777389751*z^18+170340943895386*z^16-\ 32446128595137466069766746*z^50+20941737826133308911944257*z^48+ 36309671175853536*z^20+103334965586237515961983*z^36-26886830336058659370517*z^ 34-11915133880345161099196413*z^66+6095721076014747636244*z^80+348817658110*z^ 100-402621309612823545*z^90+3767642188831286519*z^88+204464343236274612953*z^84 -2736684777389751*z^94-29992700261291386236*z^86+170340943895386*z^96-\ 8626700834565*z^98+36309671175853536*z^92-1200278477078919250291*z^82+ 20941737826133308911944257*z^64+z^112-341*z^110-4450268*z^106+50775*z^108-\ 1200278477078919250291*z^30-2633296164742206636701211*z^42+ 5971396193959915069279080*z^44-11915133880345161099196413*z^46-\ 53471735893102075918729571*z^58+56913913605096582903833834*z^56-\ 53471735893102075918729571*z^54+44341185746608886722144235*z^52+ 44341185746608886722144235*z^60-2633296164742206636701211*z^70+ 5971396193959915069279080*z^68-26886830336058659370517*z^78+ 6095721076014747636244*z^32-347008728444113499975088*z^38+ 1020505244272702401607653*z^40-32446128595137466069766746*z^62+ 103334965586237515961983*z^76-347008728444113499975088*z^74+ 1020505244272702401607653*z^72+260804575*z^104)/(-1-751272969152094399076*z^28+ 104071064164374686464*z^26+421*z^2-12330695474456429352*z^24+ 1240835505133947956*z^22-73332*z^4+7292936*z^6+760972965460*z^102-475739514*z^8 +22020872958*z^10-760972965460*z^12+20342861196816*z^14+7427953419779269*z^18-\ 431860571719817*z^16+221581782911465560658632866*z^50-\ 134883128668041555527567062*z^48-105155145893683208*z^20-\ 474511083772565437165048*z^36+116818120092202618332132*z^34+ 134883128668041555527567062*z^66-116818120092202618332132*z^80-20342861196816*z ^100+12330695474456429352*z^90-104071064164374686464*z^88-\ 4666003731843976906428*z^84+105155145893683208*z^94+751272969152094399076*z^86-\ 7427953419779269*z^96+431860571719817*z^98-1240835505133947956*z^92+ 25056197079619227793808*z^82-221581782911465560658632866*z^64-421*z^112+z^114+ 73332*z^110+475739514*z^106-7292936*z^108+4666003731843976906428*z^30+ 14290004286383901482657260*z^42-34287824936243704873604880*z^44+ 72435851128017250569239492*z^46+465812688452396843239608980*z^58-\ 465812688452396843239608980*z^56+411611825621708953453994384*z^54-\ 321346888807270873052322732*z^52-411611825621708953453994384*z^60+ 34287824936243704873604880*z^70-72435851128017250569239492*z^68+ 474511083772565437165048*z^78-25056197079619227793808*z^32+ 1684214146160794312006748*z^38-5236363024472323697141920*z^40+ 321346888807270873052322732*z^62-1684214146160794312006748*z^76+ 5236363024472323697141920*z^74-14290004286383901482657260*z^72-22020872958*z^ 104) The first , 40, terms are: [0, 80, 0, 11123, 0, 1658891, 0, 251221216, 0, 38207981193, 0, 5819055679585, 0, 886671509308696, 0, 135130051582336267, 0, 20595466189751585287, 0, 3139087113836542181896, 0, 478453733991339844758917, 0, 72925344477459288363839405, 0, 11115214028136777556341806936, 0, 1694171989708807544781566923967, 0, 258224412980553264391195831185899, 0, 39358374668049448768788611820227512, 0, 5998974765715093719683576661336243425, 0, 914359378810109135836003617981917534713, 0, 139365993966959043039030235847657884594272, 0, 21242063892135225715233563765131677642489275] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 3111519775601627439 z + 749666925723933378 z + 308 z 24 22 4 6 - 149609119923597599 z + 24602952781350602 z - 40165 z + 2985733 z 8 10 12 14 - 143708937 z + 4820930985 z - 118319552698 z + 2198370894497 z 18 16 50 + 361720569164002 z - 31706275363082 z + 150057144153284698144 z 48 20 - 253633192478689017263 z - 3311637178691154 z 36 34 - 150057144153284698144 z + 74436796202102010163 z 66 80 84 86 82 + 3311637178691154 z - 2985733 z - 308 z + z + 40165 z 64 30 - 24602952781350602 z + 10739979333171476303 z 42 44 + 428381249211625789340 z - 428381249211625789340 z 46 58 + 359741622657971852156 z + 3111519775601627439 z 56 54 - 10739979333171476303 z + 30922812819366632531 z 52 60 70 - 74436796202102010163 z - 749666925723933378 z + 31706275363082 z 68 78 32 - 361720569164002 z + 143708937 z - 30922812819366632531 z 38 40 + 253633192478689017263 z - 359741622657971852156 z 62 76 74 + 149609119923597599 z - 4820930985 z + 118319552698 z 72 / 28 - 2198370894497 z ) / (1 + 13866510804538204214 z / 26 2 24 - 3104272659633931454 z - 382 z + 575490563935659851 z 22 4 6 8 - 87835961898126236 z + 58988 z - 5048508 z + 274678889 z 10 12 14 - 10281061240 z + 278798030414 z - 5680717407066 z 18 16 50 - 1106177457653724 z + 89329131531197 z - 1653933187595239640506 z 48 20 + 2544549670654975529751 z + 10955618869136918 z 36 34 + 904004886196551818256 z - 415060637951907523660 z 66 80 88 84 86 - 87835961898126236 z + 274678889 z + z + 58988 z - 382 z 82 64 30 - 5048508 z + 575490563935659851 z - 51527042062482576720 z 42 44 - 3294218705998195561624 z + 3590164655524812746068 z 46 58 - 3294218705998195561624 z - 51527042062482576720 z 56 54 + 159842862907635907207 z - 415060637951907523660 z 52 60 + 904004886196551818256 z + 13866510804538204214 z 70 68 78 - 1106177457653724 z + 10955618869136918 z - 10281061240 z 32 38 + 159842862907635907207 z - 1653933187595239640506 z 40 62 76 + 2544549670654975529751 z - 3104272659633931454 z + 278798030414 z 74 72 - 5680717407066 z + 89329131531197 z ) And in Maple-input format, it is: -(-1-3111519775601627439*z^28+749666925723933378*z^26+308*z^2-\ 149609119923597599*z^24+24602952781350602*z^22-40165*z^4+2985733*z^6-143708937* z^8+4820930985*z^10-118319552698*z^12+2198370894497*z^14+361720569164002*z^18-\ 31706275363082*z^16+150057144153284698144*z^50-253633192478689017263*z^48-\ 3311637178691154*z^20-150057144153284698144*z^36+74436796202102010163*z^34+ 3311637178691154*z^66-2985733*z^80-308*z^84+z^86+40165*z^82-24602952781350602*z ^64+10739979333171476303*z^30+428381249211625789340*z^42-428381249211625789340* z^44+359741622657971852156*z^46+3111519775601627439*z^58-10739979333171476303*z ^56+30922812819366632531*z^54-74436796202102010163*z^52-749666925723933378*z^60 +31706275363082*z^70-361720569164002*z^68+143708937*z^78-30922812819366632531*z ^32+253633192478689017263*z^38-359741622657971852156*z^40+149609119923597599*z^ 62-4820930985*z^76+118319552698*z^74-2198370894497*z^72)/(1+ 13866510804538204214*z^28-3104272659633931454*z^26-382*z^2+575490563935659851*z ^24-87835961898126236*z^22+58988*z^4-5048508*z^6+274678889*z^8-10281061240*z^10 +278798030414*z^12-5680717407066*z^14-1106177457653724*z^18+89329131531197*z^16 -1653933187595239640506*z^50+2544549670654975529751*z^48+10955618869136918*z^20 +904004886196551818256*z^36-415060637951907523660*z^34-87835961898126236*z^66+ 274678889*z^80+z^88+58988*z^84-382*z^86-5048508*z^82+575490563935659851*z^64-\ 51527042062482576720*z^30-3294218705998195561624*z^42+3590164655524812746068*z^ 44-3294218705998195561624*z^46-51527042062482576720*z^58+159842862907635907207* z^56-415060637951907523660*z^54+904004886196551818256*z^52+13866510804538204214 *z^60-1106177457653724*z^70+10955618869136918*z^68-10281061240*z^78+ 159842862907635907207*z^32-1653933187595239640506*z^38+2544549670654975529751*z ^40-3104272659633931454*z^62+278798030414*z^76-5680717407066*z^74+ 89329131531197*z^72) The first , 40, terms are: [0, 74, 0, 9445, 0, 1305653, 0, 184237426, 0, 26177888097, 0, 3729733531329, 0, 532021665176466, 0, 75928901832168781, 0, 10838954118058278245, 0, 1547441966376834539370, 0, 220934097526519840493913, 0, 31544301976484650389703945, 0, 4503846577691372380425022314, 0, 643055288489808740180172844965, 0, 91815073284785713758299872548509, 0, 13109317727252872322477710055472242, 0, 1871743753699294286651999384995030593, 0, 267246967976055724715928335752334611681, 0, 38157439354004088955764227698656872641682, 0, 5448107618847754599609415576716210761114277] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 206584035853550489282 z - 29884329583641515291 z - 335 z 24 22 4 6 + 3704132527424910544 z - 390925512568219613 z + 49078 z - 4245405 z 102 8 10 12 - 10325531899 z + 246461552 z - 10325531899 z + 327217358674 z 14 18 16 - 8106059436997 z - 2602780777588143 z + 160820543200820 z 50 48 - 36965469499247226589531139 z + 23731353478664369268476196 z 20 36 + 34862447302387606 z + 110357073349747505698358 z 34 66 - 28334420951607312740931 z - 13410925195435298140077117 z 80 100 90 + 6335693106413526344388 z + 327217358674 z - 390925512568219613 z 88 84 + 3704132527424910544 z + 206584035853550489282 z 94 86 96 - 2602780777588143 z - 29884329583641515291 z + 160820543200820 z 98 92 82 - 8106059436997 z + 34862447302387606 z - 1230030448588098477277 z 64 112 110 106 + 23731353478664369268476196 z + z - 335 z - 4245405 z 108 30 42 + 49078 z - 1230030448588098477277 z - 2912107922150161734958811 z 44 46 + 6666364429345091834357098 z - 13410925195435298140077117 z 58 56 - 61297593421525117317692537 z + 65294340786013026129950554 z 54 52 - 61297593421525117317692537 z + 50712560540182911099719318 z 60 70 + 50712560540182911099719318 z - 2912107922150161734958811 z 68 78 + 6666364429345091834357098 z - 28334420951607312740931 z 32 38 + 6335693106413526344388 z - 375281476407537264230745 z 40 62 + 1116620313184529711122288 z - 36965469499247226589531139 z 76 74 + 110357073349747505698358 z - 375281476407537264230745 z 72 104 / + 1116620313184529711122288 z + 246461552 z ) / (-1 / 28 26 2 - 748483561329006231348 z + 102138578996536272870 z + 412 z 24 22 4 - 11929490208358011928 z + 1184669874165025134 z - 70419 z 6 102 8 10 + 6893124 z + 702822866554 z - 444310034 z + 20408315060 z 12 14 18 - 702822866554 z + 18798463510043 z + 6941656543561357 z 16 50 - 400711747137155 z + 252153898951900386588732133 z 48 20 - 152520167900716708501860995 z - 99227076950527553 z 36 34 - 502160642207117361244315 z + 121838018378219318208319 z 66 80 + 152520167900716708501860995 z - 121838018378219318208319 z 100 90 - 18798463510043 z + 11929490208358011928 z 88 84 - 102138578996536272870 z - 4720594145499771013293 z 94 86 96 + 99227076950527553 z + 748483561329006231348 z - 6941656543561357 z 98 92 + 400711747137155 z - 1184669874165025134 z 82 64 112 + 25741687701553887625873 z - 252153898951900386588732133 z - 412 z 114 110 106 108 + z + 70419 z + 444310034 z - 6893124 z 30 42 + 4720594145499771013293 z + 15716816601622719412596476 z 44 46 - 38113138149908747600370022 z + 81266939318535800673478799 z 58 56 + 535231021823153292377863554 z - 535231021823153292377863554 z 54 52 + 472186527719255495999115382 z - 367448739457198745101849445 z 60 70 - 472186527719255495999115382 z + 38113138149908747600370022 z 68 78 - 81266939318535800673478799 z + 502160642207117361244315 z 32 38 - 25741687701553887625873 z + 1807160315575527604649864 z 40 62 - 5691552741951769837125738 z + 367448739457198745101849445 z 76 74 - 1807160315575527604649864 z + 5691552741951769837125738 z 72 104 - 15716816601622719412596476 z - 20408315060 z ) And in Maple-input format, it is: -(1+206584035853550489282*z^28-29884329583641515291*z^26-335*z^2+ 3704132527424910544*z^24-390925512568219613*z^22+49078*z^4-4245405*z^6-\ 10325531899*z^102+246461552*z^8-10325531899*z^10+327217358674*z^12-\ 8106059436997*z^14-2602780777588143*z^18+160820543200820*z^16-\ 36965469499247226589531139*z^50+23731353478664369268476196*z^48+ 34862447302387606*z^20+110357073349747505698358*z^36-28334420951607312740931*z^ 34-13410925195435298140077117*z^66+6335693106413526344388*z^80+327217358674*z^ 100-390925512568219613*z^90+3704132527424910544*z^88+206584035853550489282*z^84 -2602780777588143*z^94-29884329583641515291*z^86+160820543200820*z^96-\ 8106059436997*z^98+34862447302387606*z^92-1230030448588098477277*z^82+ 23731353478664369268476196*z^64+z^112-335*z^110-4245405*z^106+49078*z^108-\ 1230030448588098477277*z^30-2912107922150161734958811*z^42+ 6666364429345091834357098*z^44-13410925195435298140077117*z^46-\ 61297593421525117317692537*z^58+65294340786013026129950554*z^56-\ 61297593421525117317692537*z^54+50712560540182911099719318*z^52+ 50712560540182911099719318*z^60-2912107922150161734958811*z^70+ 6666364429345091834357098*z^68-28334420951607312740931*z^78+ 6335693106413526344388*z^32-375281476407537264230745*z^38+ 1116620313184529711122288*z^40-36965469499247226589531139*z^62+ 110357073349747505698358*z^76-375281476407537264230745*z^74+ 1116620313184529711122288*z^72+246461552*z^104)/(-1-748483561329006231348*z^28+ 102138578996536272870*z^26+412*z^2-11929490208358011928*z^24+ 1184669874165025134*z^22-70419*z^4+6893124*z^6+702822866554*z^102-444310034*z^8 +20408315060*z^10-702822866554*z^12+18798463510043*z^14+6941656543561357*z^18-\ 400711747137155*z^16+252153898951900386588732133*z^50-\ 152520167900716708501860995*z^48-99227076950527553*z^20-\ 502160642207117361244315*z^36+121838018378219318208319*z^34+ 152520167900716708501860995*z^66-121838018378219318208319*z^80-18798463510043*z ^100+11929490208358011928*z^90-102138578996536272870*z^88-\ 4720594145499771013293*z^84+99227076950527553*z^94+748483561329006231348*z^86-\ 6941656543561357*z^96+400711747137155*z^98-1184669874165025134*z^92+ 25741687701553887625873*z^82-252153898951900386588732133*z^64-412*z^112+z^114+ 70419*z^110+444310034*z^106-6893124*z^108+4720594145499771013293*z^30+ 15716816601622719412596476*z^42-38113138149908747600370022*z^44+ 81266939318535800673478799*z^46+535231021823153292377863554*z^58-\ 535231021823153292377863554*z^56+472186527719255495999115382*z^54-\ 367448739457198745101849445*z^52-472186527719255495999115382*z^60+ 38113138149908747600370022*z^70-81266939318535800673478799*z^68+ 502160642207117361244315*z^78-25741687701553887625873*z^32+ 1807160315575527604649864*z^38-5691552741951769837125738*z^40+ 367448739457198745101849445*z^62-1807160315575527604649864*z^76+ 5691552741951769837125738*z^74-15716816601622719412596476*z^72-20408315060*z^ 104) The first , 40, terms are: [0, 77, 0, 10383, 0, 1503252, 0, 221101413, 0, 32678496603, 0, 4838466306355, 0, 716905153701615, 0, 106253338481914279, 0, 15749851653597179701, 0, 2334708336543407217492, 0, 346097290133503009361279, 0, 51305950037035439853517481, 0, 7605695733730246890752897805, 0, 1127485255019499020160337598053, 0, 167141037984169234473691059979729, 0, 24777383677470691691160690774129487, 0, 3673058545864389032889510998908189684, 0, 544503001771944269620893527519622570821, 0, 80718431740908436781515676979962797156623, 0, 11965894150570453950445068412556353356137383] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 192028963493582089702 z - 28163201272821658263 z - 341 z 24 22 4 6 + 3540092530847698568 z - 378926060278598169 z + 50614 z - 4410253 z 102 8 10 12 - 10711954607 z + 256462732 z - 10711954607 z + 337186719570 z 14 18 16 - 8274263325079 z - 2593350667196245 z + 162292618554912 z 50 48 - 30997617713719915181597361 z + 19986851630708762199688860 z 20 36 + 34268360600522290 z + 97613060804121424428994 z 34 66 - 25349958298889706181253 z - 11357488626542270074247703 z 80 100 90 + 5737582075949582578496 z + 337186719570 z - 378926060278598169 z 88 84 + 3540092530847698568 z + 192028963493582089702 z 94 86 96 - 2593350667196245 z - 28163201272821658263 z + 162292618554912 z 98 92 82 - 8274263325079 z + 34268360600522290 z - 1128238331066616800815 z 64 112 110 106 + 19986851630708762199688860 z + z - 341 z - 4410253 z 108 30 42 + 50614 z - 1128238331066616800815 z - 2501998704031660113812375 z 44 46 + 5683374557652652380899010 z - 11357488626542270074247703 z 58 56 - 51143664531878436907207233 z + 54443893972195400188618190 z 54 52 - 51143664531878436907207233 z + 42392190596461159674337950 z 60 70 + 42392190596461159674337950 z - 2501998704031660113812375 z 68 78 + 5683374557652652380899010 z - 25349958298889706181253 z 32 38 + 5737582075949582578496 z - 328444015167489335616961 z 40 62 + 967810315411353867203688 z - 30997617713719915181597361 z 76 74 + 97613060804121424428994 z - 328444015167489335616961 z 72 104 / 2 + 967810315411353867203688 z + 256462732 z ) / ((-1 + z ) (1 / 28 26 2 + 618958281596031811716 z - 87198789467991184652 z - 425 z 24 22 4 + 10503187182440900654 z - 1074488363863959702 z + 73976 z 6 102 8 10 - 7288912 z - 21292534934 z + 468405818 z - 21292534934 z 12 14 18 + 721684359456 z - 18920093735637 z - 6656237390061735 z 16 50 + 394104136731844 z - 132428222486752296248734171 z 48 20 + 84391858807210247700107260 z + 92602046631062512 z 36 34 + 361171104489737653157296 z - 90927495561252677574661 z 66 80 - 47246570954659364508395637 z + 19904768341027566584824 z 100 90 88 + 721684359456 z - 1074488363863959702 z + 10503187182440900654 z 84 94 + 618958281596031811716 z - 6656237390061735 z 86 96 98 - 87198789467991184652 z + 394104136731844 z - 18920093735637 z 92 82 + 92602046631062512 z - 3777067030093159408191 z 64 112 110 106 + 84391858807210247700107260 z + z - 425 z - 7288912 z 108 30 42 + 73976 z - 3777067030093159408191 z - 10012655369665818170104846 z 44 46 + 23222526832065573600208448 z - 47246570954659364508395637 z 58 56 - 221483762271415865410026706 z + 236179493014449214557738444 z 54 52 - 221483762271415865410026706 z + 182647933084644373652227256 z 60 70 + 182647933084644373652227256 z - 10012655369665818170104846 z 68 78 + 23222526832065573600208448 z - 90927495561252677574661 z 32 38 + 19904768341027566584824 z - 1250597915977393360344340 z 40 62 + 3782831182574985591120050 z - 132428222486752296248734171 z 76 74 + 361171104489737653157296 z - 1250597915977393360344340 z 72 104 + 3782831182574985591120050 z + 468405818 z )) And in Maple-input format, it is: -(1+192028963493582089702*z^28-28163201272821658263*z^26-341*z^2+ 3540092530847698568*z^24-378926060278598169*z^22+50614*z^4-4410253*z^6-\ 10711954607*z^102+256462732*z^8-10711954607*z^10+337186719570*z^12-\ 8274263325079*z^14-2593350667196245*z^18+162292618554912*z^16-\ 30997617713719915181597361*z^50+19986851630708762199688860*z^48+ 34268360600522290*z^20+97613060804121424428994*z^36-25349958298889706181253*z^ 34-11357488626542270074247703*z^66+5737582075949582578496*z^80+337186719570*z^ 100-378926060278598169*z^90+3540092530847698568*z^88+192028963493582089702*z^84 -2593350667196245*z^94-28163201272821658263*z^86+162292618554912*z^96-\ 8274263325079*z^98+34268360600522290*z^92-1128238331066616800815*z^82+ 19986851630708762199688860*z^64+z^112-341*z^110-4410253*z^106+50614*z^108-\ 1128238331066616800815*z^30-2501998704031660113812375*z^42+ 5683374557652652380899010*z^44-11357488626542270074247703*z^46-\ 51143664531878436907207233*z^58+54443893972195400188618190*z^56-\ 51143664531878436907207233*z^54+42392190596461159674337950*z^52+ 42392190596461159674337950*z^60-2501998704031660113812375*z^70+ 5683374557652652380899010*z^68-25349958298889706181253*z^78+ 5737582075949582578496*z^32-328444015167489335616961*z^38+ 967810315411353867203688*z^40-30997617713719915181597361*z^62+ 97613060804121424428994*z^76-328444015167489335616961*z^74+ 967810315411353867203688*z^72+256462732*z^104)/(-1+z^2)/(1+ 618958281596031811716*z^28-87198789467991184652*z^26-425*z^2+ 10503187182440900654*z^24-1074488363863959702*z^22+73976*z^4-7288912*z^6-\ 21292534934*z^102+468405818*z^8-21292534934*z^10+721684359456*z^12-\ 18920093735637*z^14-6656237390061735*z^18+394104136731844*z^16-\ 132428222486752296248734171*z^50+84391858807210247700107260*z^48+ 92602046631062512*z^20+361171104489737653157296*z^36-90927495561252677574661*z^ 34-47246570954659364508395637*z^66+19904768341027566584824*z^80+721684359456*z^ 100-1074488363863959702*z^90+10503187182440900654*z^88+618958281596031811716*z^ 84-6656237390061735*z^94-87198789467991184652*z^86+394104136731844*z^96-\ 18920093735637*z^98+92602046631062512*z^92-3777067030093159408191*z^82+ 84391858807210247700107260*z^64+z^112-425*z^110-7288912*z^106+73976*z^108-\ 3777067030093159408191*z^30-10012655369665818170104846*z^42+ 23222526832065573600208448*z^44-47246570954659364508395637*z^46-\ 221483762271415865410026706*z^58+236179493014449214557738444*z^56-\ 221483762271415865410026706*z^54+182647933084644373652227256*z^52+ 182647933084644373652227256*z^60-10012655369665818170104846*z^70+ 23222526832065573600208448*z^68-90927495561252677574661*z^78+ 19904768341027566584824*z^32-1250597915977393360344340*z^38+ 3782831182574985591120050*z^40-132428222486752296248734171*z^62+ 361171104489737653157296*z^76-1250597915977393360344340*z^74+ 3782831182574985591120050*z^72+468405818*z^104) The first , 40, terms are: [0, 85, 0, 12423, 0, 1920748, 0, 300568507, 0, 47220703753, 0, 7430008868005, 0, 1169829395662313, 0, 184235231989041749, 0, 29018357001937807643, 0, 4570823020453194306748, 0, 719987940678988886857847, 0, 113412262568325988180288689, 0, 17864732639653546187040329253, 0, 2814062679588923305390465170301, 0, 443273053046974361373737425362489, 0, 69824692686253270094456550087336503, 0, 10998837717789076154397004605168029724, 0, 1732545230393745996018845973692943220443, 0, 272911841121230118460629114861593297986109, 0, 42989281050538643574078656030303752896125217] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 223000610197063460227 z - 32373684532975719875 z - 338 z 24 22 4 6 + 4024066971751893423 z - 425479359908934960 z + 50099 z - 4391719 z 102 8 10 12 - 10953559916 z + 258356199 z - 10953559916 z + 350546995461 z 14 18 16 - 8748886329346 z - 2831668640175302 z + 174457053175110 z 50 48 - 38368660133871083459043300 z + 24677740990716007142056410 z 20 36 + 37966136410565893 z + 117105896393213970002728 z 34 66 - 30199012401717208303780 z - 13978495651662223452797894 z 80 100 90 + 6782466451131524523369 z + 350546995461 z - 425479359908934960 z 88 84 + 4024066971751893423 z + 223000610197063460227 z 94 86 96 - 2831668640175302 z - 32373684532975719875 z + 174457053175110 z 98 92 82 - 8748886329346 z + 37966136410565893 z - 1322445294629638527846 z 64 112 110 106 + 24677740990716007142056410 z + z - 338 z - 4391719 z 108 30 42 + 50099 z - 1322445294629638527846 z - 3053839688101293369965160 z 44 46 + 6968102674208589975736314 z - 13978495651662223452797894 z 58 56 - 63488739489879749800681928 z + 67610200340495121482480260 z 54 52 - 63488739489879749800681928 z + 52567544409855442620354998 z 60 70 + 52567544409855442620354998 z - 3053839688101293369965160 z 68 78 + 6968102674208589975736314 z - 30199012401717208303780 z 32 38 + 6782466451131524523369 z - 396554238630088132008364 z 40 62 + 1175239791155090791420342 z - 38368660133871083459043300 z 76 74 + 117105896393213970002728 z - 396554238630088132008364 z 72 104 / + 1175239791155090791420342 z + 258356199 z ) / (-1 / 28 26 2 - 816161964193390762362 z + 111727595746507102187 z + 425 z 24 22 4 - 13083867986414118803 z + 1301725189889372740 z - 73678 z 6 102 8 10 + 7292350 z + 763662122112 z - 474835171 z + 22008877947 z 12 14 18 - 763662122112 z + 20542399981160 z + 7630195063011405 z 16 50 - 439559480839005 z + 266474116830316156445855322 z 48 20 - 161406370761982440120934842 z - 109121615517779916 z 36 34 - 539756381572356539527972 z + 131426128180583569135793 z 66 80 + 161406370761982440120934842 z - 131426128180583569135793 z 100 90 - 20542399981160 z + 13083867986414118803 z 88 84 - 111727595746507102187 z - 5129456530368821043130 z 94 86 + 109121615517779916 z + 816161964193390762362 z 96 98 92 - 7630195063011405 z + 439559480839005 z - 1301725189889372740 z 82 64 112 + 27869280020243285771321 z - 266474116830316156445855322 z - 425 z 114 110 106 108 + z + 73678 z + 474835171 z - 7292350 z 30 42 + 5129456530368821043130 z + 16737892915930188481184582 z 44 46 - 40489555230610112116851360 z + 86152317265519074752661904 z 58 56 + 564460572680823373615754750 z - 564460572680823373615754750 z 54 52 + 498143559749214119001643348 z - 387914513082972295688183076 z 60 70 - 498143559749214119001643348 z + 40489555230610112116851360 z 68 78 - 86152317265519074752661904 z + 539756381572356539527972 z 32 38 - 27869280020243285771321 z + 1935943049836934592574532 z 40 62 - 6078275933533368865488726 z + 387914513082972295688183076 z 76 74 - 1935943049836934592574532 z + 6078275933533368865488726 z 72 104 - 16737892915930188481184582 z - 22008877947 z ) And in Maple-input format, it is: -(1+223000610197063460227*z^28-32373684532975719875*z^26-338*z^2+ 4024066971751893423*z^24-425479359908934960*z^22+50099*z^4-4391719*z^6-\ 10953559916*z^102+258356199*z^8-10953559916*z^10+350546995461*z^12-\ 8748886329346*z^14-2831668640175302*z^18+174457053175110*z^16-\ 38368660133871083459043300*z^50+24677740990716007142056410*z^48+ 37966136410565893*z^20+117105896393213970002728*z^36-30199012401717208303780*z^ 34-13978495651662223452797894*z^66+6782466451131524523369*z^80+350546995461*z^ 100-425479359908934960*z^90+4024066971751893423*z^88+223000610197063460227*z^84 -2831668640175302*z^94-32373684532975719875*z^86+174457053175110*z^96-\ 8748886329346*z^98+37966136410565893*z^92-1322445294629638527846*z^82+ 24677740990716007142056410*z^64+z^112-338*z^110-4391719*z^106+50099*z^108-\ 1322445294629638527846*z^30-3053839688101293369965160*z^42+ 6968102674208589975736314*z^44-13978495651662223452797894*z^46-\ 63488739489879749800681928*z^58+67610200340495121482480260*z^56-\ 63488739489879749800681928*z^54+52567544409855442620354998*z^52+ 52567544409855442620354998*z^60-3053839688101293369965160*z^70+ 6968102674208589975736314*z^68-30199012401717208303780*z^78+ 6782466451131524523369*z^32-396554238630088132008364*z^38+ 1175239791155090791420342*z^40-38368660133871083459043300*z^62+ 117105896393213970002728*z^76-396554238630088132008364*z^74+ 1175239791155090791420342*z^72+258356199*z^104)/(-1-816161964193390762362*z^28+ 111727595746507102187*z^26+425*z^2-13083867986414118803*z^24+ 1301725189889372740*z^22-73678*z^4+7292350*z^6+763662122112*z^102-474835171*z^8 +22008877947*z^10-763662122112*z^12+20542399981160*z^14+7630195063011405*z^18-\ 439559480839005*z^16+266474116830316156445855322*z^50-\ 161406370761982440120934842*z^48-109121615517779916*z^20-\ 539756381572356539527972*z^36+131426128180583569135793*z^34+ 161406370761982440120934842*z^66-131426128180583569135793*z^80-20542399981160*z ^100+13083867986414118803*z^90-111727595746507102187*z^88-\ 5129456530368821043130*z^84+109121615517779916*z^94+816161964193390762362*z^86-\ 7630195063011405*z^96+439559480839005*z^98-1301725189889372740*z^92+ 27869280020243285771321*z^82-266474116830316156445855322*z^64-425*z^112+z^114+ 73678*z^110+474835171*z^106-7292350*z^108+5129456530368821043130*z^30+ 16737892915930188481184582*z^42-40489555230610112116851360*z^44+ 86152317265519074752661904*z^46+564460572680823373615754750*z^58-\ 564460572680823373615754750*z^56+498143559749214119001643348*z^54-\ 387914513082972295688183076*z^52-498143559749214119001643348*z^60+ 40489555230610112116851360*z^70-86152317265519074752661904*z^68+ 539756381572356539527972*z^78-27869280020243285771321*z^32+ 1935943049836934592574532*z^38-6078275933533368865488726*z^40+ 387914513082972295688183076*z^62-1935943049836934592574532*z^76+ 6078275933533368865488726*z^74-16737892915930188481184582*z^72-22008877947*z^ 104) The first , 40, terms are: [0, 87, 0, 13396, 0, 2183945, 0, 359141615, 0, 59159465419, 0, 9748793517877, 0, 1606644448578380, 0, 264788986843018967, 0, 43639839996201103229, 0, 7192291043631067230077, 0, 1185363656524619780754667, 0, 195360174710139650114573652, 0, 32197376351270415190674544025, 0, 5306460545740103223520847420627, 0, 874559571010625205368975106230247, 0, 144136461147690110361937163380864461, 0, 23755179326146818962273476069581348108, 0, 3915099208037423362827203925546644847707, 0, 645248836008483821890974402894231834198737, 0, 106343680773282828198820519522889683866654849] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1979594381184461837 z - 474603601047982881 z - 293 z 24 22 4 6 + 94930258317722637 z - 15751471485522340 z + 36117 z - 2535272 z 8 10 12 14 + 115561284 z - 3688981144 z + 86635110011 z - 1549140680703 z 18 16 50 - 240162275971988 z + 21625668898459 z - 187996856650050420379 z 48 20 + 283117085348764266151 z + 2152700347698476 z 36 34 + 105852190641087515283 z - 50474791924677382296 z 66 80 88 84 86 - 15751471485522340 z + 115561284 z + z + 36117 z - 293 z 82 64 30 - 2535272 z + 94930258317722637 z - 6920215482135598904 z 42 44 - 361837454145957791976 z + 392650403791721697240 z 46 58 - 361837454145957791976 z - 6920215482135598904 z 56 54 + 20348738052721130100 z - 50474791924677382296 z 52 60 + 105852190641087515283 z + 1979594381184461837 z 70 68 78 - 240162275971988 z + 2152700347698476 z - 3688981144 z 32 38 + 20348738052721130100 z - 187996856650050420379 z 40 62 76 + 283117085348764266151 z - 474603601047982881 z + 86635110011 z 74 72 / 2 - 1549140680703 z + 21625668898459 z ) / ((-1 + z ) (1 / 28 26 2 + 7148918814167339676 z - 1642933504738887436 z - 370 z 24 22 4 6 + 313796745590482115 z - 49529964954025466 z + 54162 z - 4349338 z 8 10 12 14 + 221451639 z - 7770981152 z + 198343053320 z - 3821840197620 z 18 16 50 - 675256643520806 z + 57111314122713 z - 786035227686173501102 z 48 20 + 1200782349703753177723 z + 6414192593778498 z 36 34 + 433952266364267921106 z - 201858329796339438934 z 66 80 88 84 86 - 49529964954025466 z + 221451639 z + z + 54162 z - 370 z 82 64 30 - 4349338 z + 313796745590482115 z - 25967207936313419296 z 42 44 - 1548025962311900798736 z + 1684742737964563221220 z 46 58 - 1548025962311900798736 z - 25967207936313419296 z 56 54 + 79004698202255212429 z - 201858329796339438934 z 52 60 + 433952266364267921106 z + 7148918814167339676 z 70 68 78 - 675256643520806 z + 6414192593778498 z - 7770981152 z 32 38 + 79004698202255212429 z - 786035227686173501102 z 40 62 76 + 1200782349703753177723 z - 1642933504738887436 z + 198343053320 z 74 72 - 3821840197620 z + 57111314122713 z )) And in Maple-input format, it is: -(1+1979594381184461837*z^28-474603601047982881*z^26-293*z^2+94930258317722637* z^24-15751471485522340*z^22+36117*z^4-2535272*z^6+115561284*z^8-3688981144*z^10 +86635110011*z^12-1549140680703*z^14-240162275971988*z^18+21625668898459*z^16-\ 187996856650050420379*z^50+283117085348764266151*z^48+2152700347698476*z^20+ 105852190641087515283*z^36-50474791924677382296*z^34-15751471485522340*z^66+ 115561284*z^80+z^88+36117*z^84-293*z^86-2535272*z^82+94930258317722637*z^64-\ 6920215482135598904*z^30-361837454145957791976*z^42+392650403791721697240*z^44-\ 361837454145957791976*z^46-6920215482135598904*z^58+20348738052721130100*z^56-\ 50474791924677382296*z^54+105852190641087515283*z^52+1979594381184461837*z^60-\ 240162275971988*z^70+2152700347698476*z^68-3688981144*z^78+20348738052721130100 *z^32-187996856650050420379*z^38+283117085348764266151*z^40-474603601047982881* z^62+86635110011*z^76-1549140680703*z^74+21625668898459*z^72)/(-1+z^2)/(1+ 7148918814167339676*z^28-1642933504738887436*z^26-370*z^2+313796745590482115*z^ 24-49529964954025466*z^22+54162*z^4-4349338*z^6+221451639*z^8-7770981152*z^10+ 198343053320*z^12-3821840197620*z^14-675256643520806*z^18+57111314122713*z^16-\ 786035227686173501102*z^50+1200782349703753177723*z^48+6414192593778498*z^20+ 433952266364267921106*z^36-201858329796339438934*z^34-49529964954025466*z^66+ 221451639*z^80+z^88+54162*z^84-370*z^86-4349338*z^82+313796745590482115*z^64-\ 25967207936313419296*z^30-1548025962311900798736*z^42+1684742737964563221220*z^ 44-1548025962311900798736*z^46-25967207936313419296*z^58+79004698202255212429*z ^56-201858329796339438934*z^54+433952266364267921106*z^52+7148918814167339676*z ^60-675256643520806*z^70+6414192593778498*z^68-7770981152*z^78+ 79004698202255212429*z^32-786035227686173501102*z^38+1200782349703753177723*z^ 40-1642933504738887436*z^62+198343053320*z^76-3821840197620*z^74+57111314122713 *z^72) The first , 40, terms are: [0, 78, 0, 10523, 0, 1518765, 0, 222854886, 0, 32886875667, 0, 4864017058871, 0, 720064612777006, 0, 106639855108752437, 0, 15795790968974211699, 0, 2339887914827973034878, 0, 346627103258322402041089, 0, 51349469245042156571057341, 0, 7606975092324904765264766502, 0, 1126909799727444951054519085895, 0, 166942458894974196560148581316305, 0, 24731170336712634123222064068613062, 0, 3663722938957059710208417881529831539, 0, 542750983035657338786711043071310110399, 0, 80404180747300827865093677655025517032510, 0, 11911231110131102441340738545531537396122281] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2101420268795676557 z - 504408094596074471 z - 295 z 24 22 4 6 + 100964315138515609 z - 16753937163632760 z + 36669 z - 2596214 z 8 10 12 14 + 119342522 z - 3839456306 z + 90777537911 z - 1632104349333 z 18 16 50 - 254811234530120 z + 22878564976491 z - 198086663117919049277 z 48 20 + 298027751802227869179 z + 2287990824226388 z 36 34 + 111673576758442720767 z - 53329706844310653650 z 66 80 88 84 86 - 16753937163632760 z + 119342522 z + z + 36669 z - 295 z 82 64 30 - 2596214 z + 100964315138515609 z - 7335239082879179462 z 42 44 - 380667778339114616192 z + 413000935528059328696 z 46 58 - 380667778339114616192 z - 7335239082879179462 z 56 54 + 21534313994550205110 z - 53329706844310653650 z 52 60 + 111673576758442720767 z + 2101420268795676557 z 70 68 78 - 254811234530120 z + 2287990824226388 z - 3839456306 z 32 38 + 21534313994550205110 z - 198086663117919049277 z 40 62 76 + 298027751802227869179 z - 504408094596074471 z + 90777537911 z 74 72 / 2 - 1632104349333 z + 22878564976491 z ) / ((-1 + z ) (1 / 28 26 2 + 7662985262762643752 z - 1760317842566282560 z - 380 z 24 22 4 6 + 335928686939145167 z - 52950954961696552 z + 56194 z - 4533100 z 8 10 12 14 + 231604055 z - 8156115300 z + 208953496300 z - 4041323660216 z 18 16 50 - 718704168441896 z + 60601330901153 z - 841655405136404109884 z 48 20 + 1285366139812262407599 z + 6844013327506962 z 36 34 + 464825877064711638122 z - 216298064180527578500 z 66 80 88 84 86 - 52950954961696552 z + 231604055 z + z + 56194 z - 380 z 82 64 30 - 4533100 z + 335928686939145167 z - 27837072185001501788 z 42 44 - 1656744377189590743840 z + 1802939573675498233820 z 46 58 - 1656744377189590743840 z - 27837072185001501788 z 56 54 + 84681287769017812441 z - 216298064180527578500 z 52 60 + 464825877064711638122 z + 7662985262762643752 z 70 68 78 - 718704168441896 z + 6844013327506962 z - 8156115300 z 32 38 + 84681287769017812441 z - 841655405136404109884 z 40 62 76 + 1285366139812262407599 z - 1760317842566282560 z + 208953496300 z 74 72 - 4041323660216 z + 60601330901153 z )) And in Maple-input format, it is: -(1+2101420268795676557*z^28-504408094596074471*z^26-295*z^2+100964315138515609 *z^24-16753937163632760*z^22+36669*z^4-2596214*z^6+119342522*z^8-3839456306*z^ 10+90777537911*z^12-1632104349333*z^14-254811234530120*z^18+22878564976491*z^16 -198086663117919049277*z^50+298027751802227869179*z^48+2287990824226388*z^20+ 111673576758442720767*z^36-53329706844310653650*z^34-16753937163632760*z^66+ 119342522*z^80+z^88+36669*z^84-295*z^86-2596214*z^82+100964315138515609*z^64-\ 7335239082879179462*z^30-380667778339114616192*z^42+413000935528059328696*z^44-\ 380667778339114616192*z^46-7335239082879179462*z^58+21534313994550205110*z^56-\ 53329706844310653650*z^54+111673576758442720767*z^52+2101420268795676557*z^60-\ 254811234530120*z^70+2287990824226388*z^68-3839456306*z^78+21534313994550205110 *z^32-198086663117919049277*z^38+298027751802227869179*z^40-504408094596074471* z^62+90777537911*z^76-1632104349333*z^74+22878564976491*z^72)/(-1+z^2)/(1+ 7662985262762643752*z^28-1760317842566282560*z^26-380*z^2+335928686939145167*z^ 24-52950954961696552*z^22+56194*z^4-4533100*z^6+231604055*z^8-8156115300*z^10+ 208953496300*z^12-4041323660216*z^14-718704168441896*z^18+60601330901153*z^16-\ 841655405136404109884*z^50+1285366139812262407599*z^48+6844013327506962*z^20+ 464825877064711638122*z^36-216298064180527578500*z^34-52950954961696552*z^66+ 231604055*z^80+z^88+56194*z^84-380*z^86-4533100*z^82+335928686939145167*z^64-\ 27837072185001501788*z^30-1656744377189590743840*z^42+1802939573675498233820*z^ 44-1656744377189590743840*z^46-27837072185001501788*z^58+84681287769017812441*z ^56-216298064180527578500*z^54+464825877064711638122*z^52+7662985262762643752*z ^60-718704168441896*z^70+6844013327506962*z^68-8156115300*z^78+ 84681287769017812441*z^32-841655405136404109884*z^38+1285366139812262407599*z^ 40-1760317842566282560*z^62+208953496300*z^76-4041323660216*z^74+60601330901153 *z^72) The first , 40, terms are: [0, 86, 0, 12861, 0, 2027757, 0, 322861854, 0, 51555419709, 0, 8241053254877, 0, 1317845380705150, 0, 210772811550885461, 0, 33712579881827793925, 0, 5392377808732266712918, 0, 862527644181927647078513, 0, 137964546130515873035511233, 0, 22067984939364746512100326710, 0, 3529865409168387395427611948677, 0, 564616712073284874549751281423317, 0, 90312810400331397629557830299810622, 0, 14445913301557670444278145621184896397, 0, 2310684539457126489738089680863156873709, 0, 369603704964536289756451628781874870550302, 0, 59119666389491115238152537662144640883712333] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2203056812181780395 z - 536073402724384877 z - 297 z 24 22 4 6 + 108593944797544065 z - 18192245447641988 z + 37331 z - 2681804 z 8 10 12 14 + 125233142 z - 4089547532 z + 97915003321 z - 1776772091343 z 18 16 50 - 279228442722508 z + 25039781001019 z - 194428256391944886043 z 48 20 + 290249215844840872379 z + 2500354294360540 z 36 34 + 110755668811228131777 z - 53554209655857271116 z 66 80 88 84 86 - 18192245447641988 z + 125233142 z + z + 37331 z - 297 z 82 64 30 - 2681804 z + 108593944797544065 z - 7579039214397868052 z 42 44 - 368942262980146017272 z + 399623621736770085976 z 46 58 - 368942262980146017272 z - 7579039214397868052 z 56 54 + 21927443126307705490 z - 53554209655857271116 z 52 60 + 110755668811228131777 z + 2203056812181780395 z 70 68 78 - 279228442722508 z + 2500354294360540 z - 4089547532 z 32 38 + 21927443126307705490 z - 194428256391944886043 z 40 62 76 + 290249215844840872379 z - 536073402724384877 z + 97915003321 z 74 72 / - 1776772091343 z + 25039781001019 z ) / (-1 / 28 26 2 - 10035010767741647326 z + 2263121650875678709 z + 391 z 24 22 4 6 - 424738871262916191 z + 65882502352671726 z - 58496 z + 4790566 z 8 10 12 14 - 249658839 z + 8993148191 z - 235856854944 z + 4666552378534 z 18 16 50 + 864232681808585 z - 71475894542155 z + 2089844636683688095113 z 48 20 - 2871729282033407137959 z - 8376392122082326 z 36 34 - 683699800543008589470 z + 306314103290711025617 z 66 80 90 88 84 + 424738871262916191 z - 8993148191 z + z - 391 z - 4790566 z 86 82 64 + 58496 z + 249658839 z - 2263121650875678709 z 30 42 + 37242923075211515964 z + 2871729282033407137959 z 44 46 - 3365563865891640368508 z + 3365563865891640368508 z 58 56 + 116240394703802549657 z - 306314103290711025617 z 54 52 + 683699800543008589470 z - 1295873094068542699612 z 60 70 68 - 37242923075211515964 z + 8376392122082326 z - 65882502352671726 z 78 32 + 235856854944 z - 116240394703802549657 z 38 40 + 1295873094068542699612 z - 2089844636683688095113 z 62 76 74 + 10035010767741647326 z - 4666552378534 z + 71475894542155 z 72 - 864232681808585 z ) And in Maple-input format, it is: -(1+2203056812181780395*z^28-536073402724384877*z^26-297*z^2+108593944797544065 *z^24-18192245447641988*z^22+37331*z^4-2681804*z^6+125233142*z^8-4089547532*z^ 10+97915003321*z^12-1776772091343*z^14-279228442722508*z^18+25039781001019*z^16 -194428256391944886043*z^50+290249215844840872379*z^48+2500354294360540*z^20+ 110755668811228131777*z^36-53554209655857271116*z^34-18192245447641988*z^66+ 125233142*z^80+z^88+37331*z^84-297*z^86-2681804*z^82+108593944797544065*z^64-\ 7579039214397868052*z^30-368942262980146017272*z^42+399623621736770085976*z^44-\ 368942262980146017272*z^46-7579039214397868052*z^58+21927443126307705490*z^56-\ 53554209655857271116*z^54+110755668811228131777*z^52+2203056812181780395*z^60-\ 279228442722508*z^70+2500354294360540*z^68-4089547532*z^78+21927443126307705490 *z^32-194428256391944886043*z^38+290249215844840872379*z^40-536073402724384877* z^62+97915003321*z^76-1776772091343*z^74+25039781001019*z^72)/(-1-\ 10035010767741647326*z^28+2263121650875678709*z^26+391*z^2-424738871262916191*z ^24+65882502352671726*z^22-58496*z^4+4790566*z^6-249658839*z^8+8993148191*z^10-\ 235856854944*z^12+4666552378534*z^14+864232681808585*z^18-71475894542155*z^16+ 2089844636683688095113*z^50-2871729282033407137959*z^48-8376392122082326*z^20-\ 683699800543008589470*z^36+306314103290711025617*z^34+424738871262916191*z^66-\ 8993148191*z^80+z^90-391*z^88-4790566*z^84+58496*z^86+249658839*z^82-\ 2263121650875678709*z^64+37242923075211515964*z^30+2871729282033407137959*z^42-\ 3365563865891640368508*z^44+3365563865891640368508*z^46+116240394703802549657*z ^58-306314103290711025617*z^56+683699800543008589470*z^54-\ 1295873094068542699612*z^52-37242923075211515964*z^60+8376392122082326*z^70-\ 65882502352671726*z^68+235856854944*z^78-116240394703802549657*z^32+ 1295873094068542699612*z^38-2089844636683688095113*z^40+10035010767741647326*z^ 62-4666552378534*z^76+71475894542155*z^74-864232681808585*z^72) The first , 40, terms are: [0, 94, 0, 15589, 0, 2705437, 0, 471819230, 0, 82339879345, 0, 14371412090317, 0, 2508426869197318, 0, 437831412636286917, 0, 76421121199276112221, 0, 13338905616019047683646, 0, 2328236471534087390826801, 0, 406381571784641962128001317, 0, 70931792080225974672945111206, 0, 12380775875340903546839598540249, 0, 2161000125432523173212159724245073, 0, 377191348268484097544109388812425214, 0, 65836790827446556531800316592990996849, 0, 11491469904111552379570438292275908106813, 0, 2005776388853547249814588527210805632992438, 0, 350097851334619372168346403262872420728804225] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 192627581284250449300 z - 28300683881956827757 z - 341 z 24 22 4 6 + 3562804533474991080 z - 381814417848825859 z + 50632 z - 4415851 z 102 8 10 12 - 10760806477 z + 257175356 z - 10760806477 z + 339295495084 z 14 18 16 - 8337092220067 z - 2615899350930013 z + 163660124714612 z 50 48 - 30613583680609515954368857 z + 19750192300624520667554236 z 20 36 + 34556520585677116 z + 97174236221918555080212 z 34 66 - 25282552529172582305793 z - 11231312456718897658839631 z 80 100 90 + 5733370759713171939312 z + 339295495084 z - 381814417848825859 z 88 84 + 3562804533474991080 z + 192627581284250449300 z 94 86 96 - 2615899350930013 z - 28300683881956827757 z + 163660124714612 z 98 92 82 - 8337092220067 z + 34556520585677116 z - 1129611020126952144959 z 64 112 110 106 + 19750192300624520667554236 z + z - 341 z - 4415851 z 108 30 42 + 50632 z - 1129611020126952144959 z - 2479345762239210121432193 z 44 46 + 5625499658338765391363044 z - 11231312456718897658839631 z 58 56 - 50479297324591680344708039 z + 53732681489946090414375326 z 54 52 - 50479297324591680344708039 z + 41850915360464667595140136 z 60 70 + 41850915360464667595140136 z - 2479345762239210121432193 z 68 78 + 5625499658338765391363044 z - 25282552529172582305793 z 32 38 + 5733370759713171939312 z - 326409430172043194598479 z 40 62 + 960337606514354806601540 z - 30613583680609515954368857 z 76 74 + 97174236221918555080212 z - 326409430172043194598479 z 72 104 / + 960337606514354806601540 z + 257175356 z ) / (-1 / 28 26 2 - 707208526509453383068 z + 98104878476072293474 z + 420 z 24 22 4 - 11647500002809420612 z + 1175302414203779738 z - 72943 z 6 102 8 10 + 7220060 z + 739037554482 z - 468180650 z + 21528917192 z 12 14 18 - 739037554482 z + 19631423804951 z + 7090725693348053 z 16 50 - 414352157118467 z + 209374951811316805897089201 z 48 20 - 127374988536470877346655043 z - 99950960016939797 z 36 34 - 446237532851750295659003 z + 109826645981989109880903 z 66 80 + 127374988536470877346655043 z - 109826645981989109880903 z 100 90 - 19631423804951 z + 11647500002809420612 z 88 84 - 98104878476072293474 z - 4388558255505540357889 z 94 86 96 + 99950960016939797 z + 707208526509453383068 z - 7090725693348053 z 98 92 + 414352157118467 z - 1175302414203779738 z 82 64 112 + 23557002538081247159849 z - 209374951811316805897089201 z - 420 z 114 110 106 108 + z + 72943 z + 468180650 z - 7220060 z 30 42 + 4388558255505540357889 z + 13464230563576444970865328 z 44 46 - 32331689039963980661029414 z + 68355596276151619624951091 z 58 56 + 440586017400942145330887854 z - 440586017400942145330887854 z 54 52 + 389253978742986847455145846 z - 303790532548891147080570633 z 60 70 - 389253978742986847455145846 z + 32331689039963980661029414 z 68 78 - 68355596276151619624951091 z + 446237532851750295659003 z 32 38 - 23557002538081247159849 z + 1584659968013469710894800 z 40 62 - 4930082206468192952660918 z + 303790532548891147080570633 z 76 74 - 1584659968013469710894800 z + 4930082206468192952660918 z 72 104 - 13464230563576444970865328 z - 21528917192 z ) And in Maple-input format, it is: -(1+192627581284250449300*z^28-28300683881956827757*z^26-341*z^2+ 3562804533474991080*z^24-381814417848825859*z^22+50632*z^4-4415851*z^6-\ 10760806477*z^102+257175356*z^8-10760806477*z^10+339295495084*z^12-\ 8337092220067*z^14-2615899350930013*z^18+163660124714612*z^16-\ 30613583680609515954368857*z^50+19750192300624520667554236*z^48+ 34556520585677116*z^20+97174236221918555080212*z^36-25282552529172582305793*z^ 34-11231312456718897658839631*z^66+5733370759713171939312*z^80+339295495084*z^ 100-381814417848825859*z^90+3562804533474991080*z^88+192627581284250449300*z^84 -2615899350930013*z^94-28300683881956827757*z^86+163660124714612*z^96-\ 8337092220067*z^98+34556520585677116*z^92-1129611020126952144959*z^82+ 19750192300624520667554236*z^64+z^112-341*z^110-4415851*z^106+50632*z^108-\ 1129611020126952144959*z^30-2479345762239210121432193*z^42+ 5625499658338765391363044*z^44-11231312456718897658839631*z^46-\ 50479297324591680344708039*z^58+53732681489946090414375326*z^56-\ 50479297324591680344708039*z^54+41850915360464667595140136*z^52+ 41850915360464667595140136*z^60-2479345762239210121432193*z^70+ 5625499658338765391363044*z^68-25282552529172582305793*z^78+ 5733370759713171939312*z^32-326409430172043194598479*z^38+ 960337606514354806601540*z^40-30613583680609515954368857*z^62+ 97174236221918555080212*z^76-326409430172043194598479*z^74+ 960337606514354806601540*z^72+257175356*z^104)/(-1-707208526509453383068*z^28+ 98104878476072293474*z^26+420*z^2-11647500002809420612*z^24+1175302414203779738 *z^22-72943*z^4+7220060*z^6+739037554482*z^102-468180650*z^8+21528917192*z^10-\ 739037554482*z^12+19631423804951*z^14+7090725693348053*z^18-414352157118467*z^ 16+209374951811316805897089201*z^50-127374988536470877346655043*z^48-\ 99950960016939797*z^20-446237532851750295659003*z^36+109826645981989109880903*z ^34+127374988536470877346655043*z^66-109826645981989109880903*z^80-\ 19631423804951*z^100+11647500002809420612*z^90-98104878476072293474*z^88-\ 4388558255505540357889*z^84+99950960016939797*z^94+707208526509453383068*z^86-\ 7090725693348053*z^96+414352157118467*z^98-1175302414203779738*z^92+ 23557002538081247159849*z^82-209374951811316805897089201*z^64-420*z^112+z^114+ 72943*z^110+468180650*z^106-7220060*z^108+4388558255505540357889*z^30+ 13464230563576444970865328*z^42-32331689039963980661029414*z^44+ 68355596276151619624951091*z^46+440586017400942145330887854*z^58-\ 440586017400942145330887854*z^56+389253978742986847455145846*z^54-\ 303790532548891147080570633*z^52-389253978742986847455145846*z^60+ 32331689039963980661029414*z^70-68355596276151619624951091*z^68+ 446237532851750295659003*z^78-23557002538081247159849*z^32+ 1584659968013469710894800*z^38-4930082206468192952660918*z^40+ 303790532548891147080570633*z^62-1584659968013469710894800*z^76+ 4930082206468192952660918*z^74-13464230563576444970865328*z^72-21528917192*z^ 104) The first , 40, terms are: [0, 79, 0, 10869, 0, 1606692, 0, 241372619, 0, 36436236929, 0, 5509576117903, 0, 833664391575207, 0, 126177791128933081, 0, 19099619967982524019, 0, 2891265895605585038852, 0, 437683961631636729036309, 0, 66257838486702591792641591, 0, 10030339856481817330233139265, 0, 1518430072338206356500360770929, 0, 229865755632702616752651821061911, 0, 34797969437193585042771642701243893, 0, 5267852315884904076285144047711954500, 0, 797468077341505329212906594851734875491, 0, 120723838451580016779297739214017388624057, 0, 18275647339820765977123726352381703733918567] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 210344492963710291848 z - 30603857766823740054 z - 338 z 24 22 4 6 + 3814650149355667020 z - 404697849917665406 z + 49966 z - 4360830 z 102 8 10 12 - 10751939760 z + 255136151 z - 10751939760 z + 341977317390 z 14 18 16 - 8483905505940 z - 2716478093144834 z + 168222555814401 z 50 48 - 36153442693208313367645282 z + 23240530311718977557794598 z 20 36 + 36255542967799656 z + 110023097869877513053176 z 34 66 - 28382649088472301656210 z - 13156493717139852955135638 z 80 100 90 + 6379265936559643670692 z + 341977317390 z - 404697849917665406 z 88 84 + 3814650149355667020 z + 210344492963710291848 z 94 86 96 - 2716478093144834 z - 30603857766823740054 z + 168222555814401 z 98 92 82 - 8483905505940 z + 36255542967799656 z - 1245303750171609498630 z 64 112 110 106 + 23240530311718977557794598 z + z - 338 z - 4360830 z 108 30 42 + 49966 z - 1245303750171609498630 z - 2870931355094849100143114 z 44 46 + 6554358976190782327329000 z - 13156493717139852955135638 z 58 56 - 59864380535346696921310554 z + 63756420866825755562777514 z 54 52 - 59864380535346696921310554 z + 49553369577113506412682084 z 60 70 + 49553369577113506412682084 z - 2870931355094849100143114 z 68 78 + 6554358976190782327329000 z - 28382649088472301656210 z 32 38 + 6379265936559643670692 z - 372558333790564446216834 z 40 62 + 1104379740889879714565804 z - 36153442693208313367645282 z 76 74 + 110023097869877513053176 z - 372558333790564446216834 z 72 104 / 2 + 1104379740889879714565804 z + 255136151 z ) / ((-1 + z ) (1 / 28 26 2 + 674562753933939571270 z - 94219677431599831561 z - 421 z 24 22 4 + 11245260809880095600 z - 1139173593060548991 z + 72488 z 6 102 8 10 - 7122469 z - 21078056230 z + 459664363 z - 21078056230 z 12 14 18 + 722360051038 z - 19166568895524 z - 6905805205718237 z 16 50 + 404103503103217 z - 153647616531644447061114883 z 48 20 + 97633664246314567489275982 z + 97152502612224286 z 36 34 + 405412809273926847931850 z - 101381532979791823050149 z 66 80 - 54464159189610528850102079 z + 22035153070036274855832 z 100 90 88 + 722360051038 z - 1139173593060548991 z + 11245260809880095600 z 84 94 + 674562753933939571270 z - 6905805205718237 z 86 96 98 - 94219677431599831561 z + 404103503103217 z - 19166568895524 z 92 82 + 97152502612224286 z - 4149710356501429910695 z 64 112 110 106 + 97633664246314567489275982 z + z - 421 z - 7122469 z 108 30 42 + 72488 z - 4149710356501429910695 z - 11437866004717424015342933 z 44 46 + 26656696242569734015035058 z - 54464159189610528850102079 z 58 56 - 257840368690794634679597785 z + 275066241461057168545396410 z 54 52 - 257840368690794634679597785 z + 212358549679334264078455754 z 60 70 + 212358549679334264078455754 z - 11437866004717424015342933 z 68 78 + 26656696242569734015035058 z - 101381532979791823050149 z 32 38 + 22035153070036274855832 z - 1412662031731644526388187 z 40 62 + 4298134290956555469758544 z - 153647616531644447061114883 z 76 74 + 405412809273926847931850 z - 1412662031731644526388187 z 72 104 + 4298134290956555469758544 z + 459664363 z )) And in Maple-input format, it is: -(1+210344492963710291848*z^28-30603857766823740054*z^26-338*z^2+ 3814650149355667020*z^24-404697849917665406*z^22+49966*z^4-4360830*z^6-\ 10751939760*z^102+255136151*z^8-10751939760*z^10+341977317390*z^12-\ 8483905505940*z^14-2716478093144834*z^18+168222555814401*z^16-\ 36153442693208313367645282*z^50+23240530311718977557794598*z^48+ 36255542967799656*z^20+110023097869877513053176*z^36-28382649088472301656210*z^ 34-13156493717139852955135638*z^66+6379265936559643670692*z^80+341977317390*z^ 100-404697849917665406*z^90+3814650149355667020*z^88+210344492963710291848*z^84 -2716478093144834*z^94-30603857766823740054*z^86+168222555814401*z^96-\ 8483905505940*z^98+36255542967799656*z^92-1245303750171609498630*z^82+ 23240530311718977557794598*z^64+z^112-338*z^110-4360830*z^106+49966*z^108-\ 1245303750171609498630*z^30-2870931355094849100143114*z^42+ 6554358976190782327329000*z^44-13156493717139852955135638*z^46-\ 59864380535346696921310554*z^58+63756420866825755562777514*z^56-\ 59864380535346696921310554*z^54+49553369577113506412682084*z^52+ 49553369577113506412682084*z^60-2870931355094849100143114*z^70+ 6554358976190782327329000*z^68-28382649088472301656210*z^78+ 6379265936559643670692*z^32-372558333790564446216834*z^38+ 1104379740889879714565804*z^40-36153442693208313367645282*z^62+ 110023097869877513053176*z^76-372558333790564446216834*z^74+ 1104379740889879714565804*z^72+255136151*z^104)/(-1+z^2)/(1+ 674562753933939571270*z^28-94219677431599831561*z^26-421*z^2+ 11245260809880095600*z^24-1139173593060548991*z^22+72488*z^4-7122469*z^6-\ 21078056230*z^102+459664363*z^8-21078056230*z^10+722360051038*z^12-\ 19166568895524*z^14-6905805205718237*z^18+404103503103217*z^16-\ 153647616531644447061114883*z^50+97633664246314567489275982*z^48+ 97152502612224286*z^20+405412809273926847931850*z^36-101381532979791823050149*z ^34-54464159189610528850102079*z^66+22035153070036274855832*z^80+722360051038*z ^100-1139173593060548991*z^90+11245260809880095600*z^88+674562753933939571270*z ^84-6905805205718237*z^94-94219677431599831561*z^86+404103503103217*z^96-\ 19166568895524*z^98+97152502612224286*z^92-4149710356501429910695*z^82+ 97633664246314567489275982*z^64+z^112-421*z^110-7122469*z^106+72488*z^108-\ 4149710356501429910695*z^30-11437866004717424015342933*z^42+ 26656696242569734015035058*z^44-54464159189610528850102079*z^46-\ 257840368690794634679597785*z^58+275066241461057168545396410*z^56-\ 257840368690794634679597785*z^54+212358549679334264078455754*z^52+ 212358549679334264078455754*z^60-11437866004717424015342933*z^70+ 26656696242569734015035058*z^68-101381532979791823050149*z^78+ 22035153070036274855832*z^32-1412662031731644526388187*z^38+ 4298134290956555469758544*z^40-153647616531644447061114883*z^62+ 405412809273926847931850*z^76-1412662031731644526388187*z^74+ 4298134290956555469758544*z^72+459664363*z^104) The first , 40, terms are: [0, 84, 0, 12505, 0, 1986881, 0, 319462444, 0, 51500268769, 0, 8307487855813, 0, 1340280858063556, 0, 216241063940492101, 0, 34888682106965245821, 0, 5629009338750138050540, 0, 908196087831694799567589, 0, 146530269382472162582883917, 0, 23641503112651707373065723404, 0, 3814370075083856815037712899189, 0, 615418530478283100345407644621069, 0, 99292926597503594326240438042423364, 0, 16020130668012724146085190958396300669, 0, 2584721746296197414876603552363367943817, 0, 417024470292258461976863046456040408511180, 0, 67283609569257462326179264531142556883413177] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1461028142337164164 z - 354772570743864534 z - 281 z 24 22 4 6 + 71904249159297192 z - 12093723815573962 z + 33250 z - 2254831 z 8 10 12 14 + 99940765 z - 3117773184 z + 71800697102 z - 1261823239772 z 18 16 50 - 189678307271000 z + 17337166426826 z - 131978513030513522976 z 48 20 + 197677623533093146034 z + 1675968537027148 z 36 34 + 74856630737666330466 z - 36015366301460832404 z 66 80 88 84 86 - 12093723815573962 z + 99940765 z + z + 33250 z - 281 z 82 64 30 - 2254831 z + 71904249159297192 z - 5045741336378447272 z 42 44 - 251796448648573274200 z + 272929424125872097356 z 46 58 - 251796448648573274200 z - 5045741336378447272 z 56 54 + 14669534451947767174 z - 36015366301460832404 z 52 60 70 + 74856630737666330466 z + 1461028142337164164 z - 189678307271000 z 68 78 32 + 1675968537027148 z - 3117773184 z + 14669534451947767174 z 38 40 - 131978513030513522976 z + 197677623533093146034 z 62 76 74 - 354772570743864534 z + 71800697102 z - 1261823239772 z 72 / 28 + 17337166426826 z ) / (-1 - 6537293753294055738 z / 26 2 24 + 1473505277240943330 z + 360 z - 277124664457585558 z 22 4 6 8 + 43222506786803502 z - 50554 z + 3926424 z - 195344230 z 10 12 14 + 6753418137 z - 170868924070 z + 3278975278266 z 18 16 50 + 580748421093318 z - 48980219862762 z + 1390940458842555926414 z 48 20 - 1915589667471848811738 z - 5548488504619308 z 36 34 - 451878387130535771294 z + 201599013618443001390 z 66 80 90 88 84 + 277124664457585558 z - 6753418137 z + z - 360 z - 3926424 z 86 82 64 + 50554 z + 195344230 z - 1473505277240943330 z 30 42 + 24320875580881575548 z + 1915589667471848811738 z 44 46 - 2247576654348615676326 z + 2247576654348615676326 z 58 56 + 76183796563312979874 z - 201599013618443001390 z 54 52 + 451878387130535771294 z - 859784250307832461058 z 60 70 68 - 24320875580881575548 z + 5548488504619308 z - 43222506786803502 z 78 32 38 + 170868924070 z - 76183796563312979874 z + 859784250307832461058 z 40 62 76 - 1390940458842555926414 z + 6537293753294055738 z - 3278975278266 z 74 72 + 48980219862762 z - 580748421093318 z ) And in Maple-input format, it is: -(1+1461028142337164164*z^28-354772570743864534*z^26-281*z^2+71904249159297192* z^24-12093723815573962*z^22+33250*z^4-2254831*z^6+99940765*z^8-3117773184*z^10+ 71800697102*z^12-1261823239772*z^14-189678307271000*z^18+17337166426826*z^16-\ 131978513030513522976*z^50+197677623533093146034*z^48+1675968537027148*z^20+ 74856630737666330466*z^36-36015366301460832404*z^34-12093723815573962*z^66+ 99940765*z^80+z^88+33250*z^84-281*z^86-2254831*z^82+71904249159297192*z^64-\ 5045741336378447272*z^30-251796448648573274200*z^42+272929424125872097356*z^44-\ 251796448648573274200*z^46-5045741336378447272*z^58+14669534451947767174*z^56-\ 36015366301460832404*z^54+74856630737666330466*z^52+1461028142337164164*z^60-\ 189678307271000*z^70+1675968537027148*z^68-3117773184*z^78+14669534451947767174 *z^32-131978513030513522976*z^38+197677623533093146034*z^40-354772570743864534* z^62+71800697102*z^76-1261823239772*z^74+17337166426826*z^72)/(-1-\ 6537293753294055738*z^28+1473505277240943330*z^26+360*z^2-277124664457585558*z^ 24+43222506786803502*z^22-50554*z^4+3926424*z^6-195344230*z^8+6753418137*z^10-\ 170868924070*z^12+3278975278266*z^14+580748421093318*z^18-48980219862762*z^16+ 1390940458842555926414*z^50-1915589667471848811738*z^48-5548488504619308*z^20-\ 451878387130535771294*z^36+201599013618443001390*z^34+277124664457585558*z^66-\ 6753418137*z^80+z^90-360*z^88-3926424*z^84+50554*z^86+195344230*z^82-\ 1473505277240943330*z^64+24320875580881575548*z^30+1915589667471848811738*z^42-\ 2247576654348615676326*z^44+2247576654348615676326*z^46+76183796563312979874*z^ 58-201599013618443001390*z^56+451878387130535771294*z^54-859784250307832461058* z^52-24320875580881575548*z^60+5548488504619308*z^70-43222506786803502*z^68+ 170868924070*z^78-76183796563312979874*z^32+859784250307832461058*z^38-\ 1390940458842555926414*z^40+6537293753294055738*z^62-3278975278266*z^76+ 48980219862762*z^74-580748421093318*z^72) The first , 40, terms are: [0, 79, 0, 11136, 0, 1686787, 0, 259058007, 0, 39915160969, 0, 6155178883225, 0, 949385390132728, 0, 146444363849180053, 0, 22589731544134410463, 0, 3484592192318716330175, 0, 537518729338666328349149, 0, 82915452313600028097046888, 0, 12790202427380229773621808433, 0, 1972965012508138938752661138441, 0, 304341625849325608859369545885367, 0, 46946512020932001291214654718168763, 0, 7241779655109588148567364830131886576, 0, 1117087731130772704105483861552375463095, 0, 172317449391930848606844459719614142807425, 0, 26580994973643033426936551113227522755449601] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2773609194323426834 z - 668386289544050938 z - 309 z 24 22 4 6 + 134132089419324992 z - 22272606797611974 z + 40312 z - 2986859 z 8 10 12 14 + 142816463 z - 4744963278 z + 115044402210 z - 2107740308368 z 18 16 50 - 336451887681730 z + 29944348758142 z - 255245468293105310506 z 48 20 + 382842605923897188624 z + 3036083140986126 z 36 34 + 144482353332791202502 z - 69327741969389216624 z 66 80 88 84 86 - 22272606797611974 z + 142816463 z + z + 40312 z - 309 z 82 64 30 - 2986859 z + 134132089419324992 z - 9635164098348761158 z 42 44 - 488060193269007665568 z + 529168583492905123472 z 46 58 - 488060193269007665568 z - 9635164098348761158 z 56 54 + 28139506661447006106 z - 69327741969389216624 z 52 60 + 144482353332791202502 z + 2773609194323426834 z 70 68 78 - 336451887681730 z + 3036083140986126 z - 4744963278 z 32 38 + 28139506661447006106 z - 255245468293105310506 z 40 62 76 + 382842605923897188624 z - 668386289544050938 z + 115044402210 z 74 72 / 2 - 2107740308368 z + 29944348758142 z ) / ((-1 + z ) (1 / 28 26 2 + 10177996331460242712 z - 2349693824233937572 z - 397 z 24 22 4 6 + 449881281248036860 z - 70985049067977780 z + 61569 z - 5202645 z 8 10 12 14 + 277010509 z - 10095973276 z + 265788944976 z - 5247686659352 z 18 16 50 - 956422220028372 z + 79863576407024 z - 1084203508097662566196 z 48 20 + 1649164836660992779114 z + 9157632362160304 z 36 34 + 601924663011845445040 z - 281815725369372776232 z 66 80 88 84 86 - 70985049067977780 z + 277010509 z + z + 61569 z - 397 z 82 64 30 - 5202645 z + 449881281248036860 z - 36750183024293426908 z 42 44 - 2120324833509863619142 z + 2305455359455600062342 z 46 58 - 2120324833509863619142 z - 36750183024293426908 z 56 54 + 111061595794095441808 z - 281815725369372776232 z 52 60 + 601924663011845445040 z + 10177996331460242712 z 70 68 78 - 956422220028372 z + 9157632362160304 z - 10095973276 z 32 38 + 111061595794095441808 z - 1084203508097662566196 z 40 62 76 + 1649164836660992779114 z - 2349693824233937572 z + 265788944976 z 74 72 - 5247686659352 z + 79863576407024 z )) And in Maple-input format, it is: -(1+2773609194323426834*z^28-668386289544050938*z^26-309*z^2+134132089419324992 *z^24-22272606797611974*z^22+40312*z^4-2986859*z^6+142816463*z^8-4744963278*z^ 10+115044402210*z^12-2107740308368*z^14-336451887681730*z^18+29944348758142*z^ 16-255245468293105310506*z^50+382842605923897188624*z^48+3036083140986126*z^20+ 144482353332791202502*z^36-69327741969389216624*z^34-22272606797611974*z^66+ 142816463*z^80+z^88+40312*z^84-309*z^86-2986859*z^82+134132089419324992*z^64-\ 9635164098348761158*z^30-488060193269007665568*z^42+529168583492905123472*z^44-\ 488060193269007665568*z^46-9635164098348761158*z^58+28139506661447006106*z^56-\ 69327741969389216624*z^54+144482353332791202502*z^52+2773609194323426834*z^60-\ 336451887681730*z^70+3036083140986126*z^68-4744963278*z^78+28139506661447006106 *z^32-255245468293105310506*z^38+382842605923897188624*z^40-668386289544050938* z^62+115044402210*z^76-2107740308368*z^74+29944348758142*z^72)/(-1+z^2)/(1+ 10177996331460242712*z^28-2349693824233937572*z^26-397*z^2+449881281248036860*z ^24-70985049067977780*z^22+61569*z^4-5202645*z^6+277010509*z^8-10095973276*z^10 +265788944976*z^12-5247686659352*z^14-956422220028372*z^18+79863576407024*z^16-\ 1084203508097662566196*z^50+1649164836660992779114*z^48+9157632362160304*z^20+ 601924663011845445040*z^36-281815725369372776232*z^34-70985049067977780*z^66+ 277010509*z^80+z^88+61569*z^84-397*z^86-5202645*z^82+449881281248036860*z^64-\ 36750183024293426908*z^30-2120324833509863619142*z^42+2305455359455600062342*z^ 44-2120324833509863619142*z^46-36750183024293426908*z^58+111061595794095441808* z^56-281815725369372776232*z^54+601924663011845445040*z^52+10177996331460242712 *z^60-956422220028372*z^70+9157632362160304*z^68-10095973276*z^78+ 111061595794095441808*z^32-1084203508097662566196*z^38+1649164836660992779114*z ^40-2349693824233937572*z^62+265788944976*z^76-5247686659352*z^74+ 79863576407024*z^72) The first , 40, terms are: [0, 89, 0, 13768, 0, 2242045, 0, 368304377, 0, 60643329729, 0, 9993145221141, 0, 1647212913155064, 0, 271548473453558961, 0, 44767710075869719273, 0, 7380577604359032593257, 0, 1216799480568435473171153, 0, 200608330756187142458523608, 0, 33073444618834313901559049461, 0, 5452681144222041915202094704513, 0, 898961016318866254305950806691353, 0, 148208000824185876506785441934443389, 0, 24434443535416527044500342917807131496, 0, 4028406260558618214504670490230486768665, 0, 664146783561335005111944054950524099909473, 0, 109495150807203669614577289050322766861565153] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 1917645375964154689 z + 478605877327759769 z + 304 z 24 22 4 6 - 99182175972799356 z + 16968337838376638 z - 38803 z + 2806721 z 8 10 12 14 - 130881178 z + 4238977063 z - 100170575661 z + 1788458247146 z 18 16 50 + 270902676134657 z - 24755680175246 z + 83145112997622923466 z 48 20 - 138349248211373005273 z - 2379220567356013 z 36 34 - 83145112997622923466 z + 42096774553488268645 z 66 80 84 86 82 + 2379220567356013 z - 2806721 z - 304 z + z + 38803 z 64 30 - 16968337838376638 z + 6408731224939558876 z 42 44 + 229964866812357940082 z - 229964866812357940082 z 46 58 + 194155456540760175380 z + 1917645375964154689 z 56 54 - 6408731224939558876 z + 17928031651289647657 z 52 60 70 - 42096774553488268645 z - 478605877327759769 z + 24755680175246 z 68 78 32 - 270902676134657 z + 130881178 z - 17928031651289647657 z 38 40 + 138349248211373005273 z - 194155456540760175380 z 62 76 74 + 99182175972799356 z - 4238977063 z + 100170575661 z 72 / 28 - 1788458247146 z ) / (1 + 8680412316793196858 z / 26 2 24 - 2009775938700897232 z - 384 z + 386423345437189247 z 22 4 6 8 - 61317723352889924 z + 58456 z - 4874660 z + 256523886 z 10 12 14 - 9243716648 z + 240623279946 z - 4698506993072 z 18 16 50 - 839174880642996 z + 70754885566609 z - 922495802713371634816 z 48 20 + 1403743495618517651151 z + 7966978507074460 z 36 34 + 511959915835358990852 z - 239654189083652519492 z 66 80 88 84 86 - 61317723352889924 z + 256523886 z + z + 58456 z - 384 z 82 64 30 - 4874660 z + 386423345437189247 z - 31287203040751832048 z 42 44 - 1805306073268661393032 z + 1963136751380568893192 z 46 58 - 1805306073268661393032 z - 31287203040751832048 z 56 54 + 94468544874747258802 z - 239654189083652519492 z 52 60 + 511959915835358990852 z + 8680412316793196858 z 70 68 78 - 839174880642996 z + 7966978507074460 z - 9243716648 z 32 38 + 94468544874747258802 z - 922495802713371634816 z 40 62 76 + 1403743495618517651151 z - 2009775938700897232 z + 240623279946 z 74 72 - 4698506993072 z + 70754885566609 z ) And in Maple-input format, it is: -(-1-1917645375964154689*z^28+478605877327759769*z^26+304*z^2-99182175972799356 *z^24+16968337838376638*z^22-38803*z^4+2806721*z^6-130881178*z^8+4238977063*z^ 10-100170575661*z^12+1788458247146*z^14+270902676134657*z^18-24755680175246*z^ 16+83145112997622923466*z^50-138349248211373005273*z^48-2379220567356013*z^20-\ 83145112997622923466*z^36+42096774553488268645*z^34+2379220567356013*z^66-\ 2806721*z^80-304*z^84+z^86+38803*z^82-16968337838376638*z^64+ 6408731224939558876*z^30+229964866812357940082*z^42-229964866812357940082*z^44+ 194155456540760175380*z^46+1917645375964154689*z^58-6408731224939558876*z^56+ 17928031651289647657*z^54-42096774553488268645*z^52-478605877327759769*z^60+ 24755680175246*z^70-270902676134657*z^68+130881178*z^78-17928031651289647657*z^ 32+138349248211373005273*z^38-194155456540760175380*z^40+99182175972799356*z^62 -4238977063*z^76+100170575661*z^74-1788458247146*z^72)/(1+8680412316793196858*z ^28-2009775938700897232*z^26-384*z^2+386423345437189247*z^24-61317723352889924* z^22+58456*z^4-4874660*z^6+256523886*z^8-9243716648*z^10+240623279946*z^12-\ 4698506993072*z^14-839174880642996*z^18+70754885566609*z^16-\ 922495802713371634816*z^50+1403743495618517651151*z^48+7966978507074460*z^20+ 511959915835358990852*z^36-239654189083652519492*z^34-61317723352889924*z^66+ 256523886*z^80+z^88+58456*z^84-384*z^86-4874660*z^82+386423345437189247*z^64-\ 31287203040751832048*z^30-1805306073268661393032*z^42+1963136751380568893192*z^ 44-1805306073268661393032*z^46-31287203040751832048*z^58+94468544874747258802*z ^56-239654189083652519492*z^54+511959915835358990852*z^52+8680412316793196858*z ^60-839174880642996*z^70+7966978507074460*z^68-9243716648*z^78+ 94468544874747258802*z^32-922495802713371634816*z^38+1403743495618517651151*z^ 40-2009775938700897232*z^62+240623279946*z^76-4698506993072*z^74+70754885566609 *z^72) The first , 40, terms are: [0, 80, 0, 11067, 0, 1641187, 0, 247613348, 0, 37576989285, 0, 5715401417365, 0, 870101273186860, 0, 132513263093126763, 0, 20184562446434887267, 0, 3074746698771546653176, 0, 468394931225310358334753, 0, 71354363119589725080433137, 0, 10870042581547253001846928616, 0, 1655933806314616965545429869379, 0, 252263917883492988942839961962443, 0, 38429743724700539594270555572286108, 0, 5854366688989760062124251479539644741, 0, 891851174453561284997634591992947891669, 0, 135864148429779959469000021196481028823220, 0, 20697474680937387995426716773946492124955875] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3353745971933057970 z - 793441237431232745 z - 305 z 24 22 4 6 + 156079213794655270 z - 25372156426516011 z + 39511 z - 2930428 z 8 10 12 14 + 141261247 z - 4758233525 z + 117433272980 z - 2195828258331 z 18 16 50 - 366478524327039 z + 31887346382996 z - 328446790808867196619 z 48 20 + 495494620648131322517 z + 3382796780527734 z 36 34 + 184422630417210814227 z - 87582501001203236078 z 66 80 88 84 86 - 25372156426516011 z + 141261247 z + z + 39511 z - 305 z 82 64 30 - 2930428 z + 156079213794655270 z - 11845759738458551729 z 42 44 - 633871016825105459974 z + 688057493671801308084 z 46 58 - 633871016825105459974 z - 11845759738458551729 z 56 54 + 35106012558515389681 z - 87582501001203236078 z 52 60 + 184422630417210814227 z + 3353745971933057970 z 70 68 78 - 366478524327039 z + 3382796780527734 z - 4758233525 z 32 38 + 35106012558515389681 z - 328446790808867196619 z 40 62 76 + 495494620648131322517 z - 793441237431232745 z + 117433272980 z 74 72 / - 2195828258331 z + 31887346382996 z ) / (-1 / 28 26 2 - 15098147492733409490 z + 3309686361917604539 z + 397 z 24 22 4 6 - 603020835801270015 z + 90733287091705724 z - 60976 z + 5149680 z 8 10 12 14 - 277001290 z + 10295620178 z - 278487228914 z + 5681636135074 z 18 16 50 + 1118966137917265 z - 89735225536309 z + 3542133187893866674851 z 48 20 - 4906548447335080516879 z - 11185814946967340 z 36 34 - 1127579489802887053612 z + 495758491803515813142 z 66 80 90 88 84 + 603020835801270015 z - 10295620178 z + z - 397 z - 5149680 z 86 82 64 + 60976 z + 277001290 z - 3309686361917604539 z 30 42 + 57547823082627783058 z + 4906548447335080516879 z 44 46 - 5773596647227439723736 z + 5773596647227439723736 z 58 56 + 184065037907512619214 z - 495758491803515813142 z 54 52 + 1127579489802887053612 z - 2170532899476911679036 z 60 70 68 - 57547823082627783058 z + 11185814946967340 z - 90733287091705724 z 78 32 + 278487228914 z - 184065037907512619214 z 38 40 + 2170532899476911679036 z - 3542133187893866674851 z 62 76 74 + 15098147492733409490 z - 5681636135074 z + 89735225536309 z 72 - 1118966137917265 z ) And in Maple-input format, it is: -(1+3353745971933057970*z^28-793441237431232745*z^26-305*z^2+156079213794655270 *z^24-25372156426516011*z^22+39511*z^4-2930428*z^6+141261247*z^8-4758233525*z^ 10+117433272980*z^12-2195828258331*z^14-366478524327039*z^18+31887346382996*z^ 16-328446790808867196619*z^50+495494620648131322517*z^48+3382796780527734*z^20+ 184422630417210814227*z^36-87582501001203236078*z^34-25372156426516011*z^66+ 141261247*z^80+z^88+39511*z^84-305*z^86-2930428*z^82+156079213794655270*z^64-\ 11845759738458551729*z^30-633871016825105459974*z^42+688057493671801308084*z^44 -633871016825105459974*z^46-11845759738458551729*z^58+35106012558515389681*z^56 -87582501001203236078*z^54+184422630417210814227*z^52+3353745971933057970*z^60-\ 366478524327039*z^70+3382796780527734*z^68-4758233525*z^78+35106012558515389681 *z^32-328446790808867196619*z^38+495494620648131322517*z^40-793441237431232745* z^62+117433272980*z^76-2195828258331*z^74+31887346382996*z^72)/(-1-\ 15098147492733409490*z^28+3309686361917604539*z^26+397*z^2-603020835801270015*z ^24+90733287091705724*z^22-60976*z^4+5149680*z^6-277001290*z^8+10295620178*z^10 -278487228914*z^12+5681636135074*z^14+1118966137917265*z^18-89735225536309*z^16 +3542133187893866674851*z^50-4906548447335080516879*z^48-11185814946967340*z^20 -1127579489802887053612*z^36+495758491803515813142*z^34+603020835801270015*z^66 -10295620178*z^80+z^90-397*z^88-5149680*z^84+60976*z^86+277001290*z^82-\ 3309686361917604539*z^64+57547823082627783058*z^30+4906548447335080516879*z^42-\ 5773596647227439723736*z^44+5773596647227439723736*z^46+184065037907512619214*z ^58-495758491803515813142*z^56+1127579489802887053612*z^54-\ 2170532899476911679036*z^52-57547823082627783058*z^60+11185814946967340*z^70-\ 90733287091705724*z^68+278487228914*z^78-184065037907512619214*z^32+ 2170532899476911679036*z^38-3542133187893866674851*z^40+15098147492733409490*z^ 62-5681636135074*z^76+89735225536309*z^74-1118966137917265*z^72) The first , 40, terms are: [0, 92, 0, 15059, 0, 2587883, 0, 447182484, 0, 77334991433, 0, 13376142456289, 0, 2313656609441332, 0, 400193329180978563, 0, 69221572408293460683, 0, 11973282574991499167196, 0, 2071023566475083476455481, 0, 358225797459411004119673065, 0, 61962463706349890654236397084, 0, 10717672875891734395130408787067, 0, 1853840293532607681065565150810611, 0, 320659519463008337013647311497780276, 0, 55464609212854438892270565017263399889, 0, 9593736310395042881313155438434087088281, 0, 1659432522822937940535299765852894474517268, 0, 287032727261785551940140654007451761330937275] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 237889151594668593010 z - 34108232841660811331 z - 335 z 24 22 4 6 + 4186865330799803876 z - 437239746398325825 z + 49242 z - 4291761 z 102 8 10 12 - 10686298567 z + 251846440 z - 10686298567 z + 343307948758 z 14 18 16 - 8623259582889 z - 2843245270467891 z + 173415780402048 z 50 48 - 44939187972878443153738623 z + 28792409346940798541156588 z 20 36 + 38551245087415854 z + 130724882876782308118366 z 34 66 - 33361322053001195160367 z - 16229153560168456520002705 z 80 100 90 + 7409874872301031709904 z + 343307948758 z - 437239746398325825 z 88 84 + 4186865330799803876 z + 237889151594668593010 z 94 86 96 - 2843245270467891 z - 34108232841660811331 z + 173415780402048 z 98 92 82 - 8623259582889 z + 38551245087415854 z - 1427977761849286710549 z 64 112 110 106 + 28792409346940798541156588 z + z - 335 z - 4291761 z 108 30 42 + 49242 z - 1427977761849286710549 z - 3500077460998276090714215 z 44 46 + 8042006583557582291629270 z - 16229153560168456520002705 z 58 56 - 74692277571193027760412793 z + 79585447888492429624645690 z 54 52 - 74692277571193027760412793 z + 61740605934707748341501458 z 60 70 + 61740605934707748341501458 z - 3500077460998276090714215 z 68 78 + 8042006583557582291629270 z - 33361322053001195160367 z 32 38 + 7409874872301031709904 z - 446966598537974630589077 z 40 62 + 1336364329972299117531892 z - 44939187972878443153738623 z 76 74 + 130724882876782308118366 z - 446966598537974630589077 z 72 104 / + 1336364329972299117531892 z + 251846440 z ) / (-1 / 28 26 2 - 859839509277507523520 z + 116067022760936928470 z + 424 z 24 22 4 - 13406867623188382868 z + 1316535888133548358 z - 72811 z 6 102 8 10 + 7141168 z + 741044344202 z - 462006370 z + 21348852088 z 12 14 18 - 741044344202 z + 20005487737503 z + 7543990823999653 z 16 50 - 430813143886823 z + 315266882958868545508172437 z 48 20 - 189942800377521620234706511 z - 109040724370374901 z 36 34 - 600210692054008633782467 z + 144292294456862212122867 z 66 80 + 189942800377521620234706511 z - 144292294456862212122867 z 100 90 - 20005487737503 z + 13406867623188382868 z 88 84 - 116067022760936928470 z - 5480434050049497947761 z 94 86 + 109040724370374901 z + 859839509277507523520 z 96 98 92 - 7543990823999653 z + 430813143886823 z - 1316535888133548358 z 82 64 112 + 30190730116075633637825 z - 315266882958868545508172437 z - 424 z 114 110 106 108 + z + 72811 z + 462006370 z - 7141168 z 30 42 + 5480434050049497947761 z + 19240596856195299922093352 z 44 46 - 46964482987000670841263698 z + 100715606473338377382062847 z 58 56 + 673250609749740013410112550 z - 673250609749740013410112550 z 54 52 + 593345814963852869670564862 z - 460801190083278706206395721 z 60 70 - 593345814963852869670564862 z + 46964482987000670841263698 z 68 78 - 100715606473338377382062847 z + 600210692054008633782467 z 32 38 - 30190730116075633637825 z + 2178753123806296963965252 z 40 62 - 6916957702561766891541970 z + 460801190083278706206395721 z 76 74 - 2178753123806296963965252 z + 6916957702561766891541970 z 72 104 - 19240596856195299922093352 z - 21348852088 z ) And in Maple-input format, it is: -(1+237889151594668593010*z^28-34108232841660811331*z^26-335*z^2+ 4186865330799803876*z^24-437239746398325825*z^22+49242*z^4-4291761*z^6-\ 10686298567*z^102+251846440*z^8-10686298567*z^10+343307948758*z^12-\ 8623259582889*z^14-2843245270467891*z^18+173415780402048*z^16-\ 44939187972878443153738623*z^50+28792409346940798541156588*z^48+ 38551245087415854*z^20+130724882876782308118366*z^36-33361322053001195160367*z^ 34-16229153560168456520002705*z^66+7409874872301031709904*z^80+343307948758*z^ 100-437239746398325825*z^90+4186865330799803876*z^88+237889151594668593010*z^84 -2843245270467891*z^94-34108232841660811331*z^86+173415780402048*z^96-\ 8623259582889*z^98+38551245087415854*z^92-1427977761849286710549*z^82+ 28792409346940798541156588*z^64+z^112-335*z^110-4291761*z^106+49242*z^108-\ 1427977761849286710549*z^30-3500077460998276090714215*z^42+ 8042006583557582291629270*z^44-16229153560168456520002705*z^46-\ 74692277571193027760412793*z^58+79585447888492429624645690*z^56-\ 74692277571193027760412793*z^54+61740605934707748341501458*z^52+ 61740605934707748341501458*z^60-3500077460998276090714215*z^70+ 8042006583557582291629270*z^68-33361322053001195160367*z^78+ 7409874872301031709904*z^32-446966598537974630589077*z^38+ 1336364329972299117531892*z^40-44939187972878443153738623*z^62+ 130724882876782308118366*z^76-446966598537974630589077*z^74+ 1336364329972299117531892*z^72+251846440*z^104)/(-1-859839509277507523520*z^28+ 116067022760936928470*z^26+424*z^2-13406867623188382868*z^24+ 1316535888133548358*z^22-72811*z^4+7141168*z^6+741044344202*z^102-462006370*z^8 +21348852088*z^10-741044344202*z^12+20005487737503*z^14+7543990823999653*z^18-\ 430813143886823*z^16+315266882958868545508172437*z^50-\ 189942800377521620234706511*z^48-109040724370374901*z^20-\ 600210692054008633782467*z^36+144292294456862212122867*z^34+ 189942800377521620234706511*z^66-144292294456862212122867*z^80-20005487737503*z ^100+13406867623188382868*z^90-116067022760936928470*z^88-\ 5480434050049497947761*z^84+109040724370374901*z^94+859839509277507523520*z^86-\ 7543990823999653*z^96+430813143886823*z^98-1316535888133548358*z^92+ 30190730116075633637825*z^82-315266882958868545508172437*z^64-424*z^112+z^114+ 72811*z^110+462006370*z^106-7141168*z^108+5480434050049497947761*z^30+ 19240596856195299922093352*z^42-46964482987000670841263698*z^44+ 100715606473338377382062847*z^46+673250609749740013410112550*z^58-\ 673250609749740013410112550*z^56+593345814963852869670564862*z^54-\ 460801190083278706206395721*z^52-593345814963852869670564862*z^60+ 46964482987000670841263698*z^70-100715606473338377382062847*z^68+ 600210692054008633782467*z^78-30190730116075633637825*z^32+ 2178753123806296963965252*z^38-6916957702561766891541970*z^40+ 460801190083278706206395721*z^62-2178753123806296963965252*z^76+ 6916957702561766891541970*z^74-19240596856195299922093352*z^72-21348852088*z^ 104) The first , 40, terms are: [0, 89, 0, 14167, 0, 2376036, 0, 401329849, 0, 67875212427, 0, 11482601880155, 0, 1942659681989067, 0, 328669684875761735, 0, 55606320544847420309, 0, 9407821439372317425652, 0, 1591673822554401197516739, 0, 269289306324570816159277077, 0, 45560045415936218434161386849, 0, 7708132845413295739997996634433, 0, 1304110026386235460432593650118685, 0, 220637474152294433233059559163284187, 0, 37328825037324929770897982560852325428, 0, 6315523616526205691702430004356554304845, 0, 1068499705288740424393628027305747034629199, 0, 180775449436470757351562985413658748252118027] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 476049385384278 z + 361934002086042 z + 283 z 24 22 4 6 - 209008327393152 z + 91489947319202 z - 31456 z + 1810727 z 8 10 12 14 - 60822877 z + 1299399741 z - 18741406431 z + 190063074752 z 18 16 50 48 + 7519408140689 z - 1393165530211 z + 60822877 z - 1299399741 z 20 36 34 - 30255470662238 z - 91489947319202 z + 209008327393152 z 30 42 44 + 476049385384278 z + 1393165530211 z - 190063074752 z 46 58 56 54 52 + 18741406431 z + z - 283 z + 31456 z - 1810727 z 32 38 40 / - 361934002086042 z + 30255470662238 z - 7519408140689 z ) / (1 / 28 26 2 + 3869037540052536 z - 2565379364176056 z - 388 z 24 22 4 6 + 1292963229589736 z - 494869885640264 z + 53540 z - 3625988 z 8 10 12 14 + 139269404 z - 3360203590 z + 54509694252 z - 621848579300 z 18 16 50 - 31393891830964 z + 5139889740244 z - 3360203590 z 48 20 36 + 54509694252 z + 143512364757487 z + 1292963229589736 z 34 30 42 - 2565379364176056 z - 4436880629666004 z - 31393891830964 z 44 46 58 56 54 + 5139889740244 z - 621848579300 z - 388 z + 53540 z - 3625988 z 52 60 32 38 + 139269404 z + z + 3869037540052536 z - 494869885640264 z 40 + 143512364757487 z ) And in Maple-input format, it is: -(-1-476049385384278*z^28+361934002086042*z^26+283*z^2-209008327393152*z^24+ 91489947319202*z^22-31456*z^4+1810727*z^6-60822877*z^8+1299399741*z^10-\ 18741406431*z^12+190063074752*z^14+7519408140689*z^18-1393165530211*z^16+ 60822877*z^50-1299399741*z^48-30255470662238*z^20-91489947319202*z^36+ 209008327393152*z^34+476049385384278*z^30+1393165530211*z^42-190063074752*z^44+ 18741406431*z^46+z^58-283*z^56+31456*z^54-1810727*z^52-361934002086042*z^32+ 30255470662238*z^38-7519408140689*z^40)/(1+3869037540052536*z^28-\ 2565379364176056*z^26-388*z^2+1292963229589736*z^24-494869885640264*z^22+53540* z^4-3625988*z^6+139269404*z^8-3360203590*z^10+54509694252*z^12-621848579300*z^ 14-31393891830964*z^18+5139889740244*z^16-3360203590*z^50+54509694252*z^48+ 143512364757487*z^20+1292963229589736*z^36-2565379364176056*z^34-\ 4436880629666004*z^30-31393891830964*z^42+5139889740244*z^44-621848579300*z^46-\ 388*z^58+53540*z^56-3625988*z^54+139269404*z^52+z^60+3869037540052536*z^32-\ 494869885640264*z^38+143512364757487*z^40) The first , 40, terms are: [0, 105, 0, 18656, 0, 3432089, 0, 635090505, 0, 117745019437, 0, 21845878520893, 0, 4054374311771616, 0, 752539380679346189, 0, 139686730402587008405, 0, 25929213255916897224093, 0, 4813121882430297661603093, 0, 893440681183244669694911392, 0, 165846049050375254560660598085, 0, 30785396762809112988760246379477, 0, 5714582000799783720206505355611137, 0, 1060777302327815626537849376016591921, 0, 196908281716691731129681917949803849376, 0, 36551377790341224593165069522601387110401, 0, 6784901142630431694404209142837563970735577, 0, 1259456863161651702129111969910286897113767401] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 7}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2654453253661359034 z - 638473684753850843 z - 311 z 24 22 4 6 + 127976809606523246 z - 21245956795428425 z + 40511 z - 2983412 z 8 10 12 14 + 141469099 z - 4658709087 z + 112014974148 z - 2037535094349 z 18 16 50 - 322075282489133 z + 28782268993860 z - 246948611968722750881 z 48 20 + 370873713919867362641 z + 2899118592410466 z 36 34 + 139544089553405344079 z - 66817835877280749662 z 66 80 88 84 86 - 21245956795428425 z + 141469099 z + z + 40511 z - 311 z 82 64 30 - 2983412 z + 127976809606523246 z - 9242236675161879715 z 42 44 - 473173740357938006902 z + 513164023011694640100 z 46 58 - 473173740357938006902 z - 9242236675161879715 z 56 54 + 27057295582780863537 z - 66817835877280749662 z 52 60 + 139544089553405344079 z + 2654453253661359034 z 70 68 78 - 322075282489133 z + 2899118592410466 z - 4658709087 z 32 38 + 27057295582780863537 z - 246948611968722750881 z 40 62 76 + 370873713919867362641 z - 638473684753850843 z + 112014974148 z 74 72 / 2 - 2037535094349 z + 28782268993860 z ) / ((-1 + z ) (1 / 28 26 2 + 9902539948831642646 z - 2271707642089243382 z - 414 z 24 22 4 6 + 432795836560847359 z - 68080107416449586 z + 64380 z - 5378580 z 8 10 12 14 + 281768918 z - 10100351266 z + 262002913118 z - 5111592109354 z 18 16 50 - 918669561421310 z + 77121446865653 z - 1090758231544694938578 z 48 20 + 1665892658941958013475 z + 8777052160565076 z 36 34 + 602301410315192245784 z - 280179156598041966636 z 66 80 88 84 86 - 68080107416449586 z + 281768918 z + z + 64380 z - 414 z 82 64 30 - 5378580 z + 432795836560847359 z - 36010799572199833102 z 42 44 - 2147238266539198058768 z + 2336713454687003796200 z 46 58 - 2147238266539198058768 z - 36010799572199833102 z 56 54 + 109632178987344303650 z - 280179156598041966636 z 52 60 + 602301410315192245784 z + 9902539948831642646 z 70 68 78 - 918669561421310 z + 8777052160565076 z - 10100351266 z 32 38 + 109632178987344303650 z - 1090758231544694938578 z 40 62 76 + 1665892658941958013475 z - 2271707642089243382 z + 262002913118 z 74 72 - 5111592109354 z + 77121446865653 z )) And in Maple-input format, it is: -(1+2654453253661359034*z^28-638473684753850843*z^26-311*z^2+127976809606523246 *z^24-21245956795428425*z^22+40511*z^4-2983412*z^6+141469099*z^8-4658709087*z^ 10+112014974148*z^12-2037535094349*z^14-322075282489133*z^18+28782268993860*z^ 16-246948611968722750881*z^50+370873713919867362641*z^48+2899118592410466*z^20+ 139544089553405344079*z^36-66817835877280749662*z^34-21245956795428425*z^66+ 141469099*z^80+z^88+40511*z^84-311*z^86-2983412*z^82+127976809606523246*z^64-\ 9242236675161879715*z^30-473173740357938006902*z^42+513164023011694640100*z^44-\ 473173740357938006902*z^46-9242236675161879715*z^58+27057295582780863537*z^56-\ 66817835877280749662*z^54+139544089553405344079*z^52+2654453253661359034*z^60-\ 322075282489133*z^70+2899118592410466*z^68-4658709087*z^78+27057295582780863537 *z^32-246948611968722750881*z^38+370873713919867362641*z^40-638473684753850843* z^62+112014974148*z^76-2037535094349*z^74+28782268993860*z^72)/(-1+z^2)/(1+ 9902539948831642646*z^28-2271707642089243382*z^26-414*z^2+432795836560847359*z^ 24-68080107416449586*z^22+64380*z^4-5378580*z^6+281768918*z^8-10100351266*z^10+ 262002913118*z^12-5111592109354*z^14-918669561421310*z^18+77121446865653*z^16-\ 1090758231544694938578*z^50+1665892658941958013475*z^48+8777052160565076*z^20+ 602301410315192245784*z^36-280179156598041966636*z^34-68080107416449586*z^66+ 281768918*z^80+z^88+64380*z^84-414*z^86-5378580*z^82+432795836560847359*z^64-\ 36010799572199833102*z^30-2147238266539198058768*z^42+2336713454687003796200*z^ 44-2147238266539198058768*z^46-36010799572199833102*z^58+109632178987344303650* z^56-280179156598041966636*z^54+602301410315192245784*z^52+9902539948831642646* z^60-918669561421310*z^70+8777052160565076*z^68-10100351266*z^78+ 109632178987344303650*z^32-1090758231544694938578*z^38+1665892658941958013475*z ^40-2271707642089243382*z^62+262002913118*z^76-5111592109354*z^74+ 77121446865653*z^72) The first , 40, terms are: [0, 104, 0, 18877, 0, 3554927, 0, 672567808, 0, 127384527507, 0, 24134714716927, 0, 4573145417990736, 0, 866569956784469479, 0, 164209217716471059585, 0, 31116676979540853357016, 0, 5896434623410648313068873, 0, 1117341590048131309577855081, 0, 211730054276143361926087119464, 0, 40121676867599886698185576649297, 0, 7602836491335535874705634441617943, 0, 1440695593219100186355713176071394832, 0, 273003871491065612777677372695105560639, 0, 51732728447113532893051835245552059855203, 0, 9803066815748067970166364498146728414102368, 0, 1857627113163677766395869536334552818451781743] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 128876515798 z + 258968918705 z + 216 z - 366567567797 z 22 4 6 8 10 + 366567567797 z - 16991 z + 657725 z - 14574971 z + 200714599 z 12 14 18 16 - 1802102270 z + 10869594023 z + 128876515798 z - 44919891721 z 20 36 34 30 - 258968918705 z - 200714599 z + 1802102270 z + 44919891721 z 42 44 46 32 38 40 + 16991 z - 216 z + z - 10869594023 z + 14574971 z - 657725 z ) / 28 26 2 / (1 + 1559313439082 z - 2583706524072 z - 288 z / 24 22 4 6 + 3055958698269 z - 2583706524072 z + 29322 z - 1410520 z 8 10 12 14 + 37380137 z - 600203832 z + 6197771126 z - 42779113008 z 18 16 48 20 - 668959270736 z + 202520905934 z + z + 1559313439082 z 36 34 30 42 + 6197771126 z - 42779113008 z - 668959270736 z - 1410520 z 44 46 32 38 40 + 29322 z - 288 z + 202520905934 z - 600203832 z + 37380137 z ) And in Maple-input format, it is: -(-1-128876515798*z^28+258968918705*z^26+216*z^2-366567567797*z^24+366567567797 *z^22-16991*z^4+657725*z^6-14574971*z^8+200714599*z^10-1802102270*z^12+ 10869594023*z^14+128876515798*z^18-44919891721*z^16-258968918705*z^20-200714599 *z^36+1802102270*z^34+44919891721*z^30+16991*z^42-216*z^44+z^46-10869594023*z^ 32+14574971*z^38-657725*z^40)/(1+1559313439082*z^28-2583706524072*z^26-288*z^2+ 3055958698269*z^24-2583706524072*z^22+29322*z^4-1410520*z^6+37380137*z^8-\ 600203832*z^10+6197771126*z^12-42779113008*z^14-668959270736*z^18+202520905934* z^16+z^48+1559313439082*z^20+6197771126*z^36-42779113008*z^34-668959270736*z^30 -1410520*z^42+29322*z^44-288*z^46+202520905934*z^32-600203832*z^38+37380137*z^ 40) The first , 40, terms are: [0, 72, 0, 8405, 0, 1062251, 0, 138229152, 0, 18226211923, 0, 2418959074963, 0, 322129506359664, 0, 42974388987548447, 0, 5738562093850428005, 0, 766686455251022839992, 0, 102459181792942593936709, 0, 13694535786243020725367893, 0, 1830533308079168712420263352, 0, 244695558808053368852880511205, 0, 32710278254441659746531061824815, 0, 4372678825291180331157111465282096, 0, 584539275015575269876754728119571027, 0, 78141419984408805825960071489099293363, 0, 10445992167293108082908280647532575919648, 0, 1396427901852895885888044944276666911479899] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {3, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6694657330804280 z - 3102266700648262 z - 243 z 24 22 4 6 + 1147773580671234 z - 337532120539648 z + 23932 z - 1307377 z 8 10 12 14 + 45176259 z - 1061496192 z + 17755859863 z - 218263276301 z 18 16 50 - 14282759620183 z + 2018053682508 z - 14282759620183 z 48 20 36 + 78423452151213 z + 78423452151213 z + 16056370410896118 z 34 66 64 30 - 17906173054083840 z - 243 z + 23932 z - 11571873203064434 z 42 44 46 - 3102266700648262 z + 1147773580671234 z - 337532120539648 z 58 56 54 52 - 1061496192 z + 17755859863 z - 218263276301 z + 2018053682508 z 60 68 32 38 + 45176259 z + z + 16056370410896118 z - 11571873203064434 z 40 62 / 2 + 6694657330804280 z - 1307377 z ) / ((-1 + z ) (1 / 28 26 2 + 27099740450967382 z - 12177471235270052 z - 318 z 24 22 4 6 + 4333861875098898 z - 1216380390242008 z + 37467 z - 2362664 z 8 10 12 14 + 92425557 z - 2425442462 z + 44804542833 z - 602072813744 z 18 16 50 - 45792688156654 z + 6028097965071 z - 45792688156654 z 48 20 36 + 267637169616909 z + 267637169616909 z + 67375879527903130 z 34 66 64 30 - 75482160948378004 z - 318 z + 37467 z - 47901863657307096 z 42 44 46 - 12177471235270052 z + 4333861875098898 z - 1216380390242008 z 58 56 54 52 - 2425442462 z + 44804542833 z - 602072813744 z + 6028097965071 z 60 68 32 38 + 92425557 z + z + 67375879527903130 z - 47901863657307096 z 40 62 + 27099740450967382 z - 2362664 z )) And in Maple-input format, it is: -(1+6694657330804280*z^28-3102266700648262*z^26-243*z^2+1147773580671234*z^24-\ 337532120539648*z^22+23932*z^4-1307377*z^6+45176259*z^8-1061496192*z^10+ 17755859863*z^12-218263276301*z^14-14282759620183*z^18+2018053682508*z^16-\ 14282759620183*z^50+78423452151213*z^48+78423452151213*z^20+16056370410896118*z ^36-17906173054083840*z^34-243*z^66+23932*z^64-11571873203064434*z^30-\ 3102266700648262*z^42+1147773580671234*z^44-337532120539648*z^46-1061496192*z^ 58+17755859863*z^56-218263276301*z^54+2018053682508*z^52+45176259*z^60+z^68+ 16056370410896118*z^32-11571873203064434*z^38+6694657330804280*z^40-1307377*z^ 62)/(-1+z^2)/(1+27099740450967382*z^28-12177471235270052*z^26-318*z^2+ 4333861875098898*z^24-1216380390242008*z^22+37467*z^4-2362664*z^6+92425557*z^8-\ 2425442462*z^10+44804542833*z^12-602072813744*z^14-45792688156654*z^18+ 6028097965071*z^16-45792688156654*z^50+267637169616909*z^48+267637169616909*z^ 20+67375879527903130*z^36-75482160948378004*z^34-318*z^66+37467*z^64-\ 47901863657307096*z^30-12177471235270052*z^42+4333861875098898*z^44-\ 1216380390242008*z^46-2425442462*z^58+44804542833*z^56-602072813744*z^54+ 6028097965071*z^52+92425557*z^60+z^68+67375879527903130*z^32-47901863657307096* z^38+27099740450967382*z^40-2362664*z^62) The first , 40, terms are: [0, 76, 0, 10391, 0, 1535823, 0, 230101596, 0, 34563565321, 0, 5194504364953, 0, 780760922317500, 0, 117355333852237567, 0, 17639656036504210023, 0, 2651416750690354960364, 0, 398534596344432335754353, 0, 59903762076870880919912977, 0, 9004138719828484074446502828, 0, 1353412736313932927844263746631, 0, 203431565697752627689300151771679, 0, 30577813277437987861758547734604028, 0, 4596153314350892927003582375511380985, 0, 690848135473899273917217327903985893289, 0, 103841432965461931435714439866455558256156, 0, 15608413262842439775028946811420963490713391] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {3, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3866777589945538687 z - 903644666730058031 z - 311 z 24 22 4 6 + 175604888907386029 z - 28214205340335192 z + 40887 z - 3064186 z 8 10 12 14 + 148798916 z - 5041586918 z + 125127878385 z - 2354297687041 z 18 16 50 - 399193830578120 z + 34438502325163 z - 397942837411231231769 z 48 20 + 603525862531776288839 z + 3721023846871384 z 36 34 + 221843658434058973065 z - 104424615533812944150 z 66 80 88 84 86 - 28214205340335192 z + 148798916 z + z + 40887 z - 311 z 82 64 30 - 3064186 z + 175604888907386029 z - 13822601275305254890 z 42 44 - 774581351786599816320 z + 841715569140812818384 z 46 58 - 774581351786599816320 z - 13822601275305254890 z 56 54 + 41430222829681949100 z - 104424615533812944150 z 52 60 + 221843658434058973065 z + 3866777589945538687 z 70 68 78 - 399193830578120 z + 3721023846871384 z - 5041586918 z 32 38 + 41430222829681949100 z - 397942837411231231769 z 40 62 76 + 603525862531776288839 z - 903644666730058031 z + 125127878385 z 74 72 / 2 - 2354297687041 z + 34438502325163 z ) / ((-1 + z ) (1 / 28 26 2 + 14038392955037554460 z - 3142168761032367116 z - 394 z 24 22 4 6 + 582597447927837787 z - 88973467151782210 z + 61590 z - 5277278 z 8 10 12 14 + 285917335 z - 10637614844 z + 286751733056 z - 5812938136144 z 18 16 50 - 1124075535724250 z + 91041390977305 z - 1678994814155445707346 z 48 20 + 2584242320705373916691 z + 11110837184559350 z 36 34 + 917197189294226109702 z - 420861073214662073854 z 66 80 88 84 86 - 88973467151782210 z + 285917335 z + z + 61590 z - 394 z 82 64 30 - 5277278 z + 582597447927837787 z - 52186970420489017264 z 42 44 - 3346554411770872678860 z + 3647559580057210728556 z 46 58 - 3346554411770872678860 z - 52186970420489017264 z 56 54 + 161978049145957557333 z - 420861073214662073854 z 52 60 + 917197189294226109702 z + 14038392955037554460 z 70 68 78 - 1124075535724250 z + 11110837184559350 z - 10637614844 z 32 38 + 161978049145957557333 z - 1678994814155445707346 z 40 62 76 + 2584242320705373916691 z - 3142168761032367116 z + 286751733056 z 74 72 - 5812938136144 z + 91041390977305 z )) And in Maple-input format, it is: -(1+3866777589945538687*z^28-903644666730058031*z^26-311*z^2+175604888907386029 *z^24-28214205340335192*z^22+40887*z^4-3064186*z^6+148798916*z^8-5041586918*z^ 10+125127878385*z^12-2354297687041*z^14-399193830578120*z^18+34438502325163*z^ 16-397942837411231231769*z^50+603525862531776288839*z^48+3721023846871384*z^20+ 221843658434058973065*z^36-104424615533812944150*z^34-28214205340335192*z^66+ 148798916*z^80+z^88+40887*z^84-311*z^86-3064186*z^82+175604888907386029*z^64-\ 13822601275305254890*z^30-774581351786599816320*z^42+841715569140812818384*z^44 -774581351786599816320*z^46-13822601275305254890*z^58+41430222829681949100*z^56 -104424615533812944150*z^54+221843658434058973065*z^52+3866777589945538687*z^60 -399193830578120*z^70+3721023846871384*z^68-5041586918*z^78+ 41430222829681949100*z^32-397942837411231231769*z^38+603525862531776288839*z^40 -903644666730058031*z^62+125127878385*z^76-2354297687041*z^74+34438502325163*z^ 72)/(-1+z^2)/(1+14038392955037554460*z^28-3142168761032367116*z^26-394*z^2+ 582597447927837787*z^24-88973467151782210*z^22+61590*z^4-5277278*z^6+285917335* z^8-10637614844*z^10+286751733056*z^12-5812938136144*z^14-1124075535724250*z^18 +91041390977305*z^16-1678994814155445707346*z^50+2584242320705373916691*z^48+ 11110837184559350*z^20+917197189294226109702*z^36-420861073214662073854*z^34-\ 88973467151782210*z^66+285917335*z^80+z^88+61590*z^84-394*z^86-5277278*z^82+ 582597447927837787*z^64-52186970420489017264*z^30-3346554411770872678860*z^42+ 3647559580057210728556*z^44-3346554411770872678860*z^46-52186970420489017264*z^ 58+161978049145957557333*z^56-420861073214662073854*z^54+917197189294226109702* z^52+14038392955037554460*z^60-1124075535724250*z^70+11110837184559350*z^68-\ 10637614844*z^78+161978049145957557333*z^32-1678994814155445707346*z^38+ 2584242320705373916691*z^40-3142168761032367116*z^62+286751733056*z^76-\ 5812938136144*z^74+91041390977305*z^72) The first , 40, terms are: [0, 84, 0, 12083, 0, 1840811, 0, 284236888, 0, 44103881549, 0, 6857551612653, 0, 1067220598353776, 0, 166154969292826139, 0, 25873177367758283959, 0, 4029216044965762056716, 0, 627489725156807545269409, 0, 97723603881160760172903885, 0, 15219323562629727710061501164, 0, 2370241340203423192997394939875, 0, 369139384602204731657283777917279, 0, 57489491152251497952672710606270080, 0, 8953372707130432374772820307595980537, 0, 1394392103910081955559457651150118639817, 0, 217161678344754109605248829907156147127528, 0, 33820612965362937810286882688372743320816719] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2498179613622093337 z - 582817124330584093 z - 285 z 24 22 4 6 + 113222505841810305 z - 18219876420532104 z + 34481 z - 2407994 z 8 10 12 14 + 110518382 z - 3585173542 z + 86122915659 z - 1582170593911 z 18 16 50 - 260738868819080 z + 22754847808163 z - 261123636299122725647 z 48 20 + 396924404921222999835 z + 2412712874959996 z 36 34 + 145146211985257492587 z - 68095763309196750302 z 66 80 88 84 86 - 18219876420532104 z + 110518382 z + z + 34481 z - 285 z 82 64 30 - 2407994 z + 113222505841810305 z - 8954009931102222978 z 42 44 - 510157789796801187120 z + 554647553175988442312 z 46 58 - 510157789796801187120 z - 8954009931102222978 z 56 54 + 26924267494064242098 z - 68095763309196750302 z 52 60 + 145146211985257492587 z + 2498179613622093337 z 70 68 78 - 260738868819080 z + 2412712874959996 z - 3585173542 z 32 38 + 26924267494064242098 z - 261123636299122725647 z 40 62 76 + 396924404921222999835 z - 582817124330584093 z + 86122915659 z 74 72 / - 1582170593911 z + 22754847808163 z ) / (-1 / 28 26 2 - 11011750738699388472 z + 2376600035354840271 z + 369 z 24 22 4 6 - 427170687090402099 z + 63594468108329806 z - 52790 z + 4189958 z 8 10 12 14 - 214132539 z + 7645871443 z - 200766129344 z + 4014160097588 z 18 16 50 + 777802810548645 z - 62657286854497 z + 2819111016708718027743 z 48 20 - 3932430412692444148807 z - 7788036178581118 z 36 34 - 877158756137636219718 z + 379989119928671437157 z 66 80 90 88 84 + 427170687090402099 z - 7645871443 z + z - 369 z - 4189958 z 86 82 64 + 52790 z + 214132539 z - 2376600035354840271 z 30 42 + 42676389693170151956 z + 3932430412692444148807 z 44 46 - 4643927219198145422004 z + 4643927219198145422004 z 58 56 + 138815682819183923213 z - 379989119928671437157 z 54 52 + 877158756137636219718 z - 1710132179227811886430 z 60 70 68 - 42676389693170151956 z + 7788036178581118 z - 63594468108329806 z 78 32 + 200766129344 z - 138815682819183923213 z 38 40 + 1710132179227811886430 z - 2819111016708718027743 z 62 76 74 + 11011750738699388472 z - 4014160097588 z + 62657286854497 z 72 - 777802810548645 z ) And in Maple-input format, it is: -(1+2498179613622093337*z^28-582817124330584093*z^26-285*z^2+113222505841810305 *z^24-18219876420532104*z^22+34481*z^4-2407994*z^6+110518382*z^8-3585173542*z^ 10+86122915659*z^12-1582170593911*z^14-260738868819080*z^18+22754847808163*z^16 -261123636299122725647*z^50+396924404921222999835*z^48+2412712874959996*z^20+ 145146211985257492587*z^36-68095763309196750302*z^34-18219876420532104*z^66+ 110518382*z^80+z^88+34481*z^84-285*z^86-2407994*z^82+113222505841810305*z^64-\ 8954009931102222978*z^30-510157789796801187120*z^42+554647553175988442312*z^44-\ 510157789796801187120*z^46-8954009931102222978*z^58+26924267494064242098*z^56-\ 68095763309196750302*z^54+145146211985257492587*z^52+2498179613622093337*z^60-\ 260738868819080*z^70+2412712874959996*z^68-3585173542*z^78+26924267494064242098 *z^32-261123636299122725647*z^38+396924404921222999835*z^40-582817124330584093* z^62+86122915659*z^76-1582170593911*z^74+22754847808163*z^72)/(-1-\ 11011750738699388472*z^28+2376600035354840271*z^26+369*z^2-427170687090402099*z ^24+63594468108329806*z^22-52790*z^4+4189958*z^6-214132539*z^8+7645871443*z^10-\ 200766129344*z^12+4014160097588*z^14+777802810548645*z^18-62657286854497*z^16+ 2819111016708718027743*z^50-3932430412692444148807*z^48-7788036178581118*z^20-\ 877158756137636219718*z^36+379989119928671437157*z^34+427170687090402099*z^66-\ 7645871443*z^80+z^90-369*z^88-4189958*z^84+52790*z^86+214132539*z^82-\ 2376600035354840271*z^64+42676389693170151956*z^30+3932430412692444148807*z^42-\ 4643927219198145422004*z^44+4643927219198145422004*z^46+138815682819183923213*z ^58-379989119928671437157*z^56+877158756137636219718*z^54-\ 1710132179227811886430*z^52-42676389693170151956*z^60+7788036178581118*z^70-\ 63594468108329806*z^68+200766129344*z^78-138815682819183923213*z^32+ 1710132179227811886430*z^38-2819111016708718027743*z^40+11011750738699388472*z^ 62-4014160097588*z^76+62657286854497*z^74-777802810548645*z^72) The first , 40, terms are: [0, 84, 0, 12687, 0, 2029107, 0, 327336068, 0, 52902012333, 0, 8553555093897, 0, 1383161947217516, 0, 223672949952331267, 0, 36170768751011791811, 0, 5849288814921514565836, 0, 945907527355694606240985, 0, 152965812178404505365592937, 0, 24736605004970310084671214892, 0, 4000237876402872599231353638587, 0, 646891646037494461601471630793555, 0, 104610979417744234418768924602812060, 0, 16916986152094032488577527231242704001, 0, 2735701568668630018389359489433088282629, 0, 442399314256993926713437075732101871891348, 0, 71541850725881808907363612050127361699046835] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4151579519204212 z - 1945086757959498 z - 245 z 24 22 4 6 + 730613938942298 z - 219186682151840 z + 23670 z - 1236459 z 8 10 12 14 + 40314551 z - 890474928 z + 14035912755 z - 163521336431 z 18 16 50 - 9823813657889 z + 1443259677142 z - 9823813657889 z 48 20 36 + 52244761810085 z + 52244761810085 z + 9841253544728790 z 34 66 64 30 - 10959721994894176 z - 245 z + 23670 z - 7123026442575046 z 42 44 46 - 1945086757959498 z + 730613938942298 z - 219186682151840 z 58 56 54 52 - 890474928 z + 14035912755 z - 163521336431 z + 1443259677142 z 60 68 32 38 + 40314551 z + z + 9841253544728790 z - 7123026442575046 z 40 62 / 2 + 4151579519204212 z - 1236459 z ) / ((-1 + z ) (1 / 28 26 2 + 17709782758935658 z - 7992145291734632 z - 340 z 24 22 4 6 + 2866984884320118 z - 815333101966800 z + 40007 z - 2402284 z 8 10 12 14 + 87590585 z - 2131561772 z + 36677361601 z - 463413053628 z 18 16 50 - 32233001130244 z + 4412051928879 z - 32233001130244 z 48 20 36 + 183039634133353 z + 183039634133353 z + 43911006409450982 z 34 66 64 30 - 49185926200671816 z - 340 z + 40007 z - 31242124468156824 z 42 44 46 - 7992145291734632 z + 2866984884320118 z - 815333101966800 z 58 56 54 52 - 2131561772 z + 36677361601 z - 463413053628 z + 4412051928879 z 60 68 32 38 + 87590585 z + z + 43911006409450982 z - 31242124468156824 z 40 62 + 17709782758935658 z - 2402284 z )) And in Maple-input format, it is: -(1+4151579519204212*z^28-1945086757959498*z^26-245*z^2+730613938942298*z^24-\ 219186682151840*z^22+23670*z^4-1236459*z^6+40314551*z^8-890474928*z^10+ 14035912755*z^12-163521336431*z^14-9823813657889*z^18+1443259677142*z^16-\ 9823813657889*z^50+52244761810085*z^48+52244761810085*z^20+9841253544728790*z^ 36-10959721994894176*z^34-245*z^66+23670*z^64-7123026442575046*z^30-\ 1945086757959498*z^42+730613938942298*z^44-219186682151840*z^46-890474928*z^58+ 14035912755*z^56-163521336431*z^54+1443259677142*z^52+40314551*z^60+z^68+ 9841253544728790*z^32-7123026442575046*z^38+4151579519204212*z^40-1236459*z^62) /(-1+z^2)/(1+17709782758935658*z^28-7992145291734632*z^26-340*z^2+ 2866984884320118*z^24-815333101966800*z^22+40007*z^4-2402284*z^6+87590585*z^8-\ 2131561772*z^10+36677361601*z^12-463413053628*z^14-32233001130244*z^18+ 4412051928879*z^16-32233001130244*z^50+183039634133353*z^48+183039634133353*z^ 20+43911006409450982*z^36-49185926200671816*z^34-340*z^66+40007*z^64-\ 31242124468156824*z^30-7992145291734632*z^42+2866984884320118*z^44-\ 815333101966800*z^46-2131561772*z^58+36677361601*z^56-463413053628*z^54+ 4412051928879*z^52+87590585*z^60+z^68+43911006409450982*z^32-31242124468156824* z^38+17709782758935658*z^40-2402284*z^62) The first , 40, terms are: [0, 96, 0, 16059, 0, 2808639, 0, 494595044, 0, 87246865445, 0, 15398338760809, 0, 2718120060473020, 0, 479828515721654091, 0, 84705333449500511239, 0, 14953326079330379086888, 0, 2639767067581691478514165, 0, 466008300589197020861107021, 0, 82266264348677549541366420376, 0, 14522785612749971914762548534063, 0, 2563764230358545112208315945299123, 0, 452591344647548713551344412960993100, 0, 79897723520045320804174001662532528033, 0, 14104658215793002561725318977190713737069, 0, 2489950585069191134766787216584419227810900, 0, 439560733882011414512018490365777999688437079] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 14027878575793776 z - 5707262736328318 z - 259 z 24 22 4 6 + 1888301814885175 z - 505580403881847 z + 26856 z - 1510783 z 8 10 12 14 + 52920867 z - 1253131954 z + 21173044416 z - 265018218018 z 18 16 50 - 18671767565019 z + 2524606106423 z - 505580403881847 z 48 20 36 + 1888301814885175 z + 108828689986368 z + 68600700116760912 z 34 66 64 - 62147142174137300 z - 1510783 z + 52920867 z 30 42 44 - 28140542104092678 z - 28140542104092678 z + 14027878575793776 z 46 58 56 - 5707262736328318 z - 265018218018 z + 2524606106423 z 54 52 60 70 - 18671767565019 z + 108828689986368 z + 21173044416 z - 259 z 68 32 38 + 26856 z + 46191560497994606 z - 62147142174137300 z 40 62 72 / 2 + 46191560497994606 z - 1253131954 z + z ) / ((-1 + z ) (1 / 28 26 2 + 55761785359742650 z - 21962024733151804 z - 332 z 24 22 4 6 + 6987114934905071 z - 1786915214349448 z + 41571 z - 2721352 z 8 10 12 14 + 108273875 z - 2861822016 z + 53244952754 z - 725713498388 z 18 16 50 - 58983892043944 z + 7456474723147 z - 1786915214349448 z 48 20 36 + 6987114934905071 z + 364927516586515 z + 289277583455723324 z 34 66 64 - 261082546236562396 z - 2721352 z + 108273875 z 30 42 44 - 114759940834868432 z - 114759940834868432 z + 55761785359742650 z 46 58 56 - 21962024733151804 z - 725713498388 z + 7456474723147 z 54 52 60 70 - 58983892043944 z + 364927516586515 z + 53244952754 z - 332 z 68 32 38 + 41571 z + 191891010486418034 z - 261082546236562396 z 40 62 72 + 191891010486418034 z - 2861822016 z + z )) And in Maple-input format, it is: -(1+14027878575793776*z^28-5707262736328318*z^26-259*z^2+1888301814885175*z^24-\ 505580403881847*z^22+26856*z^4-1510783*z^6+52920867*z^8-1253131954*z^10+ 21173044416*z^12-265018218018*z^14-18671767565019*z^18+2524606106423*z^16-\ 505580403881847*z^50+1888301814885175*z^48+108828689986368*z^20+ 68600700116760912*z^36-62147142174137300*z^34-1510783*z^66+52920867*z^64-\ 28140542104092678*z^30-28140542104092678*z^42+14027878575793776*z^44-\ 5707262736328318*z^46-265018218018*z^58+2524606106423*z^56-18671767565019*z^54+ 108828689986368*z^52+21173044416*z^60-259*z^70+26856*z^68+46191560497994606*z^ 32-62147142174137300*z^38+46191560497994606*z^40-1253131954*z^62+z^72)/(-1+z^2) /(1+55761785359742650*z^28-21962024733151804*z^26-332*z^2+6987114934905071*z^24 -1786915214349448*z^22+41571*z^4-2721352*z^6+108273875*z^8-2861822016*z^10+ 53244952754*z^12-725713498388*z^14-58983892043944*z^18+7456474723147*z^16-\ 1786915214349448*z^50+6987114934905071*z^48+364927516586515*z^20+ 289277583455723324*z^36-261082546236562396*z^34-2721352*z^66+108273875*z^64-\ 114759940834868432*z^30-114759940834868432*z^42+55761785359742650*z^44-\ 21962024733151804*z^46-725713498388*z^58+7456474723147*z^56-58983892043944*z^54 +364927516586515*z^52+53244952754*z^60-332*z^70+41571*z^68+191891010486418034*z ^32-261082546236562396*z^38+191891010486418034*z^40-2861822016*z^62+z^72) The first , 40, terms are: [0, 74, 0, 9595, 0, 1346453, 0, 192691506, 0, 27759414763, 0, 4009533064795, 0, 579773800189242, 0, 83875234876778957, 0, 12136743753615171747, 0, 1756354786710286824338, 0, 254179757443629518154017, 0, 36785616477712410761976833, 0, 5323764950393722250378921730, 0, 770480043212235421669290481267, 0, 111507652754542028564363014958845, 0, 16137947358958686944108606518549674, 0, 2335565589975485370458395836088253723, 0, 338014951232889815670028623694012012011, 0, 48919249318063836752666528041311181359554, 0, 7079843722062206597657996356510738997980709] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 225843428518532785625 z - 31894013116805660673 z - 326 z 24 22 4 6 + 3863149908091291001 z - 398930839383666264 z + 46749 z - 3992749 z 102 8 10 12 - 9678342176 z + 230669789 z - 9678342176 z + 308694496549 z 14 18 16 - 7726904678526 z - 2555711740586022 z + 155385424749378 z 50 48 - 50973123862501866455220360 z + 32339159599592589449756890 z 20 36 + 34865179885575881 z + 133317165921843328305112 z 34 66 - 33389834560609467079264 z - 18007554199670370859484194 z 80 100 90 + 7280580395103157074533 z + 308694496549 z - 398930839383666264 z 88 84 + 3863149908091291001 z + 225843428518532785625 z 94 86 96 - 2555711740586022 z - 31894013116805660673 z + 155385424749378 z 98 92 82 - 7726904678526 z + 34865179885575881 z - 1378399499956442426870 z 64 112 110 106 + 32339159599592589449756890 z + z - 326 z - 3992749 z 108 30 42 + 46749 z - 1378399499956442426870 z - 3768125319137979391717476 z 44 46 + 8797072881562657996643258 z - 18007554199670370859484194 z 58 56 - 85710103246934928471913612 z + 91460323109499126138140140 z 54 52 - 85710103246934928471913612 z + 70536844645309769431973234 z 60 70 + 70536844645309769431973234 z - 3768125319137979391717476 z 68 78 + 8797072881562657996643258 z - 33389834560609467079264 z 32 38 + 7280580395103157074533 z - 464431734505432115654352 z 40 62 + 1414072723188537905904282 z - 50973123862501866455220360 z 76 74 + 133317165921843328305112 z - 464431734505432115654352 z 72 104 / + 1414072723188537905904282 z + 230669789 z ) / (-1 / 28 26 2 - 798216422488516878444 z + 106236952617513675785 z + 399 z 24 22 4 - 12113688673805426421 z + 1175924041985584066 z - 66124 z 6 102 8 10 + 6345208 z + 643530302822 z - 405162665 z + 18589618949 z 12 14 18 - 643530302822 z + 17385710000548 z + 6618563655842605 z 16 50 - 375726188439389 z + 342631286339500643586228622 z 48 20 - 204545853161324449370050358 z - 96436475339887384 z 36 34 - 593742388495876660426340 z + 140415372249543804134161 z 66 80 + 204545853161324449370050358 z - 140415372249543804134161 z 100 90 - 17385710000548 z + 12113688673805426421 z 88 84 - 106236952617513675785 z - 5165015102965574813040 z 94 86 96 + 96436475339887384 z + 798216422488516878444 z - 6618563655842605 z 98 92 + 375726188439389 z - 1175924041985584066 z 82 64 112 + 28906244569807088807831 z - 342631286339500643586228622 z - 399 z 114 110 106 108 + z + 66124 z + 405162665 z - 6345208 z 30 42 + 5165015102965574813040 z + 19952488651455437532211266 z 44 46 - 49389857755201607559226636 z + 107267527639704638401564676 z 58 56 + 742244876770215948246437502 z - 742244876770215948246437502 z 54 52 + 652559195053195390529676780 z - 504353179829542342814546816 z 60 70 - 652559195053195390529676780 z + 49389857755201607559226636 z 68 78 - 107267527639704638401564676 z + 593742388495876660426340 z 32 38 - 28906244569807088807831 z + 2190615830656732221380068 z 40 62 - 7065550869811881049439318 z + 504353179829542342814546816 z 76 74 - 2190615830656732221380068 z + 7065550869811881049439318 z 72 104 - 19952488651455437532211266 z - 18589618949 z ) And in Maple-input format, it is: -(1+225843428518532785625*z^28-31894013116805660673*z^26-326*z^2+ 3863149908091291001*z^24-398930839383666264*z^22+46749*z^4-3992749*z^6-\ 9678342176*z^102+230669789*z^8-9678342176*z^10+308694496549*z^12-7726904678526* z^14-2555711740586022*z^18+155385424749378*z^16-50973123862501866455220360*z^50 +32339159599592589449756890*z^48+34865179885575881*z^20+ 133317165921843328305112*z^36-33389834560609467079264*z^34-\ 18007554199670370859484194*z^66+7280580395103157074533*z^80+308694496549*z^100-\ 398930839383666264*z^90+3863149908091291001*z^88+225843428518532785625*z^84-\ 2555711740586022*z^94-31894013116805660673*z^86+155385424749378*z^96-\ 7726904678526*z^98+34865179885575881*z^92-1378399499956442426870*z^82+ 32339159599592589449756890*z^64+z^112-326*z^110-3992749*z^106+46749*z^108-\ 1378399499956442426870*z^30-3768125319137979391717476*z^42+ 8797072881562657996643258*z^44-18007554199670370859484194*z^46-\ 85710103246934928471913612*z^58+91460323109499126138140140*z^56-\ 85710103246934928471913612*z^54+70536844645309769431973234*z^52+ 70536844645309769431973234*z^60-3768125319137979391717476*z^70+ 8797072881562657996643258*z^68-33389834560609467079264*z^78+ 7280580395103157074533*z^32-464431734505432115654352*z^38+ 1414072723188537905904282*z^40-50973123862501866455220360*z^62+ 133317165921843328305112*z^76-464431734505432115654352*z^74+ 1414072723188537905904282*z^72+230669789*z^104)/(-1-798216422488516878444*z^28+ 106236952617513675785*z^26+399*z^2-12113688673805426421*z^24+ 1175924041985584066*z^22-66124*z^4+6345208*z^6+643530302822*z^102-405162665*z^8 +18589618949*z^10-643530302822*z^12+17385710000548*z^14+6618563655842605*z^18-\ 375726188439389*z^16+342631286339500643586228622*z^50-\ 204545853161324449370050358*z^48-96436475339887384*z^20-\ 593742388495876660426340*z^36+140415372249543804134161*z^34+ 204545853161324449370050358*z^66-140415372249543804134161*z^80-17385710000548*z ^100+12113688673805426421*z^90-106236952617513675785*z^88-\ 5165015102965574813040*z^84+96436475339887384*z^94+798216422488516878444*z^86-\ 6618563655842605*z^96+375726188439389*z^98-1175924041985584066*z^92+ 28906244569807088807831*z^82-342631286339500643586228622*z^64-399*z^112+z^114+ 66124*z^110+405162665*z^106-6345208*z^108+5165015102965574813040*z^30+ 19952488651455437532211266*z^42-49389857755201607559226636*z^44+ 107267527639704638401564676*z^46+742244876770215948246437502*z^58-\ 742244876770215948246437502*z^56+652559195053195390529676780*z^54-\ 504353179829542342814546816*z^52-652559195053195390529676780*z^60+ 49389857755201607559226636*z^70-107267527639704638401564676*z^68+ 593742388495876660426340*z^78-28906244569807088807831*z^32+ 2190615830656732221380068*z^38-7065550869811881049439318*z^40+ 504353179829542342814546816*z^62-2190615830656732221380068*z^76+ 7065550869811881049439318*z^74-19952488651455437532211266*z^72-18589618949*z^ 104) The first , 40, terms are: [0, 73, 0, 9752, 0, 1416455, 0, 209031605, 0, 30954810619, 0, 4587725253525, 0, 680067865638648, 0, 100815710814505523, 0, 14945464937250934307, 0, 2215603059553281964895, 0, 328454193858648323358583, 0, 48692015269132050251430872, 0, 7218396077459284738488276537, 0, 1070098295830920998559413052639, 0, 158637785172504562048824890894969, 0, 23517416111974433199969496778298075, 0, 3486362723039993314910940531864227768, 0, 516839306636656586974180533655899079661, 0, 76619356650029596436808825668979978609437, 0, 11358512671347322546789285021951202786403861] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 194210818040027835146 z - 28093671074078603519 z - 335 z 24 22 4 6 + 3485607405046989240 z - 368616262502077201 z + 48862 z - 4199321 z 102 8 10 12 - 10061616975 z + 241992784 z - 10061616975 z + 316510826890 z 14 18 16 - 7787275374677 z - 2472290376855007 z + 153555045355268 z 50 48 - 35872572523022343768000395 z + 22972265377398482697929020 z 20 36 + 32975951636437926 z + 104607413595028485394870 z 34 66 - 26776697913557144444819 z - 12942757541427407695116269 z 80 100 90 + 5972546537989505837780 z + 316510826890 z - 368616262502077201 z 88 84 + 3485607405046989240 z + 194210818040027835146 z 94 86 96 - 2472290376855007 z - 28093671074078603519 z + 153555045355268 z 98 92 82 - 7787275374677 z + 32975951636437926 z - 1157464901856553027549 z 64 112 110 106 + 22972265377398482697929020 z + z - 335 z - 4199321 z 108 30 42 + 48862 z - 1157464901856553027549 z - 2790596577776528762774607 z 44 46 + 6411681523071422668178530 z - 12942757541427407695116269 z 58 56 - 59665099075119973559593373 z + 63580179265104618324846930 z 54 52 - 59665099075119973559593373 z + 49305116491856929928985470 z 60 70 + 49305116491856929928985470 z - 2790596577776528762774607 z 68 78 + 6411681523071422668178530 z - 26776697913557144444819 z 32 38 + 5972546537989505837780 z - 356955792652854536594093 z 40 62 + 1066013423014662632854264 z - 35872572523022343768000395 z 76 74 + 104607413595028485394870 z - 356955792652854536594093 z 72 104 / + 1066013423014662632854264 z + 241992784 z ) / (-1 / 28 26 2 - 701830906454904932944 z + 95855664734352382886 z + 408 z 24 22 4 - 11215061710536273856 z + 1116629230634110482 z - 69395 z 6 102 8 10 + 6762336 z + 678373400486 z - 433701938 z + 19811841244 z 12 14 18 - 678373400486 z + 18041186474811 z + 6593541214504153 z 16 50 - 382491080060751 z + 241503504984511480012179665 z 48 20 - 145779246471307073004697543 z - 93851978362792793 z 36 34 - 472674134931062640844795 z + 114469172573659585786627 z 66 80 + 145779246471307073004697543 z - 114469172573659585786627 z 100 90 - 18041186474811 z + 11215061710536273856 z 88 84 - 95855664734352382886 z - 4426120535860761022645 z 94 86 96 + 93851978362792793 z + 701830906454904932944 z - 6593541214504153 z 98 92 + 382491080060751 z - 1116629230634110482 z 82 64 112 + 24152518907644892937821 z - 241503504984511480012179665 z - 408 z 114 110 106 108 + z + 69395 z + 433701938 z - 6762336 z 30 42 + 4426120535860761022645 z + 14906393658076012524950716 z 44 46 - 36246768979974059468199882 z + 77490040444849485464056775 z 58 56 + 514324373259036987482525810 z - 514324373259036987482525810 z 54 52 + 453482851154563182661466618 z - 352498815645599838158668781 z 60 70 - 453482851154563182661466618 z + 36246768979974059468199882 z 68 78 - 77490040444849485464056775 z + 472674134931062640844795 z 32 38 - 24152518907644892937821 z + 1704939102163532258573844 z 40 62 - 5383425011401406677818058 z + 352498815645599838158668781 z 76 74 - 1704939102163532258573844 z + 5383425011401406677818058 z 72 104 - 14906393658076012524950716 z - 19811841244 z ) And in Maple-input format, it is: -(1+194210818040027835146*z^28-28093671074078603519*z^26-335*z^2+ 3485607405046989240*z^24-368616262502077201*z^22+48862*z^4-4199321*z^6-\ 10061616975*z^102+241992784*z^8-10061616975*z^10+316510826890*z^12-\ 7787275374677*z^14-2472290376855007*z^18+153555045355268*z^16-\ 35872572523022343768000395*z^50+22972265377398482697929020*z^48+ 32975951636437926*z^20+104607413595028485394870*z^36-26776697913557144444819*z^ 34-12942757541427407695116269*z^66+5972546537989505837780*z^80+316510826890*z^ 100-368616262502077201*z^90+3485607405046989240*z^88+194210818040027835146*z^84 -2472290376855007*z^94-28093671074078603519*z^86+153555045355268*z^96-\ 7787275374677*z^98+32975951636437926*z^92-1157464901856553027549*z^82+ 22972265377398482697929020*z^64+z^112-335*z^110-4199321*z^106+48862*z^108-\ 1157464901856553027549*z^30-2790596577776528762774607*z^42+ 6411681523071422668178530*z^44-12942757541427407695116269*z^46-\ 59665099075119973559593373*z^58+63580179265104618324846930*z^56-\ 59665099075119973559593373*z^54+49305116491856929928985470*z^52+ 49305116491856929928985470*z^60-2790596577776528762774607*z^70+ 6411681523071422668178530*z^68-26776697913557144444819*z^78+ 5972546537989505837780*z^32-356955792652854536594093*z^38+ 1066013423014662632854264*z^40-35872572523022343768000395*z^62+ 104607413595028485394870*z^76-356955792652854536594093*z^74+ 1066013423014662632854264*z^72+241992784*z^104)/(-1-701830906454904932944*z^28+ 95855664734352382886*z^26+408*z^2-11215061710536273856*z^24+1116629230634110482 *z^22-69395*z^4+6762336*z^6+678373400486*z^102-433701938*z^8+19811841244*z^10-\ 678373400486*z^12+18041186474811*z^14+6593541214504153*z^18-382491080060751*z^ 16+241503504984511480012179665*z^50-145779246471307073004697543*z^48-\ 93851978362792793*z^20-472674134931062640844795*z^36+114469172573659585786627*z ^34+145779246471307073004697543*z^66-114469172573659585786627*z^80-\ 18041186474811*z^100+11215061710536273856*z^90-95855664734352382886*z^88-\ 4426120535860761022645*z^84+93851978362792793*z^94+701830906454904932944*z^86-\ 6593541214504153*z^96+382491080060751*z^98-1116629230634110482*z^92+ 24152518907644892937821*z^82-241503504984511480012179665*z^64-408*z^112+z^114+ 69395*z^110+433701938*z^106-6762336*z^108+4426120535860761022645*z^30+ 14906393658076012524950716*z^42-36246768979974059468199882*z^44+ 77490040444849485464056775*z^46+514324373259036987482525810*z^58-\ 514324373259036987482525810*z^56+453482851154563182661466618*z^54-\ 352498815645599838158668781*z^52-453482851154563182661466618*z^60+ 36246768979974059468199882*z^70-77490040444849485464056775*z^68+ 472674134931062640844795*z^78-24152518907644892937821*z^32+ 1704939102163532258573844*z^38-5383425011401406677818058*z^40+ 352498815645599838158668781*z^62-1704939102163532258573844*z^76+ 5383425011401406677818058*z^74-14906393658076012524950716*z^72-19811841244*z^ 104) The first , 40, terms are: [0, 73, 0, 9251, 0, 1271588, 0, 178776133, 0, 25347166135, 0, 3606604551891, 0, 513994159635247, 0, 73304730788985643, 0, 10458033485183813093, 0, 1492225736500154520164, 0, 212936313299688423344403, 0, 30386391959482260322060093, 0, 4336257974890371391547892621, 0, 618805448776022330781515327205, 0, 88306882851148083035061668827989, 0, 12601888495121903661671190239989635, 0, 1798361564869964655106053796957587748, 0, 256636563655315727727844574730583328149, 0, 36623522715102943092630340526317610670947, 0, 5226388981860853168830897149454833678797287] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 12503705132769509 z - 5042460822102103 z - 244 z 24 22 4 6 + 1651291233421857 z - 437113179337942 z + 23835 z - 1286469 z 8 10 12 14 + 44013975 z - 1031696590 z + 17416886793 z - 219161348562 z 18 16 50 - 15758086469062 z + 2106877186570 z - 437113179337942 z 48 20 36 + 1651291233421857 z + 92962991242619 z + 62172209379904932 z 34 66 64 - 56262832007372720 z - 1286469 z + 44013975 z 30 42 44 - 25262278479455104 z - 25262278479455104 z + 12503705132769509 z 46 58 56 - 5042460822102103 z - 219161348562 z + 2106877186570 z 54 52 60 70 - 15758086469062 z + 92962991242619 z + 17416886793 z - 244 z 68 32 38 + 23835 z + 41684474430589379 z - 56262832007372720 z 40 62 72 / + 41684474430589379 z - 1031696590 z + z ) / (-1 / 28 26 2 - 68929134278896338 z + 25441425109103477 z + 319 z 24 22 4 6 - 7630613507548625 z + 1849918309512008 z - 37564 z + 2361548 z 8 10 12 14 - 92263977 z + 2437325733 z - 45931880950 z + 640585137468 z 18 16 50 + 55742388326125 z - 6788318891945 z + 7630613507548625 z 48 20 36 - 25441425109103477 z - 360013608483506 z - 497489882819035990 z 34 66 64 + 408654911428857533 z + 92263977 z - 2437325733 z 30 42 44 + 152362949491470106 z + 275573946858230351 z - 152362949491470106 z 46 58 56 + 68929134278896338 z + 6788318891945 z - 55742388326125 z 54 52 60 + 360013608483506 z - 1849918309512008 z - 640585137468 z 70 68 32 + 37564 z - 2361548 z - 275573946858230351 z 38 40 62 74 + 497489882819035990 z - 408654911428857533 z + 45931880950 z + z 72 - 319 z ) And in Maple-input format, it is: -(1+12503705132769509*z^28-5042460822102103*z^26-244*z^2+1651291233421857*z^24-\ 437113179337942*z^22+23835*z^4-1286469*z^6+44013975*z^8-1031696590*z^10+ 17416886793*z^12-219161348562*z^14-15758086469062*z^18+2106877186570*z^16-\ 437113179337942*z^50+1651291233421857*z^48+92962991242619*z^20+ 62172209379904932*z^36-56262832007372720*z^34-1286469*z^66+44013975*z^64-\ 25262278479455104*z^30-25262278479455104*z^42+12503705132769509*z^44-\ 5042460822102103*z^46-219161348562*z^58+2106877186570*z^56-15758086469062*z^54+ 92962991242619*z^52+17416886793*z^60-244*z^70+23835*z^68+41684474430589379*z^32 -56262832007372720*z^38+41684474430589379*z^40-1031696590*z^62+z^72)/(-1-\ 68929134278896338*z^28+25441425109103477*z^26+319*z^2-7630613507548625*z^24+ 1849918309512008*z^22-37564*z^4+2361548*z^6-92263977*z^8+2437325733*z^10-\ 45931880950*z^12+640585137468*z^14+55742388326125*z^18-6788318891945*z^16+ 7630613507548625*z^50-25441425109103477*z^48-360013608483506*z^20-\ 497489882819035990*z^36+408654911428857533*z^34+92263977*z^66-2437325733*z^64+ 152362949491470106*z^30+275573946858230351*z^42-152362949491470106*z^44+ 68929134278896338*z^46+6788318891945*z^58-55742388326125*z^56+360013608483506*z ^54-1849918309512008*z^52-640585137468*z^60+37564*z^70-2361548*z^68-\ 275573946858230351*z^32+497489882819035990*z^38-408654911428857533*z^40+ 45931880950*z^62+z^74-319*z^72) The first , 40, terms are: [0, 75, 0, 10196, 0, 1510303, 0, 227650211, 0, 34451569693, 0, 5218812161433, 0, 790750160672196, 0, 119821104247942553, 0, 18156578734287569815, 0, 2751290224022404940051, 0, 416907031809454304299229, 0, 63174548069713554143676404, 0, 9572934662678892756435426445, 0, 1450601267799355174865249248081, 0, 219811804929299091352686005665159, 0, 33308415419502307515479434521871763, 0, 5047274592025654470381043493597970404, 0, 764821156700005707948829007122580368383, 0, 115894507239589047178876657992482718372109, 0, 17561670059334459786088117577557595560896581] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 5}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 22443998950693770 z - 10203224304445538 z - 288 z 24 22 4 6 + 3678771983591720 z - 1046526669783584 z + 33671 z - 2144250 z 8 10 12 14 + 84455450 z - 2216949926 z + 40753406348 z - 543178388340 z 18 16 50 - 40426819530078 z + 5383394279471 z - 40426819530078 z 48 20 36 + 233333626381511 z + 233333626381511 z + 54958750282324266 z 34 66 64 30 - 61444171295226100 z - 288 z + 33671 z - 39307470282645106 z 42 44 46 - 10203224304445538 z + 3678771983591720 z - 1046526669783584 z 58 56 54 52 - 2216949926 z + 40753406348 z - 543178388340 z + 5383394279471 z 60 68 32 38 + 84455450 z + z + 54958750282324266 z - 39307470282645106 z 40 62 / 28 + 22443998950693770 z - 2144250 z ) / (-1 - 132448649215991474 z / 26 2 24 + 54415407618542340 z + 370 z - 17763226954618852 z 22 4 6 8 + 4581868312383867 z - 52524 z + 3927721 z - 177658320 z 10 12 14 + 5274495214 z - 108499802508 z + 1606501594024 z 18 16 50 + 145804499117986 z - 17605240277147 z + 927143704056106 z 48 20 36 - 4581868312383867 z - 927143704056106 z - 498099597802467430 z 34 66 64 + 498099597802467430 z + 52524 z - 3927721 z 30 42 44 + 257219321342681212 z + 132448649215991474 z - 54415407618542340 z 46 58 56 + 17763226954618852 z + 108499802508 z - 1606501594024 z 54 52 60 70 + 17605240277147 z - 145804499117986 z - 5274495214 z + z 68 32 38 - 370 z - 399743834935072104 z + 399743834935072104 z 40 62 - 257219321342681212 z + 177658320 z ) And in Maple-input format, it is: -(1+22443998950693770*z^28-10203224304445538*z^26-288*z^2+3678771983591720*z^24 -1046526669783584*z^22+33671*z^4-2144250*z^6+84455450*z^8-2216949926*z^10+ 40753406348*z^12-543178388340*z^14-40426819530078*z^18+5383394279471*z^16-\ 40426819530078*z^50+233333626381511*z^48+233333626381511*z^20+54958750282324266 *z^36-61444171295226100*z^34-288*z^66+33671*z^64-39307470282645106*z^30-\ 10203224304445538*z^42+3678771983591720*z^44-1046526669783584*z^46-2216949926*z ^58+40753406348*z^56-543178388340*z^54+5383394279471*z^52+84455450*z^60+z^68+ 54958750282324266*z^32-39307470282645106*z^38+22443998950693770*z^40-2144250*z^ 62)/(-1-132448649215991474*z^28+54415407618542340*z^26+370*z^2-\ 17763226954618852*z^24+4581868312383867*z^22-52524*z^4+3927721*z^6-177658320*z^ 8+5274495214*z^10-108499802508*z^12+1606501594024*z^14+145804499117986*z^18-\ 17605240277147*z^16+927143704056106*z^50-4581868312383867*z^48-927143704056106* z^20-498099597802467430*z^36+498099597802467430*z^34+52524*z^66-3927721*z^64+ 257219321342681212*z^30+132448649215991474*z^42-54415407618542340*z^44+ 17763226954618852*z^46+108499802508*z^58-1606501594024*z^56+17605240277147*z^54 -145804499117986*z^52-5274495214*z^60+z^70-370*z^68-399743834935072104*z^32+ 399743834935072104*z^38-257219321342681212*z^40+177658320*z^62) The first , 40, terms are: [0, 82, 0, 11487, 0, 1726693, 0, 264403474, 0, 40743756423, 0, 6293631254335, 0, 973114667432538, 0, 150522977693696021, 0, 23287151922222875967, 0, 3602982279539663711514, 0, 557470425880956362354537, 0, 86255665024038010135422905, 0, 13346152686583757097656801738, 0, 2065026934729696990804446871391, 0, 319518395648975697167343799350613, 0, 49438608620480463413942113647019530, 0, 7649564523797618123170851176081631151, 0, 1183606192894575421735778001693759022295, 0, 183137703861451486983294047353087867036642, 0, 28336637166656778922237201265159165102716133] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3150207199223704458 z - 745640005236248439 z - 303 z 24 22 4 6 + 146842857201674750 z - 23915203716775001 z + 38967 z - 2869992 z 8 10 12 14 + 137458659 z - 4602137935 z + 112934160704 z - 2100597684425 z 18 16 50 - 347509535660745 z + 30360590384248 z - 309563262380032798945 z 48 20 + 467381381277534167565 z + 3196907895900462 z 36 34 + 173652935828729212587 z - 82387632228156826630 z 66 80 88 84 86 - 23915203716775001 z + 137458659 z + z + 38967 z - 303 z 82 64 30 - 2869992 z + 146842857201674750 z - 11127882477177744663 z 42 44 - 598223707829811575162 z + 649483172610735219340 z 46 58 - 598223707829811575162 z - 11127882477177744663 z 56 54 + 32996208891373408441 z - 82387632228156826630 z 52 60 + 173652935828729212587 z + 3150207199223704458 z 70 68 78 - 347509535660745 z + 3196907895900462 z - 4602137935 z 32 38 + 32996208891373408441 z - 309563262380032798945 z 40 62 76 + 467381381277534167565 z - 745640005236248439 z + 112934160704 z 74 72 / - 2100597684425 z + 30360590384248 z ) / (-1 / 28 26 2 - 14119018683366171022 z + 3103044124027609299 z + 393 z 24 22 4 6 - 567104684906499687 z + 85627613576607280 z - 59844 z + 5021652 z 8 10 12 14 - 268829674 z + 9951912862 z - 268153483450 z + 5449155674030 z 18 16 50 + 1064366499629225 z - 85711251607761 z + 3291426019523891702891 z 48 20 - 4558663219497261073967 z - 10596918233876948 z 36 34 - 1048561860195216581148 z + 461376119363790463586 z 66 80 90 88 84 + 567104684906499687 z - 9951912862 z + z - 393 z - 5021652 z 86 82 64 + 59844 z + 268829674 z - 3103044124027609299 z 30 42 + 53704023652407269430 z + 4558663219497261073967 z 44 46 - 5363960079569078181540 z + 5363960079569078181540 z 58 56 + 171497184713092481502 z - 461376119363790463586 z 54 52 + 1048561860195216581148 z - 2017452392369157241624 z 60 70 68 - 53704023652407269430 z + 10596918233876948 z - 85627613576607280 z 78 32 + 268153483450 z - 171497184713092481502 z 38 40 + 2017452392369157241624 z - 3291426019523891702891 z 62 76 74 + 14119018683366171022 z - 5449155674030 z + 85711251607761 z 72 - 1064366499629225 z ) And in Maple-input format, it is: -(1+3150207199223704458*z^28-745640005236248439*z^26-303*z^2+146842857201674750 *z^24-23915203716775001*z^22+38967*z^4-2869992*z^6+137458659*z^8-4602137935*z^ 10+112934160704*z^12-2100597684425*z^14-347509535660745*z^18+30360590384248*z^ 16-309563262380032798945*z^50+467381381277534167565*z^48+3196907895900462*z^20+ 173652935828729212587*z^36-82387632228156826630*z^34-23915203716775001*z^66+ 137458659*z^80+z^88+38967*z^84-303*z^86-2869992*z^82+146842857201674750*z^64-\ 11127882477177744663*z^30-598223707829811575162*z^42+649483172610735219340*z^44 -598223707829811575162*z^46-11127882477177744663*z^58+32996208891373408441*z^56 -82387632228156826630*z^54+173652935828729212587*z^52+3150207199223704458*z^60-\ 347509535660745*z^70+3196907895900462*z^68-4602137935*z^78+32996208891373408441 *z^32-309563262380032798945*z^38+467381381277534167565*z^40-745640005236248439* z^62+112934160704*z^76-2100597684425*z^74+30360590384248*z^72)/(-1-\ 14119018683366171022*z^28+3103044124027609299*z^26+393*z^2-567104684906499687*z ^24+85627613576607280*z^22-59844*z^4+5021652*z^6-268829674*z^8+9951912862*z^10-\ 268153483450*z^12+5449155674030*z^14+1064366499629225*z^18-85711251607761*z^16+ 3291426019523891702891*z^50-4558663219497261073967*z^48-10596918233876948*z^20-\ 1048561860195216581148*z^36+461376119363790463586*z^34+567104684906499687*z^66-\ 9951912862*z^80+z^90-393*z^88-5021652*z^84+59844*z^86+268829674*z^82-\ 3103044124027609299*z^64+53704023652407269430*z^30+4558663219497261073967*z^42-\ 5363960079569078181540*z^44+5363960079569078181540*z^46+171497184713092481502*z ^58-461376119363790463586*z^56+1048561860195216581148*z^54-\ 2017452392369157241624*z^52-53704023652407269430*z^60+10596918233876948*z^70-\ 85627613576607280*z^68+268153483450*z^78-171497184713092481502*z^32+ 2017452392369157241624*z^38-3291426019523891702891*z^40+14119018683366171022*z^ 62-5449155674030*z^76+85711251607761*z^74-1064366499629225*z^72) The first , 40, terms are: [0, 90, 0, 14493, 0, 2461449, 0, 420608030, 0, 71929908537, 0, 12302431771021, 0, 2104166679939030, 0, 359890678665074345, 0, 61554708963498475609, 0, 10528148614970638255018, 0, 1800705698211585431398345, 0, 307987770766714244879469545, 0, 52677384814365223564030399306, 0, 9009795630268572545043791484473, 0, 1541010769372124495010926466526201, 0, 263570261611258854302322787086252614, 0, 45080335703440254301040435234504434445, 0, 7710417156745882057752290291176586856697, 0, 1318768633892594151446147446111704466226958, 0, 225558575416036878845692736590813274242453785] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 11921712235267399 z - 4808916661303187 z - 250 z 24 22 4 6 + 1577164604105965 z - 418863713558064 z + 24905 z - 1348449 z 8 10 12 14 + 45771067 z - 1058480092 z + 17595388505 z - 218102029104 z 18 16 50 - 15308195040008 z + 2069014796006 z - 418863713558064 z 48 20 36 + 1577164604105965 z + 89577028003899 z + 59397723723790620 z 34 66 64 - 53740022475789376 z - 1348449 z + 45771067 z 30 42 44 - 24097923245725490 z - 24097923245725490 z + 11921712235267399 z 46 58 56 - 4808916661303187 z - 218102029104 z + 2069014796006 z 54 52 60 70 - 15308195040008 z + 89577028003899 z + 17595388505 z - 250 z 68 32 38 + 24905 z + 39791851127253923 z - 53740022475789376 z 40 62 72 / + 39791851127253923 z - 1058480092 z + z ) / (-1 / 28 26 2 - 67424388880769524 z + 24666730767538947 z + 335 z 24 22 4 6 - 7342362049837929 z + 1771095068420636 z - 40744 z + 2552234 z 8 10 12 14 - 97629067 z + 2511524021 z - 46141361930 z + 630074357680 z 18 16 50 + 53493304609345 z - 6575242845453 z + 7342362049837929 z 48 20 36 - 24666730767538947 z - 344241869013174 z - 499169485366177310 z 34 66 64 + 408787064376244453 z + 97629067 z - 2511524021 z 30 42 44 + 150365455947125190 z + 274083949961609527 z - 150365455947125190 z 46 58 56 + 67424388880769524 z + 6575242845453 z - 53493304609345 z 54 52 60 + 344241869013174 z - 1771095068420636 z - 630074357680 z 70 68 32 + 40744 z - 2552234 z - 274083949961609527 z 38 40 62 74 + 499169485366177310 z - 408787064376244453 z + 46141361930 z + z 72 - 335 z ) And in Maple-input format, it is: -(1+11921712235267399*z^28-4808916661303187*z^26-250*z^2+1577164604105965*z^24-\ 418863713558064*z^22+24905*z^4-1348449*z^6+45771067*z^8-1058480092*z^10+ 17595388505*z^12-218102029104*z^14-15308195040008*z^18+2069014796006*z^16-\ 418863713558064*z^50+1577164604105965*z^48+89577028003899*z^20+ 59397723723790620*z^36-53740022475789376*z^34-1348449*z^66+45771067*z^64-\ 24097923245725490*z^30-24097923245725490*z^42+11921712235267399*z^44-\ 4808916661303187*z^46-218102029104*z^58+2069014796006*z^56-15308195040008*z^54+ 89577028003899*z^52+17595388505*z^60-250*z^70+24905*z^68+39791851127253923*z^32 -53740022475789376*z^38+39791851127253923*z^40-1058480092*z^62+z^72)/(-1-\ 67424388880769524*z^28+24666730767538947*z^26+335*z^2-7342362049837929*z^24+ 1771095068420636*z^22-40744*z^4+2552234*z^6-97629067*z^8+2511524021*z^10-\ 46141361930*z^12+630074357680*z^14+53493304609345*z^18-6575242845453*z^16+ 7342362049837929*z^50-24666730767538947*z^48-344241869013174*z^20-\ 499169485366177310*z^36+408787064376244453*z^34+97629067*z^66-2511524021*z^64+ 150365455947125190*z^30+274083949961609527*z^42-150365455947125190*z^44+ 67424388880769524*z^46+6575242845453*z^58-53493304609345*z^56+344241869013174*z ^54-1771095068420636*z^52-630074357680*z^60+40744*z^70-2552234*z^68-\ 274083949961609527*z^32+499169485366177310*z^38-408787064376244453*z^40+ 46141361930*z^62+z^74-335*z^72) The first , 40, terms are: [0, 85, 0, 12636, 0, 1973605, 0, 311398381, 0, 49310497573, 0, 7819795512809, 0, 1240830478527204, 0, 196941621201582577, 0, 31261316795320077097, 0, 4962442986822075407785, 0, 787755514788967822719473, 0, 125051971945168986780249572, 0, 19851391257604027629376904313, 0, 3151315581889812859359514420741, 0, 500256879170458467395122625389869, 0, 79413499625614572613213214350302501, 0, 12606532253185601103947575445418743388, 0, 2001229791336810375499720771079207037413, 0, 317686153349600130046370337852858177897377, 0, 50431236383627558879789145448583459561379425] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 237758772444396062292 z - 34109627235127847477 z - 335 z 24 22 4 6 + 4189151472104169388 z - 437656302265656119 z + 49248 z - 4293179 z 102 8 10 12 - 10696041957 z + 251999676 z - 10696041957 z + 343714606280 z 14 18 16 - 8635147308465 z - 2847280061768647 z + 173669301362720 z 50 48 - 44583019548898232053202127 z + 28576452830231002907830840 z 20 36 + 38599332422112656 z + 130276102061236197343088 z 34 66 - 33272437965017406741907 z - 16115938263872200159625837 z 80 100 90 + 7395674960014948715256 z + 343714606280 z - 437656302265656119 z 88 84 + 4189151472104169388 z + 237758772444396062292 z 94 86 96 - 2847280061768647 z - 34109627235127847477 z + 173669301362720 z 98 92 82 - 8635147308465 z + 38599332422112656 z - 1426256128929274979005 z 64 112 110 106 + 28576452830231002907830840 z + z - 335 z - 4293179 z 108 30 42 + 49248 z - 1426256128929274979005 z - 3480159652209090065184377 z 44 46 + 7990801262704742973080024 z - 16115938263872200159625837 z 58 56 - 74062569080467080891408531 z + 78909342347605383946549706 z 54 52 - 74062569080467080891408531 z + 61231937106200928778273648 z 60 70 + 61231937106200928778273648 z - 3480159652209090065184377 z 68 78 + 7990801262704742973080024 z - 33272437965017406741907 z 32 38 + 7395674960014948715256 z - 445086884825127247866171 z 40 62 + 1329730537847370246205236 z - 44583019548898232053202127 z 76 74 + 130276102061236197343088 z - 445086884825127247866171 z 72 104 / 2 + 1329730537847370246205236 z + 251999676 z ) / ((-1 + z ) (1 / 28 26 2 + 754921837217527586600 z - 103823404715185833164 z - 421 z 24 22 4 + 12193018830113216878 z - 1214944699453688390 z + 71908 z 6 102 8 10 - 7019176 z - 20782440030 z + 452001450 z - 20782440030 z 12 14 18 + 717289950068 z - 19235121461177 z - 7128981904502683 z 16 50 + 410950003008964 z - 189677365008847203910123575 z 48 20 + 120131391825187803989112268 z + 101916086480111820 z 36 34 + 477917658786481163319836 z - 118170887888771885975305 z 66 80 - 66725451576340340028701537 z + 25365863342828578659992 z 100 90 88 + 717289950068 z - 1214944699453688390 z + 12193018830113216878 z 84 94 + 754921837217527586600 z - 7128981904502683 z 86 96 98 - 103823404715185833164 z + 410950003008964 z - 19235121461177 z 92 82 + 101916086480111820 z - 4712441036323641836275 z 64 112 110 106 + 120131391825187803989112268 z + z - 421 z - 7019176 z 108 30 42 + 71908 z - 4712441036323641836275 z - 13847102051712764551282142 z 44 46 + 32482272006784890931055284 z - 66725451576340340028701537 z 58 56 - 319482649876289488108605130 z + 340982478322586106543446028 z 54 52 - 319482649876289488108605130 z + 262765053802643560208132468 z 60 70 + 262765053802643560208132468 z - 13847102051712764551282142 z 68 78 + 32482272006784890931055284 z - 118170887888771885975305 z 32 38 + 25365863342828578659992 z - 1682175638516303884451284 z 40 62 + 5163724990910707045683570 z - 189677365008847203910123575 z 76 74 + 477917658786481163319836 z - 1682175638516303884451284 z 72 104 + 5163724990910707045683570 z + 452001450 z )) And in Maple-input format, it is: -(1+237758772444396062292*z^28-34109627235127847477*z^26-335*z^2+ 4189151472104169388*z^24-437656302265656119*z^22+49248*z^4-4293179*z^6-\ 10696041957*z^102+251999676*z^8-10696041957*z^10+343714606280*z^12-\ 8635147308465*z^14-2847280061768647*z^18+173669301362720*z^16-\ 44583019548898232053202127*z^50+28576452830231002907830840*z^48+ 38599332422112656*z^20+130276102061236197343088*z^36-33272437965017406741907*z^ 34-16115938263872200159625837*z^66+7395674960014948715256*z^80+343714606280*z^ 100-437656302265656119*z^90+4189151472104169388*z^88+237758772444396062292*z^84 -2847280061768647*z^94-34109627235127847477*z^86+173669301362720*z^96-\ 8635147308465*z^98+38599332422112656*z^92-1426256128929274979005*z^82+ 28576452830231002907830840*z^64+z^112-335*z^110-4293179*z^106+49248*z^108-\ 1426256128929274979005*z^30-3480159652209090065184377*z^42+ 7990801262704742973080024*z^44-16115938263872200159625837*z^46-\ 74062569080467080891408531*z^58+78909342347605383946549706*z^56-\ 74062569080467080891408531*z^54+61231937106200928778273648*z^52+ 61231937106200928778273648*z^60-3480159652209090065184377*z^70+ 7990801262704742973080024*z^68-33272437965017406741907*z^78+ 7395674960014948715256*z^32-445086884825127247866171*z^38+ 1329730537847370246205236*z^40-44583019548898232053202127*z^62+ 130276102061236197343088*z^76-445086884825127247866171*z^74+ 1329730537847370246205236*z^72+251999676*z^104)/(-1+z^2)/(1+ 754921837217527586600*z^28-103823404715185833164*z^26-421*z^2+ 12193018830113216878*z^24-1214944699453688390*z^22+71908*z^4-7019176*z^6-\ 20782440030*z^102+452001450*z^8-20782440030*z^10+717289950068*z^12-\ 19235121461177*z^14-7128981904502683*z^18+410950003008964*z^16-\ 189677365008847203910123575*z^50+120131391825187803989112268*z^48+ 101916086480111820*z^20+477917658786481163319836*z^36-118170887888771885975305* z^34-66725451576340340028701537*z^66+25365863342828578659992*z^80+717289950068* z^100-1214944699453688390*z^90+12193018830113216878*z^88+754921837217527586600* z^84-7128981904502683*z^94-103823404715185833164*z^86+410950003008964*z^96-\ 19235121461177*z^98+101916086480111820*z^92-4712441036323641836275*z^82+ 120131391825187803989112268*z^64+z^112-421*z^110-7019176*z^106+71908*z^108-\ 4712441036323641836275*z^30-13847102051712764551282142*z^42+ 32482272006784890931055284*z^44-66725451576340340028701537*z^46-\ 319482649876289488108605130*z^58+340982478322586106543446028*z^56-\ 319482649876289488108605130*z^54+262765053802643560208132468*z^52+ 262765053802643560208132468*z^60-13847102051712764551282142*z^70+ 32482272006784890931055284*z^68-118170887888771885975305*z^78+ 25365863342828578659992*z^32-1682175638516303884451284*z^38+ 5163724990910707045683570*z^40-189677365008847203910123575*z^62+ 477917658786481163319836*z^76-1682175638516303884451284*z^74+ 5163724990910707045683570*z^72+452001450*z^104) The first , 40, terms are: [0, 87, 0, 13633, 0, 2258408, 0, 376890277, 0, 62975657895, 0, 10525402051513, 0, 1759252171778985, 0, 294051116303131371, 0, 49149468630392890113, 0, 8215142974390968041832, 0, 1373129511530741101292229, 0, 229513320068258797662578867, 0, 38362269728199486630919406897, 0, 6412106028179210502392366493521, 0, 1071758893343264568017113738622059, 0, 179140382374576206532881227778974461, 0, 29942626833872409519128449846751890408, 0, 5004795065430870536272212778501748455305, 0, 836532271735896876193866555390983899540851, 0, 139823156094864914560719921374377839309279561] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 250758837378136745061 z - 35552439388572236428 z - 329 z 24 22 4 6 + 4314442329048367295 z - 445390345526254345 z + 47739 z - 4131732 z 102 8 10 12 - 10294148769 z + 242024759 z - 10294148769 z + 332548697150 z 14 18 16 - 8418679044705 z - 2831591328406779 z + 170904670579818 z 50 48 - 51495461715281669970503514 z + 32877983484678681029787841 z 20 36 + 38822651904357112 z + 143480550943029526889987 z 34 66 - 36278784204600719849073 z - 18449860790117212347012049 z 80 100 90 + 7978683611935818408644 z + 332548697150 z - 445390345526254345 z 88 84 + 4314442329048367295 z + 250758837378136745061 z 94 86 96 - 2831591328406779 z - 35552439388572236428 z + 170904670579818 z 98 92 82 - 8418679044705 z + 38822651904357112 z - 1521670129797225287599 z 64 112 110 106 + 32877983484678681029787841 z + z - 329 z - 4131732 z 108 30 42 + 47739 z - 1521670129797225287599 z - 3933098583282774404175107 z 44 46 + 9093533619198026589523088 z - 18449860790117212347012049 z 58 56 - 85934903294424714114003971 z + 91610997187222486529676410 z 54 52 - 85934903294424714114003971 z + 70926141279464580705294071 z 60 70 + 70926141279464580705294071 z - 3933098583282774404175107 z 68 78 + 9093533619198026589523088 z - 36278784204600719849073 z 32 38 + 7978683611935818408644 z - 494815190751575039858240 z 40 62 + 1491104705081190616961853 z - 51495461715281669970503514 z 76 74 + 143480550943029526889987 z - 494815190751575039858240 z 72 104 / 2 + 1491104705081190616961853 z + 242024759 z ) / ((-1 + z ) (1 / 28 26 2 + 788122772968443984188 z - 107095924828009539292 z - 408 z 24 22 4 + 12423056888320395392 z - 1222436664966061388 z + 68424 z 6 102 8 10 - 6620596 z - 19628328392 z + 425549962 z - 19628328392 z 12 14 18 + 682016372792 z - 18457240519720 z - 6996587365146244 z 16 50 + 398597914656273 z - 216426634576941439481171984 z 48 20 + 136554452499054064618436510 z + 101264570300996844 z 36 34 + 520788425956149078588612 z - 127514820621743403786824 z 66 80 - 75484180647237096751421904 z + 27086951948008633960112 z 100 90 88 + 682016372792 z - 1222436664966061388 z + 12423056888320395392 z 84 94 + 788122772968443984188 z - 6996587365146244 z 86 96 98 - 107095924828009539292 z + 398597914656273 z - 18457240519720 z 92 82 + 101264570300996844 z - 4976919209008054630516 z 64 112 110 106 + 136554452499054064618436510 z + z - 408 z - 6620596 z 108 30 42 + 68424 z - 4976919209008054630516 z - 15471030841628708709555408 z 44 46 + 36534833241321457362488612 z - 75484180647237096751421904 z 58 56 - 366143250066886587363061576 z + 390999808407847535421498540 z 54 52 - 366143250066886587363061576 z + 300643690406050028051503424 z 60 70 + 300643690406050028051503424 z - 15471030841628708709555408 z 68 78 + 36534833241321457362488612 z - 127514820621743403786824 z 32 38 + 27086951948008633960112 z - 1849833070871108880658276 z 40 62 + 5725972893034307714714864 z - 216426634576941439481171984 z 76 74 + 520788425956149078588612 z - 1849833070871108880658276 z 72 104 + 5725972893034307714714864 z + 425549962 z )) And in Maple-input format, it is: -(1+250758837378136745061*z^28-35552439388572236428*z^26-329*z^2+ 4314442329048367295*z^24-445390345526254345*z^22+47739*z^4-4131732*z^6-\ 10294148769*z^102+242024759*z^8-10294148769*z^10+332548697150*z^12-\ 8418679044705*z^14-2831591328406779*z^18+170904670579818*z^16-\ 51495461715281669970503514*z^50+32877983484678681029787841*z^48+ 38822651904357112*z^20+143480550943029526889987*z^36-36278784204600719849073*z^ 34-18449860790117212347012049*z^66+7978683611935818408644*z^80+332548697150*z^ 100-445390345526254345*z^90+4314442329048367295*z^88+250758837378136745061*z^84 -2831591328406779*z^94-35552439388572236428*z^86+170904670579818*z^96-\ 8418679044705*z^98+38822651904357112*z^92-1521670129797225287599*z^82+ 32877983484678681029787841*z^64+z^112-329*z^110-4131732*z^106+47739*z^108-\ 1521670129797225287599*z^30-3933098583282774404175107*z^42+ 9093533619198026589523088*z^44-18449860790117212347012049*z^46-\ 85934903294424714114003971*z^58+91610997187222486529676410*z^56-\ 85934903294424714114003971*z^54+70926141279464580705294071*z^52+ 70926141279464580705294071*z^60-3933098583282774404175107*z^70+ 9093533619198026589523088*z^68-36278784204600719849073*z^78+ 7978683611935818408644*z^32-494815190751575039858240*z^38+ 1491104705081190616961853*z^40-51495461715281669970503514*z^62+ 143480550943029526889987*z^76-494815190751575039858240*z^74+ 1491104705081190616961853*z^72+242024759*z^104)/(-1+z^2)/(1+ 788122772968443984188*z^28-107095924828009539292*z^26-408*z^2+ 12423056888320395392*z^24-1222436664966061388*z^22+68424*z^4-6620596*z^6-\ 19628328392*z^102+425549962*z^8-19628328392*z^10+682016372792*z^12-\ 18457240519720*z^14-6996587365146244*z^18+398597914656273*z^16-\ 216426634576941439481171984*z^50+136554452499054064618436510*z^48+ 101264570300996844*z^20+520788425956149078588612*z^36-127514820621743403786824* z^34-75484180647237096751421904*z^66+27086951948008633960112*z^80+682016372792* z^100-1222436664966061388*z^90+12423056888320395392*z^88+788122772968443984188* z^84-6996587365146244*z^94-107095924828009539292*z^86+398597914656273*z^96-\ 18457240519720*z^98+101264570300996844*z^92-4976919209008054630516*z^82+ 136554452499054064618436510*z^64+z^112-408*z^110-6620596*z^106+68424*z^108-\ 4976919209008054630516*z^30-15471030841628708709555408*z^42+ 36534833241321457362488612*z^44-75484180647237096751421904*z^46-\ 366143250066886587363061576*z^58+390999808407847535421498540*z^56-\ 366143250066886587363061576*z^54+300643690406050028051503424*z^52+ 300643690406050028051503424*z^60-15471030841628708709555408*z^70+ 36534833241321457362488612*z^68-127514820621743403786824*z^78+ 27086951948008633960112*z^32-1849833070871108880658276*z^38+ 5725972893034307714714864*z^40-216426634576941439481171984*z^62+ 520788425956149078588612*z^76-1849833070871108880658276*z^74+ 5725972893034307714714864*z^72+425549962*z^104) The first , 40, terms are: [0, 80, 0, 11627, 0, 1806171, 0, 283390076, 0, 44543499297, 0, 7003866630081, 0, 1101346710719172, 0, 173187867840103439, 0, 27234067294546734103, 0, 4282604047512872774600, 0, 673447007560912570414649, 0, 105900729644407563950247225, 0, 16653076690661247101590428456, 0, 2618725715359353007753183904575, 0, 411799243104142382445454056235127, 0, 64756158177143594706467971861191028, 0, 10183020227046705327225631074756043889, 0, 1601297913036502330188032761455677862577, 0, 251806924579185016857541242917673047821852, 0, 39597083559436873474517373208750776607187539] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 201791645773448283832 z - 29178554526214896642 z - 338 z 24 22 4 6 + 3618374538211692332 z - 382421925189514642 z + 49666 z - 4292466 z 102 8 10 12 - 10357849152 z + 248377355 z - 10357849152 z + 326542279730 z 14 18 16 - 8047332864176 z - 2560800647681798 z + 158887218465153 z 50 48 - 37249115838092946572789030 z + 23858076490731771822609702 z 20 36 + 34186051182664392 z + 108754255747460099095528 z 34 66 - 27838115224127690884518 z - 13444622276535952600798578 z 80 100 90 + 6208716117627023531748 z + 326542279730 z - 382421925189514642 z 88 84 + 3618374538211692332 z + 201791645773448283832 z 94 86 96 - 2560800647681798 z - 29178554526214896642 z + 158887218465153 z 98 92 82 - 8047332864176 z + 34186051182664392 z - 1203002101920609137938 z 64 112 110 106 + 23858076490731771822609702 z + z - 338 z - 4292466 z 108 30 42 + 49666 z - 1203002101920609137938 z - 2900053764075036467352790 z 44 46 + 6661757986650121124014376 z - 13444622276535952600798578 z 58 56 - 61940629546646200735935094 z + 66003055695132386200893826 z 54 52 - 61940629546646200735935094 z + 51190007111269456917938700 z 60 70 + 51190007111269456917938700 z - 2900053764075036467352790 z 68 78 + 6661757986650121124014376 z - 27838115224127690884518 z 32 38 + 6208716117627023531748 z - 371077764543175207403126 z 40 62 + 1108032018316651714714124 z - 37249115838092946572789030 z 76 74 + 108754255747460099095528 z - 371077764543175207403126 z 72 104 / + 1108032018316651714714124 z + 248377355 z ) / (-1 / 28 26 2 - 733167656380312558159 z + 99924375690442480085 z + 422 z 24 22 4 - 11672593721522924531 z + 1161099139163105373 z - 72853 z 6 102 8 10 + 7132013 z + 710801354068 z - 457184960 z + 20828165965 z 12 14 18 - 710801354068 z + 18845032950938 z + 6859679008997274 z 16 50 - 398560457850697 z + 259676404937336870946003181 z 48 20 - 156488154662012567151583281 z - 97573181148942859 z 36 34 - 499428672311756046137775 z + 120577894990490079465833 z 66 80 + 156488154662012567151583281 z - 120577894990490079465833 z 100 90 - 18845032950938 z + 11672593721522924531 z 88 84 - 99924375690442480085 z - 4635399600150829937997 z 94 86 96 + 97573181148942859 z + 733167656380312558159 z - 6859679008997274 z 98 92 + 398560457850697 z - 1161099139163105373 z 82 64 112 + 25365580196425079719971 z - 259676404937336870946003181 z - 422 z 114 110 106 108 + z + 72853 z + 457184960 z - 7132013 z 30 42 + 4635399600150829937997 z + 15890340715188546722975345 z 44 46 - 38740230442773566317812647 z + 83014490922253251895428801 z 58 56 + 554455612253523585243263319 z - 554455612253523585243263319 z 54 52 + 488653223902028680590845907 z - 379508605277276463176075789 z 60 70 - 488653223902028680590845907 z + 38740230442773566317812647 z 68 78 - 83014490922253251895428801 z + 499428672311756046137775 z 32 38 - 25365580196425079719971 z + 1806967596107240505238725 z 40 62 - 5722598161224121271351231 z + 379508605277276463176075789 z 76 74 - 1806967596107240505238725 z + 5722598161224121271351231 z 72 104 - 15890340715188546722975345 z - 20828165965 z ) And in Maple-input format, it is: -(1+201791645773448283832*z^28-29178554526214896642*z^26-338*z^2+ 3618374538211692332*z^24-382421925189514642*z^22+49666*z^4-4292466*z^6-\ 10357849152*z^102+248377355*z^8-10357849152*z^10+326542279730*z^12-\ 8047332864176*z^14-2560800647681798*z^18+158887218465153*z^16-\ 37249115838092946572789030*z^50+23858076490731771822609702*z^48+ 34186051182664392*z^20+108754255747460099095528*z^36-27838115224127690884518*z^ 34-13444622276535952600798578*z^66+6208716117627023531748*z^80+326542279730*z^ 100-382421925189514642*z^90+3618374538211692332*z^88+201791645773448283832*z^84 -2560800647681798*z^94-29178554526214896642*z^86+158887218465153*z^96-\ 8047332864176*z^98+34186051182664392*z^92-1203002101920609137938*z^82+ 23858076490731771822609702*z^64+z^112-338*z^110-4292466*z^106+49666*z^108-\ 1203002101920609137938*z^30-2900053764075036467352790*z^42+ 6661757986650121124014376*z^44-13444622276535952600798578*z^46-\ 61940629546646200735935094*z^58+66003055695132386200893826*z^56-\ 61940629546646200735935094*z^54+51190007111269456917938700*z^52+ 51190007111269456917938700*z^60-2900053764075036467352790*z^70+ 6661757986650121124014376*z^68-27838115224127690884518*z^78+ 6208716117627023531748*z^32-371077764543175207403126*z^38+ 1108032018316651714714124*z^40-37249115838092946572789030*z^62+ 108754255747460099095528*z^76-371077764543175207403126*z^74+ 1108032018316651714714124*z^72+248377355*z^104)/(-1-733167656380312558159*z^28+ 99924375690442480085*z^26+422*z^2-11672593721522924531*z^24+1161099139163105373 *z^22-72853*z^4+7132013*z^6+710801354068*z^102-457184960*z^8+20828165965*z^10-\ 710801354068*z^12+18845032950938*z^14+6859679008997274*z^18-398560457850697*z^ 16+259676404937336870946003181*z^50-156488154662012567151583281*z^48-\ 97573181148942859*z^20-499428672311756046137775*z^36+120577894990490079465833*z ^34+156488154662012567151583281*z^66-120577894990490079465833*z^80-\ 18845032950938*z^100+11672593721522924531*z^90-99924375690442480085*z^88-\ 4635399600150829937997*z^84+97573181148942859*z^94+733167656380312558159*z^86-\ 6859679008997274*z^96+398560457850697*z^98-1161099139163105373*z^92+ 25365580196425079719971*z^82-259676404937336870946003181*z^64-422*z^112+z^114+ 72853*z^110+457184960*z^106-7132013*z^108+4635399600150829937997*z^30+ 15890340715188546722975345*z^42-38740230442773566317812647*z^44+ 83014490922253251895428801*z^46+554455612253523585243263319*z^58-\ 554455612253523585243263319*z^56+488653223902028680590845907*z^54-\ 379508605277276463176075789*z^52-488653223902028680590845907*z^60+ 38740230442773566317812647*z^70-83014490922253251895428801*z^68+ 499428672311756046137775*z^78-25365580196425079719971*z^32+ 1806967596107240505238725*z^38-5722598161224121271351231*z^40+ 379508605277276463176075789*z^62-1806967596107240505238725*z^76+ 5722598161224121271351231*z^74-15890340715188546722975345*z^72-20828165965*z^ 104) The first , 40, terms are: [0, 84, 0, 12261, 0, 1894037, 0, 296314468, 0, 46570819501, 0, 7333546470861, 0, 1155790634619468, 0, 182223449775679501, 0, 28734232818950612301, 0, 4531331242952073364172, 0, 714604216774369199190145, 0, 112696725263261394180337313, 0, 17772954792200177261460950380, 0, 2802909826116199134901987546685, 0, 442037507384735600969539158808413, 0, 69712288285365708456498121232631980, 0, 10994099252896679770954678935461002349, 0, 1733843964751825529897938157098725060493, 0, 273438956236901044325695150219963255363364, 0, 43123179324446694253028164925968170648013573] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 238308579855001787033 z - 34222888641383953652 z - 335 z 24 22 4 6 + 4206268416256429371 z - 439658770364362603 z + 49231 z - 4292720 z 102 8 10 12 - 10713237851 z + 252157467 z - 10713237851 z + 344617252094 z 14 18 16 - 8665639934623 z - 2860507723153373 z + 174402929322538 z 50 48 - 44040879285501371462677498 z + 28252303983711843599337585 z 20 36 + 38783270316783164 z + 129849125389822650862195 z 34 66 - 33214537258692404641423 z - 15949517102478678753757807 z 80 100 90 + 7393720930855762850212 z + 344617252094 z - 439658770364362603 z 88 84 + 4206268416256429371 z + 238308579855001787033 z 94 86 96 - 2860507723153373 z - 34222888641383953652 z + 174402929322538 z 98 92 82 - 8665639934623 z + 38783270316783164 z - 1427828575757583332765 z 64 112 110 106 + 28252303983711843599337585 z + z - 335 z - 4292720 z 108 30 42 + 49231 z - 1427828575757583332765 z - 3453021890921274206798369 z 44 46 + 7917806692879905460803324 z - 15949517102478678753757807 z 58 56 - 73090691264616946486323969 z + 77864203724304106874658770 z 54 52 - 73090691264616946486323969 z + 60450739162000115346583151 z 60 70 + 60450739162000115346583151 z - 3453021890921274206798369 z 68 78 + 7917806692879905460803324 z - 33214537258692404641423 z 32 38 + 7393720930855762850212 z - 442937564649424627097684 z 40 62 + 1321287038547960677793921 z - 44040879285501371462677498 z 76 74 + 129849125389822650862195 z - 442937564649424627097684 z 72 104 / 2 + 1321287038547960677793921 z + 252157467 z ) / ((-1 + z ) (1 / 28 26 2 + 760226219173670330924 z - 104725348157633625676 z - 418 z 24 22 4 + 12311814371262787156 z - 1227239859468458174 z + 71288 z 6 102 8 10 - 6967104 z - 20746689818 z + 449805950 z - 20746689818 z 12 14 18 + 718356951876 z - 19320273062686 z - 7190852201261950 z 16 50 + 413773869467165 z - 185516120741356362024256416 z 48 20 + 117699176589895622183493538 z + 102911833356660152 z 36 34 + 476158434167550358070388 z - 118098369259169519727702 z 66 80 - 65515027886162239761232842 z + 25423402695695539070352 z 100 90 88 + 718356951876 z - 1227239859468458174 z + 12311814371262787156 z 84 94 + 760226219173670330924 z - 7190852201261950 z 86 96 98 - 104725348157633625676 z + 413773869467165 z - 19320273062686 z 92 82 + 102911833356660152 z - 4735306080346899529218 z 64 112 110 106 + 117699176589895622183493538 z + z - 418 z - 6967104 z 108 30 42 + 71288 z - 4735306080346899529218 z - 13668307120554331480981274 z 44 46 + 31973149544633514614800616 z - 65515027886162239761232842 z 58 56 - 311837357849118608192903374 z + 332736333804066028058067460 z 54 52 - 311837357849118608192903374 z + 256675166911674289046423172 z 60 70 + 256675166911674289046423172 z - 13668307120554331480981274 z 68 78 + 31973149544633514614800616 z - 118098369259169519727702 z 32 38 + 25423402695695539070352 z - 1670698992668440262137268 z 40 62 + 5112437798543539524384348 z - 185516120741356362024256416 z 76 74 + 476158434167550358070388 z - 1670698992668440262137268 z 72 104 + 5112437798543539524384348 z + 449805950 z )) And in Maple-input format, it is: -(1+238308579855001787033*z^28-34222888641383953652*z^26-335*z^2+ 4206268416256429371*z^24-439658770364362603*z^22+49231*z^4-4292720*z^6-\ 10713237851*z^102+252157467*z^8-10713237851*z^10+344617252094*z^12-\ 8665639934623*z^14-2860507723153373*z^18+174402929322538*z^16-\ 44040879285501371462677498*z^50+28252303983711843599337585*z^48+ 38783270316783164*z^20+129849125389822650862195*z^36-33214537258692404641423*z^ 34-15949517102478678753757807*z^66+7393720930855762850212*z^80+344617252094*z^ 100-439658770364362603*z^90+4206268416256429371*z^88+238308579855001787033*z^84 -2860507723153373*z^94-34222888641383953652*z^86+174402929322538*z^96-\ 8665639934623*z^98+38783270316783164*z^92-1427828575757583332765*z^82+ 28252303983711843599337585*z^64+z^112-335*z^110-4292720*z^106+49231*z^108-\ 1427828575757583332765*z^30-3453021890921274206798369*z^42+ 7917806692879905460803324*z^44-15949517102478678753757807*z^46-\ 73090691264616946486323969*z^58+77864203724304106874658770*z^56-\ 73090691264616946486323969*z^54+60450739162000115346583151*z^52+ 60450739162000115346583151*z^60-3453021890921274206798369*z^70+ 7917806692879905460803324*z^68-33214537258692404641423*z^78+ 7393720930855762850212*z^32-442937564649424627097684*z^38+ 1321287038547960677793921*z^40-44040879285501371462677498*z^62+ 129849125389822650862195*z^76-442937564649424627097684*z^74+ 1321287038547960677793921*z^72+252157467*z^104)/(-1+z^2)/(1+ 760226219173670330924*z^28-104725348157633625676*z^26-418*z^2+ 12311814371262787156*z^24-1227239859468458174*z^22+71288*z^4-6967104*z^6-\ 20746689818*z^102+449805950*z^8-20746689818*z^10+718356951876*z^12-\ 19320273062686*z^14-7190852201261950*z^18+413773869467165*z^16-\ 185516120741356362024256416*z^50+117699176589895622183493538*z^48+ 102911833356660152*z^20+476158434167550358070388*z^36-118098369259169519727702* z^34-65515027886162239761232842*z^66+25423402695695539070352*z^80+718356951876* z^100-1227239859468458174*z^90+12311814371262787156*z^88+760226219173670330924* z^84-7190852201261950*z^94-104725348157633625676*z^86+413773869467165*z^96-\ 19320273062686*z^98+102911833356660152*z^92-4735306080346899529218*z^82+ 117699176589895622183493538*z^64+z^112-418*z^110-6967104*z^106+71288*z^108-\ 4735306080346899529218*z^30-13668307120554331480981274*z^42+ 31973149544633514614800616*z^44-65515027886162239761232842*z^46-\ 311837357849118608192903374*z^58+332736333804066028058067460*z^56-\ 311837357849118608192903374*z^54+256675166911674289046423172*z^52+ 256675166911674289046423172*z^60-13668307120554331480981274*z^70+ 31973149544633514614800616*z^68-118098369259169519727702*z^78+ 25423402695695539070352*z^32-1670698992668440262137268*z^38+ 5112437798543539524384348*z^40-185516120741356362024256416*z^62+ 476158434167550358070388*z^76-1670698992668440262137268*z^74+ 5112437798543539524384348*z^72+449805950*z^104) The first , 40, terms are: [0, 84, 0, 12721, 0, 2052467, 0, 334420988, 0, 54597901283, 0, 8918005820091, 0, 1456850622808780, 0, 238000345579519335, 0, 38881627593253659221, 0, 6352029315002625525212, 0, 1037721687191434198745397, 0, 169531103650343633368162945, 0, 27696054997785546787961028604, 0, 4524665147929582367231631789641, 0, 739188119262621864619015237938515, 0, 120760113418614328368328800973074012, 0, 19728408261448218242018171960116337599, 0, 3223002045602512761130078939590118408655, 0, 526537267915195562199524226965265167436812, 0, 86019645840574425340825747505161876495749527] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 15581398687683932 z - 6239236215161348 z - 250 z 24 22 4 6 + 2026595103332319 z - 531627207986594 z + 25070 z - 1388252 z 8 10 12 14 + 48610746 z - 1162510514 z + 19962674506 z - 254892586554 z 18 16 50 - 18778343684092 z + 2481941478514 z - 531627207986594 z 48 20 36 + 2026595103332319 z + 111959363314806 z + 78540856797657900 z 34 66 64 - 71011971637359284 z - 1388252 z + 48610746 z 30 42 44 - 31663506666890792 z - 31663506666890792 z + 15581398687683932 z 46 58 56 - 6239236215161348 z - 254892586554 z + 2481941478514 z 54 52 60 70 - 18778343684092 z + 111959363314806 z + 19962674506 z - 250 z 68 32 38 + 25070 z + 52472492679363732 z - 71011971637359284 z 40 62 72 / 2 + 52472492679363732 z - 1162510514 z + z ) / ((-1 + z ) (1 / 28 26 2 + 62720200266937948 z - 24146444332304402 z - 329 z 24 22 4 6 + 7489374424454007 z - 1864531813819093 z + 39746 z - 2537191 z 8 10 12 14 + 99827072 z - 2639479169 z + 49559569256 z - 686238756877 z 18 16 50 - 58337460235883 z + 7198532040568 z - 1864531813819093 z 48 20 36 + 7489374424454007 z + 370555448777778 z + 340527471206897696 z 34 66 64 - 306400346562733218 z - 2537191 z + 99827072 z 30 42 44 - 131565599118316094 z - 131565599118316094 z + 62720200266937948 z 46 58 56 - 24146444332304402 z - 686238756877 z + 7198532040568 z 54 52 60 70 - 58337460235883 z + 370555448777778 z + 49559569256 z - 329 z 68 32 38 + 39746 z + 223184836912454872 z - 306400346562733218 z 40 62 72 + 223184836912454872 z - 2639479169 z + z )) And in Maple-input format, it is: -(1+15581398687683932*z^28-6239236215161348*z^26-250*z^2+2026595103332319*z^24-\ 531627207986594*z^22+25070*z^4-1388252*z^6+48610746*z^8-1162510514*z^10+ 19962674506*z^12-254892586554*z^14-18778343684092*z^18+2481941478514*z^16-\ 531627207986594*z^50+2026595103332319*z^48+111959363314806*z^20+ 78540856797657900*z^36-71011971637359284*z^34-1388252*z^66+48610746*z^64-\ 31663506666890792*z^30-31663506666890792*z^42+15581398687683932*z^44-\ 6239236215161348*z^46-254892586554*z^58+2481941478514*z^56-18778343684092*z^54+ 111959363314806*z^52+19962674506*z^60-250*z^70+25070*z^68+52472492679363732*z^ 32-71011971637359284*z^38+52472492679363732*z^40-1162510514*z^62+z^72)/(-1+z^2) /(1+62720200266937948*z^28-24146444332304402*z^26-329*z^2+7489374424454007*z^24 -1864531813819093*z^22+39746*z^4-2537191*z^6+99827072*z^8-2639479169*z^10+ 49559569256*z^12-686238756877*z^14-58337460235883*z^18+7198532040568*z^16-\ 1864531813819093*z^50+7489374424454007*z^48+370555448777778*z^20+ 340527471206897696*z^36-306400346562733218*z^34-2537191*z^66+99827072*z^64-\ 131565599118316094*z^30-131565599118316094*z^42+62720200266937948*z^44-\ 24146444332304402*z^46-686238756877*z^58+7198532040568*z^56-58337460235883*z^54 +370555448777778*z^52+49559569256*z^60-329*z^70+39746*z^68+223184836912454872*z ^32-306400346562733218*z^38+223184836912454872*z^40-2639479169*z^62+z^72) The first , 40, terms are: [0, 80, 0, 11395, 0, 1743035, 0, 270948368, 0, 42312685617, 0, 6617089138281, 0, 1035270297443840, 0, 161994405608048739, 0, 25349237819584493003, 0, 3966757315379129622144, 0, 620737790803091184595225, 0, 97136245853890093666528809, 0, 15200386227582094364264858080, 0, 2378636001886879236129152389307, 0, 372221427483581373317008623937811, 0, 58247160447400988267607036192727264, 0, 9114821080333058502193422001979994553, 0, 1426334997152572464570546609635234586209, 0, 223200379576707057179708558154379006055344, 0, 34927565788400817759910157291214836805821707] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 239810177757867244657 z - 33868899752999703444 z - 329 z 24 22 4 6 + 4100103750186446355 z - 422887306114190529 z + 47559 z - 4091200 z 102 8 10 12 - 10040162033 z + 237895623 z - 10040162033 z + 321918833142 z 14 18 16 - 8095221464653 z - 2697034297855347 z + 163439150188130 z 50 48 - 53057853430244043786789990 z + 33714096165858591646291661 z 20 36 + 36888771808673356 z + 140817583505817472280995 z 34 66 - 35335818309703200216761 z - 18808505205004616141036573 z 80 100 90 + 7717018835147973732032 z + 321918833142 z - 422887306114190529 z 88 84 + 4100103750186446355 z + 239810177757867244657 z 94 86 96 - 2697034297855347 z - 33868899752999703444 z + 163439150188130 z 98 92 82 - 8095221464653 z + 36888771808673356 z - 1462703360566208438091 z 64 112 110 106 + 33714096165858591646291661 z + z - 329 z - 4091200 z 108 30 42 + 47559 z - 1462703360566208438091 z - 3953137282846573716658375 z 44 46 + 9207932783003797526073148 z - 18808505205004616141036573 z 58 56 - 89048445888128388777586675 z + 94999640860638412306613858 z 54 52 - 89048445888128388777586675 z + 73336665912188659792397127 z 60 70 + 73336665912188659792397127 z - 3953137282846573716658375 z 68 78 + 9207932783003797526073148 z - 35335818309703200216761 z 32 38 + 7717018835147973732032 z - 489503980947553342395952 z 40 62 + 1486984762290802333306213 z - 53057853430244043786789990 z 76 74 + 140817583505817472280995 z - 489503980947553342395952 z 72 104 / + 1486984762290802333306213 z + 237895623 z ) / (-1 / 28 26 2 - 850407598475070148420 z + 113072899504857709768 z + 413 z 24 22 4 - 12882116143504506232 z + 1249658050449306668 z - 69504 z 6 102 8 10 + 6716644 z + 683784085160 z - 430131738 z + 19752680278 z 12 14 18 - 683784085160 z + 18467916910316 z + 7028675777750949 z 16 50 - 399020220066289 z + 368317479505096056322115842 z 48 20 - 219796519772305731968302486 z - 102434906215476840 z 36 34 - 635228040950364334979624 z + 150072919261539655098308 z 66 80 + 219796519772305731968302486 z - 150072919261539655098308 z 100 90 - 18467916910316 z + 12882116143504506232 z 88 84 - 113072899504857709768 z - 5508484311514520283708 z 94 86 + 102434906215476840 z + 850407598475070148420 z 96 98 92 - 7028675777750949 z + 399020220066289 z - 1249658050449306668 z 82 64 112 + 30861565143924530232624 z - 368317479505096056322115842 z - 413 z 114 110 106 108 + z + 69504 z + 430131738 z - 6716644 z 30 42 + 5508484311514520283708 z + 21402622916025269855686524 z 44 46 - 53015826026038949584535608 z + 115209808063043098840714292 z 58 56 + 798336777282412034816541404 z - 798336777282412034816541404 z 54 52 + 701808506573615112037894880 z - 542316186557623941688456260 z 60 70 - 701808506573615112037894880 z + 53015826026038949584535608 z 68 78 - 115209808063043098840714292 z + 635228040950364334979624 z 32 38 - 30861565143924530232624 z + 2345924799067369395425628 z 40 62 - 7573122957652715617555224 z + 542316186557623941688456260 z 76 74 - 2345924799067369395425628 z + 7573122957652715617555224 z 72 104 - 21402622916025269855686524 z - 19752680278 z ) And in Maple-input format, it is: -(1+239810177757867244657*z^28-33868899752999703444*z^26-329*z^2+ 4100103750186446355*z^24-422887306114190529*z^22+47559*z^4-4091200*z^6-\ 10040162033*z^102+237895623*z^8-10040162033*z^10+321918833142*z^12-\ 8095221464653*z^14-2697034297855347*z^18+163439150188130*z^16-\ 53057853430244043786789990*z^50+33714096165858591646291661*z^48+ 36888771808673356*z^20+140817583505817472280995*z^36-35335818309703200216761*z^ 34-18808505205004616141036573*z^66+7717018835147973732032*z^80+321918833142*z^ 100-422887306114190529*z^90+4100103750186446355*z^88+239810177757867244657*z^84 -2697034297855347*z^94-33868899752999703444*z^86+163439150188130*z^96-\ 8095221464653*z^98+36888771808673356*z^92-1462703360566208438091*z^82+ 33714096165858591646291661*z^64+z^112-329*z^110-4091200*z^106+47559*z^108-\ 1462703360566208438091*z^30-3953137282846573716658375*z^42+ 9207932783003797526073148*z^44-18808505205004616141036573*z^46-\ 89048445888128388777586675*z^58+94999640860638412306613858*z^56-\ 89048445888128388777586675*z^54+73336665912188659792397127*z^52+ 73336665912188659792397127*z^60-3953137282846573716658375*z^70+ 9207932783003797526073148*z^68-35335818309703200216761*z^78+ 7717018835147973732032*z^32-489503980947553342395952*z^38+ 1486984762290802333306213*z^40-53057853430244043786789990*z^62+ 140817583505817472280995*z^76-489503980947553342395952*z^74+ 1486984762290802333306213*z^72+237895623*z^104)/(-1-850407598475070148420*z^28+ 113072899504857709768*z^26+413*z^2-12882116143504506232*z^24+ 1249658050449306668*z^22-69504*z^4+6716644*z^6+683784085160*z^102-430131738*z^8 +19752680278*z^10-683784085160*z^12+18467916910316*z^14+7028675777750949*z^18-\ 399020220066289*z^16+368317479505096056322115842*z^50-\ 219796519772305731968302486*z^48-102434906215476840*z^20-\ 635228040950364334979624*z^36+150072919261539655098308*z^34+ 219796519772305731968302486*z^66-150072919261539655098308*z^80-18467916910316*z ^100+12882116143504506232*z^90-113072899504857709768*z^88-\ 5508484311514520283708*z^84+102434906215476840*z^94+850407598475070148420*z^86-\ 7028675777750949*z^96+399020220066289*z^98-1249658050449306668*z^92+ 30861565143924530232624*z^82-368317479505096056322115842*z^64-413*z^112+z^114+ 69504*z^110+430131738*z^106-6716644*z^108+5508484311514520283708*z^30+ 21402622916025269855686524*z^42-53015826026038949584535608*z^44+ 115209808063043098840714292*z^46+798336777282412034816541404*z^58-\ 798336777282412034816541404*z^56+701808506573615112037894880*z^54-\ 542316186557623941688456260*z^52-701808506573615112037894880*z^60+ 53015826026038949584535608*z^70-115209808063043098840714292*z^68+ 635228040950364334979624*z^78-30861565143924530232624*z^32+ 2345924799067369395425628*z^38-7573122957652715617555224*z^40+ 542316186557623941688456260*z^62-2345924799067369395425628*z^76+ 7573122957652715617555224*z^74-21402622916025269855686524*z^72-19752680278*z^ 104) The first , 40, terms are: [0, 84, 0, 12747, 0, 2051619, 0, 333313140, 0, 54261113165, 0, 8837708328269, 0, 1439610633669164, 0, 234511634400261043, 0, 38202109175822618827, 0, 6223163614533046456396, 0, 1013760434918829232148613, 0, 165142752185023804291392857, 0, 26901947117834419846759953452, 0, 4382358632779433544852167744303, 0, 713891345854655232020869837857431, 0, 116293735113914662231338369021631820, 0, 18944385455203495868859351128105624297, 0, 3086062546183774591518702203540265452265, 0, 502723197940913879927074850609582849221620, 0, 81894196882432629640819803292497026678690359] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 225691606700317650137 z - 32866922027137419100 z - 341 z 24 22 4 6 + 4095939453371938567 z - 433946138543343301 z + 50791 z - 4465808 z 102 8 10 12 - 11177686893 z + 263252755 z - 11177686893 z + 358099904550 z 14 18 16 - 8943323718849 z - 2894460386477415 z + 178374269518430 z 50 48 - 37176146490086770274381590 z + 23972346974642628441670177 z 20 36 + 38776126322049988 z + 116582947222159718337371 z 34 66 - 30200439168561376619901 z - 13622155008989924521323001 z 80 100 90 + 6812174381241776160880 z + 358099904550 z - 433946138543343301 z 88 84 + 4095939453371938567 z + 225691606700317650137 z 94 86 96 - 2894460386477415 z - 32866922027137419100 z + 178374269518430 z 98 92 82 - 8943323718849 z + 38776126322049988 z - 1333580762431773692679 z 64 112 110 106 + 23972346974642628441670177 z + z - 341 z - 4465808 z 108 30 42 + 50791 z - 1333580762431773692679 z - 2999268538797276584530659 z 44 46 + 6815585926819916256085828 z - 13622155008989924521323001 z 58 56 - 61328039466221395031185303 z + 65283772298334911308750890 z 54 52 - 61328039466221395031185303 z + 50837384299063061807195159 z 60 70 + 50837384299063061807195159 z - 2999268538797276584530659 z 68 78 + 6815585926819916256085828 z - 30200439168561376619901 z 32 38 + 6812174381241776160880 z - 392970079517417494769432 z 40 62 + 1159318609517893913202113 z - 37176146490086770274381590 z 76 74 + 116582947222159718337371 z - 392970079517417494769432 z 72 104 / + 1159318609517893913202113 z + 263252755 z ) / (-1 / 28 26 2 - 830657414595002963576 z + 114013912456953389576 z + 429 z 24 22 4 - 13379903100806497276 z + 1333289020244364344 z - 74788 z 6 102 8 10 + 7431172 z + 782811608384 z - 485213118 z + 22532696254 z 12 14 18 - 782811608384 z + 21072841283732 z + 7828256607847449 z 16 50 - 451036325634961 z + 259498520516871374685403666 z 48 20 - 157661884764972429142796130 z - 111887596177854844 z 36 34 - 541057082100766389783400 z + 132317528309779132888304 z 66 80 + 157661884764972429142796130 z - 132317528309779132888304 z 100 90 - 21072841283732 z + 13379903100806497276 z 88 84 - 114013912456953389576 z - 5204050391196881549372 z 94 86 + 111887596177854844 z + 830657414595002963576 z 96 98 92 - 7828256607847449 z + 451036325634961 z - 1333289020244364344 z 82 64 112 + 28171894005058929979652 z - 259498520516871374685403666 z - 429 z 114 110 106 108 + z + 74788 z + 485213118 z - 7431172 z 30 42 + 5204050391196881549372 z + 16548580176850751383290760 z 44 46 - 39855889165143199162462684 z + 84459107979546285338312704 z 58 56 + 547024504639231263838508384 z - 547024504639231263838508384 z 54 52 + 483157636824332473112292952 z - 376860408879112194198866136 z 60 70 - 483157636824332473112292952 z + 39855889165143199162462684 z 68 78 - 84459107979546285338312704 z + 541057082100766389783400 z 32 38 - 28171894005058929979652 z + 1931785394460383406864840 z 40 62 - 6037156413357175680160632 z + 376860408879112194198866136 z 76 74 - 1931785394460383406864840 z + 6037156413357175680160632 z 72 104 - 16548580176850751383290760 z - 22532696254 z ) And in Maple-input format, it is: -(1+225691606700317650137*z^28-32866922027137419100*z^26-341*z^2+ 4095939453371938567*z^24-433946138543343301*z^22+50791*z^4-4465808*z^6-\ 11177686893*z^102+263252755*z^8-11177686893*z^10+358099904550*z^12-\ 8943323718849*z^14-2894460386477415*z^18+178374269518430*z^16-\ 37176146490086770274381590*z^50+23972346974642628441670177*z^48+ 38776126322049988*z^20+116582947222159718337371*z^36-30200439168561376619901*z^ 34-13622155008989924521323001*z^66+6812174381241776160880*z^80+358099904550*z^ 100-433946138543343301*z^90+4095939453371938567*z^88+225691606700317650137*z^84 -2894460386477415*z^94-32866922027137419100*z^86+178374269518430*z^96-\ 8943323718849*z^98+38776126322049988*z^92-1333580762431773692679*z^82+ 23972346974642628441670177*z^64+z^112-341*z^110-4465808*z^106+50791*z^108-\ 1333580762431773692679*z^30-2999268538797276584530659*z^42+ 6815585926819916256085828*z^44-13622155008989924521323001*z^46-\ 61328039466221395031185303*z^58+65283772298334911308750890*z^56-\ 61328039466221395031185303*z^54+50837384299063061807195159*z^52+ 50837384299063061807195159*z^60-2999268538797276584530659*z^70+ 6815585926819916256085828*z^68-30200439168561376619901*z^78+ 6812174381241776160880*z^32-392970079517417494769432*z^38+ 1159318609517893913202113*z^40-37176146490086770274381590*z^62+ 116582947222159718337371*z^76-392970079517417494769432*z^74+ 1159318609517893913202113*z^72+263252755*z^104)/(-1-830657414595002963576*z^28+ 114013912456953389576*z^26+429*z^2-13379903100806497276*z^24+ 1333289020244364344*z^22-74788*z^4+7431172*z^6+782811608384*z^102-485213118*z^8 +22532696254*z^10-782811608384*z^12+21072841283732*z^14+7828256607847449*z^18-\ 451036325634961*z^16+259498520516871374685403666*z^50-\ 157661884764972429142796130*z^48-111887596177854844*z^20-\ 541057082100766389783400*z^36+132317528309779132888304*z^34+ 157661884764972429142796130*z^66-132317528309779132888304*z^80-21072841283732*z ^100+13379903100806497276*z^90-114013912456953389576*z^88-\ 5204050391196881549372*z^84+111887596177854844*z^94+830657414595002963576*z^86-\ 7828256607847449*z^96+451036325634961*z^98-1333289020244364344*z^92+ 28171894005058929979652*z^82-259498520516871374685403666*z^64-429*z^112+z^114+ 74788*z^110+485213118*z^106-7431172*z^108+5204050391196881549372*z^30+ 16548580176850751383290760*z^42-39855889165143199162462684*z^44+ 84459107979546285338312704*z^46+547024504639231263838508384*z^58-\ 547024504639231263838508384*z^56+483157636824332473112292952*z^54-\ 376860408879112194198866136*z^52-483157636824332473112292952*z^60+ 39855889165143199162462684*z^70-84459107979546285338312704*z^68+ 541057082100766389783400*z^78-28171894005058929979652*z^32+ 1931785394460383406864840*z^38-6037156413357175680160632*z^40+ 376860408879112194198866136*z^62-1931785394460383406864840*z^76+ 6037156413357175680160632*z^74-16548580176850751383290760*z^72-22532696254*z^ 104) The first , 40, terms are: [0, 88, 0, 13755, 0, 2284915, 0, 383502368, 0, 64510318689, 0, 10857207118405, 0, 1827524800184000, 0, 307625697487511319, 0, 51782804044296151447, 0, 8716646809396071079176, 0, 1467281955102508908358257, 0, 246989088391257496310965165, 0, 41575930179307120839244711032, 0, 6998519647229929406120111653379, 0, 1178068107511342835265381493034259, 0, 198305432704249848048155323351309584, 0, 33380960232168693286141161240157823545, 0, 5619051837786609169696450403128373795621, 0, 945860854109128602217176400608253930107568, 0, 159217743698897512369675487166634167393825487] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 7}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4428424571648584 z - 2090313455012482 z - 239 z 24 22 4 6 + 791032734667670 z - 238753512370496 z + 22892 z - 1210211 z 8 10 12 14 + 40364257 z - 914631808 z + 14758067945 z - 175151152803 z 18 16 50 - 10723845683023 z + 1565374244620 z - 10723845683023 z 48 20 36 + 57091031115465 z + 57091031115465 z + 10388878878193686 z 34 66 64 30 - 11552855203678336 z - 239 z + 22892 z - 7550612907427898 z 42 44 46 - 2090313455012482 z + 791032734667670 z - 238753512370496 z 58 56 54 52 - 914631808 z + 14758067945 z - 175151152803 z + 1565374244620 z 60 68 32 38 + 40364257 z + z + 10388878878193686 z - 7550612907427898 z 40 62 / 2 + 4428424571648584 z - 1210211 z ) / ((-1 + z ) (1 / 28 26 2 + 17878942231427090 z - 8206462076833056 z - 316 z 24 22 4 6 + 2996509318767238 z - 865890872769056 z + 36291 z - 2218444 z 8 10 12 14 + 83895909 z - 2124960832 z + 37863206365 z - 490799344180 z 18 16 50 - 34795775134852 z + 4742417227051 z - 34795775134852 z 48 20 36 + 196659693910585 z + 196659693910585 z + 43318297958860798 z 34 66 64 30 - 48367705566888640 z - 316 z + 36291 z - 31103695161798752 z 42 44 46 - 8206462076833056 z + 2996509318767238 z - 865890872769056 z 58 56 54 52 - 2124960832 z + 37863206365 z - 490799344180 z + 4742417227051 z 60 68 32 38 + 83895909 z + z + 43318297958860798 z - 31103695161798752 z 40 62 + 17878942231427090 z - 2218444 z )) And in Maple-input format, it is: -(1+4428424571648584*z^28-2090313455012482*z^26-239*z^2+791032734667670*z^24-\ 238753512370496*z^22+22892*z^4-1210211*z^6+40364257*z^8-914631808*z^10+ 14758067945*z^12-175151152803*z^14-10723845683023*z^18+1565374244620*z^16-\ 10723845683023*z^50+57091031115465*z^48+57091031115465*z^20+10388878878193686*z ^36-11552855203678336*z^34-239*z^66+22892*z^64-7550612907427898*z^30-\ 2090313455012482*z^42+791032734667670*z^44-238753512370496*z^46-914631808*z^58+ 14758067945*z^56-175151152803*z^54+1565374244620*z^52+40364257*z^60+z^68+ 10388878878193686*z^32-7550612907427898*z^38+4428424571648584*z^40-1210211*z^62 )/(-1+z^2)/(1+17878942231427090*z^28-8206462076833056*z^26-316*z^2+ 2996509318767238*z^24-865890872769056*z^22+36291*z^4-2218444*z^6+83895909*z^8-\ 2124960832*z^10+37863206365*z^12-490799344180*z^14-34795775134852*z^18+ 4742417227051*z^16-34795775134852*z^50+196659693910585*z^48+196659693910585*z^ 20+43318297958860798*z^36-48367705566888640*z^34-316*z^66+36291*z^64-\ 31103695161798752*z^30-8206462076833056*z^42+2996509318767238*z^44-\ 865890872769056*z^46-2124960832*z^58+37863206365*z^56-490799344180*z^54+ 4742417227051*z^52+83895909*z^60+z^68+43318297958860798*z^32-31103695161798752* z^38+17878942231427090*z^40-2218444*z^62) The first , 40, terms are: [0, 78, 0, 11011, 0, 1679665, 0, 259493362, 0, 40176091583, 0, 6222602578515, 0, 963840489579226, 0, 149294356969368253, 0, 23125042713233915839, 0, 3581969335187789798742, 0, 554831634338958364840549, 0, 85941033499885674341541181, 0, 13311896440281513802025802502, 0, 2061955502616160848077785937575, 0, 319388038681256205259989804132613, 0, 49471833473252251545519987809109834, 0, 7662974221954566228982505672303539467, 0, 1186961747802167727190440345615410617863, 0, 183855269499584251707757541728185420665474, 0, 28478390466547668934431511763597368309608041] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2234902563548262614 z - 542360708949701419 z - 309 z 24 22 4 6 + 109791636997231910 z - 18424853511138285 z + 39749 z - 2885046 z 8 10 12 14 + 134768515 z - 4372586599 z + 103621389480 z - 1858543692605 z 18 16 50 - 286085788393057 z + 25900356109668 z - 202113198822476296515 z 48 20 + 302790879883454850809 z + 2543554455236210 z 36 34 + 114609170131252601817 z - 55129318054948655620 z 66 80 88 84 86 - 18424853511138285 z + 134768515 z + z + 39749 z - 309 z 82 64 30 - 2885046 z + 109791636997231910 z - 7720680769008311987 z 42 44 - 385745649685149054718 z + 418143792111627223076 z 46 58 - 385745649685149054718 z - 7720680769008311987 z 56 54 + 22450711013216525529 z - 55129318054948655620 z 52 60 + 114609170131252601817 z + 2234902563548262614 z 70 68 78 - 286085788393057 z + 2543554455236210 z - 4372586599 z 32 38 + 22450711013216525529 z - 202113198822476296515 z 40 62 76 + 302790879883454850809 z - 542360708949701419 z + 103621389480 z 74 72 / 2 - 1858543692605 z + 25900356109668 z ) / ((-1 + z ) (1 / 28 26 2 + 8102809480684059878 z - 1896968568693861772 z - 384 z 24 22 4 6 + 368903735951460427 z - 59207379630330144 z + 58880 z - 4938072 z 8 10 12 14 + 260514874 z - 9379518956 z + 243213377670 z - 4719090358276 z 18 16 50 - 827995741056288 z + 70483841186361 z - 825222661601781516400 z 48 20 + 1250171588838657141635 z + 7778637277489388 z 36 34 + 460749174650052152588 z - 217301706041436523128 z 66 80 88 84 86 - 59207379630330144 z + 260514874 z + z + 58880 z - 384 z 82 64 30 - 4938072 z + 368903735951460427 z - 28898697248411371124 z 42 44 - 1603453949534956083728 z + 1742049675676443301144 z 46 58 - 1603453949534956083728 z - 28898697248411371124 z 56 54 + 86408859199179185614 z - 217301706041436523128 z 52 60 + 460749174650052152588 z + 8102809480684059878 z 70 68 78 - 827995741056288 z + 7778637277489388 z - 9379518956 z 32 38 + 86408859199179185614 z - 825222661601781516400 z 40 62 76 + 1250171588838657141635 z - 1896968568693861772 z + 243213377670 z 74 72 - 4719090358276 z + 70483841186361 z )) And in Maple-input format, it is: -(1+2234902563548262614*z^28-542360708949701419*z^26-309*z^2+109791636997231910 *z^24-18424853511138285*z^22+39749*z^4-2885046*z^6+134768515*z^8-4372586599*z^ 10+103621389480*z^12-1858543692605*z^14-286085788393057*z^18+25900356109668*z^ 16-202113198822476296515*z^50+302790879883454850809*z^48+2543554455236210*z^20+ 114609170131252601817*z^36-55129318054948655620*z^34-18424853511138285*z^66+ 134768515*z^80+z^88+39749*z^84-309*z^86-2885046*z^82+109791636997231910*z^64-\ 7720680769008311987*z^30-385745649685149054718*z^42+418143792111627223076*z^44-\ 385745649685149054718*z^46-7720680769008311987*z^58+22450711013216525529*z^56-\ 55129318054948655620*z^54+114609170131252601817*z^52+2234902563548262614*z^60-\ 286085788393057*z^70+2543554455236210*z^68-4372586599*z^78+22450711013216525529 *z^32-202113198822476296515*z^38+302790879883454850809*z^40-542360708949701419* z^62+103621389480*z^76-1858543692605*z^74+25900356109668*z^72)/(-1+z^2)/(1+ 8102809480684059878*z^28-1896968568693861772*z^26-384*z^2+368903735951460427*z^ 24-59207379630330144*z^22+58880*z^4-4938072*z^6+260514874*z^8-9379518956*z^10+ 243213377670*z^12-4719090358276*z^14-827995741056288*z^18+70483841186361*z^16-\ 825222661601781516400*z^50+1250171588838657141635*z^48+7778637277489388*z^20+ 460749174650052152588*z^36-217301706041436523128*z^34-59207379630330144*z^66+ 260514874*z^80+z^88+58880*z^84-384*z^86-4938072*z^82+368903735951460427*z^64-\ 28898697248411371124*z^30-1603453949534956083728*z^42+1742049675676443301144*z^ 44-1603453949534956083728*z^46-28898697248411371124*z^58+86408859199179185614*z ^56-217301706041436523128*z^54+460749174650052152588*z^52+8102809480684059878*z ^60-827995741056288*z^70+7778637277489388*z^68-9379518956*z^78+ 86408859199179185614*z^32-825222661601781516400*z^38+1250171588838657141635*z^ 40-1896968568693861772*z^62+243213377670*z^76-4719090358276*z^74+70483841186361 *z^72) The first , 40, terms are: [0, 76, 0, 9745, 0, 1359667, 0, 195028036, 0, 28294809347, 0, 4126382913239, 0, 603260793846812, 0, 88300036716553979, 0, 12932152562506702101, 0, 1894546762503772799108, 0, 277588194812112470869473, 0, 40674919143305226263848321, 0, 5960286942029668583047489492, 0, 873403434569064269420223887093, 0, 127987097532672223363205521870059, 0, 18755093744137593174699277515332684, 0, 2748357034258283328775004047227077639, 0, 402742522365712412413020508377311095635, 0, 59017665882693227406668159868287330864372, 0, 8648418070530640111870758688518198516880963] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 240228664247800750294 z - 33920692748123488071 z - 329 z 24 22 4 6 + 4104777841295919188 z - 423143168864179989 z + 47514 z - 4083473 z 102 8 10 12 - 10013354899 z + 237304244 z - 10013354899 z + 321116358958 z 14 18 16 - 8078646549447 z - 2695045937865533 z + 163205012250036 z 50 48 - 52783667444817616837449377 z + 33564148250407403569498492 z 20 36 + 36887396134425026 z + 140914151838484282205906 z 34 66 - 35379349979498538968125 z - 18740739408078369708829015 z 80 100 90 + 7729351382109582438108 z + 321116358958 z - 423143168864179989 z 88 84 + 4104777841295919188 z + 240228664247800750294 z 94 86 96 - 2695045937865533 z - 33920692748123488071 z + 163205012250036 z 98 92 82 - 8078646549447 z + 36887396134425026 z - 1465292816705090307491 z 64 112 110 106 + 33564148250407403569498492 z + z - 329 z - 4083473 z 108 30 42 + 47514 z - 1465292816705090307491 z - 3946232361426853194820667 z 44 46 + 9183211630083484514515190 z - 18740739408078369708829015 z 58 56 - 88508518440479842691297617 z + 94412449743808767592072470 z 54 52 - 88508518440479842691297617 z + 72917325976141910647839354 z 60 70 + 72917325976141910647839354 z - 3946232361426853194820667 z 68 78 + 9183211630083484514515190 z - 35379349979498538968125 z 32 38 + 7729351382109582438108 z - 489495947416580464728413 z 40 62 + 1485733989303942542260588 z - 52783667444817616837449377 z 76 74 + 140914151838484282205906 z - 489495947416580464728413 z 72 104 / + 1485733989303942542260588 z + 237304244 z ) / (-1 / 28 26 2 - 855470596361050426544 z + 113834748505210332694 z + 406 z 24 22 4 - 12967188740219632396 z + 1256543450459183306 z - 67913 z 6 102 8 10 + 6558696 z + 675212669178 z - 421024238 z + 19412657792 z 12 14 18 - 675212669178 z + 18324550577377 z + 7031969477418003 z 16 50 - 397691079463581 z + 356992399086695791588359015 z 48 20 - 213671868784560614939532901 z - 102788350754384315 z 36 34 - 632321773830805634083709 z + 149913259815659284021625 z 66 80 + 213671868784560614939532901 z - 149913259815659284021625 z 100 90 - 18324550577377 z + 12967188740219632396 z 88 84 - 113834748505210332694 z - 5532304535033544006015 z 94 86 + 102788350754384315 z + 855470596361050426544 z 96 98 92 - 7031969477418003 z + 397691079463581 z - 1256543450459183306 z 82 64 112 + 30921634124403067033551 z - 356992399086695791588359015 z - 406 z 114 110 106 108 + z + 67913 z + 421024238 z - 6558696 z 30 42 + 5532304535033544006015 z + 21046197135146108349993952 z 44 46 - 51919181803340651312899902 z + 112390747150012701309703709 z 58 56 + 770105774684583425945027286 z - 770105774684583425945027286 z 54 52 + 677548072630351575094411374 z - 524413869545717379756295727 z 60 70 - 677548072630351575094411374 z + 51919181803340651312899902 z 68 78 - 112390747150012701309703709 z + 632321773830805634083709 z 32 38 - 30921634124403067033551 z + 2326114295105214244066812 z 40 62 - 7478292100383922620761202 z + 524413869545717379756295727 z 76 74 - 2326114295105214244066812 z + 7478292100383922620761202 z 72 104 - 21046197135146108349993952 z - 19412657792 z ) And in Maple-input format, it is: -(1+240228664247800750294*z^28-33920692748123488071*z^26-329*z^2+ 4104777841295919188*z^24-423143168864179989*z^22+47514*z^4-4083473*z^6-\ 10013354899*z^102+237304244*z^8-10013354899*z^10+321116358958*z^12-\ 8078646549447*z^14-2695045937865533*z^18+163205012250036*z^16-\ 52783667444817616837449377*z^50+33564148250407403569498492*z^48+ 36887396134425026*z^20+140914151838484282205906*z^36-35379349979498538968125*z^ 34-18740739408078369708829015*z^66+7729351382109582438108*z^80+321116358958*z^ 100-423143168864179989*z^90+4104777841295919188*z^88+240228664247800750294*z^84 -2695045937865533*z^94-33920692748123488071*z^86+163205012250036*z^96-\ 8078646549447*z^98+36887396134425026*z^92-1465292816705090307491*z^82+ 33564148250407403569498492*z^64+z^112-329*z^110-4083473*z^106+47514*z^108-\ 1465292816705090307491*z^30-3946232361426853194820667*z^42+ 9183211630083484514515190*z^44-18740739408078369708829015*z^46-\ 88508518440479842691297617*z^58+94412449743808767592072470*z^56-\ 88508518440479842691297617*z^54+72917325976141910647839354*z^52+ 72917325976141910647839354*z^60-3946232361426853194820667*z^70+ 9183211630083484514515190*z^68-35379349979498538968125*z^78+ 7729351382109582438108*z^32-489495947416580464728413*z^38+ 1485733989303942542260588*z^40-52783667444817616837449377*z^62+ 140914151838484282205906*z^76-489495947416580464728413*z^74+ 1485733989303942542260588*z^72+237304244*z^104)/(-1-855470596361050426544*z^28+ 113834748505210332694*z^26+406*z^2-12967188740219632396*z^24+ 1256543450459183306*z^22-67913*z^4+6558696*z^6+675212669178*z^102-421024238*z^8 +19412657792*z^10-675212669178*z^12+18324550577377*z^14+7031969477418003*z^18-\ 397691079463581*z^16+356992399086695791588359015*z^50-\ 213671868784560614939532901*z^48-102788350754384315*z^20-\ 632321773830805634083709*z^36+149913259815659284021625*z^34+ 213671868784560614939532901*z^66-149913259815659284021625*z^80-18324550577377*z ^100+12967188740219632396*z^90-113834748505210332694*z^88-\ 5532304535033544006015*z^84+102788350754384315*z^94+855470596361050426544*z^86-\ 7031969477418003*z^96+397691079463581*z^98-1256543450459183306*z^92+ 30921634124403067033551*z^82-356992399086695791588359015*z^64-406*z^112+z^114+ 67913*z^110+421024238*z^106-6558696*z^108+5532304535033544006015*z^30+ 21046197135146108349993952*z^42-51919181803340651312899902*z^44+ 112390747150012701309703709*z^46+770105774684583425945027286*z^58-\ 770105774684583425945027286*z^56+677548072630351575094411374*z^54-\ 524413869545717379756295727*z^52-677548072630351575094411374*z^60+ 51919181803340651312899902*z^70-112390747150012701309703709*z^68+ 632321773830805634083709*z^78-30921634124403067033551*z^32+ 2326114295105214244066812*z^38-7478292100383922620761202*z^40+ 524413869545717379756295727*z^62-2326114295105214244066812*z^76+ 7478292100383922620761202*z^74-21046197135146108349993952*z^72-19412657792*z^ 104) The first , 40, terms are: [0, 77, 0, 10863, 0, 1656300, 0, 256018479, 0, 39686751789, 0, 6156098489177, 0, 955068771199209, 0, 148177013790867865, 0, 22989594927191563835, 0, 3566833863202692121868, 0, 553394344692054575403027, 0, 85859157271741221231975353, 0, 13321052621492116895027724409, 0, 2066762013622306319746073469001, 0, 320658235764000730819314330868081, 0, 49750142277622921723072626133850795, 0, 7718737213050625382906362429092106764, 0, 1197562488047354886844654883393846908643, 0, 185801883547619274486573163880901554160145, 0, 28827172088722312361924461872926351107685257] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1192321840337171676 z - 299575904084685546 z - 293 z 24 22 4 6 + 62917555282101352 z - 10972451717958582 z + 35558 z - 2442175 z 8 10 12 14 + 108406965 z - 3353926632 z + 75973751914 z - 1304620484348 z 18 16 50 - 184641904595536 z + 17425736967870 z - 95057187701211846424 z 48 20 + 140618405952871392794 z + 1576126896030772 z 36 34 + 54845470904145523750 z - 26960559006498010100 z 66 80 88 84 86 - 10972451717958582 z + 108406965 z + z + 35558 z - 293 z 82 64 30 - 2442175 z + 62917555282101352 z - 3988509313809005712 z 42 44 - 177770649308703410556 z + 192204460884152325572 z 46 58 - 177770649308703410556 z - 3988509313809005712 z 56 54 + 11264699312480202098 z - 26960559006498010100 z 52 60 70 + 54845470904145523750 z + 1192321840337171676 z - 184641904595536 z 68 78 32 + 1576126896030772 z - 3353926632 z + 11264699312480202098 z 38 40 - 95057187701211846424 z + 140618405952871392794 z 62 76 74 - 299575904084685546 z + 75973751914 z - 1304620484348 z 72 / 2 28 + 17425736967870 z ) / ((-1 + z ) (1 + 4354371439879958100 z / 26 2 24 - 1054931239516864170 z - 371 z + 212752794875534604 z 22 4 6 8 - 35473155222581302 z + 53719 z - 4252981 z + 212561073 z 10 12 14 - 7276914504 z + 179961477018 z - 3337797831180 z 18 16 50 - 537646242551544 z + 47734621908674 z - 391678089607853273784 z 48 20 + 586009632636699548918 z + 4848687745416916 z 36 34 + 222460567430252268270 z - 107186686437937939396 z 66 80 88 84 86 - 35473155222581302 z + 212561073 z + z + 53719 z - 371 z 82 64 30 - 4252981 z + 212752794875534604 z - 15045257170704004488 z 42 44 - 745915895373762838328 z + 808323390176972581738 z 46 58 - 745915895373762838328 z - 15045257170704004488 z 56 54 + 43712957983737543110 z - 107186686437937939396 z 52 60 + 222460567430252268270 z + 4354371439879958100 z 70 68 78 - 537646242551544 z + 4848687745416916 z - 7276914504 z 32 38 + 43712957983737543110 z - 391678089607853273784 z 40 62 76 + 586009632636699548918 z - 1054931239516864170 z + 179961477018 z 74 72 - 3337797831180 z + 47734621908674 z )) And in Maple-input format, it is: -(1+1192321840337171676*z^28-299575904084685546*z^26-293*z^2+62917555282101352* z^24-10972451717958582*z^22+35558*z^4-2442175*z^6+108406965*z^8-3353926632*z^10 +75973751914*z^12-1304620484348*z^14-184641904595536*z^18+17425736967870*z^16-\ 95057187701211846424*z^50+140618405952871392794*z^48+1576126896030772*z^20+ 54845470904145523750*z^36-26960559006498010100*z^34-10972451717958582*z^66+ 108406965*z^80+z^88+35558*z^84-293*z^86-2442175*z^82+62917555282101352*z^64-\ 3988509313809005712*z^30-177770649308703410556*z^42+192204460884152325572*z^44-\ 177770649308703410556*z^46-3988509313809005712*z^58+11264699312480202098*z^56-\ 26960559006498010100*z^54+54845470904145523750*z^52+1192321840337171676*z^60-\ 184641904595536*z^70+1576126896030772*z^68-3353926632*z^78+11264699312480202098 *z^32-95057187701211846424*z^38+140618405952871392794*z^40-299575904084685546*z ^62+75973751914*z^76-1304620484348*z^74+17425736967870*z^72)/(-1+z^2)/(1+ 4354371439879958100*z^28-1054931239516864170*z^26-371*z^2+212752794875534604*z^ 24-35473155222581302*z^22+53719*z^4-4252981*z^6+212561073*z^8-7276914504*z^10+ 179961477018*z^12-3337797831180*z^14-537646242551544*z^18+47734621908674*z^16-\ 391678089607853273784*z^50+586009632636699548918*z^48+4848687745416916*z^20+ 222460567430252268270*z^36-107186686437937939396*z^34-35473155222581302*z^66+ 212561073*z^80+z^88+53719*z^84-371*z^86-4252981*z^82+212752794875534604*z^64-\ 15045257170704004488*z^30-745915895373762838328*z^42+808323390176972581738*z^44 -745915895373762838328*z^46-15045257170704004488*z^58+43712957983737543110*z^56 -107186686437937939396*z^54+222460567430252268270*z^52+4354371439879958100*z^60 -537646242551544*z^70+4848687745416916*z^68-7276914504*z^78+ 43712957983737543110*z^32-391678089607853273784*z^38+586009632636699548918*z^40 -1054931239516864170*z^62+179961477018*z^76-3337797831180*z^74+47734621908674*z ^72) The first , 40, terms are: [0, 79, 0, 10856, 0, 1629847, 0, 250924255, 0, 38946172509, 0, 6061415856049, 0, 944257033734704, 0, 147145504691609553, 0, 22932562869997877875, 0, 3574169817396856748683, 0, 557062212933414398951041, 0, 86822899697033291608375616, 0, 13532112366558290408014439153, 0, 2109099930004798406561637894373, 0, 328722027997232628830883491091431, 0, 51234262468528135617051498696771991, 0, 7985317357970789742849243522390907128, 0, 1244583034700610191929558942215378928319, 0, 193979383104684943679783290976149136472089, 0, 30233419595782805414200106795335448103187497] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4836099785978452 z - 2234197944979494 z - 237 z 24 22 4 6 + 824722985420494 z - 242443277041256 z + 22288 z - 1155931 z 8 10 12 14 + 37970791 z - 852698764 z + 13736201571 z - 163992589399 z 18 16 50 - 10366536400225 z + 1484809651412 z - 10366536400225 z 48 20 36 + 56492621755317 z + 56492621755317 z + 11655739245211074 z 34 66 64 30 - 13007921106154120 z - 237 z + 22288 z - 8383420093704234 z 42 44 46 - 2234197944979494 z + 824722985420494 z - 242443277041256 z 58 56 54 52 - 852698764 z + 13736201571 z - 163992589399 z + 1484809651412 z 60 68 32 38 + 37970791 z + z + 11655739245211074 z - 8383420093704234 z 40 62 / 28 + 4836099785978452 z - 1155931 z ) / (-1 - 28769504618114546 z / 26 2 24 + 12003275550972062 z + 315 z - 4007810790236994 z 22 4 6 8 + 1067036815645025 z - 35917 z + 2168367 z - 81094777 z 10 12 14 18 + 2046139379 z - 36731999863 z + 486218711885 z + 37494127191205 z 16 50 48 - 4866181345531 z + 225367244344355 z - 1067036815645025 z 20 36 34 - 225367244344355 z - 106227220952033658 z + 106227220952033658 z 66 64 30 42 + 35917 z - 2168367 z + 55320019216241594 z + 28769504618114546 z 44 46 58 - 12003275550972062 z + 4007810790236994 z + 36731999863 z 56 54 52 - 486218711885 z + 4866181345531 z - 37494127191205 z 60 70 68 32 - 2046139379 z + z - 315 z - 85476556069977526 z 38 40 62 + 85476556069977526 z - 55320019216241594 z + 81094777 z ) And in Maple-input format, it is: -(1+4836099785978452*z^28-2234197944979494*z^26-237*z^2+824722985420494*z^24-\ 242443277041256*z^22+22288*z^4-1155931*z^6+37970791*z^8-852698764*z^10+ 13736201571*z^12-163992589399*z^14-10366536400225*z^18+1484809651412*z^16-\ 10366536400225*z^50+56492621755317*z^48+56492621755317*z^20+11655739245211074*z ^36-13007921106154120*z^34-237*z^66+22288*z^64-8383420093704234*z^30-\ 2234197944979494*z^42+824722985420494*z^44-242443277041256*z^46-852698764*z^58+ 13736201571*z^56-163992589399*z^54+1484809651412*z^52+37970791*z^60+z^68+ 11655739245211074*z^32-8383420093704234*z^38+4836099785978452*z^40-1155931*z^62 )/(-1-28769504618114546*z^28+12003275550972062*z^26+315*z^2-4007810790236994*z^ 24+1067036815645025*z^22-35917*z^4+2168367*z^6-81094777*z^8+2046139379*z^10-\ 36731999863*z^12+486218711885*z^14+37494127191205*z^18-4866181345531*z^16+ 225367244344355*z^50-1067036815645025*z^48-225367244344355*z^20-\ 106227220952033658*z^36+106227220952033658*z^34+35917*z^66-2168367*z^64+ 55320019216241594*z^30+28769504618114546*z^42-12003275550972062*z^44+ 4007810790236994*z^46+36731999863*z^58-486218711885*z^56+4866181345531*z^54-\ 37494127191205*z^52-2046139379*z^60+z^70-315*z^68-85476556069977526*z^32+ 85476556069977526*z^38-55320019216241594*z^40+81094777*z^62) The first , 40, terms are: [0, 78, 0, 10941, 0, 1657325, 0, 255098118, 0, 39421916501, 0, 6098578549997, 0, 943725148903230, 0, 146048610773102385, 0, 22602637894978284581, 0, 3498030395120291453662, 0, 541363294684690725311677, 0, 83782681004433639033187789, 0, 12966410368346563167438023358, 0, 2006713113044445162853117396757, 0, 310563791332374880721768786902337, 0, 48063606187999772369651912822727038, 0, 7438440366762332399184671633703998269, 0, 1151191088070379890497987136646827707397, 0, 178161127334032100644149065589305162665222, 0, 27572648557237346501752810374179356982994973] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 4}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 109370783951 z + 216312301586 z + 206 z - 303742353035 z 22 4 6 8 10 + 303742353035 z - 15845 z + 617435 z - 13751326 z + 187745331 z 12 14 18 16 - 1651078561 z + 9695207234 z + 109370783951 z - 38998772881 z 20 36 34 30 - 216312301586 z - 187745331 z + 1651078561 z + 38998772881 z 42 44 46 32 38 40 + 15845 z - 206 z + z - 9695207234 z + 13751326 z - 617435 z ) / 28 26 2 / (1 + 1404582165094 z - 2337408060154 z - 294 z / 24 22 4 6 + 2769514651606 z - 2337408060154 z + 28138 z - 1306924 z 8 10 12 14 + 34129194 z - 543039142 z + 5568774724 z - 38279272986 z 18 16 48 20 - 599799152676 z + 181102475430 z + z + 1404582165094 z 36 34 30 42 + 5568774724 z - 38279272986 z - 599799152676 z - 1306924 z 44 46 32 38 40 + 28138 z - 294 z + 181102475430 z - 543039142 z + 34129194 z ) And in Maple-input format, it is: -(-1-109370783951*z^28+216312301586*z^26+206*z^2-303742353035*z^24+303742353035 *z^22-15845*z^4+617435*z^6-13751326*z^8+187745331*z^10-1651078561*z^12+ 9695207234*z^14+109370783951*z^18-38998772881*z^16-216312301586*z^20-187745331* z^36+1651078561*z^34+38998772881*z^30+15845*z^42-206*z^44+z^46-9695207234*z^32+ 13751326*z^38-617435*z^40)/(1+1404582165094*z^28-2337408060154*z^26-294*z^2+ 2769514651606*z^24-2337408060154*z^22+28138*z^4-1306924*z^6+34129194*z^8-\ 543039142*z^10+5568774724*z^12-38279272986*z^14-599799152676*z^18+181102475430* z^16+z^48+1404582165094*z^20+5568774724*z^36-38279272986*z^34-599799152676*z^30 -1306924*z^42+28138*z^44-294*z^46+181102475430*z^32-543039142*z^38+34129194*z^ 40) The first , 40, terms are: [0, 88, 0, 13579, 0, 2205571, 0, 360983416, 0, 59167413241, 0, 9700811230057, 0, 1590601817693368, 0, 260808066821443411, 0, 42764354734421620411, 0, 7012019735313163813144, 0, 1149752609557846075148497, 0, 188523586563946912043008561, 0, 30911991567322720329841204120, 0, 5068603042031737215154522404571, 0, 831092902959446871195916071272755, 0, 136273329689419427787510079064820280, 0, 22344578228124617315813535758892949897, 0, 3663814315926393826063869287563047560665, 0, 600751341312186195475465968691982707367416, 0, 98504493669248650478463827469408800166990499] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 18370481460956452 z - 7357004082757196 z - 254 z 24 22 4 6 + 2387163039278495 z - 624540959570134 z + 26106 z - 1478228 z 8 10 12 14 + 52783574 z - 1284509750 z + 22403224970 z - 289965053358 z 18 16 50 - 21807324140948 z + 2855949199086 z - 624540959570134 z 48 20 36 + 2387163039278495 z + 130915386168802 z + 92335352638847020 z 34 66 64 - 83508349696364604 z - 1478228 z + 52783574 z 30 42 44 - 37301370106828568 z - 37301370106828568 z + 18370481460956452 z 46 58 56 - 7357004082757196 z - 289965053358 z + 2855949199086 z 54 52 60 70 - 21807324140948 z + 130915386168802 z + 22403224970 z - 254 z 68 32 38 + 26106 z + 61754248838244924 z - 83508349696364604 z 40 62 72 / + 61754248838244924 z - 1284509750 z + z ) / (-1 / 28 26 2 - 104376751483747166 z + 37930184919245801 z + 346 z 24 22 4 6 - 11169088131245340 z + 2651689293885767 z - 42587 z + 2758841 z 8 10 12 14 - 110624271 z + 2998135225 z - 57996328337 z + 830605685757 z 18 16 50 + 76168738596555 z - 9038489588877 z + 11169088131245340 z 48 20 36 - 37930184919245801 z - 504272710504589 z - 777361442307244242 z 34 66 64 + 636549718875950778 z + 110624271 z - 2998135225 z 30 42 44 + 233630046966081754 z + 426563800585339206 z - 233630046966081754 z 46 58 56 + 104376751483747166 z + 9038489588877 z - 76168738596555 z 54 52 60 + 504272710504589 z - 2651689293885767 z - 830605685757 z 70 68 32 + 42587 z - 2758841 z - 426563800585339206 z 38 40 62 74 + 777361442307244242 z - 636549718875950778 z + 57996328337 z + z 72 - 346 z ) And in Maple-input format, it is: -(1+18370481460956452*z^28-7357004082757196*z^26-254*z^2+2387163039278495*z^24-\ 624540959570134*z^22+26106*z^4-1478228*z^6+52783574*z^8-1284509750*z^10+ 22403224970*z^12-289965053358*z^14-21807324140948*z^18+2855949199086*z^16-\ 624540959570134*z^50+2387163039278495*z^48+130915386168802*z^20+ 92335352638847020*z^36-83508349696364604*z^34-1478228*z^66+52783574*z^64-\ 37301370106828568*z^30-37301370106828568*z^42+18370481460956452*z^44-\ 7357004082757196*z^46-289965053358*z^58+2855949199086*z^56-21807324140948*z^54+ 130915386168802*z^52+22403224970*z^60-254*z^70+26106*z^68+61754248838244924*z^ 32-83508349696364604*z^38+61754248838244924*z^40-1284509750*z^62+z^72)/(-1-\ 104376751483747166*z^28+37930184919245801*z^26+346*z^2-11169088131245340*z^24+ 2651689293885767*z^22-42587*z^4+2758841*z^6-110624271*z^8+2998135225*z^10-\ 57996328337*z^12+830605685757*z^14+76168738596555*z^18-9038489588877*z^16+ 11169088131245340*z^50-37930184919245801*z^48-504272710504589*z^20-\ 777361442307244242*z^36+636549718875950778*z^34+110624271*z^66-2998135225*z^64+ 233630046966081754*z^30+426563800585339206*z^42-233630046966081754*z^44+ 104376751483747166*z^46+9038489588877*z^58-76168738596555*z^56+504272710504589* z^54-2651689293885767*z^52-830605685757*z^60+42587*z^70-2758841*z^68-\ 426563800585339206*z^32+777361442307244242*z^38-636549718875950778*z^40+ 57996328337*z^62+z^74-346*z^72) The first , 40, terms are: [0, 92, 0, 15351, 0, 2674055, 0, 467442668, 0, 81742343577, 0, 14295204698993, 0, 2499993604841132, 0, 437208639309021951, 0, 76460815615571614703, 0, 13371779749686278796700, 0, 2338511605390235919395225, 0, 408968493637063732420748969, 0, 71522086607531400074064777436, 0, 12508075704920187682376369479935, 0, 2187463555904399940563277745503247, 0, 382552594161470825373170660115873260, 0, 66902365942817086730349693619961879041, 0, 11700160022680154598277988324373689779337, 0, 2046171949638024576230210065086149092394284, 0, 357842938846304278355803863986633652584047223] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 5}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 13435972927202453 z - 5419762219372131 z - 250 z 24 22 4 6 + 1774880677044277 z - 469717392473810 z + 24935 z - 1362033 z 8 10 12 14 + 46890327 z - 1102622322 z + 18646257211 z - 234885178722 z 18 16 50 - 16915114601794 z + 2259926126682 z - 469717392473810 z 48 20 36 + 1774880677044277 z + 99850995584241 z + 66724384180231060 z 34 66 64 - 60389438942776708 z - 1362033 z + 46890327 z 30 42 44 - 27135124310880346 z - 27135124310880346 z + 13435972927202453 z 46 58 56 - 5419762219372131 z - 234885178722 z + 2259926126682 z 54 52 60 70 - 16915114601794 z + 99850995584241 z + 18646257211 z - 250 z 68 32 38 + 24935 z + 44756018672086875 z - 60389438942776708 z 40 62 72 / + 44756018672086875 z - 1102622322 z + z ) / (-1 / 28 26 2 - 76901684646416334 z + 28193142758516073 z + 345 z 24 22 4 6 - 8395119628538407 z + 2020785830233170 z - 41602 z + 2613272 z 8 10 12 14 - 101231517 z + 2650371143 z - 49622123252 z + 689829437792 z 18 16 50 + 60201005621601 z - 7310496466197 z + 8395119628538407 z 48 20 36 - 28193142758516073 z - 390752334773750 z - 563568514186982006 z 34 66 64 + 462185087600350997 z + 101231517 z - 2650371143 z 30 42 44 + 170981994385437292 z + 310681510881988941 z - 170981994385437292 z 46 58 56 + 76901684646416334 z + 7310496466197 z - 60201005621601 z 54 52 60 + 390752334773750 z - 2020785830233170 z - 689829437792 z 70 68 32 + 41602 z - 2613272 z - 310681510881988941 z 38 40 62 74 + 563568514186982006 z - 462185087600350997 z + 49622123252 z + z 72 - 345 z ) And in Maple-input format, it is: -(1+13435972927202453*z^28-5419762219372131*z^26-250*z^2+1774880677044277*z^24-\ 469717392473810*z^22+24935*z^4-1362033*z^6+46890327*z^8-1102622322*z^10+ 18646257211*z^12-234885178722*z^14-16915114601794*z^18+2259926126682*z^16-\ 469717392473810*z^50+1774880677044277*z^48+99850995584241*z^20+ 66724384180231060*z^36-60389438942776708*z^34-1362033*z^66+46890327*z^64-\ 27135124310880346*z^30-27135124310880346*z^42+13435972927202453*z^44-\ 5419762219372131*z^46-234885178722*z^58+2259926126682*z^56-16915114601794*z^54+ 99850995584241*z^52+18646257211*z^60-250*z^70+24935*z^68+44756018672086875*z^32 -60389438942776708*z^38+44756018672086875*z^40-1102622322*z^62+z^72)/(-1-\ 76901684646416334*z^28+28193142758516073*z^26+345*z^2-8395119628538407*z^24+ 2020785830233170*z^22-41602*z^4+2613272*z^6-101231517*z^8+2650371143*z^10-\ 49622123252*z^12+689829437792*z^14+60201005621601*z^18-7310496466197*z^16+ 8395119628538407*z^50-28193142758516073*z^48-390752334773750*z^20-\ 563568514186982006*z^36+462185087600350997*z^34+101231517*z^66-2650371143*z^64+ 170981994385437292*z^30+310681510881988941*z^42-170981994385437292*z^44+ 76901684646416334*z^46+7310496466197*z^58-60201005621601*z^56+390752334773750*z ^54-2020785830233170*z^52-689829437792*z^60+41602*z^70-2613272*z^68-\ 310681510881988941*z^32+563568514186982006*z^38-462185087600350997*z^40+ 49622123252*z^62+z^74-345*z^72) The first , 40, terms are: [0, 95, 0, 16108, 0, 2856309, 0, 509221239, 0, 90878500519, 0, 16222945143933, 0, 2896206829580276, 0, 517056843636991855, 0, 92310164083831163213, 0, 16480161819717623808513, 0, 2942209712989590004827691, 0, 525273917287071671050688516, 0, 93777373353332591865512052385, 0, 16742114084844301365732757106011, 0, 2988976710257940909873226169933051, 0, 533623277031759817602880586036783961, 0, 95267989504550463118258945203991550684, 0, 17008234489655216266973760299912269669259, 0, 3036487302455737916906157679940723405119477, 0, 542105363354264427013803009440600700246700461] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 7}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3563814871960740948 z - 831240787002527644 z - 303 z 24 22 4 6 + 161226402173407552 z - 25856154325868892 z + 38904 z - 2864063 z 8 10 12 14 + 137437249 z - 4623672932 z + 114336774148 z - 2148362288608 z 18 16 50 - 364669057906164 z + 31429443626324 z - 369810095310130911068 z 48 20 + 561400843576645109622 z + 3404174471687284 z 36 34 + 205896787055343911868 z - 96769923070324676512 z 66 80 88 84 86 - 25856154325868892 z + 137437249 z + z + 38904 z - 303 z 82 64 30 - 2864063 z + 161226402173407552 z - 12763891587218546636 z 42 44 - 720949981990875677978 z + 783595736945536932496 z 46 58 - 720949981990875677978 z - 12763891587218546636 z 56 54 + 38327379626157229036 z - 96769923070324676512 z 52 60 + 205896787055343911868 z + 3563814871960740948 z 70 68 78 - 364669057906164 z + 3404174471687284 z - 4623672932 z 32 38 + 38327379626157229036 z - 369810095310130911068 z 40 62 76 + 561400843576645109622 z - 831240787002527644 z + 114336774148 z 74 72 / - 2148362288608 z + 31429443626324 z ) / (-1 / 28 26 2 - 15846952409528825612 z + 3421297258338064436 z + 394 z 24 22 4 6 - 614193102298825260 z + 91139336407428788 z - 59842 z + 5007134 z 8 10 12 14 - 267671934 z + 9920281489 z - 268423776620 z + 5494638826372 z 18 16 50 + 1098138291594948 z - 87312746492460 z + 4000689316615100197174 z 48 20 - 5571778765276653721400 z - 11096072523193260 z 36 34 - 1250813382502801009900 z + 543366294346322964324 z 66 80 90 88 84 + 614193102298825260 z - 9920281489 z + z - 394 z - 5007134 z 86 82 64 + 59842 z + 267671934 z - 3421297258338064436 z 30 42 + 61324999195950080908 z + 5571778765276653721400 z 44 46 - 6574414615289450539712 z + 6574414615289450539712 z 58 56 + 199029205701591031980 z - 543366294346322964324 z 54 52 + 1250813382502801009900 z - 2432252329875377350436 z 60 70 68 - 61324999195950080908 z + 11096072523193260 z - 91139336407428788 z 78 32 + 268423776620 z - 199029205701591031980 z 38 40 + 2432252329875377350436 z - 4000689316615100197174 z 62 76 74 + 15846952409528825612 z - 5494638826372 z + 87312746492460 z 72 - 1098138291594948 z ) And in Maple-input format, it is: -(1+3563814871960740948*z^28-831240787002527644*z^26-303*z^2+161226402173407552 *z^24-25856154325868892*z^22+38904*z^4-2864063*z^6+137437249*z^8-4623672932*z^ 10+114336774148*z^12-2148362288608*z^14-364669057906164*z^18+31429443626324*z^ 16-369810095310130911068*z^50+561400843576645109622*z^48+3404174471687284*z^20+ 205896787055343911868*z^36-96769923070324676512*z^34-25856154325868892*z^66+ 137437249*z^80+z^88+38904*z^84-303*z^86-2864063*z^82+161226402173407552*z^64-\ 12763891587218546636*z^30-720949981990875677978*z^42+783595736945536932496*z^44 -720949981990875677978*z^46-12763891587218546636*z^58+38327379626157229036*z^56 -96769923070324676512*z^54+205896787055343911868*z^52+3563814871960740948*z^60-\ 364669057906164*z^70+3404174471687284*z^68-4623672932*z^78+38327379626157229036 *z^32-369810095310130911068*z^38+561400843576645109622*z^40-831240787002527644* z^62+114336774148*z^76-2148362288608*z^74+31429443626324*z^72)/(-1-\ 15846952409528825612*z^28+3421297258338064436*z^26+394*z^2-614193102298825260*z ^24+91139336407428788*z^22-59842*z^4+5007134*z^6-267671934*z^8+9920281489*z^10-\ 268423776620*z^12+5494638826372*z^14+1098138291594948*z^18-87312746492460*z^16+ 4000689316615100197174*z^50-5571778765276653721400*z^48-11096072523193260*z^20-\ 1250813382502801009900*z^36+543366294346322964324*z^34+614193102298825260*z^66-\ 9920281489*z^80+z^90-394*z^88-5007134*z^84+59842*z^86+267671934*z^82-\ 3421297258338064436*z^64+61324999195950080908*z^30+5571778765276653721400*z^42-\ 6574414615289450539712*z^44+6574414615289450539712*z^46+199029205701591031980*z ^58-543366294346322964324*z^56+1250813382502801009900*z^54-\ 2432252329875377350436*z^52-61324999195950080908*z^60+11096072523193260*z^70-\ 91139336407428788*z^68+268423776620*z^78-199029205701591031980*z^32+ 2432252329875377350436*z^38-4000689316615100197174*z^40+15846952409528825612*z^ 62-5494638826372*z^76+87312746492460*z^74-1098138291594948*z^72) The first , 40, terms are: [0, 91, 0, 14916, 0, 2574353, 0, 447106319, 0, 77730330767, 0, 13516208460385, 0, 2350377516359188, 0, 408718789853823147, 0, 71074299862073226641, 0, 12359497655735057794913, 0, 2149260750624364207675723, 0, 373746737641985635465561012, 0, 64992870221644140895259591905, 0, 11301966713577279418394083730943, 0, 1965360988407790740591737841846207, 0, 341767403242386766466352862375010513, 0, 59431808513332681350879996723698826148, 0, 10334923195374032750477433900463992942395, 0, 1797196486634320889778632595883982210047569, 0, 312524355577023875613675200292183943446594161] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 2359125236481046 z + 1222187428436220 z + 263 z 24 22 4 6 - 506079676003830 z + 166731157639006 z - 26768 z + 1435514 z 8 10 12 14 - 46856132 z + 1013955714 z - 15379007394 z + 169879402226 z 18 16 50 + 8870728345785 z - 1404657794917 z + 1404657794917 z 48 20 36 - 8870728345785 z - 43432546043834 z - 3651211181899828 z 34 66 64 30 + 4540130615194018 z + z - 263 z + 3651211181899828 z 42 44 46 + 506079676003830 z - 166731157639006 z + 43432546043834 z 58 56 54 52 + 46856132 z - 1013955714 z + 15379007394 z - 169879402226 z 60 32 38 - 1435514 z - 4540130615194018 z + 2359125236481046 z 40 62 / 28 - 1222187428436220 z + 26768 z ) / (1 + 15162070393844446 z / 26 2 24 - 7082945466275046 z - 359 z + 2648271993521006 z 22 4 6 8 - 788872639864640 z + 45209 z - 2855820 z + 106826550 z 10 12 14 18 - 2605566126 z + 44090592694 z - 540095205784 z - 34415964474035 z 16 50 48 + 4936480265545 z - 34415964474035 z + 186016693906721 z 20 36 34 + 186016693906721 z + 36039224846126532 z - 40147009276523172 z 66 64 30 42 - 359 z + 45209 z - 26060283198048412 z - 7082945466275046 z 44 46 58 + 2648271993521006 z - 788872639864640 z - 2605566126 z 56 54 52 60 + 44090592694 z - 540095205784 z + 4936480265545 z + 106826550 z 68 32 38 + z + 36039224846126532 z - 26060283198048412 z 40 62 + 15162070393844446 z - 2855820 z ) And in Maple-input format, it is: -(-1-2359125236481046*z^28+1222187428436220*z^26+263*z^2-506079676003830*z^24+ 166731157639006*z^22-26768*z^4+1435514*z^6-46856132*z^8+1013955714*z^10-\ 15379007394*z^12+169879402226*z^14+8870728345785*z^18-1404657794917*z^16+ 1404657794917*z^50-8870728345785*z^48-43432546043834*z^20-3651211181899828*z^36 +4540130615194018*z^34+z^66-263*z^64+3651211181899828*z^30+506079676003830*z^42 -166731157639006*z^44+43432546043834*z^46+46856132*z^58-1013955714*z^56+ 15379007394*z^54-169879402226*z^52-1435514*z^60-4540130615194018*z^32+ 2359125236481046*z^38-1222187428436220*z^40+26768*z^62)/(1+15162070393844446*z^ 28-7082945466275046*z^26-359*z^2+2648271993521006*z^24-788872639864640*z^22+ 45209*z^4-2855820*z^6+106826550*z^8-2605566126*z^10+44090592694*z^12-\ 540095205784*z^14-34415964474035*z^18+4936480265545*z^16-34415964474035*z^50+ 186016693906721*z^48+186016693906721*z^20+36039224846126532*z^36-\ 40147009276523172*z^34-359*z^66+45209*z^64-26060283198048412*z^30-\ 7082945466275046*z^42+2648271993521006*z^44-788872639864640*z^46-2605566126*z^ 58+44090592694*z^56-540095205784*z^54+4936480265545*z^52+106826550*z^60+z^68+ 36039224846126532*z^32-26060283198048412*z^38+15162070393844446*z^40-2855820*z^ 62) The first , 40, terms are: [0, 96, 0, 16023, 0, 2832499, 0, 506671636, 0, 90935735505, 0, 16338659300697, 0, 2936728422919580, 0, 527923282012949395, 0, 94907307559946742167, 0, 17062258420118452757048, 0, 3067441856175976732160473, 0, 551464115929872293900331273, 0, 99142206724067067807810984584, 0, 17823789906685481992311605048183, 0, 3204362074096116523173349516992243, 0, 576080443070220299784433985107174540, 0, 103567785103949611553048854801837249641, 0, 18619424265658377313889295517285765486465, 0, 3347401515499033018568817705867622858467364, 0, 601796100459980822799420186152924906321951699] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2621069599528105915 z - 613228090153566577 z - 289 z 24 22 4 6 + 119443084685412517 z - 19263548225373912 z + 35367 z - 2493374 z 8 10 12 14 + 115333168 z - 3764504842 z + 90841901737 z - 1673818346323 z 18 16 50 - 276335138642136 z + 24109609679943 z - 270569313708021864859 z 48 20 + 410710244234304537299 z + 2555025863839376 z 36 34 + 150678912588915656245 z - 70853229886476632186 z 66 80 88 84 86 - 19263548225373912 z + 115333168 z + z + 35367 z - 289 z 82 64 30 - 2493374 z + 119443084685412517 z - 9367250306748826654 z 42 44 - 527424443434169269936 z + 573254254945845742208 z 46 58 - 527424443434169269936 z - 9367250306748826654 z 56 54 + 28087630781100104640 z - 70853229886476632186 z 52 60 + 150678912588915656245 z + 2621069599528105915 z 70 68 78 - 276335138642136 z + 2555025863839376 z - 3764504842 z 32 38 + 28087630781100104640 z - 270569313708021864859 z 40 62 76 + 410710244234304537299 z - 613228090153566577 z + 90841901737 z 74 72 / 2 - 1673818346323 z + 24109609679943 z ) / ((-1 + z ) (1 / 28 26 2 + 9476497630969087518 z - 2119160139648091318 z - 374 z 24 22 4 6 + 392961887214043427 z - 60106071977894964 z + 53860 z - 4291812 z 8 10 12 14 + 219713553 z - 7837181048 z + 204918250918 z - 4065358185186 z 18 16 50 - 766855929448372 z + 62729481092213 z - 1144285684042487339330 z 48 20 + 1763712571143213423943 z + 7532618625371790 z 36 34 + 623969712646191569688 z - 285723210473451909492 z 66 80 88 84 86 - 60106071977894964 z + 219713553 z + z + 53860 z - 374 z 82 64 30 - 4291812 z + 392961887214043427 z - 35284466348939246336 z 42 44 - 2286011234959219348776 z + 2492383807753797805860 z 46 58 - 2286011234959219348776 z - 35284466348939246336 z 56 54 + 109733390308399345151 z - 285723210473451909492 z 52 60 + 623969712646191569688 z + 9476497630969087518 z 70 68 78 - 766855929448372 z + 7532618625371790 z - 7837181048 z 32 38 + 109733390308399345151 z - 1144285684042487339330 z 40 62 76 + 1763712571143213423943 z - 2119160139648091318 z + 204918250918 z 74 72 - 4065358185186 z + 62729481092213 z )) And in Maple-input format, it is: -(1+2621069599528105915*z^28-613228090153566577*z^26-289*z^2+119443084685412517 *z^24-19263548225373912*z^22+35367*z^4-2493374*z^6+115333168*z^8-3764504842*z^ 10+90841901737*z^12-1673818346323*z^14-276335138642136*z^18+24109609679943*z^16 -270569313708021864859*z^50+410710244234304537299*z^48+2555025863839376*z^20+ 150678912588915656245*z^36-70853229886476632186*z^34-19263548225373912*z^66+ 115333168*z^80+z^88+35367*z^84-289*z^86-2493374*z^82+119443084685412517*z^64-\ 9367250306748826654*z^30-527424443434169269936*z^42+573254254945845742208*z^44-\ 527424443434169269936*z^46-9367250306748826654*z^58+28087630781100104640*z^56-\ 70853229886476632186*z^54+150678912588915656245*z^52+2621069599528105915*z^60-\ 276335138642136*z^70+2555025863839376*z^68-3764504842*z^78+28087630781100104640 *z^32-270569313708021864859*z^38+410710244234304537299*z^40-613228090153566577* z^62+90841901737*z^76-1673818346323*z^74+24109609679943*z^72)/(-1+z^2)/(1+ 9476497630969087518*z^28-2119160139648091318*z^26-374*z^2+392961887214043427*z^ 24-60106071977894964*z^22+53860*z^4-4291812*z^6+219713553*z^8-7837181048*z^10+ 204918250918*z^12-4065358185186*z^14-766855929448372*z^18+62729481092213*z^16-\ 1144285684042487339330*z^50+1763712571143213423943*z^48+7532618625371790*z^20+ 623969712646191569688*z^36-285723210473451909492*z^34-60106071977894964*z^66+ 219713553*z^80+z^88+53860*z^84-374*z^86-4291812*z^82+392961887214043427*z^64-\ 35284466348939246336*z^30-2286011234959219348776*z^42+2492383807753797805860*z^ 44-2286011234959219348776*z^46-35284466348939246336*z^58+109733390308399345151* z^56-285723210473451909492*z^54+623969712646191569688*z^52+9476497630969087518* z^60-766855929448372*z^70+7532618625371790*z^68-7837181048*z^78+ 109733390308399345151*z^32-1144285684042487339330*z^38+1763712571143213423943*z ^40-2119160139648091318*z^62+204918250918*z^76-4065358185186*z^74+ 62729481092213*z^72) The first , 40, terms are: [0, 86, 0, 13383, 0, 2206799, 0, 366791598, 0, 61049369029, 0, 10164078089533, 0, 1692323876829198, 0, 281777290067784455, 0, 46917004781350304415, 0, 7811870460785595450230, 0, 1300708294138897271231305, 0, 216573252341425297954408233, 0, 36060333259803926254996676854, 0, 6004193166207234752813629596479, 0, 999722752653858792387079496157927, 0, 166457932795480262063796703743238318, 0, 27715927560707781471823325000357764093, 0, 4614815453193183892557419361239985996517, 0, 768385673563971700088789940664491805471246, 0, 127939361677175534618951048168948856292235983] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {3, 7}, {4, 5}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 64604263693 z + 123414714394 z + 214 z - 170192239475 z 22 4 6 8 10 + 170192239475 z - 16087 z + 575065 z - 11588110 z + 144211283 z 12 14 18 16 - 1168991481 z + 6392224770 z + 64604263693 z - 24192972931 z 20 36 34 30 - 123414714394 z - 144211283 z + 1168991481 z + 24192972931 z 42 44 46 32 38 40 + 16087 z - 214 z + z - 6392224770 z + 11588110 z - 575065 z ) / 28 26 2 24 / (1 + 763554295954 z - 1221671931154 z - 286 z + 1427864237366 z / 22 4 6 8 10 - 1221671931154 z + 27726 z - 1231724 z + 29867782 z - 437406046 z 12 14 18 16 + 4115444516 z - 25930270066 z - 346372973140 z + 112782090522 z 48 20 36 34 + z + 763554295954 z + 4115444516 z - 25930270066 z 30 42 44 46 32 - 346372973140 z - 1231724 z + 27726 z - 286 z + 112782090522 z 38 40 - 437406046 z + 29867782 z ) And in Maple-input format, it is: -(-1-64604263693*z^28+123414714394*z^26+214*z^2-170192239475*z^24+170192239475* z^22-16087*z^4+575065*z^6-11588110*z^8+144211283*z^10-1168991481*z^12+ 6392224770*z^14+64604263693*z^18-24192972931*z^16-123414714394*z^20-144211283*z ^36+1168991481*z^34+24192972931*z^30+16087*z^42-214*z^44+z^46-6392224770*z^32+ 11588110*z^38-575065*z^40)/(1+763554295954*z^28-1221671931154*z^26-286*z^2+ 1427864237366*z^24-1221671931154*z^22+27726*z^4-1231724*z^6+29867782*z^8-\ 437406046*z^10+4115444516*z^12-25930270066*z^14-346372973140*z^18+112782090522* z^16+z^48+763554295954*z^20+4115444516*z^36-25930270066*z^34-346372973140*z^30-\ 1231724*z^42+27726*z^44-286*z^46+112782090522*z^32-437406046*z^38+29867782*z^40 ) The first , 40, terms are: [0, 72, 0, 8953, 0, 1220945, 0, 171363848, 0, 24328478889, 0, 3471718701817, 0, 496625510633192, 0, 71124718131253153, 0, 10191947187093638825, 0, 1460872843051403885736, 0, 209423362334053224923121, 0, 30023800830880135077931537, 0, 4304470062770401798187361640, 0, 617135089929164605558364160457, 0, 88479763159740062320728554673537, 0, 12685546837350306197977958137712296, 0, 1818759087561007577238786146156158553, 0, 260760327962048890160978333427711204169, 0, 37385916766841158862593791960194672472392, 0, 5360121512611950930321563986021374912407153] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {3, 7}, {4, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2017210561062122209 z - 490316466713850425 z - 297 z 24 22 4 6 + 99311921575266217 z - 16655019347331888 z + 36897 z - 2614754 z 8 10 12 14 + 120527430 z - 3890036118 z + 92185139763 z - 1658189975067 z 18 16 50 - 257284872073072 z + 23201123510683 z - 179943284500386428867 z 48 20 + 269047123142820897819 z + 2294705092171228 z 36 34 + 102297383885913598251 z - 49350456282844149718 z 66 80 88 84 86 - 16655019347331888 z + 120527430 z + z + 36897 z - 297 z 82 64 30 - 2614754 z + 99311921575266217 z - 6952100162870532626 z 42 44 - 342331009156910474240 z + 370925059396598385096 z 46 58 - 342331009156910474240 z - 6952100162870532626 z 56 54 + 20158203327912726138 z - 49350456282844149718 z 52 60 + 102297383885913598251 z + 2017210561062122209 z 70 68 78 - 257284872073072 z + 2294705092171228 z - 3890036118 z 32 38 + 20158203327912726138 z - 179943284500386428867 z 40 62 76 + 269047123142820897819 z - 490316466713850425 z + 92185139763 z 74 72 / - 1658189975067 z + 23201123510683 z ) / (-1 / 28 26 2 - 8986914209129570988 z + 2039578598945566115 z + 373 z 24 22 4 6 - 385450307994197027 z + 60231901931685230 z - 54954 z + 4491862 z 8 10 12 14 - 234616715 z + 8467156675 z - 221992884272 z + 4379828990304 z 18 16 50 + 801919386428037 z - 66751902932045 z + 1830204484835747429531 z 48 20 - 2511741832972539561079 z - 7716014862209454 z 36 34 - 601489097109709556166 z + 270379781170412382885 z 66 80 90 88 84 + 385450307994197027 z - 8467156675 z + z - 373 z - 4491862 z 86 82 64 + 54954 z + 234616715 z - 2039578598945566115 z 30 42 + 33167404814247356244 z + 2511741832972539561079 z 44 46 - 2941806673817690583460 z + 2941806673817690583460 z 58 56 + 103021958477607806157 z - 270379781170412382885 z 54 52 + 601489097109709556166 z - 1137072856385106118858 z 60 70 68 - 33167404814247356244 z + 7716014862209454 z - 60231901931685230 z 78 32 + 221992884272 z - 103021958477607806157 z 38 40 + 1137072856385106118858 z - 1830204484835747429531 z 62 76 74 + 8986914209129570988 z - 4379828990304 z + 66751902932045 z 72 - 801919386428037 z ) And in Maple-input format, it is: -(1+2017210561062122209*z^28-490316466713850425*z^26-297*z^2+99311921575266217* z^24-16655019347331888*z^22+36897*z^4-2614754*z^6+120527430*z^8-3890036118*z^10 +92185139763*z^12-1658189975067*z^14-257284872073072*z^18+23201123510683*z^16-\ 179943284500386428867*z^50+269047123142820897819*z^48+2294705092171228*z^20+ 102297383885913598251*z^36-49350456282844149718*z^34-16655019347331888*z^66+ 120527430*z^80+z^88+36897*z^84-297*z^86-2614754*z^82+99311921575266217*z^64-\ 6952100162870532626*z^30-342331009156910474240*z^42+370925059396598385096*z^44-\ 342331009156910474240*z^46-6952100162870532626*z^58+20158203327912726138*z^56-\ 49350456282844149718*z^54+102297383885913598251*z^52+2017210561062122209*z^60-\ 257284872073072*z^70+2294705092171228*z^68-3890036118*z^78+20158203327912726138 *z^32-179943284500386428867*z^38+269047123142820897819*z^40-490316466713850425* z^62+92185139763*z^76-1658189975067*z^74+23201123510683*z^72)/(-1-\ 8986914209129570988*z^28+2039578598945566115*z^26+373*z^2-385450307994197027*z^ 24+60231901931685230*z^22-54954*z^4+4491862*z^6-234616715*z^8+8467156675*z^10-\ 221992884272*z^12+4379828990304*z^14+801919386428037*z^18-66751902932045*z^16+ 1830204484835747429531*z^50-2511741832972539561079*z^48-7716014862209454*z^20-\ 601489097109709556166*z^36+270379781170412382885*z^34+385450307994197027*z^66-\ 8467156675*z^80+z^90-373*z^88-4491862*z^84+54954*z^86+234616715*z^82-\ 2039578598945566115*z^64+33167404814247356244*z^30+2511741832972539561079*z^42-\ 2941806673817690583460*z^44+2941806673817690583460*z^46+103021958477607806157*z ^58-270379781170412382885*z^56+601489097109709556166*z^54-\ 1137072856385106118858*z^52-33167404814247356244*z^60+7716014862209454*z^70-\ 60231901931685230*z^68+221992884272*z^78-103021958477607806157*z^32+ 1137072856385106118858*z^38-1830204484835747429531*z^40+8986914209129570988*z^ 62-4379828990304*z^76+66751902932045*z^74-801919386428037*z^72) The first , 40, terms are: [0, 76, 0, 10291, 0, 1539147, 0, 235862444, 0, 36366409433, 0, 5615977441373, 0, 867617547192660, 0, 134053527841687939, 0, 20712888236047853263, 0, 3200415938515478133204, 0, 494507736471514796028673, 0, 76408205681046322410207425, 0, 11806114165069672989240924756, 0, 1824206396405396607528966408135, 0, 281864885317412546972871292609571, 0, 43551987310992874716137681448827076, 0, 6729378859244483879082500710118550709, 0, 1039781250831656600444941810871294430545, 0, 160660452066905565410565717133310646322108, 0, 24824241481751518811119452968602059621896603] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 343345 z + 4792454 z + 202 z - 36698675 z + 160814876 z 4 6 8 10 12 - 12571 z + 343345 z - 4792454 z + 36698675 z - 160814876 z 14 18 16 20 34 + 419779616 z + 672841476 z - 672841476 z - 419779616 z + z 30 32 / 36 34 32 30 + 12571 z - 202 z ) / (z - 272 z + 23912 z - 873974 z / 28 26 24 22 + 15165672 z - 138738256 z + 719858443 z - 2244849280 z 20 18 16 14 + 4376977808 z - 5455350060 z + 4376977808 z - 2244849280 z 12 10 8 6 4 + 719858443 z - 138738256 z + 15165672 z - 873974 z + 23912 z 2 - 272 z + 1) And in Maple-input format, it is: -(-1-343345*z^28+4792454*z^26+202*z^2-36698675*z^24+160814876*z^22-12571*z^4+ 343345*z^6-4792454*z^8+36698675*z^10-160814876*z^12+419779616*z^14+672841476*z^ 18-672841476*z^16-419779616*z^20+z^34+12571*z^30-202*z^32)/(z^36-272*z^34+23912 *z^32-873974*z^30+15165672*z^28-138738256*z^26+719858443*z^24-2244849280*z^22+ 4376977808*z^20-5455350060*z^18+4376977808*z^16-2244849280*z^14+719858443*z^12-\ 138738256*z^10+15165672*z^8-873974*z^6+23912*z^4-272*z^2+1) The first , 40, terms are: [0, 70, 0, 7699, 0, 950917, 0, 125355898, 0, 17127645319, 0, 2384678153575, 0, 335232283993498, 0, 47355307306981669, 0, 6705687780724217875, 0, 950694143984988365350, 0, 134864479329402996520321, 0, 19137392538212141587532353, 0, 2716010994960826685891084518, 0, 385488781664405823194289837139, 0, 54715142102164634739898549376677, 0, 7766243825025547936985181536611546, 0, 1102346993230714890202155359573020903, 0, 156468712036698343937767262299903153543, 0, 22209440927276987192406207758809526815546, 0, 3152449994578122133638758012651345472937093] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 71141886 z + 429873491 z + 197 z - 1615245543 z 22 4 6 8 10 + 3845502388 z - 13390 z + 421186 z - 7160274 z + 71141886 z 12 14 18 16 - 429873491 z + 1615245543 z + 5902935532 z - 3845502388 z 20 36 34 30 32 38 - 5902935532 z - 197 z + 13390 z + 7160274 z - 421186 z + z ) / 26 32 34 12 / (-7651564079 z + 20350978 z - 980990 z + 1716834635 z + 1 / 6 14 30 16 36 - 980990 z - 7651564079 z - 240658878 z + 21835935847 z + 24525 z 38 40 8 20 22 2 - 273 z + z + 20350978 z + 49919754620 z - 40639559876 z - 273 z 10 18 24 4 - 240658878 z - 40639559876 z + 21835935847 z + 24525 z 28 + 1716834635 z ) And in Maple-input format, it is: -(-1-71141886*z^28+429873491*z^26+197*z^2-1615245543*z^24+3845502388*z^22-13390 *z^4+421186*z^6-7160274*z^8+71141886*z^10-429873491*z^12+1615245543*z^14+ 5902935532*z^18-3845502388*z^16-5902935532*z^20-197*z^36+13390*z^34+7160274*z^ 30-421186*z^32+z^38)/(-7651564079*z^26+20350978*z^32-980990*z^34+1716834635*z^ 12+1-980990*z^6-7651564079*z^14-240658878*z^30+21835935847*z^16+24525*z^36-273* z^38+z^40+20350978*z^8+49919754620*z^20-40639559876*z^22-273*z^2-240658878*z^10 -40639559876*z^18+21835935847*z^24+24525*z^4+1716834635*z^28) The first , 40, terms are: [0, 76, 0, 9613, 0, 1320253, 0, 186034780, 0, 26461389649, 0, 3777980547217, 0, 540241938003772, 0, 77305235274036925, 0, 11065117873725155917, 0, 1584011296718490120556, 0, 226769448851874409416193, 0, 32465441637887037715693441, 0, 4647963127863250292212389676, 0, 665435611128673495856009398861, 0, 95268708920272556278417392670525, 0, 13639388627436234596517216550335868, 0, 1952718746597538061931723152081645969, 0, 279566133809592996137499898116746401105, 0, 40024826447001612511540256270574712851484, 0, 5730260535789793091362205501891546665570237] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 4645072449253372 z + 2789008359384698 z + 293 z 24 22 4 6 - 1292576814773146 z + 460225600934780 z - 34856 z + 2229086 z 8 10 12 14 - 86208958 z + 2167129500 z - 37190040689 z + 451061920405 z 18 16 50 + 25735113765872 z - 3964958526724 z + 37190040689 z 48 20 36 - 451061920405 z - 125085301791952 z - 2789008359384698 z 34 30 42 + 4645072449253372 z + 5990034577547172 z + 125085301791952 z 44 46 58 56 - 25735113765872 z + 3964958526724 z + 34856 z - 2229086 z 54 52 60 32 + 86208958 z - 2167129500 z - 293 z - 5990034577547172 z 38 40 62 / + 1292576814773146 z - 460225600934780 z + z ) / (1 / 28 26 2 + 31747368690632602 z - 16927140257422866 z - 389 z 24 22 4 6 + 6986215866269618 z - 2221145040940520 z + 56565 z - 4244178 z 8 10 12 14 + 188437862 z - 5370043618 z + 103624281233 z - 1405792699945 z 18 16 50 - 99687958103944 z + 13782315801645 z - 1405792699945 z 48 20 36 + 13782315801645 z + 540351789751224 z + 31747368690632602 z 34 64 30 - 46241995083662492 z + z - 46241995083662492 z 42 44 46 - 2221145040940520 z + 540351789751224 z - 99687958103944 z 58 56 54 52 - 4244178 z + 188437862 z - 5370043618 z + 103624281233 z 60 32 38 + 56565 z + 52407115428042500 z - 16927140257422866 z 40 62 + 6986215866269618 z - 389 z ) And in Maple-input format, it is: -(-1-4645072449253372*z^28+2789008359384698*z^26+293*z^2-1292576814773146*z^24+ 460225600934780*z^22-34856*z^4+2229086*z^6-86208958*z^8+2167129500*z^10-\ 37190040689*z^12+451061920405*z^14+25735113765872*z^18-3964958526724*z^16+ 37190040689*z^50-451061920405*z^48-125085301791952*z^20-2789008359384698*z^36+ 4645072449253372*z^34+5990034577547172*z^30+125085301791952*z^42-25735113765872 *z^44+3964958526724*z^46+34856*z^58-2229086*z^56+86208958*z^54-2167129500*z^52-\ 293*z^60-5990034577547172*z^32+1292576814773146*z^38-460225600934780*z^40+z^62) /(1+31747368690632602*z^28-16927140257422866*z^26-389*z^2+6986215866269618*z^24 -2221145040940520*z^22+56565*z^4-4244178*z^6+188437862*z^8-5370043618*z^10+ 103624281233*z^12-1405792699945*z^14-99687958103944*z^18+13782315801645*z^16-\ 1405792699945*z^50+13782315801645*z^48+540351789751224*z^20+31747368690632602*z ^36-46241995083662492*z^34+z^64-46241995083662492*z^30-2221145040940520*z^42+ 540351789751224*z^44-99687958103944*z^46-4244178*z^58+188437862*z^56-5370043618 *z^54+103624281233*z^52+56565*z^60+52407115428042500*z^32-16927140257422866*z^ 38+6986215866269618*z^40-389*z^62) The first , 40, terms are: [0, 96, 0, 15635, 0, 2666867, 0, 458229672, 0, 78870612949, 0, 13582429265221, 0, 2339485771590648, 0, 402989119607802515, 0, 69418884221752079027, 0, 11958214164983934316368, 0, 2059950238724549691245953, 0, 354852440758546391148545665, 0, 61127850506316975381092835696, 0, 10530052942822386754507730397587, 0, 1813936272065016133946020170907955, 0, 312473729940566057097240420570343896, 0, 53827598476017109102684934668334804773, 0, 9272492682164260495069601920418652104053, 0, 1597305530420634623439783143442786487318728, 0, 275156319583908679566686597246847727234353299] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 27461299389458088 z - 12371128538455454 z - 287 z 24 22 4 6 + 4410534658027650 z - 1238351961993664 z + 33796 z - 2185221 z 8 10 12 14 + 87783643 z - 2353809440 z + 44190176687 z - 600936994609 z 18 16 50 - 46381717664211 z + 6069106546516 z - 46381717664211 z 48 20 36 + 272066723114413 z + 272066723114413 z + 67972527392864582 z 34 66 64 30 - 76098038897207360 z - 287 z + 33796 z - 48417056588407354 z 42 44 46 - 12371128538455454 z + 4410534658027650 z - 1238351961993664 z 58 56 54 52 - 2353809440 z + 44190176687 z - 600936994609 z + 6069106546516 z 60 68 32 38 + 87783643 z + z + 67972527392864582 z - 48417056588407354 z 40 62 / 28 + 27461299389458088 z - 2185221 z ) / (-1 - 164110015609378266 z / 26 2 24 + 66600895677215846 z + 375 z - 21421134413818986 z 22 4 6 8 + 5431880965447525 z - 53401 z + 4012723 z - 183401149 z 10 12 14 + 5529996595 z - 115944887247 z + 1753303165985 z 18 16 50 + 166164753736221 z - 19637072242191 z + 1078483045037523 z 48 20 36 - 5431880965447525 z - 1078483045037523 z - 629080368891833134 z 34 66 64 + 629080368891833134 z + 53401 z - 4012723 z 30 42 44 + 321733271900288686 z + 164110015609378266 z - 66600895677215846 z 46 58 56 + 21421134413818986 z + 115944887247 z - 1753303165985 z 54 52 60 70 + 19637072242191 z - 166164753736221 z - 5529996595 z + z 68 32 38 - 375 z - 503225129936236914 z + 503225129936236914 z 40 62 - 321733271900288686 z + 183401149 z ) And in Maple-input format, it is: -(1+27461299389458088*z^28-12371128538455454*z^26-287*z^2+4410534658027650*z^24 -1238351961993664*z^22+33796*z^4-2185221*z^6+87783643*z^8-2353809440*z^10+ 44190176687*z^12-600936994609*z^14-46381717664211*z^18+6069106546516*z^16-\ 46381717664211*z^50+272066723114413*z^48+272066723114413*z^20+67972527392864582 *z^36-76098038897207360*z^34-287*z^66+33796*z^64-48417056588407354*z^30-\ 12371128538455454*z^42+4410534658027650*z^44-1238351961993664*z^46-2353809440*z ^58+44190176687*z^56-600936994609*z^54+6069106546516*z^52+87783643*z^60+z^68+ 67972527392864582*z^32-48417056588407354*z^38+27461299389458088*z^40-2185221*z^ 62)/(-1-164110015609378266*z^28+66600895677215846*z^26+375*z^2-\ 21421134413818986*z^24+5431880965447525*z^22-53401*z^4+4012723*z^6-183401149*z^ 8+5529996595*z^10-115944887247*z^12+1753303165985*z^14+166164753736221*z^18-\ 19637072242191*z^16+1078483045037523*z^50-5431880965447525*z^48-\ 1078483045037523*z^20-629080368891833134*z^36+629080368891833134*z^34+53401*z^ 66-4012723*z^64+321733271900288686*z^30+164110015609378266*z^42-\ 66600895677215846*z^44+21421134413818986*z^46+115944887247*z^58-1753303165985*z ^56+19637072242191*z^54-166164753736221*z^52-5529996595*z^60+z^70-375*z^68-\ 503225129936236914*z^32+503225129936236914*z^38-321733271900288686*z^40+ 183401149*z^62) The first , 40, terms are: [0, 88, 0, 13395, 0, 2151339, 0, 348947848, 0, 56759099689, 0, 9241452437369, 0, 1505248512986344, 0, 245211037571930971, 0, 39948188767233493411, 0, 6508250139770772348792, 0, 1060316211090540129422001, 0, 172746070358605666951700689, 0, 28143726253830901407491234104, 0, 4585168265971770794437170812035, 0, 747014552053751699051863869845115, 0, 121703449525224530177275640928632360, 0, 19827900483877958243145445444884662617, 0, 3230357440383931197487486322486479771209, 0, 526289168443042866515945482316559169697864, 0, 85742923074308171720922832230280517909825419] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4012901933692018505 z - 938259348872383125 z - 313 z 24 22 4 6 + 182300551555529385 z - 29265689759811104 z + 41425 z - 3120370 z 8 10 12 14 + 152056034 z - 5165402254 z + 128498279451 z - 2423250159495 z 18 16 50 - 412712966536720 z + 35528040537947 z - 409563397864107615403 z 48 20 + 620241154686645733115 z + 3854240033859060 z 36 34 + 228735666663562746315 z - 107879121844963980582 z 66 80 88 84 86 - 29265689759811104 z + 152056034 z + z + 41425 z - 313 z 82 64 30 - 3120370 z + 182300551555529385 z - 14329171832866933786 z 42 44 - 795277885297086346512 z + 863921355926875074520 z 46 58 - 795277885297086346512 z - 14329171832866933786 z 56 54 + 42880925323460679998 z - 107879121844963980582 z 52 60 + 228735666663562746315 z + 4012901933692018505 z 70 68 78 - 412712966536720 z + 3854240033859060 z - 5165402254 z 32 38 + 42880925323460679998 z - 409563397864107615403 z 40 62 76 + 620241154686645733115 z - 938259348872383125 z + 128498279451 z 74 72 / - 2423250159495 z + 35528040537947 z ) / (-1 / 28 26 2 - 17992367174200712696 z + 3892477175662502371 z + 405 z 24 22 4 6 - 700217846811152103 z + 104119324409372106 z - 64166 z + 5544374 z 8 10 12 14 - 302375831 z + 11328835611 z - 308055670060 z + 6314849894580 z 18 16 50 + 1259513827876393 z - 100293622206029 z + 4491896993003910700763 z 48 20 - 6249683481861023762447 z - 12702356101929594 z 36 34 - 1408748093755101625134 z + 613144810717499402881 z 66 80 90 88 84 + 700217846811152103 z - 11328835611 z + z - 405 z - 5544374 z 86 82 64 + 64166 z + 302375831 z - 3892477175662502371 z 30 42 + 69483736551691385040 z + 6249683481861023762447 z 44 46 - 7370505295584598917788 z + 7370505295584598917788 z 58 56 + 225043131618431885037 z - 613144810717499402881 z 54 52 + 1408748093755101625134 z - 2734699097737064586766 z 60 70 - 69483736551691385040 z + 12702356101929594 z 68 78 32 - 104119324409372106 z + 308055670060 z - 225043131618431885037 z 38 40 + 2734699097737064586766 z - 4491896993003910700763 z 62 76 74 + 17992367174200712696 z - 6314849894580 z + 100293622206029 z 72 - 1259513827876393 z ) And in Maple-input format, it is: -(1+4012901933692018505*z^28-938259348872383125*z^26-313*z^2+182300551555529385 *z^24-29265689759811104*z^22+41425*z^4-3120370*z^6+152056034*z^8-5165402254*z^ 10+128498279451*z^12-2423250159495*z^14-412712966536720*z^18+35528040537947*z^ 16-409563397864107615403*z^50+620241154686645733115*z^48+3854240033859060*z^20+ 228735666663562746315*z^36-107879121844963980582*z^34-29265689759811104*z^66+ 152056034*z^80+z^88+41425*z^84-313*z^86-3120370*z^82+182300551555529385*z^64-\ 14329171832866933786*z^30-795277885297086346512*z^42+863921355926875074520*z^44 -795277885297086346512*z^46-14329171832866933786*z^58+42880925323460679998*z^56 -107879121844963980582*z^54+228735666663562746315*z^52+4012901933692018505*z^60 -412712966536720*z^70+3854240033859060*z^68-5165402254*z^78+ 42880925323460679998*z^32-409563397864107615403*z^38+620241154686645733115*z^40 -938259348872383125*z^62+128498279451*z^76-2423250159495*z^74+35528040537947*z^ 72)/(-1-17992367174200712696*z^28+3892477175662502371*z^26+405*z^2-\ 700217846811152103*z^24+104119324409372106*z^22-64166*z^4+5544374*z^6-302375831 *z^8+11328835611*z^10-308055670060*z^12+6314849894580*z^14+1259513827876393*z^ 18-100293622206029*z^16+4491896993003910700763*z^50-6249683481861023762447*z^48 -12702356101929594*z^20-1408748093755101625134*z^36+613144810717499402881*z^34+ 700217846811152103*z^66-11328835611*z^80+z^90-405*z^88-5544374*z^84+64166*z^86+ 302375831*z^82-3892477175662502371*z^64+69483736551691385040*z^30+ 6249683481861023762447*z^42-7370505295584598917788*z^44+7370505295584598917788* z^46+225043131618431885037*z^58-613144810717499402881*z^56+ 1408748093755101625134*z^54-2734699097737064586766*z^52-69483736551691385040*z^ 60+12702356101929594*z^70-104119324409372106*z^68+308055670060*z^78-\ 225043131618431885037*z^32+2734699097737064586766*z^38-4491896993003910700763*z ^40+17992367174200712696*z^62-6314849894580*z^76+100293622206029*z^74-\ 1259513827876393*z^72) The first , 40, terms are: [0, 92, 0, 14519, 0, 2400927, 0, 400511892, 0, 66993057389, 0, 11217080210485, 0, 1878880231049796, 0, 314764021820427639, 0, 52734825709636212207, 0, 8835282673029873943820, 0, 1480292578084320963098041, 0, 248014081077976614129945961, 0, 41553324815025817547741697452, 0, 6962023471791039765274663366975, 0, 1166447758264343499280896796277703, 0, 195431760588621774579405887103068132, 0, 32743493439076217483047154490502310149, 0, 5485988438393013678504553345885597252445, 0, 919146559201857402300211536051814008282356, 0, 153997845392223629328676036218527892274401103] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 35078941617656 z - 30645026148162 z - 219 z 24 22 4 6 + 20417814413706 z - 10353685905332 z + 18396 z - 816835 z 8 10 12 14 + 21974693 z - 386672268 z + 4673533732 z - 40114833280 z 18 16 50 48 - 1154367290208 z + 250306503400 z - 816835 z + 21974693 z 20 36 34 + 3981294905068 z + 3981294905068 z - 10353685905332 z 30 42 44 46 - 30645026148162 z - 40114833280 z + 4673533732 z - 386672268 z 56 54 52 32 38 + z - 219 z + 18396 z + 20417814413706 z - 1154367290208 z 40 / 2 28 + 250306503400 z ) / ((-1 + z ) (1 + 155643577586618 z / 26 2 24 22 - 134770323645836 z - 310 z + 87497584035898 z - 42580996268904 z 4 6 8 10 12 + 31743 z - 1637502 z + 50089293 z - 988562968 z + 13251505256 z 14 18 16 50 - 124885003112 z - 4213246351832 z + 847622083236 z - 1637502 z 48 20 36 + 50089293 z + 15510320410000 z + 15510320410000 z 34 30 42 - 42580996268904 z - 134770323645836 z - 124885003112 z 44 46 56 54 52 + 13251505256 z - 988562968 z + z - 310 z + 31743 z 32 38 40 + 87497584035898 z - 4213246351832 z + 847622083236 z )) And in Maple-input format, it is: -(1+35078941617656*z^28-30645026148162*z^26-219*z^2+20417814413706*z^24-\ 10353685905332*z^22+18396*z^4-816835*z^6+21974693*z^8-386672268*z^10+4673533732 *z^12-40114833280*z^14-1154367290208*z^18+250306503400*z^16-816835*z^50+ 21974693*z^48+3981294905068*z^20+3981294905068*z^36-10353685905332*z^34-\ 30645026148162*z^30-40114833280*z^42+4673533732*z^44-386672268*z^46+z^56-219*z^ 54+18396*z^52+20417814413706*z^32-1154367290208*z^38+250306503400*z^40)/(-1+z^2 )/(1+155643577586618*z^28-134770323645836*z^26-310*z^2+87497584035898*z^24-\ 42580996268904*z^22+31743*z^4-1637502*z^6+50089293*z^8-988562968*z^10+ 13251505256*z^12-124885003112*z^14-4213246351832*z^18+847622083236*z^16-1637502 *z^50+50089293*z^48+15510320410000*z^20+15510320410000*z^36-42580996268904*z^34 -134770323645836*z^30-124885003112*z^42+13251505256*z^44-988562968*z^46+z^56-\ 310*z^54+31743*z^52+87497584035898*z^32-4213246351832*z^38+847622083236*z^40) The first , 40, terms are: [0, 92, 0, 14955, 0, 2554539, 0, 438927452, 0, 75482472833, 0, 12982674139457, 0, 2233032275057948, 0, 384086117231711019, 0, 66063692890347703915, 0, 11363110474809811142172, 0, 1954481867292416003737409, 0, 336175508388756614751826625, 0, 57822983590021261248647351708, 0, 9945690127099853951631210206571, 0, 1710682257945033695172886743662891, 0, 294241400086142788701197837874588316, 0, 50610217720893744328746398919200841665, 0, 8705077317518621242597108379520338171649, 0, 1497293916456579575983000969105903546669276, 0, 257538099948473958754581628277369890377233067] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4009638374788096047 z - 937394796045312753 z - 313 z 24 22 4 6 + 182168628317285589 z - 29256694506890592 z + 41355 z - 3112178 z 8 10 12 14 + 151713420 z - 5159125582 z + 128476980389 z - 2424536939667 z 18 16 50 - 412964333141104 z + 35555387373719 z - 410998844675850683267 z 48 20 + 622957860352521334099 z + 3854981340489640 z 36 34 + 229296691421724603361 z - 108026413442711532598 z 66 80 88 84 86 - 29256694506890592 z + 151713420 z + z + 41355 z - 313 z 82 64 30 - 3112178 z + 182168628317285589 z - 14323891081691163434 z 42 44 - 799221478907676736336 z + 868380051409328079888 z 46 58 - 799221478907676736336 z - 14323891081691163434 z 56 54 + 42897419845285971268 z - 108026413442711532598 z 52 60 + 229296691421724603361 z + 4009638374788096047 z 70 68 78 - 412964333141104 z + 3854981340489640 z - 5159125582 z 32 38 + 42897419845285971268 z - 410998844675850683267 z 40 62 76 + 622957860352521334099 z - 937394796045312753 z + 128476980389 z 74 72 / 2 - 2424536939667 z + 35555387373719 z ) / ((-1 + z ) (1 / 28 26 2 + 14659535218169243882 z - 3275776996731644542 z - 402 z 24 22 4 6 + 606489167835166379 z - 92513228922089128 z + 63456 z - 5462308 z 8 10 12 14 + 296567577 z - 11042787260 z + 297721443146 z - 6034893666290 z 18 16 50 - 1167203410197240 z + 94515589968029 z - 1767562612788429812430 z 48 20 + 2723226642782214282903 z + 11543030931712438 z 36 34 + 964328488218087796396 z - 441804063982090942652 z 66 80 88 84 86 - 92513228922089128 z + 296567577 z + z + 63456 z - 402 z 82 64 30 - 5462308 z + 606489167835166379 z - 54592461997733941300 z 42 44 - 3528685043770378280432 z + 3846866755992853290196 z 46 58 - 3528685043770378280432 z - 54592461997733941300 z 56 54 + 169747474818858734071 z - 441804063982090942652 z 52 60 + 964328488218087796396 z + 14659535218169243882 z 70 68 78 - 1167203410197240 z + 11543030931712438 z - 11042787260 z 32 38 + 169747474818858734071 z - 1767562612788429812430 z 40 62 76 + 2723226642782214282903 z - 3275776996731644542 z + 297721443146 z 74 72 - 6034893666290 z + 94515589968029 z )) And in Maple-input format, it is: -(1+4009638374788096047*z^28-937394796045312753*z^26-313*z^2+182168628317285589 *z^24-29256694506890592*z^22+41355*z^4-3112178*z^6+151713420*z^8-5159125582*z^ 10+128476980389*z^12-2424536939667*z^14-412964333141104*z^18+35555387373719*z^ 16-410998844675850683267*z^50+622957860352521334099*z^48+3854981340489640*z^20+ 229296691421724603361*z^36-108026413442711532598*z^34-29256694506890592*z^66+ 151713420*z^80+z^88+41355*z^84-313*z^86-3112178*z^82+182168628317285589*z^64-\ 14323891081691163434*z^30-799221478907676736336*z^42+868380051409328079888*z^44 -799221478907676736336*z^46-14323891081691163434*z^58+42897419845285971268*z^56 -108026413442711532598*z^54+229296691421724603361*z^52+4009638374788096047*z^60 -412964333141104*z^70+3854981340489640*z^68-5159125582*z^78+ 42897419845285971268*z^32-410998844675850683267*z^38+622957860352521334099*z^40 -937394796045312753*z^62+128476980389*z^76-2424536939667*z^74+35555387373719*z^ 72)/(-1+z^2)/(1+14659535218169243882*z^28-3275776996731644542*z^26-402*z^2+ 606489167835166379*z^24-92513228922089128*z^22+63456*z^4-5462308*z^6+296567577* z^8-11042787260*z^10+297721443146*z^12-6034893666290*z^14-1167203410197240*z^18 +94515589968029*z^16-1767562612788429812430*z^50+2723226642782214282903*z^48+ 11543030931712438*z^20+964328488218087796396*z^36-441804063982090942652*z^34-\ 92513228922089128*z^66+296567577*z^80+z^88+63456*z^84-402*z^86-5462308*z^82+ 606489167835166379*z^64-54592461997733941300*z^30-3528685043770378280432*z^42+ 3846866755992853290196*z^44-3528685043770378280432*z^46-54592461997733941300*z^ 58+169747474818858734071*z^56-441804063982090942652*z^54+964328488218087796396* z^52+14659535218169243882*z^60-1167203410197240*z^70+11543030931712438*z^68-\ 11042787260*z^78+169747474818858734071*z^32-1767562612788429812430*z^38+ 2723226642782214282903*z^40-3275776996731644542*z^62+297721443146*z^76-\ 6034893666290*z^74+94515589968029*z^72) The first , 40, terms are: [0, 90, 0, 13767, 0, 2214467, 0, 360299410, 0, 58859961137, 0, 9631395900737, 0, 1577095640837618, 0, 258317738944078643, 0, 42316023788471395143, 0, 6932322536783824427162, 0, 1135697420437761289670225, 0, 186059045410363085713674481, 0, 30481811066920735520445909242, 0, 4993804416048481335723788907175, 0, 818130581913618619094299638086643, 0, 134033657509389663806894464829312370, 0, 21958626522448573705311020230774930785, 0, 3597464375963242898771337623241893727761, 0, 589369752258071428507586015362122967101010, 0, 96555982895774758017725055532809968176993667] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4592873521832481573 z - 1072751847785306249 z - 317 z 24 22 4 6 + 208092568274645893 z - 33320401722901720 z + 42537 z - 3253372 z 8 10 12 14 + 161171688 z - 5565160112 z + 140519712419 z - 2683813032351 z 18 16 50 - 465409533487176 z + 39752583406227 z - 470110293316448634683 z 48 20 + 712237368080071929591 z + 4371092359220376 z 36 34 + 262416453009587109135 z - 123697353256599509868 z 66 80 88 84 86 - 33320401722901720 z + 161171688 z + z + 42537 z - 317 z 82 64 30 - 3253372 z + 208092568274645893 z - 16411821873023089976 z 42 44 - 913497314164494614288 z + 992444703056874406928 z 46 58 - 913497314164494614288 z - 16411821873023089976 z 56 54 + 49141773265359757064 z - 123697353256599509868 z 52 60 + 262416453009587109135 z + 4592873521832481573 z 70 68 78 - 465409533487176 z + 4371092359220376 z - 5565160112 z 32 38 + 49141773265359757064 z - 470110293316448634683 z 40 62 76 + 712237368080071929591 z - 1072751847785306249 z + 140519712419 z 74 72 / 2 - 2683813032351 z + 39752583406227 z ) / ((-1 + z ) (1 / 28 26 2 + 16973028133687560592 z - 3782942540218955628 z - 414 z 24 22 4 6 + 697890741814832011 z - 105954616822104602 z + 66434 z - 5789054 z 8 10 12 14 + 317951199 z - 11977362636 z + 326653329868 z - 6694079124392 z 18 16 50 - 1319068797136306 z + 105887227740521 z - 2054765771099153665558 z 48 20 + 3165752520303848637555 z + 13141137279105286 z 36 34 + 1120881896436310338938 z - 513359516543362719758 z 66 80 88 84 86 - 105954616822104602 z + 317951199 z + z + 66434 z - 414 z 82 64 30 - 5789054 z + 697890741814832011 z - 63320360182735340656 z 42 44 - 4102007911504274127548 z + 4471830830175340137324 z 46 58 - 4102007911504274127548 z - 63320360182735340656 z 56 54 + 197109784264287127101 z - 513359516543362719758 z 52 60 + 1120881896436310338938 z + 16973028133687560592 z 70 68 78 - 1319068797136306 z + 13141137279105286 z - 11977362636 z 32 38 + 197109784264287127101 z - 2054765771099153665558 z 40 62 76 + 3165752520303848637555 z - 3782942540218955628 z + 326653329868 z 74 72 - 6694079124392 z + 105887227740521 z )) And in Maple-input format, it is: -(1+4592873521832481573*z^28-1072751847785306249*z^26-317*z^2+ 208092568274645893*z^24-33320401722901720*z^22+42537*z^4-3253372*z^6+161171688* z^8-5565160112*z^10+140519712419*z^12-2683813032351*z^14-465409533487176*z^18+ 39752583406227*z^16-470110293316448634683*z^50+712237368080071929591*z^48+ 4371092359220376*z^20+262416453009587109135*z^36-123697353256599509868*z^34-\ 33320401722901720*z^66+161171688*z^80+z^88+42537*z^84-317*z^86-3253372*z^82+ 208092568274645893*z^64-16411821873023089976*z^30-913497314164494614288*z^42+ 992444703056874406928*z^44-913497314164494614288*z^46-16411821873023089976*z^58 +49141773265359757064*z^56-123697353256599509868*z^54+262416453009587109135*z^ 52+4592873521832481573*z^60-465409533487176*z^70+4371092359220376*z^68-\ 5565160112*z^78+49141773265359757064*z^32-470110293316448634683*z^38+ 712237368080071929591*z^40-1072751847785306249*z^62+140519712419*z^76-\ 2683813032351*z^74+39752583406227*z^72)/(-1+z^2)/(1+16973028133687560592*z^28-\ 3782942540218955628*z^26-414*z^2+697890741814832011*z^24-105954616822104602*z^ 22+66434*z^4-5789054*z^6+317951199*z^8-11977362636*z^10+326653329868*z^12-\ 6694079124392*z^14-1319068797136306*z^18+105887227740521*z^16-\ 2054765771099153665558*z^50+3165752520303848637555*z^48+13141137279105286*z^20+ 1120881896436310338938*z^36-513359516543362719758*z^34-105954616822104602*z^66+ 317951199*z^80+z^88+66434*z^84-414*z^86-5789054*z^82+697890741814832011*z^64-\ 63320360182735340656*z^30-4102007911504274127548*z^42+4471830830175340137324*z^ 44-4102007911504274127548*z^46-63320360182735340656*z^58+197109784264287127101* z^56-513359516543362719758*z^54+1120881896436310338938*z^52+ 16973028133687560592*z^60-1319068797136306*z^70+13141137279105286*z^68-\ 11977362636*z^78+197109784264287127101*z^32-2054765771099153665558*z^38+ 3165752520303848637555*z^40-3782942540218955628*z^62+326653329868*z^76-\ 6694079124392*z^74+105887227740521*z^72) The first , 40, terms are: [0, 98, 0, 16359, 0, 2839997, 0, 496301582, 0, 86910574195, 0, 15231451467843, 0, 2670211056433542, 0, 468170743212347333, 0, 82088972044242800627, 0, 14393756937748687117818, 0, 2523870429424942901160881, 0, 442549009305627666537621005, 0, 77599025091945817179334750770, 0, 13606655483532185195065895116223, 0, 2385869061703465266186673919827377, 0, 418352010056545642415188090068033486, 0, 73356251280268143473525522536190018359, 0, 12862707834555809665735173850031777018679, 0, 2255421329254099759995141676038440501746342, 0, 395478576475216062935019357427705781199192697] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 5724846963475323281 z - 1297891853153291887 z - 307 z 24 22 4 6 + 243925310303836593 z - 37794319636541388 z + 40305 z - 3054858 z 8 10 12 14 + 151749482 z - 5305487250 z + 136673269715 z - 2678105466885 z 18 16 50 - 493859642019292 z + 40856078955251 z - 651561685437889081753 z 48 20 + 997464092433171736667 z + 4794866940517388 z 36 34 + 358523645781451632427 z - 165973138916908422982 z 66 80 88 84 86 - 37794319636541388 z + 151749482 z + z + 40305 z - 307 z 82 64 30 - 3054858 z + 243925310303836593 z - 21026168556231795070 z 42 44 - 1287410739199569701816 z + 1401629408260667807176 z 46 58 - 1287410739199569701816 z - 21026168556231795070 z 56 54 + 64531864349272847014 z - 165973138916908422982 z 52 60 + 358523645781451632427 z + 5724846963475323281 z 70 68 78 - 493859642019292 z + 4794866940517388 z - 5305487250 z 32 38 + 64531864349272847014 z - 651561685437889081753 z 40 62 76 + 997464092433171736667 z - 1297891853153291887 z + 136673269715 z 74 72 / - 2678105466885 z + 40856078955251 z ) / (-1 / 28 26 2 - 25235815312793040368 z + 5290722798284748627 z + 401 z 24 22 4 6 - 919961310482427007 z + 131910897657436162 z - 62294 z + 5358090 z 8 10 12 14 - 295888567 z + 11371550191 z - 319870311836 z + 6815297269548 z 18 16 50 + 1475125922143809 z - 112741924604341 z + 7140596604275950333015 z 48 20 - 10018043536567291054575 z - 15485710411565778 z 36 34 - 2176874493114686297986 z + 929134100213645975129 z 66 80 90 88 84 + 919961310482427007 z - 11371550191 z + z - 401 z - 5358090 z 86 82 64 + 62294 z + 295888567 z - 5290722798284748627 z 30 42 + 100294343859145429544 z + 10018043536567291054575 z 44 46 - 11864554675214097106108 z + 11864554675214097106108 z 58 56 + 333334966502732560689 z - 929134100213645975129 z 54 52 + 2176874493114686297986 z - 4294123225062347136734 z 60 70 - 100294343859145429544 z + 15485710411565778 z 68 78 32 - 131910897657436162 z + 319870311836 z - 333334966502732560689 z 38 40 + 4294123225062347136734 z - 7140596604275950333015 z 62 76 74 + 25235815312793040368 z - 6815297269548 z + 112741924604341 z 72 - 1475125922143809 z ) And in Maple-input format, it is: -(1+5724846963475323281*z^28-1297891853153291887*z^26-307*z^2+ 243925310303836593*z^24-37794319636541388*z^22+40305*z^4-3054858*z^6+151749482* z^8-5305487250*z^10+136673269715*z^12-2678105466885*z^14-493859642019292*z^18+ 40856078955251*z^16-651561685437889081753*z^50+997464092433171736667*z^48+ 4794866940517388*z^20+358523645781451632427*z^36-165973138916908422982*z^34-\ 37794319636541388*z^66+151749482*z^80+z^88+40305*z^84-307*z^86-3054858*z^82+ 243925310303836593*z^64-21026168556231795070*z^30-1287410739199569701816*z^42+ 1401629408260667807176*z^44-1287410739199569701816*z^46-21026168556231795070*z^ 58+64531864349272847014*z^56-165973138916908422982*z^54+358523645781451632427*z ^52+5724846963475323281*z^60-493859642019292*z^70+4794866940517388*z^68-\ 5305487250*z^78+64531864349272847014*z^32-651561685437889081753*z^38+ 997464092433171736667*z^40-1297891853153291887*z^62+136673269715*z^76-\ 2678105466885*z^74+40856078955251*z^72)/(-1-25235815312793040368*z^28+ 5290722798284748627*z^26+401*z^2-919961310482427007*z^24+131910897657436162*z^ 22-62294*z^4+5358090*z^6-295888567*z^8+11371550191*z^10-319870311836*z^12+ 6815297269548*z^14+1475125922143809*z^18-112741924604341*z^16+ 7140596604275950333015*z^50-10018043536567291054575*z^48-15485710411565778*z^20 -2176874493114686297986*z^36+929134100213645975129*z^34+919961310482427007*z^66 -11371550191*z^80+z^90-401*z^88-5358090*z^84+62294*z^86+295888567*z^82-\ 5290722798284748627*z^64+100294343859145429544*z^30+10018043536567291054575*z^ 42-11864554675214097106108*z^44+11864554675214097106108*z^46+ 333334966502732560689*z^58-929134100213645975129*z^56+2176874493114686297986*z^ 54-4294123225062347136734*z^52-100294343859145429544*z^60+15485710411565778*z^ 70-131910897657436162*z^68+319870311836*z^78-333334966502732560689*z^32+ 4294123225062347136734*z^38-7140596604275950333015*z^40+25235815312793040368*z^ 62-6815297269548*z^76+112741924604341*z^74-1475125922143809*z^72) The first , 40, terms are: [0, 94, 0, 15705, 0, 2745301, 0, 482059806, 0, 84691542805, 0, 14880243676029, 0, 2614478971889566, 0, 459368608516776421, 0, 80711929301866942705, 0, 14181240118453452733566, 0, 2491671078015958668698257, 0, 437791390862301974739064385, 0, 76920787922816469813422757822, 0, 13515130137066641245427538687473, 0, 2374634316739490314838211920300917, 0, 417227809239819591555927549882682878, 0, 73307727248704985383488828350042432221, 0, 12880308443955727554957911478929712208789, 0, 2263094926526584697278394847061619558968894, 0, 397630124213089679630835816656012660146580453] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4197649083902 z - 4838160503312 z - 216 z 24 22 4 6 + 4197649083902 z - 2737134291744 z + 17769 z - 760608 z 8 10 12 14 + 19259392 z - 309921152 z + 3321740340 z - 24532307488 z 18 16 50 48 - 483013235776 z + 128133664868 z - 216 z + 17769 z 20 36 34 + 1334946854912 z + 128133664868 z - 483013235776 z 30 42 44 46 52 - 2737134291744 z - 309921152 z + 19259392 z - 760608 z + z 32 38 40 / 2 + 1334946854912 z - 24532307488 z + 3321740340 z ) / ((-1 + z ) (1 / 28 26 2 24 + 18520591151094 z - 21571142732250 z - 303 z + 18520591151094 z 22 4 6 8 10 - 11711516274848 z + 30737 z - 1546656 z + 44889424 z - 814721408 z 12 14 18 16 + 9730985740 z - 79292574180 z - 1844199844928 z + 452564523244 z 50 48 20 36 - 303 z + 30737 z + 5438435947184 z + 452564523244 z 34 30 42 44 - 1844199844928 z - 11711516274848 z - 814721408 z + 44889424 z 46 52 32 38 40 - 1546656 z + z + 5438435947184 z - 79292574180 z + 9730985740 z )) And in Maple-input format, it is: -(1+4197649083902*z^28-4838160503312*z^26-216*z^2+4197649083902*z^24-\ 2737134291744*z^22+17769*z^4-760608*z^6+19259392*z^8-309921152*z^10+3321740340* z^12-24532307488*z^14-483013235776*z^18+128133664868*z^16-216*z^50+17769*z^48+ 1334946854912*z^20+128133664868*z^36-483013235776*z^34-2737134291744*z^30-\ 309921152*z^42+19259392*z^44-760608*z^46+z^52+1334946854912*z^32-24532307488*z^ 38+3321740340*z^40)/(-1+z^2)/(1+18520591151094*z^28-21571142732250*z^26-303*z^2 +18520591151094*z^24-11711516274848*z^22+30737*z^4-1546656*z^6+44889424*z^8-\ 814721408*z^10+9730985740*z^12-79292574180*z^14-1844199844928*z^18+452564523244 *z^16-303*z^50+30737*z^48+5438435947184*z^20+452564523244*z^36-1844199844928*z^ 34-11711516274848*z^30-814721408*z^42+44889424*z^44-1546656*z^46+z^52+ 5438435947184*z^32-79292574180*z^38+9730985740*z^40) The first , 40, terms are: [0, 88, 0, 13481, 0, 2183489, 0, 356964312, 0, 58469801961, 0, 9581284807769, 0, 1570213148457432, 0, 257337764141131345, 0, 42174586446736841753, 0, 6911920244095453462616, 0, 1132783053374027851289617, 0, 185649935346740174258793713, 0, 30425860496273834704155493464, 0, 4986443919671535331742850676281, 0, 817220042008129990241754232366385, 0, 133932840318703664249134181528271320, 0, 21950031564367609251931262010391571577, 0, 3597354349678268385784305698119435405769, 0, 589564451388474552987642048782567908597720, 0, 96622742314073271219239336345001052827427937] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 10778106770226 z - 12482353992084 z - 254 z 24 22 4 6 + 10778106770226 z - 6927773623312 z + 24533 z - 1197088 z 8 10 12 14 + 33671032 z - 590261704 z + 6797130480 z - 53363337144 z 18 16 50 48 - 1154525649032 z + 293510515592 z - 254 z + 24533 z 20 36 34 + 3298982936328 z + 293510515592 z - 1154525649032 z 30 42 44 46 52 - 6927773623312 z - 590261704 z + 33671032 z - 1197088 z + z 32 38 40 / 2 + 3298982936328 z - 53363337144 z + 6797130480 z ) / ((-1 + z ) (1 / 28 26 2 24 + 48434469166706 z - 56725336199030 z - 353 z + 48434469166706 z 22 4 6 8 - 30127642260200 z + 42229 z - 2431784 z + 78535820 z 10 12 14 18 - 1553964768 z + 19960481664 z - 173191616996 z - 4451744231952 z 16 50 48 20 + 1043604765208 z - 353 z + 42229 z + 13618668321908 z 36 34 30 + 1043604765208 z - 4451744231952 z - 30127642260200 z 42 44 46 52 32 - 1553964768 z + 78535820 z - 2431784 z + z + 13618668321908 z 38 40 - 173191616996 z + 19960481664 z )) And in Maple-input format, it is: -(1+10778106770226*z^28-12482353992084*z^26-254*z^2+10778106770226*z^24-\ 6927773623312*z^22+24533*z^4-1197088*z^6+33671032*z^8-590261704*z^10+6797130480 *z^12-53363337144*z^14-1154525649032*z^18+293510515592*z^16-254*z^50+24533*z^48 +3298982936328*z^20+293510515592*z^36-1154525649032*z^34-6927773623312*z^30-\ 590261704*z^42+33671032*z^44-1197088*z^46+z^52+3298982936328*z^32-53363337144*z ^38+6797130480*z^40)/(-1+z^2)/(1+48434469166706*z^28-56725336199030*z^26-353*z^ 2+48434469166706*z^24-30127642260200*z^22+42229*z^4-2431784*z^6+78535820*z^8-\ 1553964768*z^10+19960481664*z^12-173191616996*z^14-4451744231952*z^18+ 1043604765208*z^16-353*z^50+42229*z^48+13618668321908*z^20+1043604765208*z^36-\ 4451744231952*z^34-30127642260200*z^30-1553964768*z^42+78535820*z^44-2431784*z^ 46+z^52+13618668321908*z^32-173191616996*z^38+19960481664*z^40) The first , 40, terms are: [0, 100, 0, 17351, 0, 3160979, 0, 580251012, 0, 106680128517, 0, 19620483886605, 0, 3608923576194820, 0, 663831386586179051, 0, 122107312379455686927, 0, 22460882317682340434532, 0, 4131543961540066046033753, 0, 759972875439050530816327657, 0, 139792495464232428053966997860, 0, 25714000125702445583718287093567, 0, 4729937828116331014223744924810491, 0, 870044017567688264537377394835484420, 0, 160039438455399931206149829322640151101, 0, 29438305834236425562353188184423477803605, 0, 5415001820565467973534122908116791862852484, 0, 996057479813755352979981300491285377766865411] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 15255688623431272 z - 7129007539743750 z - 299 z 24 22 4 6 + 2660706480557554 z - 788441082191336 z + 35906 z - 2299147 z 8 10 12 14 + 88928393 z - 2253435172 z + 39568981389 z - 500993115799 z 18 16 50 - 33535584300667 z + 4707476805894 z - 33535584300667 z 48 20 36 + 184086288349153 z + 184086288349153 z + 36175101143746886 z 34 66 64 30 - 40279886880386952 z - 299 z + 35906 z - 26188898784885122 z 42 44 46 - 7129007539743750 z + 2660706480557554 z - 788441082191336 z 58 56 54 52 - 2253435172 z + 39568981389 z - 500993115799 z + 4707476805894 z 60 68 32 38 + 88928393 z + z + 36175101143746886 z - 26188898784885122 z 40 62 / 2 + 15255688623431272 z - 2299147 z ) / ((-1 + z ) (1 / 28 26 2 + 67231477030686526 z - 30172393360871716 z - 406 z 24 22 4 6 + 10713008552484202 z - 2995615513821320 z + 59739 z - 4433568 z 8 10 12 14 + 192649021 z - 5391616830 z + 103556087973 z - 1425491570752 z 18 16 50 - 111310478461566 z + 14496878391843 z - 111310478461566 z 48 20 36 + 655537854963337 z + 655537854963337 z + 167276396635315650 z 34 66 64 30 - 187407858895573020 z - 406 z + 59739 z - 118906341683366256 z 42 44 46 - 30172393360871716 z + 10713008552484202 z - 2995615513821320 z 58 56 54 - 5391616830 z + 103556087973 z - 1425491570752 z 52 60 68 32 + 14496878391843 z + 192649021 z + z + 167276396635315650 z 38 40 62 - 118906341683366256 z + 67231477030686526 z - 4433568 z )) And in Maple-input format, it is: -(1+15255688623431272*z^28-7129007539743750*z^26-299*z^2+2660706480557554*z^24-\ 788441082191336*z^22+35906*z^4-2299147*z^6+88928393*z^8-2253435172*z^10+ 39568981389*z^12-500993115799*z^14-33535584300667*z^18+4707476805894*z^16-\ 33535584300667*z^50+184086288349153*z^48+184086288349153*z^20+36175101143746886 *z^36-40279886880386952*z^34-299*z^66+35906*z^64-26188898784885122*z^30-\ 7129007539743750*z^42+2660706480557554*z^44-788441082191336*z^46-2253435172*z^ 58+39568981389*z^56-500993115799*z^54+4707476805894*z^52+88928393*z^60+z^68+ 36175101143746886*z^32-26188898784885122*z^38+15255688623431272*z^40-2299147*z^ 62)/(-1+z^2)/(1+67231477030686526*z^28-30172393360871716*z^26-406*z^2+ 10713008552484202*z^24-2995615513821320*z^22+59739*z^4-4433568*z^6+192649021*z^ 8-5391616830*z^10+103556087973*z^12-1425491570752*z^14-111310478461566*z^18+ 14496878391843*z^16-111310478461566*z^50+655537854963337*z^48+655537854963337*z ^20+167276396635315650*z^36-187407858895573020*z^34-406*z^66+59739*z^64-\ 118906341683366256*z^30-30172393360871716*z^42+10713008552484202*z^44-\ 2995615513821320*z^46-5391616830*z^58+103556087973*z^56-1425491570752*z^54+ 14496878391843*z^52+192649021*z^60+z^68+167276396635315650*z^32-\ 118906341683366256*z^38+67231477030686526*z^40-4433568*z^62) The first , 40, terms are: [0, 108, 0, 19717, 0, 3723319, 0, 706634828, 0, 134301798927, 0, 25538140340343, 0, 4857133965166860, 0, 923853668406918499, 0, 175727189793341967469, 0, 33425641490752967033596, 0, 6358029361979549917007277, 0, 1209389243821007053777833245, 0, 230043513266943033311210269916, 0, 43757651841944515363361503426061, 0, 8323348447576275289721641468208147, 0, 1583223269907868741715622728417332748, 0, 301152352969341913742655732941623042871, 0, 57283607503515968442956331283794329932463, 0, 10896184825397148376003927573265532664109388, 0, 2072614646104829577236497385620394070658301703] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 22445208292148 z - 19767345575482 z - 237 z 24 22 4 6 + 13487876377026 z - 7106564160768 z + 20958 z - 924945 z 8 10 12 14 + 23833429 z - 394097408 z + 4438167240 z - 35402940168 z 18 16 50 48 - 885762755776 z + 205543860360 z - 924945 z + 23833429 z 20 36 34 + 2874984139808 z + 2874984139808 z - 7106564160768 z 30 42 44 46 - 19767345575482 z - 35402940168 z + 4438167240 z - 394097408 z 56 54 52 32 38 + z - 237 z + 20958 z + 13487876377026 z - 885762755776 z 40 / 2 28 + 205543860360 z ) / ((-1 + z ) (1 + 103748996423134 z / 26 2 24 22 - 90578026459248 z - 344 z + 60232397223670 z - 30439289930192 z 4 6 8 10 12 + 37509 z - 1918008 z + 55934097 z - 1034260016 z + 12914156236 z 14 18 16 50 - 113351780832 z - 3350152931360 z + 718458572428 z - 1918008 z 48 20 36 + 55934097 z + 11643548975532 z + 11643548975532 z 34 30 42 - 30439289930192 z - 90578026459248 z - 113351780832 z 44 46 56 54 52 + 12914156236 z - 1034260016 z + z - 344 z + 37509 z 32 38 40 + 60232397223670 z - 3350152931360 z + 718458572428 z )) And in Maple-input format, it is: -(1+22445208292148*z^28-19767345575482*z^26-237*z^2+13487876377026*z^24-\ 7106564160768*z^22+20958*z^4-924945*z^6+23833429*z^8-394097408*z^10+4438167240* z^12-35402940168*z^14-885762755776*z^18+205543860360*z^16-924945*z^50+23833429* z^48+2874984139808*z^20+2874984139808*z^36-7106564160768*z^34-19767345575482*z^ 30-35402940168*z^42+4438167240*z^44-394097408*z^46+z^56-237*z^54+20958*z^52+ 13487876377026*z^32-885762755776*z^38+205543860360*z^40)/(-1+z^2)/(1+ 103748996423134*z^28-90578026459248*z^26-344*z^2+60232397223670*z^24-\ 30439289930192*z^22+37509*z^4-1918008*z^6+55934097*z^8-1034260016*z^10+ 12914156236*z^12-113351780832*z^14-3350152931360*z^18+718458572428*z^16-1918008 *z^50+55934097*z^48+11643548975532*z^20+11643548975532*z^36-30439289930192*z^34 -90578026459248*z^30-113351780832*z^42+12914156236*z^44-1034260016*z^46+z^56-\ 344*z^54+37509*z^52+60232397223670*z^32-3350152931360*z^38+718458572428*z^40) The first , 40, terms are: [0, 108, 0, 20365, 0, 3968373, 0, 775389500, 0, 151566727401, 0, 29629995670553, 0, 5792588028579996, 0, 1132447288880306341, 0, 221393443336324662109, 0, 43282463724492839664396, 0, 8461733505163832074863057, 0, 1654271405025118512560129585, 0, 323410561466745940040815309260, 0, 63226863642017911031393427938237, 0, 12360871221954738356822369065158213, 0, 2416554115870539606259628132494015964, 0, 472437071190518845296921541797154982329, 0, 92361592401762273801416869196202044943177, 0, 18056719659807173754818821441739173280189436, 0, 3530094235053928132470502963391023148295909333] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 5680087018731418073 z - 1288724838028761137 z - 309 z 24 22 4 6 + 242580558988601953 z - 37674921582539796 z + 40745 z - 3097770 z 8 10 12 14 + 154120158 z - 5387621450 z + 138555656459 z - 2707247106251 z 18 16 50 - 495587876871716 z + 41154121280771 z - 648745961131745859343 z 48 20 + 994046580869392236379 z + 4794439336148020 z 36 34 + 356591202178655172915 z - 164900164495877367894 z 66 80 88 84 86 - 37674921582539796 z + 154120158 z + z + 40745 z - 309 z 82 64 30 - 3097770 z + 242580558988601953 z - 20860743892210289110 z 42 44 - 1283762221288429940968 z + 1397947044383781774392 z 46 58 - 1283762221288429940968 z - 20860743892210289110 z 56 54 + 64057180325509093714 z - 164900164495877367894 z 52 60 + 356591202178655172915 z + 5680087018731418073 z 70 68 78 - 495587876871716 z + 4794439336148020 z - 5387621450 z 32 38 + 64057180325509093714 z - 648745961131745859343 z 40 62 76 + 994046580869392236379 z - 1288724838028761137 z + 138555656459 z 74 72 / 2 - 2707247106251 z + 41154121280771 z ) / ((-1 + z ) (1 / 28 26 2 + 20635557705744135166 z - 4467687116462628566 z - 398 z 24 22 4 6 + 799574851330894155 z - 117663127813000228 z + 61876 z - 5328524 z 8 10 12 14 + 293771209 z - 11227367152 z + 312824294182 z - 6578996392690 z 18 16 50 - 1375531105376772 z + 107104691110781 z - 2784657711585167625466 z 48 20 + 4338731962842387871831 z + 14140021096815478 z 36 34 + 1496000387904892999208 z - 672299214638535590604 z 66 80 88 84 86 - 117663127813000228 z + 293771209 z + z + 61876 z - 398 z 82 64 30 - 5328524 z + 799574851330894155 z - 79108121586090678024 z 42 44 - 5660634074091564527464 z + 6185260409151803308308 z 46 58 - 5660634074091564527464 z - 79108121586090678024 z 56 54 + 252473919957044650215 z - 672299214638535590604 z 52 60 + 1496000387904892999208 z + 20635557705744135166 z 70 68 78 - 1375531105376772 z + 14140021096815478 z - 11227367152 z 32 38 + 252473919957044650215 z - 2784657711585167625466 z 40 62 76 + 4338731962842387871831 z - 4467687116462628566 z + 312824294182 z 74 72 - 6578996392690 z + 107104691110781 z )) And in Maple-input format, it is: -(1+5680087018731418073*z^28-1288724838028761137*z^26-309*z^2+ 242580558988601953*z^24-37674921582539796*z^22+40745*z^4-3097770*z^6+154120158* z^8-5387621450*z^10+138555656459*z^12-2707247106251*z^14-495587876871716*z^18+ 41154121280771*z^16-648745961131745859343*z^50+994046580869392236379*z^48+ 4794439336148020*z^20+356591202178655172915*z^36-164900164495877367894*z^34-\ 37674921582539796*z^66+154120158*z^80+z^88+40745*z^84-309*z^86-3097770*z^82+ 242580558988601953*z^64-20860743892210289110*z^30-1283762221288429940968*z^42+ 1397947044383781774392*z^44-1283762221288429940968*z^46-20860743892210289110*z^ 58+64057180325509093714*z^56-164900164495877367894*z^54+356591202178655172915*z ^52+5680087018731418073*z^60-495587876871716*z^70+4794439336148020*z^68-\ 5387621450*z^78+64057180325509093714*z^32-648745961131745859343*z^38+ 994046580869392236379*z^40-1288724838028761137*z^62+138555656459*z^76-\ 2707247106251*z^74+41154121280771*z^72)/(-1+z^2)/(1+20635557705744135166*z^28-\ 4467687116462628566*z^26-398*z^2+799574851330894155*z^24-117663127813000228*z^ 22+61876*z^4-5328524*z^6+293771209*z^8-11227367152*z^10+312824294182*z^12-\ 6578996392690*z^14-1375531105376772*z^18+107104691110781*z^16-\ 2784657711585167625466*z^50+4338731962842387871831*z^48+14140021096815478*z^20+ 1496000387904892999208*z^36-672299214638535590604*z^34-117663127813000228*z^66+ 293771209*z^80+z^88+61876*z^84-398*z^86-5328524*z^82+799574851330894155*z^64-\ 79108121586090678024*z^30-5660634074091564527464*z^42+6185260409151803308308*z^ 44-5660634074091564527464*z^46-79108121586090678024*z^58+252473919957044650215* z^56-672299214638535590604*z^54+1496000387904892999208*z^52+ 20635557705744135166*z^60-1375531105376772*z^70+14140021096815478*z^68-\ 11227367152*z^78+252473919957044650215*z^32-2784657711585167625466*z^38+ 4338731962842387871831*z^40-4467687116462628566*z^62+312824294182*z^76-\ 6578996392690*z^74+107104691110781*z^72) The first , 40, terms are: [0, 90, 0, 14381, 0, 2425989, 0, 412563642, 0, 70270737513, 0, 11973140318721, 0, 2040218887791082, 0, 347659406628193333, 0, 59242492798243864885, 0, 10095159826579087634442, 0, 1720256510950502887923009, 0, 293138766875311643946542913, 0, 49952049535017857022875085706, 0, 8512034395836457427657035566053, 0, 1450485622395295580195228883115861, 0, 247168707706096901925753776960602826, 0, 42118563005334182975366544071967463473, 0, 7177176132629848739221962102213927586041, 0, 1223020292326443375194962311631342373630362, 0, 208407681211103754074575682970267134305504165] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 395091378909 z + 793540371910 z + 266 z - 1121516338751 z 22 4 6 8 10 + 1121516338751 z - 26643 z + 1303387 z - 34451902 z + 534634637 z 12 14 18 16 - 5169549763 z + 32499851810 z + 395091378909 z - 136919280821 z 20 36 34 30 - 793540371910 z - 534634637 z + 5169549763 z + 136919280821 z 42 44 46 32 38 40 + 26643 z - 266 z + z - 32499851810 z + 34451902 z - 1303387 z / 28 26 2 ) / (1 + 5267581688404 z - 8846169175428 z - 372 z / 24 22 4 6 + 10508312649742 z - 8846169175428 z + 47108 z - 2763124 z 8 10 12 14 + 85607624 z - 1548854508 z + 17508454728 z - 129385052172 z 18 16 48 20 - 2206549637036 z + 644324396704 z + z + 5267581688404 z 36 34 30 42 + 17508454728 z - 129385052172 z - 2206549637036 z - 2763124 z 44 46 32 38 40 + 47108 z - 372 z + 644324396704 z - 1548854508 z + 85607624 z ) And in Maple-input format, it is: -(-1-395091378909*z^28+793540371910*z^26+266*z^2-1121516338751*z^24+ 1121516338751*z^22-26643*z^4+1303387*z^6-34451902*z^8+534634637*z^10-5169549763 *z^12+32499851810*z^14+395091378909*z^18-136919280821*z^16-793540371910*z^20-\ 534634637*z^36+5169549763*z^34+136919280821*z^30+26643*z^42-266*z^44+z^46-\ 32499851810*z^32+34451902*z^38-1303387*z^40)/(1+5267581688404*z^28-\ 8846169175428*z^26-372*z^2+10508312649742*z^24-8846169175428*z^22+47108*z^4-\ 2763124*z^6+85607624*z^8-1548854508*z^10+17508454728*z^12-129385052172*z^14-\ 2206549637036*z^18+644324396704*z^16+z^48+5267581688404*z^20+17508454728*z^36-\ 129385052172*z^34-2206549637036*z^30-2763124*z^42+47108*z^44-372*z^46+ 644324396704*z^32-1548854508*z^38+85607624*z^40) The first , 40, terms are: [0, 106, 0, 18967, 0, 3522013, 0, 658426822, 0, 123367774015, 0, 23135519719891, 0, 4340226139438078, 0, 814346372566344697, 0, 152803095302579897923, 0, 28672523072863555309762, 0, 5380270466019973591087141, 0, 1009587895279506686665445725, 0, 189445771378567453367563757074, 0, 35548886997582264321672911530219, 0, 6670635776162653181523210531043969, 0, 1251723776493753810845530850140973710, 0, 234882032840262530754841442653790364331, 0, 44074876208323789308643529752467789299623, 0, 8270512247872785811138565745113348428906678, 0, 1551935682372291377291120354216142678393330325] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 27394596165566400 z - 12218395259421574 z - 279 z 24 22 4 6 + 4303186780572666 z - 1191538583344520 z + 31902 z - 2016335 z 8 10 12 14 + 79851969 z - 2128357692 z + 40004557117 z - 547757352419 z 18 16 50 - 43326442940543 z + 5592983702306 z - 43326442940543 z 48 20 36 + 257929734551817 z + 257929734551817 z + 68619927316269806 z 34 66 64 30 - 76940072786088152 z - 279 z + 31902 z - 48657426535301618 z 42 44 46 - 12218395259421574 z + 4303186780572666 z - 1191538583344520 z 58 56 54 52 - 2128357692 z + 40004557117 z - 547757352419 z + 5592983702306 z 60 68 32 38 + 79851969 z + z + 68619927316269806 z - 48657426535301618 z 40 62 / 28 + 27394596165566400 z - 2016335 z ) / (-1 - 168033067597570678 z / 26 2 24 + 66935255599397618 z + 381 z - 21079016800914818 z 22 4 6 8 + 5227654995841441 z - 52399 z + 3795581 z - 168685743 z 10 12 14 + 4999399647 z - 104109338319 z + 1578066270727 z 18 16 50 + 153433884542263 z - 17850538154845 z + 1015582021234901 z 48 20 36 - 5227654995841441 z - 1015582021234901 z - 664301624072246302 z 34 66 64 + 664301624072246302 z + 52399 z - 3795581 z 30 42 44 + 334397684893622186 z + 168033067597570678 z - 66935255599397618 z 46 58 56 + 21079016800914818 z + 104109338319 z - 1578066270727 z 54 52 60 70 + 17850538154845 z - 153433884542263 z - 4999399647 z + z 68 32 38 - 381 z - 528544176596299686 z + 528544176596299686 z 40 62 - 334397684893622186 z + 168685743 z ) And in Maple-input format, it is: -(1+27394596165566400*z^28-12218395259421574*z^26-279*z^2+4303186780572666*z^24 -1191538583344520*z^22+31902*z^4-2016335*z^6+79851969*z^8-2128357692*z^10+ 40004557117*z^12-547757352419*z^14-43326442940543*z^18+5592983702306*z^16-\ 43326442940543*z^50+257929734551817*z^48+257929734551817*z^20+68619927316269806 *z^36-76940072786088152*z^34-279*z^66+31902*z^64-48657426535301618*z^30-\ 12218395259421574*z^42+4303186780572666*z^44-1191538583344520*z^46-2128357692*z ^58+40004557117*z^56-547757352419*z^54+5592983702306*z^52+79851969*z^60+z^68+ 68619927316269806*z^32-48657426535301618*z^38+27394596165566400*z^40-2016335*z^ 62)/(-1-168033067597570678*z^28+66935255599397618*z^26+381*z^2-\ 21079016800914818*z^24+5227654995841441*z^22-52399*z^4+3795581*z^6-168685743*z^ 8+4999399647*z^10-104109338319*z^12+1578066270727*z^14+153433884542263*z^18-\ 17850538154845*z^16+1015582021234901*z^50-5227654995841441*z^48-\ 1015582021234901*z^20-664301624072246302*z^36+664301624072246302*z^34+52399*z^ 66-3795581*z^64+334397684893622186*z^30+168033067597570678*z^42-\ 66935255599397618*z^44+21079016800914818*z^46+104109338319*z^58-1578066270727*z ^56+17850538154845*z^54-153433884542263*z^52-4999399647*z^60+z^70-381*z^68-\ 528544176596299686*z^32+528544176596299686*z^38-334397684893622186*z^40+ 168685743*z^62) The first , 40, terms are: [0, 102, 0, 18365, 0, 3431613, 0, 643452406, 0, 120713218333, 0, 22648358977629, 0, 4249409511997134, 0, 797301954792238689, 0, 149595201161182378093, 0, 28068075853429513384934, 0, 5266325001792484854633421, 0, 988104056242781088093738093, 0, 185394868468811267218568330982, 0, 34785058397191738618250837609261, 0, 6526611540428976422345428835144145, 0, 1224567678360145240473587006348696110, 0, 229761797469520482865177115911154926205, 0, 43109486318780528583534945274753002454765, 0, 8088497875361388488932698948187234735161846, 0, 1517619518728624433325398772065203533344995805] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 4}, {3, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 15049936543476013 z - 6082916578632827 z - 252 z 24 22 4 6 + 1995040338657779 z - 528252183884762 z + 25491 z - 1416673 z 8 10 12 14 + 49657413 z - 1187404466 z + 20369110989 z - 259511327376 z 18 16 50 - 18947355829144 z + 2517655830034 z - 528252183884762 z 48 20 36 + 1995040338657779 z + 112183025280231 z + 74340930950935380 z 34 66 64 - 67310042882493008 z - 1416673 z + 49657413 z 30 42 44 - 30332446500057376 z - 30332446500057376 z + 15049936543476013 z 46 58 56 - 6082916578632827 z - 259511327376 z + 2517655830034 z 54 52 60 70 - 18947355829144 z + 112183025280231 z + 20369110989 z - 252 z 68 32 38 + 25491 z + 49942702746666707 z - 67310042882493008 z 40 62 72 / 2 + 49942702746666707 z - 1187404466 z + z ) / ((-1 + z ) (1 / 28 26 2 + 63343722601534260 z - 24481141907510604 z - 350 z 24 22 4 6 + 7622563461595839 z - 1904631729017046 z + 42766 z - 2712964 z 8 10 12 14 + 105683107 z - 2769037626 z + 51623567436 z - 711021525732 z 18 16 50 - 59990349137108 z + 7428029327397 z - 1904631729017046 z 48 20 36 + 7622563461595839 z + 379812029747366 z + 340805215769871466 z 34 66 64 - 306855250626970552 z - 2712964 z + 105683107 z 30 42 44 - 132399168393408306 z - 132399168393408306 z + 63343722601534260 z 46 58 56 - 24481141907510604 z - 711021525732 z + 7428029327397 z 54 52 60 70 - 59990349137108 z + 379812029747366 z + 51623567436 z - 350 z 68 32 38 + 42766 z + 223944712382340161 z - 306855250626970552 z 40 62 72 + 223944712382340161 z - 2769037626 z + z )) And in Maple-input format, it is: -(1+15049936543476013*z^28-6082916578632827*z^26-252*z^2+1995040338657779*z^24-\ 528252183884762*z^22+25491*z^4-1416673*z^6+49657413*z^8-1187404466*z^10+ 20369110989*z^12-259511327376*z^14-18947355829144*z^18+2517655830034*z^16-\ 528252183884762*z^50+1995040338657779*z^48+112183025280231*z^20+ 74340930950935380*z^36-67310042882493008*z^34-1416673*z^66+49657413*z^64-\ 30332446500057376*z^30-30332446500057376*z^42+15049936543476013*z^44-\ 6082916578632827*z^46-259511327376*z^58+2517655830034*z^56-18947355829144*z^54+ 112183025280231*z^52+20369110989*z^60-252*z^70+25491*z^68+49942702746666707*z^ 32-67310042882493008*z^38+49942702746666707*z^40-1187404466*z^62+z^72)/(-1+z^2) /(1+63343722601534260*z^28-24481141907510604*z^26-350*z^2+7622563461595839*z^24 -1904631729017046*z^22+42766*z^4-2712964*z^6+105683107*z^8-2769037626*z^10+ 51623567436*z^12-711021525732*z^14-59990349137108*z^18+7428029327397*z^16-\ 1904631729017046*z^50+7622563461595839*z^48+379812029747366*z^20+ 340805215769871466*z^36-306855250626970552*z^34-2712964*z^66+105683107*z^64-\ 132399168393408306*z^30-132399168393408306*z^42+63343722601534260*z^44-\ 24481141907510604*z^46-711021525732*z^58+7428029327397*z^56-59990349137108*z^54 +379812029747366*z^52+51623567436*z^60-350*z^70+42766*z^68+223944712382340161*z ^32-306855250626970552*z^38+223944712382340161*z^40-2769037626*z^62+z^72) The first , 40, terms are: [0, 99, 0, 17124, 0, 3081097, 0, 557225275, 0, 100886719031, 0, 18270984397481, 0, 3309212327377644, 0, 599373046317568303, 0, 108560696823159962561, 0, 19662958050607076183457, 0, 3561437428872103360460479, 0, 645062584731880722912422988, 0, 116836464294462192899503685969, 0, 21161914967052998042767037777839, 0, 3832935628948703596346744059800595, 0, 694237528808543726711309081046801409, 0, 125743240485638938731895251336210412292, 0, 22775148091447917510863741546599337394771, 0, 4125131248388053953760302687534713927997233, 0, 747161236810800092738810635948350451785877713] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 4}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 30462630058156 z - 26687459094522 z - 221 z 24 22 4 6 + 17929109399790 z - 9212867769620 z + 18722 z - 834233 z 8 10 12 14 + 22332993 z - 388137468 z + 4608924840 z - 38742223148 z 18 16 50 48 - 1066599144964 z + 236421834008 z - 834233 z + 22332993 z 20 36 34 + 3604579685224 z + 3604579685224 z - 9212867769620 z 30 42 44 46 - 26687459094522 z - 38742223148 z + 4608924840 z - 388137468 z 56 54 52 32 38 + z - 221 z + 18722 z + 17929109399790 z - 1066599144964 z 40 / 2 28 + 236421834008 z ) / ((-1 + z ) (1 + 139864833002018 z / 26 2 24 22 - 121262354907722 z - 314 z + 79029589833678 z - 38707026120394 z 4 6 8 10 12 + 32631 z - 1689236 z + 51146913 z - 990597934 z + 12992801528 z 14 18 16 50 - 119903038882 z - 3911614481398 z + 798979233448 z - 1689236 z 48 20 36 + 51146913 z + 14227990848184 z + 14227990848184 z 34 30 42 - 38707026120394 z - 121262354907722 z - 119903038882 z 44 46 56 54 52 + 12992801528 z - 990597934 z + z - 314 z + 32631 z 32 38 40 + 79029589833678 z - 3911614481398 z + 798979233448 z )) And in Maple-input format, it is: -(1+30462630058156*z^28-26687459094522*z^26-221*z^2+17929109399790*z^24-\ 9212867769620*z^22+18722*z^4-834233*z^6+22332993*z^8-388137468*z^10+4608924840* z^12-38742223148*z^14-1066599144964*z^18+236421834008*z^16-834233*z^50+22332993 *z^48+3604579685224*z^20+3604579685224*z^36-9212867769620*z^34-26687459094522*z ^30-38742223148*z^42+4608924840*z^44-388137468*z^46+z^56-221*z^54+18722*z^52+ 17929109399790*z^32-1066599144964*z^38+236421834008*z^40)/(-1+z^2)/(1+ 139864833002018*z^28-121262354907722*z^26-314*z^2+79029589833678*z^24-\ 38707026120394*z^22+32631*z^4-1689236*z^6+51146913*z^8-990597934*z^10+ 12992801528*z^12-119903038882*z^14-3911614481398*z^18+798979233448*z^16-1689236 *z^50+51146913*z^48+14227990848184*z^20+14227990848184*z^36-38707026120394*z^34 -121262354907722*z^30-119903038882*z^42+12992801528*z^44-990597934*z^46+z^56-\ 314*z^54+32631*z^52+79029589833678*z^32-3911614481398*z^38+798979233448*z^40) The first , 40, terms are: [0, 94, 0, 15387, 0, 2637709, 0, 455305962, 0, 78703431927, 0, 13608869937951, 0, 2353328432776586, 0, 406958931731530213, 0, 70375326713418661507, 0, 12170003197108800090654, 0, 2104558795647965622219577, 0, 363941397869230202303624777, 0, 62936394632295790154501018654, 0, 10883592263786215773215444902323, 0, 1882099877982782727643466725716533, 0, 325471578273463265146459237203652042, 0, 56283808052865079982512594662725445967, 0, 9733160313941097772349717486330844588999, 0, 1683155653010415226172821538836184272854442, 0, 291068148564752554855632596393071674791178557] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 211167157601355712912 z - 30476329878885996044 z - 338 z 24 22 4 6 + 3770293837225606976 z - 397317518279441648 z + 49724 z - 4307896 z 102 8 10 12 - 10478712504 z + 250165757 z - 10478712504 z + 331968831988 z 14 18 16 - 8221961139382 z - 2640782020423480 z + 163119612112925 z 50 48 - 39126841639634136375206568 z + 25063812127118783243466330 z 20 36 + 35394855096769476 z + 114238463039249387291936 z 34 66 - 29227556022398206088584 z - 14125889024440024722402776 z 80 100 90 + 6513680542141567400636 z + 331968831988 z - 397317518279441648 z 88 84 + 3770293837225606976 z + 211167157601355712912 z 94 86 96 - 2640782020423480 z - 30476329878885996044 z + 163119612112925 z 98 92 82 - 8221961139382 z + 35394855096769476 z - 1260733107738118196424 z 64 112 110 106 + 25063812127118783243466330 z + z - 338 z - 4307896 z 108 30 42 + 49724 z - 1260733107738118196424 z - 3047554444489856551495552 z 44 46 + 7000105271740973853271548 z - 14125889024440024722402776 z 58 56 - 65052453797618363741520064 z + 69317436841345461787790350 z 54 52 - 65052453797618363741520064 z + 53765176600046656145883936 z 60 70 + 53765176600046656145883936 z - 3047554444489856551495552 z 68 78 + 7000105271740973853271548 z - 29227556022398206088584 z 32 38 + 6513680542141567400636 z - 389900306065748536000196 z 40 62 + 1164378841504496386476616 z - 39126841639634136375206568 z 76 74 + 114238463039249387291936 z - 389900306065748536000196 z 72 104 / 2 + 1164378841504496386476616 z + 250165757 z ) / ((-1 + z ) (1 / 28 26 2 + 673199554131115728946 z - 93350592312693122299 z - 415 z 24 22 4 + 11064936766745606772 z - 1113823528962987937 z + 71096 z 6 102 8 10 - 6948919 z - 20369415846 z + 446083811 z - 20369415846 z 12 14 18 + 696387862434 z - 18469389046104 z - 6684955942068511 z 16 50 + 389969488672149 z - 163288744337339061774189989 z 48 20 + 103469803156566098438273898 z + 94466162548926910 z 36 34 + 416505308308746041621550 z - 103431628903874347834999 z 66 80 - 57514918290640721965433853 z + 22318426148641305685648 z 100 90 88 + 696387862434 z - 1113823528962987937 z + 11064936766745606772 z 84 94 + 673199554131115728946 z - 6684955942068511 z 86 96 98 - 93350592312693122299 z + 389969488672149 z - 18469389046104 z 92 82 + 94466162548926910 z - 4172089593256425003445 z 64 112 110 106 + 103469803156566098438273898 z + z - 415 z - 6948919 z 108 30 42 + 71096 z - 4172089593256425003445 z - 11967630268165245679002179 z 44 46 + 28029748255745345802635762 z - 57514918290640721965433853 z 58 56 - 274908036163207085327196151 z + 293393678884773041585229434 z 54 52 - 274908036163207085327196151 z + 226139807203733925175066534 z 60 70 + 226139807203733925175066534 z - 11967630268165245679002179 z 68 78 + 28029748255745345802635762 z - 103431628903874347834999 z 32 38 + 22318426148641305685648 z - 1460914551232104236198249 z 40 62 + 4472235647969744286426092 z - 163288744337339061774189989 z 76 74 + 416505308308746041621550 z - 1460914551232104236198249 z 72 104 + 4472235647969744286426092 z + 446083811 z )) And in Maple-input format, it is: -(1+211167157601355712912*z^28-30476329878885996044*z^26-338*z^2+ 3770293837225606976*z^24-397317518279441648*z^22+49724*z^4-4307896*z^6-\ 10478712504*z^102+250165757*z^8-10478712504*z^10+331968831988*z^12-\ 8221961139382*z^14-2640782020423480*z^18+163119612112925*z^16-\ 39126841639634136375206568*z^50+25063812127118783243466330*z^48+ 35394855096769476*z^20+114238463039249387291936*z^36-29227556022398206088584*z^ 34-14125889024440024722402776*z^66+6513680542141567400636*z^80+331968831988*z^ 100-397317518279441648*z^90+3770293837225606976*z^88+211167157601355712912*z^84 -2640782020423480*z^94-30476329878885996044*z^86+163119612112925*z^96-\ 8221961139382*z^98+35394855096769476*z^92-1260733107738118196424*z^82+ 25063812127118783243466330*z^64+z^112-338*z^110-4307896*z^106+49724*z^108-\ 1260733107738118196424*z^30-3047554444489856551495552*z^42+ 7000105271740973853271548*z^44-14125889024440024722402776*z^46-\ 65052453797618363741520064*z^58+69317436841345461787790350*z^56-\ 65052453797618363741520064*z^54+53765176600046656145883936*z^52+ 53765176600046656145883936*z^60-3047554444489856551495552*z^70+ 7000105271740973853271548*z^68-29227556022398206088584*z^78+ 6513680542141567400636*z^32-389900306065748536000196*z^38+ 1164378841504496386476616*z^40-39126841639634136375206568*z^62+ 114238463039249387291936*z^76-389900306065748536000196*z^74+ 1164378841504496386476616*z^72+250165757*z^104)/(-1+z^2)/(1+ 673199554131115728946*z^28-93350592312693122299*z^26-415*z^2+ 11064936766745606772*z^24-1113823528962987937*z^22+71096*z^4-6948919*z^6-\ 20369415846*z^102+446083811*z^8-20369415846*z^10+696387862434*z^12-\ 18469389046104*z^14-6684955942068511*z^18+389969488672149*z^16-\ 163288744337339061774189989*z^50+103469803156566098438273898*z^48+ 94466162548926910*z^20+416505308308746041621550*z^36-103431628903874347834999*z ^34-57514918290640721965433853*z^66+22318426148641305685648*z^80+696387862434*z ^100-1113823528962987937*z^90+11064936766745606772*z^88+673199554131115728946*z ^84-6684955942068511*z^94-93350592312693122299*z^86+389969488672149*z^96-\ 18469389046104*z^98+94466162548926910*z^92-4172089593256425003445*z^82+ 103469803156566098438273898*z^64+z^112-415*z^110-6948919*z^106+71096*z^108-\ 4172089593256425003445*z^30-11967630268165245679002179*z^42+ 28029748255745345802635762*z^44-57514918290640721965433853*z^46-\ 274908036163207085327196151*z^58+293393678884773041585229434*z^56-\ 274908036163207085327196151*z^54+226139807203733925175066534*z^52+ 226139807203733925175066534*z^60-11967630268165245679002179*z^70+ 28029748255745345802635762*z^68-103431628903874347834999*z^78+ 22318426148641305685648*z^32-1460914551232104236198249*z^38+ 4472235647969744286426092*z^40-163288744337339061774189989*z^62+ 416505308308746041621550*z^76-1460914551232104236198249*z^74+ 4472235647969744286426092*z^72+446083811*z^104) The first , 40, terms are: [0, 78, 0, 10661, 0, 1569237, 0, 235118018, 0, 35432002509, 0, 5351294329525, 0, 808927590571154, 0, 122327214630195549, 0, 18501449233996342525, 0, 2798453429227811166102, 0, 423295045914209481765721, 0, 64028568112067099461618857, 0, 9685159280719860094249515302, 0, 1465010609461302036026993119949, 0, 221602785226749854237147814591613, 0, 33520450775029749226592675973038578, 0, 5070427397502433453836131092389441973, 0, 766971669762948407139129176800817517741, 0, 116014985905603923738178539007000901690946, 0, 17548858310169319379070133457096933737689269] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 15194963895601104 z - 6193466532613858 z - 263 z 24 22 4 6 + 2052728268643895 z - 550343255979599 z + 27680 z - 1577739 z 8 10 12 14 + 55924315 z - 1337773262 z + 22786471088 z - 286888683282 z 18 16 50 - 20327952013839 z + 2743229587871 z - 550343255979599 z 48 20 36 + 2052728268643895 z + 118537393821608 z + 73995422897999984 z 34 66 64 - 67054456930176264 z - 1577739 z + 55924315 z 30 42 44 - 30430181756410054 z - 30430181756410054 z + 15194963895601104 z 46 58 56 - 6193466532613858 z - 286888683282 z + 2743229587871 z 54 52 60 70 - 20327952013839 z + 118537393821608 z + 22786471088 z - 263 z 68 32 38 + 27680 z + 49882357581106766 z - 67054456930176264 z 40 62 72 / 2 + 49882357581106766 z - 1337773262 z + z ) / ((-1 + z ) (1 / 28 26 2 + 62114147974066214 z - 24412879316702116 z - 350 z 24 22 4 6 + 7746504763944719 z - 1974907109494078 z + 44671 z - 2938754 z 8 10 12 14 + 117066499 z - 3098044272 z + 57768349598 z - 789872880064 z 18 16 50 - 64700974390770 z + 8146467141047 z - 1974907109494078 z 48 20 36 + 7746504763944719 z + 401870206686635 z + 323370036770826148 z 34 66 64 - 291790212703642256 z - 2938754 z + 117066499 z 30 42 44 - 128034945210392204 z - 128034945210392204 z + 62114147974066214 z 46 58 56 - 24412879316702116 z - 789872880064 z + 8146467141047 z 54 52 60 70 - 64700974390770 z + 401870206686635 z + 57768349598 z - 350 z 68 32 38 + 44671 z + 214322693227242998 z - 291790212703642256 z 40 62 72 + 214322693227242998 z - 3098044272 z + z )) And in Maple-input format, it is: -(1+15194963895601104*z^28-6193466532613858*z^26-263*z^2+2052728268643895*z^24-\ 550343255979599*z^22+27680*z^4-1577739*z^6+55924315*z^8-1337773262*z^10+ 22786471088*z^12-286888683282*z^14-20327952013839*z^18+2743229587871*z^16-\ 550343255979599*z^50+2052728268643895*z^48+118537393821608*z^20+ 73995422897999984*z^36-67054456930176264*z^34-1577739*z^66+55924315*z^64-\ 30430181756410054*z^30-30430181756410054*z^42+15194963895601104*z^44-\ 6193466532613858*z^46-286888683282*z^58+2743229587871*z^56-20327952013839*z^54+ 118537393821608*z^52+22786471088*z^60-263*z^70+27680*z^68+49882357581106766*z^ 32-67054456930176264*z^38+49882357581106766*z^40-1337773262*z^62+z^72)/(-1+z^2) /(1+62114147974066214*z^28-24412879316702116*z^26-350*z^2+7746504763944719*z^24 -1974907109494078*z^22+44671*z^4-2938754*z^6+117066499*z^8-3098044272*z^10+ 57768349598*z^12-789872880064*z^14-64700974390770*z^18+8146467141047*z^16-\ 1974907109494078*z^50+7746504763944719*z^48+401870206686635*z^20+ 323370036770826148*z^36-291790212703642256*z^34-2938754*z^66+117066499*z^64-\ 128034945210392204*z^30-128034945210392204*z^42+62114147974066214*z^44-\ 24412879316702116*z^46-789872880064*z^58+8146467141047*z^56-64700974390770*z^54 +401870206686635*z^52+57768349598*z^60-350*z^70+44671*z^68+214322693227242998*z ^32-291790212703642256*z^38+214322693227242998*z^40-3098044272*z^62+z^72) The first , 40, terms are: [0, 88, 0, 13547, 0, 2198835, 0, 360352060, 0, 59223156245, 0, 9743115721285, 0, 1603519110490692, 0, 263947440057300763, 0, 43449768508444857555, 0, 7152669484985131160624, 0, 1177478879722137175998561, 0, 193838397253913725913616257, 0, 31910027804007356421139447616, 0, 5253089874221962708205556644451, 0, 864773933636141554953321677515179, 0, 142360791602132841548808759924176372, 0, 23435715427822456344881001925411934661, 0, 3858033946609406827784258253860489406549, 0, 635117202949704022208224670524216738543884, 0, 104554254352224630973472621449593757275394915] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 23097009148919755 z - 9209602609980145 z - 260 z 24 22 4 6 + 2972262543214403 z - 772576958026334 z + 27445 z - 1599631 z 8 10 12 14 + 58722557 z - 1464658214 z + 26085861261 z - 343544101902 z 18 16 50 - 26512522835326 z + 3432110779730 z - 772576958026334 z 48 20 36 + 2972262543214403 z + 160681650061835 z + 117001705248311388 z 34 66 64 - 105764600505156020 z - 1599631 z + 58722557 z 30 42 44 - 47058286019024632 z - 47058286019024632 z + 23097009148919755 z 46 58 56 - 9209602609980145 z - 343544101902 z + 3432110779730 z 54 52 60 70 - 26512522835326 z + 160681650061835 z + 26085861261 z - 260 z 68 32 38 + 27445 z + 78097659418398183 z - 105764600505156020 z 40 62 72 / + 78097659418398183 z - 1464658214 z + z ) / (-1 / 28 26 2 - 130265173387496136 z + 47130751692375323 z + 349 z 24 22 4 6 - 13806675845977891 z + 3258048794725086 z - 44134 z + 2954266 z 8 10 12 14 - 122200003 z + 3401521467 z - 67251915468 z + 980045723020 z 18 16 50 + 92106216862193 z - 10811759527781 z + 13806675845977891 z 48 20 36 - 47130751692375323 z - 615137397487294 z - 979203919640939114 z 34 66 64 + 801054455321568245 z + 122200003 z - 3401521467 z 30 42 44 + 292631036925609852 z + 535779788767379897 z - 292631036925609852 z 46 58 56 + 130265173387496136 z + 10811759527781 z - 92106216862193 z 54 52 60 + 615137397487294 z - 3258048794725086 z - 980045723020 z 70 68 32 + 44134 z - 2954266 z - 535779788767379897 z 38 40 62 74 + 979203919640939114 z - 801054455321568245 z + 67251915468 z + z 72 - 349 z ) And in Maple-input format, it is: -(1+23097009148919755*z^28-9209602609980145*z^26-260*z^2+2972262543214403*z^24-\ 772576958026334*z^22+27445*z^4-1599631*z^6+58722557*z^8-1464658214*z^10+ 26085861261*z^12-343544101902*z^14-26512522835326*z^18+3432110779730*z^16-\ 772576958026334*z^50+2972262543214403*z^48+160681650061835*z^20+ 117001705248311388*z^36-105764600505156020*z^34-1599631*z^66+58722557*z^64-\ 47058286019024632*z^30-47058286019024632*z^42+23097009148919755*z^44-\ 9209602609980145*z^46-343544101902*z^58+3432110779730*z^56-26512522835326*z^54+ 160681650061835*z^52+26085861261*z^60-260*z^70+27445*z^68+78097659418398183*z^ 32-105764600505156020*z^38+78097659418398183*z^40-1464658214*z^62+z^72)/(-1-\ 130265173387496136*z^28+47130751692375323*z^26+349*z^2-13806675845977891*z^24+ 3258048794725086*z^22-44134*z^4+2954266*z^6-122200003*z^8+3401521467*z^10-\ 67251915468*z^12+980045723020*z^14+92106216862193*z^18-10811759527781*z^16+ 13806675845977891*z^50-47130751692375323*z^48-615137397487294*z^20-\ 979203919640939114*z^36+801054455321568245*z^34+122200003*z^66-3401521467*z^64+ 292631036925609852*z^30+535779788767379897*z^42-292631036925609852*z^44+ 130265173387496136*z^46+10811759527781*z^58-92106216862193*z^56+615137397487294 *z^54-3258048794725086*z^52-980045723020*z^60+44134*z^70-2954266*z^68-\ 535779788767379897*z^32+979203919640939114*z^38-801054455321568245*z^40+ 67251915468*z^62+z^74-349*z^72) The first , 40, terms are: [0, 89, 0, 14372, 0, 2442537, 0, 417603793, 0, 71464569737, 0, 12231823964033, 0, 2093661648746276, 0, 358364402335990977, 0, 61340015767131471369, 0, 10499365236761179067161, 0, 1797141338205500629225137, 0, 307610695031197703921926756, 0, 52652697922704245094130040401, 0, 9012386907483007489107704160633, 0, 1542620245252911028488435993131041, 0, 264045168674396313976861615694000217, 0, 45195731947227491527645242847495117412, 0, 7736002883536764574932001360478581957609, 0, 1324145843771051392997961737606128648959153, 0, 226649633146813657514538114881673588926255057] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6060717209781642924 z - 1402178597806228678 z - 325 z 24 22 4 6 + 268784388945704920 z - 42428148627410282 z + 44662 z - 3495195 z 8 10 12 14 + 177157949 z - 6260144372 z + 161774831706 z - 3161128110836 z 18 16 50 - 572012259548268 z + 47862412213550 z - 633297554766300274636 z 48 20 + 960331849070712934386 z + 5474095739832964 z 36 34 + 352954450396198550182 z - 165955714119674119644 z 66 80 88 84 86 - 42428148627410282 z + 177157949 z + z + 44662 z - 325 z 82 64 30 - 3495195 z + 268784388945704920 z - 21816477253091634868 z 42 44 - 1232243073823104654480 z + 1338914005994343038724 z 46 58 - 1232243073823104654480 z - 21816477253091634868 z 56 54 + 65679675705434949106 z - 165955714119674119644 z 52 60 + 352954450396198550182 z + 6060717209781642924 z 70 68 78 - 572012259548268 z + 5474095739832964 z - 6260144372 z 32 38 + 65679675705434949106 z - 633297554766300274636 z 40 62 76 + 960331849070712934386 z - 1402178597806228678 z + 161774831706 z 74 72 / - 3161128110836 z + 47862412213550 z ) / (-1 / 28 26 2 - 26972376035756769030 z + 5798969792461829362 z + 410 z 24 22 4 6 - 1033100558779833846 z + 151506012648293058 z - 67104 z + 6051734 z 8 10 12 14 - 346119156 z + 13602338713 z - 386893773086 z + 8258215424102 z 18 16 50 + 1756702297812494 z - 135835513529654 z + 6738687382258408658478 z 48 20 - 9365197354158298763642 z - 18143311806295004 z 36 34 - 2119446996477746299266 z + 923427541890965546674 z 66 80 90 88 84 + 1033100558779833846 z - 13602338713 z + z - 410 z - 6051734 z 86 82 64 + 67104 z + 346119156 z - 5798969792461829362 z 30 42 + 104522168886112940908 z + 9365197354158298763642 z 44 46 - 11038017492636711058610 z + 11038017492636711058610 z 58 56 + 338975178540727545978 z - 923427541890965546674 z 54 52 + 2119446996477746299266 z - 4108428203429217276842 z 60 70 - 104522168886112940908 z + 18143311806295004 z 68 78 32 - 151506012648293058 z + 386893773086 z - 338975178540727545978 z 38 40 + 4108428203429217276842 z - 6738687382258408658478 z 62 76 74 + 26972376035756769030 z - 8258215424102 z + 135835513529654 z 72 - 1756702297812494 z ) And in Maple-input format, it is: -(1+6060717209781642924*z^28-1402178597806228678*z^26-325*z^2+ 268784388945704920*z^24-42428148627410282*z^22+44662*z^4-3495195*z^6+177157949* z^8-6260144372*z^10+161774831706*z^12-3161128110836*z^14-572012259548268*z^18+ 47862412213550*z^16-633297554766300274636*z^50+960331849070712934386*z^48+ 5474095739832964*z^20+352954450396198550182*z^36-165955714119674119644*z^34-\ 42428148627410282*z^66+177157949*z^80+z^88+44662*z^84-325*z^86-3495195*z^82+ 268784388945704920*z^64-21816477253091634868*z^30-1232243073823104654480*z^42+ 1338914005994343038724*z^44-1232243073823104654480*z^46-21816477253091634868*z^ 58+65679675705434949106*z^56-165955714119674119644*z^54+352954450396198550182*z ^52+6060717209781642924*z^60-572012259548268*z^70+5474095739832964*z^68-\ 6260144372*z^78+65679675705434949106*z^32-633297554766300274636*z^38+ 960331849070712934386*z^40-1402178597806228678*z^62+161774831706*z^76-\ 3161128110836*z^74+47862412213550*z^72)/(-1-26972376035756769030*z^28+ 5798969792461829362*z^26+410*z^2-1033100558779833846*z^24+151506012648293058*z^ 22-67104*z^4+6051734*z^6-346119156*z^8+13602338713*z^10-386893773086*z^12+ 8258215424102*z^14+1756702297812494*z^18-135835513529654*z^16+ 6738687382258408658478*z^50-9365197354158298763642*z^48-18143311806295004*z^20-\ 2119446996477746299266*z^36+923427541890965546674*z^34+1033100558779833846*z^66 -13602338713*z^80+z^90-410*z^88-6051734*z^84+67104*z^86+346119156*z^82-\ 5798969792461829362*z^64+104522168886112940908*z^30+9365197354158298763642*z^42 -11038017492636711058610*z^44+11038017492636711058610*z^46+ 338975178540727545978*z^58-923427541890965546674*z^56+2119446996477746299266*z^ 54-4108428203429217276842*z^52-104522168886112940908*z^60+18143311806295004*z^ 70-151506012648293058*z^68+386893773086*z^78-338975178540727545978*z^32+ 4108428203429217276842*z^38-6738687382258408658478*z^40+26972376035756769030*z^ 62-8258215424102*z^76+135835513529654*z^74-1756702297812494*z^72) The first , 40, terms are: [0, 85, 0, 12408, 0, 1939979, 0, 308201141, 0, 49194098547, 0, 7864721273769, 0, 1258091298390032, 0, 201300104687380151, 0, 32212062823351821287, 0, 5154792385484920257351, 0, 824919553400923919659263, 0, 132012600164792703350646688, 0, 21126163491860497475262733377, 0, 3380854850789664452231752223219, 0, 541044134960321022459775657865429, 0, 86584266666493501580859629602360995, 0, 13856237721241017745234000912679805640, 0, 2217439050419867153663632253223428143901, 0, 354860832805771519486664363166983403950209, 0, 56789029634537661088709105182950325191680577] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 314085342991818 z - 181336900864046 z - 213 z 24 22 4 6 + 83719367036550 z - 30779698872672 z + 17295 z - 759192 z 8 10 12 14 + 20837250 z - 387110408 z + 5115531719 z - 49734916157 z 18 16 50 - 2049117993856 z + 364447890969 z - 49734916157 z 48 20 36 + 364447890969 z + 8959789522784 z + 314085342991818 z 34 64 30 42 - 436236088095424 z + z - 436236088095424 z - 30779698872672 z 44 46 58 56 + 8959789522784 z - 2049117993856 z - 759192 z + 20837250 z 54 52 60 32 - 387110408 z + 5115531719 z + 17295 z + 486635691157692 z 38 40 62 / - 181336900864046 z + 83719367036550 z - 213 z ) / (-1 / 28 26 2 - 2122669403387764 z + 1103733350983202 z + 309 z 24 22 4 6 - 459387085508386 z + 152375459036276 z - 30612 z + 1560948 z 8 10 12 14 - 48749165 z + 1019253801 z - 15057481968 z + 162961390712 z 18 16 50 + 8263945500253 z - 1325791606057 z + 1325791606057 z 48 20 36 - 8263945500253 z - 40034417484428 z - 3277553581008156 z 34 66 64 30 + 4070869206962282 z + z - 309 z + 3277553581008156 z 42 44 46 + 459387085508386 z - 152375459036276 z + 40034417484428 z 58 56 54 52 + 48749165 z - 1019253801 z + 15057481968 z - 162961390712 z 60 32 38 - 1560948 z - 4070869206962282 z + 2122669403387764 z 40 62 - 1103733350983202 z + 30612 z ) And in Maple-input format, it is: -(1+314085342991818*z^28-181336900864046*z^26-213*z^2+83719367036550*z^24-\ 30779698872672*z^22+17295*z^4-759192*z^6+20837250*z^8-387110408*z^10+5115531719 *z^12-49734916157*z^14-2049117993856*z^18+364447890969*z^16-49734916157*z^50+ 364447890969*z^48+8959789522784*z^20+314085342991818*z^36-436236088095424*z^34+ z^64-436236088095424*z^30-30779698872672*z^42+8959789522784*z^44-2049117993856* z^46-759192*z^58+20837250*z^56-387110408*z^54+5115531719*z^52+17295*z^60+ 486635691157692*z^32-181336900864046*z^38+83719367036550*z^40-213*z^62)/(-1-\ 2122669403387764*z^28+1103733350983202*z^26+309*z^2-459387085508386*z^24+ 152375459036276*z^22-30612*z^4+1560948*z^6-48749165*z^8+1019253801*z^10-\ 15057481968*z^12+162961390712*z^14+8263945500253*z^18-1325791606057*z^16+ 1325791606057*z^50-8263945500253*z^48-40034417484428*z^20-3277553581008156*z^36 +4070869206962282*z^34+z^66-309*z^64+3277553581008156*z^30+459387085508386*z^42 -152375459036276*z^44+40034417484428*z^46+48749165*z^58-1019253801*z^56+ 15057481968*z^54-162961390712*z^52-1560948*z^60-4070869206962282*z^32+ 2122669403387764*z^38-1103733350983202*z^40+30612*z^62) The first , 40, terms are: [0, 96, 0, 16347, 0, 2914227, 0, 522020872, 0, 93563173033, 0, 16770878155121, 0, 3006156250372968, 0, 538850200716415659, 0, 96588332745497159203, 0, 17313358050269973090944, 0, 3103401424044434635150681, 0, 556281478398190274349619017, 0, 99712876615765605296966414336, 0, 17873429458863740998388628474547, 0, 3203793646977446707541895641167835, 0, 574276680144158517187992676700807400, 0, 102938497823857468650394149922798437377, 0, 18451618706810119854731894895536729576281, 0, 3307433468517167255297962701723228257172744, 0, 592854010398020404282218715485396215214903747] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 638626714279546 z - 362461148638296 z - 227 z 24 22 4 6 + 163394105426216 z - 58272083286332 z + 19940 z - 944035 z 8 10 12 14 + 27747077 z - 548733480 z + 7685420504 z - 78910994756 z 18 16 50 - 3585740383856 z + 608503774784 z - 78910994756 z 48 20 36 + 608503774784 z + 16353346167744 z + 638626714279546 z 34 64 30 42 - 896137801144890 z + z - 896137801144890 z - 58272083286332 z 44 46 58 56 + 16353346167744 z - 3585740383856 z - 944035 z + 27747077 z 54 52 60 32 - 548733480 z + 7685420504 z + 19940 z + 1003091702918808 z 38 40 62 / - 362461148638296 z + 163394105426216 z - 227 z ) / (-1 / 28 26 2 - 4202106748065791 z + 2144129893097823 z + 308 z 24 22 4 6 - 870290901248545 z + 279815474484743 z - 33129 z + 1843144 z 8 10 12 14 - 62217933 z + 1393001536 z - 21882888609 z + 250509885855 z 18 16 50 + 14021201128575 z - 2145982199929 z + 2145982199929 z 48 20 36 - 14021201128575 z - 70854351054977 z - 6570517463704447 z 34 66 64 30 + 8212340593173967 z + z - 308 z + 6570517463704447 z 42 44 46 + 870290901248545 z - 279815474484743 z + 70854351054977 z 58 56 54 52 + 62217933 z - 1393001536 z + 21882888609 z - 250509885855 z 60 32 38 - 1843144 z - 8212340593173967 z + 4202106748065791 z 40 62 - 2144129893097823 z + 33129 z ) And in Maple-input format, it is: -(1+638626714279546*z^28-362461148638296*z^26-227*z^2+163394105426216*z^24-\ 58272083286332*z^22+19940*z^4-944035*z^6+27747077*z^8-548733480*z^10+7685420504 *z^12-78910994756*z^14-3585740383856*z^18+608503774784*z^16-78910994756*z^50+ 608503774784*z^48+16353346167744*z^20+638626714279546*z^36-896137801144890*z^34 +z^64-896137801144890*z^30-58272083286332*z^42+16353346167744*z^44-\ 3585740383856*z^46-944035*z^58+27747077*z^56-548733480*z^54+7685420504*z^52+ 19940*z^60+1003091702918808*z^32-362461148638296*z^38+163394105426216*z^40-227* z^62)/(-1-4202106748065791*z^28+2144129893097823*z^26+308*z^2-870290901248545*z ^24+279815474484743*z^22-33129*z^4+1843144*z^6-62217933*z^8+1393001536*z^10-\ 21882888609*z^12+250509885855*z^14+14021201128575*z^18-2145982199929*z^16+ 2145982199929*z^50-14021201128575*z^48-70854351054977*z^20-6570517463704447*z^ 36+8212340593173967*z^34+z^66-308*z^64+6570517463704447*z^30+870290901248545*z^ 42-279815474484743*z^44+70854351054977*z^46+62217933*z^58-1393001536*z^56+ 21882888609*z^54-250509885855*z^52-1843144*z^60-8212340593173967*z^32+ 4202106748065791*z^38-2144129893097823*z^40+33129*z^62) The first , 40, terms are: [0, 81, 0, 11759, 0, 1837432, 0, 291188953, 0, 46292058575, 0, 7364821965535, 0, 1171916897349375, 0, 186487951262935567, 0, 29676286010352265961, 0, 4722473346807327534104, 0, 751501379343153962308607, 0, 119588694119751844700326097, 0, 19030512482528883268855481953, 0, 3028383343032220617877502839073, 0, 481915855176372779018407699472721, 0, 76688736307084047772517561237187295, 0, 12203711943131644347236589305333862104, 0, 1942013812838294719648651546470399018441, 0, 309038566861545450523067927881226499935759, 0, 49178247434050395396096592738538938918177599] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 217297092631641962464 z - 31374833750624460765 z - 341 z 24 22 4 6 + 3883314022430439052 z - 409430061291602791 z + 50552 z - 4405343 z 102 8 10 12 - 10786000841 z + 256854652 z - 10786000841 z + 342184727652 z 14 18 16 - 8480361962903 z - 2723588136928729 z + 168267139586568 z 50 48 - 40204326755692468941041069 z + 25753485354229029563714132 z 20 36 + 36490945937854008 z + 117417170877963117632880 z 34 66 - 30046824043097274359525 z - 14514371474016597237262667 z 80 100 90 + 6698009928769185017604 z + 342184727652 z - 409430061291602791 z 88 84 + 3883314022430439052 z + 217297092631641962464 z 94 86 96 - 2723588136928729 z - 31374833750624460765 z + 168267139586568 z 98 92 82 - 8480361962903 z + 36490945937854008 z - 1296830463871684032199 z 64 112 110 106 + 25753485354229029563714132 z + z - 341 z - 4405343 z 108 30 42 + 50552 z - 1296830463871684032199 z - 3131434399615351485982613 z 44 46 + 7192613651237404164540756 z - 14514371474016597237262667 z 58 56 - 66846203608528750104032607 z + 71229156989381531345589686 z 54 52 - 66846203608528750104032607 z + 55246897639841202985738736 z 60 70 + 55246897639841202985738736 z - 3131434399615351485982613 z 68 78 + 7192613651237404164540756 z - 30046824043097274359525 z 32 38 + 6698009928769185017604 z - 400691643792238859745003 z 40 62 + 1196491785277071266615224 z - 40204326755692468941041069 z 76 74 + 117417170877963117632880 z - 400691643792238859745003 z 72 104 / + 1196491785277071266615224 z + 256854652 z ) / (-1 / 28 26 2 - 792994233411336657788 z + 107840506543587464082 z + 432 z 24 22 4 - 12567921456564527560 z + 1247061606706544898 z - 75499 z 6 102 8 10 + 7446932 z + 752110056094 z - 479923054 z + 21957256276 z 12 14 18 - 752110056094 z + 20007664715535 z + 7327606754464757 z 16 50 - 424485855673543 z + 285607921152983296195087125 z 48 20 - 171969339813401999651429023 z - 104521170948941589 z 36 34 - 544346022345787106781067 z + 131190505564314759479115 z 66 80 + 171969339813401999651429023 z - 131190505564314759479115 z 100 90 - 20007664715535 z + 12567921456564527560 z 88 84 - 107840506543587464082 z - 5024087020615880691225 z 94 86 + 104521170948941589 z + 792994233411336657788 z 96 98 92 - 7327606754464757 z + 424485855673543 z - 1247061606706544898 z 82 64 112 + 27546701702783055066929 z - 285607921152983296195087125 z - 432 z 114 110 106 108 + z + 75499 z + 479923054 z - 7446932 z 30 42 + 5024087020615880691225 z + 17400737418079175331352948 z 44 46 - 42478803543814972677170118 z + 91133544871631490728013295 z 58 56 + 610631282853872024916871626 z - 610631282853872024916871626 z 54 52 + 538040934873919023727241314 z - 417679491430329496507085417 z 60 70 - 538040934873919023727241314 z + 42478803543814972677170118 z 68 78 - 91133544871631490728013295 z + 544346022345787106781067 z 32 38 - 27546701702783055066929 z + 1972767123218811556617480 z 40 62 - 6257453411383959232113038 z + 417679491430329496507085417 z 76 74 - 1972767123218811556617480 z + 6257453411383959232113038 z 72 104 - 17400737418079175331352948 z - 21957256276 z ) And in Maple-input format, it is: -(1+217297092631641962464*z^28-31374833750624460765*z^26-341*z^2+ 3883314022430439052*z^24-409430061291602791*z^22+50552*z^4-4405343*z^6-\ 10786000841*z^102+256854652*z^8-10786000841*z^10+342184727652*z^12-\ 8480361962903*z^14-2723588136928729*z^18+168267139586568*z^16-\ 40204326755692468941041069*z^50+25753485354229029563714132*z^48+ 36490945937854008*z^20+117417170877963117632880*z^36-30046824043097274359525*z^ 34-14514371474016597237262667*z^66+6698009928769185017604*z^80+342184727652*z^ 100-409430061291602791*z^90+3883314022430439052*z^88+217297092631641962464*z^84 -2723588136928729*z^94-31374833750624460765*z^86+168267139586568*z^96-\ 8480361962903*z^98+36490945937854008*z^92-1296830463871684032199*z^82+ 25753485354229029563714132*z^64+z^112-341*z^110-4405343*z^106+50552*z^108-\ 1296830463871684032199*z^30-3131434399615351485982613*z^42+ 7192613651237404164540756*z^44-14514371474016597237262667*z^46-\ 66846203608528750104032607*z^58+71229156989381531345589686*z^56-\ 66846203608528750104032607*z^54+55246897639841202985738736*z^52+ 55246897639841202985738736*z^60-3131434399615351485982613*z^70+ 7192613651237404164540756*z^68-30046824043097274359525*z^78+ 6698009928769185017604*z^32-400691643792238859745003*z^38+ 1196491785277071266615224*z^40-40204326755692468941041069*z^62+ 117417170877963117632880*z^76-400691643792238859745003*z^74+ 1196491785277071266615224*z^72+256854652*z^104)/(-1-792994233411336657788*z^28+ 107840506543587464082*z^26+432*z^2-12567921456564527560*z^24+ 1247061606706544898*z^22-75499*z^4+7446932*z^6+752110056094*z^102-479923054*z^8 +21957256276*z^10-752110056094*z^12+20007664715535*z^14+7327606754464757*z^18-\ 424485855673543*z^16+285607921152983296195087125*z^50-\ 171969339813401999651429023*z^48-104521170948941589*z^20-\ 544346022345787106781067*z^36+131190505564314759479115*z^34+ 171969339813401999651429023*z^66-131190505564314759479115*z^80-20007664715535*z ^100+12567921456564527560*z^90-107840506543587464082*z^88-\ 5024087020615880691225*z^84+104521170948941589*z^94+792994233411336657788*z^86-\ 7327606754464757*z^96+424485855673543*z^98-1247061606706544898*z^92+ 27546701702783055066929*z^82-285607921152983296195087125*z^64-432*z^112+z^114+ 75499*z^110+479923054*z^106-7446932*z^108+5024087020615880691225*z^30+ 17400737418079175331352948*z^42-42478803543814972677170118*z^44+ 91133544871631490728013295*z^46+610631282853872024916871626*z^58-\ 610631282853872024916871626*z^56+538040934873919023727241314*z^54-\ 417679491430329496507085417*z^52-538040934873919023727241314*z^60+ 42478803543814972677170118*z^70-91133544871631490728013295*z^68+ 544346022345787106781067*z^78-27546701702783055066929*z^32+ 1972767123218811556617480*z^38-6257453411383959232113038*z^40+ 417679491430329496507085417*z^62-1972767123218811556617480*z^76+ 6257453411383959232113038*z^74-17400737418079175331352948*z^72-21957256276*z^ 104) The first , 40, terms are: [0, 91, 0, 14365, 0, 2376860, 0, 396862795, 0, 66467610001, 0, 11145668476211, 0, 1869892305874971, 0, 313772561106533945, 0, 52656187747581270779, 0, 8836874419637070007308, 0, 1483044013246904641389941, 0, 248892549947065182092355075, 0, 41770606517024262496626417121, 0, 7010194818051656665808722008833, 0, 1176493594356416335725113497419363, 0, 197446353548049292255437215186847781, 0, 33136657387696499040714802917805998156, 0, 5561197122545593274329835800405502825755, 0, 933314227366445825020695692104635936256281, 0, 156634521646451349749696390561728806099716363] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 293293572886546 z - 169816320628550 z - 213 z 24 22 4 6 + 78693411861158 z - 29060922647880 z + 17363 z - 760412 z 8 10 12 14 + 20726110 z - 381571604 z + 4995473275 z - 48150046565 z 18 16 50 - 1955838528056 z + 350162291833 z - 48150046565 z 48 20 36 + 350162291833 z + 8502666858640 z + 293293572886546 z 34 64 30 42 - 406625337751952 z + z - 406625337751952 z - 29060922647880 z 44 46 58 56 + 8502666858640 z - 1955838528056 z - 760412 z + 20726110 z 54 52 60 32 - 381571604 z + 4995473275 z + 17363 z + 453324244080676 z 38 40 62 / - 169816320628550 z + 78693411861158 z - 213 z ) / (-1 / 28 26 2 - 1962404804879720 z + 1023468508762974 z + 297 z 24 22 4 6 - 427544578087118 z + 142406944551992 z - 29576 z + 1516012 z 8 10 12 14 - 47340393 z + 985656329 z - 14472276708 z + 155608478048 z 18 16 50 + 7798481233313 z - 1258163118889 z + 1258163118889 z 48 20 36 - 7798481233313 z - 37588313490840 z - 3023589871022272 z 34 66 64 30 + 3751214910786750 z + z - 297 z + 3023589871022272 z 42 44 46 + 427544578087118 z - 142406944551992 z + 37588313490840 z 58 56 54 52 + 47340393 z - 985656329 z + 14472276708 z - 155608478048 z 60 32 38 - 1516012 z - 3751214910786750 z + 1962404804879720 z 40 62 - 1023468508762974 z + 29576 z ) And in Maple-input format, it is: -(1+293293572886546*z^28-169816320628550*z^26-213*z^2+78693411861158*z^24-\ 29060922647880*z^22+17363*z^4-760412*z^6+20726110*z^8-381571604*z^10+4995473275 *z^12-48150046565*z^14-1955838528056*z^18+350162291833*z^16-48150046565*z^50+ 350162291833*z^48+8502666858640*z^20+293293572886546*z^36-406625337751952*z^34+ z^64-406625337751952*z^30-29060922647880*z^42+8502666858640*z^44-1955838528056* z^46-760412*z^58+20726110*z^56-381571604*z^54+4995473275*z^52+17363*z^60+ 453324244080676*z^32-169816320628550*z^38+78693411861158*z^40-213*z^62)/(-1-\ 1962404804879720*z^28+1023468508762974*z^26+297*z^2-427544578087118*z^24+ 142406944551992*z^22-29576*z^4+1516012*z^6-47340393*z^8+985656329*z^10-\ 14472276708*z^12+155608478048*z^14+7798481233313*z^18-1258163118889*z^16+ 1258163118889*z^50-7798481233313*z^48-37588313490840*z^20-3023589871022272*z^36 +3751214910786750*z^34+z^66-297*z^64+3023589871022272*z^30+427544578087118*z^42 -142406944551992*z^44+37588313490840*z^46+47340393*z^58-985656329*z^56+ 14472276708*z^54-155608478048*z^52-1516012*z^60-3751214910786750*z^32+ 1962404804879720*z^38-1023468508762974*z^40+29576*z^62) The first , 40, terms are: [0, 84, 0, 12735, 0, 2053511, 0, 333973132, 0, 54389283401, 0, 8859613559545, 0, 1443225159874476, 0, 235102243613886103, 0, 38298355028083378863, 0, 6238836496386554675188, 0, 1016312146492533749688977, 0, 165558176508295968091939313, 0, 26969578184557516126863728564, 0, 4393368925049418085946944454863, 0, 715683811682570351075895624879991, 0, 116585546777558052832379144739744876, 0, 18991892083704211751759419747820723225, 0, 3093796571610204654438669325292787570729, 0, 503982288037731144570681642603506779194316, 0, 82099175164439907411476886212210739050802343] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 252560304306266082476 z - 35526462804907596102 z - 332 z 24 22 4 6 + 4284669458896093368 z - 440408600881080558 z + 48320 z - 4176274 z 102 8 10 12 - 10307018212 z + 243632477 z - 10307018212 z + 331146168324 z 14 18 16 - 8343738334566 z - 2792462205190226 z + 168815843861909 z 50 48 - 58287275007998290128203558 z + 36954209069796609448865954 z 20 36 + 38298656369067440 z + 150935970008007219025252 z 34 66 - 37705523353651487819938 z - 20558803848221513760706630 z 80 100 90 + 8197771153369743641324 z + 331146168324 z - 440408600881080558 z 88 84 + 4284669458896093368 z + 252560304306266082476 z 94 86 96 - 2792462205190226 z - 35526462804907596102 z + 168815843861909 z 98 92 82 - 8343738334566 z + 38298656369067440 z - 1547015820269999777238 z 64 112 110 106 + 36954209069796609448865954 z + z - 332 z - 4176274 z 108 30 42 + 48320 z - 1547015820269999777238 z - 4291137370205401841086458 z 44 46 + 10032005742825492576472336 z - 20558803848221513760706630 z 58 56 - 98083473176089485076870546 z + 104673636403693698310121198 z 54 52 - 98083473176089485076870546 z + 80696868519590311607299116 z 60 70 + 80696868519590311607299116 z - 4291137370205401841086458 z 68 78 + 10032005742825492576472336 z - 37705523353651487819938 z 32 38 + 8197771153369743641324 z - 526993528890646758206050 z 40 62 + 1607679281591155078189328 z - 58287275007998290128203558 z 76 74 + 150935970008007219025252 z - 526993528890646758206050 z 72 104 / 2 + 1607679281591155078189328 z + 243632477 z ) / ((-1 + z ) (1 / 28 26 2 + 793749913117631566622 z - 106933476225963601029 z - 421 z 24 22 4 + 12328677424176485380 z - 1209097435791379655 z + 71344 z 6 102 8 10 - 6897781 z - 20110839982 z + 440308891 z - 20110839982 z 12 14 18 + 691500044006 z - 18534588029468 z - 6934544380560185 z 16 50 + 397166594273781 z - 252842759277290851918091283 z 48 20 + 158046038374054676053613034 z + 100115868062497714 z 36 34 + 552987234474429875749474 z - 133360343989128860759873 z 66 80 - 86367592848545054239951163 z + 27930516789194458896336 z 100 90 88 + 691500044006 z - 1209097435791379655 z + 12328677424176485380 z 84 94 + 793749913117631566622 z - 6934544380560185 z 86 96 98 - 106933476225963601029 z + 397166594273781 z - 18534588029468 z 92 82 + 100115868062497714 z - 5067131535023633328643 z 64 112 110 106 + 158046038374054676053613034 z + z - 421 z - 6897781 z 108 30 42 + 71344 z - 5067131535023633328643 z - 17217037815842255723103621 z 44 46 + 41253552167178471599018078 z - 86367592848545054239951163 z 58 56 - 432555624746056850435952473 z + 462583355265108674127613322 z 54 52 - 432555624746056850435952473 z + 353669142743087667064085770 z 60 70 + 353669142743087667064085770 z - 17217037815842255723103621 z 68 78 + 41253552167178471599018078 z - 133360343989128860759873 z 32 38 + 27930516789194458896336 z - 1995331948636748533653607 z 40 62 + 6274747414398676921691516 z - 252842759277290851918091283 z 76 74 + 552987234474429875749474 z - 1995331948636748533653607 z 72 104 + 6274747414398676921691516 z + 440308891 z )) And in Maple-input format, it is: -(1+252560304306266082476*z^28-35526462804907596102*z^26-332*z^2+ 4284669458896093368*z^24-440408600881080558*z^22+48320*z^4-4176274*z^6-\ 10307018212*z^102+243632477*z^8-10307018212*z^10+331146168324*z^12-\ 8343738334566*z^14-2792462205190226*z^18+168815843861909*z^16-\ 58287275007998290128203558*z^50+36954209069796609448865954*z^48+ 38298656369067440*z^20+150935970008007219025252*z^36-37705523353651487819938*z^ 34-20558803848221513760706630*z^66+8197771153369743641324*z^80+331146168324*z^ 100-440408600881080558*z^90+4284669458896093368*z^88+252560304306266082476*z^84 -2792462205190226*z^94-35526462804907596102*z^86+168815843861909*z^96-\ 8343738334566*z^98+38298656369067440*z^92-1547015820269999777238*z^82+ 36954209069796609448865954*z^64+z^112-332*z^110-4176274*z^106+48320*z^108-\ 1547015820269999777238*z^30-4291137370205401841086458*z^42+ 10032005742825492576472336*z^44-20558803848221513760706630*z^46-\ 98083473176089485076870546*z^58+104673636403693698310121198*z^56-\ 98083473176089485076870546*z^54+80696868519590311607299116*z^52+ 80696868519590311607299116*z^60-4291137370205401841086458*z^70+ 10032005742825492576472336*z^68-37705523353651487819938*z^78+ 8197771153369743641324*z^32-526993528890646758206050*z^38+ 1607679281591155078189328*z^40-58287275007998290128203558*z^62+ 150935970008007219025252*z^76-526993528890646758206050*z^74+ 1607679281591155078189328*z^72+243632477*z^104)/(-1+z^2)/(1+ 793749913117631566622*z^28-106933476225963601029*z^26-421*z^2+ 12328677424176485380*z^24-1209097435791379655*z^22+71344*z^4-6897781*z^6-\ 20110839982*z^102+440308891*z^8-20110839982*z^10+691500044006*z^12-\ 18534588029468*z^14-6934544380560185*z^18+397166594273781*z^16-\ 252842759277290851918091283*z^50+158046038374054676053613034*z^48+ 100115868062497714*z^20+552987234474429875749474*z^36-133360343989128860759873* z^34-86367592848545054239951163*z^66+27930516789194458896336*z^80+691500044006* z^100-1209097435791379655*z^90+12328677424176485380*z^88+793749913117631566622* z^84-6934544380560185*z^94-106933476225963601029*z^86+397166594273781*z^96-\ 18534588029468*z^98+100115868062497714*z^92-5067131535023633328643*z^82+ 158046038374054676053613034*z^64+z^112-421*z^110-6897781*z^106+71344*z^108-\ 5067131535023633328643*z^30-17217037815842255723103621*z^42+ 41253552167178471599018078*z^44-86367592848545054239951163*z^46-\ 432555624746056850435952473*z^58+462583355265108674127613322*z^56-\ 432555624746056850435952473*z^54+353669142743087667064085770*z^52+ 353669142743087667064085770*z^60-17217037815842255723103621*z^70+ 41253552167178471599018078*z^68-133360343989128860759873*z^78+ 27930516789194458896336*z^32-1995331948636748533653607*z^38+ 6274747414398676921691516*z^40-252842759277290851918091283*z^62+ 552987234474429875749474*z^76-1995331948636748533653607*z^74+ 6274747414398676921691516*z^72+440308891*z^104) The first , 40, terms are: [0, 90, 0, 14535, 0, 2467771, 0, 421942142, 0, 72251760165, 0, 12376759244761, 0, 2120357026294598, 0, 363264472450459215, 0, 62235778849929997643, 0, 10662480185108637652698, 0, 1826739401367810374869053, 0, 312964462153402281938142693, 0, 53618353484425516272849732170, 0, 9186116041286608497900482389187, 0, 1573803049377798961194057247412407, 0, 269630388892439586273043684423800838, 0, 46194183358971167359783566729889868001, 0, 7914176829725792839404441457272809396797, 0, 1355889212430824673850357721173489980641278, 0, 232296497279644535027237651446617449259574483] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 244954862657482 z - 143851503163078 z - 217 z 24 22 4 6 + 67933227795950 z - 25663907804264 z + 17875 z - 785900 z 8 10 12 14 + 21341670 z - 388620372 z + 5001591811 z - 47170414817 z 18 16 50 - 1819687586328 z + 334577763057 z - 47170414817 z 48 20 36 + 334577763057 z + 7701969706288 z + 244954862657482 z 34 64 30 42 - 336644346089152 z + z - 336644346089152 z - 25663907804264 z 44 46 58 56 + 7701969706288 z - 1819687586328 z - 785900 z + 21341670 z 54 52 60 32 - 388620372 z + 5001591811 z + 17875 z + 374194687233748 z 38 40 62 / - 143851503163078 z + 67933227795950 z - 217 z ) / (-1 / 28 26 2 - 1671910131434064 z + 887011267287350 z + 313 z 24 22 4 6 - 378578227006662 z + 129251685481088 z - 31504 z + 1612924 z 8 10 12 14 - 50020449 z + 1029523513 z - 14874810620 z + 156708209136 z 18 16 50 + 7469622563881 z - 1237138909505 z + 1237138909505 z 48 20 36 - 7469622563881 z - 35037763956816 z - 2545538098444696 z 34 66 64 30 + 3138916610963862 z + z - 313 z + 2545538098444696 z 42 44 46 + 378578227006662 z - 129251685481088 z + 35037763956816 z 58 56 54 52 + 50020449 z - 1029523513 z + 14874810620 z - 156708209136 z 60 32 38 - 1612924 z - 3138916610963862 z + 1671910131434064 z 40 62 - 887011267287350 z + 31504 z ) And in Maple-input format, it is: -(1+244954862657482*z^28-143851503163078*z^26-217*z^2+67933227795950*z^24-\ 25663907804264*z^22+17875*z^4-785900*z^6+21341670*z^8-388620372*z^10+5001591811 *z^12-47170414817*z^14-1819687586328*z^18+334577763057*z^16-47170414817*z^50+ 334577763057*z^48+7701969706288*z^20+244954862657482*z^36-336644346089152*z^34+ z^64-336644346089152*z^30-25663907804264*z^42+7701969706288*z^44-1819687586328* z^46-785900*z^58+21341670*z^56-388620372*z^54+5001591811*z^52+17875*z^60+ 374194687233748*z^32-143851503163078*z^38+67933227795950*z^40-217*z^62)/(-1-\ 1671910131434064*z^28+887011267287350*z^26+313*z^2-378578227006662*z^24+ 129251685481088*z^22-31504*z^4+1612924*z^6-50020449*z^8+1029523513*z^10-\ 14874810620*z^12+156708209136*z^14+7469622563881*z^18-1237138909505*z^16+ 1237138909505*z^50-7469622563881*z^48-35037763956816*z^20-2545538098444696*z^36 +3138916610963862*z^34+z^66-313*z^64+2545538098444696*z^30+378578227006662*z^42 -129251685481088*z^44+35037763956816*z^46+50020449*z^58-1029523513*z^56+ 14874810620*z^54-156708209136*z^52-1612924*z^60-3138916610963862*z^32+ 1671910131434064*z^38-887011267287350*z^40+31504*z^62) The first , 40, terms are: [0, 96, 0, 16419, 0, 2941787, 0, 529677080, 0, 95432407585, 0, 17195950987281, 0, 3098594760975416, 0, 558348006285122763, 0, 100611003151463706291, 0, 18129509503880204252736, 0, 3266830811518303840109457, 0, 588663671087150195151479921, 0, 106073726528954553047279755520, 0, 19113860792609333856486801952275, 0, 3444205142734047060856893764088939, 0, 620625481899601606871385787877816696, 0, 111833056633291307551103328610676825905, 0, 20151658158912712727407739662952409458945, 0, 3631210116033150817217592580649087249593560, 0, 654322676714818294366735618781902992235956411] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 237369080047621034029 z - 34399032719390890683 z - 344 z 24 22 4 6 + 4268151231033478457 z - 450478222270585334 z + 51609 z - 4559855 z 102 8 10 12 - 11468361054 z + 269602373 z - 11468361054 z + 367892798473 z 14 18 16 - 9199304078426 z - 2987619761465130 z + 183754714455142 z 50 48 - 41283317763101274306016044 z + 26544915978222697461517986 z 20 36 + 40125980881834453 z + 125403989579407881071320 z 34 66 - 32296895383427137290632 z - 15030291466212243405273798 z 80 100 90 + 7243119014955898541965 z + 367892798473 z - 450478222270585334 z 88 84 + 4268151231033478457 z + 237369080047621034029 z 94 86 96 - 2987619761465130 z - 34399032719390890683 z + 183754714455142 z 98 92 82 - 9199304078426 z + 40125980881834453 z - 1410030490835190279368 z 64 112 110 106 + 26544915978222697461517986 z + z - 344 z - 4559855 z 108 30 42 + 51609 z - 1410030490835190279368 z - 3279788431285508445501632 z 44 46 + 7488560743183353238985850 z - 15030291466212243405273798 z 58 56 - 68331822713296877053548220 z + 72770219060803189485566180 z 54 52 - 68331822713296877053548220 z + 56571450475839800910000130 z 60 70 + 56571450475839800910000130 z - 3279788431285508445501632 z 68 78 + 7488560743183353238985850 z - 32296895383427137290632 z 32 38 + 7243119014955898541965 z - 425137318421952565969560 z 40 62 + 1261173956916700242345266 z - 41283317763101274306016044 z 76 74 + 125403989579407881071320 z - 425137318421952565969560 z 72 104 / 2 + 1261173956916700242345266 z + 269602373 z ) / ((-1 + z ) (1 / 28 26 2 + 767776441060905338782 z - 106703922991926406896 z - 440 z 24 22 4 + 12670639789986946055 z - 1277064058095251640 z + 77378 z 6 102 8 10 - 7688048 z - 23008255864 z + 499428863 z - 23008255864 z 12 14 18 + 791705595288 z - 21091390116744 z - 7666430106432328 z 16 50 + 446581877707357 z - 180824710951616939751403856 z 48 20 + 114780012610708105230324418 z + 108371275182334732 z 36 34 + 469637984517386202425204 z - 116983308184686706521856 z 66 80 - 63937598398114218385772128 z + 25317600362675027912657 z 100 90 88 + 791705595288 z - 1277064058095251640 z + 12670639789986946055 z 84 94 + 767776441060905338782 z - 7666430106432328 z 86 96 98 - 106703922991926406896 z + 446581877707357 z - 21091390116744 z 92 82 + 108371275182334732 z - 4746030875412191002200 z 64 112 110 106 + 114780012610708105230324418 z + z - 440 z - 7688048 z 108 30 42 + 77378 z - 4746030875412191002200 z - 13373127941560434722988080 z 44 46 + 31236506583914860934617448 z - 63937598398114218385772128 z 58 56 - 303803555157943663345514160 z + 324146403140743370451830382 z 54 52 - 303803555157943663345514160 z + 250105554985431463391184292 z 60 70 + 250105554985431463391184292 z - 13373127941560434722988080 z 68 78 + 31236506583914860934617448 z - 116983308184686706521856 z 32 38 + 25317600362675027912657 z - 1642216290203980094176640 z 40 62 + 5012034175969538327346934 z - 180824710951616939751403856 z 76 74 + 469637984517386202425204 z - 1642216290203980094176640 z 72 104 + 5012034175969538327346934 z + 499428863 z )) And in Maple-input format, it is: -(1+237369080047621034029*z^28-34399032719390890683*z^26-344*z^2+ 4268151231033478457*z^24-450478222270585334*z^22+51609*z^4-4559855*z^6-\ 11468361054*z^102+269602373*z^8-11468361054*z^10+367892798473*z^12-\ 9199304078426*z^14-2987619761465130*z^18+183754714455142*z^16-\ 41283317763101274306016044*z^50+26544915978222697461517986*z^48+ 40125980881834453*z^20+125403989579407881071320*z^36-32296895383427137290632*z^ 34-15030291466212243405273798*z^66+7243119014955898541965*z^80+367892798473*z^ 100-450478222270585334*z^90+4268151231033478457*z^88+237369080047621034029*z^84 -2987619761465130*z^94-34399032719390890683*z^86+183754714455142*z^96-\ 9199304078426*z^98+40125980881834453*z^92-1410030490835190279368*z^82+ 26544915978222697461517986*z^64+z^112-344*z^110-4559855*z^106+51609*z^108-\ 1410030490835190279368*z^30-3279788431285508445501632*z^42+ 7488560743183353238985850*z^44-15030291466212243405273798*z^46-\ 68331822713296877053548220*z^58+72770219060803189485566180*z^56-\ 68331822713296877053548220*z^54+56571450475839800910000130*z^52+ 56571450475839800910000130*z^60-3279788431285508445501632*z^70+ 7488560743183353238985850*z^68-32296895383427137290632*z^78+ 7243119014955898541965*z^32-425137318421952565969560*z^38+ 1261173956916700242345266*z^40-41283317763101274306016044*z^62+ 125403989579407881071320*z^76-425137318421952565969560*z^74+ 1261173956916700242345266*z^72+269602373*z^104)/(-1+z^2)/(1+ 767776441060905338782*z^28-106703922991926406896*z^26-440*z^2+ 12670639789986946055*z^24-1277064058095251640*z^22+77378*z^4-7688048*z^6-\ 23008255864*z^102+499428863*z^8-23008255864*z^10+791705595288*z^12-\ 21091390116744*z^14-7666430106432328*z^18+446581877707357*z^16-\ 180824710951616939751403856*z^50+114780012610708105230324418*z^48+ 108371275182334732*z^20+469637984517386202425204*z^36-116983308184686706521856* z^34-63937598398114218385772128*z^66+25317600362675027912657*z^80+791705595288* z^100-1277064058095251640*z^90+12670639789986946055*z^88+767776441060905338782* z^84-7666430106432328*z^94-106703922991926406896*z^86+446581877707357*z^96-\ 21091390116744*z^98+108371275182334732*z^92-4746030875412191002200*z^82+ 114780012610708105230324418*z^64+z^112-440*z^110-7688048*z^106+77378*z^108-\ 4746030875412191002200*z^30-13373127941560434722988080*z^42+ 31236506583914860934617448*z^44-63937598398114218385772128*z^46-\ 303803555157943663345514160*z^58+324146403140743370451830382*z^56-\ 303803555157943663345514160*z^54+250105554985431463391184292*z^52+ 250105554985431463391184292*z^60-13373127941560434722988080*z^70+ 31236506583914860934617448*z^68-116983308184686706521856*z^78+ 25317600362675027912657*z^32-1642216290203980094176640*z^38+ 5012034175969538327346934*z^40-180824710951616939751403856*z^62+ 469637984517386202425204*z^76-1642216290203980094176640*z^74+ 5012034175969538327346934*z^72+499428863*z^104) The first , 40, terms are: [0, 97, 0, 16568, 0, 2963713, 0, 533440593, 0, 96123644553, 0, 17325252552929, 0, 3122862941332208, 0, 562900950336254321, 0, 101464107124889649057, 0, 18289138356988117276029, 0, 3296659882149508284753009, 0, 594230671848325449778385080, 0, 107111473182184138827930194545, 0, 19307094495447097679053761441093, 0, 3480149111949124207139050972362029, 0, 627305048312766109961772936828845185, 0, 113073207789671550924740623797163957664, 0, 20381711185923571430130147934020049898305, 0, 3673851295015048740847631157781624293420141, 0, 662220321679166280980977340583192458318570901] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 5}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 24490017700013090 z - 11133988387833870 z - 292 z 24 22 4 6 + 4013196593780392 z - 1140658421238952 z + 34579 z - 2225790 z 8 10 12 14 + 88551686 z - 2346303146 z + 43489606424 z - 583687706652 z 18 16 50 - 43862560646518 z + 5816907255563 z - 43862560646518 z 48 20 36 + 253881096889103 z + 253881096889103 z + 59946148290174954 z 34 66 64 30 - 67015311113347420 z - 292 z + 34579 z - 42882353204037310 z 42 44 46 - 11133988387833870 z + 4013196593780392 z - 1140658421238952 z 58 56 54 52 - 2346303146 z + 43489606424 z - 583687706652 z + 5816907255563 z 60 68 32 38 + 88551686 z + z + 59946148290174954 z - 42882353204037310 z 40 62 / 2 + 24490017700013090 z - 2225790 z ) / ((-1 + z ) (1 / 28 26 2 + 102359156060089894 z - 44947539896801814 z - 383 z 24 22 4 6 + 15515395458477086 z - 4190504220889792 z + 54977 z - 4109684 z 8 10 12 14 + 185198350 z - 5466736958 z + 111523111078 z - 1631837686552 z 18 16 50 - 142479948276907 z + 17590798793561 z - 142479948276907 z 48 20 36 + 879986372481841 z + 879986372481841 z + 261049012955933884 z 34 66 64 30 - 293367009164863236 z - 383 z + 54977 z - 183855589854739380 z 42 44 46 - 44947539896801814 z + 15515395458477086 z - 4190504220889792 z 58 56 54 - 5466736958 z + 111523111078 z - 1631837686552 z 52 60 68 32 + 17590798793561 z + 185198350 z + z + 261049012955933884 z 38 40 62 - 183855589854739380 z + 102359156060089894 z - 4109684 z )) And in Maple-input format, it is: -(1+24490017700013090*z^28-11133988387833870*z^26-292*z^2+4013196593780392*z^24 -1140658421238952*z^22+34579*z^4-2225790*z^6+88551686*z^8-2346303146*z^10+ 43489606424*z^12-583687706652*z^14-43862560646518*z^18+5816907255563*z^16-\ 43862560646518*z^50+253881096889103*z^48+253881096889103*z^20+59946148290174954 *z^36-67015311113347420*z^34-292*z^66+34579*z^64-42882353204037310*z^30-\ 11133988387833870*z^42+4013196593780392*z^44-1140658421238952*z^46-2346303146*z ^58+43489606424*z^56-583687706652*z^54+5816907255563*z^52+88551686*z^60+z^68+ 59946148290174954*z^32-42882353204037310*z^38+24490017700013090*z^40-2225790*z^ 62)/(-1+z^2)/(1+102359156060089894*z^28-44947539896801814*z^26-383*z^2+ 15515395458477086*z^24-4190504220889792*z^22+54977*z^4-4109684*z^6+185198350*z^ 8-5466736958*z^10+111523111078*z^12-1631837686552*z^14-142479948276907*z^18+ 17590798793561*z^16-142479948276907*z^50+879986372481841*z^48+879986372481841*z ^20+261049012955933884*z^36-293367009164863236*z^34-383*z^66+54977*z^64-\ 183855589854739380*z^30-44947539896801814*z^42+15515395458477086*z^44-\ 4190504220889792*z^46-5466736958*z^58+111523111078*z^56-1631837686552*z^54+ 17590798793561*z^52+185198350*z^60+z^68+261049012955933884*z^32-\ 183855589854739380*z^38+102359156060089894*z^40-4109684*z^62) The first , 40, terms are: [0, 92, 0, 14547, 0, 2431799, 0, 410881360, 0, 69626666201, 0, 11810480022849, 0, 2004129885729592, 0, 340135523648556119, 0, 57730656322680456211, 0, 9798802553523380878548, 0, 1663200571261873975200633, 0, 282304902831537332029303465, 0, 47917388853904166921967022564, 0, 8133327568137850165294010127155, 0, 1380522700904313809444928424232055, 0, 234325155157933228520857669395415912, 0, 39773545337280449380651458811038494929, 0, 6751024897490270946073413744686075025801, 0, 1145895779943092461634359517868963932871200, 0, 194500415221490607555387077621097398327501143] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 5}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 30158315819772 z - 26375915266820 z - 216 z 24 22 4 6 + 17630555608400 z - 8985321459568 z + 17672 z - 766761 z 8 10 12 14 + 20260998 z - 351770411 z + 4208187328 z - 35814455824 z 18 16 50 48 - 1015217713860 z + 221759597195 z - 766761 z + 20260998 z 20 36 34 + 3476920630924 z + 3476920630924 z - 8985321459568 z 30 42 44 46 - 26375915266820 z - 35814455824 z + 4208187328 z - 351770411 z 56 54 52 32 38 + z - 216 z + 17672 z + 17630555608400 z - 1015217713860 z 40 / 28 26 + 221759597195 z ) / (-1 - 246001849239866 z + 188767253344114 z / 2 24 22 4 + 305 z - 111006275079466 z + 49891360282050 z - 30553 z 6 8 10 12 + 1556417 z - 47416949 z + 939235701 z - 12734258017 z 14 18 16 50 + 122373804289 z + 4411516309705 z - 854387992481 z + 47416949 z 48 20 36 - 939235701 z - 17062661814298 z - 49891360282050 z 34 30 42 + 111006275079466 z + 246001849239866 z + 854387992481 z 44 46 58 56 54 - 122373804289 z + 12734258017 z + z - 305 z + 30553 z 52 32 38 - 1556417 z - 188767253344114 z + 17062661814298 z 40 - 4411516309705 z ) And in Maple-input format, it is: -(1+30158315819772*z^28-26375915266820*z^26-216*z^2+17630555608400*z^24-\ 8985321459568*z^22+17672*z^4-766761*z^6+20260998*z^8-351770411*z^10+4208187328* z^12-35814455824*z^14-1015217713860*z^18+221759597195*z^16-766761*z^50+20260998 *z^48+3476920630924*z^20+3476920630924*z^36-8985321459568*z^34-26375915266820*z ^30-35814455824*z^42+4208187328*z^44-351770411*z^46+z^56-216*z^54+17672*z^52+ 17630555608400*z^32-1015217713860*z^38+221759597195*z^40)/(-1-246001849239866*z ^28+188767253344114*z^26+305*z^2-111006275079466*z^24+49891360282050*z^22-30553 *z^4+1556417*z^6-47416949*z^8+939235701*z^10-12734258017*z^12+122373804289*z^14 +4411516309705*z^18-854387992481*z^16+47416949*z^50-939235701*z^48-\ 17062661814298*z^20-49891360282050*z^36+111006275079466*z^34+246001849239866*z^ 30+854387992481*z^42-122373804289*z^44+12734258017*z^46+z^58-305*z^56+30553*z^ 54-1556417*z^52-188767253344114*z^32+17062661814298*z^38-4411516309705*z^40) The first , 40, terms are: [0, 89, 0, 14264, 0, 2420959, 0, 413949665, 0, 70855176415, 0, 12130156981897, 0, 2076693949211320, 0, 355533407505275735, 0, 60867945394634819623, 0, 10420700700693165323895, 0, 1784042564502926280146999, 0, 305431273314210927840592984, 0, 52290379521374723674410670137, 0, 8952206370826785403294717362495, 0, 1532633720411582050011416560588929, 0, 262389630405129340534360862023174735, 0, 44921573385257060752995968471332608600, 0, 7690653598968764944217661847426396696057, 0, 1316653632589457232354306843557588522735377, 0, 225413453603407006970652812857889258015523441] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 6}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 218358366645132150601 z - 31508736404749318384 z - 341 z 24 22 4 6 + 3897216819706309883 z - 410583646585363849 z + 50495 z - 4396040 z 102 8 10 12 - 10761608657 z + 256218059 z - 10761608657 z + 341622694638 z 14 18 16 - 8473582995073 z - 2726526617180827 z + 168290025994398 z 50 48 - 40531810360444706787906574 z + 25960877540293132012041293 z 20 36 + 36563226026649560 z + 118207279182732469855183 z 34 66 - 30237673317772362870805 z - 14629455990349366410340257 z 80 100 90 + 6737669865478166418372 z + 341622694638 z - 410583646585363849 z 88 84 + 3897216819706309883 z + 218358366645132150601 z 94 86 96 - 2726526617180827 z - 31508736404749318384 z + 168290025994398 z 98 92 82 - 8473582995073 z + 36563226026649560 z - 1303876682001038618811 z 64 112 110 106 + 25960877540293132012041293 z + z - 341 z - 4396040 z 108 30 42 + 50495 z - 1303876682001038618811 z - 3155152861905429295485411 z 44 46 + 7248503865978773156544624 z - 14629455990349366410340257 z 58 56 - 67397458132904284733963871 z + 71817430309582773406230226 z 54 52 - 67397458132904284733963871 z + 55700435320839539294677587 z 60 70 + 55700435320839539294677587 z - 3155152861905429295485411 z 68 78 + 7248503865978773156544624 z - 30237673317772362870805 z 32 38 + 6737669865478166418372 z - 403519156032212480951100 z 40 62 + 1205270850424564459280385 z - 40531810360444706787906574 z 76 74 + 118207279182732469855183 z - 403519156032212480951100 z 72 104 / + 1205270850424564459280385 z + 256218059 z ) / (-1 / 28 26 2 - 797229296447715698308 z + 108432297523910262564 z + 429 z 24 22 4 - 12635188269788203908 z + 1253149393618537784 z - 74808 z 6 102 8 10 + 7378428 z + 749607459252 z - 476156450 z + 21831507814 z 12 14 18 - 749607459252 z + 19987853176244 z + 7347849079048405 z 16 50 - 424942647833509 z + 284972685040263465771708158 z 48 20 - 171678639064226401620871278 z - 104942310489475360 z 36 34 - 545952609731192544906872 z + 131679213042882339124624 z 66 80 + 171678639064226401620871278 z - 131679213042882339124624 z 100 90 - 19987853176244 z + 12635188269788203908 z 88 84 - 108432297523910262564 z - 5049029128752029416756 z 94 86 + 104942310489475360 z + 797229296447715698308 z 96 98 92 - 7347849079048405 z + 424942647833509 z - 1253149393618537784 z 82 64 112 + 27668192614704649472252 z - 284972685040263465771708158 z - 429 z 114 110 106 108 + z + 74808 z + 476156450 z - 7378428 z 30 42 + 5049029128752029416756 z + 17408707379267329008903864 z 44 46 - 42464806337398569643615940 z + 91037605013380243808201540 z 58 56 + 608759422957369353499560508 z - 608759422957369353499560508 z 54 52 + 536468906093326071133920372 z - 416577406646764343620891700 z 60 70 - 536468906093326071133920372 z + 42464806337398569643615940 z 68 78 - 91037605013380243808201540 z + 545952609731192544906872 z 32 38 - 27668192614704649472252 z + 1976963249420466188398412 z 40 62 - 6265499534476649768328668 z + 416577406646764343620891700 z 76 74 - 1976963249420466188398412 z + 6265499534476649768328668 z 72 104 - 17408707379267329008903864 z - 21831507814 z ) And in Maple-input format, it is: -(1+218358366645132150601*z^28-31508736404749318384*z^26-341*z^2+ 3897216819706309883*z^24-410583646585363849*z^22+50495*z^4-4396040*z^6-\ 10761608657*z^102+256218059*z^8-10761608657*z^10+341622694638*z^12-\ 8473582995073*z^14-2726526617180827*z^18+168290025994398*z^16-\ 40531810360444706787906574*z^50+25960877540293132012041293*z^48+ 36563226026649560*z^20+118207279182732469855183*z^36-30237673317772362870805*z^ 34-14629455990349366410340257*z^66+6737669865478166418372*z^80+341622694638*z^ 100-410583646585363849*z^90+3897216819706309883*z^88+218358366645132150601*z^84 -2726526617180827*z^94-31508736404749318384*z^86+168290025994398*z^96-\ 8473582995073*z^98+36563226026649560*z^92-1303876682001038618811*z^82+ 25960877540293132012041293*z^64+z^112-341*z^110-4396040*z^106+50495*z^108-\ 1303876682001038618811*z^30-3155152861905429295485411*z^42+ 7248503865978773156544624*z^44-14629455990349366410340257*z^46-\ 67397458132904284733963871*z^58+71817430309582773406230226*z^56-\ 67397458132904284733963871*z^54+55700435320839539294677587*z^52+ 55700435320839539294677587*z^60-3155152861905429295485411*z^70+ 7248503865978773156544624*z^68-30237673317772362870805*z^78+ 6737669865478166418372*z^32-403519156032212480951100*z^38+ 1205270850424564459280385*z^40-40531810360444706787906574*z^62+ 118207279182732469855183*z^76-403519156032212480951100*z^74+ 1205270850424564459280385*z^72+256218059*z^104)/(-1-797229296447715698308*z^28+ 108432297523910262564*z^26+429*z^2-12635188269788203908*z^24+ 1253149393618537784*z^22-74808*z^4+7378428*z^6+749607459252*z^102-476156450*z^8 +21831507814*z^10-749607459252*z^12+19987853176244*z^14+7347849079048405*z^18-\ 424942647833509*z^16+284972685040263465771708158*z^50-\ 171678639064226401620871278*z^48-104942310489475360*z^20-\ 545952609731192544906872*z^36+131679213042882339124624*z^34+ 171678639064226401620871278*z^66-131679213042882339124624*z^80-19987853176244*z ^100+12635188269788203908*z^90-108432297523910262564*z^88-\ 5049029128752029416756*z^84+104942310489475360*z^94+797229296447715698308*z^86-\ 7347849079048405*z^96+424942647833509*z^98-1253149393618537784*z^92+ 27668192614704649472252*z^82-284972685040263465771708158*z^64-429*z^112+z^114+ 74808*z^110+476156450*z^106-7378428*z^108+5049029128752029416756*z^30+ 17408707379267329008903864*z^42-42464806337398569643615940*z^44+ 91037605013380243808201540*z^46+608759422957369353499560508*z^58-\ 608759422957369353499560508*z^56+536468906093326071133920372*z^54-\ 416577406646764343620891700*z^52-536468906093326071133920372*z^60+ 42464806337398569643615940*z^70-91037605013380243808201540*z^68+ 545952609731192544906872*z^78-27668192614704649472252*z^32+ 1976963249420466188398412*z^38-6265499534476649768328668*z^40+ 416577406646764343620891700*z^62-1976963249420466188398412*z^76+ 6265499534476649768328668*z^74-17408707379267329008903864*z^72-21831507814*z^ 104) The first , 40, terms are: [0, 88, 0, 13439, 0, 2164615, 0, 352638396, 0, 57678178413, 0, 9449342727897, 0, 1549142927732188, 0, 254044238126594783, 0, 41666011783700254335, 0, 6834046420119296829704, 0, 1120944037003405313376581, 0, 183862949191641279932571645, 0, 30158264498989107320865898888, 0, 4946741927688545723250714950511, 0, 811395317546112022879567225463407, 0, 133090140642860591102681835897498988, 0, 21830281315666381378867232527054482113, 0, 3580740157301918835063034669279138877349, 0, 587335555617639633809010673274962424823260, 0, 96338478514613903473336156515773990263631863] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 827816070783952 z - 464920619053912 z - 231 z 24 22 4 6 + 206750043413100 z - 72579020149750 z + 20820 z - 1010027 z 8 10 12 14 + 30365079 z - 613105944 z + 8754466964 z - 91560578146 z 18 16 50 - 4313116175428 z + 718921538000 z - 91560578146 z 48 20 36 + 718921538000 z + 20022758038760 z + 827816070783952 z 34 64 30 42 - 1169381046037414 z + z - 1169381046037414 z - 72579020149750 z 44 46 58 56 + 20022758038760 z - 4313116175428 z - 1010027 z + 30365079 z 54 52 60 32 - 613105944 z + 8754466964 z + 20820 z + 1311938801506272 z 38 40 62 / - 464920619053912 z + 206750043413100 z - 231 z ) / (-1 / 28 26 2 - 5541138876688083 z + 2782875818619795 z + 322 z 24 22 4 6 - 1107634850210917 z + 348281978673687 z - 34997 z + 1957024 z 8 10 12 14 - 66671083 z + 1514190140 z - 24227724125 z + 283313655979 z 18 16 50 + 16627403551507 z - 2483894723401 z + 2483894723401 z 48 20 36 - 16627403551507 z - 86109605847213 z - 8761706348715475 z 34 66 64 30 + 11014612442313291 z + z - 322 z + 8761706348715475 z 42 44 46 + 1107634850210917 z - 348281978673687 z + 86109605847213 z 58 56 54 52 + 66671083 z - 1514190140 z + 24227724125 z - 283313655979 z 60 32 38 - 1957024 z - 11014612442313291 z + 5541138876688083 z 40 62 - 2782875818619795 z + 34997 z ) And in Maple-input format, it is: -(1+827816070783952*z^28-464920619053912*z^26-231*z^2+206750043413100*z^24-\ 72579020149750*z^22+20820*z^4-1010027*z^6+30365079*z^8-613105944*z^10+ 8754466964*z^12-91560578146*z^14-4313116175428*z^18+718921538000*z^16-\ 91560578146*z^50+718921538000*z^48+20022758038760*z^20+827816070783952*z^36-\ 1169381046037414*z^34+z^64-1169381046037414*z^30-72579020149750*z^42+ 20022758038760*z^44-4313116175428*z^46-1010027*z^58+30365079*z^56-613105944*z^ 54+8754466964*z^52+20820*z^60+1311938801506272*z^32-464920619053912*z^38+ 206750043413100*z^40-231*z^62)/(-1-5541138876688083*z^28+2782875818619795*z^26+ 322*z^2-1107634850210917*z^24+348281978673687*z^22-34997*z^4+1957024*z^6-\ 66671083*z^8+1514190140*z^10-24227724125*z^12+283313655979*z^14+16627403551507* z^18-2483894723401*z^16+2483894723401*z^50-16627403551507*z^48-86109605847213*z ^20-8761706348715475*z^36+11014612442313291*z^34+z^66-322*z^64+8761706348715475 *z^30+1107634850210917*z^42-348281978673687*z^44+86109605847213*z^46+66671083*z ^58-1514190140*z^56+24227724125*z^54-283313655979*z^52-1957024*z^60-\ 11014612442313291*z^32+5541138876688083*z^38-2782875818619795*z^40+34997*z^62) The first , 40, terms are: [0, 91, 0, 15125, 0, 2632520, 0, 460124995, 0, 80463949593, 0, 14072220054615, 0, 2461109868525487, 0, 430428449239588169, 0, 75278560465898301683, 0, 13165632604689229848360, 0, 2302566486416035361559093, 0, 402700929577167083385815963, 0, 70429253668540447147497996409, 0, 12317527503829361478304937728137, 0, 2154239551485413364381178005928027, 0, 376759706356804211326858996404460021, 0, 65892335992851955891274244059778310248, 0, 11524055968167456841603562848265568838099, 0, 2015467564724780128211289545203300021289321, 0, 352489567534113597567424110290012535107132543] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 247399439037969408004 z - 35404660815362370710 z - 338 z 24 22 4 6 + 4337970995886710044 z - 452201643724247278 z + 49930 z - 4366830 z 102 8 10 12 - 10929077808 z + 256960559 z - 10929077808 z + 351846844118 z 14 18 16 - 8854853494588 z - 2930155367195866 z + 178399402608233 z 50 48 - 47617206721948104004644378 z + 30477562252361832576742510 z 20 36 + 39799705233332188 z + 137019690934147863229596 z 34 66 - 34899246019349745220650 z - 17157666151152639372492094 z 80 100 90 + 7736186081543728366508 z + 351846844118 z - 452201643724247278 z 88 84 + 4337970995886710044 z + 247399439037969408004 z 94 86 96 - 2930155367195866 z - 35404660815362370710 z + 178399402608233 z 98 92 82 - 8854853494588 z + 39799705233332188 z - 1487940075551023513774 z 64 112 110 106 + 30477562252361832576742510 z + z - 338 z - 4366830 z 108 30 42 + 49930 z - 1487940075551023513774 z - 3688985445228673137315162 z 44 46 + 8489799341507236431963444 z - 17157666151152639372492094 z 58 56 - 79237738736080190565941098 z + 84441553920206337652940394 z 54 52 - 79237738736080190565941098 z + 65468245453477285242655136 z 60 70 + 65468245453477285242655136 z - 3688985445228673137315162 z 68 78 + 8489799341507236431963444 z - 34899246019349745220650 z 32 38 + 7736186081543728366508 z - 469391741587955132556258 z 40 62 + 1406020180040690148110764 z - 47617206721948104004644378 z 76 74 + 137019690934147863229596 z - 469391741587955132556258 z 72 104 / + 1406020180040690148110764 z + 256960559 z ) / (-1 / 28 26 2 - 896857566786922931199 z + 120905824500426903721 z + 430 z 24 22 4 - 13950138121587221999 z + 1368568684254025437 z - 74417 z 6 102 8 10 + 7335101 z + 766803464388 z - 476156436 z + 22054329877 z 12 14 18 - 766803464388 z + 20726003780046 z + 7829403681130966 z 16 50 - 446750196135973 z + 335643673062055113856171929 z 48 20 - 201927034927110702832574181 z - 113254890278766515 z 36 34 - 630472288289983960251895 z + 151265610882632301020213 z 66 80 + 201927034927110702832574181 z - 151265610882632301020213 z 100 90 - 20726003780046 z + 13950138121587221999 z 88 84 - 120905824500426903721 z - 5724965207084547007141 z 94 86 + 113254890278766515 z + 896857566786922931199 z 96 98 92 - 7829403681130966 z + 446750196135973 z - 1368568684254025437 z 82 64 112 + 31591094893568063526151 z - 335643673062055113856171929 z - 430 z 114 110 106 108 + z + 74417 z + 476156436 z - 7335101 z 30 42 + 5724965207084547007141 z + 20339153851117910551543237 z 44 46 - 49747567717422956072859015 z + 106887584814647231508773753 z 58 56 + 718427369702983334865016011 z - 718427369702983334865016011 z 54 52 + 632908377776520070936610099 z - 491141823220050060173344317 z 60 70 - 632908377776520070936610099 z + 49747567717422956072859015 z 68 78 - 106887584814647231508773753 z + 630472288289983960251895 z 32 38 - 31591094893568063526151 z + 2293383154126334941465685 z 40 62 - 7296411499115081130595483 z + 491141823220050060173344317 z 76 74 - 2293383154126334941465685 z + 7296411499115081130595483 z 72 104 - 20339153851117910551543237 z - 22054329877 z ) And in Maple-input format, it is: -(1+247399439037969408004*z^28-35404660815362370710*z^26-338*z^2+ 4337970995886710044*z^24-452201643724247278*z^22+49930*z^4-4366830*z^6-\ 10929077808*z^102+256960559*z^8-10929077808*z^10+351846844118*z^12-\ 8854853494588*z^14-2930155367195866*z^18+178399402608233*z^16-\ 47617206721948104004644378*z^50+30477562252361832576742510*z^48+ 39799705233332188*z^20+137019690934147863229596*z^36-34899246019349745220650*z^ 34-17157666151152639372492094*z^66+7736186081543728366508*z^80+351846844118*z^ 100-452201643724247278*z^90+4337970995886710044*z^88+247399439037969408004*z^84 -2930155367195866*z^94-35404660815362370710*z^86+178399402608233*z^96-\ 8854853494588*z^98+39799705233332188*z^92-1487940075551023513774*z^82+ 30477562252361832576742510*z^64+z^112-338*z^110-4366830*z^106+49930*z^108-\ 1487940075551023513774*z^30-3688985445228673137315162*z^42+ 8489799341507236431963444*z^44-17157666151152639372492094*z^46-\ 79237738736080190565941098*z^58+84441553920206337652940394*z^56-\ 79237738736080190565941098*z^54+65468245453477285242655136*z^52+ 65468245453477285242655136*z^60-3688985445228673137315162*z^70+ 8489799341507236431963444*z^68-34899246019349745220650*z^78+ 7736186081543728366508*z^32-469391741587955132556258*z^38+ 1406020180040690148110764*z^40-47617206721948104004644378*z^62+ 137019690934147863229596*z^76-469391741587955132556258*z^74+ 1406020180040690148110764*z^72+256960559*z^104)/(-1-896857566786922931199*z^28+ 120905824500426903721*z^26+430*z^2-13950138121587221999*z^24+ 1368568684254025437*z^22-74417*z^4+7335101*z^6+766803464388*z^102-476156436*z^8 +22054329877*z^10-766803464388*z^12+20726003780046*z^14+7829403681130966*z^18-\ 446750196135973*z^16+335643673062055113856171929*z^50-\ 201927034927110702832574181*z^48-113254890278766515*z^20-\ 630472288289983960251895*z^36+151265610882632301020213*z^34+ 201927034927110702832574181*z^66-151265610882632301020213*z^80-20726003780046*z ^100+13950138121587221999*z^90-120905824500426903721*z^88-\ 5724965207084547007141*z^84+113254890278766515*z^94+896857566786922931199*z^86-\ 7829403681130966*z^96+446750196135973*z^98-1368568684254025437*z^92+ 31591094893568063526151*z^82-335643673062055113856171929*z^64-430*z^112+z^114+ 74417*z^110+476156436*z^106-7335101*z^108+5724965207084547007141*z^30+ 20339153851117910551543237*z^42-49747567717422956072859015*z^44+ 106887584814647231508773753*z^46+718427369702983334865016011*z^58-\ 718427369702983334865016011*z^56+632908377776520070936610099*z^54-\ 491141823220050060173344317*z^52-632908377776520070936610099*z^60+ 49747567717422956072859015*z^70-106887584814647231508773753*z^68+ 630472288289983960251895*z^78-31591094893568063526151*z^32+ 2293383154126334941465685*z^38-7296411499115081130595483*z^40+ 491141823220050060173344317*z^62-2293383154126334941465685*z^76+ 7296411499115081130595483*z^74-20339153851117910551543237*z^72-22054329877*z^ 104) The first , 40, terms are: [0, 92, 0, 15073, 0, 2603297, 0, 453363684, 0, 79097668601, 0, 13806414422785, 0, 2410188933817308, 0, 420761035249965209, 0, 73455411899628999257, 0, 12823693489802791768068, 0, 2238735286182675190889929, 0, 390834090090091345819642153, 0, 68231065259358444256528712132, 0, 11911648564538135526630466896105, 0, 2079512777635855329585724025458057, 0, 363037355676766842111707945907363676, 0, 63378365869977933790922633911117992705, 0, 11064473663202640497233202738314804756569, 0, 1931614609586443117673006349376508594250692, 0, 337217577045437958779699994533047560025596657] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 270180769487734928929 z - 38561091017161217860 z - 341 z 24 22 4 6 + 4708556693631258707 z - 488798257114962137 z + 50855 z - 4485292 z 102 8 10 12 - 11392348809 z + 265941731 z - 11392348809 z + 369308375646 z 14 18 16 - 9356771012997 z - 3134830287271887 z + 189721681888662 z 50 48 - 51867634418646155654306938 z + 33230865859635360805571061 z 20 36 + 42812925919270200 z + 150233695564348206490535 z 34 66 - 38261284628757223299693 z - 18728919297451689691568053 z 80 100 90 + 8475965020978214183844 z + 369308375646 z - 488798257114962137 z 88 84 + 4708556693631258707 z + 270180769487734928929 z 94 86 96 - 3134830287271887 z - 38561091017161217860 z + 189721681888662 z 98 92 82 - 9356771012997 z + 42812925919270200 z - 1628146521356015800875 z 64 112 110 106 + 33230865859635360805571061 z + z - 341 z - 4485292 z 108 30 42 + 50855 z - 1628146521356015800875 z - 4036247048237388517340267 z 44 46 + 9278327740639982984933280 z - 18728919297451689691568053 z 58 56 - 86203230123767290971614155 z + 91849356281333711593755682 z 54 52 - 86203230123767290971614155 z + 71257662442949185214365435 z 60 70 + 71257662442949185214365435 z - 4036247048237388517340267 z 68 78 + 9278327740639982984933280 z - 38261284628757223299693 z 32 38 + 8475965020978214183844 z - 514468532371308459532040 z 40 62 + 1539909127851537505719529 z - 51867634418646155654306938 z 76 74 + 150233695564348206490535 z - 514468532371308459532040 z 72 104 / + 1539909127851537505719529 z + 265941731 z ) / (-1 / 28 26 2 - 986224111875218327344 z + 132456340522623414192 z + 441 z 24 22 4 - 15216583679872170072 z + 1485547903643380956 z - 77156 z 6 102 8 10 + 7647848 z + 809919919488 z - 498511798 z + 23185974986 z 12 14 18 - 809919919488 z + 22005368788108 z + 8406548253916157 z 16 50 - 476966215322317 z + 372237305677565393184738058 z 48 20 - 224050190179781194444679710 z - 122284788873283680 z 36 34 - 699419517742843429880476 z + 167586893245856784484764 z 66 80 + 224050190179781194444679710 z - 167586893245856784484764 z 100 90 - 22005368788108 z + 15216583679872170072 z 88 84 - 132456340522623414192 z - 6315067116920874918616 z 94 86 + 122284788873283680 z + 986224111875218327344 z 96 98 92 - 8406548253916157 z + 476966215322317 z - 1485547903643380956 z 82 64 112 + 34934041191770481929688 z - 372237305677565393184738058 z - 441 z 114 110 106 108 + z + 77156 z + 498511798 z - 7647848 z 30 42 + 6315067116920874918616 z + 22591675276815991468343596 z 44 46 - 55245232092198169698444456 z + 118654208197148136101251644 z 58 56 + 796031736421994595705846460 z - 796031736421994595705846460 z 54 52 + 701392111886586283619234836 z - 544457094066222345501813360 z 60 70 - 701392111886586283619234836 z + 55245232092198169698444456 z 68 78 - 118654208197148136101251644 z + 699419517742843429880476 z 32 38 - 34934041191770481929688 z + 2546261917503614601542144 z 40 62 - 8104107724693338935962448 z + 544457094066222345501813360 z 76 74 - 2546261917503614601542144 z + 8104107724693338935962448 z 72 104 - 22591675276815991468343596 z - 23185974986 z ) And in Maple-input format, it is: -(1+270180769487734928929*z^28-38561091017161217860*z^26-341*z^2+ 4708556693631258707*z^24-488798257114962137*z^22+50855*z^4-4485292*z^6-\ 11392348809*z^102+265941731*z^8-11392348809*z^10+369308375646*z^12-\ 9356771012997*z^14-3134830287271887*z^18+189721681888662*z^16-\ 51867634418646155654306938*z^50+33230865859635360805571061*z^48+ 42812925919270200*z^20+150233695564348206490535*z^36-38261284628757223299693*z^ 34-18728919297451689691568053*z^66+8475965020978214183844*z^80+369308375646*z^ 100-488798257114962137*z^90+4708556693631258707*z^88+270180769487734928929*z^84 -3134830287271887*z^94-38561091017161217860*z^86+189721681888662*z^96-\ 9356771012997*z^98+42812925919270200*z^92-1628146521356015800875*z^82+ 33230865859635360805571061*z^64+z^112-341*z^110-4485292*z^106+50855*z^108-\ 1628146521356015800875*z^30-4036247048237388517340267*z^42+ 9278327740639982984933280*z^44-18728919297451689691568053*z^46-\ 86203230123767290971614155*z^58+91849356281333711593755682*z^56-\ 86203230123767290971614155*z^54+71257662442949185214365435*z^52+ 71257662442949185214365435*z^60-4036247048237388517340267*z^70+ 9278327740639982984933280*z^68-38261284628757223299693*z^78+ 8475965020978214183844*z^32-514468532371308459532040*z^38+ 1539909127851537505719529*z^40-51867634418646155654306938*z^62+ 150233695564348206490535*z^76-514468532371308459532040*z^74+ 1539909127851537505719529*z^72+265941731*z^104)/(-1-986224111875218327344*z^28+ 132456340522623414192*z^26+441*z^2-15216583679872170072*z^24+ 1485547903643380956*z^22-77156*z^4+7647848*z^6+809919919488*z^102-498511798*z^8 +23185974986*z^10-809919919488*z^12+22005368788108*z^14+8406548253916157*z^18-\ 476966215322317*z^16+372237305677565393184738058*z^50-\ 224050190179781194444679710*z^48-122284788873283680*z^20-\ 699419517742843429880476*z^36+167586893245856784484764*z^34+ 224050190179781194444679710*z^66-167586893245856784484764*z^80-22005368788108*z ^100+15216583679872170072*z^90-132456340522623414192*z^88-\ 6315067116920874918616*z^84+122284788873283680*z^94+986224111875218327344*z^86-\ 8406548253916157*z^96+476966215322317*z^98-1485547903643380956*z^92+ 34934041191770481929688*z^82-372237305677565393184738058*z^64-441*z^112+z^114+ 77156*z^110+498511798*z^106-7647848*z^108+6315067116920874918616*z^30+ 22591675276815991468343596*z^42-55245232092198169698444456*z^44+ 118654208197148136101251644*z^46+796031736421994595705846460*z^58-\ 796031736421994595705846460*z^56+701392111886586283619234836*z^54-\ 544457094066222345501813360*z^52-701392111886586283619234836*z^60+ 55245232092198169698444456*z^70-118654208197148136101251644*z^68+ 699419517742843429880476*z^78-34934041191770481929688*z^32+ 2546261917503614601542144*z^38-8104107724693338935962448*z^40+ 544457094066222345501813360*z^62-2546261917503614601542144*z^76+ 8104107724693338935962448*z^74-22591675276815991468343596*z^72-23185974986*z^ 104) The first , 40, terms are: [0, 100, 0, 17799, 0, 3296315, 0, 612590004, 0, 113888204553, 0, 21174394401525, 0, 3936836714670404, 0, 731955554643356839, 0, 136088756278321526967, 0, 25302290618390374938468, 0, 4704326446867568332669241, 0, 874651544158761565938385737, 0, 162619523690250444497574343524, 0, 30235022947454156871835460873751, 0, 5621444412823580716342276028128479, 0, 1045166638162464324385903395415872212, 0, 194322530177222982388378919873584106493, 0, 36129402102894460214160759201336180868225, 0, 6717356423490294454078118929059570894453988, 0, 1248923998014717880660327227541502659660007587] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 270724700602722529776 z - 38073135125338169613 z - 335 z 24 22 4 6 + 4586722648924950176 z - 470507544902198359 z + 49136 z - 4277091 z 102 8 10 12 - 10698902473 z + 251224672 z - 10698902473 z + 345920648448 z 14 18 16 - 8767487387573 z - 2963419491872995 z + 178331678099476 z 50 48 - 60623972625814970216468983 z + 38536735590346630491143500 z 20 36 + 40797160147983928 z + 160684461791278893717424 z 34 66 - 40249807848971944375139 z - 21506536763688718610769305 z 80 100 90 + 8769464684653456391500 z + 345920648448 z - 470507544902198359 z 88 84 + 4586722648924950176 z + 270724700602722529776 z 94 86 96 - 2963419491872995 z - 38073135125338169613 z + 178331678099476 z 98 92 82 - 8767487387573 z + 40797160147983928 z - 1657240286963190748501 z 64 112 110 106 + 38536735590346630491143500 z + z - 335 z - 4277091 z 108 30 42 + 49136 z - 1657240286963190748501 z - 4521185524128900349255189 z 44 46 + 10531125400270239439117064 z - 21506536763688718610769305 z 58 56 - 101688263011664114013235683 z + 108475246128097739896447794 z 54 52 - 101688263011664114013235683 z + 83765836870541264142252600 z 60 70 + 83765836870541264142252600 z - 4521185524128900349255189 z 68 78 + 10531125400270239439117064 z - 40249807848971944375139 z 32 38 + 8769464684653456391500 z - 559247758975475476643919 z 40 62 + 1700101074941187191796984 z - 60623972625814970216468983 z 76 74 + 160684461791278893717424 z - 559247758975475476643919 z 72 104 / 2 + 1700101074941187191796984 z + 251224672 z ) / ((-1 + z ) (1 / 28 26 2 + 854126215871684312532 z - 114947433876464872944 z - 433 z 24 22 4 + 13232156756722396330 z - 1295115633031891070 z + 74212 z 6 102 8 10 - 7215584 z - 21180345194 z + 462291338 z - 21180345194 z 12 14 18 + 730457930084 z - 19637713849121 z - 7390461147048207 z 16 50 + 422062249325072 z - 267788111056072892888206383 z 48 20 + 167707552928090115611330884 z + 106983778802909224 z 36 34 + 594297184720024104195224 z - 143483074126915781778433 z 66 80 - 91850162631145851427799509 z + 30068621857681493494180 z 100 90 88 + 730457930084 z - 1295115633031891070 z + 13232156756722396330 z 84 94 + 854126215871684312532 z - 7390461147048207 z 86 96 98 - 114947433876464872944 z + 422062249325072 z - 19637713849121 z 92 82 + 106983778802909224 z - 5455306224687835698767 z 64 112 110 106 + 167707552928090115611330884 z + z - 433 z - 7215584 z 108 30 42 + 74212 z - 5455306224687835698767 z - 18396870799350179129023526 z 44 46 + 43976226452622114378090956 z - 91850162631145851427799509 z 58 56 - 457026326486773855456194974 z + 488597824312762112705570276 z 54 52 - 457026326486773855456194974 z + 374023917705787713522131548 z 60 70 + 374023917705787713522131548 z - 18396870799350179129023526 z 68 78 + 43976226452622114378090956 z - 143483074126915781778433 z 32 38 + 30068621857681493494180 z - 2141006991415774643194688 z 40 62 + 6719690079857243453409614 z - 267788111056072892888206383 z 76 74 + 594297184720024104195224 z - 2141006991415774643194688 z 72 104 + 6719690079857243453409614 z + 462291338 z )) And in Maple-input format, it is: -(1+270724700602722529776*z^28-38073135125338169613*z^26-335*z^2+ 4586722648924950176*z^24-470507544902198359*z^22+49136*z^4-4277091*z^6-\ 10698902473*z^102+251224672*z^8-10698902473*z^10+345920648448*z^12-\ 8767487387573*z^14-2963419491872995*z^18+178331678099476*z^16-\ 60623972625814970216468983*z^50+38536735590346630491143500*z^48+ 40797160147983928*z^20+160684461791278893717424*z^36-40249807848971944375139*z^ 34-21506536763688718610769305*z^66+8769464684653456391500*z^80+345920648448*z^ 100-470507544902198359*z^90+4586722648924950176*z^88+270724700602722529776*z^84 -2963419491872995*z^94-38073135125338169613*z^86+178331678099476*z^96-\ 8767487387573*z^98+40797160147983928*z^92-1657240286963190748501*z^82+ 38536735590346630491143500*z^64+z^112-335*z^110-4277091*z^106+49136*z^108-\ 1657240286963190748501*z^30-4521185524128900349255189*z^42+ 10531125400270239439117064*z^44-21506536763688718610769305*z^46-\ 101688263011664114013235683*z^58+108475246128097739896447794*z^56-\ 101688263011664114013235683*z^54+83765836870541264142252600*z^52+ 83765836870541264142252600*z^60-4521185524128900349255189*z^70+ 10531125400270239439117064*z^68-40249807848971944375139*z^78+ 8769464684653456391500*z^32-559247758975475476643919*z^38+ 1700101074941187191796984*z^40-60623972625814970216468983*z^62+ 160684461791278893717424*z^76-559247758975475476643919*z^74+ 1700101074941187191796984*z^72+251224672*z^104)/(-1+z^2)/(1+ 854126215871684312532*z^28-114947433876464872944*z^26-433*z^2+ 13232156756722396330*z^24-1295115633031891070*z^22+74212*z^4-7215584*z^6-\ 21180345194*z^102+462291338*z^8-21180345194*z^10+730457930084*z^12-\ 19637713849121*z^14-7390461147048207*z^18+422062249325072*z^16-\ 267788111056072892888206383*z^50+167707552928090115611330884*z^48+ 106983778802909224*z^20+594297184720024104195224*z^36-143483074126915781778433* z^34-91850162631145851427799509*z^66+30068621857681493494180*z^80+730457930084* z^100-1295115633031891070*z^90+13232156756722396330*z^88+854126215871684312532* z^84-7390461147048207*z^94-114947433876464872944*z^86+422062249325072*z^96-\ 19637713849121*z^98+106983778802909224*z^92-5455306224687835698767*z^82+ 167707552928090115611330884*z^64+z^112-433*z^110-7215584*z^106+74212*z^108-\ 5455306224687835698767*z^30-18396870799350179129023526*z^42+ 43976226452622114378090956*z^44-91850162631145851427799509*z^46-\ 457026326486773855456194974*z^58+488597824312762112705570276*z^56-\ 457026326486773855456194974*z^54+374023917705787713522131548*z^52+ 374023917705787713522131548*z^60-18396870799350179129023526*z^70+ 43976226452622114378090956*z^68-143483074126915781778433*z^78+ 30068621857681493494180*z^32-2141006991415774643194688*z^38+ 6719690079857243453409614*z^40-267788111056072892888206383*z^62+ 594297184720024104195224*z^76-2141006991415774643194688*z^74+ 6719690079857243453409614*z^72+462291338*z^104) The first , 40, terms are: [0, 99, 0, 17457, 0, 3199188, 0, 588777381, 0, 108446512647, 0, 19978647822185, 0, 3680773305496149, 0, 678137940540922619, 0, 124939172117981901557, 0, 23018638760545201424260, 0, 4240926713357703232635065, 0, 781343365944851636082240471, 0, 143953788562699339828662083525, 0, 26521877901102075288926361451357, 0, 4886359820192050504429255813793375, 0, 900257228788142984965198342425879929, 0, 165862340861520299496818432356821643076, 0, 30558284051613606188267551128949294997861, 0, 5630022579783391312603353834982826270777923, 0, 1037268787586291908573718706743684539810542781] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 257939830364941757197 z - 36423027486695144127 z - 332 z 24 22 4 6 + 4404648383668404129 z - 453390181317270954 z + 48361 z - 4190627 z 102 8 10 12 - 10434589922 z + 245458541 z - 10434589922 z + 336909139825 z 14 18 16 - 8527282693178 z - 2871531860649690 z + 173159690559470 z 50 48 - 55657698727282478257453524 z + 35437748134890962648530274 z 20 36 + 39431733826814269 z + 150522272667423524947208 z 34 66 - 37860971356058828326344 z - 19818447741280647641871950 z 80 100 90 + 8284297039995085992845 z + 336909139825 z - 453390181317270954 z 88 84 + 4404648383668404129 z + 257939830364941757197 z 94 86 96 - 2871531860649690 z - 36423027486695144127 z + 173159690559470 z 98 92 82 - 8527282693178 z + 39431733826814269 z - 1572303812620194853388 z 64 112 110 106 + 35437748134890962648530274 z + z - 332 z - 4190627 z 108 30 42 + 48361 z - 1572303812620194853388 z - 4189192889996915287723816 z 44 46 + 9729107196887650454300730 z - 19818447741280647641871950 z 58 56 - 93183019873099185482921468 z + 99378989208187616103891700 z 54 52 - 93183019873099185482921468 z + 76813714538044929489161618 z 60 70 + 76813714538044929489161618 z - 4189192889996915287723816 z 68 78 + 9729107196887650454300730 z - 37860971356058828326344 z 32 38 + 8284297039995085992845 z - 521818454671860642168520 z 40 62 + 1580514413089054116962642 z - 55657698727282478257453524 z 76 74 + 150522272667423524947208 z - 521818454671860642168520 z 72 104 / + 1580514413089054116962642 z + 245458541 z ) / (-1 / 28 26 2 - 925349772603227026030 z + 123166914003808300195 z + 413 z 24 22 4 - 14026628545331036659 z + 1357980745488624736 z - 69882 z 6 102 8 10 + 6812730 z + 717042075236 z - 440973555 z + 20482940083 z 12 14 18 - 717042075236 z + 19566227992372 z + 7568016549476841 z 16 50 - 426531478543057 z + 381336685987212125708680962 z 48 20 - 228421335315499177906402162 z - 110902313950906864 z 36 34 - 681055100644159263259996 z + 161686190647691223721945 z 66 80 + 228421335315499177906402162 z - 161686190647691223721945 z 100 90 - 19566227992372 z + 14026628545331036659 z 88 84 - 123166914003808300195 z - 5980076237108850303534 z 94 86 + 110902313950906864 z + 925349772603227026030 z 96 98 92 - 7568016549476841 z + 426531478543057 z - 1357980745488624736 z 82 64 112 + 33390778881503252985781 z - 381336685987212125708680962 z - 413 z 114 110 106 108 + z + 69882 z + 440973555 z - 6812730 z 30 42 + 5980076237108850303534 z + 22573875018888814878404182 z 44 46 - 55618304554502582821362176 z + 120264055665625296395416160 z 58 56 + 821638504614900407739383918 z - 821638504614900407739383918 z 54 52 + 723032398829602021586782916 z - 559842018165110124834753444 z 60 70 - 723032398829602021586782916 z + 55618304554502582821362176 z 68 78 - 120264055665625296395416160 z + 681055100644159263259996 z 32 38 - 33390778881503252985781 z + 2501849911879427396024076 z 40 62 - 8031952562126207356156494 z + 559842018165110124834753444 z 76 74 - 2501849911879427396024076 z + 8031952562126207356156494 z 72 104 - 22573875018888814878404182 z - 20482940083 z ) And in Maple-input format, it is: -(1+257939830364941757197*z^28-36423027486695144127*z^26-332*z^2+ 4404648383668404129*z^24-453390181317270954*z^22+48361*z^4-4190627*z^6-\ 10434589922*z^102+245458541*z^8-10434589922*z^10+336909139825*z^12-\ 8527282693178*z^14-2871531860649690*z^18+173159690559470*z^16-\ 55657698727282478257453524*z^50+35437748134890962648530274*z^48+ 39431733826814269*z^20+150522272667423524947208*z^36-37860971356058828326344*z^ 34-19818447741280647641871950*z^66+8284297039995085992845*z^80+336909139825*z^ 100-453390181317270954*z^90+4404648383668404129*z^88+257939830364941757197*z^84 -2871531860649690*z^94-36423027486695144127*z^86+173159690559470*z^96-\ 8527282693178*z^98+39431733826814269*z^92-1572303812620194853388*z^82+ 35437748134890962648530274*z^64+z^112-332*z^110-4190627*z^106+48361*z^108-\ 1572303812620194853388*z^30-4189192889996915287723816*z^42+ 9729107196887650454300730*z^44-19818447741280647641871950*z^46-\ 93183019873099185482921468*z^58+99378989208187616103891700*z^56-\ 93183019873099185482921468*z^54+76813714538044929489161618*z^52+ 76813714538044929489161618*z^60-4189192889996915287723816*z^70+ 9729107196887650454300730*z^68-37860971356058828326344*z^78+ 8284297039995085992845*z^32-521818454671860642168520*z^38+ 1580514413089054116962642*z^40-55657698727282478257453524*z^62+ 150522272667423524947208*z^76-521818454671860642168520*z^74+ 1580514413089054116962642*z^72+245458541*z^104)/(-1-925349772603227026030*z^28+ 123166914003808300195*z^26+413*z^2-14026628545331036659*z^24+ 1357980745488624736*z^22-69882*z^4+6812730*z^6+717042075236*z^102-440973555*z^8 +20482940083*z^10-717042075236*z^12+19566227992372*z^14+7568016549476841*z^18-\ 426531478543057*z^16+381336685987212125708680962*z^50-\ 228421335315499177906402162*z^48-110902313950906864*z^20-\ 681055100644159263259996*z^36+161686190647691223721945*z^34+ 228421335315499177906402162*z^66-161686190647691223721945*z^80-19566227992372*z ^100+14026628545331036659*z^90-123166914003808300195*z^88-\ 5980076237108850303534*z^84+110902313950906864*z^94+925349772603227026030*z^86-\ 7568016549476841*z^96+426531478543057*z^98-1357980745488624736*z^92+ 33390778881503252985781*z^82-381336685987212125708680962*z^64-413*z^112+z^114+ 69882*z^110+440973555*z^106-6812730*z^108+5980076237108850303534*z^30+ 22573875018888814878404182*z^42-55618304554502582821362176*z^44+ 120264055665625296395416160*z^46+821638504614900407739383918*z^58-\ 821638504614900407739383918*z^56+723032398829602021586782916*z^54-\ 559842018165110124834753444*z^52-723032398829602021586782916*z^60+ 55618304554502582821362176*z^70-120264055665625296395416160*z^68+ 681055100644159263259996*z^78-33390778881503252985781*z^32+ 2501849911879427396024076*z^38-8031952562126207356156494*z^40+ 559842018165110124834753444*z^62-2501849911879427396024076*z^76+ 8031952562126207356156494*z^74-22573875018888814878404182*z^72-20482940083*z^ 104) The first , 40, terms are: [0, 81, 0, 11932, 0, 1889577, 0, 302879393, 0, 48660755961, 0, 7821541138529, 0, 1257328560357964, 0, 202122385986784453, 0, 32492418286315818341, 0, 5223361516080851466473, 0, 839688544399714131006285, 0, 134985273873475219810234388, 0, 21699741495409353495519578641, 0, 3488371497980969338325874780997, 0, 560777911409674662748708691518861, 0, 90148616952355333241358258230711737, 0, 14491963704822986441079256017440981972, 0, 2329675364115677931338698493690329919161, 0, 374510136287982701638531209190154140629381, 0, 60204887059755402398667419576846781135906493] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 7}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3435022066624049150 z - 810738236851181735 z - 309 z 24 22 4 6 + 159139657383600806 z - 25823827918442889 z + 40273 z - 2992802 z 8 10 12 14 + 144215539 z - 4850978467 z + 119532933208 z - 2232246474841 z 18 16 50 - 372286416642965 z + 32393486141492 z - 340066513956053339755 z 48 20 + 513654354325353874937 z + 3438655417829722 z 36 34 + 190632417828909799397 z - 90350440823193249568 z 66 80 88 84 86 - 25823827918442889 z + 144215539 z + z + 40273 z - 309 z 82 64 30 - 2992802 z + 159139657383600806 z - 12162854339063999839 z 42 44 - 657599524462115945854 z + 713996846527316921460 z 46 58 - 657599524462115945854 z - 12162854339063999839 z 56 54 + 36133441858568573721 z - 90350440823193249568 z 52 60 + 190632417828909799397 z + 3435022066624049150 z 70 68 78 - 372286416642965 z + 3438655417829722 z - 4850978467 z 32 38 + 36133441858568573721 z - 340066513956053339755 z 40 62 76 + 513654354325353874937 z - 810738236851181735 z + 119532933208 z 74 72 / - 2232246474841 z + 32393486141492 z ) / (-1 / 28 26 2 - 15666604594138581262 z + 3417701347001959923 z + 413 z 24 22 4 6 - 620278862877762079 z + 93085152584223216 z - 64212 z + 5430792 z 8 10 12 14 - 291046570 z + 10754360938 z - 289080170982 z + 5865148429822 z 18 16 50 + 1147476282835493 z - 92248247836593 z + 3787602149738268590503 z 48 20 - 5259716175091120477251 z - 11463334347501272 z 36 34 - 1195979375275828590836 z + 523106513419906666142 z 66 80 90 88 84 + 620278862877762079 z - 10754360938 z + z - 413 z - 5430792 z 86 82 64 + 64212 z + 291046570 z - 3417701347001959923 z 30 42 + 60040520398229443542 z + 5259716175091120477251 z 44 46 - 6197116674023328641384 z + 6197116674023328641384 z 58 56 + 193134060347990697598 z - 523106513419906666142 z 54 52 + 1195979375275828590836 z - 2312613651416802401176 z 60 70 68 - 60040520398229443542 z + 11463334347501272 z - 93085152584223216 z 78 32 + 289080170982 z - 193134060347990697598 z 38 40 + 2312613651416802401176 z - 3787602149738268590503 z 62 76 74 + 15666604594138581262 z - 5865148429822 z + 92248247836593 z 72 - 1147476282835493 z ) And in Maple-input format, it is: -(1+3435022066624049150*z^28-810738236851181735*z^26-309*z^2+159139657383600806 *z^24-25823827918442889*z^22+40273*z^4-2992802*z^6+144215539*z^8-4850978467*z^ 10+119532933208*z^12-2232246474841*z^14-372286416642965*z^18+32393486141492*z^ 16-340066513956053339755*z^50+513654354325353874937*z^48+3438655417829722*z^20+ 190632417828909799397*z^36-90350440823193249568*z^34-25823827918442889*z^66+ 144215539*z^80+z^88+40273*z^84-309*z^86-2992802*z^82+159139657383600806*z^64-\ 12162854339063999839*z^30-657599524462115945854*z^42+713996846527316921460*z^44 -657599524462115945854*z^46-12162854339063999839*z^58+36133441858568573721*z^56 -90350440823193249568*z^54+190632417828909799397*z^52+3435022066624049150*z^60-\ 372286416642965*z^70+3438655417829722*z^68-4850978467*z^78+36133441858568573721 *z^32-340066513956053339755*z^38+513654354325353874937*z^40-810738236851181735* z^62+119532933208*z^76-2232246474841*z^74+32393486141492*z^72)/(-1-\ 15666604594138581262*z^28+3417701347001959923*z^26+413*z^2-620278862877762079*z ^24+93085152584223216*z^22-64212*z^4+5430792*z^6-291046570*z^8+10754360938*z^10 -289080170982*z^12+5865148429822*z^14+1147476282835493*z^18-92248247836593*z^16 +3787602149738268590503*z^50-5259716175091120477251*z^48-11463334347501272*z^20 -1195979375275828590836*z^36+523106513419906666142*z^34+620278862877762079*z^66 -10754360938*z^80+z^90-413*z^88-5430792*z^84+64212*z^86+291046570*z^82-\ 3417701347001959923*z^64+60040520398229443542*z^30+5259716175091120477251*z^42-\ 6197116674023328641384*z^44+6197116674023328641384*z^46+193134060347990697598*z ^58-523106513419906666142*z^56+1195979375275828590836*z^54-\ 2312613651416802401176*z^52-60040520398229443542*z^60+11463334347501272*z^70-\ 93085152584223216*z^68+289080170982*z^78-193134060347990697598*z^32+ 2312613651416802401176*z^38-3787602149738268590503*z^40+15666604594138581262*z^ 62-5865148429822*z^76+92248247836593*z^74-1147476282835493*z^72) The first , 40, terms are: [0, 104, 0, 19013, 0, 3612311, 0, 688993024, 0, 131490592467, 0, 25096942176463, 0, 4790231500547000, 0, 914311695645218367, 0, 174514899681488211729, 0, 33309710148320925444496, 0, 6357834508071663115352417, 0, 1213521821881428072779815201, 0, 231625282894131838469249876832, 0, 44210388923751664871380440238129, 0, 8438450520971505628511756929421743, 0, 1610649644373426854828888183842060744, 0, 307425192634277977951603741963708756287, 0, 58678341001468090966683983159183119850275, 0, 11199952981025055335500138816807207211471248, 0, 2137738467657953064630603144297219570753379943] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 18845812971260 z - 16619444120456 z - 236 z 24 22 4 6 + 11385149202644 z - 6038428373436 z + 20520 z - 890797 z 8 10 12 14 + 22556934 z - 366095059 z + 4045146604 z - 31684349952 z 18 16 50 48 - 768624277892 z + 180932936151 z - 890797 z + 22556934 z 20 36 34 + 2465491761968 z + 2465491761968 z - 6038428373436 z 30 42 44 46 - 16619444120456 z - 31684349952 z + 4045146604 z - 366095059 z 56 54 52 32 38 + z - 236 z + 20520 z + 11385149202644 z - 768624277892 z 40 / 2 28 + 180932936151 z ) / ((-1 + z ) (1 + 86750771777498 z / 26 2 24 22 - 75833629562152 z - 340 z + 50619655292890 z - 25745050294952 z 4 6 8 10 12 + 36761 z - 1858204 z + 53281669 z - 964850668 z + 11783795409 z 14 18 16 50 - 101279790132 z - 2894145472904 z + 630214910185 z - 1858204 z 48 20 36 + 53281669 z + 9937979742234 z + 9937979742234 z 34 30 42 - 25745050294952 z - 75833629562152 z - 101279790132 z 44 46 56 54 52 + 11783795409 z - 964850668 z + z - 340 z + 36761 z 32 38 40 + 50619655292890 z - 2894145472904 z + 630214910185 z )) And in Maple-input format, it is: -(1+18845812971260*z^28-16619444120456*z^26-236*z^2+11385149202644*z^24-\ 6038428373436*z^22+20520*z^4-890797*z^6+22556934*z^8-366095059*z^10+4045146604* z^12-31684349952*z^14-768624277892*z^18+180932936151*z^16-890797*z^50+22556934* z^48+2465491761968*z^20+2465491761968*z^36-6038428373436*z^34-16619444120456*z^ 30-31684349952*z^42+4045146604*z^44-366095059*z^46+z^56-236*z^54+20520*z^52+ 11385149202644*z^32-768624277892*z^38+180932936151*z^40)/(-1+z^2)/(1+ 86750771777498*z^28-75833629562152*z^26-340*z^2+50619655292890*z^24-\ 25745050294952*z^22+36761*z^4-1858204*z^6+53281669*z^8-964850668*z^10+ 11783795409*z^12-101279790132*z^14-2894145472904*z^18+630214910185*z^16-1858204 *z^50+53281669*z^48+9937979742234*z^20+9937979742234*z^36-25745050294952*z^34-\ 75833629562152*z^30-101279790132*z^42+11783795409*z^44-964850668*z^46+z^56-340* z^54+36761*z^52+50619655292890*z^32-2894145472904*z^38+630214910185*z^40) The first , 40, terms are: [0, 105, 0, 19224, 0, 3663947, 0, 702564689, 0, 134929619075, 0, 25926390382201, 0, 4982496594105112, 0, 957580530968804299, 0, 184039648161660520507, 0, 35371223545333269177011, 0, 6798132751188778210137011, 0, 1306560395314394586395187000, 0, 251113138077648843120981765473, 0, 48262455415002205162557114937163, 0, 9275757846304089392030377693815801, 0, 1782745689195061918780561628273065555, 0, 342633157728497959004370096550477895480, 0, 65852062689537713769249852543649712840241, 0, 12656376255485366706739476253482398459476441, 0, 2432480523689491156127921383960462806971596137] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 21740389957143652 z - 8697074477462066 z - 259 z 24 22 4 6 + 2817561958117911 z - 735433145311267 z + 27116 z - 1569699 z 8 10 12 14 + 57358871 z - 1425867586 z + 25318948172 z - 332381112846 z 18 16 50 - 25456439946087 z + 3308643736911 z - 735433145311267 z 48 20 36 + 2817561958117911 z + 153622150543528 z + 109465571188037240 z 34 66 64 - 98990154743955000 z - 1569699 z + 57358871 z 30 42 44 - 44178434939616534 z - 44178434939616534 z + 21740389957143652 z 46 58 56 - 8697074477462066 z - 332381112846 z + 3308643736911 z 54 52 60 70 - 25456439946087 z + 153622150543528 z + 25318948172 z - 259 z 68 32 38 + 27116 z + 73179193878874626 z - 98990154743955000 z 40 62 72 / 2 + 73179193878874626 z - 1425867586 z + z ) / ((-1 + z ) (1 / 28 26 2 + 90328306775280610 z - 34595968123052204 z - 352 z 24 22 4 6 + 10653504997464655 z - 2626684547173252 z + 44179 z - 2920764 z 8 10 12 14 + 119044643 z - 3257116320 z + 63120713850 z - 898874600220 z 18 16 50 - 79850484188140 z + 9658354181763 z - 2626684547173252 z 48 20 36 + 10653504997464655 z + 515450315107483 z + 494110667593282668 z 34 66 64 - 444412637795005428 z - 2920764 z + 119044643 z 30 42 44 - 190152273830303384 z - 190152273830303384 z + 90328306775280610 z 46 58 56 - 34595968123052204 z - 898874600220 z + 9658354181763 z 54 52 60 70 - 79850484188140 z + 515450315107483 z + 63120713850 z - 352 z 68 32 38 + 44179 z + 323306003266183882 z - 444412637795005428 z 40 62 72 + 323306003266183882 z - 3257116320 z + z )) And in Maple-input format, it is: -(1+21740389957143652*z^28-8697074477462066*z^26-259*z^2+2817561958117911*z^24-\ 735433145311267*z^22+27116*z^4-1569699*z^6+57358871*z^8-1425867586*z^10+ 25318948172*z^12-332381112846*z^14-25456439946087*z^18+3308643736911*z^16-\ 735433145311267*z^50+2817561958117911*z^48+153622150543528*z^20+ 109465571188037240*z^36-98990154743955000*z^34-1569699*z^66+57358871*z^64-\ 44178434939616534*z^30-44178434939616534*z^42+21740389957143652*z^44-\ 8697074477462066*z^46-332381112846*z^58+3308643736911*z^56-25456439946087*z^54+ 153622150543528*z^52+25318948172*z^60-259*z^70+27116*z^68+73179193878874626*z^ 32-98990154743955000*z^38+73179193878874626*z^40-1425867586*z^62+z^72)/(-1+z^2) /(1+90328306775280610*z^28-34595968123052204*z^26-352*z^2+10653504997464655*z^ 24-2626684547173252*z^22+44179*z^4-2920764*z^6+119044643*z^8-3257116320*z^10+ 63120713850*z^12-898874600220*z^14-79850484188140*z^18+9658354181763*z^16-\ 2626684547173252*z^50+10653504997464655*z^48+515450315107483*z^20+ 494110667593282668*z^36-444412637795005428*z^34-2920764*z^66+119044643*z^64-\ 190152273830303384*z^30-190152273830303384*z^42+90328306775280610*z^44-\ 34595968123052204*z^46-898874600220*z^58+9658354181763*z^56-79850484188140*z^54 +515450315107483*z^52+63120713850*z^60-352*z^70+44179*z^68+323306003266183882*z ^32-444412637795005428*z^38+323306003266183882*z^40-3257116320*z^62+z^72) The first , 40, terms are: [0, 94, 0, 15767, 0, 2775081, 0, 491581422, 0, 87184911355, 0, 15466890066971, 0, 2744049946626102, 0, 486841633874037537, 0, 86374406427268298287, 0, 15324378127017243720774, 0, 2718821836936746857982849, 0, 482368196500067885263283361, 0, 85580848846691417222119093366, 0, 15183591646204814996121923470943, 0, 2693843992223114331781541016805457, 0, 477936684904665982330610283184330406, 0, 84794618935739180760268224951697715643, 0, 15044100249416291821680003939740781836315, 0, 2669095694468387987983081162417143016150398, 0, 473545888961620200859016842745752817794679865] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 22241780255704 z - 19517801861514 z - 222 z 24 22 4 6 + 13178445939568 z - 6829531710732 z + 18340 z - 781375 z 8 10 12 14 + 19943524 z - 331825175 z + 3795626828 z - 30928673398 z 18 16 50 48 - 813320573520 z + 184004309561 z - 781375 z + 19943524 z 20 36 34 + 2704760417702 z + 2704760417702 z - 6829531710732 z 30 42 44 46 - 19517801861514 z - 30928673398 z + 3795626828 z - 331825175 z 56 54 52 32 38 + z - 222 z + 18340 z + 13178445939568 z - 813320573520 z 40 / 28 26 + 184004309561 z ) / (-1 - 188268246671506 z + 145032982767866 z / 2 24 22 4 6 + 317 z - 85970945118638 z + 39118814086098 z - 32571 z + 1624737 z 8 10 12 14 - 47402365 z + 893509537 z - 11551149909 z + 106438001543 z 18 16 50 48 + 3599206141077 z - 717345578465 z + 47402365 z - 893509537 z 20 36 34 - 13609977368858 z - 39118814086098 z + 85970945118638 z 30 42 44 + 188268246671506 z + 717345578465 z - 106438001543 z 46 58 56 54 52 + 11551149909 z + z - 317 z + 32571 z - 1624737 z 32 38 40 - 145032982767866 z + 13609977368858 z - 3599206141077 z ) And in Maple-input format, it is: -(1+22241780255704*z^28-19517801861514*z^26-222*z^2+13178445939568*z^24-\ 6829531710732*z^22+18340*z^4-781375*z^6+19943524*z^8-331825175*z^10+3795626828* z^12-30928673398*z^14-813320573520*z^18+184004309561*z^16-781375*z^50+19943524* z^48+2704760417702*z^20+2704760417702*z^36-6829531710732*z^34-19517801861514*z^ 30-30928673398*z^42+3795626828*z^44-331825175*z^46+z^56-222*z^54+18340*z^52+ 13178445939568*z^32-813320573520*z^38+184004309561*z^40)/(-1-188268246671506*z^ 28+145032982767866*z^26+317*z^2-85970945118638*z^24+39118814086098*z^22-32571*z ^4+1624737*z^6-47402365*z^8+893509537*z^10-11551149909*z^12+106438001543*z^14+ 3599206141077*z^18-717345578465*z^16+47402365*z^50-893509537*z^48-\ 13609977368858*z^20-39118814086098*z^36+85970945118638*z^34+188268246671506*z^ 30+717345578465*z^42-106438001543*z^44+11551149909*z^46+z^58-317*z^56+32571*z^ 54-1624737*z^52-145032982767866*z^32+13609977368858*z^38-3599206141077*z^40) The first , 40, terms are: [0, 95, 0, 15884, 0, 2784345, 0, 492170775, 0, 87195016875, 0, 15458343096389, 0, 2741100052977436, 0, 486087873830436483, 0, 86201200344102091437, 0, 15286726763146794256917, 0, 2710918834677537428915835, 0, 480749433032877254826876476, 0, 85255248559754686954685317869, 0, 15119014839332229550484551562563, 0, 2681179370260118711090049502380543, 0, 475475613331262198639250824212254257, 0, 84320005498774745529435534623473967980, 0, 14953160859504154322476962420637766884199, 0, 2651767138800945954636176576448246493819673, 0, 470259701265407055087466834425624830494273513] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 282486109604201903837 z - 39247792196757469008 z - 329 z 24 22 4 6 + 4672188312003817691 z - 473765693722493433 z + 47619 z - 4112680 z 102 8 10 12 - 10263974901 z + 240825315 z - 10263974901 z + 333120690678 z 14 18 16 - 8494976079573 z - 2921052247864371 z + 174161328244310 z 50 48 - 70578056593001755133537178 z + 44619504999153546823268141 z 20 36 + 40629280039394420 z + 175934916447397146418247 z 34 66 - 43556400678354177491949 z - 24731994513011567152945521 z 80 100 90 + 9376538019557751460924 z + 333120690678 z - 473765693722493433 z 88 84 + 4672188312003817691 z + 282486109604201903837 z 94 86 96 - 2921052247864371 z - 39247792196757469008 z + 174161328244310 z 98 92 82 - 8494976079573 z + 40629280039394420 z - 1750506805600952498455 z 64 112 110 106 + 44619504999153546823268141 z + z - 329 z - 4112680 z 108 30 42 + 47619 z - 1750506805600952498455 z - 5111078799623264899717283 z 44 46 + 12013731743743411962704460 z - 24731994513011567152945521 z 58 56 - 119149699274736409018996639 z + 127206339288069821244366402 z 54 52 - 119149699274736409018996639 z + 97910598253868514404592163 z 60 70 + 97910598253868514404592163 z - 5111078799623264899717283 z 68 78 + 12013731743743411962704460 z - 43556400678354177491949 z 32 38 + 9376538019557751460924 z - 619275597522919860499192 z 40 62 + 1902848115500764008921697 z - 70578056593001755133537178 z 76 74 + 175934916447397146418247 z - 619275597522919860499192 z 72 104 / 2 + 1902848115500764008921697 z + 240825315 z ) / ((-1 + z ) (1 / 28 26 2 + 883242632400862578116 z - 117487891986725485928 z - 416 z 24 22 4 + 13362739342362502916 z - 1291863516276450564 z + 69800 z 6 102 8 10 - 6723700 z - 19786444912 z + 430175070 z - 19786444912 z 12 14 18 + 687515385708 z - 18665541648872 z - 7190403460664444 z 16 50 + 405709926934913 z - 303380284935551716205120568 z 48 20 + 189171711867215658320980214 z + 105390410591722136 z 36 34 + 640954104680246205064508 z - 153242453022613249044620 z 66 80 - 103045508952319536772137460 z + 31785246506967748869240 z 100 90 88 + 687515385708 z - 1291863516276450564 z + 13362739342362502916 z 84 94 + 883242632400862578116 z - 7190403460664444 z 86 96 98 - 117487891986725485928 z + 405709926934913 z - 18665541648872 z 92 82 + 105390410591722136 z - 5704982787210069233924 z 64 112 110 106 + 189171711867215658320980214 z + z - 416 z - 6723700 z 108 30 42 + 69800 z - 5704982787210069233924 z - 20360420118267144528461708 z 44 46 + 49022771185011607672542568 z - 103045508952319536772137460 z 58 56 - 520441962930599932409753740 z + 556759666032271010550248532 z 54 52 - 520441962930599932409753740 z + 425091936325592726914532588 z 60 70 + 425091936325592726914532588 z - 20360420118267144528461708 z 68 78 + 49022771185011607672542568 z - 153242453022613249044620 z 32 38 + 31785246506967748869240 z - 2330501704528252175213784 z 40 62 + 7377856407985353253162116 z - 303380284935551716205120568 z 76 74 + 640954104680246205064508 z - 2330501704528252175213784 z 72 104 + 7377856407985353253162116 z + 430175070 z )) And in Maple-input format, it is: -(1+282486109604201903837*z^28-39247792196757469008*z^26-329*z^2+ 4672188312003817691*z^24-473765693722493433*z^22+47619*z^4-4112680*z^6-\ 10263974901*z^102+240825315*z^8-10263974901*z^10+333120690678*z^12-\ 8494976079573*z^14-2921052247864371*z^18+174161328244310*z^16-\ 70578056593001755133537178*z^50+44619504999153546823268141*z^48+ 40629280039394420*z^20+175934916447397146418247*z^36-43556400678354177491949*z^ 34-24731994513011567152945521*z^66+9376538019557751460924*z^80+333120690678*z^ 100-473765693722493433*z^90+4672188312003817691*z^88+282486109604201903837*z^84 -2921052247864371*z^94-39247792196757469008*z^86+174161328244310*z^96-\ 8494976079573*z^98+40629280039394420*z^92-1750506805600952498455*z^82+ 44619504999153546823268141*z^64+z^112-329*z^110-4112680*z^106+47619*z^108-\ 1750506805600952498455*z^30-5111078799623264899717283*z^42+ 12013731743743411962704460*z^44-24731994513011567152945521*z^46-\ 119149699274736409018996639*z^58+127206339288069821244366402*z^56-\ 119149699274736409018996639*z^54+97910598253868514404592163*z^52+ 97910598253868514404592163*z^60-5111078799623264899717283*z^70+ 12013731743743411962704460*z^68-43556400678354177491949*z^78+ 9376538019557751460924*z^32-619275597522919860499192*z^38+ 1902848115500764008921697*z^40-70578056593001755133537178*z^62+ 175934916447397146418247*z^76-619275597522919860499192*z^74+ 1902848115500764008921697*z^72+240825315*z^104)/(-1+z^2)/(1+ 883242632400862578116*z^28-117487891986725485928*z^26-416*z^2+ 13362739342362502916*z^24-1291863516276450564*z^22+69800*z^4-6723700*z^6-\ 19786444912*z^102+430175070*z^8-19786444912*z^10+687515385708*z^12-\ 18665541648872*z^14-7190403460664444*z^18+405709926934913*z^16-\ 303380284935551716205120568*z^50+189171711867215658320980214*z^48+ 105390410591722136*z^20+640954104680246205064508*z^36-153242453022613249044620* z^34-103045508952319536772137460*z^66+31785246506967748869240*z^80+687515385708 *z^100-1291863516276450564*z^90+13362739342362502916*z^88+883242632400862578116 *z^84-7190403460664444*z^94-117487891986725485928*z^86+405709926934913*z^96-\ 18665541648872*z^98+105390410591722136*z^92-5704982787210069233924*z^82+ 189171711867215658320980214*z^64+z^112-416*z^110-6723700*z^106+69800*z^108-\ 5704982787210069233924*z^30-20360420118267144528461708*z^42+ 49022771185011607672542568*z^44-103045508952319536772137460*z^46-\ 520441962930599932409753740*z^58+556759666032271010550248532*z^56-\ 520441962930599932409753740*z^54+425091936325592726914532588*z^52+ 425091936325592726914532588*z^60-20360420118267144528461708*z^70+ 49022771185011607672542568*z^68-153242453022613249044620*z^78+ 31785246506967748869240*z^32-2330501704528252175213784*z^38+ 7377856407985353253162116*z^40-303380284935551716205120568*z^62+ 640954104680246205064508*z^76-2330501704528252175213784*z^74+ 7377856407985353253162116*z^72+430175070*z^104) The first , 40, terms are: [0, 88, 0, 14099, 0, 2381095, 0, 404695776, 0, 68854281893, 0, 11717131484509, 0, 1994024494130880, 0, 339346886142625471, 0, 57750825943675243675, 0, 9828172020497453023256, 0, 1672581731240573404314105, 0, 284643951761036678166852985, 0, 48441387504786905372252008280, 0, 8243871026415652661308782222811, 0, 1402961661992264370642559833058271, 0, 238759366669673598952316371855635200, 0, 40632639307183512220512645019934437149, 0, 6914959610187160455943635042935000180549, 0, 1176804343155561348834588213169937012870688, 0, 200271373968680052382172828082206855089439015] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6798690432443368962 z - 1572726408134909324 z - 331 z 24 22 4 6 + 301220941031435688 z - 47465600200761684 z + 46234 z - 3671423 z 8 10 12 14 + 188574579 z - 6742855682 z + 176040632092 z - 3469450499420 z 18 16 50 - 635583703451838 z + 52895565794676 z - 707145936146935686670 z 48 20 + 1071387786134311961356 z + 6106932468812990 z 36 34 + 394535270686337028772 z - 185720496076178644660 z 66 80 88 84 86 - 47465600200761684 z + 188574579 z + z + 46234 z - 331 z 82 64 30 - 3671423 z + 301220941031435688 z - 24461945149462007602 z 42 44 - 1373973653881562754966 z + 1492624490577642083996 z 46 58 - 1373973653881562754966 z - 24461945149462007602 z 56 54 + 73581488835384258252 z - 185720496076178644660 z 52 60 + 394535270686337028772 z + 6798690432443368962 z 70 68 78 - 635583703451838 z + 6106932468812990 z - 6742855682 z 32 38 + 73581488835384258252 z - 707145936146935686670 z 40 62 76 + 1071387786134311961356 z - 1572726408134909324 z + 176040632092 z 74 72 / - 3469450499420 z + 52895565794676 z ) / (-1 / 28 26 2 - 30880177471492281806 z + 6613109975123394878 z + 434 z 24 22 4 6 - 1173003617560891898 z + 171212708535463834 z - 72444 z + 6586518 z 8 10 12 14 - 378511404 z + 14936572641 z - 426650912554 z + 9148511819774 z 18 16 50 + 1965408659610906 z - 151211627367478 z + 7814530163720788249734 z 48 20 - 10866950496789146509002 z - 20401853617859604 z 36 34 - 2452092665587019057850 z + 1066399864238783568194 z 66 80 90 88 84 + 1173003617560891898 z - 14936572641 z + z - 434 z - 6586518 z 86 82 64 + 72444 z + 378511404 z - 6613109975123394878 z 30 42 + 120075592862438773652 z + 10866950496789146509002 z 44 46 - 12811751131321636475814 z + 12811751131321636475814 z 58 56 + 390540813070910465166 z - 1066399864238783568194 z 54 52 + 2452092665587019057850 z - 4759764689534267492414 z 60 70 - 120075592862438773652 z + 20401853617859604 z 68 78 32 - 171212708535463834 z + 426650912554 z - 390540813070910465166 z 38 40 + 4759764689534267492414 z - 7814530163720788249734 z 62 76 74 + 30880177471492281806 z - 9148511819774 z + 151211627367478 z 72 - 1965408659610906 z ) And in Maple-input format, it is: -(1+6798690432443368962*z^28-1572726408134909324*z^26-331*z^2+ 301220941031435688*z^24-47465600200761684*z^22+46234*z^4-3671423*z^6+188574579* z^8-6742855682*z^10+176040632092*z^12-3469450499420*z^14-635583703451838*z^18+ 52895565794676*z^16-707145936146935686670*z^50+1071387786134311961356*z^48+ 6106932468812990*z^20+394535270686337028772*z^36-185720496076178644660*z^34-\ 47465600200761684*z^66+188574579*z^80+z^88+46234*z^84-331*z^86-3671423*z^82+ 301220941031435688*z^64-24461945149462007602*z^30-1373973653881562754966*z^42+ 1492624490577642083996*z^44-1373973653881562754966*z^46-24461945149462007602*z^ 58+73581488835384258252*z^56-185720496076178644660*z^54+394535270686337028772*z ^52+6798690432443368962*z^60-635583703451838*z^70+6106932468812990*z^68-\ 6742855682*z^78+73581488835384258252*z^32-707145936146935686670*z^38+ 1071387786134311961356*z^40-1572726408134909324*z^62+176040632092*z^76-\ 3469450499420*z^74+52895565794676*z^72)/(-1-30880177471492281806*z^28+ 6613109975123394878*z^26+434*z^2-1173003617560891898*z^24+171212708535463834*z^ 22-72444*z^4+6586518*z^6-378511404*z^8+14936572641*z^10-426650912554*z^12+ 9148511819774*z^14+1965408659610906*z^18-151211627367478*z^16+ 7814530163720788249734*z^50-10866950496789146509002*z^48-20401853617859604*z^20 -2452092665587019057850*z^36+1066399864238783568194*z^34+1173003617560891898*z^ 66-14936572641*z^80+z^90-434*z^88-6586518*z^84+72444*z^86+378511404*z^82-\ 6613109975123394878*z^64+120075592862438773652*z^30+10866950496789146509002*z^ 42-12811751131321636475814*z^44+12811751131321636475814*z^46+ 390540813070910465166*z^58-1066399864238783568194*z^56+2452092665587019057850*z ^54-4759764689534267492414*z^52-120075592862438773652*z^60+20401853617859604*z^ 70-171212708535463834*z^68+426650912554*z^78-390540813070910465166*z^32+ 4759764689534267492414*z^38-7814530163720788249734*z^40+30880177471492281806*z^ 62-9148511819774*z^76+151211627367478*z^74-1965408659610906*z^72) The first , 40, terms are: [0, 103, 0, 18492, 0, 3478891, 0, 658678775, 0, 124846741949, 0, 23668362840097, 0, 4487212908435892, 0, 850724469963920025, 0, 161288010980454923455, 0, 30578451747580462466359, 0, 5797342660375486117621441, 0, 1099113322062552747668588468, 0, 208379974476386452229175821137, 0, 39506585121414618790500720436477, 0, 7490020454207370856205970169019399, 0, 1420026718855465443658278620002545355, 0, 269221679042639017750381284517474583804, 0, 51041513170239345743307651490958224660255, 0, 9676917832349795287651184834526903311325257, 0, 1834638765952356245690068661580158081717089945] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 16513741671379772 z + 7865179396255152 z + 276 z 24 22 4 6 - 2922280752354894 z + 845523491394368 z - 30740 z + 1871543 z 8 10 12 14 - 71266720 z + 1831046628 z - 33288552519 z + 442039707840 z 18 16 50 + 32950724666541 z - 4382713746884 z + 4382713746884 z 48 20 36 - 32950724666541 z - 189918034842840 z - 27070644007834494 z 34 66 64 30 + 34658293349757948 z + z - 276 z + 27070644007834494 z 42 44 46 + 2922280752354894 z - 845523491394368 z + 189918034842840 z 58 56 54 52 + 71266720 z - 1831046628 z + 33288552519 z - 442039707840 z 60 32 38 - 1871543 z - 34658293349757948 z + 16513741671379772 z 40 62 / 28 - 7865179396255152 z + 30740 z ) / (1 + 100300093510705442 z / 26 2 24 - 42771934061440432 z - 358 z + 14272244833172302 z 22 4 6 8 - 3718021609693492 z + 48339 z - 3433512 z + 149152621 z 10 12 14 - 4312783686 z + 87517098497 z - 1290948943136 z 18 16 50 - 118038771791962 z + 14182295723119 z - 118038771791962 z 48 20 36 + 753337518249837 z + 753337518249837 z + 265398544861643726 z 34 66 64 30 - 299711547785192956 z - 358 z + 48339 z - 184273438563146600 z 42 44 46 - 42771934061440432 z + 14272244833172302 z - 3718021609693492 z 58 56 54 - 4312783686 z + 87517098497 z - 1290948943136 z 52 60 68 32 + 14182295723119 z + 149152621 z + z + 265398544861643726 z 38 40 62 - 184273438563146600 z + 100300093510705442 z - 3433512 z ) And in Maple-input format, it is: -(-1-16513741671379772*z^28+7865179396255152*z^26+276*z^2-2922280752354894*z^24 +845523491394368*z^22-30740*z^4+1871543*z^6-71266720*z^8+1831046628*z^10-\ 33288552519*z^12+442039707840*z^14+32950724666541*z^18-4382713746884*z^16+ 4382713746884*z^50-32950724666541*z^48-189918034842840*z^20-27070644007834494*z ^36+34658293349757948*z^34+z^66-276*z^64+27070644007834494*z^30+ 2922280752354894*z^42-845523491394368*z^44+189918034842840*z^46+71266720*z^58-\ 1831046628*z^56+33288552519*z^54-442039707840*z^52-1871543*z^60-\ 34658293349757948*z^32+16513741671379772*z^38-7865179396255152*z^40+30740*z^62) /(1+100300093510705442*z^28-42771934061440432*z^26-358*z^2+14272244833172302*z^ 24-3718021609693492*z^22+48339*z^4-3433512*z^6+149152621*z^8-4312783686*z^10+ 87517098497*z^12-1290948943136*z^14-118038771791962*z^18+14182295723119*z^16-\ 118038771791962*z^50+753337518249837*z^48+753337518249837*z^20+ 265398544861643726*z^36-299711547785192956*z^34-358*z^66+48339*z^64-\ 184273438563146600*z^30-42771934061440432*z^42+14272244833172302*z^44-\ 3718021609693492*z^46-4312783686*z^58+87517098497*z^56-1290948943136*z^54+ 14182295723119*z^52+149152621*z^60+z^68+265398544861643726*z^32-\ 184273438563146600*z^38+100300093510705442*z^40-3433512*z^62) The first , 40, terms are: [0, 82, 0, 11757, 0, 1807177, 0, 282309826, 0, 44328811425, 0, 6973936077937, 0, 1098005594511210, 0, 172930245442783301, 0, 27239352387397236669, 0, 4290896545972112579010, 0, 675943264322284895091045, 0, 106482233576838552867346805, 0, 16774364938225563862332168002, 0, 2642505472518748036175788759757, 0, 416280511902812103905089136667349, 0, 65577737152018880546444908983625194, 0, 10330630858365325904420079218953810817, 0, 1627411160667809925514285213332875958801, 0, 256370322521797697322460526869118042565186, 0, 40386685740431131105917877065610404735534137] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {4, 6}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 241848619983911663045 z - 35304826139891138236 z - 353 z 24 22 4 6 + 4411310053737091303 z - 468673298985888069 z + 53955 z - 4820520 z 102 8 10 12 - 12214149149 z + 286614291 z - 12214149149 z + 391524613934 z 14 18 16 - 9765628669517 z - 3144589469198379 z + 194329439866390 z 50 48 - 39276456266755009929479262 z + 25339546618776473631494441 z 20 36 + 42002533734318876 z + 123990573437759078186271 z 34 66 - 32170444608345043323841 z - 14408433636124439479705261 z 80 100 90 + 7269454628366826561136 z + 391524613934 z - 468673298985888069 z 88 84 + 4411310053737091303 z + 241848619983911663045 z 94 86 96 - 3144589469198379 z - 35304826139891138236 z + 194329439866390 z 98 92 82 - 9765628669517 z + 42002533734318876 z - 1425920473232680614187 z 64 112 110 106 + 25339546618776473631494441 z + z - 353 z - 4820520 z 108 30 42 + 53955 z - 1425920473232680614187 z - 3177898698793528073324267 z 44 46 + 7214717456271106320438780 z - 14408433636124439479705261 z 58 56 - 64755140996601332375929735 z + 68926897468455302560485450 z 54 52 - 64755140996601332375929735 z + 53689985985862789997676851 z 60 70 + 53689985985862789997676851 z - 3177898698793528073324267 z 68 78 + 7214717456271106320438780 z - 32170444608345043323841 z 32 38 + 7269454628366826561136 z - 417349215293392559208056 z 40 62 + 1229705829971656747574289 z - 39276456266755009929479262 z 76 74 + 123990573437759078186271 z - 417349215293392559208056 z 72 104 / + 1229705829971656747574289 z + 286614291 z ) / (-1 / 28 26 2 - 907143953096303963684 z + 124614366216707772772 z + 465 z 24 22 4 - 14652009658694296928 z + 1464608122514196420 z - 84224 z 6 102 8 10 + 8494432 z + 887634022772 z - 556017338 z + 25721983098 z 12 14 18 - 887634022772 z + 23712774265148 z + 8685240306655905 z 16 50 - 503766808016361 z + 290167104551141095622561526 z 48 20 - 175928950978703227213873458 z - 123444791620363124 z 36 34 - 593882794501064525300308 z + 144913145366470929404404 z 66 80 + 175928950978703227213873458 z - 144913145366470929404404 z 100 90 - 23712774265148 z + 14652009658694296928 z 88 84 - 124614366216707772772 z - 5684139397930439430248 z 94 86 + 123444791620363124 z + 907143953096303963684 z 96 98 92 - 8685240306655905 z + 503766808016361 z - 1464608122514196420 z 82 64 112 + 30801568580109804584136 z - 290167104551141095622561526 z - 465 z 114 110 106 108 + z + 84224 z + 556017338 z - 8494432 z 30 42 + 5684139397930439430248 z + 18317462772248748290021340 z 44 46 - 44243987166559502577564344 z + 94014070890428100389724592 z 58 56 + 613707009907349820223584368 z - 613707009907349820223584368 z 54 52 + 541745933786348477299500640 z - 422086208089551944971624148 z 60 70 - 541745933786348477299500640 z + 44243987166559502577564344 z 68 78 - 94014070890428100389724592 z + 593882794501064525300308 z 32 38 - 30801568580109804584136 z + 2125939646691831127999708 z 40 62 - 6662863595842931445699036 z + 422086208089551944971624148 z 76 74 - 2125939646691831127999708 z + 6662863595842931445699036 z 72 104 - 18317462772248748290021340 z - 25721983098 z ) And in Maple-input format, it is: -(1+241848619983911663045*z^28-35304826139891138236*z^26-353*z^2+ 4411310053737091303*z^24-468673298985888069*z^22+53955*z^4-4820520*z^6-\ 12214149149*z^102+286614291*z^8-12214149149*z^10+391524613934*z^12-\ 9765628669517*z^14-3144589469198379*z^18+194329439866390*z^16-\ 39276456266755009929479262*z^50+25339546618776473631494441*z^48+ 42002533734318876*z^20+123990573437759078186271*z^36-32170444608345043323841*z^ 34-14408433636124439479705261*z^66+7269454628366826561136*z^80+391524613934*z^ 100-468673298985888069*z^90+4411310053737091303*z^88+241848619983911663045*z^84 -3144589469198379*z^94-35304826139891138236*z^86+194329439866390*z^96-\ 9765628669517*z^98+42002533734318876*z^92-1425920473232680614187*z^82+ 25339546618776473631494441*z^64+z^112-353*z^110-4820520*z^106+53955*z^108-\ 1425920473232680614187*z^30-3177898698793528073324267*z^42+ 7214717456271106320438780*z^44-14408433636124439479705261*z^46-\ 64755140996601332375929735*z^58+68926897468455302560485450*z^56-\ 64755140996601332375929735*z^54+53689985985862789997676851*z^52+ 53689985985862789997676851*z^60-3177898698793528073324267*z^70+ 7214717456271106320438780*z^68-32170444608345043323841*z^78+ 7269454628366826561136*z^32-417349215293392559208056*z^38+ 1229705829971656747574289*z^40-39276456266755009929479262*z^62+ 123990573437759078186271*z^76-417349215293392559208056*z^74+ 1229705829971656747574289*z^72+286614291*z^104)/(-1-907143953096303963684*z^28+ 124614366216707772772*z^26+465*z^2-14652009658694296928*z^24+ 1464608122514196420*z^22-84224*z^4+8494432*z^6+887634022772*z^102-556017338*z^8 +25721983098*z^10-887634022772*z^12+23712774265148*z^14+8685240306655905*z^18-\ 503766808016361*z^16+290167104551141095622561526*z^50-\ 175928950978703227213873458*z^48-123444791620363124*z^20-\ 593882794501064525300308*z^36+144913145366470929404404*z^34+ 175928950978703227213873458*z^66-144913145366470929404404*z^80-23712774265148*z ^100+14652009658694296928*z^90-124614366216707772772*z^88-\ 5684139397930439430248*z^84+123444791620363124*z^94+907143953096303963684*z^86-\ 8685240306655905*z^96+503766808016361*z^98-1464608122514196420*z^92+ 30801568580109804584136*z^82-290167104551141095622561526*z^64-465*z^112+z^114+ 84224*z^110+556017338*z^106-8494432*z^108+5684139397930439430248*z^30+ 18317462772248748290021340*z^42-44243987166559502577564344*z^44+ 94014070890428100389724592*z^46+613707009907349820223584368*z^58-\ 613707009907349820223584368*z^56+541745933786348477299500640*z^54-\ 422086208089551944971624148*z^52-541745933786348477299500640*z^60+ 44243987166559502577564344*z^70-94014070890428100389724592*z^68+ 593882794501064525300308*z^78-30801568580109804584136*z^32+ 2125939646691831127999708*z^38-6662863595842931445699036*z^40+ 422086208089551944971624148*z^62-2125939646691831127999708*z^76+ 6662863595842931445699036*z^74-18317462772248748290021340*z^72-25721983098*z^ 104) The first , 40, terms are: [0, 112, 0, 21811, 0, 4382939, 0, 883030308, 0, 177966387329, 0, 35870061281661, 0, 7229942627818988, 0, 1457270585330016915, 0, 293728696718886336911, 0, 59204242175931235719832, 0, 11933267111005616070965965, 0, 2405281581627433229346772321, 0, 484811037615938213572202497128, 0, 97719013699765263517788843173931, 0, 19696345419502473773929406513864663, 0, 3970015747012192294002843509926283820, 0, 800200478800935762144457506629176782289, 0, 161289235894171939721204121955905128466293, 0, 32509625156650001636638203126812852735316388, 0, 6552673660926419422384314483457771002972899871] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 240170929984927834059 z - 34286235318005826675 z - 338 z 24 22 4 6 + 4195188449006036987 z - 437215527826531052 z + 49847 z - 4345575 z 102 8 10 12 - 10776134208 z + 254607447 z - 10776134208 z + 345203831377 z 14 18 16 - 8647557798134 z - 2841454675214550 z + 173532865994978 z 50 48 - 48844592739159564222951988 z + 31147795152305794050123646 z 20 36 + 38516132093236049 z + 135464555277822501524008 z 34 66 - 34312980628483506152648 z - 17455909682044938257985558 z 80 100 90 + 7568333529939118421545 z + 345203831377 z - 437215527826531052 z 88 84 + 4195188449006036987 z + 240170929984927834059 z 94 86 96 - 2841454675214550 z - 34286235318005826675 z + 173532865994978 z 98 92 82 - 8647557798134 z + 38516132093236049 z - 1449427832696557798414 z 64 112 110 106 + 31147795152305794050123646 z + z - 338 z - 4345575 z 108 30 42 + 49847 z - 1449427832696557798414 z - 3712651981726555181874592 z 44 46 + 8592742819866538402621966 z - 17455909682044938257985558 z 58 56 - 81640741390010772238580484 z + 87051864874369786888972364 z 54 52 - 81640741390010772238580484 z + 67340074351788302530818622 z 60 70 + 67340074351788302530818622 z - 3712651981726555181874592 z 68 78 + 8592742819866538402621966 z - 34312980628483506152648 z 32 38 + 7568333529939118421545 z - 466795403369629868802696 z 40 62 + 1406666276312218703502550 z - 48844592739159564222951988 z 76 74 + 135464555277822501524008 z - 466795403369629868802696 z 72 104 / + 1406666276312218703502550 z + 254607447 z ) / (-1 / 28 26 2 - 871538358799508495926 z + 117307200332319768227 z + 425 z 24 22 4 - 13523896766742508019 z + 1326683075596123868 z - 73266 z 6 102 8 10 + 7208818 z + 749482024992 z - 467246675 z + 21602270371 z 12 14 18 - 749482024992 z + 20211951681324 z + 7604867358347801 z 16 50 - 434730893623377 z + 340155609941600991585481906 z 48 20 - 203987663910358727523769938 z - 109862464927372712 z 36 34 - 620601101151919742687628 z + 148299911412175757994985 z 66 80 + 203987663910358727523769938 z - 148299911412175757994985 z 100 90 - 20211951681324 z + 13523896766742508019 z 88 84 - 117307200332319768227 z - 5576200729080568512902 z 94 86 + 109862464927372712 z + 871538358799508495926 z 96 98 92 - 7604867358347801 z + 434730893623377 z - 1326683075596123868 z 82 64 112 + 30861874516952559033761 z - 340155609941600991585481906 z - 425 z 114 110 106 108 + z + 73266 z + 467246675 z - 7208818 z 30 42 + 5576200729080568512902 z + 20292811124154564611645918 z 44 46 - 49856684296327297084760208 z + 107572374985706001108905824 z 58 56 + 731827481653743147746748254 z - 731827481653743147746748254 z 54 52 + 644144594241350555807082228 z - 498996615195498214027763180 z 60 70 - 644144594241350555807082228 z + 49856684296327297084760208 z 68 78 - 107572374985706001108905824 z + 620601101151919742687628 z 32 38 - 30861874516952559033761 z + 2267334462923034996794684 z 40 62 - 7246441988536799392202654 z + 498996615195498214027763180 z 76 74 - 2267334462923034996794684 z + 7246441988536799392202654 z 72 104 - 20292811124154564611645918 z - 21602270371 z ) And in Maple-input format, it is: -(1+240170929984927834059*z^28-34286235318005826675*z^26-338*z^2+ 4195188449006036987*z^24-437215527826531052*z^22+49847*z^4-4345575*z^6-\ 10776134208*z^102+254607447*z^8-10776134208*z^10+345203831377*z^12-\ 8647557798134*z^14-2841454675214550*z^18+173532865994978*z^16-\ 48844592739159564222951988*z^50+31147795152305794050123646*z^48+ 38516132093236049*z^20+135464555277822501524008*z^36-34312980628483506152648*z^ 34-17455909682044938257985558*z^66+7568333529939118421545*z^80+345203831377*z^ 100-437215527826531052*z^90+4195188449006036987*z^88+240170929984927834059*z^84 -2841454675214550*z^94-34286235318005826675*z^86+173532865994978*z^96-\ 8647557798134*z^98+38516132093236049*z^92-1449427832696557798414*z^82+ 31147795152305794050123646*z^64+z^112-338*z^110-4345575*z^106+49847*z^108-\ 1449427832696557798414*z^30-3712651981726555181874592*z^42+ 8592742819866538402621966*z^44-17455909682044938257985558*z^46-\ 81640741390010772238580484*z^58+87051864874369786888972364*z^56-\ 81640741390010772238580484*z^54+67340074351788302530818622*z^52+ 67340074351788302530818622*z^60-3712651981726555181874592*z^70+ 8592742819866538402621966*z^68-34312980628483506152648*z^78+ 7568333529939118421545*z^32-466795403369629868802696*z^38+ 1406666276312218703502550*z^40-48844592739159564222951988*z^62+ 135464555277822501524008*z^76-466795403369629868802696*z^74+ 1406666276312218703502550*z^72+254607447*z^104)/(-1-871538358799508495926*z^28+ 117307200332319768227*z^26+425*z^2-13523896766742508019*z^24+ 1326683075596123868*z^22-73266*z^4+7208818*z^6+749482024992*z^102-467246675*z^8 +21602270371*z^10-749482024992*z^12+20211951681324*z^14+7604867358347801*z^18-\ 434730893623377*z^16+340155609941600991585481906*z^50-\ 203987663910358727523769938*z^48-109862464927372712*z^20-\ 620601101151919742687628*z^36+148299911412175757994985*z^34+ 203987663910358727523769938*z^66-148299911412175757994985*z^80-20211951681324*z ^100+13523896766742508019*z^90-117307200332319768227*z^88-\ 5576200729080568512902*z^84+109862464927372712*z^94+871538358799508495926*z^86-\ 7604867358347801*z^96+434730893623377*z^98-1326683075596123868*z^92+ 30861874516952559033761*z^82-340155609941600991585481906*z^64-425*z^112+z^114+ 73266*z^110+467246675*z^106-7208818*z^108+5576200729080568512902*z^30+ 20292811124154564611645918*z^42-49856684296327297084760208*z^44+ 107572374985706001108905824*z^46+731827481653743147746748254*z^58-\ 731827481653743147746748254*z^56+644144594241350555807082228*z^54-\ 498996615195498214027763180*z^52-644144594241350555807082228*z^60+ 49856684296327297084760208*z^70-107572374985706001108905824*z^68+ 620601101151919742687628*z^78-30861874516952559033761*z^32+ 2267334462923034996794684*z^38-7246441988536799392202654*z^40+ 498996615195498214027763180*z^62-2267334462923034996794684*z^76+ 7246441988536799392202654*z^74-20292811124154564611645918*z^72-21602270371*z^ 104) The first , 40, terms are: [0, 87, 0, 13556, 0, 2250401, 0, 377754467, 0, 63566181055, 0, 10702922807533, 0, 1802373023507388, 0, 303531738365346895, 0, 51117316785530728097, 0, 8608612998882177593241, 0, 1449768431033947510143375, 0, 244154188005757905673392276, 0, 41117787772540273473509892381, 0, 6924609850619454829319290507439, 0, 1166167353955810805139518568099411, 0, 196393201664826318350675386055726913, 0, 33074403549510201958799914093500347100, 0, 5570030739180946192267962694716435428535, 0, 938043898195183731550961556020473136160761, 0, 157975134456001777949504945361763680412683961] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 273405160104626917990 z - 38498415820117400803 z - 335 z 24 22 4 6 + 4641879770829711868 z - 476357283757064745 z + 49182 z - 4287849 z 102 8 10 12 - 10766001715 z + 252322808 z - 10766001715 z + 348712053642 z 14 18 16 - 8852160902433 z - 2998607672217455 z + 180284307213488 z 50 48 - 59924906626487668019410823 z + 38139723284000138859691508 z 20 36 + 41302199863348674 z + 160992974854431352402770 z 34 66 - 40418880180382923184455 z - 21318020586966693812212197 z 80 100 90 + 8825104856352071284288 z + 348712053642 z - 476357283757064745 z 88 84 + 4641879770829711868 z + 273405160104626917990 z 94 86 96 - 2998607672217455 z - 38498415820117400803 z + 180284307213488 z 98 92 82 - 8852160902433 z + 41302199863348674 z - 1670946387976671908373 z 64 112 110 106 + 38139723284000138859691508 z + z - 335 z - 4287849 z 108 30 42 + 49182 z - 1670946387976671908373 z - 4498928681878442965374415 z 44 46 + 10457860149757772815841066 z - 21318020586966693812212197 z 58 56 - 100369349492913024205579801 z + 107048523897983524560536498 z 54 52 - 100369349492913024205579801 z + 82724931408930752102464166 z 60 70 + 82724931408930752102464166 z - 4498928681878442965374415 z 68 78 + 10457860149757772815841066 z - 40418880180382923184455 z 32 38 + 8825104856352071284288 z - 559016451652665307118297 z 40 62 + 1695477684334040926260636 z - 59924906626487668019410823 z 76 74 + 160992974854431352402770 z - 559016451652665307118297 z 72 104 / 2 + 1695477684334040926260636 z + 252322808 z ) / ((-1 + z ) (1 / 28 26 2 + 867283395924850777940 z - 116947240685879171844 z - 427 z 24 22 4 + 13472948169587152758 z - 1318056534132656482 z + 72856 z 6 102 8 10 - 7090408 z - 21007698374 z + 456054942 z - 21007698374 z 12 14 18 + 728793329556 z - 19705829603667 z - 7485370272359441 z 16 50 + 425700393459152 z - 258848577056767811326314929 z 48 20 + 162621268134242631909360784 z + 108686392349077636 z 36 34 + 593541806227433067684844 z - 144030172499302460267367 z 66 80 - 89412315518614729360741475 z + 30321974122409559614392 z 100 90 88 + 728793329556 z - 1318056534132656482 z + 13472948169587152758 z 84 94 + 867283395924850777940 z - 7485370272359441 z 86 96 98 - 116947240685879171844 z + 425700393459152 z - 19705829603667 z 92 82 + 108686392349077636 z - 5522710151084607769845 z 64 112 110 106 + 162621268134242631909360784 z + z - 427 z - 7090408 z 108 30 42 + 72856 z - 5522710151084607769845 z - 18079895774256190637206622 z 44 46 + 43002994747504843257662332 z - 89412315518614729360741475 z 58 56 - 440121324951517092302028734 z + 470300597009565887059276020 z 54 52 - 440121324951517092302028734 z + 360700254219926158476356312 z 60 70 + 360700254219926158476356312 z - 18079895774256190637206622 z 68 78 + 43002994747504843257662332 z - 144030172499302460267367 z 32 38 + 30321974122409559614392 z - 2126852877146000781478952 z 40 62 + 6639077081543076505464510 z - 258848577056767811326314929 z 76 74 + 593541806227433067684844 z - 2126852877146000781478952 z 72 104 + 6639077081543076505464510 z + 456054942 z )) And in Maple-input format, it is: -(1+273405160104626917990*z^28-38498415820117400803*z^26-335*z^2+ 4641879770829711868*z^24-476357283757064745*z^22+49182*z^4-4287849*z^6-\ 10766001715*z^102+252322808*z^8-10766001715*z^10+348712053642*z^12-\ 8852160902433*z^14-2998607672217455*z^18+180284307213488*z^16-\ 59924906626487668019410823*z^50+38139723284000138859691508*z^48+ 41302199863348674*z^20+160992974854431352402770*z^36-40418880180382923184455*z^ 34-21318020586966693812212197*z^66+8825104856352071284288*z^80+348712053642*z^ 100-476357283757064745*z^90+4641879770829711868*z^88+273405160104626917990*z^84 -2998607672217455*z^94-38498415820117400803*z^86+180284307213488*z^96-\ 8852160902433*z^98+41302199863348674*z^92-1670946387976671908373*z^82+ 38139723284000138859691508*z^64+z^112-335*z^110-4287849*z^106+49182*z^108-\ 1670946387976671908373*z^30-4498928681878442965374415*z^42+ 10457860149757772815841066*z^44-21318020586966693812212197*z^46-\ 100369349492913024205579801*z^58+107048523897983524560536498*z^56-\ 100369349492913024205579801*z^54+82724931408930752102464166*z^52+ 82724931408930752102464166*z^60-4498928681878442965374415*z^70+ 10457860149757772815841066*z^68-40418880180382923184455*z^78+ 8825104856352071284288*z^32-559016451652665307118297*z^38+ 1695477684334040926260636*z^40-59924906626487668019410823*z^62+ 160992974854431352402770*z^76-559016451652665307118297*z^74+ 1695477684334040926260636*z^72+252322808*z^104)/(-1+z^2)/(1+ 867283395924850777940*z^28-116947240685879171844*z^26-427*z^2+ 13472948169587152758*z^24-1318056534132656482*z^22+72856*z^4-7090408*z^6-\ 21007698374*z^102+456054942*z^8-21007698374*z^10+728793329556*z^12-\ 19705829603667*z^14-7485370272359441*z^18+425700393459152*z^16-\ 258848577056767811326314929*z^50+162621268134242631909360784*z^48+ 108686392349077636*z^20+593541806227433067684844*z^36-144030172499302460267367* z^34-89412315518614729360741475*z^66+30321974122409559614392*z^80+728793329556* z^100-1318056534132656482*z^90+13472948169587152758*z^88+867283395924850777940* z^84-7485370272359441*z^94-116947240685879171844*z^86+425700393459152*z^96-\ 19705829603667*z^98+108686392349077636*z^92-5522710151084607769845*z^82+ 162621268134242631909360784*z^64+z^112-427*z^110-7090408*z^106+72856*z^108-\ 5522710151084607769845*z^30-18079895774256190637206622*z^42+ 43002994747504843257662332*z^44-89412315518614729360741475*z^46-\ 440121324951517092302028734*z^58+470300597009565887059276020*z^56-\ 440121324951517092302028734*z^54+360700254219926158476356312*z^52+ 360700254219926158476356312*z^60-18079895774256190637206622*z^70+ 43002994747504843257662332*z^68-144030172499302460267367*z^78+ 30321974122409559614392*z^32-2126852877146000781478952*z^38+ 6639077081543076505464510*z^40-258848577056767811326314929*z^62+ 593541806227433067684844*z^76-2126852877146000781478952*z^74+ 6639077081543076505464510*z^72+456054942*z^104) The first , 40, terms are: [0, 93, 0, 15703, 0, 2780980, 0, 494857501, 0, 88110421731, 0, 15689780826855, 0, 2793923409535779, 0, 497523713628703379, 0, 88595862499276219281, 0, 15776591407093286977012, 0, 2809395837360881911397563, 0, 500279486722855739123405421, 0, 89086615031945407664712476013, 0, 15863982419800427957973336105669, 0, 2824957914941613643825310169044077, 0, 503050684895500863714633694251999715, 0, 89580092587342373256313203146845539892, 0, 15951857792697353303496091190195333859193, 0, 2840606207126331123973083909875820924348403, 0, 505837234059313922191472683197667485370176923] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2887858004389914137 z - 676534504830550881 z - 293 z 24 22 4 6 + 131890178620210797 z - 21275016728384260 z + 36393 z - 2603948 z 8 10 12 14 + 122121556 z - 4034569100 z + 98336042735 z - 1826034012511 z 18 16 50 - 304303243166100 z + 26450905415867 z - 296086426387528478939 z 48 20 + 449113380157080280503 z + 2819558156337948 z 36 34 + 165056636250895971407 z - 77712090986457946804 z 66 80 88 84 86 - 21275016728384260 z + 122121556 z + z + 36393 z - 293 z 82 64 30 - 2603948 z + 131890178620210797 z - 10305000705358665140 z 42 44 - 576484892360188073432 z + 626484908953188870232 z 46 58 - 576484892360188073432 z - 10305000705358665140 z 56 54 + 30851460922273535700 z - 77712090986457946804 z 52 60 + 165056636250895971407 z + 2887858004389914137 z 70 68 78 - 304303243166100 z + 2819558156337948 z - 4034569100 z 32 38 + 30851460922273535700 z - 296086426387528478939 z 40 62 76 + 449113380157080280503 z - 676534504830550881 z + 98336042735 z 74 72 / - 1826034012511 z + 26450905415867 z ) / (-1 / 28 26 2 - 12884038455622319036 z + 2793355693956561611 z + 383 z 24 22 4 6 - 504106483325051981 z + 75276127144522140 z - 56328 z + 4584132 z 8 10 12 14 - 239656937 z + 8724957551 z - 232654265216 z + 4703943917544 z 18 16 50 + 921335510613263 z - 73950852092529 z + 3226842174512718890685 z 48 20 - 4494522553903806989227 z - 9232114332479464 z 36 34 - 1009175414310856122224 z + 438736011925690518667 z 66 80 90 88 84 + 504106483325051981 z - 8724957551 z + z - 383 z - 4584132 z 86 82 64 + 56328 z + 239656937 z - 2793355693956561611 z 30 42 + 49699759622989220612 z + 4494522553903806989227 z 44 46 - 5303783536170726946132 z + 5303783536170726946132 z 58 56 + 160935583916601534741 z - 438736011925690518667 z 54 52 + 1009175414310856122224 z - 1961799036321704176124 z 60 70 68 - 49699759622989220612 z + 9232114332479464 z - 75276127144522140 z 78 32 + 232654265216 z - 160935583916601534741 z 38 40 + 1961799036321704176124 z - 3226842174512718890685 z 62 76 74 + 12884038455622319036 z - 4703943917544 z + 73950852092529 z 72 - 921335510613263 z ) And in Maple-input format, it is: -(1+2887858004389914137*z^28-676534504830550881*z^26-293*z^2+131890178620210797 *z^24-21275016728384260*z^22+36393*z^4-2603948*z^6+122121556*z^8-4034569100*z^ 10+98336042735*z^12-1826034012511*z^14-304303243166100*z^18+26450905415867*z^16 -296086426387528478939*z^50+449113380157080280503*z^48+2819558156337948*z^20+ 165056636250895971407*z^36-77712090986457946804*z^34-21275016728384260*z^66+ 122121556*z^80+z^88+36393*z^84-293*z^86-2603948*z^82+131890178620210797*z^64-\ 10305000705358665140*z^30-576484892360188073432*z^42+626484908953188870232*z^44 -576484892360188073432*z^46-10305000705358665140*z^58+30851460922273535700*z^56 -77712090986457946804*z^54+165056636250895971407*z^52+2887858004389914137*z^60-\ 304303243166100*z^70+2819558156337948*z^68-4034569100*z^78+30851460922273535700 *z^32-296086426387528478939*z^38+449113380157080280503*z^40-676534504830550881* z^62+98336042735*z^76-1826034012511*z^74+26450905415867*z^72)/(-1-\ 12884038455622319036*z^28+2793355693956561611*z^26+383*z^2-504106483325051981*z ^24+75276127144522140*z^22-56328*z^4+4584132*z^6-239656937*z^8+8724957551*z^10-\ 232654265216*z^12+4703943917544*z^14+921335510613263*z^18-73950852092529*z^16+ 3226842174512718890685*z^50-4494522553903806989227*z^48-9232114332479464*z^20-\ 1009175414310856122224*z^36+438736011925690518667*z^34+504106483325051981*z^66-\ 8724957551*z^80+z^90-383*z^88-4584132*z^84+56328*z^86+239656937*z^82-\ 2793355693956561611*z^64+49699759622989220612*z^30+4494522553903806989227*z^42-\ 5303783536170726946132*z^44+5303783536170726946132*z^46+160935583916601534741*z ^58-438736011925690518667*z^56+1009175414310856122224*z^54-\ 1961799036321704176124*z^52-49699759622989220612*z^60+9232114332479464*z^70-\ 75276127144522140*z^68+232654265216*z^78-160935583916601534741*z^32+ 1961799036321704176124*z^38-3226842174512718890685*z^40+12884038455622319036*z^ 62-4703943917544*z^76+73950852092529*z^74-921335510613263*z^72) The first , 40, terms are: [0, 90, 0, 14535, 0, 2477569, 0, 425217946, 0, 73053589427, 0, 12552866001175, 0, 2157029972732066, 0, 370656419569238097, 0, 63692344453711419439, 0, 10944678506370512753674, 0, 1880696838810489101809753, 0, 323172637306334512637522629, 0, 55532902192407421849156642178, 0, 9542587677548816206774362738259, 0, 1639766264538649239027664890489949, 0, 281771935788983947179749328904296810, 0, 48418744497547632838922409676151809331, 0, 8320114677691903461697208817733305345839, 0, 1429700603935641877569444128483844991966722, 0, 245674957146265567884069447878622535023604733] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {4, 6}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3637549296582121186 z - 853417941639512697 z - 305 z 24 22 4 6 + 166388710086380930 z - 26797188228860039 z + 39391 z - 2918552 z 8 10 12 14 + 140967155 z - 4770206653 z + 118507500516 z - 2233714943851 z 18 16 50 - 379703082675179 z + 32727987266144 z - 367055414005658164139 z 48 20 + 555385898863143959389 z + 3538762124387174 z 36 34 + 205262508859422909339 z - 96981863615051239754 z 66 80 88 84 86 - 26797188228860039 z + 140967155 z + z + 39391 z - 305 z 82 64 30 - 2918552 z + 166388710086380930 z - 12947518713173469825 z 42 44 - 711771219678239681710 z + 773084883167733715124 z 46 58 - 711771219678239681710 z - 12947518713173469825 z 56 54 + 38638254161769210829 z - 96981863615051239754 z 52 60 + 205262508859422909339 z + 3637549296582121186 z 70 68 78 - 379703082675179 z + 3538762124387174 z - 4770206653 z 32 38 + 38638254161769210829 z - 367055414005658164139 z 40 62 76 + 555385898863143959389 z - 853417941639512697 z + 118507500516 z 74 72 / 2 - 2233714943851 z + 32727987266144 z ) / ((-1 + z ) (1 / 28 26 2 + 13332888209961726826 z - 2993836230104944260 z - 396 z 24 22 4 6 + 556470572889071647 z - 85109219439990904 z + 60392 z - 5074020 z 8 10 12 14 + 272065230 z - 10088008284 z + 272123656794 z - 5530069436476 z 18 16 50 - 1074527808490440 z + 86847909558417 z - 1567643461326869810868 z 48 20 + 2407932864026427976367 z + 10631347533775388 z 36 34 + 858708778124267792436 z - 395317090613169132924 z 66 80 88 84 86 - 85109219439990904 z + 272065230 z + z + 60392 z - 396 z 82 64 30 - 5074020 z + 556470572889071647 z - 49384374890512521860 z 42 44 - 3114307066727385181696 z + 3392976720477077974024 z 46 58 - 3114307066727385181696 z - 49384374890512521860 z 56 54 + 152701939539927285714 z - 395317090613169132924 z 52 60 + 858708778124267792436 z + 13332888209961726826 z 70 68 78 - 1074527808490440 z + 10631347533775388 z - 10088008284 z 32 38 + 152701939539927285714 z - 1567643461326869810868 z 40 62 76 + 2407932864026427976367 z - 2993836230104944260 z + 272123656794 z 74 72 - 5530069436476 z + 86847909558417 z )) And in Maple-input format, it is: -(1+3637549296582121186*z^28-853417941639512697*z^26-305*z^2+166388710086380930 *z^24-26797188228860039*z^22+39391*z^4-2918552*z^6+140967155*z^8-4770206653*z^ 10+118507500516*z^12-2233714943851*z^14-379703082675179*z^18+32727987266144*z^ 16-367055414005658164139*z^50+555385898863143959389*z^48+3538762124387174*z^20+ 205262508859422909339*z^36-96981863615051239754*z^34-26797188228860039*z^66+ 140967155*z^80+z^88+39391*z^84-305*z^86-2918552*z^82+166388710086380930*z^64-\ 12947518713173469825*z^30-711771219678239681710*z^42+773084883167733715124*z^44 -711771219678239681710*z^46-12947518713173469825*z^58+38638254161769210829*z^56 -96981863615051239754*z^54+205262508859422909339*z^52+3637549296582121186*z^60-\ 379703082675179*z^70+3538762124387174*z^68-4770206653*z^78+38638254161769210829 *z^32-367055414005658164139*z^38+555385898863143959389*z^40-853417941639512697* z^62+118507500516*z^76-2233714943851*z^74+32727987266144*z^72)/(-1+z^2)/(1+ 13332888209961726826*z^28-2993836230104944260*z^26-396*z^2+556470572889071647*z ^24-85109219439990904*z^22+60392*z^4-5074020*z^6+272065230*z^8-10088008284*z^10 +272123656794*z^12-5530069436476*z^14-1074527808490440*z^18+86847909558417*z^16 -1567643461326869810868*z^50+2407932864026427976367*z^48+10631347533775388*z^20 +858708778124267792436*z^36-395317090613169132924*z^34-85109219439990904*z^66+ 272065230*z^80+z^88+60392*z^84-396*z^86-5074020*z^82+556470572889071647*z^64-\ 49384374890512521860*z^30-3114307066727385181696*z^42+3392976720477077974024*z^ 44-3114307066727385181696*z^46-49384374890512521860*z^58+152701939539927285714* z^56-395317090613169132924*z^54+858708778124267792436*z^52+13332888209961726826 *z^60-1074527808490440*z^70+10631347533775388*z^68-10088008284*z^78+ 152701939539927285714*z^32-1567643461326869810868*z^38+2407932864026427976367*z ^40-2993836230104944260*z^62+272123656794*z^76-5530069436476*z^74+ 86847909558417*z^72) The first , 40, terms are: [0, 92, 0, 15127, 0, 2628783, 0, 460280584, 0, 80694237029, 0, 14150468104893, 0, 2481547598956048, 0, 435190905766105967, 0, 76319987190882263655, 0, 13384343450076121287108, 0, 2347231526891332833579257, 0, 411637379780209481877346489, 0, 72189441980861975077503739636, 0, 12659966762341717685586044393303, 0, 2220196667547911386810722562032511, 0, 389359098316044034193474221236297920, 0, 68282467793355027976987537618482784253, 0, 11974795062334437369854937013988471214597, 0, 2100037116689537585345190760420975075332440, 0, 368286544238875976501086725529915827976005183] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 18790454031847235 z - 7456441738500471 z - 250 z 24 22 4 6 + 2394582411040033 z - 619605807083992 z + 25149 z - 1402893 z 8 10 12 14 + 49735639 z - 1209273948 z + 21169293417 z - 275890198840 z 18 16 50 - 21155598435440 z + 2741985534622 z - 619605807083992 z 48 20 36 + 2394582411040033 z + 128420889762707 z + 96174242772785356 z 34 66 64 - 86875381366456080 z - 1402893 z + 49735639 z 30 42 44 - 38446441326796106 z - 38446441326796106 z + 18790454031847235 z 46 58 56 - 7456441738500471 z - 275890198840 z + 2741985534622 z 54 52 60 70 - 21155598435440 z + 128420889762707 z + 21169293417 z - 250 z 68 32 38 + 25149 z + 64015961037897787 z - 86875381366456080 z 40 62 72 / 2 + 64015961037897787 z - 1209273948 z + z ) / ((-1 + z ) (1 / 28 26 2 + 78103973101248840 z - 29628894293546320 z - 346 z 24 22 4 6 + 9030762652800703 z - 2204125236151414 z + 41458 z - 2621276 z 8 10 12 14 + 103086491 z - 2749500230 z + 52439346064 z - 740887923920 z 18 16 50 - 65945457230924 z + 7949205130557 z - 2204125236151414 z 48 20 36 + 9030762652800703 z + 428627674290466 z + 435777481177022042 z 34 66 64 - 391419748443168060 z - 2621276 z + 103086491 z 30 42 44 - 165773492061254786 z - 165773492061254786 z + 78103973101248840 z 46 58 56 - 29628894293546320 z - 740887923920 z + 7949205130557 z 54 52 60 70 - 65945457230924 z + 428627674290466 z + 52439346064 z - 346 z 68 32 38 + 41458 z + 283632850678588449 z - 391419748443168060 z 40 62 72 + 283632850678588449 z - 2749500230 z + z )) And in Maple-input format, it is: -(1+18790454031847235*z^28-7456441738500471*z^26-250*z^2+2394582411040033*z^24-\ 619605807083992*z^22+25149*z^4-1402893*z^6+49735639*z^8-1209273948*z^10+ 21169293417*z^12-275890198840*z^14-21155598435440*z^18+2741985534622*z^16-\ 619605807083992*z^50+2394582411040033*z^48+128420889762707*z^20+ 96174242772785356*z^36-86875381366456080*z^34-1402893*z^66+49735639*z^64-\ 38446441326796106*z^30-38446441326796106*z^42+18790454031847235*z^44-\ 7456441738500471*z^46-275890198840*z^58+2741985534622*z^56-21155598435440*z^54+ 128420889762707*z^52+21169293417*z^60-250*z^70+25149*z^68+64015961037897787*z^ 32-86875381366456080*z^38+64015961037897787*z^40-1209273948*z^62+z^72)/(-1+z^2) /(1+78103973101248840*z^28-29628894293546320*z^26-346*z^2+9030762652800703*z^24 -2204125236151414*z^22+41458*z^4-2621276*z^6+103086491*z^8-2749500230*z^10+ 52439346064*z^12-740887923920*z^14-65945457230924*z^18+7949205130557*z^16-\ 2204125236151414*z^50+9030762652800703*z^48+428627674290466*z^20+ 435777481177022042*z^36-391419748443168060*z^34-2621276*z^66+103086491*z^64-\ 165773492061254786*z^30-165773492061254786*z^42+78103973101248840*z^44-\ 29628894293546320*z^46-740887923920*z^58+7949205130557*z^56-65945457230924*z^54 +428627674290466*z^52+52439346064*z^60-346*z^70+41458*z^68+283632850678588449*z ^32-391419748443168060*z^38+283632850678588449*z^40-2749500230*z^62+z^72) The first , 40, terms are: [0, 97, 0, 17004, 0, 3105241, 0, 568996481, 0, 104297072453, 0, 19118412527353, 0, 3504562017482996, 0, 642415458994223553, 0, 117760126795779280413, 0, 21586416584789892883421, 0, 3956970794697370397364665, 0, 725345858976878896455728548, 0, 132961965730389384582244450993, 0, 24373040960459508671714726116277, 0, 4467782364680097856065247884542401, 0, 818981894403420317149896145505161233, 0, 150126234586362795970883463644329704828, 0, 27519395074659531697872895063673680731417, 0, 5044535402901507511620409855213626757151057, 0, 924705552650727006960738796632452211543654769] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}, {4, 5}, {4, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 31258604456452 z - 27223518871430 z - 211 z 24 22 4 6 + 17975862474554 z - 8987948537168 z + 16694 z - 702927 z 8 10 12 14 + 18207197 z - 313667248 z + 3767608208 z - 32516709800 z 18 16 50 48 - 965723886640 z + 205662418376 z - 702927 z + 18207197 z 20 36 34 + 3395410343768 z + 3395410343768 z - 8987948537168 z 30 42 44 46 - 27223518871430 z - 32516709800 z + 3767608208 z - 313667248 z 56 54 52 32 38 + z - 211 z + 16694 z + 17975862474554 z - 965723886640 z 40 / 2 28 + 205662418376 z ) / ((-1 + z ) (1 + 133901860660890 z / 26 2 24 22 - 115669518697380 z - 286 z + 74561716375270 z - 35858018324672 z 4 6 8 10 12 + 27503 z - 1349806 z + 39872225 z - 772387648 z + 10318153724 z 14 18 16 50 - 98063166680 z - 3426483025336 z + 676266008388 z - 1349806 z 48 20 36 + 39872225 z + 12853704083004 z + 12853704083004 z 34 30 42 - 35858018324672 z - 115669518697380 z - 98063166680 z 44 46 56 54 52 + 10318153724 z - 772387648 z + z - 286 z + 27503 z 32 38 40 + 74561716375270 z - 3426483025336 z + 676266008388 z )) And in Maple-input format, it is: -(1+31258604456452*z^28-27223518871430*z^26-211*z^2+17975862474554*z^24-\ 8987948537168*z^22+16694*z^4-702927*z^6+18207197*z^8-313667248*z^10+3767608208* z^12-32516709800*z^14-965723886640*z^18+205662418376*z^16-702927*z^50+18207197* z^48+3395410343768*z^20+3395410343768*z^36-8987948537168*z^34-27223518871430*z^ 30-32516709800*z^42+3767608208*z^44-313667248*z^46+z^56-211*z^54+16694*z^52+ 17975862474554*z^32-965723886640*z^38+205662418376*z^40)/(-1+z^2)/(1+ 133901860660890*z^28-115669518697380*z^26-286*z^2+74561716375270*z^24-\ 35858018324672*z^22+27503*z^4-1349806*z^6+39872225*z^8-772387648*z^10+ 10318153724*z^12-98063166680*z^14-3426483025336*z^18+676266008388*z^16-1349806* z^50+39872225*z^48+12853704083004*z^20+12853704083004*z^36-35858018324672*z^34-\ 115669518697380*z^30-98063166680*z^42+10318153724*z^44-772387648*z^46+z^56-286* z^54+27503*z^52+74561716375270*z^32-3426483025336*z^38+676266008388*z^40) The first , 40, terms are: [0, 76, 0, 10717, 0, 1638197, 0, 254008476, 0, 39502915001, 0, 6147630848553, 0, 956879873650172, 0, 148944437841216869, 0, 23184373972664017389, 0, 3608838871068881946092, 0, 561745839775219209612753, 0, 87440433137844101864946097, 0, 13610834433507991363912379052, 0, 2118640186803739733979215672653, 0, 329784060833702514573602078377605, 0, 51333646701862072030086747097282364, 0, 7990511358841639649228916519960943305, 0, 1243789909348511054058208274226889077785, 0, 193606299914420717755644698476072168166236, 0, 30136439510318623651217207395096612490568981] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}, {4, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 5626690086 z - 12901096282 z - 182 z + 21151402927 z 22 4 6 8 10 - 24926600976 z + 11098 z - 324238 z + 5364853 z - 55261112 z 12 14 18 16 + 375127707 z - 1738616162 z - 12901096282 z + 5626690086 z 20 36 34 30 42 + 21151402927 z + 5364853 z - 55261112 z - 1738616162 z - 182 z 44 32 38 40 / 46 44 + z + 375127707 z - 324238 z + 11098 z ) / (z - 270 z / 42 40 38 36 34 + 20607 z - 724203 z + 14256282 z - 173716181 z + 1388768995 z 32 30 28 26 - 7555497454 z + 28651552477 z - 76951559193 z + 147912893930 z 24 22 20 18 - 204764768551 z + 204764768551 z - 147912893930 z + 76951559193 z 16 14 12 10 - 28651552477 z + 7555497454 z - 1388768995 z + 173716181 z 8 6 4 2 - 14256282 z + 724203 z - 20607 z + 270 z - 1) And in Maple-input format, it is: -(1+5626690086*z^28-12901096282*z^26-182*z^2+21151402927*z^24-24926600976*z^22+ 11098*z^4-324238*z^6+5364853*z^8-55261112*z^10+375127707*z^12-1738616162*z^14-\ 12901096282*z^18+5626690086*z^16+21151402927*z^20+5364853*z^36-55261112*z^34-\ 1738616162*z^30-182*z^42+z^44+375127707*z^32-324238*z^38+11098*z^40)/(z^46-270* z^44+20607*z^42-724203*z^40+14256282*z^38-173716181*z^36+1388768995*z^34-\ 7555497454*z^32+28651552477*z^30-76951559193*z^28+147912893930*z^26-\ 204764768551*z^24+204764768551*z^22-147912893930*z^20+76951559193*z^18-\ 28651552477*z^16+7555497454*z^14-1388768995*z^12+173716181*z^10-14256282*z^8+ 724203*z^6-20607*z^4+270*z^2-1) The first , 40, terms are: [0, 88, 0, 14251, 0, 2434319, 0, 418434208, 0, 71997743733, 0, 12390765314269, 0, 2132535334846128, 0, 367027436626269975, 0, 63168674820757347587, 0, 10871894113082575176584, 0, 1871150455090702942619001, 0, 322041778217871486626624073, 0, 55426279174386718794113951656, 0, 9539359905480508291057225107475, 0, 1641809423037520116653229943891655, 0, 282570131377579887684154303237658896, 0, 48632854719635873026038419762480562125, 0, 8370150612375374182607251112542394835717, 0, 1440578014138521301988932582554063921299264, 0, 247936400541148050445390097740345293731147999] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {2, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 39121996 z + 199932324 z + 192 z - 658234035 z 22 4 6 8 10 + 1435133632 z - 12020 z + 330865 z - 4730523 z + 39121996 z 12 14 18 16 - 199932324 z + 658234035 z + 2109598812 z - 1435133632 z 20 36 34 30 32 38 - 2109598812 z - 192 z + 12020 z + 4730523 z - 330865 z + z ) / 24 12 40 8 6 / (9470089511 z + 883522131 z + z + 14222528 z - 829496 z + 1 / 18 28 38 36 26 - 16958635368 z + 883522131 z - 312 z + 25045 z - 3553975816 z 16 2 32 10 20 + 9470089511 z - 312 z + 14222528 z - 142101776 z + 20575528832 z 34 4 22 30 14 - 829496 z + 25045 z - 16958635368 z - 142101776 z - 3553975816 z ) And in Maple-input format, it is: -(-1-39121996*z^28+199932324*z^26+192*z^2-658234035*z^24+1435133632*z^22-12020* z^4+330865*z^6-4730523*z^8+39121996*z^10-199932324*z^12+658234035*z^14+ 2109598812*z^18-1435133632*z^16-2109598812*z^20-192*z^36+12020*z^34+4730523*z^ 30-330865*z^32+z^38)/(9470089511*z^24+883522131*z^12+z^40+14222528*z^8-829496*z ^6+1-16958635368*z^18+883522131*z^28-312*z^38+25045*z^36-3553975816*z^26+ 9470089511*z^16-312*z^2+14222528*z^32-142101776*z^10+20575528832*z^20-829496*z^ 34+25045*z^4-16958635368*z^22-142101776*z^30-3553975816*z^14) The first , 40, terms are: [0, 120, 0, 24415, 0, 5110711, 0, 1073115672, 0, 225462753929, 0, 47376637154457, 0, 9955620335149784, 0, 2092068790233224423, 0, 439627093806629662063, 0, 92383230583785107520312, 0, 19413412850790765612416945, 0, 4079534852569258712347956945, 0, 857273517676734426218943605176, 0, 180147470693821605597093185109199, 0, 37856192383031286025865365490398727, 0, 7955100875725545503737444321561497944, 0, 1671685025902824071114067586942784382713, 0, 351287918217405799396872200596388955913513, 0, 73819648781645613336823125544947150703125400, 0, 15512462181733106627286101646575553488930130327] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2707019346092026205 z - 629552954323660219 z - 291 z 24 22 4 6 + 121906823102368845 z - 19553327841424604 z + 35637 z - 2507542 z 8 10 12 14 + 115704574 z - 3769610542 z + 90896319807 z - 1675716610345 z 18 16 50 - 277995647377300 z + 24181191141231 z - 286410081381050521705 z 48 20 + 435914406908675800555 z + 2580827001497316 z 36 34 + 158926888189225092583 z - 74401647941650743834 z 66 80 88 84 86 - 19553327841424604 z + 115704574 z + z + 35637 z - 291 z 82 64 30 - 2507542 z + 121906823102368845 z - 9731696194311141018 z 42 44 - 560705703599568993256 z + 609762839190936066952 z 46 58 - 560705703599568993256 z - 9731696194311141018 z 56 54 + 29344269114453776314 z - 74401647941650743834 z 52 60 + 158926888189225092583 z + 2707019346092026205 z 70 68 78 - 277995647377300 z + 2580827001497316 z - 3769610542 z 32 38 + 29344269114453776314 z - 286410081381050521705 z 40 62 76 + 435914406908675800555 z - 629552954323660219 z + 90896319807 z 74 72 / 2 - 1675716610345 z + 24181191141231 z ) / ((-1 + z ) (1 / 28 26 2 + 9818854708886571024 z - 2180802495807732066 z - 382 z 24 22 4 6 + 401839270384760735 z - 61122265643062708 z + 55186 z - 4383520 z 8 10 12 14 + 223311683 z - 7929560652 z + 206667057028 z - 4093667299662 z 18 16 50 - 773728587590500 z + 63177016812945 z - 1223114915642305883378 z 48 20 + 1892137563242193200575 z + 7625171998953046 z 36 34 + 663717709704757964922 z - 302158729287687420216 z 66 80 88 84 86 - 61122265643062708 z + 223311683 z + z + 55186 z - 382 z 82 64 30 - 4383520 z + 401839270384760735 z - 36816445849696134620 z 42 44 - 2458064966498592961352 z + 2682044262168268474420 z 46 58 - 2458064966498592961352 z - 36816445849696134620 z 56 54 + 115291973643578378205 z - 302158729287687420216 z 52 60 + 663717709704757964922 z + 9818854708886571024 z 70 68 78 - 773728587590500 z + 7625171998953046 z - 7929560652 z 32 38 + 115291973643578378205 z - 1223114915642305883378 z 40 62 76 + 1892137563242193200575 z - 2180802495807732066 z + 206667057028 z 74 72 - 4093667299662 z + 63177016812945 z )) And in Maple-input format, it is: -(1+2707019346092026205*z^28-629552954323660219*z^26-291*z^2+121906823102368845 *z^24-19553327841424604*z^22+35637*z^4-2507542*z^6+115704574*z^8-3769610542*z^ 10+90896319807*z^12-1675716610345*z^14-277995647377300*z^18+24181191141231*z^16 -286410081381050521705*z^50+435914406908675800555*z^48+2580827001497316*z^20+ 158926888189225092583*z^36-74401647941650743834*z^34-19553327841424604*z^66+ 115704574*z^80+z^88+35637*z^84-291*z^86-2507542*z^82+121906823102368845*z^64-\ 9731696194311141018*z^30-560705703599568993256*z^42+609762839190936066952*z^44-\ 560705703599568993256*z^46-9731696194311141018*z^58+29344269114453776314*z^56-\ 74401647941650743834*z^54+158926888189225092583*z^52+2707019346092026205*z^60-\ 277995647377300*z^70+2580827001497316*z^68-3769610542*z^78+29344269114453776314 *z^32-286410081381050521705*z^38+435914406908675800555*z^40-629552954323660219* z^62+90896319807*z^76-1675716610345*z^74+24181191141231*z^72)/(-1+z^2)/(1+ 9818854708886571024*z^28-2180802495807732066*z^26-382*z^2+401839270384760735*z^ 24-61122265643062708*z^22+55186*z^4-4383520*z^6+223311683*z^8-7929560652*z^10+ 206667057028*z^12-4093667299662*z^14-773728587590500*z^18+63177016812945*z^16-\ 1223114915642305883378*z^50+1892137563242193200575*z^48+7625171998953046*z^20+ 663717709704757964922*z^36-302158729287687420216*z^34-61122265643062708*z^66+ 223311683*z^80+z^88+55186*z^84-382*z^86-4383520*z^82+401839270384760735*z^64-\ 36816445849696134620*z^30-2458064966498592961352*z^42+2682044262168268474420*z^ 44-2458064966498592961352*z^46-36816445849696134620*z^58+115291973643578378205* z^56-302158729287687420216*z^54+663717709704757964922*z^52+9818854708886571024* z^60-773728587590500*z^70+7625171998953046*z^68-7929560652*z^78+ 115291973643578378205*z^32-1223114915642305883378*z^38+1892137563242193200575*z ^40-2180802495807732066*z^62+206667057028*z^76-4093667299662*z^74+ 63177016812945*z^72) The first , 40, terms are: [0, 92, 0, 15305, 0, 2680723, 0, 472618992, 0, 83420356719, 0, 14727934615391, 0, 2600387003372904, 0, 459135444568195291, 0, 81067241794155082501, 0, 14313651241542050418196, 0, 2527292954733236788415761, 0, 446232065052523115020795669, 0, 78789069318386464224167631252, 0, 13911410592612226845761389971537, 0, 2456271493239736074246407135940159, 0, 433692155843742434842350698889416568, 0, 76574957848352824119368535429083525803, 0, 13520475504619574775146260769929277832051, 0, 2387245948398887434426017555952087350655152, 0, 421504644286389089289579929719720529576189279] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 5}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 183740873 z - 724962522 z - 166 z + 1922655543 z 22 4 6 8 10 - 3447452220 z + 8831 z - 228192 z + 3371960 z - 30914788 z 12 14 18 16 + 183740873 z - 724962522 z - 3447452220 z + 1922655543 z 20 36 34 30 32 + 4187809168 z + 8831 z - 228192 z - 30914788 z + 3371960 z 38 40 / 2 40 38 36 34 - 166 z + z ) / ((-1 + z ) (z - 257 z + 16929 z - 514754 z / 32 30 28 26 + 8794190 z - 92384266 z + 624224151 z - 2768849663 z 24 22 20 18 + 8101655367 z - 15549780532 z + 19369289572 z - 15549780532 z 16 14 12 10 + 8101655367 z - 2768849663 z + 624224151 z - 92384266 z 8 6 4 2 + 8794190 z - 514754 z + 16929 z - 257 z + 1)) And in Maple-input format, it is: -(1+183740873*z^28-724962522*z^26-166*z^2+1922655543*z^24-3447452220*z^22+8831* z^4-228192*z^6+3371960*z^8-30914788*z^10+183740873*z^12-724962522*z^14-\ 3447452220*z^18+1922655543*z^16+4187809168*z^20+8831*z^36-228192*z^34-30914788* z^30+3371960*z^32-166*z^38+z^40)/(-1+z^2)/(z^40-257*z^38+16929*z^36-514754*z^34 +8794190*z^32-92384266*z^30+624224151*z^28-2768849663*z^26+8101655367*z^24-\ 15549780532*z^22+19369289572*z^20-15549780532*z^18+8101655367*z^16-2768849663*z ^14+624224151*z^12-92384266*z^10+8794190*z^8-514754*z^6+16929*z^4-257*z^2+1) The first , 40, terms are: [0, 92, 0, 15381, 0, 2690677, 0, 472834652, 0, 83141022337, 0, 14620429335809, 0, 2571052909487932, 0, 452129548570482901, 0, 79508750565470926581, 0, 13981925797881929208380, 0, 2458776544236803648178177, 0, 432385509096942754533535873, 0, 76036689452210515531531037180, 0, 13371350384006534724598898384693, 0, 2351404465149823410073520359403541, 0, 413503707549709559686418473465881340, 0, 72716250518174273629203728371970664065, 0, 12787438160482240403057940086028923637761, 0, 2248721208023361702122920632393482150512028, 0, 395446453617364051768662371741208252441525557] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 6}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 11105512947168 z - 12865148037852 z - 258 z 24 22 4 6 + 11105512947168 z - 7131990518386 z + 25183 z - 1230030 z 8 10 12 14 + 34493032 z - 603123330 z + 6939579714 z - 54523897712 z 18 16 50 48 - 1184374390462 z + 300436005246 z - 258 z + 25183 z 20 36 34 + 3391144124664 z + 300436005246 z - 1184374390462 z 30 42 44 46 52 - 7131990518386 z - 603123330 z + 34493032 z - 1230030 z + z 32 38 40 / 2 + 3391144124664 z - 54523897712 z + 6939579714 z ) / ((-1 + z ) (1 / 28 26 2 24 + 50879908329122 z - 59633060188250 z - 367 z + 50879908329122 z 22 4 6 8 - 31580602148426 z + 44373 z - 2542254 z + 81450166 z 10 12 14 18 - 1603517726 z + 20572223400 z - 178804926876 z - 4631447554138 z 16 50 48 20 + 1081164010448 z - 367 z + 44373 z + 14227239268546 z 36 34 30 + 1081164010448 z - 4631447554138 z - 31580602148426 z 42 44 46 52 32 - 1603517726 z + 81450166 z - 2542254 z + z + 14227239268546 z 38 40 - 178804926876 z + 20572223400 z )) And in Maple-input format, it is: -(1+11105512947168*z^28-12865148037852*z^26-258*z^2+11105512947168*z^24-\ 7131990518386*z^22+25183*z^4-1230030*z^6+34493032*z^8-603123330*z^10+6939579714 *z^12-54523897712*z^14-1184374390462*z^18+300436005246*z^16-258*z^50+25183*z^48 +3391144124664*z^20+300436005246*z^36-1184374390462*z^34-7131990518386*z^30-\ 603123330*z^42+34493032*z^44-1230030*z^46+z^52+3391144124664*z^32-54523897712*z ^38+6939579714*z^40)/(-1+z^2)/(1+50879908329122*z^28-59633060188250*z^26-367*z^ 2+50879908329122*z^24-31580602148426*z^22+44373*z^4-2542254*z^6+81450166*z^8-\ 1603517726*z^10+20572223400*z^12-178804926876*z^14-4631447554138*z^18+ 1081164010448*z^16-367*z^50+44373*z^48+14227239268546*z^20+1081164010448*z^36-\ 4631447554138*z^34-31580602148426*z^30-1603517726*z^42+81450166*z^44-2542254*z^ 46+z^52+14227239268546*z^32-178804926876*z^38+20572223400*z^40) The first , 40, terms are: [0, 110, 0, 20923, 0, 4134861, 0, 820563410, 0, 162936328823, 0, 32356642050359, 0, 6425627517688562, 0, 1276053258966328973, 0, 253409138443098557083, 0, 50324073629361611035598, 0, 9993769227898757673534625, 0, 1984645058626485278654360993, 0, 394127173338438745706333019278, 0, 78269022559386716120031536274075, 0, 15543307612763414943052660489624141, 0, 3086718137814351846634527592359620402, 0, 612985929374075525584739083476364862071, 0, 121731798252235074996017686136089980238007, 0, 24174503843668537594060994452352215059915410, 0, 4800772226146224420391724196248607766628922509] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 3989647666 z - 9028047855 z - 173 z + 14683196619 z 22 4 6 8 10 - 17257333912 z + 9678 z - 265759 z + 4207931 z - 41918132 z 12 14 18 16 + 276945657 z - 1255112773 z - 9028047855 z + 3989647666 z 20 36 34 30 42 + 14683196619 z + 4207931 z - 41918132 z - 1255112773 z - 173 z 44 32 38 40 / 2 + z + 276945657 z - 265759 z + 9678 z ) / ((-1 + z ) ( / 26 32 34 28 -39584811856 z + 921393351 z - 123692286 z + 16262579031 z 30 42 36 38 40 - 4657803322 z - 276 z + 10927177 z - 602466 z + 18857 z + 1 2 6 4 8 16 - 276 z - 602466 z + 18857 z + 10927177 z + 16262579031 z 14 12 10 24 - 4657803322 z + 921393351 z - 123692286 z + 67452454127 z 22 20 18 44 - 80562635780 z + 67452454127 z - 39584811856 z + z )) And in Maple-input format, it is: -(1+3989647666*z^28-9028047855*z^26-173*z^2+14683196619*z^24-17257333912*z^22+ 9678*z^4-265759*z^6+4207931*z^8-41918132*z^10+276945657*z^12-1255112773*z^14-\ 9028047855*z^18+3989647666*z^16+14683196619*z^20+4207931*z^36-41918132*z^34-\ 1255112773*z^30-173*z^42+z^44+276945657*z^32-265759*z^38+9678*z^40)/(-1+z^2)/(-\ 39584811856*z^26+921393351*z^32-123692286*z^34+16262579031*z^28-4657803322*z^30 -276*z^42+10927177*z^36-602466*z^38+18857*z^40+1-276*z^2-602466*z^6+18857*z^4+ 10927177*z^8+16262579031*z^16-4657803322*z^14+921393351*z^12-123692286*z^10+ 67452454127*z^24-80562635780*z^22+67452454127*z^20-39584811856*z^18+z^44) The first , 40, terms are: [0, 104, 0, 19353, 0, 3726513, 0, 719259032, 0, 138853461113, 0, 26806293008937, 0, 5175087632660088, 0, 999076518204788641, 0, 192876717007011617769, 0, 37235814655650465606152, 0, 7188560210751655440845457, 0, 1387787493923947364955533297, 0, 267919315110919354505383410632, 0, 51723163469770024486400837998537, 0, 9985415341230949022629069743409729, 0, 1927734362094194284738436657196100024, 0, 372158757929102189235029892084863814985, 0, 71847109138455302731127889744135395305049, 0, 13870443679136772051122508699431839435095128, 0, 2677758509188662945657868641667348048441522961] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 21399810288164 z - 18888355798328 z - 244 z 24 22 4 6 + 12972453289724 z - 6907277316332 z + 22118 z - 991287 z 8 10 12 14 + 25640930 z - 421617361 z + 4689984118 z - 36801214680 z 18 16 50 48 - 887713621540 z + 209786952823 z - 991287 z + 25640930 z 20 36 34 + 2833709526232 z + 2833709526232 z - 6907277316332 z 30 42 44 46 - 18888355798328 z - 36801214680 z + 4689984118 z - 421617361 z 56 54 52 32 38 + z - 244 z + 22118 z + 12972453289724 z - 887713621540 z 40 / 28 26 + 209786952823 z ) / (-1 - 190913635827594 z + 148601359816718 z / 2 24 22 4 6 + 363 z - 89864005193966 z + 42061005057438 z - 40847 z + 2134991 z 8 10 12 14 - 63122133 z + 1179532005 z - 14902096943 z + 132963665007 z 18 16 50 48 + 4162771473409 z - 863106680067 z + 63122133 z - 1179532005 z 20 36 34 - 15147756189618 z - 42061005057438 z + 89864005193966 z 30 42 44 + 190913635827594 z + 863106680067 z - 132963665007 z 46 58 56 54 52 + 14902096943 z + z - 363 z + 40847 z - 2134991 z 32 38 40 - 148601359816718 z + 15147756189618 z - 4162771473409 z ) And in Maple-input format, it is: -(1+21399810288164*z^28-18888355798328*z^26-244*z^2+12972453289724*z^24-\ 6907277316332*z^22+22118*z^4-991287*z^6+25640930*z^8-421617361*z^10+4689984118* z^12-36801214680*z^14-887713621540*z^18+209786952823*z^16-991287*z^50+25640930* z^48+2833709526232*z^20+2833709526232*z^36-6907277316332*z^34-18888355798328*z^ 30-36801214680*z^42+4689984118*z^44-421617361*z^46+z^56-244*z^54+22118*z^52+ 12972453289724*z^32-887713621540*z^38+209786952823*z^40)/(-1-190913635827594*z^ 28+148601359816718*z^26+363*z^2-89864005193966*z^24+42061005057438*z^22-40847*z ^4+2134991*z^6-63122133*z^8+1179532005*z^10-14902096943*z^12+132963665007*z^14+ 4162771473409*z^18-863106680067*z^16+63122133*z^50-1179532005*z^48-\ 15147756189618*z^20-42061005057438*z^36+89864005193966*z^34+190913635827594*z^ 30+863106680067*z^42-132963665007*z^44+14902096943*z^46+z^58-363*z^56+40847*z^ 54-2134991*z^52-148601359816718*z^32+15147756189618*z^38-4162771473409*z^40) The first , 40, terms are: [0, 119, 0, 24468, 0, 5164795, 0, 1091958915, 0, 230900045385, 0, 48825941361121, 0, 10324735741245308, 0, 2183271176077909397, 0, 461675186224582262003, 0, 97625982801271354120539, 0, 20644022092809799889441245, 0, 4365391661479651970381797212, 0, 923107150099262181908746553129, 0, 195200540347799497070000604984193, 0, 41277170213723836922580609739613275, 0, 8728483937228680306365712255717223811, 0, 1845728073158091865994966078938679412212, 0, 390298263084929722627328278738793314711999, 0, 82532598589541714136149675497338999028110473, 0, 17452370338794526979329780100931409515052304697] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 672829839604888 z - 382935556566258 z - 235 z 24 22 4 6 + 173297367365706 z - 62107896373364 z + 21206 z - 1020229 z 8 10 12 14 + 30245750 z - 600161473 z + 8402126146 z - 86002577283 z 18 16 50 - 3866665100252 z + 659981225509 z - 86002577283 z 48 20 36 + 659981225509 z + 17528873950368 z + 672829839604888 z 34 64 30 42 - 942577142116490 z + z - 942577142116490 z - 62107896373364 z 44 46 58 56 + 17528873950368 z - 3866665100252 z - 1020229 z + 30245750 z 54 52 60 32 - 600161473 z + 8402126146 z + 21206 z + 1054498739054612 z 38 40 62 / - 382935556566258 z + 173297367365706 z - 235 z ) / (-1 / 28 26 2 - 4545394315961130 z + 2321810154533130 z + 331 z 24 22 4 6 - 943908915847598 z + 304121895243352 z - 36511 z + 2041447 z 8 10 12 14 - 68855466 z + 1538332794 z - 24107650679 z + 275264194535 z 18 16 50 + 15320254152545 z - 2351521542459 z + 2351521542459 z 48 20 36 - 15320254152545 z - 77203525813272 z - 7102691844615186 z 34 66 64 30 + 8874875496819692 z + z - 331 z + 7102691844615186 z 42 44 46 + 943908915847598 z - 304121895243352 z + 77203525813272 z 58 56 54 52 + 68855466 z - 1538332794 z + 24107650679 z - 275264194535 z 60 32 38 - 2041447 z - 8874875496819692 z + 4545394315961130 z 40 62 - 2321810154533130 z + 36511 z ) And in Maple-input format, it is: -(1+672829839604888*z^28-382935556566258*z^26-235*z^2+173297367365706*z^24-\ 62107896373364*z^22+21206*z^4-1020229*z^6+30245750*z^8-600161473*z^10+ 8402126146*z^12-86002577283*z^14-3866665100252*z^18+659981225509*z^16-\ 86002577283*z^50+659981225509*z^48+17528873950368*z^20+672829839604888*z^36-\ 942577142116490*z^34+z^64-942577142116490*z^30-62107896373364*z^42+ 17528873950368*z^44-3866665100252*z^46-1020229*z^58+30245750*z^56-600161473*z^ 54+8402126146*z^52+21206*z^60+1054498739054612*z^32-382935556566258*z^38+ 173297367365706*z^40-235*z^62)/(-1-4545394315961130*z^28+2321810154533130*z^26+ 331*z^2-943908915847598*z^24+304121895243352*z^22-36511*z^4+2041447*z^6-\ 68855466*z^8+1538332794*z^10-24107650679*z^12+275264194535*z^14+15320254152545* z^18-2351521542459*z^16+2351521542459*z^50-15320254152545*z^48-77203525813272*z ^20-7102691844615186*z^36+8874875496819692*z^34+z^66-331*z^64+7102691844615186* z^30+943908915847598*z^42-304121895243352*z^44+77203525813272*z^46+68855466*z^ 58-1538332794*z^56+24107650679*z^54-275264194535*z^52-2041447*z^60-\ 8874875496819692*z^32+4545394315961130*z^38-2321810154533130*z^40+36511*z^62) The first , 40, terms are: [0, 96, 0, 16471, 0, 2968063, 0, 538425368, 0, 97804568737, 0, 17771862991265, 0, 3229556338799288, 0, 586897497638651487, 0, 106655721421894447671, 0, 19382362931015485633728, 0, 3522325290715544439985089, 0, 640106514854363467515794689, 0, 116325531629701669563059620608, 0, 21139652693631813474090414147575, 0, 3841675259416461623804359592916191, 0, 698141498373510272208044850984453880, 0, 126872137519933605440178297484515104161, 0, 23056270565460461021138824106893319932449, 0, 4189979161574675800567488617280423016090072, 0, 761438209385676190497444754279775130475875455] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 18 16 14 12 10 8 f(z) = - (z - 167 z + 6463 z - 66503 z + 216798 z - 216798 z 6 4 2 / 20 18 16 + 66503 z - 6463 z + 167 z - 1) / (z - 317 z + 16046 z / 14 12 10 8 6 - 233529 z + 1276593 z - 2581112 z + 1276593 z - 233529 z 4 2 + 16046 z - 317 z + 1) And in Maple-input format, it is: -(z^18-167*z^16+6463*z^14-66503*z^12+216798*z^10-216798*z^8+66503*z^6-6463*z^4+ 167*z^2-1)/(z^20-317*z^18+16046*z^16-233529*z^14+1276593*z^12-2581112*z^10+ 1276593*z^8-233529*z^6+16046*z^4-317*z^2+1) The first , 40, terms are: [0, 150, 0, 37967, 0, 9795665, 0, 2529976878, 0, 653498700643, 0, 168802568521507, 0, 43602789769590678, 0, 11262884725029072617, 0, 2909276654006339207255, 0, 751485158262813598863678, 0, 194113524034551215882111281, 0, 50140790945745596493410948785, 0, 12951693753598645959403051302702, 0, 3345507079637902422015024385285991, 0, 864166327033779491366870177092910489, 0, 223219805847771427467298878476937018630, 0, 57659133622744969213439713256339291615299, 0, 14893730766851474300044603662417470311455619, 0, 3847147922943382725757294356645732178842428478, 0, 993743432904582528381534814699211608115232076385] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 366759 z + 5302004 z + 212 z - 42875381 z + 199855626 z 4 6 8 10 12 - 13243 z + 366759 z - 5302004 z + 42875381 z - 199855626 z 14 18 16 20 34 + 549086264 z + 905900702 z - 905900702 z - 549086264 z + z 30 32 / 10 6 24 + 13243 z - 212 z ) / (-169163776 z - 963766 z + 934163279 z / 34 32 8 16 14 - 288 z + 25920 z + 17423840 z + 6175128544 z - 3065304032 z 30 36 20 22 28 - 963766 z + z + 6175128544 z - 3065304032 z + 17423840 z + 1 18 4 2 12 26 - 7783614772 z + 25920 z - 288 z + 934163279 z - 169163776 z ) And in Maple-input format, it is: -(-1-366759*z^28+5302004*z^26+212*z^2-42875381*z^24+199855626*z^22-13243*z^4+ 366759*z^6-5302004*z^8+42875381*z^10-199855626*z^12+549086264*z^14+905900702*z^ 18-905900702*z^16-549086264*z^20+z^34+13243*z^30-212*z^32)/(-169163776*z^10-\ 963766*z^6+934163279*z^24-288*z^34+25920*z^32+17423840*z^8+6175128544*z^16-\ 3065304032*z^14-963766*z^30+z^36+6175128544*z^20-3065304032*z^22+17423840*z^28+ 1-7783614772*z^18+25920*z^4-288*z^2+934163279*z^12-169163776*z^26) The first , 40, terms are: [0, 76, 0, 9211, 0, 1279855, 0, 190973500, 0, 29505851581, 0, 4632764018341, 0, 732687829109788, 0, 116250224495242471, 0, 18470729310909342787, 0, 2936600089609786126828, 0, 467008010556769552583641, 0, 74277274067953049715301801, 0, 11814366484683207789871846828, 0, 1879208395100668375317386950963, 0, 298912340507139221808899047947127, 0, 47546075336262407740864848134068060, 0, 7562865008511797686837057741120518229, 0, 1202979963368195314975521699835802995213, 0, 191350940738929157831746772771492059728124, 0, 30437072643121706402232230619204275313439711] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3714896958561208288 z - 901046142750247944 z - 350 z 24 22 4 6 + 182122382825868821 z - 30472929734692814 z + 49812 z - 3911082 z 8 10 12 14 + 193888243 z - 6573679900 z + 160858727120 z - 2952461446420 z 18 16 50 - 467235566513226 z + 41823010476735 z - 334867375840357569260 z 48 20 + 501339392104569853758 z + 4186067880570484 z 36 34 + 190044718937985394168 z - 91495986056713905396 z 66 80 88 84 86 - 30472929734692814 z + 193888243 z + z + 49812 z - 350 z 82 64 30 - 3911082 z + 182122382825868821 z - 12831525567208691992 z 42 44 - 638414855123507735984 z + 691931219860895982752 z 46 58 - 638414855123507735984 z - 12831525567208691992 z 56 54 + 37290763488549496314 z - 91495986056713905396 z 52 60 + 190044718937985394168 z + 3714896958561208288 z 70 68 78 - 467235566513226 z + 4186067880570484 z - 6573679900 z 32 38 + 37290763488549496314 z - 334867375840357569260 z 40 62 76 + 501339392104569853758 z - 901046142750247944 z + 160858727120 z 74 72 / 2 - 2952461446420 z + 41823010476735 z ) / ((-1 + z ) (1 / 28 26 2 + 14468366821869250032 z - 3396672507257889840 z - 436 z 24 22 4 6 + 661405156712708085 z - 106055354065213524 z + 74746 z - 6879516 z 8 10 12 14 + 390923751 z - 14902048652 z + 403160996212 z - 8064680313500 z 18 16 50 - 1465155951263916 z + 123017105102307 z - 1445543157099094280344 z 48 20 + 2184414943135131787654 z + 13877782120391714 z 36 34 + 809765331010996467300 z - 383388194648215506328 z 66 80 88 84 86 - 106055354065213524 z + 390923751 z + z + 74746 z - 436 z 82 64 30 - 6879516 z + 661405156712708085 z - 51411058309973205904 z 42 44 - 2797286712244653913112 z + 3037441286906923491896 z 46 58 - 2797286712244653913112 z - 51411058309973205904 z 56 54 + 153088632618254739898 z - 383388194648215506328 z 52 60 + 809765331010996467300 z + 14468366821869250032 z 70 68 78 - 1465155951263916 z + 13877782120391714 z - 14902048652 z 32 38 + 153088632618254739898 z - 1445543157099094280344 z 40 62 76 + 2184414943135131787654 z - 3396672507257889840 z + 403160996212 z 74 72 - 8064680313500 z + 123017105102307 z )) And in Maple-input format, it is: -(1+3714896958561208288*z^28-901046142750247944*z^26-350*z^2+182122382825868821 *z^24-30472929734692814*z^22+49812*z^4-3911082*z^6+193888243*z^8-6573679900*z^ 10+160858727120*z^12-2952461446420*z^14-467235566513226*z^18+41823010476735*z^ 16-334867375840357569260*z^50+501339392104569853758*z^48+4186067880570484*z^20+ 190044718937985394168*z^36-91495986056713905396*z^34-30472929734692814*z^66+ 193888243*z^80+z^88+49812*z^84-350*z^86-3911082*z^82+182122382825868821*z^64-\ 12831525567208691992*z^30-638414855123507735984*z^42+691931219860895982752*z^44 -638414855123507735984*z^46-12831525567208691992*z^58+37290763488549496314*z^56 -91495986056713905396*z^54+190044718937985394168*z^52+3714896958561208288*z^60-\ 467235566513226*z^70+4186067880570484*z^68-6573679900*z^78+37290763488549496314 *z^32-334867375840357569260*z^38+501339392104569853758*z^40-901046142750247944* z^62+160858727120*z^76-2952461446420*z^74+41823010476735*z^72)/(-1+z^2)/(1+ 14468366821869250032*z^28-3396672507257889840*z^26-436*z^2+661405156712708085*z ^24-106055354065213524*z^22+74746*z^4-6879516*z^6+390923751*z^8-14902048652*z^ 10+403160996212*z^12-8064680313500*z^14-1465155951263916*z^18+123017105102307*z ^16-1445543157099094280344*z^50+2184414943135131787654*z^48+13877782120391714*z ^20+809765331010996467300*z^36-383388194648215506328*z^34-106055354065213524*z^ 66+390923751*z^80+z^88+74746*z^84-436*z^86-6879516*z^82+661405156712708085*z^64 -51411058309973205904*z^30-2797286712244653913112*z^42+3037441286906923491896*z ^44-2797286712244653913112*z^46-51411058309973205904*z^58+153088632618254739898 *z^56-383388194648215506328*z^54+809765331010996467300*z^52+ 14468366821869250032*z^60-1465155951263916*z^70+13877782120391714*z^68-\ 14902048652*z^78+153088632618254739898*z^32-1445543157099094280344*z^38+ 2184414943135131787654*z^40-3396672507257889840*z^62+403160996212*z^76-\ 8064680313500*z^74+123017105102307*z^72) The first , 40, terms are: [0, 87, 0, 12649, 0, 2029959, 0, 337220735, 0, 56823951969, 0, 9637475203975, 0, 1639620031035905, 0, 279373903492885281, 0, 47638539603346287319, 0, 8126367082144443104833, 0, 1386492778699344433419407, 0, 236581452568378227262590807, 0, 40370572544282740088572832393, 0, 6889057122821772693665577000327, 0, 1175601263343562507737845127657217, 0, 200614822789335030930403953004134145, 0, 34234766020124513911647373477677790439, 0, 5842145939477227850116370150610785360137, 0, 996960125817311827763876149937782942801079, 0, 170130959835375666824422753608741443182231535] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2153061373749345769 z - 525171639236094928 z - 320 z 24 22 4 6 + 106881337431747113 z - 18036564254495396 z + 41377 z - 2988942 z 8 10 12 14 + 138548964 z - 4458282118 z + 104813323819 z - 1865969525008 z 18 16 50 - 283410522721508 z + 25824528346411 z - 190933890217714728160 z 48 20 + 285443868045351995563 z + 2504452355307624 z 36 34 + 108576747904678995531 z - 52409295972823104706 z 66 80 88 84 86 - 18036564254495396 z + 138548964 z + z + 41377 z - 320 z 82 64 30 - 2988942 z + 106881337431747113 z - 7402222808060496170 z 42 44 - 363177533733593022072 z + 393508901168122478128 z 46 58 - 363177533733593022072 z - 7402222808060496170 z 56 54 + 21428896589553298588 z - 52409295972823104706 z 52 60 + 108576747904678995531 z + 2153061373749345769 z 70 68 78 - 283410522721508 z + 2504452355307624 z - 4458282118 z 32 38 + 21428896589553298588 z - 190933890217714728160 z 40 62 76 + 285443868045351995563 z - 525171639236094928 z + 104813323819 z 74 72 / 2 - 1865969525008 z + 25824528346411 z ) / ((-1 + z ) (1 / 28 26 2 + 8371688813458338547 z - 1971531649699331683 z - 407 z 24 22 4 6 + 385630821881017933 z - 62235744263233668 z + 63383 z - 5337592 z 8 10 12 14 + 281497522 z - 10111494588 z + 261354807233 z - 5052545376225 z 18 16 50 - 879077944525204 z + 75161573630927 z - 831504224210753350373 z 48 20 + 1256132858081360035235 z + 8219050040871228 z 36 34 + 466016111768283871605 z - 220795477324884101968 z 66 80 88 84 86 - 62235744263233668 z + 281497522 z + z + 63383 z - 407 z 82 64 30 - 5337592 z + 385630821881017933 z - 29683152896436510908 z 42 44 - 1608287181278894995880 z + 1746267862806695067720 z 46 58 - 1608287181278894995880 z - 29683152896436510908 z 56 54 + 88256171885492004270 z - 220795477324884101968 z 52 60 + 466016111768283871605 z + 8371688813458338547 z 70 68 78 - 879077944525204 z + 8219050040871228 z - 10111494588 z 32 38 + 88256171885492004270 z - 831504224210753350373 z 40 62 76 + 1256132858081360035235 z - 1971531649699331683 z + 261354807233 z 74 72 - 5052545376225 z + 75161573630927 z )) And in Maple-input format, it is: -(1+2153061373749345769*z^28-525171639236094928*z^26-320*z^2+106881337431747113 *z^24-18036564254495396*z^22+41377*z^4-2988942*z^6+138548964*z^8-4458282118*z^ 10+104813323819*z^12-1865969525008*z^14-283410522721508*z^18+25824528346411*z^ 16-190933890217714728160*z^50+285443868045351995563*z^48+2504452355307624*z^20+ 108576747904678995531*z^36-52409295972823104706*z^34-18036564254495396*z^66+ 138548964*z^80+z^88+41377*z^84-320*z^86-2988942*z^82+106881337431747113*z^64-\ 7402222808060496170*z^30-363177533733593022072*z^42+393508901168122478128*z^44-\ 363177533733593022072*z^46-7402222808060496170*z^58+21428896589553298588*z^56-\ 52409295972823104706*z^54+108576747904678995531*z^52+2153061373749345769*z^60-\ 283410522721508*z^70+2504452355307624*z^68-4458282118*z^78+21428896589553298588 *z^32-190933890217714728160*z^38+285443868045351995563*z^40-525171639236094928* z^62+104813323819*z^76-1865969525008*z^74+25824528346411*z^72)/(-1+z^2)/(1+ 8371688813458338547*z^28-1971531649699331683*z^26-407*z^2+385630821881017933*z^ 24-62235744263233668*z^22+63383*z^4-5337592*z^6+281497522*z^8-10111494588*z^10+ 261354807233*z^12-5052545376225*z^14-879077944525204*z^18+75161573630927*z^16-\ 831504224210753350373*z^50+1256132858081360035235*z^48+8219050040871228*z^20+ 466016111768283871605*z^36-220795477324884101968*z^34-62235744263233668*z^66+ 281497522*z^80+z^88+63383*z^84-407*z^86-5337592*z^82+385630821881017933*z^64-\ 29683152896436510908*z^30-1608287181278894995880*z^42+1746267862806695067720*z^ 44-1608287181278894995880*z^46-29683152896436510908*z^58+88256171885492004270*z ^56-220795477324884101968*z^54+466016111768283871605*z^52+8371688813458338547*z ^60-879077944525204*z^70+8219050040871228*z^68-10111494588*z^78+ 88256171885492004270*z^32-831504224210753350373*z^38+1256132858081360035235*z^ 40-1971531649699331683*z^62+261354807233*z^76-5052545376225*z^74+75161573630927 *z^72) The first , 40, terms are: [0, 88, 0, 13491, 0, 2302841, 0, 405967888, 0, 72294444599, 0, 12915266295551, 0, 2309623533400736, 0, 413160497208767753, 0, 73916421928880434067, 0, 13224440382155847432360, 0, 2366018941079860620536729, 0, 423312005858455337394410889, 0, 75736184302520243339107857800, 0, 13550221836366467511002197885363, 0, 2424317170916621355778265756916489, 0, 433743006139068376145112329081823232, 0, 77602468756110146682190656506192846287, 0, 13884127452804723558385243837419370865191, 0, 2484057510557131746984965896160159958144944, 0, 444431365172233904883332287916617509526183097] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 67589033617 z + 127686655462 z + 242 z - 175086683167 z 22 4 6 8 10 + 175086683167 z - 19559 z + 699833 z - 13750286 z + 165711387 z 12 14 18 16 - 1302876677 z + 6941833306 z + 67589033617 z - 25725163087 z 20 36 34 30 - 127686655462 z - 165711387 z + 1302876677 z + 25725163087 z 42 44 46 32 38 40 + 19559 z - 242 z + z - 6941833306 z + 13750286 z - 699833 z ) / 28 26 2 24 / (1 + 863700317668 z - 1371598490090 z - 334 z + 1599100838718 z / 22 4 6 8 10 - 1371598490090 z + 35732 z - 1619444 z + 38702452 z - 551518626 z 12 14 18 16 + 5038328448 z - 30906321366 z - 396722645612 z + 131452816900 z 48 20 36 34 + z + 863700317668 z + 5038328448 z - 30906321366 z 30 42 44 46 32 - 396722645612 z - 1619444 z + 35732 z - 334 z + 131452816900 z 38 40 - 551518626 z + 38702452 z ) And in Maple-input format, it is: -(-1-67589033617*z^28+127686655462*z^26+242*z^2-175086683167*z^24+175086683167* z^22-19559*z^4+699833*z^6-13750286*z^8+165711387*z^10-1302876677*z^12+ 6941833306*z^14+67589033617*z^18-25725163087*z^16-127686655462*z^20-165711387*z ^36+1302876677*z^34+25725163087*z^30+19559*z^42-242*z^44+z^46-6941833306*z^32+ 13750286*z^38-699833*z^40)/(1+863700317668*z^28-1371598490090*z^26-334*z^2+ 1599100838718*z^24-1371598490090*z^22+35732*z^4-1619444*z^6+38702452*z^8-\ 551518626*z^10+5038328448*z^12-30906321366*z^14-396722645612*z^18+131452816900* z^16+z^48+863700317668*z^20+5038328448*z^36-30906321366*z^34-396722645612*z^30-\ 1619444*z^42+35732*z^44-334*z^46+131452816900*z^32-551518626*z^38+38702452*z^40 ) The first , 40, terms are: [0, 92, 0, 14555, 0, 2493637, 0, 436832180, 0, 77195499911, 0, 13696405065303, 0, 2434752867473620, 0, 433219516657480389, 0, 77118430376136168475, 0, 13731066483182907743612, 0, 2445102357127250628007441, 0, 435424175468348497456164017, 0, 77542372938446536007454490812, 0, 13809279351387671249573739280603, 0, 2459266310805037376988996644230661, 0, 437966990698136915824671979336028948, 0, 77996985422124042971892426296760117367, 0, 13890393766872492518908185971092618973671, 0, 2473725146122820331171967228697775485480564, 0, 440543098113738195466933521414796139064122117] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3508710485075536989 z - 842594409277966045 z - 329 z 24 22 4 6 + 168299912527647494 z - 27776882018559156 z + 44354 z - 3359704 z 8 10 12 14 + 163567197 z - 5523655406 z + 135960410083 z - 2525628177244 z 18 16 50 - 412620559640142 z + 36328459260679 z - 324715812808102152486 z 48 20 + 487046413854384555148 z + 3757777761375077 z 36 34 + 183774945860975155652 z - 88140073083899160400 z 66 80 88 84 86 - 27776882018559156 z + 163567197 z + z + 44354 z - 329 z 82 64 30 - 3359704 z + 168299912527647494 z - 12218628744254516642 z 42 44 - 620878147595011785478 z + 673159456177015201720 z 46 58 - 620878147595011785478 z - 12218628744254516642 z 56 54 + 35741556230591667626 z - 88140073083899160400 z 52 60 + 183774945860975155652 z + 3508710485075536989 z 70 68 78 - 412620559640142 z + 3757777761375077 z - 5523655406 z 32 38 + 35741556230591667626 z - 324715812808102152486 z 40 62 76 + 487046413854384555148 z - 842594409277966045 z + 135960410083 z 74 72 / - 2525628177244 z + 36328459260679 z ) / (-1 / 28 26 2 - 16779873071362743934 z + 3757968198348652293 z + 421 z 24 22 4 6 - 699365880911136862 z + 107364842649093613 z - 67950 z + 5977009 z 8 10 12 14 - 331610274 z + 12586743785 z - 344394609990 z + 7047186038293 z 18 16 50 + 1368015523831973 z - 110827641011242 z + 3573334750640822901703 z 48 20 - 4916366070749309859651 z - 13476048408696958 z 36 34 - 1163857730639044449087 z + 519747040828397072171 z 66 80 90 88 84 + 699365880911136862 z - 12586743785 z + z - 421 z - 5977009 z 86 82 64 + 67950 z + 331610274 z - 3757968198348652293 z 30 42 + 62637379717855689622 z + 4916366070749309859651 z 44 46 - 5765376429220025832943 z + 5765376429220025832943 z 58 56 + 196447312766902700587 z - 519747040828397072171 z 54 52 + 1163857730639044449087 z - 2211583596689600382095 z 60 70 - 62637379717855689622 z + 13476048408696958 z 68 78 32 - 107364842649093613 z + 344394609990 z - 196447312766902700587 z 38 40 + 2211583596689600382095 z - 3573334750640822901703 z 62 76 74 + 16779873071362743934 z - 7047186038293 z + 110827641011242 z 72 - 1368015523831973 z ) And in Maple-input format, it is: -(1+3508710485075536989*z^28-842594409277966045*z^26-329*z^2+168299912527647494 *z^24-27776882018559156*z^22+44354*z^4-3359704*z^6+163567197*z^8-5523655406*z^ 10+135960410083*z^12-2525628177244*z^14-412620559640142*z^18+36328459260679*z^ 16-324715812808102152486*z^50+487046413854384555148*z^48+3757777761375077*z^20+ 183774945860975155652*z^36-88140073083899160400*z^34-27776882018559156*z^66+ 163567197*z^80+z^88+44354*z^84-329*z^86-3359704*z^82+168299912527647494*z^64-\ 12218628744254516642*z^30-620878147595011785478*z^42+673159456177015201720*z^44 -620878147595011785478*z^46-12218628744254516642*z^58+35741556230591667626*z^56 -88140073083899160400*z^54+183774945860975155652*z^52+3508710485075536989*z^60-\ 412620559640142*z^70+3757777761375077*z^68-5523655406*z^78+35741556230591667626 *z^32-324715812808102152486*z^38+487046413854384555148*z^40-842594409277966045* z^62+135960410083*z^76-2525628177244*z^74+36328459260679*z^72)/(-1-\ 16779873071362743934*z^28+3757968198348652293*z^26+421*z^2-699365880911136862*z ^24+107364842649093613*z^22-67950*z^4+5977009*z^6-331610274*z^8+12586743785*z^ 10-344394609990*z^12+7047186038293*z^14+1368015523831973*z^18-110827641011242*z ^16+3573334750640822901703*z^50-4916366070749309859651*z^48-13476048408696958*z ^20-1163857730639044449087*z^36+519747040828397072171*z^34+699365880911136862*z ^66-12586743785*z^80+z^90-421*z^88-5977009*z^84+67950*z^86+331610274*z^82-\ 3757968198348652293*z^64+62637379717855689622*z^30+4916366070749309859651*z^42-\ 5765376429220025832943*z^44+5765376429220025832943*z^46+196447312766902700587*z ^58-519747040828397072171*z^56+1163857730639044449087*z^54-\ 2211583596689600382095*z^52-62637379717855689622*z^60+13476048408696958*z^70-\ 107364842649093613*z^68+344394609990*z^78-196447312766902700587*z^32+ 2211583596689600382095*z^38-3573334750640822901703*z^40+16779873071362743934*z^ 62-7047186038293*z^76+110827641011242*z^74-1368015523831973*z^72) The first , 40, terms are: [0, 92, 0, 15136, 0, 2738161, 0, 506116332, 0, 94039887217, 0, 17496493820455, 0, 3256423321633008, 0, 606136956098221241, 0, 112826576245203629143, 0, 21001724034881552501216, 0, 3909302407379568141946371, 0, 727685628860384212333432704, 0, 135452924670007740708198853872, 0, 25213491559838574086337780480923, 0, 4693292259370084272981051993918995, 0, 873619277932744723378930863648179112, 0, 162617327273468903573634509852229621264, 0, 30269930852958207416010533678734628666843, 0, 5634508506906970791714144832696419025283752, 0, 1048819248022125009694812172993710268793270183] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 3716595586976286 z + 1904660561559678 z + 283 z 24 22 4 6 - 777356173063490 z + 251542240743562 z - 30270 z + 1697210 z 8 10 12 14 - 57640458 z + 1293281846 z - 20288476574 z + 231322949386 z 18 16 50 + 12784501472926 z - 1970114989298 z + 1970114989298 z 48 20 36 - 12784501472926 z - 64142559673134 z - 5793906022215050 z 34 66 64 30 + 7230542901139796 z + z - 283 z + 5793906022215050 z 42 44 46 + 777356173063490 z - 251542240743562 z + 64142559673134 z 58 56 54 52 + 57640458 z - 1293281846 z + 20288476574 z - 231322949386 z 60 32 38 - 1697210 z - 7230542901139796 z + 3716595586976286 z 40 62 / 28 - 1904660561559678 z + 30270 z ) / (1 + 25083752254661276 z / 26 2 24 - 11600736720983336 z - 386 z + 4279973007031820 z 22 4 6 8 - 1253596827718716 z + 51497 z - 3434844 z + 135026500 z 10 12 14 18 - 3445148888 z + 60725382164 z - 771672971252 z - 52275518970536 z 16 50 48 + 7287040590700 z - 52275518970536 z + 289565867197084 z 20 36 34 + 289565867197084 z + 60286846008325326 z - 67249344077835180 z 66 64 30 42 - 386 z + 51497 z - 43415428817337972 z - 11600736720983336 z 44 46 58 + 4279973007031820 z - 1253596827718716 z - 3445148888 z 56 54 52 60 + 60725382164 z - 771672971252 z + 7287040590700 z + 135026500 z 68 32 38 + z + 60286846008325326 z - 43415428817337972 z 40 62 + 25083752254661276 z - 3434844 z ) And in Maple-input format, it is: -(-1-3716595586976286*z^28+1904660561559678*z^26+283*z^2-777356173063490*z^24+ 251542240743562*z^22-30270*z^4+1697210*z^6-57640458*z^8+1293281846*z^10-\ 20288476574*z^12+231322949386*z^14+12784501472926*z^18-1970114989298*z^16+ 1970114989298*z^50-12784501472926*z^48-64142559673134*z^20-5793906022215050*z^ 36+7230542901139796*z^34+z^66-283*z^64+5793906022215050*z^30+777356173063490*z^ 42-251542240743562*z^44+64142559673134*z^46+57640458*z^58-1293281846*z^56+ 20288476574*z^54-231322949386*z^52-1697210*z^60-7230542901139796*z^32+ 3716595586976286*z^38-1904660561559678*z^40+30270*z^62)/(1+25083752254661276*z^ 28-11600736720983336*z^26-386*z^2+4279973007031820*z^24-1253596827718716*z^22+ 51497*z^4-3434844*z^6+135026500*z^8-3445148888*z^10+60725382164*z^12-\ 771672971252*z^14-52275518970536*z^18+7287040590700*z^16-52275518970536*z^50+ 289565867197084*z^48+289565867197084*z^20+60286846008325326*z^36-\ 67249344077835180*z^34-386*z^66+51497*z^64-43415428817337972*z^30-\ 11600736720983336*z^42+4279973007031820*z^44-1253596827718716*z^46-3445148888*z ^58+60725382164*z^56-771672971252*z^54+7287040590700*z^52+135026500*z^60+z^68+ 60286846008325326*z^32-43415428817337972*z^38+25083752254661276*z^40-3434844*z^ 62) The first , 40, terms are: [0, 103, 0, 18531, 0, 3586409, 0, 706465857, 0, 139901748235, 0, 27752195374351, 0, 5508294530627001, 0, 1093500528415178441, 0, 217094261687123300735, 0, 43100958510379353760635, 0, 8557139612588576101442449, 0, 1698913827840979122518889241, 0, 337298520228638045399735626099, 0, 66966505797764540886172811357815, 0, 13295383881933701991146737491663601, 0, 2639636575120253183647378710167609105, 0, 524067705211344978720890289589356742999, 0, 104047262866275579729944903662915595203155, 0, 20657317394317521606692434548190280951655353, 0, 4101258892763231635415464713812280958794257649] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 26 2 24 22 4 6 f(z) = - (-1 + z + 224 z - 224 z + 15151 z - 15151 z + 371263 z 8 10 12 14 18 - 3861753 z + 18598233 z - 42020983 z + 42020983 z + 3861753 z 16 20 / 16 24 - 18598233 z - 371263 z ) / (262661977 z + 30456 z / 14 28 12 26 10 - 378447068 z + z + 262661977 z - 312 z - 88853448 z 22 8 20 6 18 - 1092500 z + 14646193 z + 14646193 z - 1092500 z - 88853448 z 4 2 + 30456 z - 312 z + 1) And in Maple-input format, it is: -(-1+z^26+224*z^2-224*z^24+15151*z^22-15151*z^4+371263*z^6-3861753*z^8+18598233 *z^10-42020983*z^12+42020983*z^14+3861753*z^18-18598233*z^16-371263*z^20)/( 262661977*z^16+30456*z^24-378447068*z^14+z^28+262661977*z^12-312*z^26-88853448* z^10-1092500*z^22+14646193*z^8+14646193*z^20-1092500*z^6-88853448*z^18+30456*z^ 4-312*z^2+1) The first , 40, terms are: [0, 88, 0, 12151, 0, 1832221, 0, 286937656, 0, 45778783627, 0, 7375341254275, 0, 1194569044403992, 0, 194052566044543717, 0, 31574699930306352799, 0, 5142280361736714877240, 0, 837902359677287382602905, 0, 136569695468082243418052137, 0, 22263014116708038256559127928, 0, 3629542257422053816524219091855, 0, 591753827994239223079056387181333, 0, 96481083877590316166332201587499864, 0, 15730766640183192865401823862197166803, 0, 2564845971495857466770856894510984755131, 0, 418191058837967161543292061912260056238584, 0, 68185077843462355347246786269165208653210221] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6441540609494563799 z - 1505226860160579903 z - 355 z 24 22 4 6 + 292165129757030349 z - 46814416307027108 z + 51739 z - 4191268 z 8 10 12 14 + 215611938 z - 7621487680 z + 195107103725 z - 3754338099381 z 18 16 50 - 654212182645172 z + 55807954885423 z - 659223056124885072361 z 48 20 + 998852521177787173507 z + 6144530404034940 z 36 34 + 367939309952385864737 z - 173424066810693211476 z 66 80 88 84 86 - 46814416307027108 z + 215611938 z + z + 51739 z - 355 z 82 64 30 - 4191268 z + 292165129757030349 z - 23011535836607891016 z 42 44 - 1281191825119794908968 z + 1391950912116022922312 z 46 58 - 1281191825119794908968 z - 23011535836607891016 z 56 54 + 68895989682046066046 z - 173424066810693211476 z 52 60 + 367939309952385864737 z + 6441540609494563799 z 70 68 78 - 654212182645172 z + 6144530404034940 z - 7621487680 z 32 38 + 68895989682046066046 z - 659223056124885072361 z 40 62 76 + 998852521177787173507 z - 1505226860160579903 z + 195107103725 z 74 72 / 2 - 3754338099381 z + 55807954885423 z ) / ((-1 + z ) (1 / 28 26 2 + 25175300885887188548 z - 5656373535690250410 z - 450 z 24 22 4 6 + 1050916974390657163 z - 160421756636670704 z + 78914 z - 7420224 z 8 10 12 14 + 432406111 z - 17012304064 z + 478201010984 z - 9994082126190 z 18 16 50 - 2002235498984752 z + 159902284110909 z - 2935675323662278010662 z 48 20 + 4503389015350140173591 z + 19956153462878074 z 36 34 + 1610725434162135604018 z - 742879357356708778112 z 66 80 88 84 86 - 160421756636670704 z + 432406111 z + z + 78914 z - 450 z 82 64 30 - 7420224 z + 1050916974390657163 z - 93130028737664881536 z 42 44 - 5819607939840807091936 z + 6338524680024186686188 z 46 58 - 5819607939840807091936 z - 93130028737664881536 z 56 54 + 287488333275267633041 z - 742879357356708778112 z 52 60 + 1610725434162135604018 z + 25175300885887188548 z 70 68 78 - 2002235498984752 z + 19956153462878074 z - 17012304064 z 32 38 + 287488333275267633041 z - 2935675323662278010662 z 40 62 76 + 4503389015350140173591 z - 5656373535690250410 z + 478201010984 z 74 72 - 9994082126190 z + 159902284110909 z )) And in Maple-input format, it is: -(1+6441540609494563799*z^28-1505226860160579903*z^26-355*z^2+ 292165129757030349*z^24-46814416307027108*z^22+51739*z^4-4191268*z^6+215611938* z^8-7621487680*z^10+195107103725*z^12-3754338099381*z^14-654212182645172*z^18+ 55807954885423*z^16-659223056124885072361*z^50+998852521177787173507*z^48+ 6144530404034940*z^20+367939309952385864737*z^36-173424066810693211476*z^34-\ 46814416307027108*z^66+215611938*z^80+z^88+51739*z^84-355*z^86-4191268*z^82+ 292165129757030349*z^64-23011535836607891016*z^30-1281191825119794908968*z^42+ 1391950912116022922312*z^44-1281191825119794908968*z^46-23011535836607891016*z^ 58+68895989682046066046*z^56-173424066810693211476*z^54+367939309952385864737*z ^52+6441540609494563799*z^60-654212182645172*z^70+6144530404034940*z^68-\ 7621487680*z^78+68895989682046066046*z^32-659223056124885072361*z^38+ 998852521177787173507*z^40-1505226860160579903*z^62+195107103725*z^76-\ 3754338099381*z^74+55807954885423*z^72)/(-1+z^2)/(1+25175300885887188548*z^28-\ 5656373535690250410*z^26-450*z^2+1050916974390657163*z^24-160421756636670704*z^ 22+78914*z^4-7420224*z^6+432406111*z^8-17012304064*z^10+478201010984*z^12-\ 9994082126190*z^14-2002235498984752*z^18+159902284110909*z^16-\ 2935675323662278010662*z^50+4503389015350140173591*z^48+19956153462878074*z^20+ 1610725434162135604018*z^36-742879357356708778112*z^34-160421756636670704*z^66+ 432406111*z^80+z^88+78914*z^84-450*z^86-7420224*z^82+1050916974390657163*z^64-\ 93130028737664881536*z^30-5819607939840807091936*z^42+6338524680024186686188*z^ 44-5819607939840807091936*z^46-93130028737664881536*z^58+287488333275267633041* z^56-742879357356708778112*z^54+1610725434162135604018*z^52+ 25175300885887188548*z^60-2002235498984752*z^70+19956153462878074*z^68-\ 17012304064*z^78+287488333275267633041*z^32-2935675323662278010662*z^38+ 4503389015350140173591*z^40-5656373535690250410*z^62+478201010984*z^76-\ 9994082126190*z^74+159902284110909*z^72) The first , 40, terms are: [0, 96, 0, 15671, 0, 2756547, 0, 495192304, 0, 89680018929, 0, 16299040348501, 0, 2967169154807112, 0, 540578398610697055, 0, 98522280233716348643, 0, 17959160901143358127896, 0, 3273962864922117581241845, 0, 596868525669005576841543853, 0, 108815765329380149191400897416, 0, 19838502301660165276543480248971, 0, 3616827640815232368091618557655207, 0, 659398015725114511495628398403527416, 0, 120217553189761497279596936098903666861, 0, 21917364791924331778862748530434270230873, 0, 3995847310761354392764073991176577343725440, 0, 728499871344442871442935175921309934409276539] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 5713417223096 z - 6594201228952 z - 244 z 24 22 4 6 + 5713417223096 z - 3710636571926 z + 21479 z - 954698 z 8 10 12 14 + 24666800 z - 401319318 z + 4333639282 z - 32240269312 z 18 16 50 48 - 645519506314 z + 169776285550 z - 244 z + 21479 z 20 36 34 + 1798347390864 z + 169776285550 z - 645519506314 z 30 42 44 46 52 - 3710636571926 z - 401319318 z + 24666800 z - 954698 z + z 32 38 40 / + 1798347390864 z - 32240269312 z + 4333639282 z ) / (-1 / 28 26 2 24 - 55634300515916 z + 55634300515916 z + 344 z - 42079379607224 z 22 4 6 8 + 23999172378076 z - 37900 z + 2040535 z - 62755604 z 10 12 14 18 + 1195587460 z - 14902913490 z + 126632887780 z + 3254744918462 z 16 50 48 20 - 756663750220 z + 37900 z - 2040535 z - 10257612085228 z 36 34 30 - 3254744918462 z + 10257612085228 z + 42079379607224 z 42 44 46 54 52 + 14902913490 z - 1195587460 z + 62755604 z + z - 344 z 32 38 40 - 23999172378076 z + 756663750220 z - 126632887780 z ) And in Maple-input format, it is: -(1+5713417223096*z^28-6594201228952*z^26-244*z^2+5713417223096*z^24-\ 3710636571926*z^22+21479*z^4-954698*z^6+24666800*z^8-401319318*z^10+4333639282* z^12-32240269312*z^14-645519506314*z^18+169776285550*z^16-244*z^50+21479*z^48+ 1798347390864*z^20+169776285550*z^36-645519506314*z^34-3710636571926*z^30-\ 401319318*z^42+24666800*z^44-954698*z^46+z^52+1798347390864*z^32-32240269312*z^ 38+4333639282*z^40)/(-1-55634300515916*z^28+55634300515916*z^26+344*z^2-\ 42079379607224*z^24+23999172378076*z^22-37900*z^4+2040535*z^6-62755604*z^8+ 1195587460*z^10-14902913490*z^12+126632887780*z^14+3254744918462*z^18-\ 756663750220*z^16+37900*z^50-2040535*z^48-10257612085228*z^20-3254744918462*z^ 36+10257612085228*z^34+42079379607224*z^30+14902913490*z^42-1195587460*z^44+ 62755604*z^46+z^54-344*z^52-23999172378076*z^32+756663750220*z^38-126632887780* z^40) The first , 40, terms are: [0, 100, 0, 17979, 0, 3480613, 0, 681891468, 0, 133860918799, 0, 26287488545087, 0, 5162662240790764, 0, 1013920888153545909, 0, 199129497515303147723, 0, 39108158997842944679620, 0, 7680671659478053845465777, 0, 1508450416756550350193392593, 0, 296253084947454488661414846980, 0, 58182814287959874669019748194091, 0, 11426851065929371956229067835523861, 0, 2244183731744637813263091740729723948, 0, 440747900961553049742050087816224721759, 0, 86560966223436721230211040271628903426799, 0, 17000196387083681460846767550245194878542668, 0, 3338764454794401768235740091428580021146041029] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 184355 z + 2002623 z + 182 z - 12580765 z + 48251646 z 4 6 8 10 12 - 8887 z + 184355 z - 2002623 z + 12580765 z - 48251646 z 14 18 16 20 34 + 116551571 z + 180444930 z - 180444930 z - 116551571 z + z 30 32 / 36 34 32 30 + 8887 z - 182 z ) / (z - 274 z + 18871 z - 514872 z / 28 26 24 22 + 7005037 z - 53529965 z + 246457645 z - 714519905 z 20 18 16 14 + 1340324485 z - 1650512377 z + 1340324485 z - 714519905 z 12 10 8 6 4 + 246457645 z - 53529965 z + 7005037 z - 514872 z + 18871 z 2 - 274 z + 1) And in Maple-input format, it is: -(-1-184355*z^28+2002623*z^26+182*z^2-12580765*z^24+48251646*z^22-8887*z^4+ 184355*z^6-2002623*z^8+12580765*z^10-48251646*z^12+116551571*z^14+180444930*z^ 18-180444930*z^16-116551571*z^20+z^34+8887*z^30-182*z^32)/(z^36-274*z^34+18871* z^32-514872*z^30+7005037*z^28-53529965*z^26+246457645*z^24-714519905*z^22+ 1340324485*z^20-1650512377*z^18+1340324485*z^16-714519905*z^14+246457645*z^12-\ 53529965*z^10+7005037*z^8-514872*z^6+18871*z^4-274*z^2+1) The first , 40, terms are: [0, 92, 0, 15224, 0, 2765761, 0, 512892220, 0, 95574689573, 0, 17830770624467, 0, 3327533633181712, 0, 621020009223482437, 0, 115903439847354222611, 0, 21631614272455208819252, 0, 4037216103467261068144759, 0, 753485982989568009810257464, 0, 140626894566177933983377485956, 0, 26245908003209230948912668352583, 0, 4898406465939557568337503758259991, 0, 914214357756550957814537402886057316, 0, 170624446529762255485414517482512319448, 0, 31844502887432772578201334475822032583079, 0, 5943300533933969525503228945394472641939764, 0, 1109228219440254033021140856941946460599370051] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 21855264508963424 z - 8837494982096996 z - 296 z 24 22 4 6 + 2901466859773716 z - 769634467768792 z + 33538 z - 1987976 z 8 10 12 14 + 72010441 z - 1746863836 z + 30092532192 z - 383019318364 z 18 16 50 - 27771753135128 z + 3704016773220 z - 769634467768792 z 48 20 36 + 2901466859773716 z + 163875210759272 z + 107980594095186604 z 34 66 64 - 97762731048584096 z - 1987976 z + 72010441 z 30 42 44 - 44044994545339428 z - 44044994545339428 z + 21855264508963424 z 46 58 56 - 8837494982096996 z - 383019318364 z + 3704016773220 z 54 52 60 70 - 27771753135128 z + 163875210759272 z + 30092532192 z - 296 z 68 32 38 + 33538 z + 72528373776857278 z - 97762731048584096 z 40 62 72 / + 72528373776857278 z - 1746863836 z + z ) / (-1 / 28 26 2 - 134810760081169584 z + 49631673988403992 z + 403 z 24 22 4 6 - 14843289178412380 z + 3587285113327220 z - 57088 z + 3984380 z 8 10 12 14 - 164978851 z + 4502873717 z - 86453310648 z + 1218985671808 z 18 16 50 + 107316674557084 z - 13006534265824 z + 14843289178412380 z 48 20 36 - 49631673988403992 z - 695765268436620 z - 978288854403492676 z 34 66 64 + 803150205146650898 z + 164978851 z - 4502873717 z 30 42 44 + 298623993792111064 z + 540998334616541778 z - 298623993792111064 z 46 58 56 + 134810760081169584 z + 13006534265824 z - 107316674557084 z 54 52 60 + 695765268436620 z - 3587285113327220 z - 1218985671808 z 70 68 32 + 57088 z - 3984380 z - 540998334616541778 z 38 40 62 74 + 978288854403492676 z - 803150205146650898 z + 86453310648 z + z 72 - 403 z ) And in Maple-input format, it is: -(1+21855264508963424*z^28-8837494982096996*z^26-296*z^2+2901466859773716*z^24-\ 769634467768792*z^22+33538*z^4-1987976*z^6+72010441*z^8-1746863836*z^10+ 30092532192*z^12-383019318364*z^14-27771753135128*z^18+3704016773220*z^16-\ 769634467768792*z^50+2901466859773716*z^48+163875210759272*z^20+ 107980594095186604*z^36-97762731048584096*z^34-1987976*z^66+72010441*z^64-\ 44044994545339428*z^30-44044994545339428*z^42+21855264508963424*z^44-\ 8837494982096996*z^46-383019318364*z^58+3704016773220*z^56-27771753135128*z^54+ 163875210759272*z^52+30092532192*z^60-296*z^70+33538*z^68+72528373776857278*z^ 32-97762731048584096*z^38+72528373776857278*z^40-1746863836*z^62+z^72)/(-1-\ 134810760081169584*z^28+49631673988403992*z^26+403*z^2-14843289178412380*z^24+ 3587285113327220*z^22-57088*z^4+3984380*z^6-164978851*z^8+4502873717*z^10-\ 86453310648*z^12+1218985671808*z^14+107316674557084*z^18-13006534265824*z^16+ 14843289178412380*z^50-49631673988403992*z^48-695765268436620*z^20-\ 978288854403492676*z^36+803150205146650898*z^34+164978851*z^66-4502873717*z^64+ 298623993792111064*z^30+540998334616541778*z^42-298623993792111064*z^44+ 134810760081169584*z^46+13006534265824*z^58-107316674557084*z^56+ 695765268436620*z^54-3587285113327220*z^52-1218985671808*z^60+57088*z^70-\ 3984380*z^68-540998334616541778*z^32+978288854403492676*z^38-803150205146650898 *z^40+86453310648*z^62+z^74-403*z^72) The first , 40, terms are: [0, 107, 0, 19571, 0, 3775101, 0, 737456705, 0, 144763660031, 0, 28477909156175, 0, 5607536690597261, 0, 1104647034582766869, 0, 217650418778477434551, 0, 42887788265743324186967, 0, 8451329163189763421277177, 0, 1665421458309440975232199301, 0, 328191102091346479149828467643, 0, 64674194892727760402879819408355, 0, 12744886548479205416609964281838393, 0, 2511546589999103353411926468269749193, 0, 494933270799430425859486268979931974707, 0, 97533121830958578159375208977200847838059, 0, 19220187980310087397679686105872040953028341, 0, 3787591645022499501497469575840573083147918857] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 541954881921074985729 z - 75261630662812407758 z - 377 z 24 22 4 6 + 8930254183343361307 z - 899535697160728194 z + 61090 z - 5789602 z 102 8 10 12 - 16556530508 z + 365605963 z - 16556530508 z + 563849297553 z 14 18 16 - 14930911742706 z - 5401117424741646 z + 315105663907363 z 50 48 - 127648287294616502416018216 z + 81054008497176126379973206 z 20 36 + 76316468727177967 z + 332118936366768789214064 z 34 66 - 82711153736191170490110 z - 45168057427224576282137004 z 80 100 90 + 17893455722789655090915 z + 563849297553 z - 899535697160728194 z 88 84 + 8930254183343361307 z + 541954881921074985729 z 94 86 96 - 5401117424741646 z - 75261630662812407758 z + 315105663907363 z 98 92 82 - 14930911742706 z + 76316468727177967 z - 3352434429129326528880 z 64 112 110 106 + 81054008497176126379973206 z + z - 377 z - 5789602 z 108 30 42 + 61090 z - 3352434429129326528880 z - 9454060054692680226555144 z 44 46 + 22075103281225909269211706 z - 45168057427224576282137004 z 58 56 - 214364665442366576371420088 z + 228704677839980822847687338 z 54 52 - 214364665442366576371420088 z + 176506475614725030704827178 z 60 70 + 176506475614725030704827178 z - 9454060054692680226555144 z 68 78 + 22075103281225909269211706 z - 82711153736191170490110 z 32 38 + 17893455722789655090915 z - 1161382665670183621891819 z 40 62 + 3544102972343736110078133 z - 127648287294616502416018216 z 76 74 + 332118936366768789214064 z - 1161382665670183621891819 z 72 104 / 2 + 3544102972343736110078133 z + 365605963 z ) / ((-1 + z ) (1 / 28 26 2 + 1846080368055385206298 z - 245703533390405867286 z - 477 z 24 22 4 + 27856854975820265640 z - 2672441756576862976 z + 90512 z 6 102 8 10 - 9689636 z - 33490134418 z + 677171258 z - 33490134418 z 12 14 18 + 1233848212368 z - 35089751137716 z - 14397368096389898 z 16 50 + 790651273401059 z - 577281876931044095139842914 z 48 20 + 362432884915145080459783539 z + 215180669994274747 z 36 34 + 1303800141932695069093682 z - 314660102163282301014366 z 66 80 - 199075570843719977572994218 z + 65790713802107918342153 z 100 90 88 + 1233848212368 z - 2672441756576862976 z + 27856854975820265640 z 84 94 + 1846080368055385206298 z - 14397368096389898 z 86 96 98 - 245703533390405867286 z + 790651273401059 z - 35089751137716 z 92 82 + 215180669994274747 z - 11880736668518404728816 z 64 112 110 106 + 362432884915145080459783539 z + z - 477 z - 9689636 z 108 30 + 90512 z - 11880736668518404728816 z 42 44 - 40121793536869567849378154 z + 95612119151947503468956634 z 46 58 - 199075570843719977572994218 z - 982204136879174718564879522 z 56 54 + 1049632414105154235662026215 z - 982204136879174718564879522 z 52 60 + 804774062716414588976429138 z + 804774062716414588976429138 z 70 68 - 40121793536869567849378154 z + 95612119151947503468956634 z 78 32 - 314660102163282301014366 z + 65790713802107918342153 z 38 40 - 4692063326278753970004377 z + 14695491901336721192094554 z 62 76 - 577281876931044095139842914 z + 1303800141932695069093682 z 74 72 - 4692063326278753970004377 z + 14695491901336721192094554 z 104 + 677171258 z )) And in Maple-input format, it is: -(1+541954881921074985729*z^28-75261630662812407758*z^26-377*z^2+ 8930254183343361307*z^24-899535697160728194*z^22+61090*z^4-5789602*z^6-\ 16556530508*z^102+365605963*z^8-16556530508*z^10+563849297553*z^12-\ 14930911742706*z^14-5401117424741646*z^18+315105663907363*z^16-\ 127648287294616502416018216*z^50+81054008497176126379973206*z^48+ 76316468727177967*z^20+332118936366768789214064*z^36-82711153736191170490110*z^ 34-45168057427224576282137004*z^66+17893455722789655090915*z^80+563849297553*z^ 100-899535697160728194*z^90+8930254183343361307*z^88+541954881921074985729*z^84 -5401117424741646*z^94-75261630662812407758*z^86+315105663907363*z^96-\ 14930911742706*z^98+76316468727177967*z^92-3352434429129326528880*z^82+ 81054008497176126379973206*z^64+z^112-377*z^110-5789602*z^106+61090*z^108-\ 3352434429129326528880*z^30-9454060054692680226555144*z^42+ 22075103281225909269211706*z^44-45168057427224576282137004*z^46-\ 214364665442366576371420088*z^58+228704677839980822847687338*z^56-\ 214364665442366576371420088*z^54+176506475614725030704827178*z^52+ 176506475614725030704827178*z^60-9454060054692680226555144*z^70+ 22075103281225909269211706*z^68-82711153736191170490110*z^78+ 17893455722789655090915*z^32-1161382665670183621891819*z^38+ 3544102972343736110078133*z^40-127648287294616502416018216*z^62+ 332118936366768789214064*z^76-1161382665670183621891819*z^74+ 3544102972343736110078133*z^72+365605963*z^104)/(-1+z^2)/(1+ 1846080368055385206298*z^28-245703533390405867286*z^26-477*z^2+ 27856854975820265640*z^24-2672441756576862976*z^22+90512*z^4-9689636*z^6-\ 33490134418*z^102+677171258*z^8-33490134418*z^10+1233848212368*z^12-\ 35089751137716*z^14-14397368096389898*z^18+790651273401059*z^16-\ 577281876931044095139842914*z^50+362432884915145080459783539*z^48+ 215180669994274747*z^20+1303800141932695069093682*z^36-314660102163282301014366 *z^34-199075570843719977572994218*z^66+65790713802107918342153*z^80+ 1233848212368*z^100-2672441756576862976*z^90+27856854975820265640*z^88+ 1846080368055385206298*z^84-14397368096389898*z^94-245703533390405867286*z^86+ 790651273401059*z^96-35089751137716*z^98+215180669994274747*z^92-\ 11880736668518404728816*z^82+362432884915145080459783539*z^64+z^112-477*z^110-\ 9689636*z^106+90512*z^108-11880736668518404728816*z^30-\ 40121793536869567849378154*z^42+95612119151947503468956634*z^44-\ 199075570843719977572994218*z^46-982204136879174718564879522*z^58+ 1049632414105154235662026215*z^56-982204136879174718564879522*z^54+ 804774062716414588976429138*z^52+804774062716414588976429138*z^60-\ 40121793536869567849378154*z^70+95612119151947503468956634*z^68-\ 314660102163282301014366*z^78+65790713802107918342153*z^32-\ 4692063326278753970004377*z^38+14695491901336721192094554*z^40-\ 577281876931044095139842914*z^62+1303800141932695069093682*z^76-\ 4692063326278753970004377*z^74+14695491901336721192094554*z^72+677171258*z^104) The first , 40, terms are: [0, 101, 0, 18379, 0, 3585819, 0, 708274668, 0, 140272371279, 0, 27798422679139, 0, 5509827909553385, 0, 1092128849943831481, 0, 216478309890851984005, 0, 42909762784435735416001, 0, 8505466854680592795283675, 0, 1685932887286217127939083171, 0, 334181520122278800889716122828, 0, 66240649730795776051674770912087, 0, 13130060843067380311177487987560059, 0, 2602608799860024112391373179323974525, 0, 515882801128251146843838417981552992141, 0, 102257037066302736763504076291356820484437, 0, 20269141764136736968104432096683234885342381, 0, 4017700098136331082764971820375864714929109499] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6684365643488208 z - 3087538884729758 z - 267 z 24 22 4 6 + 1138860043755642 z - 334241688630912 z + 27016 z - 1462225 z 8 10 12 14 + 49304307 z - 1126661120 z + 18379746287 z - 221558369413 z 18 16 50 - 14192431855655 z + 2021151529640 z - 14192431855655 z 48 20 36 + 77665596625525 z + 77665596625525 z + 16105306202459694 z 34 66 64 30 - 17972233183648512 z - 267 z + 27016 z - 11585924794150618 z 42 44 46 - 3087538884729758 z + 1138860043755642 z - 334241688630912 z 58 56 54 52 - 1126661120 z + 18379746287 z - 221558369413 z + 2021151529640 z 60 68 32 38 + 49304307 z + z + 16105306202459694 z - 11585924794150618 z 40 62 / 28 + 6684365643488208 z - 1462225 z ) / (-1 - 43561820601219834 z / 26 2 24 + 18155526431489678 z + 371 z - 6050302162705402 z 22 4 6 8 + 1605915760341909 z - 46433 z + 2948263 z - 113574645 z 10 12 14 18 + 2921993591 z - 53209659715 z + 712325296089 z + 55853387618997 z 16 50 48 - 7195008970867 z + 337697760915631 z - 1605915760341909 z 20 36 34 - 337697760915631 z - 160951825941729718 z + 160951825941729718 z 66 64 30 42 + 46433 z - 2948263 z + 83804226654959574 z + 43561820601219834 z 44 46 58 - 18155526431489678 z + 6050302162705402 z + 53209659715 z 56 54 52 - 712325296089 z + 7195008970867 z - 55853387618997 z 60 70 68 32 - 2921993591 z + z - 371 z - 129508223299570818 z 38 40 62 + 129508223299570818 z - 83804226654959574 z + 113574645 z ) And in Maple-input format, it is: -(1+6684365643488208*z^28-3087538884729758*z^26-267*z^2+1138860043755642*z^24-\ 334241688630912*z^22+27016*z^4-1462225*z^6+49304307*z^8-1126661120*z^10+ 18379746287*z^12-221558369413*z^14-14192431855655*z^18+2021151529640*z^16-\ 14192431855655*z^50+77665596625525*z^48+77665596625525*z^20+16105306202459694*z ^36-17972233183648512*z^34-267*z^66+27016*z^64-11585924794150618*z^30-\ 3087538884729758*z^42+1138860043755642*z^44-334241688630912*z^46-1126661120*z^ 58+18379746287*z^56-221558369413*z^54+2021151529640*z^52+49304307*z^60+z^68+ 16105306202459694*z^32-11585924794150618*z^38+6684365643488208*z^40-1462225*z^ 62)/(-1-43561820601219834*z^28+18155526431489678*z^26+371*z^2-6050302162705402* z^24+1605915760341909*z^22-46433*z^4+2948263*z^6-113574645*z^8+2921993591*z^10-\ 53209659715*z^12+712325296089*z^14+55853387618997*z^18-7195008970867*z^16+ 337697760915631*z^50-1605915760341909*z^48-337697760915631*z^20-\ 160951825941729718*z^36+160951825941729718*z^34+46433*z^66-2948263*z^64+ 83804226654959574*z^30+43561820601219834*z^42-18155526431489678*z^44+ 6050302162705402*z^46+53209659715*z^58-712325296089*z^56+7195008970867*z^54-\ 55853387618997*z^52-2921993591*z^60+z^70-371*z^68-129508223299570818*z^32+ 129508223299570818*z^38-83804226654959574*z^40+113574645*z^62) The first , 40, terms are: [0, 104, 0, 19167, 0, 3767963, 0, 750281976, 0, 149889713429, 0, 29972358788141, 0, 5994962389536120, 0, 1199184800180094531, 0, 239880971294673988727, 0, 47985324504552843645608, 0, 9598910533052826720382441, 0, 1920152437905433280073189657, 0, 384104634971231217891184920744, 0, 76835763276117690813462270029767, 0, 15370120716970769898522856003989875, 0, 3074617884091444104004616237194321144, 0, 615042348701083622856472062891920230141, 0, 123032228752509655940341099792452786739269, 0, 24611198473574484604501837794952139283315320, 0, 4923190423092073035194194732782552286953683051] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4633732414056813509 z - 1076119292392433375 z - 323 z 24 22 4 6 + 207490687515816902 z - 33022097033296998 z + 43054 z - 3251682 z 8 10 12 14 + 159241463 z - 5453925024 z + 137113675731 z - 2616305124636 z 18 16 50 - 456237934028804 z + 38822119684679 z - 483822842890854479534 z 48 20 + 734392389577241554668 z + 4306816907863523 z 36 34 + 269365155564944413912 z - 126550236971475939452 z 66 80 88 84 86 - 33022097033296998 z + 159241463 z + z + 43054 z - 323 z 82 64 30 - 3251682 z + 207490687515816902 z - 16644985189312031394 z 42 44 - 942980719191284465870 z + 1024864001057255562872 z 46 58 - 942980719191284465870 z - 16644985189312031394 z 56 54 + 50073269944249612630 z - 126550236971475939452 z 52 60 + 269365155564944413912 z + 4633732414056813509 z 70 68 78 - 456237934028804 z + 4306816907863523 z - 5453925024 z 32 38 + 50073269944249612630 z - 483822842890854479534 z 40 62 76 + 734392389577241554668 z - 1076119292392433375 z + 137113675731 z 74 72 / 2 - 2616305124636 z + 38822119684679 z ) / ((-1 + z ) (1 / 28 26 2 + 18105183775617844840 z - 4014174752911794280 z - 430 z 24 22 4 6 + 735668468898249533 z - 110808842231669439 z + 68104 z - 5844647 z 8 10 12 14 + 318128679 z - 11956031962 z + 327030262598 z - 6746008014905 z 18 16 50 - 1353978175888252 z + 107644142747531 z - 2214282882910881511318 z 48 20 + 3412387190467187398369 z + 13619959854417130 z 36 34 + 1207206904447338396609 z - 552295540159215362418 z 66 80 88 84 86 - 110808842231669439 z + 318128679 z + z + 68104 z - 430 z 82 64 30 - 5844647 z + 735668468898249533 z - 67810415575514331662 z 42 44 - 4421973412374436741942 z + 4820731824428231382933 z 46 58 - 4421973412374436741942 z - 67810415575514331662 z 56 54 + 211674008086272522817 z - 552295540159215362418 z 52 60 + 1207206904447338396609 z + 18105183775617844840 z 70 68 78 - 1353978175888252 z + 13619959854417130 z - 11956031962 z 32 38 + 211674008086272522817 z - 2214282882910881511318 z 40 62 76 + 3412387190467187398369 z - 4014174752911794280 z + 327030262598 z 74 72 - 6746008014905 z + 107644142747531 z )) And in Maple-input format, it is: -(1+4633732414056813509*z^28-1076119292392433375*z^26-323*z^2+ 207490687515816902*z^24-33022097033296998*z^22+43054*z^4-3251682*z^6+159241463* z^8-5453925024*z^10+137113675731*z^12-2616305124636*z^14-456237934028804*z^18+ 38822119684679*z^16-483822842890854479534*z^50+734392389577241554668*z^48+ 4306816907863523*z^20+269365155564944413912*z^36-126550236971475939452*z^34-\ 33022097033296998*z^66+159241463*z^80+z^88+43054*z^84-323*z^86-3251682*z^82+ 207490687515816902*z^64-16644985189312031394*z^30-942980719191284465870*z^42+ 1024864001057255562872*z^44-942980719191284465870*z^46-16644985189312031394*z^ 58+50073269944249612630*z^56-126550236971475939452*z^54+269365155564944413912*z ^52+4633732414056813509*z^60-456237934028804*z^70+4306816907863523*z^68-\ 5453925024*z^78+50073269944249612630*z^32-483822842890854479534*z^38+ 734392389577241554668*z^40-1076119292392433375*z^62+137113675731*z^76-\ 2616305124636*z^74+38822119684679*z^72)/(-1+z^2)/(1+18105183775617844840*z^28-\ 4014174752911794280*z^26-430*z^2+735668468898249533*z^24-110808842231669439*z^ 22+68104*z^4-5844647*z^6+318128679*z^8-11956031962*z^10+327030262598*z^12-\ 6746008014905*z^14-1353978175888252*z^18+107644142747531*z^16-\ 2214282882910881511318*z^50+3412387190467187398369*z^48+13619959854417130*z^20+ 1207206904447338396609*z^36-552295540159215362418*z^34-110808842231669439*z^66+ 318128679*z^80+z^88+68104*z^84-430*z^86-5844647*z^82+735668468898249533*z^64-\ 67810415575514331662*z^30-4421973412374436741942*z^42+4820731824428231382933*z^ 44-4421973412374436741942*z^46-67810415575514331662*z^58+211674008086272522817* z^56-552295540159215362418*z^54+1207206904447338396609*z^52+ 18105183775617844840*z^60-1353978175888252*z^70+13619959854417130*z^68-\ 11956031962*z^78+211674008086272522817*z^32-2214282882910881511318*z^38+ 3412387190467187398369*z^40-4014174752911794280*z^62+327030262598*z^76-\ 6746008014905*z^74+107644142747531*z^72) The first , 40, terms are: [0, 108, 0, 21068, 0, 4339705, 0, 900383788, 0, 187049024635, 0, 38869088867579, 0, 8077594157488392, 0, 1678675715565273155, 0, 348861765607696266699, 0, 72500398547738603806564, 0, 15067025682658132612304513, 0, 3131228003852010959803434636, 0, 650731550264424164891987460924, 0, 135234978685398369140582358681441, 0, 28104522476228401660216690679664521, 0, 5840679619422171029958669581728059164, 0, 1213809572749437784768510298149606482844, 0, 252253808619457576214361164079255223401617, 0, 52423366392790496080175211758826916724460660, 0, 10894619823567872282269495270240124302382900611] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 450914978868291 z - 258096295710518 z - 242 z 24 22 4 6 + 117836640352954 z - 42770503067820 z + 21014 z - 954484 z 8 10 12 14 + 26677544 z - 501293719 z + 6687653876 z - 65669681048 z 18 16 50 - 2771161212877 z + 486675728768 z - 65669681048 z 48 20 36 + 486675728768 z + 12281803819088 z + 450914978868291 z 34 64 30 42 - 629715029753404 z + z - 629715029753404 z - 42770503067820 z 44 46 58 56 + 12281803819088 z - 2771161212877 z - 954484 z + 26677544 z 54 52 60 32 - 501293719 z + 6687653876 z + 21014 z + 703784708707200 z 38 40 62 / 2 - 258096295710518 z + 117836640352954 z - 242 z ) / ((-1 + z ) (1 / 28 26 2 + 2072184547328796 z - 1156452449733362 z - 346 z 24 22 4 6 + 510011628218115 z - 177242424262894 z + 37474 z - 2001014 z 8 10 12 14 + 63727971 z - 1338080686 z + 19670414222 z - 210515801030 z 18 16 50 - 10268371804958 z + 1684498216687 z - 210515801030 z 48 20 36 + 1684498216687 z + 48324116092090 z + 2072184547328796 z 34 64 30 42 - 2938885395596972 z + z - 2938885395596972 z - 177242424262894 z 44 46 58 56 + 48324116092090 z - 10268371804958 z - 2001014 z + 63727971 z 54 52 60 32 - 1338080686 z + 19670414222 z + 37474 z + 3301672267961245 z 38 40 62 - 1156452449733362 z + 510011628218115 z - 346 z )) And in Maple-input format, it is: -(1+450914978868291*z^28-258096295710518*z^26-242*z^2+117836640352954*z^24-\ 42770503067820*z^22+21014*z^4-954484*z^6+26677544*z^8-501293719*z^10+6687653876 *z^12-65669681048*z^14-2771161212877*z^18+486675728768*z^16-65669681048*z^50+ 486675728768*z^48+12281803819088*z^20+450914978868291*z^36-629715029753404*z^34 +z^64-629715029753404*z^30-42770503067820*z^42+12281803819088*z^44-\ 2771161212877*z^46-954484*z^58+26677544*z^56-501293719*z^54+6687653876*z^52+ 21014*z^60+703784708707200*z^32-258096295710518*z^38+117836640352954*z^40-242*z ^62)/(-1+z^2)/(1+2072184547328796*z^28-1156452449733362*z^26-346*z^2+ 510011628218115*z^24-177242424262894*z^22+37474*z^4-2001014*z^6+63727971*z^8-\ 1338080686*z^10+19670414222*z^12-210515801030*z^14-10268371804958*z^18+ 1684498216687*z^16-210515801030*z^50+1684498216687*z^48+48324116092090*z^20+ 2072184547328796*z^36-2938885395596972*z^34+z^64-2938885395596972*z^30-\ 177242424262894*z^42+48324116092090*z^44-10268371804958*z^46-2001014*z^58+ 63727971*z^56-1338080686*z^54+19670414222*z^52+37474*z^60+3301672267961245*z^32 -1156452449733362*z^38+510011628218115*z^40-346*z^62) The first , 40, terms are: [0, 105, 0, 19629, 0, 3924167, 0, 794306968, 0, 161224974421, 0, 32746418755211, 0, 6652207210857003, 0, 1351403846457625087, 0, 274542039828389630343, 0, 55774236724739914445819, 0, 11330750643412799701032819, 0, 2301885911874068354253719509, 0, 467637063010260691685328622904, 0, 95002286446231458910359735146055, 0, 19300083686925922542650222440326597, 0, 3920887006763159356454648158758145593, 0, 796543433252041820593735986840992006105, 0, 161820894094991089547590857351256887650313, 0, 32874543524924795273577628381443234703756377, 0, 6678591278454925630560243019045625908324986709] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 484557285274 z + 976835060918 z + 293 z - 1383765532014 z 22 4 6 8 10 + 1383765532014 z - 31283 z + 1582147 z - 42172815 z + 653746263 z 12 14 18 16 - 6307342209 z + 39638428277 z + 484557285274 z - 167325012426 z 20 36 34 30 - 976835060918 z - 653746263 z + 6307342209 z + 167325012426 z 42 44 46 32 38 40 + 31283 z - 293 z + z - 39638428277 z + 42172815 z - 1582147 z / 28 26 2 ) / (1 + 6800693019720 z - 11404080368672 z - 408 z / 24 22 4 6 + 13541299056300 z - 11404080368672 z + 55900 z - 3456360 z 8 10 12 14 + 110287594 z - 2020378872 z + 22901044956 z - 168882113864 z 18 16 48 20 - 2856980491168 z + 837636760895 z + z + 6800693019720 z 36 34 30 42 + 22901044956 z - 168882113864 z - 2856980491168 z - 3456360 z 44 46 32 38 40 + 55900 z - 408 z + 837636760895 z - 2020378872 z + 110287594 z ) And in Maple-input format, it is: -(-1-484557285274*z^28+976835060918*z^26+293*z^2-1383765532014*z^24+ 1383765532014*z^22-31283*z^4+1582147*z^6-42172815*z^8+653746263*z^10-6307342209 *z^12+39638428277*z^14+484557285274*z^18-167325012426*z^16-976835060918*z^20-\ 653746263*z^36+6307342209*z^34+167325012426*z^30+31283*z^42-293*z^44+z^46-\ 39638428277*z^32+42172815*z^38-1582147*z^40)/(1+6800693019720*z^28-\ 11404080368672*z^26-408*z^2+13541299056300*z^24-11404080368672*z^22+55900*z^4-\ 3456360*z^6+110287594*z^8-2020378872*z^10+22901044956*z^12-168882113864*z^14-\ 2856980491168*z^18+837636760895*z^16+z^48+6800693019720*z^20+22901044956*z^36-\ 168882113864*z^34-2856980491168*z^30-3456360*z^42+55900*z^44-408*z^46+ 837636760895*z^32-2020378872*z^38+110287594*z^40) The first , 40, terms are: [0, 115, 0, 22303, 0, 4545337, 0, 937126417, 0, 194033996215, 0, 40246830397291, 0, 8354514528187033, 0, 1734833325611261353, 0, 360295556907266654587, 0, 74832198339730876614151, 0, 15542849282477355371478433, 0, 3228332764560890204190983977, 0, 670545710574868628089535187151, 0, 139277049611705226454934509947235, 0, 28928852148960138141014857733146225, 0, 6008735049124118781517442790016535185, 0, 1248058663767563716631175734626253905795, 0, 259231029720891678145978617305514438259567, 0, 53844207405907296085387619896373908303574089, 0, 11183841406366727945108789815777351506497515393] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6128476688213522880 z - 1420225639239242536 z - 338 z 24 22 4 6 + 273158863239461365 z - 43338363825432850 z + 47304 z - 3735182 z 8 10 12 14 + 189818059 z - 6692622236 z + 171972024920 z - 3334286626196 z 18 16 50 - 592901952867166 z + 50044859662359 z - 645017832663602399028 z 48 20 + 979970415319715259470 z + 5629012517526776 z 36 34 + 358674083710447682112 z - 168264746859990446716 z 66 80 88 84 86 - 43338363825432850 z + 189818059 z + z + 47304 z - 338 z 82 64 30 - 3735182 z + 273158863239461365 z - 22056581288570610360 z 42 44 - 1259033760428342338288 z + 1368629271231140986832 z 46 58 - 1259033760428342338288 z - 22056581288570610360 z 56 54 + 66470632063633737434 z - 168264746859990446716 z 52 60 + 358674083710447682112 z + 6128476688213522880 z 70 68 78 - 592901952867166 z + 5629012517526776 z - 6692622236 z 32 38 + 66470632063633737434 z - 645017832663602399028 z 40 62 76 + 979970415319715259470 z - 1420225639239242536 z + 171972024920 z 74 72 / - 3334286626196 z + 50044859662359 z ) / (-1 / 28 26 2 - 29366761044732540128 z + 6301486376569599301 z + 449 z 24 22 4 6 - 1121412438642236229 z + 164438580184858674 z - 74874 z + 6757706 z 8 10 12 14 - 384917079 z + 15048555251 z - 425851403328 z + 9048560505456 z 18 16 50 + 1912041013455891 z - 148274459648127 z + 7471571074465922402374 z 48 20 - 10402698248522745935070 z - 19709827140251122 z 36 34 - 2336738203354170991476 z + 1014675044757844643402 z 66 80 90 88 84 + 1121412438642236229 z - 15048555251 z + z - 449 z - 6757706 z 86 82 64 + 74874 z + 384917079 z - 6301486376569599301 z 30 42 + 114102633889460474240 z + 10402698248522745935070 z 44 46 - 12272489569462824486128 z + 12272489569462824486128 z 58 56 + 371210323898064013418 z - 1014675044757844643402 z 54 52 + 2336738203354170991476 z - 4543626094324665812644 z 60 70 - 114102633889460474240 z + 19709827140251122 z 68 78 32 - 164438580184858674 z + 425851403328 z - 371210323898064013418 z 38 40 + 4543626094324665812644 z - 7471571074465922402374 z 62 76 74 + 29366761044732540128 z - 9048560505456 z + 148274459648127 z 72 - 1912041013455891 z ) And in Maple-input format, it is: -(1+6128476688213522880*z^28-1420225639239242536*z^26-338*z^2+ 273158863239461365*z^24-43338363825432850*z^22+47304*z^4-3735182*z^6+189818059* z^8-6692622236*z^10+171972024920*z^12-3334286626196*z^14-592901952867166*z^18+ 50044859662359*z^16-645017832663602399028*z^50+979970415319715259470*z^48+ 5629012517526776*z^20+358674083710447682112*z^36-168264746859990446716*z^34-\ 43338363825432850*z^66+189818059*z^80+z^88+47304*z^84-338*z^86-3735182*z^82+ 273158863239461365*z^64-22056581288570610360*z^30-1259033760428342338288*z^42+ 1368629271231140986832*z^44-1259033760428342338288*z^46-22056581288570610360*z^ 58+66470632063633737434*z^56-168264746859990446716*z^54+358674083710447682112*z ^52+6128476688213522880*z^60-592901952867166*z^70+5629012517526776*z^68-\ 6692622236*z^78+66470632063633737434*z^32-645017832663602399028*z^38+ 979970415319715259470*z^40-1420225639239242536*z^62+171972024920*z^76-\ 3334286626196*z^74+50044859662359*z^72)/(-1-29366761044732540128*z^28+ 6301486376569599301*z^26+449*z^2-1121412438642236229*z^24+164438580184858674*z^ 22-74874*z^4+6757706*z^6-384917079*z^8+15048555251*z^10-425851403328*z^12+ 9048560505456*z^14+1912041013455891*z^18-148274459648127*z^16+ 7471571074465922402374*z^50-10402698248522745935070*z^48-19709827140251122*z^20 -2336738203354170991476*z^36+1014675044757844643402*z^34+1121412438642236229*z^ 66-15048555251*z^80+z^90-449*z^88-6757706*z^84+74874*z^86+384917079*z^82-\ 6301486376569599301*z^64+114102633889460474240*z^30+10402698248522745935070*z^ 42-12272489569462824486128*z^44+12272489569462824486128*z^46+ 371210323898064013418*z^58-1014675044757844643402*z^56+2336738203354170991476*z ^54-4543626094324665812644*z^52-114102633889460474240*z^60+19709827140251122*z^ 70-164438580184858674*z^68+425851403328*z^78-371210323898064013418*z^32+ 4543626094324665812644*z^38-7471571074465922402374*z^40+29366761044732540128*z^ 62-9048560505456*z^76+148274459648127*z^74-1912041013455891*z^72) The first , 40, terms are: [0, 111, 0, 22269, 0, 4710291, 0, 1002557899, 0, 213587660477, 0, 45510892999095, 0, 9697709587492833, 0, 2066455909167168385, 0, 440335618837142291703, 0, 93829985990979982607693, 0, 19993992419223014527454075, 0, 4260468938912385566857501571, 0, 907852483098526658686598290861, 0, 193451975311691388812372506924719, 0, 41222189131654792565768231574865025, 0, 8783931382328977075856679835152301825, 0, 1871745585471411968256316492332835383023, 0, 398845503710632314364283545190810489941389, 0, 84988973429455222934900317979848356306369955, 0, 18110084073643420615225315260301305441477034139] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 29884011099096 z - 26283713292898 z - 256 z 24 22 4 6 + 17864082932500 z - 9353210895248 z + 23508 z - 1076606 z 8 10 12 14 + 28560555 z - 482590768 z + 5525933374 z - 44692353938 z 18 16 50 48 - 1144655808078 z + 262673687936 z - 1076606 z + 28560555 z 20 36 34 + 3752439487042 z + 3752439487042 z - 9353210895248 z 30 42 44 46 - 26283713292898 z - 44692353938 z + 5525933374 z - 482590768 z 56 54 52 32 38 + z - 256 z + 23508 z + 17864082932500 z - 1144655808078 z 40 / 28 26 + 262673687936 z ) / (-1 - 270674088683552 z + 209634247674008 z / 2 24 22 4 + 368 z - 125539436824544 z + 57933689082814 z - 42717 z 6 8 10 12 + 2332099 z - 71923544 z + 1395248031 z - 18217353154 z 14 18 16 50 + 167396045800 z + 5513116491462 z - 1115932747258 z + 71923544 z 48 20 36 - 1395248031 z - 20492309665656 z - 57933689082814 z 34 30 42 + 125539436824544 z + 270674088683552 z + 1115932747258 z 44 46 58 56 54 - 167396045800 z + 18217353154 z + z - 368 z + 42717 z 52 32 38 - 2332099 z - 209634247674008 z + 20492309665656 z 40 - 5513116491462 z ) And in Maple-input format, it is: -(1+29884011099096*z^28-26283713292898*z^26-256*z^2+17864082932500*z^24-\ 9353210895248*z^22+23508*z^4-1076606*z^6+28560555*z^8-482590768*z^10+5525933374 *z^12-44692353938*z^14-1144655808078*z^18+262673687936*z^16-1076606*z^50+ 28560555*z^48+3752439487042*z^20+3752439487042*z^36-9353210895248*z^34-\ 26283713292898*z^30-44692353938*z^42+5525933374*z^44-482590768*z^46+z^56-256*z^ 54+23508*z^52+17864082932500*z^32-1144655808078*z^38+262673687936*z^40)/(-1-\ 270674088683552*z^28+209634247674008*z^26+368*z^2-125539436824544*z^24+ 57933689082814*z^22-42717*z^4+2332099*z^6-71923544*z^8+1395248031*z^10-\ 18217353154*z^12+167396045800*z^14+5513116491462*z^18-1115932747258*z^16+ 71923544*z^50-1395248031*z^48-20492309665656*z^20-57933689082814*z^36+ 125539436824544*z^34+270674088683552*z^30+1115932747258*z^42-167396045800*z^44+ 18217353154*z^46+z^58-368*z^56+42717*z^54-2332099*z^52-209634247674008*z^32+ 20492309665656*z^38-5513116491462*z^40) The first , 40, terms are: [0, 112, 0, 22007, 0, 4569765, 0, 959432600, 0, 202044268323, 0, 42586107682283, 0, 8978568313663624, 0, 1893136762280389213, 0, 399179065873445942159, 0, 84169904111187209395552, 0, 17747897548869959653499753, 0, 3742288798287193283173719513, 0, 789092252339671386933688926848, 0, 166386577001374540694967046954783, 0, 35083975889037698046726905716512301, 0, 7397744420206930875839306668016688552, 0, 1559875162958764967501454514476146976539, 0, 328912488342798261022869796968686260962163, 0, 69353899321678207532396336256936940211457144, 0, 14623839233361718684643170978036114969277126997] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 22882172658656180 z - 9088838759639417 z - 277 z 24 22 4 6 + 2923194095917146 z - 758014793368932 z + 29624 z - 1709999 z 8 10 12 14 + 61610170 z - 1506136910 z + 26356922624 z - 342443804904 z 18 16 50 - 26047938312584 z + 3389730090731 z - 758014793368932 z 48 20 36 + 2923194095917146 z + 157558753280250 z + 117001540368519706 z 34 66 64 - 105692188264229968 z - 1709999 z + 61610170 z 30 42 44 - 46792425562242145 z - 46792425562242145 z + 22882172658656180 z 46 58 56 - 9088838759639417 z - 342443804904 z + 3389730090731 z 54 52 60 70 - 26047938312584 z + 157558753280250 z + 26356922624 z - 277 z 68 32 38 + 29624 z + 77890747483909469 z - 105692188264229968 z 40 62 72 / + 77890747483909469 z - 1506136910 z + z ) / (-1 / 28 26 2 - 139936018314834025 z + 50385658397231289 z + 389 z 24 22 4 6 - 14695719698438259 z + 3456585416862583 z - 50700 z + 3385588 z 8 10 12 14 - 137898279 z + 3767975805 z - 73253709704 z + 1053663830716 z 18 16 50 + 97705909530509 z - 11522824098465 z + 14695719698438259 z 48 20 36 - 50385658397231289 z - 651716131444863 z - 1065885009086663903 z 34 66 64 + 870614539971824633 z + 137898279 z - 3767975805 z 30 42 44 + 315833444777338457 z + 580594598284819457 z - 315833444777338457 z 46 58 56 + 139936018314834025 z + 11522824098465 z - 97705909530509 z 54 52 60 + 651716131444863 z - 3456585416862583 z - 1053663830716 z 70 68 32 + 50700 z - 3385588 z - 580594598284819457 z 38 40 62 74 + 1065885009086663903 z - 870614539971824633 z + 73253709704 z + z 72 - 389 z ) And in Maple-input format, it is: -(1+22882172658656180*z^28-9088838759639417*z^26-277*z^2+2923194095917146*z^24-\ 758014793368932*z^22+29624*z^4-1709999*z^6+61610170*z^8-1506136910*z^10+ 26356922624*z^12-342443804904*z^14-26047938312584*z^18+3389730090731*z^16-\ 758014793368932*z^50+2923194095917146*z^48+157558753280250*z^20+ 117001540368519706*z^36-105692188264229968*z^34-1709999*z^66+61610170*z^64-\ 46792425562242145*z^30-46792425562242145*z^42+22882172658656180*z^44-\ 9088838759639417*z^46-342443804904*z^58+3389730090731*z^56-26047938312584*z^54+ 157558753280250*z^52+26356922624*z^60-277*z^70+29624*z^68+77890747483909469*z^ 32-105692188264229968*z^38+77890747483909469*z^40-1506136910*z^62+z^72)/(-1-\ 139936018314834025*z^28+50385658397231289*z^26+389*z^2-14695719698438259*z^24+ 3456585416862583*z^22-50700*z^4+3385588*z^6-137898279*z^8+3767975805*z^10-\ 73253709704*z^12+1053663830716*z^14+97705909530509*z^18-11522824098465*z^16+ 14695719698438259*z^50-50385658397231289*z^48-651716131444863*z^20-\ 1065885009086663903*z^36+870614539971824633*z^34+137898279*z^66-3767975805*z^64 +315833444777338457*z^30+580594598284819457*z^42-315833444777338457*z^44+ 139936018314834025*z^46+11522824098465*z^58-97705909530509*z^56+651716131444863 *z^54-3456585416862583*z^52-1053663830716*z^60+50700*z^70-3385588*z^68-\ 580594598284819457*z^32+1065885009086663903*z^38-870614539971824633*z^40+ 73253709704*z^62+z^74-389*z^72) The first , 40, terms are: [0, 112, 0, 22492, 0, 4746577, 0, 1008971800, 0, 214804453243, 0, 45747524595615, 0, 9743875423596576, 0, 2075419189356856823, 0, 442061243625578498679, 0, 94158533296674030635296, 0, 20055665944522949616785985, 0, 4271835587466721594316239588, 0, 909896473047859872801359419144, 0, 193806989880172063163788597673429, 0, 41280684664498101703289004450037685, 0, 8792742346644266213033337019872991352, 0, 1872844857345459484572402620871882920132, 0, 398913981723274524101304271920591649637281, 0, 84968257883782768917786788980225286949037840, 0, 18098149422157436903484370092741223264479953655] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {3, 4}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 12 10 8 6 18 4 f(z) = - (-22735 z + 57164 z - 57164 z + 22735 z + z - 3383 z 16 2 14 / 12 10 - 141 z + 141 z + 3383 z - 1) / (380241 z - 617872 z / 8 20 6 18 4 16 2 + 380241 z + z - 96597 z - 253 z + 9344 z + 9344 z - 253 z 14 - 96597 z + 1) And in Maple-input format, it is: -(-22735*z^12+57164*z^10-57164*z^8+22735*z^6+z^18-3383*z^4-141*z^16+141*z^2+ 3383*z^14-1)/(380241*z^12-617872*z^10+380241*z^8+z^20-96597*z^6-253*z^18+9344*z ^4+9344*z^16-253*z^2-96597*z^14+1) The first , 40, terms are: [0, 112, 0, 22375, 0, 4688209, 0, 987540664, 0, 208160494687, 0, 43881453067951, 0, 9250580562470536, 0, 1950103648909760113, 0, 411099072359025383623, 0, 86663317855907960677696, 0, 18269393431657427610737041, 0, 3851349622904109031526457073, 0, 811898543563115481925283003296, 0, 171155389561715029778146554918535, 0, 36081069005865068467493529716098993, 0, 7606208276233205146768477708024248040, 0, 1603455937850277057369047776012742281615, 0, 338022685055974073942970639909165013788799, 0, 71258169878765494966821572163688691341704280, 0, 15021852079632371793184464847783514852035454737] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 24 22 4 6 8 f(z) = - (1 - 146 z + z - 146 z + 3833 z - 38896 z + 189495 z 10 12 14 18 16 20 - 480638 z + 655006 z - 480638 z - 38896 z + 189495 z + 3833 z / 24 14 12 26 10 22 ) / (-262 z + 5611739 z - 5611739 z + z + 3068702 z + 9614 z / 8 20 6 18 4 16 - 904003 z - 136831 z + 136831 z + 904003 z - 9614 z - 3068702 z 2 + 262 z - 1) And in Maple-input format, it is: -(1-146*z^2+z^24-146*z^22+3833*z^4-38896*z^6+189495*z^8-480638*z^10+655006*z^12 -480638*z^14-38896*z^18+189495*z^16+3833*z^20)/(-262*z^24+5611739*z^14-5611739* z^12+z^26+3068702*z^10+9614*z^22-904003*z^8-136831*z^20+136831*z^6+904003*z^18-\ 9614*z^4-3068702*z^16+262*z^2-1) The first , 40, terms are: [0, 116, 0, 24611, 0, 5430793, 0, 1201415500, 0, 265824488555, 0, 58816810816259, 0, 13013924110002508, 0, 2879486773033280449, 0, 637120981494537921467, 0, 140970658016604634711604, 0, 31191448718718351777581497, 0, 6901482101748576790527543049, 0, 1527035683090128757082464235060, 0, 337874958313627307628411715129547, 0, 74758886592892731495697493338472209, 0, 16541300226876219815962662257466046924, 0, 3659961051662687648677012806822459523091, 0, 809810275852632647685398257486136631046619, 0, 179180235423160095978977303204322619550832076, 0, 39645775959678695276252902350622715943930058297] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 39418513308823088 z - 17918136849958630 z - 331 z 24 22 4 6 + 6455517642167938 z - 1832792678539488 z + 43240 z - 3002921 z 8 10 12 14 + 126494663 z - 3493580616 z + 66668168091 z - 912582601093 z 18 16 50 - 70001030388927 z + 9211256938728 z - 70001030388927 z 48 20 36 + 406979438429421 z + 406979438429421 z + 96491293503833390 z 34 66 64 30 - 107869725852133296 z - 331 z + 43240 z - 69024872388288354 z 42 44 46 - 17918136849958630 z + 6455517642167938 z - 1832792678539488 z 58 56 54 52 - 3493580616 z + 66668168091 z - 912582601093 z + 9211256938728 z 60 68 32 38 + 126494663 z + z + 96491293503833390 z - 69024872388288354 z 40 62 / 2 + 39418513308823088 z - 3002921 z ) / ((-1 + z ) (1 / 28 26 2 + 171268747578469010 z - 75674179468825480 z - 424 z 24 22 4 6 + 26304338749841666 z - 7152419999156328 z + 67793 z - 5596626 z 8 10 12 14 + 273714229 z - 8595071982 z + 183022068385 z - 2750612976178 z 18 16 50 - 244751202098368 z + 30071521234333 z - 244751202098368 z 48 20 36 + 1509566693955165 z + 1509566693955165 z + 433398779385887690 z 34 66 64 30 - 486556085909535060 z - 424 z + 67793 z - 306159537244164108 z 42 44 46 - 75674179468825480 z + 26304338749841666 z - 7152419999156328 z 58 56 54 - 8595071982 z + 183022068385 z - 2750612976178 z 52 60 68 32 + 30071521234333 z + 273714229 z + z + 433398779385887690 z 38 40 62 - 306159537244164108 z + 171268747578469010 z - 5596626 z )) And in Maple-input format, it is: -(1+39418513308823088*z^28-17918136849958630*z^26-331*z^2+6455517642167938*z^24 -1832792678539488*z^22+43240*z^4-3002921*z^6+126494663*z^8-3493580616*z^10+ 66668168091*z^12-912582601093*z^14-70001030388927*z^18+9211256938728*z^16-\ 70001030388927*z^50+406979438429421*z^48+406979438429421*z^20+96491293503833390 *z^36-107869725852133296*z^34-331*z^66+43240*z^64-69024872388288354*z^30-\ 17918136849958630*z^42+6455517642167938*z^44-1832792678539488*z^46-3493580616*z ^58+66668168091*z^56-912582601093*z^54+9211256938728*z^52+126494663*z^60+z^68+ 96491293503833390*z^32-69024872388288354*z^38+39418513308823088*z^40-3002921*z^ 62)/(-1+z^2)/(1+171268747578469010*z^28-75674179468825480*z^26-424*z^2+ 26304338749841666*z^24-7152419999156328*z^22+67793*z^4-5596626*z^6+273714229*z^ 8-8595071982*z^10+183022068385*z^12-2750612976178*z^14-244751202098368*z^18+ 30071521234333*z^16-244751202098368*z^50+1509566693955165*z^48+1509566693955165 *z^20+433398779385887690*z^36-486556085909535060*z^34-424*z^66+67793*z^64-\ 306159537244164108*z^30-75674179468825480*z^42+26304338749841666*z^44-\ 7152419999156328*z^46-8595071982*z^58+183022068385*z^56-2750612976178*z^54+ 30071521234333*z^52+273714229*z^60+z^68+433398779385887690*z^32-\ 306159537244164108*z^38+171268747578469010*z^40-5596626*z^62) The first , 40, terms are: [0, 94, 0, 14973, 0, 2612625, 0, 468591678, 0, 84859354437, 0, 15424905327117, 0, 2808100641142878, 0, 511552083107111097, 0, 93216682388938560309, 0, 16988463495979282608958, 0, 3096279474027042993267673, 0, 564336086009150944500283369, 0, 102858639770057781157586238718, 0, 18747620006367010535380543273349, 0, 3417060024442959609128388058612265, 0, 622815733945726198737493290213490846, 0, 113518533641478206122792202203579892029, 0, 20690647148319862810714519229260943249301, 0, 3771216143090759976487245427096552385519358, 0, 687367186355275572413247075805716918105309601] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 15425045817777213889 z - 3460416046308283181 z - 373 z 24 22 4 6 + 641269739384389440 z - 97554526828330080 z + 57668 z - 4991416 z 8 10 12 14 + 276009949 z - 10523901118 z + 290827200981 z - 6032411125820 z 18 16 50 - 1208791393697578 z + 96362173534885 z - 1791347194319091771230 z 48 20 + 2744291363116346145984 z + 12089296026736425 z 36 34 + 984387649847217519816 z - 454679704933608499148 z 66 80 88 84 86 - 97554526828330080 z + 276009949 z + z + 57668 z - 373 z 82 64 30 - 4991416 z + 641269739384389440 z - 57088201887392683970 z 42 44 - 3543182079271036405146 z + 3857893968793262383496 z 46 58 - 3543182079271036405146 z - 57088201887392683970 z 56 54 + 176151866956311862674 z - 454679704933608499148 z 52 60 + 984387649847217519816 z + 15425045817777213889 z 70 68 78 - 1208791393697578 z + 12089296026736425 z - 10523901118 z 32 38 + 176151866956311862674 z - 1791347194319091771230 z 40 62 76 + 2744291363116346145984 z - 3460416046308283181 z + 290827200981 z 74 72 / - 6032411125820 z + 96362173534885 z ) / (-1 / 28 26 2 - 73477388087804657576 z + 15343470264099738261 z + 475 z 24 22 4 6 - 2645999258598925402 z + 374219891124503979 z - 88306 z + 8898909 z 8 10 12 14 - 560906376 z + 24006383073 z - 735958602196 z + 16778910484789 z 18 16 50 + 3981454804624807 z - 292477766777922 z + 20490397114316921051711 z 48 20 - 28677051754294941254235 z - 43033961715197758 z 36 34 - 6291150003425552535751 z + 2695138256971112965063 z 66 80 90 88 84 + 2645999258598925402 z - 24006383073 z + z - 475 z - 8898909 z 86 82 64 + 88306 z + 560906376 z - 15343470264099738261 z 30 42 + 292240349445594802182 z + 28677051754294941254235 z 44 46 - 33918718346784059593927 z + 33918718346784059593927 z 58 56 + 969793434053111420819 z - 2695138256971112965063 z 54 52 + 6291150003425552535751 z - 12363230312059632033539 z 60 70 - 292240349445594802182 z + 43033961715197758 z 68 78 32 - 374219891124503979 z + 735958602196 z - 969793434053111420819 z 38 40 + 12363230312059632033539 z - 20490397114316921051711 z 62 76 74 + 73477388087804657576 z - 16778910484789 z + 292477766777922 z 72 - 3981454804624807 z ) And in Maple-input format, it is: -(1+15425045817777213889*z^28-3460416046308283181*z^26-373*z^2+ 641269739384389440*z^24-97554526828330080*z^22+57668*z^4-4991416*z^6+276009949* z^8-10523901118*z^10+290827200981*z^12-6032411125820*z^14-1208791393697578*z^18 +96362173534885*z^16-1791347194319091771230*z^50+2744291363116346145984*z^48+ 12089296026736425*z^20+984387649847217519816*z^36-454679704933608499148*z^34-\ 97554526828330080*z^66+276009949*z^80+z^88+57668*z^84-373*z^86-4991416*z^82+ 641269739384389440*z^64-57088201887392683970*z^30-3543182079271036405146*z^42+ 3857893968793262383496*z^44-3543182079271036405146*z^46-57088201887392683970*z^ 58+176151866956311862674*z^56-454679704933608499148*z^54+984387649847217519816* z^52+15425045817777213889*z^60-1208791393697578*z^70+12089296026736425*z^68-\ 10523901118*z^78+176151866956311862674*z^32-1791347194319091771230*z^38+ 2744291363116346145984*z^40-3460416046308283181*z^62+290827200981*z^76-\ 6032411125820*z^74+96362173534885*z^72)/(-1-73477388087804657576*z^28+ 15343470264099738261*z^26+475*z^2-2645999258598925402*z^24+374219891124503979*z ^22-88306*z^4+8898909*z^6-560906376*z^8+24006383073*z^10-735958602196*z^12+ 16778910484789*z^14+3981454804624807*z^18-292477766777922*z^16+ 20490397114316921051711*z^50-28677051754294941254235*z^48-43033961715197758*z^ 20-6291150003425552535751*z^36+2695138256971112965063*z^34+2645999258598925402* z^66-24006383073*z^80+z^90-475*z^88-8898909*z^84+88306*z^86+560906376*z^82-\ 15343470264099738261*z^64+292240349445594802182*z^30+28677051754294941254235*z^ 42-33918718346784059593927*z^44+33918718346784059593927*z^46+ 969793434053111420819*z^58-2695138256971112965063*z^56+6291150003425552535751*z ^54-12363230312059632033539*z^52-292240349445594802182*z^60+43033961715197758*z ^70-374219891124503979*z^68+735958602196*z^78-969793434053111420819*z^32+ 12363230312059632033539*z^38-20490397114316921051711*z^40+73477388087804657576* z^62-16778910484789*z^76+292477766777922*z^74-3981454804624807*z^72) The first , 40, terms are: [0, 102, 0, 17812, 0, 3360981, 0, 646351794, 0, 124999712675, 0, 24219841372309, 0, 4696110565877368, 0, 910804868336929179, 0, 176669433131384006037, 0, 34270305135025507987106, 0, 6647880306008994935195479, 0, 1289591325971358976958208292, 0, 250162722556574203027632690142, 0, 48528153154223851037445554449507, 0, 9413805550346177518840227882290187, 0, 1826151516038334404313574687369648454, 0, 354248848088184884743968168785767095860, 0, 68719518588975984273609071863464707309327, 0, 13330663935041535960829590794194568445203050, 0, 2585969843042892805185056790342014875288753157] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 42281457030657048 z - 19279733313639802 z - 335 z 24 22 4 6 + 6966724549473230 z - 1982354701981408 z + 44436 z - 3128059 z 8 10 12 14 + 133244393 z - 3712711608 z + 71343581401 z - 981783568147 z 18 16 50 - 75752996636503 z + 9946921604948 z - 75752996636503 z 48 20 36 + 440640035729489 z + 440640035729489 z + 103029168920781726 z 34 66 64 30 - 115104012699686096 z - 335 z + 44436 z - 73837176076868178 z 42 44 46 - 19279733313639802 z + 6966724549473230 z - 1982354701981408 z 58 56 54 52 - 3712711608 z + 71343581401 z - 981783568147 z + 9946921604948 z 60 68 32 38 + 133244393 z + z + 103029168920781726 z - 73837176076868178 z 40 62 / 28 + 42281457030657048 z - 3128059 z ) / (-1 - 274889622382727538 z / 26 2 24 + 113237046250022002 z + 449 z - 37022897264584258 z 22 4 6 8 + 9543034872607485 z - 73425 z + 6110823 z - 300604227 z 10 12 14 + 9527561927 z - 205975043595 z + 3163159760775 z 18 16 50 + 299456164233661 z - 35561263946299 z + 1922614567020141 z 48 20 36 - 9543034872607485 z - 1922614567020141 z - 1027274449385472366 z 34 66 64 + 1027274449385472366 z + 73425 z - 6110823 z 30 42 44 + 532346209967173982 z + 274889622382727538 z - 113237046250022002 z 46 58 56 + 37022897264584258 z + 205975043595 z - 3163159760775 z 54 52 60 70 + 35561263946299 z - 299456164233661 z - 9527561927 z + z 68 32 38 - 449 z - 825452310829558486 z + 825452310829558486 z 40 62 - 532346209967173982 z + 300604227 z ) And in Maple-input format, it is: -(1+42281457030657048*z^28-19279733313639802*z^26-335*z^2+6966724549473230*z^24 -1982354701981408*z^22+44436*z^4-3128059*z^6+133244393*z^8-3712711608*z^10+ 71343581401*z^12-981783568147*z^14-75752996636503*z^18+9946921604948*z^16-\ 75752996636503*z^50+440640035729489*z^48+440640035729489*z^20+ 103029168920781726*z^36-115104012699686096*z^34-335*z^66+44436*z^64-\ 73837176076868178*z^30-19279733313639802*z^42+6966724549473230*z^44-\ 1982354701981408*z^46-3712711608*z^58+71343581401*z^56-981783568147*z^54+ 9946921604948*z^52+133244393*z^60+z^68+103029168920781726*z^32-\ 73837176076868178*z^38+42281457030657048*z^40-3128059*z^62)/(-1-\ 274889622382727538*z^28+113237046250022002*z^26+449*z^2-37022897264584258*z^24+ 9543034872607485*z^22-73425*z^4+6110823*z^6-300604227*z^8+9527561927*z^10-\ 205975043595*z^12+3163159760775*z^14+299456164233661*z^18-35561263946299*z^16+ 1922614567020141*z^50-9543034872607485*z^48-1922614567020141*z^20-\ 1027274449385472366*z^36+1027274449385472366*z^34+73425*z^66-6110823*z^64+ 532346209967173982*z^30+274889622382727538*z^42-113237046250022002*z^44+ 37022897264584258*z^46+205975043595*z^58-3163159760775*z^56+35561263946299*z^54 -299456164233661*z^52-9527561927*z^60+z^70-449*z^68-825452310829558486*z^32+ 825452310829558486*z^38-532346209967173982*z^40+300604227*z^62) The first , 40, terms are: [0, 114, 0, 22197, 0, 4578767, 0, 955325646, 0, 199933154651, 0, 41884234146755, 0, 8777541980378286, 0, 1839735093488659463, 0, 385621621919540543149, 0, 80830815535254109106450, 0, 16943236737776792933325929, 0, 3551545240720385417207114137, 0, 744455862293622625521520808914, 0, 156048936555646995660317763384253, 0, 32710168157487946491450127478420375, 0, 6856536345996441911895863938817913838, 0, 1437231759317490149196064910575285577971, 0, 301265108076666990312420276792108834115627, 0, 63149638429152920667483890415632153093115278, 0, 13237101600748161220809879695532110071786192959] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 18196476644412 z - 21175212661496 z - 276 z 24 22 4 6 + 18196476644412 z - 11527911752530 z + 28083 z - 1428870 z 8 10 12 14 + 41880192 z - 767959082 z + 9285115274 z - 76630642288 z 18 16 50 48 - 1813135600126 z + 442048769414 z - 276 z + 28083 z 20 36 34 + 5358201260160 z + 442048769414 z - 1813135600126 z 30 42 44 46 52 - 11527911752530 z - 767959082 z + 41880192 z - 1428870 z + z 32 38 40 / 2 + 5358201260160 z - 76630642288 z + 9285115274 z ) / ((-1 + z ) (1 / 28 26 2 24 + 86143227629286 z - 101308753312010 z - 383 z + 86143227629286 z 22 4 6 8 - 52908903862796 z + 48753 z - 2975060 z + 101544124 z 10 12 14 18 - 2123714020 z + 28839497604 z - 263989964028 z - 7415139261628 z 16 50 48 20 + 1669416791332 z - 383 z + 48753 z + 23402831288900 z 36 34 30 + 1669416791332 z - 7415139261628 z - 52908903862796 z 42 44 46 52 32 - 2123714020 z + 101544124 z - 2975060 z + z + 23402831288900 z 38 40 - 263989964028 z + 28839497604 z )) And in Maple-input format, it is: -(1+18196476644412*z^28-21175212661496*z^26-276*z^2+18196476644412*z^24-\ 11527911752530*z^22+28083*z^4-1428870*z^6+41880192*z^8-767959082*z^10+ 9285115274*z^12-76630642288*z^14-1813135600126*z^18+442048769414*z^16-276*z^50+ 28083*z^48+5358201260160*z^20+442048769414*z^36-1813135600126*z^34-\ 11527911752530*z^30-767959082*z^42+41880192*z^44-1428870*z^46+z^52+ 5358201260160*z^32-76630642288*z^38+9285115274*z^40)/(-1+z^2)/(1+86143227629286 *z^28-101308753312010*z^26-383*z^2+86143227629286*z^24-52908903862796*z^22+ 48753*z^4-2975060*z^6+101544124*z^8-2123714020*z^10+28839497604*z^12-\ 263989964028*z^14-7415139261628*z^18+1669416791332*z^16-383*z^50+48753*z^48+ 23402831288900*z^20+1669416791332*z^36-7415139261628*z^34-52908903862796*z^30-\ 2123714020*z^42+101544124*z^44-2975060*z^46+z^52+23402831288900*z^32-\ 263989964028*z^38+28839497604*z^40) The first , 40, terms are: [0, 108, 0, 20419, 0, 4129151, 0, 846218812, 0, 173970525109, 0, 35795127133293, 0, 7366654766872348, 0, 1516160622679503591, 0, 312053225196786094235, 0, 64226577381180970203788, 0, 13219094605695664606387465, 0, 2720751671249878029210918777, 0, 559984726939630244403167688780, 0, 115255984984424480394213366380139, 0, 23721972580977545540753790670899607, 0, 4882453495152377761536867842607026012, 0, 1004905983259203261086819778998854620541, 0, 206829627125362046397096998043304082588581, 0, 42569648682170851368861054040131772525646332, 0, 8761679911311391845634246111539562323091245519] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 44055559476434552 z - 19349507267240706 z - 315 z 24 22 4 6 + 6686975430608902 z - 1811920764165232 z + 38940 z - 2568271 z 8 10 12 14 + 104349533 z - 2836314992 z + 54324085413 z - 759003701671 z 18 16 50 - 62833598956707 z + 7922820775564 z - 62833598956707 z 48 20 36 + 383149298843929 z + 383149298843929 z + 112377461507896542 z 34 66 64 30 - 126294392275641472 z - 315 z + 38940 z - 79138903352438826 z 42 44 46 - 19349507267240706 z + 6686975430608902 z - 1811920764165232 z 58 56 54 52 - 2836314992 z + 54324085413 z - 759003701671 z + 7922820775564 z 60 68 32 38 + 104349533 z + z + 112377461507896542 z - 79138903352438826 z 40 62 / 28 + 44055559476434552 z - 2568271 z ) / (-1 - 282976766896829618 z / 26 2 24 + 111575509700744302 z + 423 z - 34696031802127266 z 22 4 6 8 + 8477910772084493 z - 64505 z + 5001857 z - 232751183 z 10 12 14 + 7135118905 z - 152684929557 z + 2369776741683 z 18 16 50 + 240191907577477 z - 27390269178525 z + 1619546504676475 z 48 20 36 - 8477910772084493 z - 1619546504676475 z - 1136119295334422882 z 34 66 64 + 1136119295334422882 z + 64505 z - 5001857 z 30 42 44 + 567504877578001682 z + 282976766896829618 z - 111575509700744302 z 46 58 56 + 34696031802127266 z + 152684929557 z - 2369776741683 z 54 52 60 70 + 27390269178525 z - 240191907577477 z - 7135118905 z + z 68 32 38 - 423 z - 901619488850829326 z + 901619488850829326 z 40 62 - 567504877578001682 z + 232751183 z ) And in Maple-input format, it is: -(1+44055559476434552*z^28-19349507267240706*z^26-315*z^2+6686975430608902*z^24 -1811920764165232*z^22+38940*z^4-2568271*z^6+104349533*z^8-2836314992*z^10+ 54324085413*z^12-759003701671*z^14-62833598956707*z^18+7922820775564*z^16-\ 62833598956707*z^50+383149298843929*z^48+383149298843929*z^20+ 112377461507896542*z^36-126294392275641472*z^34-315*z^66+38940*z^64-\ 79138903352438826*z^30-19349507267240706*z^42+6686975430608902*z^44-\ 1811920764165232*z^46-2836314992*z^58+54324085413*z^56-759003701671*z^54+ 7922820775564*z^52+104349533*z^60+z^68+112377461507896542*z^32-\ 79138903352438826*z^38+44055559476434552*z^40-2568271*z^62)/(-1-\ 282976766896829618*z^28+111575509700744302*z^26+423*z^2-34696031802127266*z^24+ 8477910772084493*z^22-64505*z^4+5001857*z^6-232751183*z^8+7135118905*z^10-\ 152684929557*z^12+2369776741683*z^14+240191907577477*z^18-27390269178525*z^16+ 1619546504676475*z^50-8477910772084493*z^48-1619546504676475*z^20-\ 1136119295334422882*z^36+1136119295334422882*z^34+64505*z^66-5001857*z^64+ 567504877578001682*z^30+282976766896829618*z^42-111575509700744302*z^44+ 34696031802127266*z^46+152684929557*z^58-2369776741683*z^56+27390269178525*z^54 -240191907577477*z^52-7135118905*z^60+z^70-423*z^68-901619488850829326*z^32+ 901619488850829326*z^38-567504877578001682*z^40+232751183*z^62) The first , 40, terms are: [0, 108, 0, 20119, 0, 3977383, 0, 796455820, 0, 160133758577, 0, 32245009156021, 0, 6496900553848700, 0, 1309358446608975059, 0, 263910072987266968979, 0, 53195163518373140482908, 0, 10722502440002741292833477, 0, 2161341648661704567762698877, 0, 435664378604638117426632002748, 0, 87817536410224146805029567312795, 0, 17701525040323753167115621003307451, 0, 3568126286054119381049772724436474716, 0, 719233307444167611337832355111264253549, 0, 144977091489121012012231825711244623544505, 0, 29223281494683505908617338307619424723952172, 0, 5890587115728541592067900457827108745025766719] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 18567130156927410753 z - 4083303458082762483 z - 367 z 24 22 4 6 + 740900984499418954 z - 110259468672210860 z + 56178 z - 4862088 z 8 10 12 14 + 271033829 z - 10477641914 z + 294689693999 z - 6236357428328 z 18 16 50 - 1305820073566010 z + 101791973697551 z - 2315512096340341509138 z 48 20 + 3572337880904979528488 z + 13359317314154201 z 36 34 + 1260161719082991202708 z - 575009037144781303760 z 66 80 88 84 86 - 110259468672210860 z + 271033829 z + z + 56178 z - 367 z 82 64 30 - 4862088 z + 740900984499418954 z - 69990312335183336418 z 42 44 - 4632004110380029239262 z + 5050645007971030183416 z 46 58 - 4632004110380029239262 z - 69990312335183336418 z 56 54 + 219566834684405388194 z - 575009037144781303760 z 52 60 + 1260161719082991202708 z + 18567130156927410753 z 70 68 78 - 1305820073566010 z + 13359317314154201 z - 10477641914 z 32 38 + 219566834684405388194 z - 2315512096340341509138 z 40 62 76 + 3572337880904979528488 z - 4083303458082762483 z + 294689693999 z 74 72 / - 6236357428328 z + 101791973697551 z ) / (-1 / 28 26 2 - 88639355952004661148 z + 18015391696995410577 z + 487 z 24 22 4 6 - 3022628873065260994 z + 416037497666341247 z - 89522 z + 8910585 z 8 10 12 14 - 558160452 z + 23909592869 z - 738452310576 z + 17057422820077 z 18 16 50 + 4209331770857795 z - 302657862985434 z + 27868613483786304443875 z 48 20 - 39332344368789552561075 z - 46611324780958982 z 36 34 - 8317365407034426643971 z + 3495237767697593032043 z 66 80 90 88 84 + 3022628873065260994 z - 23909592869 z + z - 487 z - 8910585 z 86 82 64 + 89522 z + 558160452 z - 18015391696995410577 z 30 42 + 361829720541130732710 z + 39332344368789552561075 z 44 46 - 46719978410351329729015 z + 46719978410351329729015 z 58 56 + 1230272962228737419635 z - 3495237767697593032043 z 54 52 + 8317365407034426643971 z - 16608618785222555852819 z 60 70 - 361829720541130732710 z + 46611324780958982 z 68 78 32 - 416037497666341247 z + 738452310576 z - 1230272962228737419635 z 38 40 + 16608618785222555852819 z - 27868613483786304443875 z 62 76 74 + 88639355952004661148 z - 17057422820077 z + 302657862985434 z 72 - 4209331770857795 z ) And in Maple-input format, it is: -(1+18567130156927410753*z^28-4083303458082762483*z^26-367*z^2+ 740900984499418954*z^24-110259468672210860*z^22+56178*z^4-4862088*z^6+271033829 *z^8-10477641914*z^10+294689693999*z^12-6236357428328*z^14-1305820073566010*z^ 18+101791973697551*z^16-2315512096340341509138*z^50+3572337880904979528488*z^48 +13359317314154201*z^20+1260161719082991202708*z^36-575009037144781303760*z^34-\ 110259468672210860*z^66+271033829*z^80+z^88+56178*z^84-367*z^86-4862088*z^82+ 740900984499418954*z^64-69990312335183336418*z^30-4632004110380029239262*z^42+ 5050645007971030183416*z^44-4632004110380029239262*z^46-69990312335183336418*z^ 58+219566834684405388194*z^56-575009037144781303760*z^54+1260161719082991202708 *z^52+18567130156927410753*z^60-1305820073566010*z^70+13359317314154201*z^68-\ 10477641914*z^78+219566834684405388194*z^32-2315512096340341509138*z^38+ 3572337880904979528488*z^40-4083303458082762483*z^62+294689693999*z^76-\ 6236357428328*z^74+101791973697551*z^72)/(-1-88639355952004661148*z^28+ 18015391696995410577*z^26+487*z^2-3022628873065260994*z^24+416037497666341247*z ^22-89522*z^4+8910585*z^6-558160452*z^8+23909592869*z^10-738452310576*z^12+ 17057422820077*z^14+4209331770857795*z^18-302657862985434*z^16+ 27868613483786304443875*z^50-39332344368789552561075*z^48-46611324780958982*z^ 20-8317365407034426643971*z^36+3495237767697593032043*z^34+3022628873065260994* z^66-23909592869*z^80+z^90-487*z^88-8910585*z^84+89522*z^86+558160452*z^82-\ 18015391696995410577*z^64+361829720541130732710*z^30+39332344368789552561075*z^ 42-46719978410351329729015*z^44+46719978410351329729015*z^46+ 1230272962228737419635*z^58-3495237767697593032043*z^56+8317365407034426643971* z^54-16608618785222555852819*z^52-361829720541130732710*z^60+46611324780958982* z^70-416037497666341247*z^68+738452310576*z^78-1230272962228737419635*z^32+ 16608618785222555852819*z^38-27868613483786304443875*z^40+88639355952004661148* z^62-17057422820077*z^76+302657862985434*z^74-4209331770857795*z^72) The first , 40, terms are: [0, 120, 0, 25096, 0, 5527609, 0, 1227445048, 0, 272995863353, 0, 60737673531431, 0, 13514313956596496, 0, 3007031155632103661, 0, 669088960371199511599, 0, 148877929947347738038976, 0, 33126604141962108814092595, 0, 7370951481979691098498282536, 0, 1640099505907010372339119854000, 0, 364936116409821784286239208561407, 0, 81201395920372772614553321055740015, 0, 18068002608584770042116599420574558208, 0, 4020284560617983484658522542890525874616, 0, 894547576690526449231293584370635538826827, 0, 199044459393088516527592789364185170102838864, 0, 44289088526549680984053990081430305328928657327] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 17890044492536 z - 20767878912848 z - 280 z 24 22 4 6 + 17890044492536 z - 11413468452570 z + 28967 z - 1488262 z 8 10 12 14 + 43856800 z - 805152906 z + 9703848498 z - 79500340672 z 18 16 50 48 - 1837434046038 z + 453701552878 z - 280 z + 28967 z 20 36 34 + 5361655120608 z + 453701552878 z - 1837434046038 z 30 42 44 46 52 - 11413468452570 z - 805152906 z + 43856800 z - 1488262 z + z 32 38 40 / 2 + 5361655120608 z - 79500340672 z + 9703848498 z ) / ((-1 + z ) (1 / 28 26 2 24 + 86710992208966 z - 101744401837850 z - 407 z + 86710992208966 z 22 4 6 8 - 53600502304042 z + 52081 z - 3150990 z + 107166674 z 10 12 14 18 - 2242464014 z + 30448073260 z - 277720366116 z - 7665215249578 z 16 50 48 20 + 1743238400908 z - 407 z + 52081 z + 23933084975222 z 36 34 30 + 1743238400908 z - 7665215249578 z - 53600502304042 z 42 44 46 52 32 - 2242464014 z + 107166674 z - 3150990 z + z + 23933084975222 z 38 40 - 277720366116 z + 30448073260 z )) And in Maple-input format, it is: -(1+17890044492536*z^28-20767878912848*z^26-280*z^2+17890044492536*z^24-\ 11413468452570*z^22+28967*z^4-1488262*z^6+43856800*z^8-805152906*z^10+ 9703848498*z^12-79500340672*z^14-1837434046038*z^18+453701552878*z^16-280*z^50+ 28967*z^48+5361655120608*z^20+453701552878*z^36-1837434046038*z^34-\ 11413468452570*z^30-805152906*z^42+43856800*z^44-1488262*z^46+z^52+ 5361655120608*z^32-79500340672*z^38+9703848498*z^40)/(-1+z^2)/(1+86710992208966 *z^28-101744401837850*z^26-407*z^2+86710992208966*z^24-53600502304042*z^22+ 52081*z^4-3150990*z^6+107166674*z^8-2242464014*z^10+30448073260*z^12-\ 277720366116*z^14-7665215249578*z^18+1743238400908*z^16-407*z^50+52081*z^48+ 23933084975222*z^20+1743238400908*z^36-7665215249578*z^34-53600502304042*z^30-\ 2242464014*z^42+107166674*z^44-3150990*z^46+z^52+23933084975222*z^32-\ 277720366116*z^38+30448073260*z^40) The first , 40, terms are: [0, 128, 0, 28703, 0, 6707169, 0, 1573494112, 0, 369301274927, 0, 86680217865055, 0, 20345223442224096, 0, 4775352830278183409, 0, 1120852768214869479855, 0, 263082339471486241560320, 0, 61749696503262262251788465, 0, 14493656394977114239841301585, 0, 3401896491813305380389465410048, 0, 798480344013714862375605444150671, 0, 187416301866310890220328757546827473, 0, 43989649174697245387698180968923383520, 0, 10325084932612266655004316623187610651135, 0, 2423465084763166775948103869733847903144079, 0, 568826605824677273485656496555173557894310496, 0, 133512840572115709833422178016555944644260050177] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 652546458643 z + 1325006083738 z + 302 z 24 22 4 6 - 1881923533891 z + 1881923533891 z - 33137 z + 1723055 z 8 10 12 14 - 47834546 z + 777407847 z - 7851371189 z + 51229372614 z 18 16 20 36 + 652546458643 z - 221962824749 z - 1325006083738 z - 777407847 z 34 30 42 44 46 + 7851371189 z + 221962824749 z + 33137 z - 302 z + z 32 38 40 / 28 - 51229372614 z + 47834546 z - 1723055 z ) / (1 + 9853155920192 z / 26 2 24 22 - 16789077521240 z - 456 z + 20040075358550 z - 16789077521240 z 4 6 8 10 12 + 63328 z - 3898756 z + 125755864 z - 2378721728 z + 28224266708 z 14 18 16 48 - 218775118496 z - 4027992678780 z + 1136742574088 z + z 20 36 34 + 9853155920192 z + 28224266708 z - 218775118496 z 30 42 44 46 - 4027992678780 z - 3898756 z + 63328 z - 456 z 32 38 40 + 1136742574088 z - 2378721728 z + 125755864 z ) And in Maple-input format, it is: -(-1-652546458643*z^28+1325006083738*z^26+302*z^2-1881923533891*z^24+ 1881923533891*z^22-33137*z^4+1723055*z^6-47834546*z^8+777407847*z^10-7851371189 *z^12+51229372614*z^14+652546458643*z^18-221962824749*z^16-1325006083738*z^20-\ 777407847*z^36+7851371189*z^34+221962824749*z^30+33137*z^42-302*z^44+z^46-\ 51229372614*z^32+47834546*z^38-1723055*z^40)/(1+9853155920192*z^28-\ 16789077521240*z^26-456*z^2+20040075358550*z^24-16789077521240*z^22+63328*z^4-\ 3898756*z^6+125755864*z^8-2378721728*z^10+28224266708*z^12-218775118496*z^14-\ 4027992678780*z^18+1136742574088*z^16+z^48+9853155920192*z^20+28224266708*z^36-\ 218775118496*z^34-4027992678780*z^30-3898756*z^42+63328*z^44-456*z^46+ 1136742574088*z^32-2378721728*z^38+125755864*z^40) The first , 40, terms are: [0, 154, 0, 40033, 0, 10678237, 0, 2856553354, 0, 764670746461, 0, 204733455904357, 0, 54818588844410314, 0, 14678255972122928965, 0, 3930279234823058928505, 0, 1052381212441209577640986, 0, 281788323640835765633249977, 0, 75452384948504853393012585289, 0, 20203330687039220967997733656474, 0, 5409697489035478299129698695087561, 0, 1448514976225642720628125935173564917, 0, 387858220150106724336077047031282030026, 0, 103853947995767541979641259688437188399445, 0, 27808209174462815454727216607383999289681005, 0, 7446000007235983155524968280846166916349384842, 0, 1993760754630340648796472694685229107816863177261] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 81535449 z + 451861927 z + 232 z - 1575796040 z 22 4 6 8 10 + 3561230557 z - 17314 z + 548379 z - 8880476 z + 81535449 z 12 14 18 16 - 451861927 z + 1575796040 z + 5324741793 z - 3561230557 z 20 36 34 30 32 38 - 5324741793 z - 232 z + 17314 z + 8880476 z - 548379 z + z ) / 40 38 36 34 32 / (z - 324 z + 33454 z - 1434220 z + 28860564 z / 30 28 26 24 - 314243454 z + 2030113788 z - 8252286074 z + 21928294370 z 22 20 18 16 - 39045739524 z + 47252810407 z - 39045739524 z + 21928294370 z 14 12 10 8 - 8252286074 z + 2030113788 z - 314243454 z + 28860564 z 6 4 2 - 1434220 z + 33454 z - 324 z + 1) And in Maple-input format, it is: -(-1-81535449*z^28+451861927*z^26+232*z^2-1575796040*z^24+3561230557*z^22-17314 *z^4+548379*z^6-8880476*z^8+81535449*z^10-451861927*z^12+1575796040*z^14+ 5324741793*z^18-3561230557*z^16-5324741793*z^20-232*z^36+17314*z^34+8880476*z^ 30-548379*z^32+z^38)/(z^40-324*z^38+33454*z^36-1434220*z^34+28860564*z^32-\ 314243454*z^30+2030113788*z^28-8252286074*z^26+21928294370*z^24-39045739524*z^ 22+47252810407*z^20-39045739524*z^18+21928294370*z^16-8252286074*z^14+ 2030113788*z^12-314243454*z^10+28860564*z^8-1434220*z^6+33454*z^4-324*z^2+1) The first , 40, terms are: [0, 92, 0, 13668, 0, 2236505, 0, 379346500, 0, 65268682807, 0, 11296901576723, 0, 1960332123324864, 0, 340558359227683115, 0, 59193286321429222099, 0, 10290856305375069428356, 0, 1789265118090024802327433, 0, 311112698132717506446944460, 0, 54096569583020926938060327836, 0, 9406449904774737912527201806549, 0, 1635624416738526839016424490241229, 0, 284408284330271152669571988337918172, 0, 49453982314577367568731624075889575612, 0, 8599247768282499328749677096922634818881, 0, 1495270380635219817729878293671986725077700, 0, 260003403278948694270966786617072274882514027] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 461799942978565149800 z - 65764258583023217232 z - 392 z 24 22 4 6 + 8008128807713745420 z - 828266292508347344 z + 65232 z - 6248304 z 102 8 10 12 - 17613051400 z + 393592102 z - 17613051400 z + 589011815256 z 14 18 16 - 15254460738252 z - 5245580841844244 z + 314093892400875 z 50 48 - 90320578866026452132600104 z + 57801651042617454894612827 z 20 36 + 72172241253022728 z + 258668964510869693567896 z 34 66 - 65758368704425672832492 z - 32532040789214801775688064 z 80 100 90 + 14541392324106334016185 z + 589011815256 z - 828266292508347344 z 88 84 + 8008128807713745420 z + 461799942978565149800 z 94 86 96 - 5245580841844244 z - 65764258583023217232 z + 314093892400875 z 98 92 82 - 15254460738252 z + 72172241253022728 z - 2788220893083515252740 z 64 112 110 106 + 57801651042617454894612827 z + z - 392 z - 6248304 z 108 30 42 + 65232 z - 2788220893083515252740 z - 6987811889796913269873152 z 44 46 + 16090957453784730297068992 z - 32532040789214801775688064 z 58 56 - 150315436233299184571881312 z + 160188752332207646589678632 z 54 52 - 150315436233299184571881312 z + 124190079690205245761240848 z 60 70 + 124190079690205245761240848 z - 6987811889796913269873152 z 68 78 + 16090957453784730297068992 z - 65758368704425672832492 z 32 38 + 14541392324106334016185 z - 887430389058722547457032 z 40 62 + 2661166764674412507303098 z - 90320578866026452132600104 z 76 74 + 258668964510869693567896 z - 887430389058722547457032 z 72 104 / + 2661166764674412507303098 z + 393592102 z ) / (-1 / 28 26 2 - 1835187462893203755420 z + 246781363302820691904 z + 493 z 24 22 4 - 28336810636363825264 z + 2758543670175677036 z - 97504 z 6 102 8 10 + 10724260 z + 1374145607560 z - 758725518 z + 37558076082 z 12 14 18 - 1374145607560 z + 38613007685472 z + 15353739196734259 z 16 50 - 857101804348799 z + 669199642623261681271873931 z 48 20 - 403581034865007553579178815 z - 225690681835588904 z 36 34 - 1284636592354289429656512 z + 308990955832936162692821 z 66 80 + 403581034865007553579178815 z - 308990955832936162692821 z 100 90 - 38613007685472 z + 28336810636363825264 z 88 84 - 246781363302820691904 z - 11722946489856851297704 z 94 86 + 225690681835588904 z + 1835187462893203755420 z 96 98 92 - 15353739196734259 z + 857101804348799 z - 2758543670175677036 z 82 64 112 + 64644063793445029642177 z - 669199642623261681271873931 z - 493 z 114 110 106 108 + z + 97504 z + 758725518 z - 10724260 z 30 42 + 11722946489856851297704 z + 41037789462249737754938810 z 44 46 - 100033810712600673443652580 z + 214245666538139495541312768 z 58 56 + 1426817463146492210961390800 z - 1426817463146492210961390800 z 54 52 + 1257815918755558478359466040 z - 977358712640942699102403376 z 60 70 - 1257815918755558478359466040 z + 100033810712600673443652580 z 68 78 - 214245666538139495541312768 z + 1284636592354289429656512 z 32 38 - 64644063793445029642177 z + 4658803505669543899532208 z 40 62 - 14772661213550625372219142 z + 977358712640942699102403376 z 76 74 - 4658803505669543899532208 z + 14772661213550625372219142 z 72 104 - 41037789462249737754938810 z - 37558076082 z ) And in Maple-input format, it is: -(1+461799942978565149800*z^28-65764258583023217232*z^26-392*z^2+ 8008128807713745420*z^24-828266292508347344*z^22+65232*z^4-6248304*z^6-\ 17613051400*z^102+393592102*z^8-17613051400*z^10+589011815256*z^12-\ 15254460738252*z^14-5245580841844244*z^18+314093892400875*z^16-\ 90320578866026452132600104*z^50+57801651042617454894612827*z^48+ 72172241253022728*z^20+258668964510869693567896*z^36-65758368704425672832492*z^ 34-32532040789214801775688064*z^66+14541392324106334016185*z^80+589011815256*z^ 100-828266292508347344*z^90+8008128807713745420*z^88+461799942978565149800*z^84 -5245580841844244*z^94-65764258583023217232*z^86+314093892400875*z^96-\ 15254460738252*z^98+72172241253022728*z^92-2788220893083515252740*z^82+ 57801651042617454894612827*z^64+z^112-392*z^110-6248304*z^106+65232*z^108-\ 2788220893083515252740*z^30-6987811889796913269873152*z^42+ 16090957453784730297068992*z^44-32532040789214801775688064*z^46-\ 150315436233299184571881312*z^58+160188752332207646589678632*z^56-\ 150315436233299184571881312*z^54+124190079690205245761240848*z^52+ 124190079690205245761240848*z^60-6987811889796913269873152*z^70+ 16090957453784730297068992*z^68-65758368704425672832492*z^78+ 14541392324106334016185*z^32-887430389058722547457032*z^38+ 2661166764674412507303098*z^40-90320578866026452132600104*z^62+ 258668964510869693567896*z^76-887430389058722547457032*z^74+ 2661166764674412507303098*z^72+393592102*z^104)/(-1-1835187462893203755420*z^28 +246781363302820691904*z^26+493*z^2-28336810636363825264*z^24+ 2758543670175677036*z^22-97504*z^4+10724260*z^6+1374145607560*z^102-758725518*z ^8+37558076082*z^10-1374145607560*z^12+38613007685472*z^14+15353739196734259*z^ 18-857101804348799*z^16+669199642623261681271873931*z^50-\ 403581034865007553579178815*z^48-225690681835588904*z^20-\ 1284636592354289429656512*z^36+308990955832936162692821*z^34+ 403581034865007553579178815*z^66-308990955832936162692821*z^80-38613007685472*z ^100+28336810636363825264*z^90-246781363302820691904*z^88-\ 11722946489856851297704*z^84+225690681835588904*z^94+1835187462893203755420*z^ 86-15353739196734259*z^96+857101804348799*z^98-2758543670175677036*z^92+ 64644063793445029642177*z^82-669199642623261681271873931*z^64-493*z^112+z^114+ 97504*z^110+758725518*z^106-10724260*z^108+11722946489856851297704*z^30+ 41037789462249737754938810*z^42-100033810712600673443652580*z^44+ 214245666538139495541312768*z^46+1426817463146492210961390800*z^58-\ 1426817463146492210961390800*z^56+1257815918755558478359466040*z^54-\ 977358712640942699102403376*z^52-1257815918755558478359466040*z^60+ 100033810712600673443652580*z^70-214245666538139495541312768*z^68+ 1284636592354289429656512*z^78-64644063793445029642177*z^32+ 4658803505669543899532208*z^38-14772661213550625372219142*z^40+ 977358712640942699102403376*z^62-4658803505669543899532208*z^76+ 14772661213550625372219142*z^74-41037789462249737754938810*z^72-37558076082*z^ 104) The first , 40, terms are: [0, 101, 0, 17521, 0, 3265905, 0, 619740425, 0, 118306735229, 0, 22637066515097, 0, 4335650759137497, 0, 830752213270868721, 0, 159209505905833640841, 0, 30514194724605676392845, 0, 5848581160982533730323281, 0, 1121001224006998279177558985, 0, 214864536585179700383075085377, 0, 41183643587418010963856421904885, 0, 7893786964436811167565453756444393, 0, 1513025707140245017258974181318739705, 0, 290006233302051070350055594027685682421, 0, 55586382503374703159623719335863991538817, 0, 10654412677265495348266086804212635231061241, 0, 2042164054210246103271678413260734588124599457] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 41539547769624 z - 36395483723880 z - 250 z 24 22 4 6 + 24455777468219 z - 12565496564918 z + 22610 z - 1047730 z 8 10 12 14 + 28782331 z - 509553324 z + 6133833684 z - 52077592492 z 18 16 50 48 - 1450039605982 z + 320011988425 z - 1047730 z + 28782331 z 20 36 34 + 4911882174030 z + 4911882174030 z - 12565496564918 z 30 42 44 46 - 36395483723880 z - 52077592492 z + 6133833684 z - 509553324 z 56 54 52 32 38 + z - 250 z + 22610 z + 24455777468219 z - 1450039605982 z 40 / 28 26 + 320011988425 z ) / (-1 - 368221260081376 z + 283738741408691 z / 2 24 22 4 + 367 z - 168206549771229 z + 76458773876732 z - 41492 z 6 8 10 12 + 2295820 z - 73689873 z + 1500915935 z - 20556907888 z 14 18 16 50 + 197187682096 z + 6951592303971 z - 1364089973165 z + 73689873 z 48 20 36 - 1500915935 z - 26504817914980 z - 76458773876732 z 34 30 42 + 168206549771229 z + 368221260081376 z + 1364089973165 z 44 46 58 56 54 - 197187682096 z + 20556907888 z + z - 367 z + 41492 z 52 32 38 - 2295820 z - 283738741408691 z + 26504817914980 z 40 - 6951592303971 z ) And in Maple-input format, it is: -(1+41539547769624*z^28-36395483723880*z^26-250*z^2+24455777468219*z^24-\ 12565496564918*z^22+22610*z^4-1047730*z^6+28782331*z^8-509553324*z^10+ 6133833684*z^12-52077592492*z^14-1450039605982*z^18+320011988425*z^16-1047730*z ^50+28782331*z^48+4911882174030*z^20+4911882174030*z^36-12565496564918*z^34-\ 36395483723880*z^30-52077592492*z^42+6133833684*z^44-509553324*z^46+z^56-250*z^ 54+22610*z^52+24455777468219*z^32-1450039605982*z^38+320011988425*z^40)/(-1-\ 368221260081376*z^28+283738741408691*z^26+367*z^2-168206549771229*z^24+ 76458773876732*z^22-41492*z^4+2295820*z^6-73689873*z^8+1500915935*z^10-\ 20556907888*z^12+197187682096*z^14+6951592303971*z^18-1364089973165*z^16+ 73689873*z^50-1500915935*z^48-26504817914980*z^20-76458773876732*z^36+ 168206549771229*z^34+368221260081376*z^30+1364089973165*z^42-197187682096*z^44+ 20556907888*z^46+z^58-367*z^56+41492*z^54-2295820*z^52-283738741408691*z^32+ 26504817914980*z^38-6951592303971*z^40) The first , 40, terms are: [0, 117, 0, 24057, 0, 5222445, 0, 1142167669, 0, 250086035793, 0, 54768974709213, 0, 11994861476717113, 0, 2626992135809403849, 0, 575337750731473305869, 0, 126004797362595195845281, 0, 27596328838074654343345381, 0, 6043876018931182088323015101, 0, 1323670174421689261036309036713, 0, 289897199393718040290868677023557, 0, 63490428240539536788042420524438129, 0, 13905048019314149293345844540152118673, 0, 3045346610162367334995556416568553724325, 0, 666961808628919720117795303758343471859017, 0, 146071403723040850495182489649785892438387357, 0, 31991119595712055471313492598242637089985443845] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 4848130 z - 38241664 z - 205 z + 180917951 z 22 4 6 8 10 - 535160143 z + 12621 z - 345774 z + 4848130 z - 38241664 z 12 14 18 16 + 180917951 z - 535160143 z - 1256199716 z + 1016042445 z 20 36 34 30 32 / + 1016042445 z + z - 205 z - 345774 z + 12621 z ) / ( / 8 26 14 12 34 -15407011 z + 846758988 z + 3022304408 z - 846758988 z + 24985 z 16 2 24 32 - 1 - 6956892186 z + 297 z - 3022304408 z - 879735 z 22 4 28 30 + 6956892186 z - 24985 z - 148240362 z + 15407011 z 20 10 6 18 36 - 10513516380 z + 148240362 z + 879735 z + 10513516380 z - 297 z 38 + z ) And in Maple-input format, it is: -(1+4848130*z^28-38241664*z^26-205*z^2+180917951*z^24-535160143*z^22+12621*z^4-\ 345774*z^6+4848130*z^8-38241664*z^10+180917951*z^12-535160143*z^14-1256199716*z ^18+1016042445*z^16+1016042445*z^20+z^36-205*z^34-345774*z^30+12621*z^32)/(-\ 15407011*z^8+846758988*z^26+3022304408*z^14-846758988*z^12+24985*z^34-1-\ 6956892186*z^16+297*z^2-3022304408*z^24-879735*z^32+6956892186*z^22-24985*z^4-\ 148240362*z^28+15407011*z^30-10513516380*z^20+148240362*z^10+879735*z^6+ 10513516380*z^18-297*z^36+z^38) The first , 40, terms are: [0, 92, 0, 14960, 0, 2678461, 0, 492104056, 0, 91086945833, 0, 16896422348783, 0, 3136226440699024, 0, 582237322816662709, 0, 108097590049659417311, 0, 20069603597292974460800, 0, 3726177728809278476812543, 0, 691813316165441752664338624, 0, 128444182189704877540752872972, 0, 23847343540044626198518503019583, 0, 4427571596577791086322060356399279, 0, 822036651586361348120975910712988860, 0, 152621870414430258551563359624931303616, 0, 28336249123546472201207786381666013558887, 0, 5260995769511405275758475842695602375694032, 0, 976772767916010818150853667162820899610809855] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6678962839060791101 z - 1557692549914330157 z - 357 z 24 22 4 6 + 301719415675774508 z - 48238448352948618 z + 52220 z - 4240690 z 8 10 12 14 + 218610003 z - 7743669272 z + 198674445421 z - 3831895033536 z 18 16 50 - 670949413782468 z + 57097599625493 z - 688186887304421178614 z 48 20 + 1043439729696537719348 z + 6316789326085055 z 36 34 + 383750722369663209012 z - 180667111477306622488 z 66 80 88 84 86 - 48238448352948618 z + 218610003 z + z + 52220 z - 357 z 82 64 30 - 4240690 z + 301719415675774508 z - 23901831618162737402 z 42 44 - 1338929546726092903014 z + 1454879706092966379776 z 46 58 - 1338929546726092903014 z - 23901831618162737402 z 56 54 + 71674621460370373846 z - 180667111477306622488 z 52 60 + 383750722369663209012 z + 6678962839060791101 z 70 68 78 - 670949413782468 z + 6316789326085055 z - 7743669272 z 32 38 + 71674621460370373846 z - 688186887304421178614 z 40 62 76 + 1043439729696537719348 z - 1557692549914330157 z + 198674445421 z 74 72 / - 3831895033536 z + 57097599625493 z ) / (-1 / 28 26 2 - 32155106256778727680 z + 6969053813414803561 z + 459 z 24 22 4 6 - 1254283657544576600 z + 186212755602552845 z - 81370 z + 7709171 z 8 10 12 14 - 452287102 z + 17927819177 z - 508566915312 z + 10751228427987 z 18 16 50 + 2222726308802427 z - 174476135151480 z + 7914325806664721919479 z 48 20 - 10999685581720235026547 z - 22613599969791854 z 36 34 - 2491171649730496332451 z + 1086951945471409452923 z 66 80 90 88 84 + 1254283657544576600 z - 17927819177 z + z - 459 z - 7709171 z 86 82 64 + 81370 z + 452287102 z - 6969053813414803561 z 30 42 + 123863919669972630254 z + 10999685581720235026547 z 44 46 - 12965466567448532831603 z + 12965466567448532831603 z 58 56 + 400044982184839555651 z - 1086951945471409452923 z 54 52 + 2491171649730496332451 z - 4825932884674340539267 z 60 70 - 123863919669972630254 z + 22613599969791854 z 68 78 32 - 186212755602552845 z + 508566915312 z - 400044982184839555651 z 38 40 + 4825932884674340539267 z - 7914325806664721919479 z 62 76 74 + 32155106256778727680 z - 10751228427987 z + 174476135151480 z 72 - 2222726308802427 z ) And in Maple-input format, it is: -(1+6678962839060791101*z^28-1557692549914330157*z^26-357*z^2+ 301719415675774508*z^24-48238448352948618*z^22+52220*z^4-4240690*z^6+218610003* z^8-7743669272*z^10+198674445421*z^12-3831895033536*z^14-670949413782468*z^18+ 57097599625493*z^16-688186887304421178614*z^50+1043439729696537719348*z^48+ 6316789326085055*z^20+383750722369663209012*z^36-180667111477306622488*z^34-\ 48238448352948618*z^66+218610003*z^80+z^88+52220*z^84-357*z^86-4240690*z^82+ 301719415675774508*z^64-23901831618162737402*z^30-1338929546726092903014*z^42+ 1454879706092966379776*z^44-1338929546726092903014*z^46-23901831618162737402*z^ 58+71674621460370373846*z^56-180667111477306622488*z^54+383750722369663209012*z ^52+6678962839060791101*z^60-670949413782468*z^70+6316789326085055*z^68-\ 7743669272*z^78+71674621460370373846*z^32-688186887304421178614*z^38+ 1043439729696537719348*z^40-1557692549914330157*z^62+198674445421*z^76-\ 3831895033536*z^74+57097599625493*z^72)/(-1-32155106256778727680*z^28+ 6969053813414803561*z^26+459*z^2-1254283657544576600*z^24+186212755602552845*z^ 22-81370*z^4+7709171*z^6-452287102*z^8+17927819177*z^10-508566915312*z^12+ 10751228427987*z^14+2222726308802427*z^18-174476135151480*z^16+ 7914325806664721919479*z^50-10999685581720235026547*z^48-22613599969791854*z^20 -2491171649730496332451*z^36+1086951945471409452923*z^34+1254283657544576600*z^ 66-17927819177*z^80+z^90-459*z^88-7709171*z^84+81370*z^86+452287102*z^82-\ 6969053813414803561*z^64+123863919669972630254*z^30+10999685581720235026547*z^ 42-12965466567448532831603*z^44+12965466567448532831603*z^46+ 400044982184839555651*z^58-1086951945471409452923*z^56+2491171649730496332451*z ^54-4825932884674340539267*z^52-123863919669972630254*z^60+22613599969791854*z^ 70-186212755602552845*z^68+508566915312*z^78-400044982184839555651*z^32+ 4825932884674340539267*z^38-7914325806664721919479*z^40+32155106256778727680*z^ 62-10751228427987*z^76+174476135151480*z^74-2222726308802427*z^72) The first , 40, terms are: [0, 102, 0, 17668, 0, 3278353, 0, 619777210, 0, 117974654509, 0, 22520215285321, 0, 4304187111906776, 0, 823089119964668785, 0, 157437676555322103517, 0, 30117447589762876828906, 0, 5761679976978988861451841, 0, 1102274570044488502640012276, 0, 210879719592938061134098160918, 0, 40344265788279845952961860219193, 0, 7718443125545494440328984545287985, 0, 1476651472673940742407440372937671414, 0, 282505221439836776199452104337576796276, 0, 54047428728970036247218273845202062648553, 0, 10340073669137307742357235501868673152365290, 0, 1978209326887711794959171476177285154916097717] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 992994962198090 z - 562313221635012 z - 262 z 24 22 4 6 + 252766072277518 z - 89853437892624 z + 25411 z - 1286536 z 8 10 12 14 + 39516548 z - 804068328 z + 11469213171 z - 119151859894 z 18 16 50 - 5485980368016 z + 925939630449 z - 119151859894 z 48 20 36 + 925939630449 z + 25125355534912 z + 992994962198090 z 34 64 30 42 - 1395445903965424 z + z - 1395445903965424 z - 89853437892624 z 44 46 58 56 + 25125355534912 z - 5485980368016 z - 1286536 z + 39516548 z 54 52 60 32 - 804068328 z + 11469213171 z + 25411 z + 1562789326828408 z 38 40 62 / - 562313221635012 z + 252766072277518 z - 262 z ) / (-1 / 28 26 2 - 7025324296301204 z + 3583698952127800 z + 362 z 24 22 4 6 - 1454042366288346 z + 467203893156300 z - 43792 z + 2651775 z 8 10 12 14 - 94758934 z + 2201746178 z - 35411931819 z + 411289228668 z 18 16 50 + 23325115986693 z - 3553265828734 z + 3553265828734 z 48 20 36 - 23325115986693 z - 118159221468244 z - 10987099579792054 z 34 66 64 30 + 13733952101458696 z + z - 362 z + 10987099579792054 z 42 44 46 + 1454042366288346 z - 467203893156300 z + 118159221468244 z 58 56 54 52 + 94758934 z - 2201746178 z + 35411931819 z - 411289228668 z 60 32 38 - 2651775 z - 13733952101458696 z + 7025324296301204 z 40 62 - 3583698952127800 z + 43792 z ) And in Maple-input format, it is: -(1+992994962198090*z^28-562313221635012*z^26-262*z^2+252766072277518*z^24-\ 89853437892624*z^22+25411*z^4-1286536*z^6+39516548*z^8-804068328*z^10+ 11469213171*z^12-119151859894*z^14-5485980368016*z^18+925939630449*z^16-\ 119151859894*z^50+925939630449*z^48+25125355534912*z^20+992994962198090*z^36-\ 1395445903965424*z^34+z^64-1395445903965424*z^30-89853437892624*z^42+ 25125355534912*z^44-5485980368016*z^46-1286536*z^58+39516548*z^56-804068328*z^ 54+11469213171*z^52+25411*z^60+1562789326828408*z^32-562313221635012*z^38+ 252766072277518*z^40-262*z^62)/(-1-7025324296301204*z^28+3583698952127800*z^26+ 362*z^2-1454042366288346*z^24+467203893156300*z^22-43792*z^4+2651775*z^6-\ 94758934*z^8+2201746178*z^10-35411931819*z^12+411289228668*z^14+23325115986693* z^18-3553265828734*z^16+3553265828734*z^50-23325115986693*z^48-118159221468244* z^20-10987099579792054*z^36+13733952101458696*z^34+z^66-362*z^64+ 10987099579792054*z^30+1454042366288346*z^42-467203893156300*z^44+ 118159221468244*z^46+94758934*z^58-2201746178*z^56+35411931819*z^54-\ 411289228668*z^52-2651775*z^60-13733952101458696*z^32+7025324296301204*z^38-\ 3583698952127800*z^40+43792*z^62) The first , 40, terms are: [0, 100, 0, 17819, 0, 3436517, 0, 673624620, 0, 132533923151, 0, 26098503143503, 0, 5140376673759772, 0, 1012502733951109813, 0, 199435647291861051979, 0, 39283544647550211152276, 0, 7737824364748276311012961, 0, 1524148017275353530176853537, 0, 300217111717264373792768917812, 0, 59134883315807080143812546161515, 0, 11648018376713140233272024631629973, 0, 2294353596188841785344269898739774012, 0, 451927379832407437484480308917808892879, 0, 89017820526866737288397450932984068059343, 0, 17534171915794151053696641037012581464329740, 0, 3453771199449097964349228338614919204105729285] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 485874039078960951291 z - 68498226017956577982 z - 389 z 24 22 4 6 + 8257521185131568897 z - 845663241975700062 z + 64224 z - 6118826 z 102 8 10 12 - 17231882188 z + 384663421 z - 17231882188 z + 578324719307 z 14 18 16 - 15060520119962 z - 5257812203787122 z + 312279831671913 z 50 48 - 102946241788519077191209792 z + 65654435021849150791631538 z 20 36 + 72990275610040833 z + 282663979620106048250254 z 34 66 - 71223744752017839269022 z - 36789965993958649301109012 z 80 100 90 + 15603484171169884894341 z + 578324719307 z - 845663241975700062 z 88 84 + 8257521185131568897 z + 485874039078960951291 z 94 86 96 - 5257812203787122 z - 68498226017956577982 z + 312279831671913 z 98 92 82 - 15060520119962 z + 72990275610040833 z - 2962951305965363354200 z 64 112 110 106 + 65654435021849150791631538 z + z - 389 z - 6118826 z 108 30 42 + 64224 z - 2962951305965363354200 z - 7812712755570221968645904 z 44 46 + 18101116531943909190673558 z - 36789965993958649301109012 z 58 56 - 172012767455578844165345256 z + 183403332223199650956478326 z 54 52 - 172012767455578844165345256 z + 141903014413527334710394990 z 60 70 + 141903014413527334710394990 z - 7812712755570221968645904 z 68 78 + 18101116531943909190673558 z - 71223744752017839269022 z 32 38 + 15603484171169884894341 z - 977835986871523285838327 z 40 62 + 2954786278143851037752841 z - 102946241788519077191209792 z 76 74 + 282663979620106048250254 z - 977835986871523285838327 z 72 104 / 2 + 2954786278143851037752841 z + 384663421 z ) / ((-1 + z ) (1 / 28 26 2 + 1685124503802909465446 z - 227693141284795813434 z - 499 z 24 22 4 + 26244406413807536540 z - 2563218217693401664 z + 98132 z 6 102 8 10 - 10645060 z - 36234986494 z + 741878998 z - 36234986494 z 12 14 18 + 1311322911196 z - 36536877234516 z - 14366759465598766 z 16 50 + 805859051751631 z - 482738643661651582580403798 z 48 20 + 303970664899035293626097827 z + 210392221770514535 z 36 34 + 1135232604632701877939606 z - 276715636521921908963426 z 66 80 - 167604462248648778373436302 z + 58505464230837433127853 z 100 90 88 + 1311322911196 z - 2563218217693401664 z + 26244406413807536540 z 84 94 + 1685124503802909465446 z - 14366759465598766 z 86 96 98 - 227693141284795813434 z + 805859051751631 z - 36536877234516 z 92 82 + 210392221770514535 z - 10697028000170254091544 z 64 112 110 106 + 303970664899035293626097827 z + z - 499 z - 10645060 z 108 30 + 98132 z - 10697028000170254091544 z 42 44 - 34133417360889712205714622 z + 80879891697154771057781026 z 46 58 - 167604462248648778373436302 z - 818612458132753474581575734 z 56 54 + 874448105363542345496969367 z - 818612458132753474581575734 z 52 60 + 671570278530853775935487930 z + 671570278530853775935487930 z 70 68 - 34133417360889712205714622 z + 80879891697154771057781026 z 78 32 - 276715636521921908963426 z + 58505464230837433127853 z 38 40 - 4049629889905724961427743 z + 12585916771383940341913722 z 62 76 - 482738643661651582580403798 z + 1135232604632701877939606 z 74 72 - 4049629889905724961427743 z + 12585916771383940341913722 z 104 + 741878998 z )) And in Maple-input format, it is: -(1+485874039078960951291*z^28-68498226017956577982*z^26-389*z^2+ 8257521185131568897*z^24-845663241975700062*z^22+64224*z^4-6118826*z^6-\ 17231882188*z^102+384663421*z^8-17231882188*z^10+578324719307*z^12-\ 15060520119962*z^14-5257812203787122*z^18+312279831671913*z^16-\ 102946241788519077191209792*z^50+65654435021849150791631538*z^48+ 72990275610040833*z^20+282663979620106048250254*z^36-71223744752017839269022*z^ 34-36789965993958649301109012*z^66+15603484171169884894341*z^80+578324719307*z^ 100-845663241975700062*z^90+8257521185131568897*z^88+485874039078960951291*z^84 -5257812203787122*z^94-68498226017956577982*z^86+312279831671913*z^96-\ 15060520119962*z^98+72990275610040833*z^92-2962951305965363354200*z^82+ 65654435021849150791631538*z^64+z^112-389*z^110-6118826*z^106+64224*z^108-\ 2962951305965363354200*z^30-7812712755570221968645904*z^42+ 18101116531943909190673558*z^44-36789965993958649301109012*z^46-\ 172012767455578844165345256*z^58+183403332223199650956478326*z^56-\ 172012767455578844165345256*z^54+141903014413527334710394990*z^52+ 141903014413527334710394990*z^60-7812712755570221968645904*z^70+ 18101116531943909190673558*z^68-71223744752017839269022*z^78+ 15603484171169884894341*z^32-977835986871523285838327*z^38+ 2954786278143851037752841*z^40-102946241788519077191209792*z^62+ 282663979620106048250254*z^76-977835986871523285838327*z^74+ 2954786278143851037752841*z^72+384663421*z^104)/(-1+z^2)/(1+ 1685124503802909465446*z^28-227693141284795813434*z^26-499*z^2+ 26244406413807536540*z^24-2563218217693401664*z^22+98132*z^4-10645060*z^6-\ 36234986494*z^102+741878998*z^8-36234986494*z^10+1311322911196*z^12-\ 36536877234516*z^14-14366759465598766*z^18+805859051751631*z^16-\ 482738643661651582580403798*z^50+303970664899035293626097827*z^48+ 210392221770514535*z^20+1135232604632701877939606*z^36-276715636521921908963426 *z^34-167604462248648778373436302*z^66+58505464230837433127853*z^80+ 1311322911196*z^100-2563218217693401664*z^90+26244406413807536540*z^88+ 1685124503802909465446*z^84-14366759465598766*z^94-227693141284795813434*z^86+ 805859051751631*z^96-36536877234516*z^98+210392221770514535*z^92-\ 10697028000170254091544*z^82+303970664899035293626097827*z^64+z^112-499*z^110-\ 10645060*z^106+98132*z^108-10697028000170254091544*z^30-\ 34133417360889712205714622*z^42+80879891697154771057781026*z^44-\ 167604462248648778373436302*z^46-818612458132753474581575734*z^58+ 874448105363542345496969367*z^56-818612458132753474581575734*z^54+ 671570278530853775935487930*z^52+671570278530853775935487930*z^60-\ 34133417360889712205714622*z^70+80879891697154771057781026*z^68-\ 276715636521921908963426*z^78+58505464230837433127853*z^32-\ 4049629889905724961427743*z^38+12585916771383940341913722*z^40-\ 482738643661651582580403798*z^62+1135232604632701877939606*z^76-\ 4049629889905724961427743*z^74+12585916771383940341913722*z^72+741878998*z^104) The first , 40, terms are: [0, 111, 0, 21093, 0, 4222825, 0, 855622492, 0, 174130755147, 0, 35503304258283, 0, 7244462701293273, 0, 1478742401605362143, 0, 301886309469921971035, 0, 61634274791158207009441, 0, 12583843003463140206158519, 0, 2569268842199666673727004355, 0, 524575602517847021546741770380, 0, 107104474325039817717159710639385, 0, 21867924617669269468459224102954689, 0, 4464858583190879555395055473491937519, 0, 911607580968544749212562940880416580413, 0, 186126488205561568985450219996762530521365, 0, 38002174993995893463959866670497144644176831, 0, 7759053211563870370847160148397018215613122569] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 578913526709186777680 z - 80333443749224634636 z - 380 z 24 22 4 6 + 9518631231047940452 z - 956775778627127116 z + 62008 z - 5915384 z 102 8 10 12 - 17133430340 z + 375965338 z - 17133430340 z + 587027165320 z 14 18 16 - 15631847018572 z - 5707800523911616 z + 331556504962667 z 50 48 - 134740875536275186705682576 z + 85649432562581259093873403 z 20 36 + 80941208760351424 z + 354037007651160385220808 z 34 66 - 88267609163472646701444 z - 47790814754341694961431412 z 80 100 90 + 19110279322633204189337 z + 587027165320 z - 956775778627127116 z 88 84 + 9518631231047940452 z + 578913526709186777680 z 94 86 96 - 5707800523911616 z - 80333443749224634636 z + 331556504962667 z 98 92 82 - 15631847018572 z + 80941208760351424 z - 3581649554577230648048 z 64 112 110 106 + 85649432562581259093873403 z + z - 380 z - 5915384 z 108 30 42 + 62008 z - 3581649554577230648048 z - 10033281084007375188074192 z 44 46 + 23391099733541146187413432 z - 47790814754341694961431412 z 58 56 - 225982693929954511482217512 z + 241059600117819697587273080 z 54 52 - 225982693929954511482217512 z + 186164701818379436622839728 z 60 70 + 186164701818379436622839728 z - 10033281084007375188074192 z 68 78 + 23391099733541146187413432 z - 88267609163472646701444 z 32 38 + 19110279322633204189337 z - 1236349276495304917743476 z 40 62 + 3767175161918722637612870 z - 134740875536275186705682576 z 76 74 + 354037007651160385220808 z - 1236349276495304917743476 z 72 104 / 2 + 3767175161918722637612870 z + 375965338 z ) / ((-1 + z ) (1 / 28 26 2 + 1987103248741537962000 z - 264283097140415645756 z - 484 z 24 22 4 + 29924039350341379756 z - 2865068189376091292 z + 92592 z 6 102 8 10 - 9979296 z - 34938942140 z + 701963230 z - 34938942140 z 12 14 18 + 1295129969504 z - 37041057719932 z - 15339599275208024 z 16 50 + 838807165484291 z - 616624929810459998470849920 z 48 20 + 387383355364381013432187243 z + 230064651877904320 z 36 34 + 1401554168394834979735344 z - 338515651982305723314756 z 66 80 - 212949734941247806382482972 z + 70816584163937299089041 z 100 90 88 + 1295129969504 z - 2865068189376091292 z + 29924039350341379756 z 84 94 + 1987103248741537962000 z - 15339599275208024 z 86 96 98 - 264283097140415645756 z + 838807165484291 z - 37041057719932 z 92 82 + 230064651877904320 z - 12790995310517037388648 z 64 112 110 106 + 387383355364381013432187243 z + z - 484 z - 9979296 z 108 30 + 92592 z - 12790995310517037388648 z 42 44 - 43000602055739578807594344 z + 102369623332007870978737152 z 46 58 - 212949734941247806382482972 z - 1048343065870303956511445544 z 56 54 + 1120202836197442521527217640 z - 1048343065870303956511445544 z 52 60 + 859213905331043186081721328 z + 859213905331043186081721328 z 70 68 - 43000602055739578807594344 z + 102369623332007870978737152 z 78 32 - 338515651982305723314756 z + 70816584163937299089041 z 38 40 - 5039125204128396211292220 z + 15766341555874025790634370 z 62 76 - 616624929810459998470849920 z + 1401554168394834979735344 z 74 72 - 5039125204128396211292220 z + 15766341555874025790634370 z 104 + 701963230 z )) And in Maple-input format, it is: -(1+578913526709186777680*z^28-80333443749224634636*z^26-380*z^2+ 9518631231047940452*z^24-956775778627127116*z^22+62008*z^4-5915384*z^6-\ 17133430340*z^102+375965338*z^8-17133430340*z^10+587027165320*z^12-\ 15631847018572*z^14-5707800523911616*z^18+331556504962667*z^16-\ 134740875536275186705682576*z^50+85649432562581259093873403*z^48+ 80941208760351424*z^20+354037007651160385220808*z^36-88267609163472646701444*z^ 34-47790814754341694961431412*z^66+19110279322633204189337*z^80+587027165320*z^ 100-956775778627127116*z^90+9518631231047940452*z^88+578913526709186777680*z^84 -5707800523911616*z^94-80333443749224634636*z^86+331556504962667*z^96-\ 15631847018572*z^98+80941208760351424*z^92-3581649554577230648048*z^82+ 85649432562581259093873403*z^64+z^112-380*z^110-5915384*z^106+62008*z^108-\ 3581649554577230648048*z^30-10033281084007375188074192*z^42+ 23391099733541146187413432*z^44-47790814754341694961431412*z^46-\ 225982693929954511482217512*z^58+241059600117819697587273080*z^56-\ 225982693929954511482217512*z^54+186164701818379436622839728*z^52+ 186164701818379436622839728*z^60-10033281084007375188074192*z^70+ 23391099733541146187413432*z^68-88267609163472646701444*z^78+ 19110279322633204189337*z^32-1236349276495304917743476*z^38+ 3767175161918722637612870*z^40-134740875536275186705682576*z^62+ 354037007651160385220808*z^76-1236349276495304917743476*z^74+ 3767175161918722637612870*z^72+375965338*z^104)/(-1+z^2)/(1+ 1987103248741537962000*z^28-264283097140415645756*z^26-484*z^2+ 29924039350341379756*z^24-2865068189376091292*z^22+92592*z^4-9979296*z^6-\ 34938942140*z^102+701963230*z^8-34938942140*z^10+1295129969504*z^12-\ 37041057719932*z^14-15339599275208024*z^18+838807165484291*z^16-\ 616624929810459998470849920*z^50+387383355364381013432187243*z^48+ 230064651877904320*z^20+1401554168394834979735344*z^36-338515651982305723314756 *z^34-212949734941247806382482972*z^66+70816584163937299089041*z^80+ 1295129969504*z^100-2865068189376091292*z^90+29924039350341379756*z^88+ 1987103248741537962000*z^84-15339599275208024*z^94-264283097140415645756*z^86+ 838807165484291*z^96-37041057719932*z^98+230064651877904320*z^92-\ 12790995310517037388648*z^82+387383355364381013432187243*z^64+z^112-484*z^110-\ 9979296*z^106+92592*z^108-12790995310517037388648*z^30-\ 43000602055739578807594344*z^42+102369623332007870978737152*z^44-\ 212949734941247806382482972*z^46-1048343065870303956511445544*z^58+ 1120202836197442521527217640*z^56-1048343065870303956511445544*z^54+ 859213905331043186081721328*z^52+859213905331043186081721328*z^60-\ 43000602055739578807594344*z^70+102369623332007870978737152*z^68-\ 338515651982305723314756*z^78+70816584163937299089041*z^32-\ 5039125204128396211292220*z^38+15766341555874025790634370*z^40-\ 616624929810459998470849920*z^62+1401554168394834979735344*z^76-\ 5039125204128396211292220*z^74+15766341555874025790634370*z^72+701963230*z^104) The first , 40, terms are: [0, 105, 0, 19857, 0, 4014169, 0, 820232885, 0, 167941145197, 0, 34399930576101, 0, 7046880995919333, 0, 1443593524759276101, 0, 295729638235532348293, 0, 60582224996108772474685, 0, 12410683155701261058223493, 0, 2542413466670927452631997897, 0, 520830822539816682729323218721, 0, 106695763712743261494672764637609, 0, 21857358492257792576142406442434881, 0, 4477629698804338638821813508085248481, 0, 917273133782049984128654578360652235913, 0, 187909688510155023776888563987976302970273, 0, 38494587637679765696673342376938031887035145, 0, 7885880122225452124700940674182988206961005061] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 545609599872294093188 z - 76743383189971535768 z - 386 z 24 22 4 6 + 9217579513721578108 z - 938996644875694504 z + 63742 z - 6126934 z 102 8 10 12 - 17787740186 z + 390645430 z - 17787740186 z + 606875437446 z 14 18 16 - 16047649998582 z - 5743338177650032 z + 337262429835787 z 50 48 - 114624725665433100506818608 z + 73167632699673139205831835 z 20 36 + 80468005526991908 z + 317464743372233091265162 z 34 66 - 80063453800985124190434 z - 41045235286130588773903014 z 80 100 90 + 17547115247535601312089 z + 606875437446 z - 938996644875694504 z 88 84 + 9217579513721578108 z + 545609599872294093188 z 94 86 96 - 5743338177650032 z - 76743383189971535768 z + 337262429835787 z 98 92 82 - 16047649998582 z + 80468005526991908 z - 3331114506342314729056 z 64 112 110 106 + 73167632699673139205831835 z + z - 386 z - 6126934 z 108 30 42 + 63742 z - 3331114506342314729056 z - 8739561428603960243965554 z 44 46 + 20220461097728205403441202 z - 41045235286130588773903014 z 58 56 - 191324907008832663805746240 z + 203967003276161201080720968 z 54 52 - 191324907008832663805746240 z + 157897603484542965724220696 z 60 70 + 157897603484542965724220696 z - 8739561428603960243965554 z 68 78 + 20220461097728205403441202 z - 80063453800985124190434 z 32 38 + 17547115247535601312089 z - 1096906777144438743068318 z 40 62 + 3310042735962278328113066 z - 114624725665433100506818608 z 76 74 + 317464743372233091265162 z - 1096906777144438743068318 z 72 104 / 2 + 3310042735962278328113066 z + 390645430 z ) / ((-1 + z ) (1 / 28 26 2 + 1894479702788017221896 z - 255429383934757438900 z - 498 z 24 22 4 + 29330976310074805068 z - 2848569646465089908 z + 97144 z 6 102 8 10 - 10587718 z - 37156901774 z + 747778718 z - 37156901774 z 12 14 18 + 1369076608928 z - 38798049289450 z - 15682511295203964 z 16 50 + 868645976108235 z - 534476536235025307020849672 z 48 20 + 337010391778039696990089243 z + 232001325039365160 z 36 34 + 1273816826248141995389776 z - 310982943966804804266214 z 66 80 - 186135513857414177497924222 z + 65816156903322263233305 z 100 90 88 + 1369076608928 z - 2848569646465089908 z + 29330976310074805068 z 84 94 + 1894479702788017221896 z - 15682511295203964 z 86 96 98 - 255429383934757438900 z + 868645976108235 z - 38798049289450 z 92 82 + 232001325039365160 z - 12036135274036107979068 z 64 112 110 106 + 337010391778039696990089243 z + z - 498 z - 10587718 z 108 30 + 97144 z - 12036135274036107979068 z 42 44 - 38061273921791098862982474 z + 89996676080296403496049944 z 46 58 - 186135513857414177497924222 z - 904874689501025025907365912 z 56 54 + 966393263405317097242117544 z - 904874689501025025907365912 z 52 60 + 742796291830012989896079184 z + 742796291830012989896079184 z 70 68 - 38061273921791098862982474 z + 89996676080296403496049944 z 78 32 - 310982943966804804266214 z + 65816156903322263233305 z 38 40 - 4535192731463040050074018 z + 14064935201968120136994626 z 62 76 - 534476536235025307020849672 z + 1273816826248141995389776 z 74 72 - 4535192731463040050074018 z + 14064935201968120136994626 z 104 + 747778718 z )) And in Maple-input format, it is: -(1+545609599872294093188*z^28-76743383189971535768*z^26-386*z^2+ 9217579513721578108*z^24-938996644875694504*z^22+63742*z^4-6126934*z^6-\ 17787740186*z^102+390645430*z^8-17787740186*z^10+606875437446*z^12-\ 16047649998582*z^14-5743338177650032*z^18+337262429835787*z^16-\ 114624725665433100506818608*z^50+73167632699673139205831835*z^48+ 80468005526991908*z^20+317464743372233091265162*z^36-80063453800985124190434*z^ 34-41045235286130588773903014*z^66+17547115247535601312089*z^80+606875437446*z^ 100-938996644875694504*z^90+9217579513721578108*z^88+545609599872294093188*z^84 -5743338177650032*z^94-76743383189971535768*z^86+337262429835787*z^96-\ 16047649998582*z^98+80468005526991908*z^92-3331114506342314729056*z^82+ 73167632699673139205831835*z^64+z^112-386*z^110-6126934*z^106+63742*z^108-\ 3331114506342314729056*z^30-8739561428603960243965554*z^42+ 20220461097728205403441202*z^44-41045235286130588773903014*z^46-\ 191324907008832663805746240*z^58+203967003276161201080720968*z^56-\ 191324907008832663805746240*z^54+157897603484542965724220696*z^52+ 157897603484542965724220696*z^60-8739561428603960243965554*z^70+ 20220461097728205403441202*z^68-80063453800985124190434*z^78+ 17547115247535601312089*z^32-1096906777144438743068318*z^38+ 3310042735962278328113066*z^40-114624725665433100506818608*z^62+ 317464743372233091265162*z^76-1096906777144438743068318*z^74+ 3310042735962278328113066*z^72+390645430*z^104)/(-1+z^2)/(1+ 1894479702788017221896*z^28-255429383934757438900*z^26-498*z^2+ 29330976310074805068*z^24-2848569646465089908*z^22+97144*z^4-10587718*z^6-\ 37156901774*z^102+747778718*z^8-37156901774*z^10+1369076608928*z^12-\ 38798049289450*z^14-15682511295203964*z^18+868645976108235*z^16-\ 534476536235025307020849672*z^50+337010391778039696990089243*z^48+ 232001325039365160*z^20+1273816826248141995389776*z^36-310982943966804804266214 *z^34-186135513857414177497924222*z^66+65816156903322263233305*z^80+ 1369076608928*z^100-2848569646465089908*z^90+29330976310074805068*z^88+ 1894479702788017221896*z^84-15682511295203964*z^94-255429383934757438900*z^86+ 868645976108235*z^96-38798049289450*z^98+232001325039365160*z^92-\ 12036135274036107979068*z^82+337010391778039696990089243*z^64+z^112-498*z^110-\ 10587718*z^106+97144*z^108-12036135274036107979068*z^30-\ 38061273921791098862982474*z^42+89996676080296403496049944*z^44-\ 186135513857414177497924222*z^46-904874689501025025907365912*z^58+ 966393263405317097242117544*z^56-904874689501025025907365912*z^54+ 742796291830012989896079184*z^52+742796291830012989896079184*z^60-\ 38061273921791098862982474*z^70+89996676080296403496049944*z^68-\ 310982943966804804266214*z^78+65816156903322263233305*z^32-\ 4535192731463040050074018*z^38+14064935201968120136994626*z^40-\ 534476536235025307020849672*z^62+1273816826248141995389776*z^76-\ 4535192731463040050074018*z^74+14064935201968120136994626*z^72+747778718*z^104) The first , 40, terms are: [0, 113, 0, 22487, 0, 4745395, 0, 1011944851, 0, 216302646891, 0, 46261077143765, 0, 9895374877405129, 0, 2116726303496433529, 0, 452794607083980139357, 0, 96858744222272901969139, 0, 20719377000892355895406387, 0, 4432151744112329318878129299, 0, 948096552064881812137552467855, 0, 202810538432717965755601831464809, 0, 43383887984096164462791302828239633, 0, 9280394167406540484648705539098356721, 0, 1985200495398232981087585143145252491833, 0, 424660950389240582932347863468933480984159, 0, 90840659775045721302476852217628348403268947, 0, 19432032685906966392110244798880072613281606259] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 44967413920068 z - 39132268015863 z - 241 z 24 22 4 6 + 25781468568875 z - 12846003775869 z + 20675 z - 912988 z 8 10 12 14 + 24353186 z - 427669135 z + 5205677400 z - 45372207780 z 18 16 50 48 - 1366698398100 z + 289201650567 z - 912988 z + 24353186 z 20 36 34 + 4831165234550 z + 4831165234550 z - 12846003775869 z 30 42 44 46 - 39132268015863 z - 45372207780 z + 5205677400 z - 427669135 z 56 54 52 32 38 + z - 241 z + 20675 z + 25781468568875 z - 1366698398100 z 40 / 28 26 + 289201650567 z ) / (-1 - 397876107359005 z + 302316957162867 z / 2 24 22 4 + 343 z - 174462360949389 z + 76379814130543 z - 36825 z 6 8 10 12 + 1933431 z - 59842273 z + 1203429087 z - 16649715059 z 14 18 16 50 + 164290694685 z + 6324261112143 z - 1183694886083 z + 59842273 z 48 20 36 - 1203429087 z - 25311627832833 z - 76379814130543 z 34 30 42 + 174462360949389 z + 397876107359005 z + 1183694886083 z 44 46 58 56 54 - 164290694685 z + 16649715059 z + z - 343 z + 36825 z 52 32 38 - 1933431 z - 302316957162867 z + 25311627832833 z 40 - 6324261112143 z ) And in Maple-input format, it is: -(1+44967413920068*z^28-39132268015863*z^26-241*z^2+25781468568875*z^24-\ 12846003775869*z^22+20675*z^4-912988*z^6+24353186*z^8-427669135*z^10+5205677400 *z^12-45372207780*z^14-1366698398100*z^18+289201650567*z^16-912988*z^50+ 24353186*z^48+4831165234550*z^20+4831165234550*z^36-12846003775869*z^34-\ 39132268015863*z^30-45372207780*z^42+5205677400*z^44-427669135*z^46+z^56-241*z^ 54+20675*z^52+25781468568875*z^32-1366698398100*z^38+289201650567*z^40)/(-1-\ 397876107359005*z^28+302316957162867*z^26+343*z^2-174462360949389*z^24+ 76379814130543*z^22-36825*z^4+1933431*z^6-59842273*z^8+1203429087*z^10-\ 16649715059*z^12+164290694685*z^14+6324261112143*z^18-1183694886083*z^16+ 59842273*z^50-1203429087*z^48-25311627832833*z^20-76379814130543*z^36+ 174462360949389*z^34+397876107359005*z^30+1183694886083*z^42-164290694685*z^44+ 16649715059*z^46+z^58-343*z^56+36825*z^54-1933431*z^52-302316957162867*z^32+ 25311627832833*z^38-6324261112143*z^40) The first , 40, terms are: [0, 102, 0, 18836, 0, 3725041, 0, 745774238, 0, 149715883231, 0, 30075638054541, 0, 6042733000851992, 0, 1214143624032632115, 0, 243955896828822634677, 0, 49017792260941677170250, 0, 9849098289946046744604295, 0, 1978970279894082255274427540, 0, 397632699730745239501240949818, 0, 79895977869516670971911183402035, 0, 16053426447253055510594471759676275, 0, 3225600433950558170676101251157797642, 0, 648116973384864413059245765369019163156, 0, 130225556392031616648327143650041887602847, 0, 26166103086687417352723904664254513475541098, 0, 5257531391796496364930197612487596848672952549] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4490022317412879583 z - 1049222706678966505 z - 329 z 24 22 4 6 + 203683903213282917 z - 32653433068041552 z + 44307 z - 3360316 z 8 10 12 14 + 164572334 z - 5620690492 z + 140621620469 z - 2666469108507 z 18 16 50 - 458149690790368 z + 39283870769591 z - 459383452835476913267 z 48 20 + 696010781014505991395 z + 4291374172281076 z 36 34 + 256421305587309967513 z - 120871208445727273812 z 66 80 88 84 86 - 32653433068041552 z + 164572334 z + z + 44307 z - 329 z 82 64 30 - 3360316 z + 203683903213282917 z - 16039930841696729524 z 42 44 - 892710429456548453072 z + 969871417888779208024 z 46 58 - 892710429456548453072 z - 16039930841696729524 z 56 54 + 48021525975817574258 z - 120871208445727273812 z 52 60 + 256421305587309967513 z + 4490022317412879583 z 70 68 78 - 458149690790368 z + 4291374172281076 z - 5620690492 z 32 38 + 48021525975817574258 z - 459383452835476913267 z 40 62 76 + 696010781014505991395 z - 1049222706678966505 z + 140621620469 z 74 72 / 2 - 2666469108507 z + 39283870769591 z ) / ((-1 + z ) (1 / 28 26 2 + 17843627269900182156 z - 3983875690982976550 z - 446 z 24 22 4 6 + 735830118615749723 z - 111775518209264712 z + 71818 z - 6202576 z 8 10 12 14 + 337788175 z - 12655141928 z + 344221823712 z - 7049237016386 z 18 16 50 - 1390621092676968 z + 111548470785709 z - 2136788746052425270170 z 48 20 + 3287432921308170999511 z + 13861893830440682 z 36 34 + 1167808308486014125290 z - 536024508517113652704 z 66 80 88 84 86 - 111775518209264712 z + 337788175 z + z + 71818 z - 446 z 82 64 30 - 6202576 z + 735830118615749723 z - 66426817278453410152 z 42 44 - 4255878768548947974032 z + 4638167914236412434956 z 46 58 - 4255878768548947974032 z - 66426817278453410152 z 56 54 + 206297271283035645921 z - 536024508517113652704 z 52 60 + 1167808308486014125290 z + 17843627269900182156 z 70 68 78 - 1390621092676968 z + 13861893830440682 z - 12655141928 z 32 38 + 206297271283035645921 z - 2136788746052425270170 z 40 62 76 + 3287432921308170999511 z - 3983875690982976550 z + 344221823712 z 74 72 - 7049237016386 z + 111548470785709 z )) And in Maple-input format, it is: -(1+4490022317412879583*z^28-1049222706678966505*z^26-329*z^2+ 203683903213282917*z^24-32653433068041552*z^22+44307*z^4-3360316*z^6+164572334* z^8-5620690492*z^10+140621620469*z^12-2666469108507*z^14-458149690790368*z^18+ 39283870769591*z^16-459383452835476913267*z^50+696010781014505991395*z^48+ 4291374172281076*z^20+256421305587309967513*z^36-120871208445727273812*z^34-\ 32653433068041552*z^66+164572334*z^80+z^88+44307*z^84-329*z^86-3360316*z^82+ 203683903213282917*z^64-16039930841696729524*z^30-892710429456548453072*z^42+ 969871417888779208024*z^44-892710429456548453072*z^46-16039930841696729524*z^58 +48021525975817574258*z^56-120871208445727273812*z^54+256421305587309967513*z^ 52+4490022317412879583*z^60-458149690790368*z^70+4291374172281076*z^68-\ 5620690492*z^78+48021525975817574258*z^32-459383452835476913267*z^38+ 696010781014505991395*z^40-1049222706678966505*z^62+140621620469*z^76-\ 2666469108507*z^74+39283870769591*z^72)/(-1+z^2)/(1+17843627269900182156*z^28-\ 3983875690982976550*z^26-446*z^2+735830118615749723*z^24-111775518209264712*z^ 22+71818*z^4-6202576*z^6+337788175*z^8-12655141928*z^10+344221823712*z^12-\ 7049237016386*z^14-1390621092676968*z^18+111548470785709*z^16-\ 2136788746052425270170*z^50+3287432921308170999511*z^48+13861893830440682*z^20+ 1167808308486014125290*z^36-536024508517113652704*z^34-111775518209264712*z^66+ 337788175*z^80+z^88+71818*z^84-446*z^86-6202576*z^82+735830118615749723*z^64-\ 66426817278453410152*z^30-4255878768548947974032*z^42+4638167914236412434956*z^ 44-4255878768548947974032*z^46-66426817278453410152*z^58+206297271283035645921* z^56-536024508517113652704*z^54+1167808308486014125290*z^52+ 17843627269900182156*z^60-1390621092676968*z^70+13861893830440682*z^68-\ 12655141928*z^78+206297271283035645921*z^32-2136788746052425270170*z^38+ 3287432921308170999511*z^40-3983875690982976550*z^62+344221823712*z^76-\ 7049237016386*z^74+111548470785709*z^72) The first , 40, terms are: [0, 118, 0, 24789, 0, 5467609, 0, 1213629002, 0, 269698150977, 0, 59949044070581, 0, 13326432371747546, 0, 2962459610694892941, 0, 658556049019224776537, 0, 146397447134755370245798, 0, 32544258552027353815639845, 0, 7234612758660900657983634301, 0, 1608259798971885467623778932006, 0, 357517351572540037149887271247185, 0, 79476373725514338084168301756412005, 0, 17667657121432111491132570115124287066, 0, 3927533348010996188405221136687008177837, 0, 873093590963569227761786199388481234226569, 0, 194089356102316652449697060504682925153638538, 0, 43146208541848185544118456341379685239287704193] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {3, 7}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 544514725214511607757 z - 75483843437272902790 z - 380 z 24 22 4 6 + 8946577954029769725 z - 900773307792924328 z + 61771 z - 5858598 z 102 8 10 12 - 16717402374 z + 369720702 z - 16717402374 z + 568214863794 z 14 18 16 - 15015969674612 z - 5414359698080470 z + 316323374948363 z 50 48 - 133045042224893057529242734 z + 84270271623648860876433559 z 20 36 + 76438236060978375 z + 337778096065541295456655 z 34 66 - 83814584400363228312344 z - 46817957967008207214029792 z 80 100 90 + 18072252888263377347055 z + 568214863794 z - 900773307792924328 z 88 84 + 8946577954029769725 z + 544514725214511607757 z 94 86 96 - 5414359698080470 z - 75483843437272902790 z + 316323374948363 z 98 92 82 - 15015969674612 z + 76438236060978375 z - 3376210461010526203066 z 64 112 110 106 + 84270271623648860876433559 z + z - 380 z - 5858598 z 108 30 42 + 61771 z - 3376210461010526203066 z - 9728846183057765486222854 z 44 46 + 22802360218532563344378510 z - 46817957967008207214029792 z 58 56 - 224099517494010453674621958 z + 239183294371705447770844388 z 54 52 - 224099517494010453674621958 z + 184310881360297473220473349 z 60 70 + 184310881360297473220473349 z - 9728846183057765486222854 z 68 78 + 22802360218532563344378510 z - 83814584400363228312344 z 32 38 + 18072252888263377347055 z - 1185745126228048947006032 z 40 62 + 3632811904764847099834858 z - 133045042224893057529242734 z 76 74 + 337778096065541295456655 z - 1185745126228048947006032 z 72 104 / + 3632811904764847099834858 z + 369720702 z ) / (-1 / 28 26 2 - 2109062634161337558508 z + 275331541292086457251 z + 484 z 24 22 4 - 30699494314813055339 z + 2903750216121298755 z - 92659 z 6 102 8 10 + 9982667 z + 1288195499690 z - 701080298 z + 34818557242 z 12 14 18 - 1288195499690 z + 36813523838009 z + 15312791502857212 z 16 50 - 834514825562322 z + 992413317742823117280827636 z 48 20 - 590916177042191374823060552 z - 231067648194024909 z 36 34 - 1655364325574559766657103 z + 387398113312486046263442 z 66 80 + 590916177042191374823060552 z - 387398113312486046263442 z 100 90 - 36813523838009 z + 30699494314813055339 z 88 84 - 275331541292086457251 z - 13876129145545808639792 z 94 86 + 231067648194024909 z + 2109062634161337558508 z 96 98 92 - 15312791502857212 z + 834514825562322 z - 2903750216121298755 z 82 64 112 + 78777888295854662106620 z - 992413317742823117280827636 z - 484 z 114 110 106 108 + z + 92659 z + 701080298 z - 9982667 z 30 42 + 13876129145545808639792 z + 56906264617730228688880349 z 44 46 - 141599715917291771020306641 z + 308841347498983508616289633 z 58 56 + 2157918157808466387632427167 z - 2157918157808466387632427167 z 54 52 + 1896022206763204960965246730 z - 1463600089368073232810818627 z 60 70 - 1896022206763204960965246730 z + 141599715917291771020306641 z 68 78 - 308841347498983508616289633 z + 1655364325574559766657103 z 32 38 - 78777888295854662106620 z + 6162123736803643613746609 z 40 62 - 20025155844914435303382378 z + 1463600089368073232810818627 z 76 74 - 6162123736803643613746609 z + 20025155844914435303382378 z 72 104 - 56906264617730228688880349 z - 34818557242 z ) And in Maple-input format, it is: -(1+544514725214511607757*z^28-75483843437272902790*z^26-380*z^2+ 8946577954029769725*z^24-900773307792924328*z^22+61771*z^4-5858598*z^6-\ 16717402374*z^102+369720702*z^8-16717402374*z^10+568214863794*z^12-\ 15015969674612*z^14-5414359698080470*z^18+316323374948363*z^16-\ 133045042224893057529242734*z^50+84270271623648860876433559*z^48+ 76438236060978375*z^20+337778096065541295456655*z^36-83814584400363228312344*z^ 34-46817957967008207214029792*z^66+18072252888263377347055*z^80+568214863794*z^ 100-900773307792924328*z^90+8946577954029769725*z^88+544514725214511607757*z^84 -5414359698080470*z^94-75483843437272902790*z^86+316323374948363*z^96-\ 15015969674612*z^98+76438236060978375*z^92-3376210461010526203066*z^82+ 84270271623648860876433559*z^64+z^112-380*z^110-5858598*z^106+61771*z^108-\ 3376210461010526203066*z^30-9728846183057765486222854*z^42+ 22802360218532563344378510*z^44-46817957967008207214029792*z^46-\ 224099517494010453674621958*z^58+239183294371705447770844388*z^56-\ 224099517494010453674621958*z^54+184310881360297473220473349*z^52+ 184310881360297473220473349*z^60-9728846183057765486222854*z^70+ 22802360218532563344378510*z^68-83814584400363228312344*z^78+ 18072252888263377347055*z^32-1185745126228048947006032*z^38+ 3632811904764847099834858*z^40-133045042224893057529242734*z^62+ 337778096065541295456655*z^76-1185745126228048947006032*z^74+ 3632811904764847099834858*z^72+369720702*z^104)/(-1-2109062634161337558508*z^28 +275331541292086457251*z^26+484*z^2-30699494314813055339*z^24+ 2903750216121298755*z^22-92659*z^4+9982667*z^6+1288195499690*z^102-701080298*z^ 8+34818557242*z^10-1288195499690*z^12+36813523838009*z^14+15312791502857212*z^ 18-834514825562322*z^16+992413317742823117280827636*z^50-\ 590916177042191374823060552*z^48-231067648194024909*z^20-\ 1655364325574559766657103*z^36+387398113312486046263442*z^34+ 590916177042191374823060552*z^66-387398113312486046263442*z^80-36813523838009*z ^100+30699494314813055339*z^90-275331541292086457251*z^88-\ 13876129145545808639792*z^84+231067648194024909*z^94+2109062634161337558508*z^ 86-15312791502857212*z^96+834514825562322*z^98-2903750216121298755*z^92+ 78777888295854662106620*z^82-992413317742823117280827636*z^64-484*z^112+z^114+ 92659*z^110+701080298*z^106-9982667*z^108+13876129145545808639792*z^30+ 56906264617730228688880349*z^42-141599715917291771020306641*z^44+ 308841347498983508616289633*z^46+2157918157808466387632427167*z^58-\ 2157918157808466387632427167*z^56+1896022206763204960965246730*z^54-\ 1463600089368073232810818627*z^52-1896022206763204960965246730*z^60+ 141599715917291771020306641*z^70-308841347498983508616289633*z^68+ 1655364325574559766657103*z^78-78777888295854662106620*z^32+ 6162123736803643613746609*z^38-20025155844914435303382378*z^40+ 1463600089368073232810818627*z^62-6162123736803643613746609*z^76+ 20025155844914435303382378*z^74-56906264617730228688880349*z^72-34818557242*z^ 104) The first , 40, terms are: [0, 104, 0, 19448, 0, 3900365, 0, 792582200, 0, 161537575957, 0, 32946897348611, 0, 6720979705784744, 0, 1371102350943072353, 0, 279712566683463554743, 0, 57063090192529074556292, 0, 11641230733565921311218099, 0, 2374884994456317554597141600, 0, 484491641199975872256887621980, 0, 98839376861258637216905754155267, 0, 20163861694348643853087823536075935, 0, 4113556070678798018600861518700009844, 0, 839191609526592592512384234337021878288, 0, 171200427423433911534251106781254009172979, 0, 34925976401051818557138361721440834921596300, 0, 7125121390924760883929070679273864011910460443] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {3, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 36352809836184 z - 31839751872864 z - 248 z 24 22 4 6 + 21373492056187 z - 10966614189768 z + 21810 z - 977048 z 8 10 12 14 + 26109243 z - 453432080 z + 5393725588 z - 45498365584 z 18 16 50 48 - 1262710129832 z + 278801975881 z - 977048 z + 26109243 z 20 36 34 + 4280910947054 z + 4280910947054 z - 10966614189768 z 30 42 44 46 - 31839751872864 z - 45498365584 z + 5393725588 z - 453432080 z 56 54 52 32 38 + z - 248 z + 21810 z + 21373492056187 z - 1262710129832 z 40 / 28 26 + 278801975881 z ) / ((1 + 173200959450024 z - 150458247912896 z / 2 24 22 4 6 - 352 z + 98577960178267 z - 48627620916704 z + 38926 z - 2063104 z 8 10 12 14 + 63267755 z - 1238396256 z + 16393021388 z - 152269875424 z 18 16 50 48 - 4973499523520 z + 1017375418873 z - 2063104 z + 63267755 z 20 36 34 + 18000661170770 z + 18000661170770 z - 48627620916704 z 30 42 44 - 150458247912896 z - 152269875424 z + 16393021388 z 46 56 54 52 32 - 1238396256 z + z - 352 z + 38926 z + 98577960178267 z 38 40 2 - 4973499523520 z + 1017375418873 z ) (-1 + z )) And in Maple-input format, it is: -(1+36352809836184*z^28-31839751872864*z^26-248*z^2+21373492056187*z^24-\ 10966614189768*z^22+21810*z^4-977048*z^6+26109243*z^8-453432080*z^10+5393725588 *z^12-45498365584*z^14-1262710129832*z^18+278801975881*z^16-977048*z^50+ 26109243*z^48+4280910947054*z^20+4280910947054*z^36-10966614189768*z^34-\ 31839751872864*z^30-45498365584*z^42+5393725588*z^44-453432080*z^46+z^56-248*z^ 54+21810*z^52+21373492056187*z^32-1262710129832*z^38+278801975881*z^40)/(1+ 173200959450024*z^28-150458247912896*z^26-352*z^2+98577960178267*z^24-\ 48627620916704*z^22+38926*z^4-2063104*z^6+63267755*z^8-1238396256*z^10+ 16393021388*z^12-152269875424*z^14-4973499523520*z^18+1017375418873*z^16-\ 2063104*z^50+63267755*z^48+18000661170770*z^20+18000661170770*z^36-\ 48627620916704*z^34-150458247912896*z^30-152269875424*z^42+16393021388*z^44-\ 1238396256*z^46+z^56-352*z^54+38926*z^52+98577960178267*z^32-4973499523520*z^38 +1017375418873*z^40)/(-1+z^2) The first , 40, terms are: [0, 105, 0, 19597, 0, 3918533, 0, 795002717, 0, 161905793573, 0, 33007650520209, 0, 6731263883281321, 0, 1372826187772139737, 0, 279991612863524813761, 0, 57105441315698586764917, 0, 11646912655538742288593581, 0, 2375441772224529663834384501, 0, 484482427723491291657273462589, 0, 98812455124701558581727297204281, 0, 20153262237338851188688816752452561, 0, 4110352078669006111797329950632267313, 0, 838325529118240895831316211725354725017, 0, 170980412269705545356176858929185900357597, 0, 34872254711360200581587164906046695352368917, 0, 7112359436622173393527880413753427833078394381] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 1804073912614908 z + 1112263494739756 z + 314 z 24 22 4 6 - 536520934789814 z + 201605094396980 z - 37280 z + 2241845 z 8 10 12 14 - 78182209 z + 1738592372 z - 26296195914 z + 282644551385 z 18 16 50 + 13108601251132 z - 2225288611516 z + 26296195914 z 48 20 36 - 282644551385 z - 58658968913156 z - 1112263494739756 z 34 30 42 + 1804073912614908 z + 2295981048694538 z + 58658968913156 z 44 46 58 56 - 13108601251132 z + 2225288611516 z + 37280 z - 2241845 z 54 52 60 32 + 78182209 z - 1738592372 z - 314 z - 2295981048694538 z 38 40 62 / + 536520934789814 z - 201605094396980 z + z ) / (1 / 28 26 2 + 13782419960261530 z - 7566688387357380 z - 430 z 24 22 4 6 + 3255184693319178 z - 1092601986405108 z + 64453 z - 4640298 z 8 10 12 14 + 188211716 z - 4779438446 z + 81610411601 z - 983700218766 z 18 16 50 - 56917126006748 z + 8655594305861 z - 983700218766 z 48 20 36 + 8655594305861 z + 284390739303168 z + 13782419960261530 z 34 64 30 - 19728408004487832 z + z - 19728408004487832 z 42 44 46 - 1092601986405108 z + 284390739303168 z - 56917126006748 z 58 56 54 52 - 4640298 z + 188211716 z - 4779438446 z + 81610411601 z 60 32 38 + 64453 z + 22229742413649400 z - 7566688387357380 z 40 62 + 3255184693319178 z - 430 z ) And in Maple-input format, it is: -(-1-1804073912614908*z^28+1112263494739756*z^26+314*z^2-536520934789814*z^24+ 201605094396980*z^22-37280*z^4+2241845*z^6-78182209*z^8+1738592372*z^10-\ 26296195914*z^12+282644551385*z^14+13108601251132*z^18-2225288611516*z^16+ 26296195914*z^50-282644551385*z^48-58658968913156*z^20-1112263494739756*z^36+ 1804073912614908*z^34+2295981048694538*z^30+58658968913156*z^42-13108601251132* z^44+2225288611516*z^46+37280*z^58-2241845*z^56+78182209*z^54-1738592372*z^52-\ 314*z^60-2295981048694538*z^32+536520934789814*z^38-201605094396980*z^40+z^62)/ (1+13782419960261530*z^28-7566688387357380*z^26-430*z^2+3255184693319178*z^24-\ 1092601986405108*z^22+64453*z^4-4640298*z^6+188211716*z^8-4779438446*z^10+ 81610411601*z^12-983700218766*z^14-56917126006748*z^18+8655594305861*z^16-\ 983700218766*z^50+8655594305861*z^48+284390739303168*z^20+13782419960261530*z^ 36-19728408004487832*z^34+z^64-19728408004487832*z^30-1092601986405108*z^42+ 284390739303168*z^44-56917126006748*z^46-4640298*z^58+188211716*z^56-4779438446 *z^54+81610411601*z^52+64453*z^60+22229742413649400*z^32-7566688387357380*z^38+ 3255184693319178*z^40-430*z^62) The first , 40, terms are: [0, 116, 0, 22707, 0, 4685915, 0, 979654240, 0, 205805577409, 0, 43324162766657, 0, 9128307573018600, 0, 1924067390850553115, 0, 405625329340262640019, 0, 85519032902711778192556, 0, 18030800107648985492625857, 0, 3801662911048696921920143169, 0, 801558138256821797690272745468, 0, 169004274913344296331139661945299, 0, 35633698308006836622741967753235611, 0, 7513189590155919777820528552237811864, 0, 1584119366901496302106616647322564998977, 0, 334003875253176378035819711458984282148801, 0, 70423098349843559169270877162522907043878704, 0, 14848369292207501194838978493992817473553469915] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6321328645288994693 z - 1469210620773905108 z - 340 z 24 22 4 6 + 283219092602680285 z - 44996970621933716 z + 47753 z - 3785950 z 8 10 12 14 + 193325288 z - 6850772966 z + 176873372983 z - 3443195768604 z 18 16 50 - 615340877516916 z + 51838102261151 z - 654412544775855231420 z 48 20 + 992132408403780274947 z + 5846556681309872 z 36 34 + 364912650044321014331 z - 171751137374476292754 z 66 80 88 84 86 - 44996970621933716 z + 193325288 z + z + 47753 z - 340 z 82 64 30 - 3785950 z + 283219092602680285 z - 22674141874613095978 z 42 44 - 1272962104501080679288 z + 1383142346422658835616 z 46 58 - 1272962104501080679288 z - 22674141874613095978 z 56 54 + 68088076941181063448 z - 171751137374476292754 z 52 60 + 364912650044321014331 z + 6321328645288994693 z 70 68 78 - 615340877516916 z + 5846556681309872 z - 6850772966 z 32 38 + 68088076941181063448 z - 654412544775855231420 z 40 62 76 + 992132408403780274947 z - 1469210620773905108 z + 176873372983 z 74 72 / 2 - 3443195768604 z + 51838102261151 z ) / ((-1 + z ) (1 / 28 26 2 + 24966748051130695155 z - 5555281388851450847 z - 451 z 24 22 4 6 + 1020432978414432761 z - 153746716208234288 z + 75267 z - 6798960 z 8 10 12 14 + 387540082 z - 15142847688 z + 427286229385 z - 9024256598453 z 18 16 50 - 1862258808024304 z + 146434896782603 z - 2992437035096151296105 z 48 20 + 4600550061346557329211 z + 18850240276070476 z 36 34 + 1636703736926186036417 z - 751701974997061530304 z 66 80 88 84 86 - 153746716208234288 z + 387540082 z + z + 75267 z - 451 z 82 64 30 - 6798960 z + 1020432978414432761 z - 93111095188033004232 z 42 44 - 5952877150001813230752 z + 6486457304536462638248 z 46 58 - 5952877150001813230752 z - 93111095188033004232 z 56 54 + 289349526167692620334 z - 751701974997061530304 z 52 60 + 1636703736926186036417 z + 24966748051130695155 z 70 68 78 - 1862258808024304 z + 18850240276070476 z - 15142847688 z 32 38 + 289349526167692620334 z - 2992437035096151296105 z 40 62 76 + 4600550061346557329211 z - 5555281388851450847 z + 427286229385 z 74 72 - 9024256598453 z + 146434896782603 z )) And in Maple-input format, it is: -(1+6321328645288994693*z^28-1469210620773905108*z^26-340*z^2+ 283219092602680285*z^24-44996970621933716*z^22+47753*z^4-3785950*z^6+193325288* z^8-6850772966*z^10+176873372983*z^12-3443195768604*z^14-615340877516916*z^18+ 51838102261151*z^16-654412544775855231420*z^50+992132408403780274947*z^48+ 5846556681309872*z^20+364912650044321014331*z^36-171751137374476292754*z^34-\ 44996970621933716*z^66+193325288*z^80+z^88+47753*z^84-340*z^86-3785950*z^82+ 283219092602680285*z^64-22674141874613095978*z^30-1272962104501080679288*z^42+ 1383142346422658835616*z^44-1272962104501080679288*z^46-22674141874613095978*z^ 58+68088076941181063448*z^56-171751137374476292754*z^54+364912650044321014331*z ^52+6321328645288994693*z^60-615340877516916*z^70+5846556681309872*z^68-\ 6850772966*z^78+68088076941181063448*z^32-654412544775855231420*z^38+ 992132408403780274947*z^40-1469210620773905108*z^62+176873372983*z^76-\ 3443195768604*z^74+51838102261151*z^72)/(-1+z^2)/(1+24966748051130695155*z^28-\ 5555281388851450847*z^26-451*z^2+1020432978414432761*z^24-153746716208234288*z^ 22+75267*z^4-6798960*z^6+387540082*z^8-15142847688*z^10+427286229385*z^12-\ 9024256598453*z^14-1862258808024304*z^18+146434896782603*z^16-\ 2992437035096151296105*z^50+4600550061346557329211*z^48+18850240276070476*z^20+ 1636703736926186036417*z^36-751701974997061530304*z^34-153746716208234288*z^66+ 387540082*z^80+z^88+75267*z^84-451*z^86-6798960*z^82+1020432978414432761*z^64-\ 93111095188033004232*z^30-5952877150001813230752*z^42+6486457304536462638248*z^ 44-5952877150001813230752*z^46-93111095188033004232*z^58+289349526167692620334* z^56-751701974997061530304*z^54+1636703736926186036417*z^52+ 24966748051130695155*z^60-1862258808024304*z^70+18850240276070476*z^68-\ 15142847688*z^78+289349526167692620334*z^32-2992437035096151296105*z^38+ 4600550061346557329211*z^40-5555281388851450847*z^62+427286229385*z^76-\ 9024256598453*z^74+146434896782603*z^72) The first , 40, terms are: [0, 112, 0, 22659, 0, 4849729, 0, 1045283016, 0, 225532894503, 0, 48670786317823, 0, 10503730299994344, 0, 2266848030054500257, 0, 489217600468007282931, 0, 105580066137470943184624, 0, 22785671410512856659000457, 0, 4917470246489559274735267481, 0, 1061259654034308603155455088496, 0, 229034848870299003881274377772851, 0, 49428960964842951651046857041815617, 0, 10667469139684206689770454294756509096, 0, 2302190772914130971789049033432872679215, 0, 496845342180983744519765828898203382683831, 0, 107226254640353549734199824004164909624786696, 0, 23140942881199951961128688717244992421422909025] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4686590198905820672 z - 1114219273254611904 z - 348 z 24 22 4 6 + 220394052472717621 z - 36040007579305836 z + 49300 z - 3889444 z 8 10 12 14 + 195286735 z - 6741880572 z + 168545604256 z - 3166748869372 z 18 16 50 - 526414090958404 z + 45967408739531 z - 452528217344057236408 z 48 20 + 681943347509127000438 z + 4833635627583620 z 36 34 + 254497396044837159656 z - 121120688887239881752 z 66 80 88 84 86 - 36040007579305836 z + 195286735 z + z + 49300 z - 348 z 82 64 30 - 3889444 z + 220394052472717621 z - 16484327773642148416 z 42 44 - 871838244028846058504 z + 946171220729592142528 z 46 58 - 871838244028846058504 z - 16484327773642148416 z 56 54 + 48684048960612770746 z - 121120688887239881752 z 52 60 + 254497396044837159656 z + 4686590198905820672 z 70 68 78 - 526414090958404 z + 4833635627583620 z - 6741880572 z 32 38 + 48684048960612770746 z - 452528217344057236408 z 40 62 76 + 681943347509127000438 z - 1114219273254611904 z + 168545604256 z 74 72 / 2 - 3166748869372 z + 45967408739531 z ) / ((-1 + z ) (1 / 28 26 2 + 18689256609001271024 z - 4252459556492671544 z - 462 z 24 22 4 6 + 801983266115505941 z - 124572909541789454 z + 79050 z - 7192534 z 8 10 12 14 + 405374099 z - 15437613912 z + 420579421084 z - 8534054164592 z 18 16 50 - 1622310341643638 z + 132848335681663 z - 2104982218658028601012 z 48 20 + 3220722150102720859358 z + 15809109882097058 z 36 34 + 1159353098604867915156 z - 537481369648408803876 z 66 80 88 84 86 - 124572909541789454 z + 405374099 z + z + 79050 z - 462 z 82 64 30 - 7192534 z + 801983266115505941 z - 68416193883740070216 z 42 44 - 4155796279373337619896 z + 4524127307983519366888 z 46 58 - 4155796279373337619896 z - 68416193883740070216 z 56 54 + 209408639994393204986 z - 537481369648408803876 z 52 60 + 1159353098604867915156 z + 18689256609001271024 z 70 68 78 - 1622310341643638 z + 15809109882097058 z - 15437613912 z 32 38 + 209408639994393204986 z - 2104982218658028601012 z 40 62 76 + 3220722150102720859358 z - 4252459556492671544 z + 420579421084 z 74 72 - 8534054164592 z + 132848335681663 z )) And in Maple-input format, it is: -(1+4686590198905820672*z^28-1114219273254611904*z^26-348*z^2+ 220394052472717621*z^24-36040007579305836*z^22+49300*z^4-3889444*z^6+195286735* z^8-6741880572*z^10+168545604256*z^12-3166748869372*z^14-526414090958404*z^18+ 45967408739531*z^16-452528217344057236408*z^50+681943347509127000438*z^48+ 4833635627583620*z^20+254497396044837159656*z^36-121120688887239881752*z^34-\ 36040007579305836*z^66+195286735*z^80+z^88+49300*z^84-348*z^86-3889444*z^82+ 220394052472717621*z^64-16484327773642148416*z^30-871838244028846058504*z^42+ 946171220729592142528*z^44-871838244028846058504*z^46-16484327773642148416*z^58 +48684048960612770746*z^56-121120688887239881752*z^54+254497396044837159656*z^ 52+4686590198905820672*z^60-526414090958404*z^70+4833635627583620*z^68-\ 6741880572*z^78+48684048960612770746*z^32-452528217344057236408*z^38+ 681943347509127000438*z^40-1114219273254611904*z^62+168545604256*z^76-\ 3166748869372*z^74+45967408739531*z^72)/(-1+z^2)/(1+18689256609001271024*z^28-\ 4252459556492671544*z^26-462*z^2+801983266115505941*z^24-124572909541789454*z^ 22+79050*z^4-7192534*z^6+405374099*z^8-15437613912*z^10+420579421084*z^12-\ 8534054164592*z^14-1622310341643638*z^18+132848335681663*z^16-\ 2104982218658028601012*z^50+3220722150102720859358*z^48+15809109882097058*z^20+ 1159353098604867915156*z^36-537481369648408803876*z^34-124572909541789454*z^66+ 405374099*z^80+z^88+79050*z^84-462*z^86-7192534*z^82+801983266115505941*z^64-\ 68416193883740070216*z^30-4155796279373337619896*z^42+4524127307983519366888*z^ 44-4155796279373337619896*z^46-68416193883740070216*z^58+209408639994393204986* z^56-537481369648408803876*z^54+1159353098604867915156*z^52+ 18689256609001271024*z^60-1622310341643638*z^70+15809109882097058*z^68-\ 15437613912*z^78+209408639994393204986*z^32-2104982218658028601012*z^38+ 3220722150102720859358*z^40-4252459556492671544*z^62+420579421084*z^76-\ 8534054164592*z^74+132848335681663*z^72) The first , 40, terms are: [0, 115, 0, 23033, 0, 4902539, 0, 1057427923, 0, 228920786297, 0, 49613164126347, 0, 10756054856659569, 0, 2332131195806312049, 0, 505668971552846123195, 0, 109643704421066997791289, 0, 23774004108758242632964483, 0, 5154913046268145941444964955, 0, 1117739166882087017147214465593, 0, 242359266716337406903221532716035, 0, 52550735758218076518061023796625473, 0, 11394570906242874378173074388879526721, 0, 2470683703280102065787002780017856707171, 0, 535718107870664540080387205777704134001337, 0, 116159705420887452471586092401221155768159675, 0, 25186897672672988266744546851149836224227188515] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1133014186668755 z - 640186325346830 z - 266 z 24 22 4 6 + 286866207221212 z - 101551383935910 z + 26136 z - 1341270 z 8 10 12 14 + 41780920 z - 861810255 z + 12447843492 z - 130760536848 z 18 16 50 - 6126956363949 z + 1025883235924 z - 130760536848 z 48 20 36 + 1025883235924 z + 28246243868440 z + 1133014186668755 z 34 64 30 42 - 1594309943763280 z + z - 1594309943763280 z - 101551383935910 z 44 46 58 56 + 28246243868440 z - 6126956363949 z - 1341270 z + 41780920 z 54 52 60 32 - 861810255 z + 12447843492 z + 26136 z + 1786282449395520 z 38 40 62 / - 640186325346830 z + 286866207221212 z - 266 z ) / (-1 / 28 26 2 - 8235566002999076 z + 4176666347667529 z + 381 z 24 22 4 6 - 1681677978371525 z + 535258298673232 z - 46336 z + 2793468 z 8 10 12 14 - 99898649 z + 2337443663 z - 38019051688 z + 447540465564 z 18 16 50 + 26095586474705 z - 3921293076821 z + 3921293076821 z 48 20 36 - 26095586474705 z - 133870961216978 z - 12930436987890456 z 34 66 64 30 + 16195090933522289 z + z - 381 z + 12930436987890456 z 42 44 46 + 1681677978371525 z - 535258298673232 z + 133870961216978 z 58 56 54 52 + 99898649 z - 2337443663 z + 38019051688 z - 447540465564 z 60 32 38 - 2793468 z - 16195090933522289 z + 8235566002999076 z 40 62 - 4176666347667529 z + 46336 z ) And in Maple-input format, it is: -(1+1133014186668755*z^28-640186325346830*z^26-266*z^2+286866207221212*z^24-\ 101551383935910*z^22+26136*z^4-1341270*z^6+41780920*z^8-861810255*z^10+ 12447843492*z^12-130760536848*z^14-6126956363949*z^18+1025883235924*z^16-\ 130760536848*z^50+1025883235924*z^48+28246243868440*z^20+1133014186668755*z^36-\ 1594309943763280*z^34+z^64-1594309943763280*z^30-101551383935910*z^42+ 28246243868440*z^44-6126956363949*z^46-1341270*z^58+41780920*z^56-861810255*z^ 54+12447843492*z^52+26136*z^60+1786282449395520*z^32-640186325346830*z^38+ 286866207221212*z^40-266*z^62)/(-1-8235566002999076*z^28+4176666347667529*z^26+ 381*z^2-1681677978371525*z^24+535258298673232*z^22-46336*z^4+2793468*z^6-\ 99898649*z^8+2337443663*z^10-38019051688*z^12+447540465564*z^14+26095586474705* z^18-3921293076821*z^16+3921293076821*z^50-26095586474705*z^48-133870961216978* z^20-12930436987890456*z^36+16195090933522289*z^34+z^66-381*z^64+ 12930436987890456*z^30+1681677978371525*z^42-535258298673232*z^44+ 133870961216978*z^46+99898649*z^58-2337443663*z^56+38019051688*z^54-\ 447540465564*z^52-2793468*z^60-16195090933522289*z^32+8235566002999076*z^38-\ 4176666347667529*z^40+46336*z^62) The first , 40, terms are: [0, 115, 0, 23615, 0, 5120873, 0, 1119959064, 0, 245378667649, 0, 53783972259243, 0, 11789972319867235, 0, 2584540877777309725, 0, 566574013217026820797, 0, 124202558274987937354579, 0, 27227300845329691986771979, 0, 5968685251610011118331210401, 0, 1308436874459355228327504718104, 0, 286831520100846848925189640283065, 0, 62878326527585219161809402551230511, 0, 13783993986486971406342405254994322019, 0, 3021684906824956153771992436557955181665, 0, 662404502299239568970229746990167492996929, 0, 145210284393778685997983398813749475566250915, 0, 31832553402877961255743819112863555986368676655] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 1796459 z + 43489487 z + 310 z - 536429428 z 22 4 6 8 10 + 3500178011 z - 35040 z + 1796459 z - 43489487 z + 536429428 z 12 14 18 16 - 3500178011 z + 12304530571 z + 23146010693 z - 23146010693 z 20 34 30 32 / 36 34 - 12304530571 z + z + 35040 z - 310 z ) / (z - 438 z / 32 30 28 26 + 63849 z - 4105378 z + 125129814 z - 1905564806 z 24 22 20 18 + 15293102502 z - 67586038236 z + 166646744255 z - 226671617124 z 16 14 12 10 + 166646744255 z - 67586038236 z + 15293102502 z - 1905564806 z 8 6 4 2 + 125129814 z - 4105378 z + 63849 z - 438 z + 1) And in Maple-input format, it is: -(-1-1796459*z^28+43489487*z^26+310*z^2-536429428*z^24+3500178011*z^22-35040*z^ 4+1796459*z^6-43489487*z^8+536429428*z^10-3500178011*z^12+12304530571*z^14+ 23146010693*z^18-23146010693*z^16-12304530571*z^20+z^34+35040*z^30-310*z^32)/(z ^36-438*z^34+63849*z^32-4105378*z^30+125129814*z^28-1905564806*z^26+15293102502 *z^24-67586038236*z^22+166646744255*z^20-226671617124*z^18+166646744255*z^16-\ 67586038236*z^14+15293102502*z^12-1905564806*z^10+125129814*z^8-4105378*z^6+ 63849*z^4-438*z^2+1) The first , 40, terms are: [0, 128, 0, 27255, 0, 6073937, 0, 1364027968, 0, 306874043047, 0, 69076522749047, 0, 15551769926903808, 0, 3501527680734064673, 0, 788398486260480329095, 0, 177516171927534579120640, 0, 39969760901244286981182353, 0, 8999652532227892722294874545, 0, 2026376547145318726485405663872, 0, 456262359440633545769471907270055, 0, 102732809631768229798213943583936961, 0, 23131494981123821854830911982844474112, 0, 5208326998293929107752185477243602658583, 0, 1172715825788115536382918948373839684999047, 0, 264050703963166923861889835789332078643914816, 0, 59454108809649248957683496205003350611385542705] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6993403129806164373 z - 1630833902325011559 z - 359 z 24 22 4 6 + 315699413864052525 z - 50415585629742440 z + 52833 z - 4312314 z 8 10 12 14 + 223292682 z - 7943157582 z + 204635529607 z - 3962311662533 z 18 16 50 - 698323011559656 z + 59249116422559 z - 718510005408460050861 z 48 20 + 1088810840471888426675 z + 6590263687582044 z 36 34 + 400934908595659534723 z - 188895881494521672030 z 66 80 88 84 86 - 50415585629742440 z + 223292682 z + z + 52833 z - 359 z 82 64 30 - 4312314 z + 315699413864052525 z - 25020682203525965210 z 42 44 - 1396645637656768177136 z + 1517404615010091920712 z 46 58 - 1396645637656768177136 z - 25020682203525965210 z 56 54 + 74990349710791961142 z - 188895881494521672030 z 52 60 + 400934908595659534723 z + 6993403129806164373 z 70 68 78 - 698323011559656 z + 6590263687582044 z - 7943157582 z 32 38 + 74990349710791961142 z - 718510005408460050861 z 40 62 76 + 1088810840471888426675 z - 1630833902325011559 z + 204635529607 z 74 72 / - 3962311662533 z + 59249116422559 z ) / (-1 / 28 26 2 - 33958727122135964284 z + 7347846233089897863 z + 469 z 24 22 4 6 - 1319766726832641059 z + 195458890741529146 z - 83706 z + 7945002 z 8 10 12 14 - 466409799 z + 18507173039 z - 526011483248 z + 11149477908352 z 18 16 50 + 2319488693817541 z - 181494081701969 z + 8390888763531796856051 z 48 20 - 11663207034224588684471 z - 23670356563089834 z 36 34 - 2639885500538053703802 z + 1151295102875485669001 z 66 80 90 88 84 + 1319766726832641059 z - 18507173039 z + z - 469 z - 7945002 z 86 82 64 + 83706 z + 466409799 z - 7347846233089897863 z 30 42 + 130979569261178975756 z + 11663207034224588684471 z 44 46 - 13748164477895499587260 z + 13748164477895499587260 z 58 56 + 423433523563107493793 z - 1151295102875485669001 z 54 52 + 2639885500538053703802 z - 5115585104272269562922 z 60 70 - 130979569261178975756 z + 23670356563089834 z 68 78 32 - 195458890741529146 z + 526011483248 z - 423433523563107493793 z 38 40 + 5115585104272269562922 z - 8390888763531796856051 z 62 76 74 + 33958727122135964284 z - 11149477908352 z + 181494081701969 z 72 - 2319488693817541 z ) And in Maple-input format, it is: -(1+6993403129806164373*z^28-1630833902325011559*z^26-359*z^2+ 315699413864052525*z^24-50415585629742440*z^22+52833*z^4-4312314*z^6+223292682* z^8-7943157582*z^10+204635529607*z^12-3962311662533*z^14-698323011559656*z^18+ 59249116422559*z^16-718510005408460050861*z^50+1088810840471888426675*z^48+ 6590263687582044*z^20+400934908595659534723*z^36-188895881494521672030*z^34-\ 50415585629742440*z^66+223292682*z^80+z^88+52833*z^84-359*z^86-4312314*z^82+ 315699413864052525*z^64-25020682203525965210*z^30-1396645637656768177136*z^42+ 1517404615010091920712*z^44-1396645637656768177136*z^46-25020682203525965210*z^ 58+74990349710791961142*z^56-188895881494521672030*z^54+400934908595659534723*z ^52+6993403129806164373*z^60-698323011559656*z^70+6590263687582044*z^68-\ 7943157582*z^78+74990349710791961142*z^32-718510005408460050861*z^38+ 1088810840471888426675*z^40-1630833902325011559*z^62+204635529607*z^76-\ 3962311662533*z^74+59249116422559*z^72)/(-1-33958727122135964284*z^28+ 7347846233089897863*z^26+469*z^2-1319766726832641059*z^24+195458890741529146*z^ 22-83706*z^4+7945002*z^6-466409799*z^8+18507173039*z^10-526011483248*z^12+ 11149477908352*z^14+2319488693817541*z^18-181494081701969*z^16+ 8390888763531796856051*z^50-11663207034224588684471*z^48-23670356563089834*z^20 -2639885500538053703802*z^36+1151295102875485669001*z^34+1319766726832641059*z^ 66-18507173039*z^80+z^90-469*z^88-7945002*z^84+83706*z^86+466409799*z^82-\ 7347846233089897863*z^64+130979569261178975756*z^30+11663207034224588684471*z^ 42-13748164477895499587260*z^44+13748164477895499587260*z^46+ 423433523563107493793*z^58-1151295102875485669001*z^56+2639885500538053703802*z ^54-5115585104272269562922*z^52-130979569261178975756*z^60+23670356563089834*z^ 70-195458890741529146*z^68+526011483248*z^78-423433523563107493793*z^32+ 5115585104272269562922*z^38-8390888763531796856051*z^40+33958727122135964284*z^ 62-11149477908352*z^76+181494081701969*z^74-2319488693817541*z^72) The first , 40, terms are: [0, 110, 0, 20717, 0, 4141301, 0, 838966070, 0, 170678889325, 0, 34776351240373, 0, 7090044365907630, 0, 1445834263572174085, 0, 294870048516810769917, 0, 60139564909811747971510, 0, 12265834510229226932745369, 0, 2501709588687068963780061945, 0, 510244009964876778164892584806, 0, 104068536166316893034750253768205, 0, 21225659221706819294032042760724373, 0, 4329153811016415701040483288412855806, 0, 882967830783622262187282682478437200949, 0, 180088823619198836045548860545396502857549, 0, 36730652833946971901408842745744645048263014, 0, 7491530226492200301415989335420614828984928005] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 7932030752318466185 z - 1850939445746394559 z - 363 z 24 22 4 6 + 358257752244313568 z - 57139107793490816 z + 54212 z - 4498832 z 8 10 12 14 + 236994901 z - 8568677634 z + 223918147921 z - 4387038352484 z 18 16 50 - 785641040662954 z + 66207228256525 z - 809660850885774158818 z 48 20 + 1225897074105872201316 z + 7448546367537569 z 36 34 + 452310203888557579040 z - 213388143332884604012 z 66 80 88 84 86 - 57139107793490816 z + 236994901 z + z + 54212 z - 363 z 82 64 30 - 4498832 z + 358257752244313568 z - 28345929922097225198 z 42 44 - 1571666873920515632026 z + 1707257312316329453848 z 46 58 - 1571666873920515632026 z - 28345929922097225198 z 56 54 + 84836929017126425454 z - 213388143332884604012 z 52 60 + 452310203888557579040 z + 7932030752318466185 z 70 68 78 - 785641040662954 z + 7448546367537569 z - 8568677634 z 32 38 + 84836929017126425454 z - 809660850885774158818 z 40 62 76 + 1225897074105872201316 z - 1850939445746394559 z + 223918147921 z 74 72 / 2 - 4387038352484 z + 66207228256525 z ) / ((-1 + z ) (1 / 28 26 2 + 31829455459027511428 z - 7098286218615941118 z - 484 z 24 22 4 6 + 1307903343280345703 z - 197827492848049023 z + 87586 z - 8367383 z 8 10 12 14 + 493223095 z - 19618002294 z + 557741981140 z - 11794033689397 z 18 16 50 - 2418221686240772 z + 190932369080569 z - 3805362841326046182674 z 48 20 + 5851555510782034901165 z + 24363932417577742 z 36 34 + 2081000566967221532737 z - 955774707386469584958 z 66 80 88 84 86 - 197827492848049023 z + 493223095 z + z + 87586 z - 484 z 82 64 30 - 8367383 z + 1307903343280345703 z - 118528937872833793522 z 42 44 - 7572853741802614922586 z + 8252142559016822562909 z 46 58 - 7572853741802614922586 z - 118528937872833793522 z 56 54 + 368029500711858020373 z - 955774707386469584958 z 52 60 + 2081000566967221532737 z + 31829455459027511428 z 70 68 78 - 2418221686240772 z + 24363932417577742 z - 19618002294 z 32 38 + 368029500711858020373 z - 3805362841326046182674 z 40 62 76 + 5851555510782034901165 z - 7098286218615941118 z + 557741981140 z 74 72 - 11794033689397 z + 190932369080569 z )) And in Maple-input format, it is: -(1+7932030752318466185*z^28-1850939445746394559*z^26-363*z^2+ 358257752244313568*z^24-57139107793490816*z^22+54212*z^4-4498832*z^6+236994901* z^8-8568677634*z^10+223918147921*z^12-4387038352484*z^14-785641040662954*z^18+ 66207228256525*z^16-809660850885774158818*z^50+1225897074105872201316*z^48+ 7448546367537569*z^20+452310203888557579040*z^36-213388143332884604012*z^34-\ 57139107793490816*z^66+236994901*z^80+z^88+54212*z^84-363*z^86-4498832*z^82+ 358257752244313568*z^64-28345929922097225198*z^30-1571666873920515632026*z^42+ 1707257312316329453848*z^44-1571666873920515632026*z^46-28345929922097225198*z^ 58+84836929017126425454*z^56-213388143332884604012*z^54+452310203888557579040*z ^52+7932030752318466185*z^60-785641040662954*z^70+7448546367537569*z^68-\ 8568677634*z^78+84836929017126425454*z^32-809660850885774158818*z^38+ 1225897074105872201316*z^40-1850939445746394559*z^62+223918147921*z^76-\ 4387038352484*z^74+66207228256525*z^72)/(-1+z^2)/(1+31829455459027511428*z^28-\ 7098286218615941118*z^26-484*z^2+1307903343280345703*z^24-197827492848049023*z^ 22+87586*z^4-8367383*z^6+493223095*z^8-19618002294*z^10+557741981140*z^12-\ 11794033689397*z^14-2418221686240772*z^18+190932369080569*z^16-\ 3805362841326046182674*z^50+5851555510782034901165*z^48+24363932417577742*z^20+ 2081000566967221532737*z^36-955774707386469584958*z^34-197827492848049023*z^66+ 493223095*z^80+z^88+87586*z^84-484*z^86-8367383*z^82+1307903343280345703*z^64-\ 118528937872833793522*z^30-7572853741802614922586*z^42+8252142559016822562909*z ^44-7572853741802614922586*z^46-118528937872833793522*z^58+ 368029500711858020373*z^56-955774707386469584958*z^54+2081000566967221532737*z^ 52+31829455459027511428*z^60-2418221686240772*z^70+24363932417577742*z^68-\ 19618002294*z^78+368029500711858020373*z^32-3805362841326046182674*z^38+ 5851555510782034901165*z^40-7098286218615941118*z^62+557741981140*z^76-\ 11794033689397*z^74+190932369080569*z^72) The first , 40, terms are: [0, 122, 0, 25312, 0, 5487917, 0, 1199322546, 0, 262711269387, 0, 57594666618857, 0, 12630562007794776, 0, 2770232151410882763, 0, 607617728453908379677, 0, 133276284392012878312506, 0, 29233346418619042089280867, 0, 6412176342311243205400096184, 0, 1406477800728949387435759103866, 0, 308503789750775402105925096718563, 0, 67668757578326629587613686959874995, 0, 14842803323140484833186521479594950258, 0, 3255694718122269404415341240323118677192, 0, 714120370888665648455709994018813176513283, 0, 156638736270758479585607682663710438053964898, 0, 34357924437793034969408021875896562794757671445] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 39920478890698752 z - 18253656670061502 z - 337 z 24 22 4 6 + 6614853049960822 z - 1887360596883424 z + 44736 z - 3133769 z 8 10 12 14 + 132105541 z - 3640350092 z + 69308527917 z - 947295967577 z 18 16 50 - 72503188580849 z + 9552169191360 z - 72503188580849 z 48 20 36 + 420604045362633 z + 420604045362633 z + 96897300460587790 z 34 66 64 30 - 108193912078460200 z - 337 z + 44736 z - 69551430622333598 z 42 44 46 - 18253656670061502 z + 6614853049960822 z - 1887360596883424 z 58 56 54 52 - 3640350092 z + 69308527917 z - 947295967577 z + 9552169191360 z 60 68 32 38 + 132105541 z + z + 96897300460587790 z - 69551430622333598 z 40 62 / 2 + 39920478890698752 z - 3133769 z ) / ((-1 + z ) (1 / 28 26 2 + 184550245756496086 z - 81137508810066804 z - 454 z 24 22 4 6 + 28018458625464114 z - 7559784856390952 z + 74579 z - 6150228 z 8 10 12 14 + 295948841 z - 9121557742 z + 191503633957 z - 2856817339036 z 18 16 50 - 254905777490102 z + 31200138401319 z - 254905777490102 z 48 20 36 + 1582886028897021 z + 1582886028897021 z + 469439507262492514 z 34 66 64 30 - 527329026603487604 z - 454 z + 74579 z - 331002911030918528 z 42 44 46 - 81137508810066804 z + 28018458625464114 z - 7559784856390952 z 58 56 54 - 9121557742 z + 191503633957 z - 2856817339036 z 52 60 68 32 + 31200138401319 z + 295948841 z + z + 469439507262492514 z 38 40 62 - 331002911030918528 z + 184550245756496086 z - 6150228 z )) And in Maple-input format, it is: -(1+39920478890698752*z^28-18253656670061502*z^26-337*z^2+6614853049960822*z^24 -1887360596883424*z^22+44736*z^4-3133769*z^6+132105541*z^8-3640350092*z^10+ 69308527917*z^12-947295967577*z^14-72503188580849*z^18+9552169191360*z^16-\ 72503188580849*z^50+420604045362633*z^48+420604045362633*z^20+96897300460587790 *z^36-108193912078460200*z^34-337*z^66+44736*z^64-69551430622333598*z^30-\ 18253656670061502*z^42+6614853049960822*z^44-1887360596883424*z^46-3640350092*z ^58+69308527917*z^56-947295967577*z^54+9552169191360*z^52+132105541*z^60+z^68+ 96897300460587790*z^32-69551430622333598*z^38+39920478890698752*z^40-3133769*z^ 62)/(-1+z^2)/(1+184550245756496086*z^28-81137508810066804*z^26-454*z^2+ 28018458625464114*z^24-7559784856390952*z^22+74579*z^4-6150228*z^6+295948841*z^ 8-9121557742*z^10+191503633957*z^12-2856817339036*z^14-254905777490102*z^18+ 31200138401319*z^16-254905777490102*z^50+1582886028897021*z^48+1582886028897021 *z^20+469439507262492514*z^36-527329026603487604*z^34-454*z^66+74579*z^64-\ 331002911030918528*z^30-81137508810066804*z^42+28018458625464114*z^44-\ 7559784856390952*z^46-9121557742*z^58+191503633957*z^56-2856817339036*z^54+ 31200138401319*z^52+295948841*z^60+z^68+469439507262492514*z^32-\ 331002911030918528*z^38+184550245756496086*z^40-6150228*z^62) The first , 40, terms are: [0, 118, 0, 23393, 0, 4880959, 0, 1030123074, 0, 218219378523, 0, 46292565958331, 0, 9825978018374946, 0, 2086132625656655031, 0, 442945605306990748649, 0, 94053864959995005418998, 0, 19971485751994783069480337, 0, 4240794780261923801393570193, 0, 900503634276339703028967223350, 0, 191216007248962830353632737700985, 0, 40603480146294284649525187781374759, 0, 8621887910943533306042514330951705634, 0, 1830802634357207269468104367709422840667, 0, 388759220614230982871366929834727841575803, 0, 82550533529159814911993670472270652314476738, 0, 17529077832566937533966357016424903279313567119] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 8840047388038736951 z - 2036486039671049385 z - 357 z 24 22 4 6 + 388783337500790361 z - 61117210377988048 z + 52767 z - 4369528 z 8 10 12 14 + 231138526 z - 8428069792 z + 222759013493 z - 4422371601331 z 18 16 50 - 815488558709424 z + 67704351558843 z - 943910919687417322879 z 48 20 + 1435409257347942739243 z + 7849117263485564 z 36 34 + 524138530088622746229 z - 245397383862886157400 z 66 80 88 84 86 - 61117210377988048 z + 231138526 z + z + 52767 z - 357 z 82 64 30 - 4369528 z + 388783337500790361 z - 31964412614087881808 z 42 44 - 1845135098595950179136 z + 2006089234364143433160 z 46 58 - 1845135098595950179136 z - 31964412614087881808 z 56 54 + 96677085541736247298 z - 245397383862886157400 z 52 60 + 524138530088622746229 z + 8840047388038736951 z 70 68 78 - 815488558709424 z + 7849117263485564 z - 8428069792 z 32 38 + 96677085541736247298 z - 943910919687417322879 z 40 62 76 + 1435409257347942739243 z - 2036486039671049385 z + 222759013493 z 74 72 / 2 - 4422371601331 z + 67704351558843 z ) / ((-1 + z ) (1 / 28 26 2 + 35298083318334716820 z - 7780987127176433150 z - 482 z 24 22 4 6 + 1415765113276372659 z - 211269372235423588 z + 85410 z - 8084532 z 8 10 12 14 + 477472799 z - 19166683008 z + 552149655384 z - 11853268908674 z 18 16 50 - 2507625547856260 z + 194928248585637 z - 4396057130536826982046 z 48 20 + 6787489225687174437575 z + 25648595276958546 z 36 34 + 2390651372122551372946 z - 1090357559968809462228 z 66 80 88 84 86 - 211269372235423588 z + 477472799 z + z + 85410 z - 482 z 82 64 30 - 8084532 z + 1415765113276372659 z - 132842376891706419448 z 42 44 - 8805966495140501303448 z + 9603888226116187866876 z 46 58 - 8805966495140501303448 z - 132842376891706419448 z 56 54 + 416393935501320483505 z - 1090357559968809462228 z 52 60 + 2390651372122551372946 z + 35298083318334716820 z 70 68 78 - 2507625547856260 z + 25648595276958546 z - 19166683008 z 32 38 + 416393935501320483505 z - 4396057130536826982046 z 40 62 76 + 6787489225687174437575 z - 7780987127176433150 z + 552149655384 z 74 72 - 11853268908674 z + 194928248585637 z )) And in Maple-input format, it is: -(1+8840047388038736951*z^28-2036486039671049385*z^26-357*z^2+ 388783337500790361*z^24-61117210377988048*z^22+52767*z^4-4369528*z^6+231138526* z^8-8428069792*z^10+222759013493*z^12-4422371601331*z^14-815488558709424*z^18+ 67704351558843*z^16-943910919687417322879*z^50+1435409257347942739243*z^48+ 7849117263485564*z^20+524138530088622746229*z^36-245397383862886157400*z^34-\ 61117210377988048*z^66+231138526*z^80+z^88+52767*z^84-357*z^86-4369528*z^82+ 388783337500790361*z^64-31964412614087881808*z^30-1845135098595950179136*z^42+ 2006089234364143433160*z^44-1845135098595950179136*z^46-31964412614087881808*z^ 58+96677085541736247298*z^56-245397383862886157400*z^54+524138530088622746229*z ^52+8840047388038736951*z^60-815488558709424*z^70+7849117263485564*z^68-\ 8428069792*z^78+96677085541736247298*z^32-943910919687417322879*z^38+ 1435409257347942739243*z^40-2036486039671049385*z^62+222759013493*z^76-\ 4422371601331*z^74+67704351558843*z^72)/(-1+z^2)/(1+35298083318334716820*z^28-\ 7780987127176433150*z^26-482*z^2+1415765113276372659*z^24-211269372235423588*z^ 22+85410*z^4-8084532*z^6+477472799*z^8-19166683008*z^10+552149655384*z^12-\ 11853268908674*z^14-2507625547856260*z^18+194928248585637*z^16-\ 4396057130536826982046*z^50+6787489225687174437575*z^48+25648595276958546*z^20+ 2390651372122551372946*z^36-1090357559968809462228*z^34-211269372235423588*z^66 +477472799*z^80+z^88+85410*z^84-482*z^86-8084532*z^82+1415765113276372659*z^64-\ 132842376891706419448*z^30-8805966495140501303448*z^42+9603888226116187866876*z ^44-8805966495140501303448*z^46-132842376891706419448*z^58+ 416393935501320483505*z^56-1090357559968809462228*z^54+2390651372122551372946*z ^52+35298083318334716820*z^60-2507625547856260*z^70+25648595276958546*z^68-\ 19166683008*z^78+416393935501320483505*z^32-4396057130536826982046*z^38+ 6787489225687174437575*z^40-7780987127176433150*z^62+552149655384*z^76-\ 11853268908674*z^74+194928248585637*z^72) The first , 40, terms are: [0, 126, 0, 27733, 0, 6373061, 0, 1471139514, 0, 339778293645, 0, 78481984272669, 0, 18127987781091650, 0, 4187262973923348389, 0, 967188440089122593477, 0, 223404548871914420308646, 0, 51602761166072033381996737, 0, 11919385691633976308287755089, 0, 2753181271680488635136723385974, 0, 635939411306327082354663603359893, 0, 146891502964270828795666535703098421, 0, 33929511618029599407854107820038457810, 0, 7837156918028052980151031702347473612269, 0, 1810253835938840905110403492599962595175005, 0, 418138744038321201991498158461298694479971178, 0, 96583145299804247697615152656370903603349600277] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 23875427363079024 z - 9648820038484628 z - 300 z 24 22 4 6 + 3164448149214828 z - 837912488579932 z + 34394 z - 2056200 z 8 10 12 14 + 75116801 z - 1838275524 z + 31936863248 z - 409597598684 z 18 16 50 - 30040833171796 z + 3986431721260 z - 837912488579932 z 48 20 36 + 3164448149214828 z + 177937131001496 z + 118000507828258588 z 34 66 64 - 106835162030265628 z - 2056200 z + 75116801 z 30 42 44 - 48128539638143692 z - 48128539638143692 z + 23875427363079024 z 46 58 56 - 9648820038484628 z - 409597598684 z + 3986431721260 z 54 52 60 70 - 30040833171796 z + 177937131001496 z + 31936863248 z - 300 z 68 32 38 + 34394 z + 79258801479309670 z - 106835162030265628 z 40 62 72 / 2 + 79258801479309670 z - 1838275524 z + z ) / ((-1 + z ) (1 / 28 26 2 + 108809932780195600 z - 42174050601827892 z - 420 z 24 22 4 6 + 13168328371105196 z - 3297988259236284 z + 59926 z - 4159552 z 8 10 12 14 + 170792833 z - 4618279524 z + 87748482928 z - 1222105325500 z 18 16 50 - 103997524135988 z + 12844212458092 z - 3297988259236284 z 48 20 36 + 13168328371105196 z + 658563672837928 z + 581641139776651396 z 34 66 64 - 523940536248847148 z - 4159552 z + 170792833 z 30 42 44 - 226837641261803692 z - 226837641261803692 z + 108809932780195600 z 46 58 56 - 42174050601827892 z - 1222105325500 z + 12844212458092 z 54 52 60 70 - 103997524135988 z + 658563672837928 z + 87748482928 z - 420 z 68 32 38 + 59926 z + 382885115126777958 z - 523940536248847148 z 40 62 72 + 382885115126777958 z - 4618279524 z + z )) And in Maple-input format, it is: -(1+23875427363079024*z^28-9648820038484628*z^26-300*z^2+3164448149214828*z^24-\ 837912488579932*z^22+34394*z^4-2056200*z^6+75116801*z^8-1838275524*z^10+ 31936863248*z^12-409597598684*z^14-30040833171796*z^18+3986431721260*z^16-\ 837912488579932*z^50+3164448149214828*z^48+177937131001496*z^20+ 118000507828258588*z^36-106835162030265628*z^34-2056200*z^66+75116801*z^64-\ 48128539638143692*z^30-48128539638143692*z^42+23875427363079024*z^44-\ 9648820038484628*z^46-409597598684*z^58+3986431721260*z^56-30040833171796*z^54+ 177937131001496*z^52+31936863248*z^60-300*z^70+34394*z^68+79258801479309670*z^ 32-106835162030265628*z^38+79258801479309670*z^40-1838275524*z^62+z^72)/(-1+z^2 )/(1+108809932780195600*z^28-42174050601827892*z^26-420*z^2+13168328371105196*z ^24-3297988259236284*z^22+59926*z^4-4159552*z^6+170792833*z^8-4618279524*z^10+ 87748482928*z^12-1222105325500*z^14-103997524135988*z^18+12844212458092*z^16-\ 3297988259236284*z^50+13168328371105196*z^48+658563672837928*z^20+ 581641139776651396*z^36-523940536248847148*z^34-4159552*z^66+170792833*z^64-\ 226837641261803692*z^30-226837641261803692*z^42+108809932780195600*z^44-\ 42174050601827892*z^46-1222105325500*z^58+12844212458092*z^56-103997524135988*z ^54+658563672837928*z^52+87748482928*z^60-420*z^70+59926*z^68+ 382885115126777958*z^32-523940536248847148*z^38+382885115126777958*z^40-\ 4618279524*z^62+z^72) The first , 40, terms are: [0, 121, 0, 24989, 0, 5381781, 0, 1168464861, 0, 254376844245, 0, 55435940162785, 0, 12086060126070713, 0, 2635419849809279433, 0, 574703148530610521809, 0, 125328207330367399990693, 0, 27331198527566069902963533, 0, 5960330832947212034978976037, 0, 1299818734891191747242917403725, 0, 283462434871543344881914968762889, 0, 61817060840443679891245003680074993, 0, 13480972984238909501452074684336579857, 0, 2939910730824336473090295621851521920489, 0, 641131412837244178993064448793407097224173, 0, 139816997531061348840464287462828784348848133, 0, 30491085699038597130045577300386102631949226093] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 594440612040214451007 z - 83407416239887616654 z - 389 z 24 22 4 + 9981452882623274193 z - 1012001440264847530 z + 64784 z 6 102 8 10 - 6272878 z - 18444595100 z + 402546925 z - 18444595100 z 12 14 18 + 633340807727 z - 16859812644638 z - 6116221402022426 z 16 50 + 356756174052113 z - 121309325152363721005457812 z 48 20 + 77686378813394294935605522 z + 86235598887903613 z 36 34 + 345063073959020260258526 z - 87220937403239480371790 z 66 80 - 43747729571808537047156408 z + 19139221653861337152765 z 100 90 88 + 633340807727 z - 1012001440264847530 z + 9981452882623274193 z 84 94 + 594440612040214451007 z - 6116221402022426 z 86 96 98 - 83407416239887616654 z + 356756174052113 z - 16859812644638 z 92 82 + 86235598887903613 z - 3633559809926572056556 z 64 112 110 106 + 77686378813394294935605522 z + z - 389 z - 6272878 z 108 30 42 + 64784 z - 3633559809926572056556 z - 9394722122207657016720580 z 44 46 + 21642601044348546709236950 z - 43747729571808537047156408 z 58 56 - 201673377093551153540290748 z + 214886815370858449600862886 z 54 52 - 201673377093551153540290748 z + 166694376976562371575031518 z 60 70 + 166694376976562371575031518 z - 9394722122207657016720580 z 68 78 + 21642601044348546709236950 z - 87220937403239480371790 z 32 38 + 19139221653861337152765 z - 1188543580482378134564139 z 40 62 + 3573065119172174432161873 z - 121309325152363721005457812 z 76 74 + 345063073959020260258526 z - 1188543580482378134564139 z 72 104 / + 3573065119172174432161873 z + 402546925 z ) / (-1 / 28 26 2 - 2353396068946998778720 z + 310511081942339180790 z + 512 z 24 22 4 - 34920070647838765916 z + 3324164319034077455 z - 101155 z 6 102 8 10 + 11112688 z + 1474402666326 z - 790475994 z + 39605692532 z 12 14 18 - 1474402666326 z + 42308145362440 z + 17638321071774397 z 16 50 - 961128308250355 z + 940103371867750458217764681 z 48 20 - 565926301024753849192847625 z - 265630948867266565 z 36 34 - 1742290799091733069004552 z + 414477902066405399646671 z 66 80 + 565926301024753849192847625 z - 414477902066405399646671 z 100 90 - 42308145362440 z + 34920070647838765916 z 88 84 - 310511081942339180790 z - 15291356032263841872018 z 94 86 + 265630948867266565 z + 2353396068946998778720 z 96 98 92 - 17638321071774397 z + 961128308250355 z - 3324164319034077455 z 82 64 112 + 85595366928767057990133 z - 940103371867750458217764681 z - 512 z 114 110 106 108 + z + 101155 z + 790475994 z - 11112688 z 30 42 + 15291356032263841872018 z + 56931274048087080676760696 z 44 46 - 139421555976351139621553032 z + 299654893318064586910993224 z 58 56 + 2009045162423308693640054749 z - 2009045162423308693640054749 z 54 52 + 1770465125603576356474495064 z - 1374684400803945135741018568 z 60 70 - 1770465125603576356474495064 z + 139421555976351139621553032 z 68 78 - 299654893318064586910993224 z + 1742290799091733069004552 z 32 38 - 85595366928767057990133 z + 6376753576718477622292273 z 40 62 - 20372019486669436262636573 z + 1374684400803945135741018568 z 76 74 - 6376753576718477622292273 z + 20372019486669436262636573 z 72 104 - 56931274048087080676760696 z - 39605692532 z ) And in Maple-input format, it is: -(1+594440612040214451007*z^28-83407416239887616654*z^26-389*z^2+ 9981452882623274193*z^24-1012001440264847530*z^22+64784*z^4-6272878*z^6-\ 18444595100*z^102+402546925*z^8-18444595100*z^10+633340807727*z^12-\ 16859812644638*z^14-6116221402022426*z^18+356756174052113*z^16-\ 121309325152363721005457812*z^50+77686378813394294935605522*z^48+ 86235598887903613*z^20+345063073959020260258526*z^36-87220937403239480371790*z^ 34-43747729571808537047156408*z^66+19139221653861337152765*z^80+633340807727*z^ 100-1012001440264847530*z^90+9981452882623274193*z^88+594440612040214451007*z^ 84-6116221402022426*z^94-83407416239887616654*z^86+356756174052113*z^96-\ 16859812644638*z^98+86235598887903613*z^92-3633559809926572056556*z^82+ 77686378813394294935605522*z^64+z^112-389*z^110-6272878*z^106+64784*z^108-\ 3633559809926572056556*z^30-9394722122207657016720580*z^42+ 21642601044348546709236950*z^44-43747729571808537047156408*z^46-\ 201673377093551153540290748*z^58+214886815370858449600862886*z^56-\ 201673377093551153540290748*z^54+166694376976562371575031518*z^52+ 166694376976562371575031518*z^60-9394722122207657016720580*z^70+ 21642601044348546709236950*z^68-87220937403239480371790*z^78+ 19139221653861337152765*z^32-1188543580482378134564139*z^38+ 3573065119172174432161873*z^40-121309325152363721005457812*z^62+ 345063073959020260258526*z^76-1188543580482378134564139*z^74+ 3573065119172174432161873*z^72+402546925*z^104)/(-1-2353396068946998778720*z^28 +310511081942339180790*z^26+512*z^2-34920070647838765916*z^24+ 3324164319034077455*z^22-101155*z^4+11112688*z^6+1474402666326*z^102-790475994* z^8+39605692532*z^10-1474402666326*z^12+42308145362440*z^14+17638321071774397*z ^18-961128308250355*z^16+940103371867750458217764681*z^50-\ 565926301024753849192847625*z^48-265630948867266565*z^20-\ 1742290799091733069004552*z^36+414477902066405399646671*z^34+ 565926301024753849192847625*z^66-414477902066405399646671*z^80-42308145362440*z ^100+34920070647838765916*z^90-310511081942339180790*z^88-\ 15291356032263841872018*z^84+265630948867266565*z^94+2353396068946998778720*z^ 86-17638321071774397*z^96+961128308250355*z^98-3324164319034077455*z^92+ 85595366928767057990133*z^82-940103371867750458217764681*z^64-512*z^112+z^114+ 101155*z^110+790475994*z^106-11112688*z^108+15291356032263841872018*z^30+ 56931274048087080676760696*z^42-139421555976351139621553032*z^44+ 299654893318064586910993224*z^46+2009045162423308693640054749*z^58-\ 2009045162423308693640054749*z^56+1770465125603576356474495064*z^54-\ 1374684400803945135741018568*z^52-1770465125603576356474495064*z^60+ 139421555976351139621553032*z^70-299654893318064586910993224*z^68+ 1742290799091733069004552*z^78-85595366928767057990133*z^32+ 6376753576718477622292273*z^38-20372019486669436262636573*z^40+ 1374684400803945135741018568*z^62-6376753576718477622292273*z^76+ 20372019486669436262636573*z^74-56931274048087080676760696*z^72-39605692532*z^ 104) The first , 40, terms are: [0, 123, 0, 26605, 0, 6019505, 0, 1369689340, 0, 311963528215, 0, 71067106740287, 0, 16190150303095133, 0, 3688390297828379879, 0, 840279314956107604935, 0, 191430287187664188454545, 0, 43611162910871538290133199, 0, 9935384808093867490314875963, 0, 2263454242262087945131319735404, 0, 515654422096941866342224797437573, 0, 117475086580905206939321837700190605, 0, 26762877183675757542977848729066766455, 0, 6097051008872082281245207674050744362025, 0, 1389014744182354679460614209320846496722393, 0, 316441826835527312674625357901191719828687503, 0, 72090976852777089599814951976072908414202282701] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 525085890558150479693 z - 73935565798775732390 z - 392 z 24 22 4 6 + 8896489031675545737 z - 908762801965345280 z + 65243 z - 6265722 z 102 8 10 12 - 17924161014 z + 397033762 z - 17924161014 z + 605976615862 z 14 18 16 - 15886593765880 z - 5607537547664706 z + 331365345307187 z 50 48 - 110848805989610703426594490 z + 70726287931748855840192687 z 20 36 + 78173350658661635 z + 305630882144582327700155 z 34 66 - 77037697369456355096100 z - 39653962769664824534469312 z 80 100 90 + 16878724834811392522891 z + 605976615862 z - 908762801965345280 z 88 84 + 8896489031675545737 z + 525085890558150479693 z 94 86 96 - 5607537547664706 z - 73935565798775732390 z + 331365345307187 z 98 92 82 - 15886593765880 z + 78173350658661635 z - 3204291645284070874362 z 64 112 110 106 + 70726287931748855840192687 z + z - 392 z - 6265722 z 108 30 42 + 65243 z - 3204291645284070874362 z - 8431937388307078096482286 z 44 46 + 19522560622063789986330366 z - 39653962769664824534469312 z 58 56 - 185117592108090248913277090 z + 197362388653138090304044900 z 54 52 - 185117592108090248913277090 z + 152745103936739823183521277 z 60 70 + 152745103936739823183521277 z - 8431937388307078096482286 z 68 78 + 19522560622063789986330366 z - 77037697369456355096100 z 32 38 + 16878724834811392522891 z - 1056728843266064262200384 z 40 62 + 3191153452920672923607006 z - 110848805989610703426594490 z 76 74 + 305630882144582327700155 z - 1056728843266064262200384 z 72 104 / 2 + 3191153452920672923607006 z + 397033762 z ) / ((-1 + z ) (1 / 28 26 2 + 1827274619255335900402 z - 246347988329790618622 z - 511 z 24 22 4 + 28325570714278819489 z - 2759153406251075014 z + 101676 z 6 102 8 10 - 11107827 z - 38185068411 z + 778241823 z - 38185068411 z 12 14 18 + 1387708016051 z - 38817303389982 z - 15371895382855748 z 16 50 + 859304671077308 z - 531534980496403985989893157 z 48 20 + 334470870399812293641426791 z + 225821102983037865 z 36 34 + 1240155399271260898682303 z - 301792005642364602570232 z 66 80 - 184265843969558551904542969 z + 63693949535452250206890 z 100 90 88 + 1387708016051 z - 2759153406251075014 z + 28325570714278819489 z 84 94 + 1827274619255335900402 z - 15371895382855748 z 86 96 98 - 246347988329790618622 z + 859304671077308 z - 38817303389982 z 92 82 + 225821102983037865 z - 11623395916737129799990 z 64 112 110 106 + 334470870399812293641426791 z + z - 511 z - 11107827 z 108 30 + 101676 z - 11623395916737129799990 z 42 44 - 37445898015247277123233945 z + 88830873518462293290463716 z 46 58 - 184265843969558551904542969 z - 902073503676768036234566956 z 56 54 + 963698739643003298305510995 z - 902073503676768036234566956 z 52 60 + 739818139807000771769180338 z + 739818139807000771769180338 z 70 68 - 37445898015247277123233945 z + 88830873518462293290463716 z 78 32 - 301792005642364602570232 z + 63693949535452250206890 z 38 40 - 4430686416122622365423150 z + 13789618375814666351872688 z 62 76 - 531534980496403985989893157 z + 1240155399271260898682303 z 74 72 - 4430686416122622365423150 z + 13789618375814666351872688 z 104 + 778241823 z )) And in Maple-input format, it is: -(1+525085890558150479693*z^28-73935565798775732390*z^26-392*z^2+ 8896489031675545737*z^24-908762801965345280*z^22+65243*z^4-6265722*z^6-\ 17924161014*z^102+397033762*z^8-17924161014*z^10+605976615862*z^12-\ 15886593765880*z^14-5607537547664706*z^18+331365345307187*z^16-\ 110848805989610703426594490*z^50+70726287931748855840192687*z^48+ 78173350658661635*z^20+305630882144582327700155*z^36-77037697369456355096100*z^ 34-39653962769664824534469312*z^66+16878724834811392522891*z^80+605976615862*z^ 100-908762801965345280*z^90+8896489031675545737*z^88+525085890558150479693*z^84 -5607537547664706*z^94-73935565798775732390*z^86+331365345307187*z^96-\ 15886593765880*z^98+78173350658661635*z^92-3204291645284070874362*z^82+ 70726287931748855840192687*z^64+z^112-392*z^110-6265722*z^106+65243*z^108-\ 3204291645284070874362*z^30-8431937388307078096482286*z^42+ 19522560622063789986330366*z^44-39653962769664824534469312*z^46-\ 185117592108090248913277090*z^58+197362388653138090304044900*z^56-\ 185117592108090248913277090*z^54+152745103936739823183521277*z^52+ 152745103936739823183521277*z^60-8431937388307078096482286*z^70+ 19522560622063789986330366*z^68-77037697369456355096100*z^78+ 16878724834811392522891*z^32-1056728843266064262200384*z^38+ 3191153452920672923607006*z^40-110848805989610703426594490*z^62+ 305630882144582327700155*z^76-1056728843266064262200384*z^74+ 3191153452920672923607006*z^72+397033762*z^104)/(-1+z^2)/(1+ 1827274619255335900402*z^28-246347988329790618622*z^26-511*z^2+ 28325570714278819489*z^24-2759153406251075014*z^22+101676*z^4-11107827*z^6-\ 38185068411*z^102+778241823*z^8-38185068411*z^10+1387708016051*z^12-\ 38817303389982*z^14-15371895382855748*z^18+859304671077308*z^16-\ 531534980496403985989893157*z^50+334470870399812293641426791*z^48+ 225821102983037865*z^20+1240155399271260898682303*z^36-301792005642364602570232 *z^34-184265843969558551904542969*z^66+63693949535452250206890*z^80+ 1387708016051*z^100-2759153406251075014*z^90+28325570714278819489*z^88+ 1827274619255335900402*z^84-15371895382855748*z^94-246347988329790618622*z^86+ 859304671077308*z^96-38817303389982*z^98+225821102983037865*z^92-\ 11623395916737129799990*z^82+334470870399812293641426791*z^64+z^112-511*z^110-\ 11107827*z^106+101676*z^108-11623395916737129799990*z^30-\ 37445898015247277123233945*z^42+88830873518462293290463716*z^44-\ 184265843969558551904542969*z^46-902073503676768036234566956*z^58+ 963698739643003298305510995*z^56-902073503676768036234566956*z^54+ 739818139807000771769180338*z^52+739818139807000771769180338*z^60-\ 37445898015247277123233945*z^70+88830873518462293290463716*z^68-\ 301792005642364602570232*z^78+63693949535452250206890*z^32-\ 4430686416122622365423150*z^38+13789618375814666351872688*z^40-\ 531534980496403985989893157*z^62+1240155399271260898682303*z^76-\ 4430686416122622365423150*z^74+13789618375814666351872688*z^72+778241823*z^104) The first , 40, terms are: [0, 120, 0, 24496, 0, 5223293, 0, 1123977736, 0, 242629135749, 0, 52440494881315, 0, 11339873681674752, 0, 2452665020471630733, 0, 530523226724517106043, 0, 114758622500237817102256, 0, 24824029514254886965972735, 0, 5369844303330115821125168416, 0, 1161587993236288641523390465600, 0, 251271334215435041661015523315419, 0, 54354304769148228838337464596617247, 0, 11757771425103237561708628895591666928, 0, 2543408445537486377739304330949933985184, 0, 550183062536312447450394299481959880139539, 0, 119014075812634503285488856869129327185754288, 0, 25744795265984557288541104764651153167227192479] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 36156156889648246 z - 14434013817364641 z - 299 z 24 22 4 6 + 4661033827935174 z - 1210876445971934 z + 34850 z - 2171251 z 8 10 12 14 + 83463154 z - 2150511788 z + 39198943048 z - 524800529366 z 18 16 50 - 41274642427646 z + 5303051411695 z - 1210876445971934 z 48 20 36 + 4661033827935174 z + 251281159000738 z + 182460614133424362 z 34 66 64 - 164986390444570444 z - 2171251 z + 83463154 z 30 42 44 - 73562548445390899 z - 73562548445390899 z + 36156156889648246 z 46 58 56 - 14434013817364641 z - 524800529366 z + 5303051411695 z 54 52 60 70 - 41274642427646 z + 251281159000738 z + 39198943048 z - 299 z 68 32 38 + 34850 z + 121931509488302493 z - 164986390444570444 z 40 62 72 / 2 + 121931509488302493 z - 2150511788 z + z ) / ((-1 + z ) (1 / 28 26 2 + 163783762342058885 z - 62675742378189138 z - 426 z 24 22 4 6 + 19257643471853419 z - 4728672007432254 z + 60076 z - 4294524 z 8 10 12 14 + 185164017 z - 5284264968 z + 105736168414 z - 1542729492226 z 18 16 50 - 141450864846168 z + 16881197390549 z - 4728672007432254 z 48 20 36 + 19257643471853419 z + 921895374066473 z + 895028986827919251 z 34 66 64 - 805144030897503858 z - 4294524 z + 185164017 z 30 42 44 - 344783850515812836 z - 344783850515812836 z + 163783762342058885 z 46 58 56 - 62675742378189138 z - 1542729492226 z + 16881197390549 z 54 52 60 70 - 141450864846168 z + 921895374066473 z + 105736168414 z - 426 z 68 32 38 + 60076 z + 585980515169066749 z - 805144030897503858 z 40 62 72 + 585980515169066749 z - 5284264968 z + z )) And in Maple-input format, it is: -(1+36156156889648246*z^28-14434013817364641*z^26-299*z^2+4661033827935174*z^24 -1210876445971934*z^22+34850*z^4-2171251*z^6+83463154*z^8-2150511788*z^10+ 39198943048*z^12-524800529366*z^14-41274642427646*z^18+5303051411695*z^16-\ 1210876445971934*z^50+4661033827935174*z^48+251281159000738*z^20+ 182460614133424362*z^36-164986390444570444*z^34-2171251*z^66+83463154*z^64-\ 73562548445390899*z^30-73562548445390899*z^42+36156156889648246*z^44-\ 14434013817364641*z^46-524800529366*z^58+5303051411695*z^56-41274642427646*z^54 +251281159000738*z^52+39198943048*z^60-299*z^70+34850*z^68+121931509488302493*z ^32-164986390444570444*z^38+121931509488302493*z^40-2150511788*z^62+z^72)/(-1+z ^2)/(1+163783762342058885*z^28-62675742378189138*z^26-426*z^2+19257643471853419 *z^24-4728672007432254*z^22+60076*z^4-4294524*z^6+185164017*z^8-5284264968*z^10 +105736168414*z^12-1542729492226*z^14-141450864846168*z^18+16881197390549*z^16-\ 4728672007432254*z^50+19257643471853419*z^48+921895374066473*z^20+ 895028986827919251*z^36-805144030897503858*z^34-4294524*z^66+185164017*z^64-\ 344783850515812836*z^30-344783850515812836*z^42+163783762342058885*z^44-\ 62675742378189138*z^46-1542729492226*z^58+16881197390549*z^56-141450864846168*z ^54+921895374066473*z^52+105736168414*z^60-426*z^70+60076*z^68+ 585980515169066749*z^32-805144030897503858*z^38+585980515169066749*z^40-\ 5284264968*z^62+z^72) The first , 40, terms are: [0, 128, 0, 29004, 0, 6823801, 0, 1610356432, 0, 380137630711, 0, 89737117195915, 0, 21183841214570960, 0, 5000776821688536775, 0, 1180511591038491060323, 0, 278678228958329423772580, 0, 65786355684611197183634145, 0, 15529898449266213854119180900, 0, 3666075485624395775724756656980, 0, 865434472122773546258303293667289, 0, 204299346393450304862087326471446313, 0, 48228056867743793060748039319877287260, 0, 11384987325214624962558916735221230366484, 0, 2687604370019725879850802879508444284083561, 0, 634451057644291777985694784308584544109804668, 0, 149772097796897151771949747062162441998088192587] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 10313398090465211225 z - 2310684571934178739 z - 347 z 24 22 4 6 + 428658909740340393 z - 65469349101605860 z + 50065 z - 4089610 z 8 10 12 14 + 215625398 z - 7905904330 z + 211587967659 z - 4276484901561 z 18 16 50 - 826501196407588 z + 66926063274459 z - 1223510706109623896633 z 48 20 + 1880812996377523349627 z + 8173258337448700 z 36 34 + 669435010024635007851 z - 307714645523078378326 z 66 80 88 84 86 - 65469349101605860 z + 215625398 z + z + 50065 z - 347 z 82 64 30 - 4089610 z + 428658909740340393 z - 38287230288918372630 z 42 44 - 2433654664378381791928 z + 2651808871661356415816 z 46 58 - 2433654664378381791928 z - 38287230288918372630 z 56 54 + 118641491380555773098 z - 307714645523078378326 z 52 60 + 669435010024635007851 z + 10313398090465211225 z 70 68 78 - 826501196407588 z + 8173258337448700 z - 7905904330 z 32 38 + 118641491380555773098 z - 1223510706109623896633 z 40 62 76 + 1880812996377523349627 z - 2310684571934178739 z + 211587967659 z 74 72 / - 4276484901561 z + 66926063274459 z ) / (-1 / 28 26 2 - 49261608092981503016 z + 10206311464852761515 z + 461 z 24 22 4 6 - 1750789268049136847 z + 247148770621839602 z - 79422 z + 7426822 z 8 10 12 14 - 439049423 z + 17836796195 z - 524963671580 z + 11607164396660 z 18 16 50 + 2656727701354033 z - 197961245918853 z + 14541594536784058830115 z 48 20 - 20454382285048749441055 z - 28491810014089026 z 36 34 - 4393366597770018106014 z + 1863223816132593541801 z 66 80 90 88 84 + 1750789268049136847 z - 17836796195 z + z - 461 z - 7426822 z 86 82 64 + 79422 z + 439049423 z - 10206311464852761515 z 30 42 + 197810849244266950944 z + 20454382285048749441055 z 44 46 - 24256013265516712182828 z + 24256013265516712182828 z 58 56 + 663348953187338590981 z - 1863223816132593541801 z 54 52 + 4393366597770018106014 z - 8711043668702159286278 z 60 70 - 197810849244266950944 z + 28491810014089026 z 68 78 32 - 247148770621839602 z + 524963671580 z - 663348953187338590981 z 38 40 + 8711043668702159286278 z - 14541594536784058830115 z 62 76 74 + 49261608092981503016 z - 11607164396660 z + 197961245918853 z 72 - 2656727701354033 z ) And in Maple-input format, it is: -(1+10313398090465211225*z^28-2310684571934178739*z^26-347*z^2+ 428658909740340393*z^24-65469349101605860*z^22+50065*z^4-4089610*z^6+215625398* z^8-7905904330*z^10+211587967659*z^12-4276484901561*z^14-826501196407588*z^18+ 66926063274459*z^16-1223510706109623896633*z^50+1880812996377523349627*z^48+ 8173258337448700*z^20+669435010024635007851*z^36-307714645523078378326*z^34-\ 65469349101605860*z^66+215625398*z^80+z^88+50065*z^84-347*z^86-4089610*z^82+ 428658909740340393*z^64-38287230288918372630*z^30-2433654664378381791928*z^42+ 2651808871661356415816*z^44-2433654664378381791928*z^46-38287230288918372630*z^ 58+118641491380555773098*z^56-307714645523078378326*z^54+669435010024635007851* z^52+10313398090465211225*z^60-826501196407588*z^70+8173258337448700*z^68-\ 7905904330*z^78+118641491380555773098*z^32-1223510706109623896633*z^38+ 1880812996377523349627*z^40-2310684571934178739*z^62+211587967659*z^76-\ 4276484901561*z^74+66926063274459*z^72)/(-1-49261608092981503016*z^28+ 10206311464852761515*z^26+461*z^2-1750789268049136847*z^24+247148770621839602*z ^22-79422*z^4+7426822*z^6-439049423*z^8+17836796195*z^10-524963671580*z^12+ 11607164396660*z^14+2656727701354033*z^18-197961245918853*z^16+ 14541594536784058830115*z^50-20454382285048749441055*z^48-28491810014089026*z^ 20-4393366597770018106014*z^36+1863223816132593541801*z^34+1750789268049136847* z^66-17836796195*z^80+z^90-461*z^88-7426822*z^84+79422*z^86+439049423*z^82-\ 10206311464852761515*z^64+197810849244266950944*z^30+20454382285048749441055*z^ 42-24256013265516712182828*z^44+24256013265516712182828*z^46+ 663348953187338590981*z^58-1863223816132593541801*z^56+4393366597770018106014*z ^54-8711043668702159286278*z^52-197810849244266950944*z^60+28491810014089026*z^ 70-247148770621839602*z^68+524963671580*z^78-663348953187338590981*z^32+ 8711043668702159286278*z^38-14541594536784058830115*z^40+49261608092981503016*z ^62-11607164396660*z^76+197961245918853*z^74-2656727701354033*z^72) The first , 40, terms are: [0, 114, 0, 23197, 0, 4976921, 0, 1075242130, 0, 232568849845, 0, 50314455301725, 0, 10885643297020306, 0, 2355156829559058897, 0, 509549706760169996293, 0, 110243630154089493711026, 0, 23851764161390151628265977, 0, 5160449346076181480988168265, 0, 1116489222364386556463134770610, 0, 241558070210391826660014235435893, 0, 52262305926795341946765788797658017, 0, 11307213286710851529432399231525149714, 0, 2446372582421296198010261430059279002445, 0, 529285037815702093335542419422812577345861, 0, 114513485504550722568117809771885824058278674, 0, 24775569731804626325843060864151541213481818569] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6962489308498392593 z - 1621906650201166919 z - 363 z 24 22 4 6 + 313736106317648968 z - 50089975215828216 z + 53772 z - 4395752 z 8 10 12 14 + 226997221 z - 8035977826 z + 205937666513 z - 3968765390276 z 18 16 50 - 695133471598986 z + 59127329068085 z - 719130340621211947322 z 48 20 + 1090371509965920965204 z + 6550865282622473 z 36 34 + 400967021163331905552 z - 188727109815892357804 z 66 80 88 84 86 - 50089975215828216 z + 226997221 z + z + 53772 z - 363 z 82 64 30 - 4395752 z + 313736106317648968 z - 24940007586192876230 z 42 44 - 1399129688993230854834 z + 1520278713187395350200 z 46 58 - 1399129688993230854834 z - 24940007586192876230 z 56 54 + 74839119415840588390 z - 188727109815892357804 z 52 60 + 400967021163331905552 z + 6962489308498392593 z 70 68 78 - 695133471598986 z + 6550865282622473 z - 8035977826 z 32 38 + 74839119415840588390 z - 719130340621211947322 z 40 62 76 + 1090371509965920965204 z - 1621906650201166919 z + 205937666513 z 74 72 / 2 - 3968765390276 z + 59127329068085 z ) / ((-1 + z ) (1 / 28 26 2 + 28333437315623679764 z - 6275896279644252582 z - 492 z 24 22 4 6 + 1150445892978567711 z - 173553961251106231 z + 89098 z - 8379391 z 8 10 12 14 + 481462479 z - 18620260950 z + 515867899116 z - 10679526730893 z 18 16 50 - 2133936237601060 z + 170200468364161 z - 3508501933079229425810 z 48 20 + 5417125336221189252237 z + 21391074782794038 z 36 34 + 1908155259240247006561 z - 870576866034367186238 z 66 80 88 84 86 - 173553961251106231 z + 481462479 z + z + 89098 z - 492 z 82 64 30 - 8379391 z + 1150445892978567711 z - 106321207544768514082 z 42 44 - 7028239418240015496602 z + 7665157490696765918861 z 46 58 - 7028239418240015496602 z - 106321207544768514082 z 56 54 + 332726234015620665701 z - 870576866034367186238 z 52 60 + 1908155259240247006561 z + 28333437315623679764 z 70 68 78 - 2133936237601060 z + 21391074782794038 z - 18620260950 z 32 38 + 332726234015620665701 z - 3508501933079229425810 z 40 62 76 + 5417125336221189252237 z - 6275896279644252582 z + 515867899116 z 74 72 - 10679526730893 z + 170200468364161 z )) And in Maple-input format, it is: -(1+6962489308498392593*z^28-1621906650201166919*z^26-363*z^2+ 313736106317648968*z^24-50089975215828216*z^22+53772*z^4-4395752*z^6+226997221* z^8-8035977826*z^10+205937666513*z^12-3968765390276*z^14-695133471598986*z^18+ 59127329068085*z^16-719130340621211947322*z^50+1090371509965920965204*z^48+ 6550865282622473*z^20+400967021163331905552*z^36-188727109815892357804*z^34-\ 50089975215828216*z^66+226997221*z^80+z^88+53772*z^84-363*z^86-4395752*z^82+ 313736106317648968*z^64-24940007586192876230*z^30-1399129688993230854834*z^42+ 1520278713187395350200*z^44-1399129688993230854834*z^46-24940007586192876230*z^ 58+74839119415840588390*z^56-188727109815892357804*z^54+400967021163331905552*z ^52+6962489308498392593*z^60-695133471598986*z^70+6550865282622473*z^68-\ 8035977826*z^78+74839119415840588390*z^32-719130340621211947322*z^38+ 1090371509965920965204*z^40-1621906650201166919*z^62+205937666513*z^76-\ 3968765390276*z^74+59127329068085*z^72)/(-1+z^2)/(1+28333437315623679764*z^28-\ 6275896279644252582*z^26-492*z^2+1150445892978567711*z^24-173553961251106231*z^ 22+89098*z^4-8379391*z^6+481462479*z^8-18620260950*z^10+515867899116*z^12-\ 10679526730893*z^14-2133936237601060*z^18+170200468364161*z^16-\ 3508501933079229425810*z^50+5417125336221189252237*z^48+21391074782794038*z^20+ 1908155259240247006561*z^36-870576866034367186238*z^34-173553961251106231*z^66+ 481462479*z^80+z^88+89098*z^84-492*z^86-8379391*z^82+1150445892978567711*z^64-\ 106321207544768514082*z^30-7028239418240015496602*z^42+7665157490696765918861*z ^44-7028239418240015496602*z^46-106321207544768514082*z^58+ 332726234015620665701*z^56-870576866034367186238*z^54+1908155259240247006561*z^ 52+28333437315623679764*z^60-2133936237601060*z^70+21391074782794038*z^68-\ 18620260950*z^78+332726234015620665701*z^32-3508501933079229425810*z^38+ 5417125336221189252237*z^40-6275896279644252582*z^62+515867899116*z^76-\ 10679526730893*z^74+170200468364161*z^72) The first , 40, terms are: [0, 130, 0, 28272, 0, 6364133, 0, 1442688010, 0, 327889936971, 0, 74599774619417, 0, 16979785260824712, 0, 3865472165043502411, 0, 880042851160736712861, 0, 200363086894473939740874, 0, 45618055068542258290421571, 0, 10386229677304658049647375160, 0, 2364721169801723834906827143210, 0, 538396598909779092924498460892603, 0, 122581468450450963492676940680346955, 0, 27909200141793116862873126957715833074, 0, 6354333206461649898913415722351637706440, 0, 1446746993901314067970241166966648917570243, 0, 329393630392199348943573479817360451976130722, 0, 74995949262424431607125979018785767774753798405] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 8448363499107098963 z - 1932299360408786249 z - 353 z 24 22 4 6 + 366468168695670829 z - 57287493393246476 z + 51423 z - 4202066 z 8 10 12 14 + 219776620 z - 7941583970 z + 208517168497 z - 4122512602219 z 18 16 50 - 759252279840172 z + 63003792409135 z - 931473058041660849899 z 48 20 + 1421571689846766478947 z + 7326230401283200 z 36 34 + 514740950048835130013 z - 239582057465793458766 z 66 80 88 84 86 - 57287493393246476 z + 219776620 z + z + 51423 z - 353 z 82 64 30 - 4202066 z + 366468168695670829 z - 30775051563733375038 z 42 44 - 1831367291351258049032 z + 1992595915588237563520 z 46 58 - 1831367291351258049032 z - 30775051563733375038 z 56 54 + 93754989858665883348 z - 239582057465793458766 z 52 60 + 514740950048835130013 z + 8448363499107098963 z 70 68 78 - 759252279840172 z + 7326230401283200 z - 7941583970 z 32 38 + 93754989858665883348 z - 931473058041660849899 z 40 62 76 + 1421571689846766478947 z - 1932299360408786249 z + 208517168497 z 74 72 / - 4122512602219 z + 63003792409135 z ) / (-1 / 28 26 2 - 41124611959489403906 z + 8680467265020513401 z + 479 z 24 22 4 6 - 1519746912582917651 z + 219331765018589870 z - 84116 z + 7881598 z 8 10 12 14 - 461050511 z + 18385338407 z - 528667796352 z + 11394297702666 z 18 16 50 + 2475360071981045 z - 189295847893335 z + 11311303426183314830417 z 48 20 - 15839876513109948402167 z - 25890998001331630 z 36 34 - 3470458910418539176158 z + 1487861677276354758145 z 66 80 90 88 84 + 1519746912582917651 z - 18385338407 z + z - 479 z - 7881598 z 86 82 64 + 84116 z + 461050511 z - 8680467265020513401 z 30 42 + 162394714018776811812 z + 15839876513109948402167 z 44 46 - 18741863875450090493860 z + 18741863875450090493860 z 58 56 + 536567856130788663113 z - 1487861677276354758145 z 54 52 + 3470458910418539176158 z - 6821072828983512792416 z 60 70 - 162394714018776811812 z + 25890998001331630 z 68 78 32 - 219331765018589870 z + 528667796352 z - 536567856130788663113 z 38 40 + 6821072828983512792416 z - 11311303426183314830417 z 62 76 74 + 41124611959489403906 z - 11394297702666 z + 189295847893335 z 72 - 2475360071981045 z ) And in Maple-input format, it is: -(1+8448363499107098963*z^28-1932299360408786249*z^26-353*z^2+ 366468168695670829*z^24-57287493393246476*z^22+51423*z^4-4202066*z^6+219776620* z^8-7941583970*z^10+208517168497*z^12-4122512602219*z^14-759252279840172*z^18+ 63003792409135*z^16-931473058041660849899*z^50+1421571689846766478947*z^48+ 7326230401283200*z^20+514740950048835130013*z^36-239582057465793458766*z^34-\ 57287493393246476*z^66+219776620*z^80+z^88+51423*z^84-353*z^86-4202066*z^82+ 366468168695670829*z^64-30775051563733375038*z^30-1831367291351258049032*z^42+ 1992595915588237563520*z^44-1831367291351258049032*z^46-30775051563733375038*z^ 58+93754989858665883348*z^56-239582057465793458766*z^54+514740950048835130013*z ^52+8448363499107098963*z^60-759252279840172*z^70+7326230401283200*z^68-\ 7941583970*z^78+93754989858665883348*z^32-931473058041660849899*z^38+ 1421571689846766478947*z^40-1932299360408786249*z^62+208517168497*z^76-\ 4122512602219*z^74+63003792409135*z^72)/(-1-41124611959489403906*z^28+ 8680467265020513401*z^26+479*z^2-1519746912582917651*z^24+219331765018589870*z^ 22-84116*z^4+7881598*z^6-461050511*z^8+18385338407*z^10-528667796352*z^12+ 11394297702666*z^14+2475360071981045*z^18-189295847893335*z^16+ 11311303426183314830417*z^50-15839876513109948402167*z^48-25890998001331630*z^ 20-3470458910418539176158*z^36+1487861677276354758145*z^34+1519746912582917651* z^66-18385338407*z^80+z^90-479*z^88-7881598*z^84+84116*z^86+461050511*z^82-\ 8680467265020513401*z^64+162394714018776811812*z^30+15839876513109948402167*z^ 42-18741863875450090493860*z^44+18741863875450090493860*z^46+ 536567856130788663113*z^58-1487861677276354758145*z^56+3470458910418539176158*z ^54-6821072828983512792416*z^52-162394714018776811812*z^60+25890998001331630*z^ 70-219331765018589870*z^68+528667796352*z^78-536567856130788663113*z^32+ 6821072828983512792416*z^38-11311303426183314830417*z^40+41124611959489403906*z ^62-11394297702666*z^76+189295847893335*z^74-2475360071981045*z^72) The first , 40, terms are: [0, 126, 0, 27661, 0, 6330535, 0, 1457401046, 0, 335960091303, 0, 77472153170387, 0, 17866711589706470, 0, 4120549039904950715, 0, 950317577684240584849, 0, 219171127383658935441278, 0, 50547324775916499566125101, 0, 11657705532149155109480545365, 0, 2688611221463981321634532876542, 0, 620073167510716240190846460708233, 0, 143007189520029517598130191178836371, 0, 32981682400388480056743209560182718662, 0, 7606550257597149407201285833302119775291, 0, 1754295191002805113976275995152302276199039, 0, 404592293881763318341638807150382516409558262, 0, 93310934846731632367437220837341279009690723727] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 586741273019112755809 z - 81150199514586626354 z - 383 z 24 22 4 6 + 9589951216119383143 z - 962129065827809334 z + 62754 z - 5994626 z 102 8 10 12 - 17320501900 z + 380756763 z - 17320501900 z + 592125056081 z 14 18 16 - 15735381683750 z - 5732894485683234 z + 333254263649311 z 50 48 - 142912413426399212165818160 z + 90605785551910826064577174 z 20 36 + 81309463041616635 z + 365164152838607827352432 z 34 66 - 90608897873292036529342 z - 50392586455468353659880284 z 80 100 90 + 19527290668805227382643 z + 592125056081 z - 962129065827809334 z 88 84 + 9589951216119383143 z + 586741273019112755809 z 94 86 96 - 5732894485683234 z - 81150199514586626354 z + 333254263649311 z 98 92 82 - 15735381683750 z + 81309463041616635 z - 3644133714584693650080 z 64 112 110 106 + 90605785551910826064577174 z + z - 383 z - 5994626 z 108 30 42 + 62754 z - 3644133714584693650080 z - 10495326660561368919431752 z 44 46 + 24571520272435584630332922 z - 50392586455468353659880284 z 58 56 - 240433742255070913944885552 z + 256576595699780416048800466 z 54 52 - 240433742255070913944885552 z + 197836026894583514918796418 z 60 70 + 197836026894583514918796418 z - 10495326660561368919431752 z 68 78 + 24571520272435584630332922 z - 90608897873292036529342 z 32 38 + 19527290668805227382643 z - 1281358973165633923939301 z 40 62 + 3922844495136068292801781 z - 142912413426399212165818160 z 76 74 + 365164152838607827352432 z - 1281358973165633923939301 z 72 104 / + 3922844495136068292801781 z + 380756763 z ) / (-1 / 28 26 2 - 2281929870774286327644 z + 296747918743900838718 z + 502 z 24 22 4 - 32969401922268151360 z + 3108659439437082915 z - 97665 z 6 102 8 10 + 10595380 z + 1370688480462 z - 745897042 z + 37057606512 z 12 14 18 - 1370688480462 z + 39167835570900 z + 16319477468877953 z 16 50 - 888312258263931 z + 1112448898301219678660057169 z 48 20 - 661298464678031453231925673 z - 246733454448851753 z 36 34 - 1820162274184022416463548 z + 424296076341751339988551 z 66 80 + 661298464678031453231925673 z - 424296076341751339988551 z 100 90 - 39167835570900 z + 32969401922268151360 z 88 84 - 296747918743900838718 z - 15074297292832450192650 z 94 86 + 246733454448851753 z + 2281929870774286327644 z 96 98 92 - 16319477468877953 z + 888312258263931 z - 3108659439437082915 z 82 64 + 85931817154107418111921 z - 1112448898301219678660057169 z 112 114 110 106 108 - 502 z + z + 97665 z + 745897042 z - 10595380 z 30 42 + 15074297292832450192650 z + 63221456872220337412862544 z 44 46 - 157756012866757519755767784 z + 344919183955541035524204768 z 58 56 + 2424941311466239885682911861 z - 2424941311466239885682911861 z 54 52 + 2129754596332518810563501312 z - 1642663787498566863524143048 z 60 70 - 2129754596332518810563501312 z + 157756012866757519755767784 z 68 78 - 344919183955541035524204768 z + 1820162274184022416463548 z 32 38 - 85931817154107418111921 z + 6800855752867235444409427 z 40 62 - 22177456633913626463948535 z + 1642663787498566863524143048 z 76 74 - 6800855752867235444409427 z + 22177456633913626463948535 z 72 104 - 63221456872220337412862544 z - 37057606512 z ) And in Maple-input format, it is: -(1+586741273019112755809*z^28-81150199514586626354*z^26-383*z^2+ 9589951216119383143*z^24-962129065827809334*z^22+62754*z^4-5994626*z^6-\ 17320501900*z^102+380756763*z^8-17320501900*z^10+592125056081*z^12-\ 15735381683750*z^14-5732894485683234*z^18+333254263649311*z^16-\ 142912413426399212165818160*z^50+90605785551910826064577174*z^48+ 81309463041616635*z^20+365164152838607827352432*z^36-90608897873292036529342*z^ 34-50392586455468353659880284*z^66+19527290668805227382643*z^80+592125056081*z^ 100-962129065827809334*z^90+9589951216119383143*z^88+586741273019112755809*z^84 -5732894485683234*z^94-81150199514586626354*z^86+333254263649311*z^96-\ 15735381683750*z^98+81309463041616635*z^92-3644133714584693650080*z^82+ 90605785551910826064577174*z^64+z^112-383*z^110-5994626*z^106+62754*z^108-\ 3644133714584693650080*z^30-10495326660561368919431752*z^42+ 24571520272435584630332922*z^44-50392586455468353659880284*z^46-\ 240433742255070913944885552*z^58+256576595699780416048800466*z^56-\ 240433742255070913944885552*z^54+197836026894583514918796418*z^52+ 197836026894583514918796418*z^60-10495326660561368919431752*z^70+ 24571520272435584630332922*z^68-90608897873292036529342*z^78+ 19527290668805227382643*z^32-1281358973165633923939301*z^38+ 3922844495136068292801781*z^40-142912413426399212165818160*z^62+ 365164152838607827352432*z^76-1281358973165633923939301*z^74+ 3922844495136068292801781*z^72+380756763*z^104)/(-1-2281929870774286327644*z^28 +296747918743900838718*z^26+502*z^2-32969401922268151360*z^24+ 3108659439437082915*z^22-97665*z^4+10595380*z^6+1370688480462*z^102-745897042*z ^8+37057606512*z^10-1370688480462*z^12+39167835570900*z^14+16319477468877953*z^ 18-888312258263931*z^16+1112448898301219678660057169*z^50-\ 661298464678031453231925673*z^48-246733454448851753*z^20-\ 1820162274184022416463548*z^36+424296076341751339988551*z^34+ 661298464678031453231925673*z^66-424296076341751339988551*z^80-39167835570900*z ^100+32969401922268151360*z^90-296747918743900838718*z^88-\ 15074297292832450192650*z^84+246733454448851753*z^94+2281929870774286327644*z^ 86-16319477468877953*z^96+888312258263931*z^98-3108659439437082915*z^92+ 85931817154107418111921*z^82-1112448898301219678660057169*z^64-502*z^112+z^114+ 97665*z^110+745897042*z^106-10595380*z^108+15074297292832450192650*z^30+ 63221456872220337412862544*z^42-157756012866757519755767784*z^44+ 344919183955541035524204768*z^46+2424941311466239885682911861*z^58-\ 2424941311466239885682911861*z^56+2129754596332518810563501312*z^54-\ 1642663787498566863524143048*z^52-2129754596332518810563501312*z^60+ 157756012866757519755767784*z^70-344919183955541035524204768*z^68+ 1820162274184022416463548*z^78-85931817154107418111921*z^32+ 6800855752867235444409427*z^38-22177456633913626463948535*z^40+ 1642663787498566863524143048*z^62-6800855752867235444409427*z^76+ 22177456633913626463948535*z^74-63221456872220337412862544*z^72-37057606512*z^ 104) The first , 40, terms are: [0, 119, 0, 24827, 0, 5441773, 0, 1202751032, 0, 266337113893, 0, 59005110331559, 0, 13073702725165063, 0, 2896814074365037069, 0, 641868385344527826145, 0, 142223777908289505207851, 0, 31513646428685247585274235, 0, 6982728605019362674510292769, 0, 1547218616646234097481183858648, 0, 342829516106868586966682844064217, 0, 75963458632647995437536111531146551, 0, 16831826840209082955511068240858568843, 0, 3729561553955122484678630043534134279725, 0, 826388574296488627077190937389386441624773, 0, 183109479721680948743146596243627979685400163, 0, 40573021707828094020446210222376790116406687983] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 663383204053078065272 z - 90510954560380637048 z - 380 z 24 22 4 + 10543834025379620108 z - 1042177906505357112 z + 61988 z 6 102 8 10 - 5918412 z - 17286429852 z + 377262110 z - 17286429852 z 12 14 18 + 597033385684 z - 16064328729948 z - 6023584850439912 z 16 50 + 344996095995739 z - 175825092711113312148980016 z 48 20 + 111154791767664220912982587 z + 86743505349863032 z 36 34 + 431826875827322079330508 z - 106116995306748491148900 z 66 80 - 61591609048674754637142212 z + 22625219305359883240041 z 100 90 88 + 597033385684 z - 1042177906505357112 z + 10543834025379620108 z 84 94 + 663383204053078065272 z - 6023584850439912 z 86 96 98 - 90510954560380637048 z + 344996095995739 z - 16064328729948 z 92 82 + 86743505349863032 z - 4172902609979281983976 z 64 112 110 106 + 111154791767664220912982587 z + z - 380 z - 5918412 z 108 30 42 + 61988 z - 4172902609979281983976 z - 12697391957541053323704148 z 44 46 + 29893359049929562300054204 z - 61591609048674754637142212 z 58 56 - 296756335878432780451202384 z + 316806532099554303319098408 z 54 52 - 296756335878432780451202384 z + 243887841939808824456703120 z 60 70 + 243887841939808824456703120 z - 12697391957541053323704148 z 68 78 + 29893359049929562300054204 z - 106116995306748491148900 z 32 38 + 22625219305359883240041 z - 1528415093031018102147396 z 40 62 + 4714841229444064439655618 z - 175825092711113312148980016 z 76 74 + 431826875827322079330508 z - 1528415093031018102147396 z 72 104 / 2 + 4714841229444064439655618 z + 377262110 z ) / ((-1 + z ) (1 / 28 26 2 + 2265478743906351989112 z - 295319763992967902360 z - 504 z 24 22 4 + 32807852299306503332 z - 3086648582183154328 z + 96876 z 6 102 8 10 - 10386900 z - 35953451044 z + 725766634 z - 35953451044 z 12 14 18 + 1330970474124 z - 38162935562952 z - 16062933859296384 z 16 50 + 869730242525899 z - 844420855459711052154547584 z 48 20 + 525929605656344924069483163 z + 244049383352110840 z 36 34 + 1734751050064768264352884 z - 410670532991704328388696 z 66 80 - 285980679086352720282612456 z + 84157672511632848531097 z 100 90 88 + 1330970474124 z - 3086648582183154328 z + 32807852299306503332 z 84 94 + 2265478743906351989112 z - 16062933859296384 z 86 96 98 - 295319763992967902360 z + 869730242525899 z - 38162935562952 z 92 82 + 244049383352110840 z - 14887300536991627879936 z 64 112 110 106 + 525929605656344924069483163 z + z - 504 z - 10386900 z 108 30 + 96876 z - 14887300536991627879936 z 42 44 - 56161758357117739138180940 z + 135705255230337776798044948 z 46 58 - 285980679086352720282612456 z - 1450140214440330024639926608 z 56 54 + 1551513421125714302580772984 z - 1450140214440330024639926608 z 52 60 + 1184020689285036027494102800 z + 1184020689285036027494102800 z 70 68 - 56161758357117739138180940 z + 135705255230337776798044948 z 78 32 - 410670532991704328388696 z + 84157672511632848531097 z 38 40 - 6357870214974363467239996 z + 20253765826114242830417206 z 62 76 - 844420855459711052154547584 z + 1734751050064768264352884 z 74 72 - 6357870214974363467239996 z + 20253765826114242830417206 z 104 + 725766634 z )) And in Maple-input format, it is: -(1+663383204053078065272*z^28-90510954560380637048*z^26-380*z^2+ 10543834025379620108*z^24-1042177906505357112*z^22+61988*z^4-5918412*z^6-\ 17286429852*z^102+377262110*z^8-17286429852*z^10+597033385684*z^12-\ 16064328729948*z^14-6023584850439912*z^18+344996095995739*z^16-\ 175825092711113312148980016*z^50+111154791767664220912982587*z^48+ 86743505349863032*z^20+431826875827322079330508*z^36-106116995306748491148900*z ^34-61591609048674754637142212*z^66+22625219305359883240041*z^80+597033385684*z ^100-1042177906505357112*z^90+10543834025379620108*z^88+663383204053078065272*z ^84-6023584850439912*z^94-90510954560380637048*z^86+344996095995739*z^96-\ 16064328729948*z^98+86743505349863032*z^92-4172902609979281983976*z^82+ 111154791767664220912982587*z^64+z^112-380*z^110-5918412*z^106+61988*z^108-\ 4172902609979281983976*z^30-12697391957541053323704148*z^42+ 29893359049929562300054204*z^44-61591609048674754637142212*z^46-\ 296756335878432780451202384*z^58+316806532099554303319098408*z^56-\ 296756335878432780451202384*z^54+243887841939808824456703120*z^52+ 243887841939808824456703120*z^60-12697391957541053323704148*z^70+ 29893359049929562300054204*z^68-106116995306748491148900*z^78+ 22625219305359883240041*z^32-1528415093031018102147396*z^38+ 4714841229444064439655618*z^40-175825092711113312148980016*z^62+ 431826875827322079330508*z^76-1528415093031018102147396*z^74+ 4714841229444064439655618*z^72+377262110*z^104)/(-1+z^2)/(1+ 2265478743906351989112*z^28-295319763992967902360*z^26-504*z^2+ 32807852299306503332*z^24-3086648582183154328*z^22+96876*z^4-10386900*z^6-\ 35953451044*z^102+725766634*z^8-35953451044*z^10+1330970474124*z^12-\ 38162935562952*z^14-16062933859296384*z^18+869730242525899*z^16-\ 844420855459711052154547584*z^50+525929605656344924069483163*z^48+ 244049383352110840*z^20+1734751050064768264352884*z^36-410670532991704328388696 *z^34-285980679086352720282612456*z^66+84157672511632848531097*z^80+ 1330970474124*z^100-3086648582183154328*z^90+32807852299306503332*z^88+ 2265478743906351989112*z^84-16062933859296384*z^94-295319763992967902360*z^86+ 869730242525899*z^96-38162935562952*z^98+244049383352110840*z^92-\ 14887300536991627879936*z^82+525929605656344924069483163*z^64+z^112-504*z^110-\ 10386900*z^106+96876*z^108-14887300536991627879936*z^30-\ 56161758357117739138180940*z^42+135705255230337776798044948*z^44-\ 285980679086352720282612456*z^46-1450140214440330024639926608*z^58+ 1551513421125714302580772984*z^56-1450140214440330024639926608*z^54+ 1184020689285036027494102800*z^52+1184020689285036027494102800*z^60-\ 56161758357117739138180940*z^70+135705255230337776798044948*z^68-\ 410670532991704328388696*z^78+84157672511632848531097*z^32-\ 6357870214974363467239996*z^38+20253765826114242830417206*z^40-\ 844420855459711052154547584*z^62+1734751050064768264352884*z^76-\ 6357870214974363467239996*z^74+20253765826114242830417206*z^72+725766634*z^104) The first , 40, terms are: [0, 125, 0, 27733, 0, 6398029, 0, 1481945681, 0, 343462660769, 0, 79611541861913, 0, 18453678469892917, 0, 4277521368741846141, 0, 991521098423705119137, 0, 229832719841728351352233, 0, 53274794409640349280243737, 0, 12348997856380814716050921381, 0, 2862474652861777982892799739021, 0, 663516282126172270990181930049717, 0, 153801835869786700741433008592437209, 0, 35650978514535470940190562912293279721, 0, 8263830284414828992689915201058120278341, 0, 1915540437185101781539228946309789960212125, 0, 444018698376942653666471137186501529033825333, 0, 102922705614141925634810635092407877147480900553] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 610692695864983580388 z - 85599757225617570416 z - 392 z 24 22 4 + 10236920319068577940 z - 1037506992538899936 z + 65616 z 6 102 8 10 - 6376448 z - 18842736776 z + 410348690 z - 18842736776 z 12 14 18 + 647998179820 z - 17266417955444 z - 6268366115489468 z 16 50 + 365545846019395 z - 127761249963863890836636696 z 48 20 + 81673710116894980756980747 z + 88392277561583140 z 36 34 + 357253210706282655276236 z - 90084386045897851659316 z 66 80 - 45894381018381326247425552 z + 19725478015054759215409 z 100 90 88 + 647998179820 z - 1037506992538899936 z + 10236920319068577940 z 84 94 + 610692695864983580388 z - 6268366115489468 z 86 96 98 - 85599757225617570416 z + 365545846019395 z - 17266417955444 z 92 82 + 88392277561583140 z - 3738170483597214945004 z 64 112 110 106 + 81673710116894980756980747 z + z - 392 z - 6376448 z 108 30 42 + 65616 z - 3738170483597214945004 z - 9805659629008388897942400 z 44 46 + 22649116880998116029700472 z - 45894381018381326247425552 z 58 56 - 212851520413224423513586160 z + 226859453873660594834019864 z 54 52 - 212851520413224423513586160 z + 175791181697970090831305336 z 60 70 + 175791181697970090831305336 z - 9805659629008388897942400 z 68 78 + 22649116880998116029700472 z - 90084386045897851659316 z 32 38 + 19725478015054759215409 z - 1233760186239660382487992 z 40 62 + 3719145071007592528526158 z - 127761249963863890836636696 z 76 74 + 357253210706282655276236 z - 1233760186239660382487992 z 72 104 / + 3719145071007592528526158 z + 410348690 z ) / (-1 / 28 26 2 - 2427562787321576063636 z + 319566011272254869704 z + 529 z 24 22 4 - 35890536397622849288 z + 3415385792415233780 z - 105688 z 6 102 8 10 + 11636120 z + 1531761862652 z - 826501446 z + 41288858642 z 12 14 18 - 1531761862652 z + 43803842493840 z + 18162821078156443 z 16 50 - 992087056029443 z + 1020246688440118917201929755 z 48 20 - 611964590581963046186902683 z - 273097539553823344 z 36 34 - 1827377773262333190158624 z + 432540305963207972384529 z 66 80 + 611964590581963046186902683 z - 432540305963207972384529 z 100 90 - 43803842493840 z + 35890536397622849288 z 88 84 - 319566011272254869704 z - 15823137235349777485168 z 94 86 + 273097539553823344 z + 2427562787321576063636 z 96 98 92 - 18162821078156443 z + 992087056029443 z - 3415385792415233780 z 82 64 + 88920021691186834584713 z - 1020246688440118917201929755 z 112 114 110 106 108 - 529 z + z + 105688 z + 826501446 z - 11636120 z 30 42 + 15823137235349777485168 z + 60688430172417824589818882 z 44 46 - 149398300499352264798057600 z + 322648743864761865303645112 z 58 56 + 2192787032796888585740896560 z - 2192787032796888585740896560 z 54 52 + 1930500948659156234724000696 z - 1496066886327774029014112224 z 60 70 - 1930500948659156234724000696 z + 149398300499352264798057600 z 68 78 - 322648743864761865303645112 z + 1827377773262333190158624 z 32 38 - 88920021691186834584713 z + 6723942215982865819640748 z 40 62 - 21598980411679101056947110 z + 1496066886327774029014112224 z 76 74 - 6723942215982865819640748 z + 21598980411679101056947110 z 72 104 - 60688430172417824589818882 z - 41288858642 z ) And in Maple-input format, it is: -(1+610692695864983580388*z^28-85599757225617570416*z^26-392*z^2+ 10236920319068577940*z^24-1037506992538899936*z^22+65616*z^4-6376448*z^6-\ 18842736776*z^102+410348690*z^8-18842736776*z^10+647998179820*z^12-\ 17266417955444*z^14-6268366115489468*z^18+365545846019395*z^16-\ 127761249963863890836636696*z^50+81673710116894980756980747*z^48+ 88392277561583140*z^20+357253210706282655276236*z^36-90084386045897851659316*z^ 34-45894381018381326247425552*z^66+19725478015054759215409*z^80+647998179820*z^ 100-1037506992538899936*z^90+10236920319068577940*z^88+610692695864983580388*z^ 84-6268366115489468*z^94-85599757225617570416*z^86+365545846019395*z^96-\ 17266417955444*z^98+88392277561583140*z^92-3738170483597214945004*z^82+ 81673710116894980756980747*z^64+z^112-392*z^110-6376448*z^106+65616*z^108-\ 3738170483597214945004*z^30-9805659629008388897942400*z^42+ 22649116880998116029700472*z^44-45894381018381326247425552*z^46-\ 212851520413224423513586160*z^58+226859453873660594834019864*z^56-\ 212851520413224423513586160*z^54+175791181697970090831305336*z^52+ 175791181697970090831305336*z^60-9805659629008388897942400*z^70+ 22649116880998116029700472*z^68-90084386045897851659316*z^78+ 19725478015054759215409*z^32-1233760186239660382487992*z^38+ 3719145071007592528526158*z^40-127761249963863890836636696*z^62+ 357253210706282655276236*z^76-1233760186239660382487992*z^74+ 3719145071007592528526158*z^72+410348690*z^104)/(-1-2427562787321576063636*z^28 +319566011272254869704*z^26+529*z^2-35890536397622849288*z^24+ 3415385792415233780*z^22-105688*z^4+11636120*z^6+1531761862652*z^102-826501446* z^8+41288858642*z^10-1531761862652*z^12+43803842493840*z^14+18162821078156443*z ^18-992087056029443*z^16+1020246688440118917201929755*z^50-\ 611964590581963046186902683*z^48-273097539553823344*z^20-\ 1827377773262333190158624*z^36+432540305963207972384529*z^34+ 611964590581963046186902683*z^66-432540305963207972384529*z^80-43803842493840*z ^100+35890536397622849288*z^90-319566011272254869704*z^88-\ 15823137235349777485168*z^84+273097539553823344*z^94+2427562787321576063636*z^ 86-18162821078156443*z^96+992087056029443*z^98-3415385792415233780*z^92+ 88920021691186834584713*z^82-1020246688440118917201929755*z^64-529*z^112+z^114+ 105688*z^110+826501446*z^106-11636120*z^108+15823137235349777485168*z^30+ 60688430172417824589818882*z^42-149398300499352264798057600*z^44+ 322648743864761865303645112*z^46+2192787032796888585740896560*z^58-\ 2192787032796888585740896560*z^56+1930500948659156234724000696*z^54-\ 1496066886327774029014112224*z^52-1930500948659156234724000696*z^60+ 149398300499352264798057600*z^70-322648743864761865303645112*z^68+ 1827377773262333190158624*z^78-88920021691186834584713*z^32+ 6723942215982865819640748*z^38-21598980411679101056947110*z^40+ 1496066886327774029014112224*z^62-6723942215982865819640748*z^76+ 21598980411679101056947110*z^74-60688430172417824589818882*z^72-41288858642*z^ 104) The first , 40, terms are: [0, 137, 0, 32401, 0, 7920545, 0, 1943567101, 0, 477277784353, 0, 117225976836925, 0, 28793731751928025, 0, 7072579841492556609, 0, 1737238114792983691149, 0, 426718305289056507064697, 0, 104814970611059814340030589, 0, 25745740815862532871506187041, 0, 6323936170068180484247509219817, 0, 1553350870541922045833427375719065, 0, 381550171580428113128346287815470825, 0, 93720315387702153483281627904108046009, 0, 23020557115758746235831898205398700377753, 0, 5654548298782236770269236219463244823635545, 0, 1388928873561184799921600590026780956062380737, 0, 341163133442867963373937694582726884676972972829] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 5247118596837933197 z - 1224019658427660809 z - 329 z 24 22 4 6 + 236859663829770568 z - 37780491751602634 z + 44688 z - 3441130 z 8 10 12 14 + 171750159 z - 5985565940 z + 152726131349 z - 2948380994980 z 18 16 50 - 521044940608984 z + 44113887857673 z - 536058825550093554630 z 48 20 + 811665651235277918496 z + 4929021091660567 z 36 34 + 299446910370592323780 z - 141256772703387569540 z 66 80 88 84 86 - 37780491751602634 z + 171750159 z + z + 44688 z - 329 z 82 64 30 - 3441130 z + 236859663829770568 z - 18756753712787983434 z 42 44 - 1040615272825672009466 z + 1130395898590916198608 z 46 58 - 1040615272825672009466 z - 18756753712787983434 z 56 54 + 56150486492652744270 z - 141256772703387569540 z 52 60 + 299446910370592323780 z + 5247118596837933197 z 70 68 78 - 521044940608984 z + 4929021091660567 z - 5985565940 z 32 38 + 56150486492652744270 z - 536058825550093554630 z 40 62 76 + 811665651235277918496 z - 1224019658427660809 z + 152726131349 z 74 72 / 2 - 2948380994980 z + 44113887857673 z ) / ((-1 + z ) (1 / 28 26 2 + 20868695777709383820 z - 4639113315799871828 z - 446 z 24 22 4 6 + 852289261269623009 z - 128610944388220527 z + 72244 z - 6327923 z 8 10 12 14 + 351051343 z - 13414952446 z + 371910862410 z - 7748454865925 z 18 16 50 - 1570858534933792 z + 124448634499135 z - 2530134436029605480130 z 48 20 + 3896858254552054236081 z + 15818884373138102 z 36 34 + 1380678716052521846869 z - 632495706217140279658 z 66 80 88 84 86 - 128610944388220527 z + 351051343 z + z + 72244 z - 446 z 82 64 30 - 6327923 z + 852289261269623009 z - 77964692804178934778 z 42 44 - 5048178498470824444774 z + 5502858140945428014197 z 46 58 - 5048178498470824444774 z - 77964692804178934778 z 56 54 + 242839666713415560645 z - 632495706217140279658 z 52 60 + 1380678716052521846869 z + 20868695777709383820 z 70 68 78 - 1570858534933792 z + 15818884373138102 z - 13414952446 z 32 38 + 242839666713415560645 z - 2530134436029605480130 z 40 62 76 + 3896858254552054236081 z - 4639113315799871828 z + 371910862410 z 74 72 - 7748454865925 z + 124448634499135 z )) And in Maple-input format, it is: -(1+5247118596837933197*z^28-1224019658427660809*z^26-329*z^2+ 236859663829770568*z^24-37780491751602634*z^22+44688*z^4-3441130*z^6+171750159* z^8-5985565940*z^10+152726131349*z^12-2948380994980*z^14-521044940608984*z^18+ 44113887857673*z^16-536058825550093554630*z^50+811665651235277918496*z^48+ 4929021091660567*z^20+299446910370592323780*z^36-141256772703387569540*z^34-\ 37780491751602634*z^66+171750159*z^80+z^88+44688*z^84-329*z^86-3441130*z^82+ 236859663829770568*z^64-18756753712787983434*z^30-1040615272825672009466*z^42+ 1130395898590916198608*z^44-1040615272825672009466*z^46-18756753712787983434*z^ 58+56150486492652744270*z^56-141256772703387569540*z^54+299446910370592323780*z ^52+5247118596837933197*z^60-521044940608984*z^70+4929021091660567*z^68-\ 5985565940*z^78+56150486492652744270*z^32-536058825550093554630*z^38+ 811665651235277918496*z^40-1224019658427660809*z^62+152726131349*z^76-\ 2948380994980*z^74+44113887857673*z^72)/(-1+z^2)/(1+20868695777709383820*z^28-\ 4639113315799871828*z^26-446*z^2+852289261269623009*z^24-128610944388220527*z^ 22+72244*z^4-6327923*z^6+351051343*z^8-13414952446*z^10+371910862410*z^12-\ 7748454865925*z^14-1570858534933792*z^18+124448634499135*z^16-\ 2530134436029605480130*z^50+3896858254552054236081*z^48+15818884373138102*z^20+ 1380678716052521846869*z^36-632495706217140279658*z^34-128610944388220527*z^66+ 351051343*z^80+z^88+72244*z^84-446*z^86-6327923*z^82+852289261269623009*z^64-\ 77964692804178934778*z^30-5048178498470824444774*z^42+5502858140945428014197*z^ 44-5048178498470824444774*z^46-77964692804178934778*z^58+242839666713415560645* z^56-632495706217140279658*z^54+1380678716052521846869*z^52+ 20868695777709383820*z^60-1570858534933792*z^70+15818884373138102*z^68-\ 13414952446*z^78+242839666713415560645*z^32-2530134436029605480130*z^38+ 3896858254552054236081*z^40-4639113315799871828*z^62+371910862410*z^76-\ 7748454865925*z^74+124448634499135*z^72) The first , 40, terms are: [0, 118, 0, 24744, 0, 5442185, 0, 1203605934, 0, 266394841557, 0, 58968067884105, 0, 13053169746806736, 0, 2889458448306272361, 0, 639612841494754245569, 0, 141585227976089255151598, 0, 31341423961918599005307241, 0, 6937763738821019045701757096, 0, 1535749167062706821541197693886, 0, 339954716452472544016373510466405, 0, 75252659563395949366142892026696285, 0, 16657991483447173880980392484843896862, 0, 3687426887942249614414127829132938708056, 0, 816251891318204440330758245494547132694145, 0, 180686199436059037289421323045539495376362222, 0, 39996847803836730969326696964159138636566264809] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 12587051063938094527 z - 2865649762871005174 z - 366 z 24 22 4 6 + 539118098938806329 z - 83264018739159008 z + 55495 z - 4728288 z 8 10 12 14 + 258107480 z - 9727261448 z + 265717904157 z - 5445107028466 z 18 16 50 - 1062391000295152 z + 85862762565451 z - 1380803479062316585642 z 48 20 + 2102753221256721679483 z + 10473117316094768 z 36 34 + 764974304566322716509 z - 356903891272691053984 z 66 80 88 84 86 - 83264018739159008 z + 258107480 z + z + 55495 z - 366 z 82 64 30 - 4728288 z + 539118098938806329 z - 45934570212358866936 z 42 44 - 2704986688396914961328 z + 2941630228852075267232 z 46 58 - 2704986688396914961328 z - 45934570212358866936 z 56 54 + 139899241914927292968 z - 356903891272691053984 z 52 60 + 764974304566322716509 z + 12587051063938094527 z 70 68 78 - 1062391000295152 z + 10473117316094768 z - 9727261448 z 32 38 + 139899241914927292968 z - 1380803479062316585642 z 40 62 76 + 2102753221256721679483 z - 2865649762871005174 z + 265717904157 z 74 72 / - 5445107028466 z + 85862762565451 z ) / (-1 / 28 26 2 - 61222463072625075772 z + 12869035859810908014 z + 490 z 24 22 4 6 - 2236033431384207649 z + 318993553599523276 z - 89440 z + 8770779 z 8 10 12 14 - 538040010 z + 22464280594 z - 673750967649 z + 15070694489948 z 18 16 50 + 3469281180857971 z - 258435559481058 z + 16645672007636219287366 z 48 20 - 23255786677669499822187 z - 37055802363857740 z 36 34 - 5140481294581133403977 z + 2210888102392487346214 z 66 80 90 88 84 + 2236033431384207649 z - 22464280594 z + z - 490 z - 8770779 z 86 82 64 + 89440 z + 538040010 z - 12869035859810908014 z 30 42 + 242081061856095482491 z + 23255786677669499822187 z 44 46 - 27482251184770552409192 z + 27482251184770552409192 z 58 56 + 799187359770191342718 z - 2210888102392487346214 z 54 52 + 5140481294581133403977 z - 10068966182640371481224 z 60 70 - 242081061856095482491 z + 37055802363857740 z 68 78 32 - 318993553599523276 z + 673750967649 z - 799187359770191342718 z 38 40 + 10068966182640371481224 z - 16645672007636219287366 z 62 76 74 + 61222463072625075772 z - 15070694489948 z + 258435559481058 z 72 - 3469281180857971 z ) And in Maple-input format, it is: -(1+12587051063938094527*z^28-2865649762871005174*z^26-366*z^2+ 539118098938806329*z^24-83264018739159008*z^22+55495*z^4-4728288*z^6+258107480* z^8-9727261448*z^10+265717904157*z^12-5445107028466*z^14-1062391000295152*z^18+ 85862762565451*z^16-1380803479062316585642*z^50+2102753221256721679483*z^48+ 10473117316094768*z^20+764974304566322716509*z^36-356903891272691053984*z^34-\ 83264018739159008*z^66+258107480*z^80+z^88+55495*z^84-366*z^86-4728288*z^82+ 539118098938806329*z^64-45934570212358866936*z^30-2704986688396914961328*z^42+ 2941630228852075267232*z^44-2704986688396914961328*z^46-45934570212358866936*z^ 58+139899241914927292968*z^56-356903891272691053984*z^54+764974304566322716509* z^52+12587051063938094527*z^60-1062391000295152*z^70+10473117316094768*z^68-\ 9727261448*z^78+139899241914927292968*z^32-1380803479062316585642*z^38+ 2102753221256721679483*z^40-2865649762871005174*z^62+265717904157*z^76-\ 5445107028466*z^74+85862762565451*z^72)/(-1-61222463072625075772*z^28+ 12869035859810908014*z^26+490*z^2-2236033431384207649*z^24+318993553599523276*z ^22-89440*z^4+8770779*z^6-538040010*z^8+22464280594*z^10-673750967649*z^12+ 15070694489948*z^14+3469281180857971*z^18-258435559481058*z^16+ 16645672007636219287366*z^50-23255786677669499822187*z^48-37055802363857740*z^ 20-5140481294581133403977*z^36+2210888102392487346214*z^34+2236033431384207649* z^66-22464280594*z^80+z^90-490*z^88-8770779*z^84+89440*z^86+538040010*z^82-\ 12869035859810908014*z^64+242081061856095482491*z^30+23255786677669499822187*z^ 42-27482251184770552409192*z^44+27482251184770552409192*z^46+ 799187359770191342718*z^58-2210888102392487346214*z^56+5140481294581133403977*z ^54-10068966182640371481224*z^52-242081061856095482491*z^60+37055802363857740*z ^70-318993553599523276*z^68+673750967649*z^78-799187359770191342718*z^32+ 10068966182640371481224*z^38-16645672007636219287366*z^40+61222463072625075772* z^62-15070694489948*z^76+258435559481058*z^74-3469281180857971*z^72) The first , 40, terms are: [0, 124, 0, 26815, 0, 6091281, 0, 1394038156, 0, 319483020591, 0, 73239181756863, 0, 16790545544495660, 0, 3849387371613766433, 0, 882509987708426225679, 0, 202324236792032890140060, 0, 46384860480922439189857873, 0, 10634194839939085818235445809, 0, 2437995924953633081870255997788, 0, 558935042281917208817611817338095, 0, 128141469992454273675989241196375041, 0, 29377718503983304951713857897548454252, 0, 6735136912221659923492137085779506604319, 0, 1544097756287619135739949203754923179447247, 0, 353999913001469653487900954826269180662796364, 0, 81158034130196981956205356664445920139169484081] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 29603266150746020 z - 13709737667032624 z - 336 z 24 22 4 6 + 5053068086659036 z - 1472523925490916 z + 44197 z - 3046276 z 8 10 12 14 + 125731468 z - 3377701328 z + 62491409396 z - 828448192876 z 18 16 50 - 59646890987904 z + 8098816273848 z - 59646890987904 z 48 20 36 + 336447820686448 z + 336447820686448 z + 70864534171312946 z 34 66 64 30 - 78994093391833696 z - 336 z + 44197 z - 51125690664619564 z 42 44 46 - 13709737667032624 z + 5053068086659036 z - 1472523925490916 z 58 56 54 52 - 3377701328 z + 62491409396 z - 828448192876 z + 8098816273848 z 60 68 32 38 + 125731468 z + z + 70864534171312946 z - 51125690664619564 z 40 62 / 28 + 29603266150746020 z - 3046276 z ) / (-1 - 200744624268392960 z / 26 2 24 + 83640177397822320 z + 461 z - 27741686014176868 z 22 4 6 8 + 7277104552433840 z - 75373 z + 6122725 z - 289832468 z 10 12 14 + 8794419424 z - 182166774704 z + 2690629832332 z 18 16 50 + 239069684431004 z - 29234022134648 z + 1497193961312732 z 48 20 36 - 7277104552433840 z - 1497193961312732 z - 737578906988944206 z 34 66 64 + 737578906988944206 z + 75373 z - 6122725 z 30 42 44 + 385472019002032892 z + 200744624268392960 z - 83640177397822320 z 46 58 56 + 27741686014176868 z + 182166774704 z - 2690629832332 z 54 52 60 70 + 29234022134648 z - 239069684431004 z - 8794419424 z + z 68 32 38 - 461 z - 594345131317786866 z + 594345131317786866 z 40 62 - 385472019002032892 z + 289832468 z ) And in Maple-input format, it is: -(1+29603266150746020*z^28-13709737667032624*z^26-336*z^2+5053068086659036*z^24 -1472523925490916*z^22+44197*z^4-3046276*z^6+125731468*z^8-3377701328*z^10+ 62491409396*z^12-828448192876*z^14-59646890987904*z^18+8098816273848*z^16-\ 59646890987904*z^50+336447820686448*z^48+336447820686448*z^20+70864534171312946 *z^36-78994093391833696*z^34-336*z^66+44197*z^64-51125690664619564*z^30-\ 13709737667032624*z^42+5053068086659036*z^44-1472523925490916*z^46-3377701328*z ^58+62491409396*z^56-828448192876*z^54+8098816273848*z^52+125731468*z^60+z^68+ 70864534171312946*z^32-51125690664619564*z^38+29603266150746020*z^40-3046276*z^ 62)/(-1-200744624268392960*z^28+83640177397822320*z^26+461*z^2-\ 27741686014176868*z^24+7277104552433840*z^22-75373*z^4+6122725*z^6-289832468*z^ 8+8794419424*z^10-182166774704*z^12+2690629832332*z^14+239069684431004*z^18-\ 29234022134648*z^16+1497193961312732*z^50-7277104552433840*z^48-\ 1497193961312732*z^20-737578906988944206*z^36+737578906988944206*z^34+75373*z^ 66-6122725*z^64+385472019002032892*z^30+200744624268392960*z^42-\ 83640177397822320*z^44+27741686014176868*z^46+182166774704*z^58-2690629832332*z ^56+29234022134648*z^54-239069684431004*z^52-8794419424*z^60+z^70-461*z^68-\ 594345131317786866*z^32+594345131317786866*z^38-385472019002032892*z^40+ 289832468*z^62) The first , 40, terms are: [0, 125, 0, 26449, 0, 5847813, 0, 1303540941, 0, 291292777673, 0, 65152578128245, 0, 14577759553512489, 0, 3262224151752047897, 0, 730067375564204704581, 0, 163388998317445378423193, 0, 36566811031842187021477629, 0, 8183765519848121531127673685, 0, 1831555205091144113609091434497, 0, 409908718736734881031628942243885, 0, 91739089300198782349187107285954705, 0, 20531550384120739807885180606088867953, 0, 4595037787282317978024739146309675449293, 0, 1028386668258424751192913961093034430253473, 0, 230156790262021567206642781587018972904392117, 0, 51509952355564645601872515326326037217323872669] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 617142100622563746573 z - 86590489551470215414 z - 392 z 24 22 4 + 10360406915732334185 z - 1049976642408910672 z + 65683 z 6 102 8 10 - 6389706 z - 18928026306 z + 411666170 z - 18928026306 z 12 14 18 + 651903312522 z - 17397905024988 z - 6334049830213882 z 16 50 + 368890415584451 z - 126047643706261316809965342 z 48 20 + 80701259478905813926041927 z + 89406056558925955 z 36 34 + 358113041496728015986043 z - 90525340970777478981988 z 66 80 - 45433670078126838008911928 z + 19866439288315128847819 z 100 90 88 + 651903312522 z - 1049976642408910672 z + 10360406915732334185 z 84 94 + 617142100622563746573 z - 6334049830213882 z 86 96 98 - 86590489551470215414 z + 368890415584451 z - 17397905024988 z 92 82 + 89406056558925955 z - 3772049656105492842818 z 64 112 110 106 + 80701259478905813926041927 z + z - 392 z - 6389706 z 108 30 42 + 65683 z - 3772049656105492842818 z - 9752275670453227688753942 z 44 46 + 22471047798847389400858854 z - 45433670078126838008911928 z 58 56 - 209618475987389755117816710 z + 223362335316420319756158900 z 54 52 - 209618475987389755117816710 z + 173239373028890886577562817 z 60 70 + 173239373028890886577562817 z - 9752275670453227688753942 z 68 78 + 22471047798847389400858854 z - 90525340970777478981988 z 32 38 + 19866439288315128847819 z - 1233495591245953619061552 z 40 62 + 3708491032999408577593654 z - 126047643706261316809965342 z 76 74 + 358113041496728015986043 z - 1233495591245953619061552 z 72 104 / + 3708491032999408577593654 z + 411666170 z ) / (-1 / 28 26 2 - 2453994536994337651152 z + 323472944521930818671 z + 524 z 24 22 4 - 36339039096925081675 z + 3455170848277624811 z - 104215 z 6 102 8 10 + 11462811 z + 1522645604066 z - 815469826 z + 40870714894 z 12 14 18 - 1522645604066 z + 43741051354377 z + 18284997282381920 z 16 50 - 994986985214426 z + 988260359604607138072092284 z 48 20 - 594583829637065032381459472 z - 275745352181341205 z 36 34 - 1822628891818301048981915 z + 433262438258243039242438 z 66 80 + 594583829637065032381459472 z - 433262438258243039242438 z 100 90 - 43741051354377 z + 36339039096925081675 z 88 84 - 323472944521930818671 z - 15958988990378352023124 z 94 86 + 275745352181341205 z + 2453994536994337651152 z 96 98 92 - 18284997282381920 z + 994986985214426 z - 3455170848277624811 z 82 64 112 + 89405318257881186485128 z - 988260359604607138072092284 z - 524 z 114 110 106 108 + z + 104215 z + 815469826 z - 11462811 z 30 42 + 15958988990378352023124 z + 59690225348040564694288629 z 44 46 - 146285159110568975869707141 z + 314626665863607211314214713 z 58 56 + 2113906671704102443884882623 z - 2113906671704102443884882623 z 54 52 + 1862573783141749765288415046 z - 1445749799716204194832062191 z 60 70 - 1862573783141749765288415046 z + 146285159110568975869707141 z 68 78 - 314626665863607211314214713 z + 1822628891818301048981915 z 32 38 - 89405318257881186485128 z + 6675788313537953980580369 z 40 62 - 21343335706525369610076698 z + 1445749799716204194832062191 z 76 74 - 6675788313537953980580369 z + 21343335706525369610076698 z 72 104 - 59690225348040564694288629 z - 40870714894 z ) And in Maple-input format, it is: -(1+617142100622563746573*z^28-86590489551470215414*z^26-392*z^2+ 10360406915732334185*z^24-1049976642408910672*z^22+65683*z^4-6389706*z^6-\ 18928026306*z^102+411666170*z^8-18928026306*z^10+651903312522*z^12-\ 17397905024988*z^14-6334049830213882*z^18+368890415584451*z^16-\ 126047643706261316809965342*z^50+80701259478905813926041927*z^48+ 89406056558925955*z^20+358113041496728015986043*z^36-90525340970777478981988*z^ 34-45433670078126838008911928*z^66+19866439288315128847819*z^80+651903312522*z^ 100-1049976642408910672*z^90+10360406915732334185*z^88+617142100622563746573*z^ 84-6334049830213882*z^94-86590489551470215414*z^86+368890415584451*z^96-\ 17397905024988*z^98+89406056558925955*z^92-3772049656105492842818*z^82+ 80701259478905813926041927*z^64+z^112-392*z^110-6389706*z^106+65683*z^108-\ 3772049656105492842818*z^30-9752275670453227688753942*z^42+ 22471047798847389400858854*z^44-45433670078126838008911928*z^46-\ 209618475987389755117816710*z^58+223362335316420319756158900*z^56-\ 209618475987389755117816710*z^54+173239373028890886577562817*z^52+ 173239373028890886577562817*z^60-9752275670453227688753942*z^70+ 22471047798847389400858854*z^68-90525340970777478981988*z^78+ 19866439288315128847819*z^32-1233495591245953619061552*z^38+ 3708491032999408577593654*z^40-126047643706261316809965342*z^62+ 358113041496728015986043*z^76-1233495591245953619061552*z^74+ 3708491032999408577593654*z^72+411666170*z^104)/(-1-2453994536994337651152*z^28 +323472944521930818671*z^26+524*z^2-36339039096925081675*z^24+ 3455170848277624811*z^22-104215*z^4+11462811*z^6+1522645604066*z^102-815469826* z^8+40870714894*z^10-1522645604066*z^12+43741051354377*z^14+18284997282381920*z ^18-994986985214426*z^16+988260359604607138072092284*z^50-\ 594583829637065032381459472*z^48-275745352181341205*z^20-\ 1822628891818301048981915*z^36+433262438258243039242438*z^34+ 594583829637065032381459472*z^66-433262438258243039242438*z^80-43741051354377*z ^100+36339039096925081675*z^90-323472944521930818671*z^88-\ 15958988990378352023124*z^84+275745352181341205*z^94+2453994536994337651152*z^ 86-18284997282381920*z^96+994986985214426*z^98-3455170848277624811*z^92+ 89405318257881186485128*z^82-988260359604607138072092284*z^64-524*z^112+z^114+ 104215*z^110+815469826*z^106-11462811*z^108+15958988990378352023124*z^30+ 59690225348040564694288629*z^42-146285159110568975869707141*z^44+ 314626665863607211314214713*z^46+2113906671704102443884882623*z^58-\ 2113906671704102443884882623*z^56+1862573783141749765288415046*z^54-\ 1445749799716204194832062191*z^52-1862573783141749765288415046*z^60+ 146285159110568975869707141*z^70-314626665863607211314214713*z^68+ 1822628891818301048981915*z^78-89405318257881186485128*z^32+ 6675788313537953980580369*z^38-21343335706525369610076698*z^40+ 1445749799716204194832062191*z^62-6675788313537953980580369*z^76+ 21343335706525369610076698*z^74-59690225348040564694288629*z^72-40870714894*z^ 104) The first , 40, terms are: [0, 132, 0, 30636, 0, 7369989, 0, 1778430892, 0, 429309733125, 0, 103641374211927, 0, 25020848941618632, 0, 6040495303806138425, 0, 1458288641851119649119, 0, 352058270976482338948816, 0, 84993485920823600261041299, 0, 20519025930080570013795900716, 0, 4953678776746447379876694843696, 0, 1195911226117250277905196412676847, 0, 288715462946492284320720749811303583, 0, 69701342986767616749322855772877136376, 0, 16827215157508068045566581125487059836492, 0, 4062406229576653704684374392299044368429879, 0, 980741270594326701603759121402051800526306072, 0, 236769388754056174564448998428381003987609131687] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 586217681021269247147 z - 81271830846524384038 z - 383 z 24 22 4 6 + 9623722788513875345 z - 967053936355989838 z + 62792 z - 6004126 z 102 8 10 12 - 17394844348 z + 381855877 z - 17394844348 z + 595411958587 z 14 18 16 - 15837144956938 z - 5772144225307322 z + 335559831727897 z 50 48 - 138347136425409660955514968 z + 87876225772643752473617314 z 20 36 + 81816832481994749 z + 360464476721690917704310 z 34 66 - 89734644473473313800106 z - 48987381873836739914683292 z 80 100 90 + 19399737321848250038125 z + 595411958587 z - 967053936355989838 z 88 84 + 9623722788513875345 z + 586217681021269247147 z 94 86 96 - 5772144225307322 z - 81271830846524384038 z + 335559831727897 z 98 92 82 - 15837144956938 z + 81816832481994749 z - 3631048284892834577072 z 64 112 110 106 + 87876225772643752473617314 z + z - 383 z - 6004126 z 108 30 42 + 62792 z - 3631048284892834577072 z - 10260114376867966376740096 z 44 46 + 23950128074083552288285462 z - 48987381873836739914683292 z 58 56 - 232232800136280310560885568 z + 247753922221750263683260878 z 54 52 - 232232800136280310560885568 z + 191250786527951322076280798 z 60 70 + 191250786527951322076280798 z - 10260114376867966376740096 z 68 78 + 23950128074083552288285462 z - 89734644473473313800106 z 32 38 + 19399737321848250038125 z - 1260700348602996768713605 z 40 62 + 3847019475275034663744137 z - 138347136425409660955514968 z 76 74 + 360464476721690917704310 z - 1260700348602996768713605 z 72 104 / 2 + 3847019475275034663744137 z + 381855877 z ) / ((-1 + z ) (1 / 28 26 2 + 2017754269659228954502 z - 267787699349762158894 z - 495 z 24 22 4 + 30281667807421156916 z - 2898379465956531168 z + 95568 z 6 102 8 10 - 10326720 z - 35986755190 z + 725481194 z - 35986755190 z 12 14 18 + 1328231652700 z - 37822970844596 z - 15558432536705842 z 16 50 + 853259818638879 z - 649447275211505806559993594 z 48 20 + 407162017096340122508706363 z + 232913806222850319 z 36 34 + 1442965738924724696122066 z - 347152412098669695366002 z 66 80 - 223245314171331994765770074 z + 72354052281145460320905 z 100 90 88 + 1328231652700 z - 2898379465956531168 z + 30281667807421156916 z 84 94 + 2017754269659228954502 z - 15558432536705842 z 86 96 98 - 267787699349762158894 z + 853259818638879 z - 37822970844596 z 92 82 + 232913806222850319 z - 13024998086189616112524 z 64 112 110 106 + 407162017096340122508706363 z + z - 495 z - 10326720 z 108 30 + 95568 z - 13024998086189616112524 z 42 44 - 44788958303351127542676714 z + 106993111655864016098657706 z 46 58 - 223245314171331994765770074 z - 1106818555821025164027788618 z 56 54 + 1183049245363312229224598783 z - 1106818555821025164027788618 z 52 60 + 906311174967361249523732362 z + 906311174967361249523732362 z 70 68 - 44788958303351127542676714 z + 106993111655864016098657706 z 78 32 - 347152412098669695366002 z + 72354052281145460320905 z 38 40 - 5208747666635303787495935 z + 16361004823610157610161914 z 62 76 - 649447275211505806559993594 z + 1442965738924724696122066 z 74 72 - 5208747666635303787495935 z + 16361004823610157610161914 z 104 + 725481194 z )) And in Maple-input format, it is: -(1+586217681021269247147*z^28-81271830846524384038*z^26-383*z^2+ 9623722788513875345*z^24-967053936355989838*z^22+62792*z^4-6004126*z^6-\ 17394844348*z^102+381855877*z^8-17394844348*z^10+595411958587*z^12-\ 15837144956938*z^14-5772144225307322*z^18+335559831727897*z^16-\ 138347136425409660955514968*z^50+87876225772643752473617314*z^48+ 81816832481994749*z^20+360464476721690917704310*z^36-89734644473473313800106*z^ 34-48987381873836739914683292*z^66+19399737321848250038125*z^80+595411958587*z^ 100-967053936355989838*z^90+9623722788513875345*z^88+586217681021269247147*z^84 -5772144225307322*z^94-81271830846524384038*z^86+335559831727897*z^96-\ 15837144956938*z^98+81816832481994749*z^92-3631048284892834577072*z^82+ 87876225772643752473617314*z^64+z^112-383*z^110-6004126*z^106+62792*z^108-\ 3631048284892834577072*z^30-10260114376867966376740096*z^42+ 23950128074083552288285462*z^44-48987381873836739914683292*z^46-\ 232232800136280310560885568*z^58+247753922221750263683260878*z^56-\ 232232800136280310560885568*z^54+191250786527951322076280798*z^52+ 191250786527951322076280798*z^60-10260114376867966376740096*z^70+ 23950128074083552288285462*z^68-89734644473473313800106*z^78+ 19399737321848250038125*z^32-1260700348602996768713605*z^38+ 3847019475275034663744137*z^40-138347136425409660955514968*z^62+ 360464476721690917704310*z^76-1260700348602996768713605*z^74+ 3847019475275034663744137*z^72+381855877*z^104)/(-1+z^2)/(1+ 2017754269659228954502*z^28-267787699349762158894*z^26-495*z^2+ 30281667807421156916*z^24-2898379465956531168*z^22+95568*z^4-10326720*z^6-\ 35986755190*z^102+725481194*z^8-35986755190*z^10+1328231652700*z^12-\ 37822970844596*z^14-15558432536705842*z^18+853259818638879*z^16-\ 649447275211505806559993594*z^50+407162017096340122508706363*z^48+ 232913806222850319*z^20+1442965738924724696122066*z^36-347152412098669695366002 *z^34-223245314171331994765770074*z^66+72354052281145460320905*z^80+ 1328231652700*z^100-2898379465956531168*z^90+30281667807421156916*z^88+ 2017754269659228954502*z^84-15558432536705842*z^94-267787699349762158894*z^86+ 853259818638879*z^96-37822970844596*z^98+232913806222850319*z^92-\ 13024998086189616112524*z^82+407162017096340122508706363*z^64+z^112-495*z^110-\ 10326720*z^106+95568*z^108-13024998086189616112524*z^30-\ 44788958303351127542676714*z^42+106993111655864016098657706*z^44-\ 223245314171331994765770074*z^46-1106818555821025164027788618*z^58+ 1183049245363312229224598783*z^56-1106818555821025164027788618*z^54+ 906311174967361249523732362*z^52+906311174967361249523732362*z^60-\ 44788958303351127542676714*z^70+106993111655864016098657706*z^68-\ 347152412098669695366002*z^78+72354052281145460320905*z^32-\ 5208747666635303787495935*z^38+16361004823610157610161914*z^40-\ 649447275211505806559993594*z^62+1442965738924724696122066*z^76-\ 5208747666635303787495935*z^74+16361004823610157610161914*z^72+725481194*z^104) The first , 40, terms are: [0, 113, 0, 22777, 0, 4860435, 0, 1046515316, 0, 225723180861, 0, 48704329686339, 0, 10509791068221475, 0, 2267923293650482767, 0, 489400459146592430095, 0, 105608961345257158197883, 0, 22789629131118023664086199, 0, 4917832851977872826086269673, 0, 1061231846768762909977966056372, 0, 229005960379490513696568715369551, 0, 49417787524357357086441720736035453, 0, 10663991976715359316080457292639209281, 0, 2301210365330085458763726553373313853609, 0, 496584126948047244193404337320041304262409, 0, 107159171039875653469748571283670180555023233, 0, 23124154226457832054654373304483832231383930949] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 5654712865959273472 z - 1337126368896591312 z - 356 z 24 22 4 6 + 262815388642820357 z - 42663426905900068 z + 51800 z - 4178296 z 8 10 12 14 + 213457899 z - 7477926112 z + 189437597040 z - 3603258165312 z 18 16 50 - 612184996011320 z + 52902484396359 z - 554749757831044676752 z 48 20 + 837101555902872547262 z + 5674333082766648 z 36 34 + 311395989375747249776 z - 147832277897500521272 z 66 80 88 84 86 - 42663426905900068 z + 213457899 z + z + 51800 z - 356 z 82 64 30 - 4178296 z + 262815388642820357 z - 19980745800276682192 z 42 44 - 1071049577561466731328 z + 1162672708836557096224 z 46 58 - 1071049577561466731328 z - 19980745800276682192 z 56 54 + 59235609305506274314 z - 147832277897500521272 z 52 60 + 311395989375747249776 z + 5654712865959273472 z 70 68 78 - 612184996011320 z + 5674333082766648 z - 7477926112 z 32 38 + 59235609305506274314 z - 554749757831044676752 z 40 62 76 + 837101555902872547262 z - 1337126368896591312 z + 189437597040 z 74 72 / - 3603258165312 z + 52902484396359 z ) / (-1 / 28 26 2 - 28388518597590547976 z + 6189653536569243405 z + 493 z 24 22 4 6 - 1121975839781211753 z + 167987802076038178 z - 86994 z + 8064750 z 8 10 12 14 - 460964263 z + 17810954255 z - 493737188420 z + 10231093139932 z 18 16 50 + 2049990000919355 z - 163236826085187 z + 6856488864724163711502 z 48 20 - 9519940302989528364862 z - 20607138134245774 z 36 34 - 2166041656845463983020 z + 947659588762059043002 z 66 80 90 88 84 + 1121975839781211753 z - 17810954255 z + z - 493 z - 8064750 z 86 82 64 + 86994 z + 460964263 z - 6189653536569243405 z 30 42 + 108808662395236886296 z + 9519940302989528364862 z 44 46 - 11215738550719526123456 z + 11215738550719526123456 z 58 56 + 349965098012507925602 z - 947659588762059043002 z 54 52 + 2166041656845463983020 z - 4187294453143684507364 z 60 70 - 108808662395236886296 z + 20607138134245774 z 68 78 32 - 167987802076038178 z + 493737188420 z - 349965098012507925602 z 38 40 + 4187294453143684507364 z - 6856488864724163711502 z 62 76 74 + 28388518597590547976 z - 10231093139932 z + 163236826085187 z 72 - 2049990000919355 z ) And in Maple-input format, it is: -(1+5654712865959273472*z^28-1337126368896591312*z^26-356*z^2+ 262815388642820357*z^24-42663426905900068*z^22+51800*z^4-4178296*z^6+213457899* z^8-7477926112*z^10+189437597040*z^12-3603258165312*z^14-612184996011320*z^18+ 52902484396359*z^16-554749757831044676752*z^50+837101555902872547262*z^48+ 5674333082766648*z^20+311395989375747249776*z^36-147832277897500521272*z^34-\ 42663426905900068*z^66+213457899*z^80+z^88+51800*z^84-356*z^86-4178296*z^82+ 262815388642820357*z^64-19980745800276682192*z^30-1071049577561466731328*z^42+ 1162672708836557096224*z^44-1071049577561466731328*z^46-19980745800276682192*z^ 58+59235609305506274314*z^56-147832277897500521272*z^54+311395989375747249776*z ^52+5654712865959273472*z^60-612184996011320*z^70+5674333082766648*z^68-\ 7477926112*z^78+59235609305506274314*z^32-554749757831044676752*z^38+ 837101555902872547262*z^40-1337126368896591312*z^62+189437597040*z^76-\ 3603258165312*z^74+52902484396359*z^72)/(-1-28388518597590547976*z^28+ 6189653536569243405*z^26+493*z^2-1121975839781211753*z^24+167987802076038178*z^ 22-86994*z^4+8064750*z^6-460964263*z^8+17810954255*z^10-493737188420*z^12+ 10231093139932*z^14+2049990000919355*z^18-163236826085187*z^16+ 6856488864724163711502*z^50-9519940302989528364862*z^48-20607138134245774*z^20-\ 2166041656845463983020*z^36+947659588762059043002*z^34+1121975839781211753*z^66 -17810954255*z^80+z^90-493*z^88-8064750*z^84+86994*z^86+460964263*z^82-\ 6189653536569243405*z^64+108808662395236886296*z^30+9519940302989528364862*z^42 -11215738550719526123456*z^44+11215738550719526123456*z^46+ 349965098012507925602*z^58-947659588762059043002*z^56+2166041656845463983020*z^ 54-4187294453143684507364*z^52-108808662395236886296*z^60+20607138134245774*z^ 70-167987802076038178*z^68+493737188420*z^78-349965098012507925602*z^32+ 4187294453143684507364*z^38-6856488864724163711502*z^40+28388518597590547976*z^ 62-10231093139932*z^76+163236826085187*z^74-2049990000919355*z^72) The first , 40, terms are: [0, 137, 0, 32347, 0, 7915347, 0, 1945635539, 0, 478662016171, 0, 117782040737081, 0, 28983311542954513, 0, 7132163073322391489, 0, 1755074364242680202457, 0, 431886880333586541661931, 0, 106278288879437996808843107, 0, 26152855114827432099199168451, 0, 6435668492131392087096345739259, 0, 1583682883628040759646386314818889, 0, 389711104579342814472071434414069201, 0, 95899719959610071395945688909896578929, 0, 23598907448200847884478746886272482739913, 0, 5807195610015521512071986480002359863020987, 0, 1429028904284653898932914666766591955435581859, 0, 351654007617631347166345817180047557549724805507] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 604861748561129695851 z - 83795199487641538078 z - 386 z 24 22 4 6 + 9914822019163844015 z - 995476297142350852 z + 63657 z - 6111176 z 102 8 10 12 - 17786009492 z + 389704706 z - 17786009492 z + 609695702188 z 14 18 16 - 16237276633388 z - 5930749064436074 z + 344423427870193 z 50 48 - 143788126755178568205131512 z + 91288709660208216319509925 z 20 36 + 84146032380480317 z + 372961269449731400465309 z 34 66 - 92781186042882161643800 z - 50860638652202703232032394 z 80 100 90 + 20044595852773137036601 z + 609695702188 z - 995476297142350852 z 88 84 + 9914822019163844015 z + 604861748561129695851 z 94 86 96 - 5930749064436074 z - 83795199487641538078 z + 344423427870193 z 98 92 82 - 16237276633388 z + 84146032380480317 z - 3749164607574762154698 z 64 112 110 106 + 91288709660208216319509925 z + z - 386 z - 6111176 z 108 30 42 + 63657 z - 3749164607574762154698 z - 10638359208458076921950084 z 44 46 + 24850057876167481183538102 z - 50860638652202703232032394 z 58 56 - 241507069612454360558946022 z + 257667612410534167213367428 z 54 52 - 241507069612454360558946022 z + 198843914332141222358387481 z 60 70 + 198843914332141222358387481 z - 10638359208458076921950084 z 68 78 + 24850057876167481183538102 z - 92781186042882161643800 z 32 38 + 20044595852773137036601 z - 1305326205177885282376396 z 40 62 + 3986026780597339400711844 z - 143788126755178568205131512 z 76 74 + 372961269449731400465309 z - 1305326205177885282376396 z 72 104 / + 3986026780597339400711844 z + 389704706 z ) / (-1 / 28 26 2 - 2370677018378116064464 z + 308474590571836895439 z + 514 z 24 22 4 - 34278279237847472563 z + 3231249758553578527 z - 100697 z 6 102 8 10 + 10941061 z + 1417296362264 z - 770407468 z + 38288222580 z 12 14 18 - 1417296362264 z + 40544973133565 z + 16935131072539260 z 16 50 - 920709531989358 z + 1133543253494106904997345494 z 48 20 - 674822000356418663435925230 z - 256295147849568605 z 36 34 - 1879840045492476967028495 z + 439033209218378371484090 z 66 80 + 674822000356418663435925230 z - 439033209218378371484090 z 100 90 - 40544973133565 z + 34278279237847472563 z 88 84 - 308474590571836895439 z - 15644873835090936902400 z 94 86 + 256295147849568605 z + 2370677018378116064464 z 96 98 92 - 16935131072539260 z + 920709531989358 z - 3231249758553578527 z 82 64 + 89063555720666431247632 z - 1133543253494106904997345494 z 112 114 110 106 108 - 514 z + z + 100697 z + 770407468 z - 10941061 z 30 42 + 15644873835090936902400 z + 64884374331414948595403831 z 44 46 - 161571238306827101092964655 z + 352580123444117554064934599 z 58 56 + 2465177341913236561028407087 z - 2465177341913236561028407087 z 54 52 + 2165958877694600740063533954 z - 1671900585272450878286203397 z 60 70 - 2165958877694600740063533954 z + 161571238306827101092964655 z 68 78 - 352580123444117554064934599 z + 1879840045492476967028495 z 32 38 - 89063555720666431247632 z + 7009516272725860763687069 z 40 62 - 22809385399827533095043310 z + 1671900585272450878286203397 z 76 74 - 7009516272725860763687069 z + 22809385399827533095043310 z 72 104 - 64884374331414948595403831 z - 38288222580 z ) And in Maple-input format, it is: -(1+604861748561129695851*z^28-83795199487641538078*z^26-386*z^2+ 9914822019163844015*z^24-995476297142350852*z^22+63657*z^4-6111176*z^6-\ 17786009492*z^102+389704706*z^8-17786009492*z^10+609695702188*z^12-\ 16237276633388*z^14-5930749064436074*z^18+344423427870193*z^16-\ 143788126755178568205131512*z^50+91288709660208216319509925*z^48+ 84146032380480317*z^20+372961269449731400465309*z^36-92781186042882161643800*z^ 34-50860638652202703232032394*z^66+20044595852773137036601*z^80+609695702188*z^ 100-995476297142350852*z^90+9914822019163844015*z^88+604861748561129695851*z^84 -5930749064436074*z^94-83795199487641538078*z^86+344423427870193*z^96-\ 16237276633388*z^98+84146032380480317*z^92-3749164607574762154698*z^82+ 91288709660208216319509925*z^64+z^112-386*z^110-6111176*z^106+63657*z^108-\ 3749164607574762154698*z^30-10638359208458076921950084*z^42+ 24850057876167481183538102*z^44-50860638652202703232032394*z^46-\ 241507069612454360558946022*z^58+257667612410534167213367428*z^56-\ 241507069612454360558946022*z^54+198843914332141222358387481*z^52+ 198843914332141222358387481*z^60-10638359208458076921950084*z^70+ 24850057876167481183538102*z^68-92781186042882161643800*z^78+ 20044595852773137036601*z^32-1305326205177885282376396*z^38+ 3986026780597339400711844*z^40-143788126755178568205131512*z^62+ 372961269449731400465309*z^76-1305326205177885282376396*z^74+ 3986026780597339400711844*z^72+389704706*z^104)/(-1-2370677018378116064464*z^28 +308474590571836895439*z^26+514*z^2-34278279237847472563*z^24+ 3231249758553578527*z^22-100697*z^4+10941061*z^6+1417296362264*z^102-770407468* z^8+38288222580*z^10-1417296362264*z^12+40544973133565*z^14+16935131072539260*z ^18-920709531989358*z^16+1133543253494106904997345494*z^50-\ 674822000356418663435925230*z^48-256295147849568605*z^20-\ 1879840045492476967028495*z^36+439033209218378371484090*z^34+ 674822000356418663435925230*z^66-439033209218378371484090*z^80-40544973133565*z ^100+34278279237847472563*z^90-308474590571836895439*z^88-\ 15644873835090936902400*z^84+256295147849568605*z^94+2370677018378116064464*z^ 86-16935131072539260*z^96+920709531989358*z^98-3231249758553578527*z^92+ 89063555720666431247632*z^82-1133543253494106904997345494*z^64-514*z^112+z^114+ 100697*z^110+770407468*z^106-10941061*z^108+15644873835090936902400*z^30+ 64884374331414948595403831*z^42-161571238306827101092964655*z^44+ 352580123444117554064934599*z^46+2465177341913236561028407087*z^58-\ 2465177341913236561028407087*z^56+2165958877694600740063533954*z^54-\ 1671900585272450878286203397*z^52-2165958877694600740063533954*z^60+ 161571238306827101092964655*z^70-352580123444117554064934599*z^68+ 1879840045492476967028495*z^78-89063555720666431247632*z^32+ 7009516272725860763687069*z^38-22809385399827533095043310*z^40+ 1671900585272450878286203397*z^62-7009516272725860763687069*z^76+ 22809385399827533095043310*z^74-64884374331414948595403831*z^72-38288222580*z^ 104) The first , 40, terms are: [0, 128, 0, 28752, 0, 6719197, 0, 1578180160, 0, 371049064987, 0, 87258892390043, 0, 20521745581023272, 0, 4826426690292129051, 0, 1135112520210782601371, 0, 266963927995354415173916, 0, 62786514916676703984740053, 0, 14766589560709646922743871504, 0, 3472914078439802647195201960100, 0, 816785230603552037896909719264501, 0, 192097500482195286693175507779136421, 0, 45178889532276515640397063008677219196, 0, 10625500354995441370794788606111849055824, 0, 2498982577166672490401986439109954051418761, 0, 587728926863084736780562942009118680563780564, 0, 138226370455087636946049062356039190398325684395] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6438733151169782736 z - 1484772106615041120 z - 342 z 24 22 4 6 + 284155025592348325 z - 44864374250774318 z + 48074 z - 3799786 z 8 10 12 14 + 193089295 z - 6808631592 z + 175098261420 z - 3400869861160 z 18 16 50 - 608470407990546 z + 51180847388523 z - 690934794932653597348 z 48 20 + 1051879303580959986166 z + 5800624255091490 z 36 34 + 383141620052267119380 z - 179127646430686993628 z 66 80 88 84 86 - 44864374250774318 z + 193089295 z + z + 48074 z - 342 z 82 64 30 - 3799786 z + 284155025592348325 z - 23283638011519074912 z 42 44 - 1353111624464022870256 z + 1471516021029980239112 z 46 58 - 1353111624464022870256 z - 23283638011519074912 z 56 54 + 70480277900524755178 z - 179127646430686993628 z 52 60 + 383141620052267119380 z + 6438733151169782736 z 70 68 78 - 608470407990546 z + 5800624255091490 z - 6808631592 z 32 38 + 70480277900524755178 z - 690934794932653597348 z 40 62 76 + 1051879303580959986166 z - 1484772106615041120 z + 175098261420 z 74 72 / 2 - 3400869861160 z + 51180847388523 z ) / ((-1 + z ) (1 / 28 26 2 + 25653286069480804464 z - 5632652024890265360 z - 468 z 24 22 4 6 + 1022709596741188293 z - 152685584486071316 z + 78670 z - 7067140 z 8 10 12 14 + 398202735 z - 15353735024 z + 427957033380 z - 8953226276384 z 18 16 50 - 1833963530691044 z + 144450156471723 z - 3280110858049207306072 z 48 20 + 5081950493742938759894 z + 18607622579725910 z 36 34 + 1775732033915997045852 z - 805612439613625725656 z 66 80 88 84 86 - 152685584486071316 z + 398202735 z + z + 78670 z - 468 z 82 64 30 - 7067140 z + 1022709596741188293 z - 97039279900220077488 z 42 44 - 6607445845551465297328 z + 7211436254114976346072 z 46 58 - 6607445845551465297328 z - 97039279900220077488 z 56 54 + 305900201535793481034 z - 805612439613625725656 z 52 60 + 1775732033915997045852 z + 25653286069480804464 z 70 68 78 - 1833963530691044 z + 18607622579725910 z - 15353735024 z 32 38 + 305900201535793481034 z - 3280110858049207306072 z 40 62 76 + 5081950493742938759894 z - 5632652024890265360 z + 427957033380 z 74 72 - 8953226276384 z + 144450156471723 z )) And in Maple-input format, it is: -(1+6438733151169782736*z^28-1484772106615041120*z^26-342*z^2+ 284155025592348325*z^24-44864374250774318*z^22+48074*z^4-3799786*z^6+193089295* z^8-6808631592*z^10+175098261420*z^12-3400869861160*z^14-608470407990546*z^18+ 51180847388523*z^16-690934794932653597348*z^50+1051879303580959986166*z^48+ 5800624255091490*z^20+383141620052267119380*z^36-179127646430686993628*z^34-\ 44864374250774318*z^66+193089295*z^80+z^88+48074*z^84-342*z^86-3799786*z^82+ 284155025592348325*z^64-23283638011519074912*z^30-1353111624464022870256*z^42+ 1471516021029980239112*z^44-1353111624464022870256*z^46-23283638011519074912*z^ 58+70480277900524755178*z^56-179127646430686993628*z^54+383141620052267119380*z ^52+6438733151169782736*z^60-608470407990546*z^70+5800624255091490*z^68-\ 6808631592*z^78+70480277900524755178*z^32-690934794932653597348*z^38+ 1051879303580959986166*z^40-1484772106615041120*z^62+175098261420*z^76-\ 3400869861160*z^74+51180847388523*z^72)/(-1+z^2)/(1+25653286069480804464*z^28-\ 5632652024890265360*z^26-468*z^2+1022709596741188293*z^24-152685584486071316*z^ 22+78670*z^4-7067140*z^6+398202735*z^8-15353735024*z^10+427957033380*z^12-\ 8953226276384*z^14-1833963530691044*z^18+144450156471723*z^16-\ 3280110858049207306072*z^50+5081950493742938759894*z^48+18607622579725910*z^20+ 1775732033915997045852*z^36-805612439613625725656*z^34-152685584486071316*z^66+ 398202735*z^80+z^88+78670*z^84-468*z^86-7067140*z^82+1022709596741188293*z^64-\ 97039279900220077488*z^30-6607445845551465297328*z^42+7211436254114976346072*z^ 44-6607445845551465297328*z^46-97039279900220077488*z^58+305900201535793481034* z^56-805612439613625725656*z^54+1775732033915997045852*z^52+ 25653286069480804464*z^60-1833963530691044*z^70+18607622579725910*z^68-\ 15353735024*z^78+305900201535793481034*z^32-3280110858049207306072*z^38+ 5081950493742938759894*z^40-5632652024890265360*z^62+427957033380*z^76-\ 8953226276384*z^74+144450156471723*z^72) The first , 40, terms are: [0, 127, 0, 28499, 0, 6661529, 0, 1564240529, 0, 367571197331, 0, 86384542528511, 0, 20302143940940633, 0, 4771447219965930953, 0, 1121395607585579119551, 0, 263552832056847045491731, 0, 61940762669817548988054017, 0, 14557453617422856241636877929, 0, 3421324621977982927671551402963, 0, 804087203996098300245065418533439, 0, 188978335342063732668484675057670993, 0, 44414102167518751643813272888837399345, 0, 10438299542599905988395009592350686587839, 0, 2453232014693850090614726530557975936573651, 0, 576563959805888846939770706037393546354423881, 0, 135505324305236720257053497192365225840385608865] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6867160541814300289 z - 1582107458182509250 z - 346 z 24 22 4 6 + 302328917725269133 z - 47635605076042764 z + 49125 z - 3911874 z 8 10 12 14 + 199915932 z - 7082469170 z + 182900837915 z - 3566137740918 z 18 16 50 - 642479571982876 z + 53861831861503 z - 735617966480799166830 z 48 20 + 1119121922619310247475 z + 6143163743313496 z 36 34 + 408241482426232080319 z - 191000072959564701198 z 66 80 88 84 86 - 47635605076042764 z + 199915932 z + z + 49125 z - 346 z 82 64 30 - 3911874 z + 302328917725269133 z - 24842028170848927806 z 42 44 - 1438922255324082018200 z + 1564569942980468099280 z 46 58 - 1438922255324082018200 z - 24842028170848927806 z 56 54 + 75188648848401010596 z - 191000072959564701198 z 52 60 + 408241482426232080319 z + 6867160541814300289 z 70 68 78 - 642479571982876 z + 6143163743313496 z - 7082469170 z 32 38 + 75188648848401010596 z - 735617966480799166830 z 40 62 76 + 1119121922619310247475 z - 1582107458182509250 z + 182900837915 z 74 72 / - 3566137740918 z + 53861831861503 z ) / (-1 / 28 26 2 - 33784416185145274110 z + 7160505938525184970 z + 486 z 24 22 4 6 - 1259322415169040341 z + 182694761201570796 z - 82898 z + 7499735 z 8 10 12 14 - 424993892 z + 16505036240 z - 464637841041 z + 9848867496402 z 18 16 50 + 2091117309959303 z - 161520249451942 z + 9117271512833806902506 z 48 20 - 12747812515718318728803 z - 21701608379849724 z 36 34 - 2811118517132259399685 z + 1208982896029621568948 z 66 80 90 88 84 + 1259322415169040341 z - 16505036240 z + z - 486 z - 7499735 z 86 82 64 + 82898 z + 424993892 z - 7160505938525184970 z 30 42 + 132896053585649268435 z + 12747812515718318728803 z 44 46 - 15071344459366860812376 z + 15071344459366860812376 z 58 56 + 437498223013699434456 z - 1208982896029621568948 z 54 52 + 2811118517132259399685 z - 5510082156642533133086 z 60 70 - 132896053585649268435 z + 21701608379849724 z 68 78 32 - 182694761201570796 z + 464637841041 z - 437498223013699434456 z 38 40 + 5510082156642533133086 z - 9117271512833806902506 z 62 76 74 + 33784416185145274110 z - 9848867496402 z + 161520249451942 z 72 - 2091117309959303 z ) And in Maple-input format, it is: -(1+6867160541814300289*z^28-1582107458182509250*z^26-346*z^2+ 302328917725269133*z^24-47635605076042764*z^22+49125*z^4-3911874*z^6+199915932* z^8-7082469170*z^10+182900837915*z^12-3566137740918*z^14-642479571982876*z^18+ 53861831861503*z^16-735617966480799166830*z^50+1119121922619310247475*z^48+ 6143163743313496*z^20+408241482426232080319*z^36-191000072959564701198*z^34-\ 47635605076042764*z^66+199915932*z^80+z^88+49125*z^84-346*z^86-3911874*z^82+ 302328917725269133*z^64-24842028170848927806*z^30-1438922255324082018200*z^42+ 1564569942980468099280*z^44-1438922255324082018200*z^46-24842028170848927806*z^ 58+75188648848401010596*z^56-191000072959564701198*z^54+408241482426232080319*z ^52+6867160541814300289*z^60-642479571982876*z^70+6143163743313496*z^68-\ 7082469170*z^78+75188648848401010596*z^32-735617966480799166830*z^38+ 1119121922619310247475*z^40-1582107458182509250*z^62+182900837915*z^76-\ 3566137740918*z^74+53861831861503*z^72)/(-1-33784416185145274110*z^28+ 7160505938525184970*z^26+486*z^2-1259322415169040341*z^24+182694761201570796*z^ 22-82898*z^4+7499735*z^6-424993892*z^8+16505036240*z^10-464637841041*z^12+ 9848867496402*z^14+2091117309959303*z^18-161520249451942*z^16+ 9117271512833806902506*z^50-12747812515718318728803*z^48-21701608379849724*z^20 -2811118517132259399685*z^36+1208982896029621568948*z^34+1259322415169040341*z^ 66-16505036240*z^80+z^90-486*z^88-7499735*z^84+82898*z^86+424993892*z^82-\ 7160505938525184970*z^64+132896053585649268435*z^30+12747812515718318728803*z^ 42-15071344459366860812376*z^44+15071344459366860812376*z^46+ 437498223013699434456*z^58-1208982896029621568948*z^56+2811118517132259399685*z ^54-5510082156642533133086*z^52-132896053585649268435*z^60+21701608379849724*z^ 70-182694761201570796*z^68+464637841041*z^78-437498223013699434456*z^32+ 5510082156642533133086*z^38-9117271512833806902506*z^40+33784416185145274110*z^ 62-9848867496402*z^76+161520249451942*z^74-2091117309959303*z^72) The first , 40, terms are: [0, 140, 0, 34267, 0, 8635903, 0, 2181268032, 0, 551114018093, 0, 139251341835477, 0, 35185439946081176, 0, 8890534908499002615, 0, 2246431103092934450019, 0, 567620912399383456892532, 0, 143424613515171098228807001, 0, 36240067096757279430906495593, 0, 9157022869593575004163671530340, 0, 2313766904824073384327785655347411, 0, 584635133792318442226565941374722695, 0, 147723713638330843255211402374835771048, 0, 37326349905953403121284030783888851492933, 0, 9431501300562727013596585732348436713272125, 0, 2383121226872546728945985884936404387698934448, 0, 602159359468270627369034762825834814807024166255] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 1934571642980131 z + 1194432989850953 z + 321 z 24 22 4 6 - 577140529288741 z + 217198315991819 z - 38659 z + 2342775 z 8 10 12 14 - 82293145 z + 1843261341 z - 28059796111 z + 303130605135 z 18 16 50 + 14130758961485 z - 2394677786989 z + 28059796111 z 48 20 36 - 303130605135 z - 63249655254487 z - 1194432989850953 z 34 30 42 + 1934571642980131 z + 2460058392107011 z + 63249655254487 z 44 46 58 56 - 14130758961485 z + 2394677786989 z + 38659 z - 2342775 z 54 52 60 32 + 82293145 z - 1843261341 z - 321 z - 2460058392107011 z 38 40 62 / + 577140529288741 z - 217198315991819 z + z ) / (1 / 28 26 2 + 15885537192804364 z - 8665182573979356 z - 476 z 24 22 4 6 + 3695295154959080 z - 1227275333105260 z + 71876 z - 5092356 z 8 10 12 14 + 203338840 z - 5121565364 z + 87382089756 z - 1058344358172 z 18 16 50 - 62443867196964 z + 9393067588012 z - 1058344358172 z 48 20 36 + 9393067588012 z + 315730448341012 z + 15885537192804364 z 34 64 30 - 22828470282127588 z + z - 22828470282127588 z 42 44 46 - 1227275333105260 z + 315730448341012 z - 62443867196964 z 58 56 54 52 - 5092356 z + 203338840 z - 5121565364 z + 87382089756 z 60 32 38 + 71876 z + 25756784156796966 z - 8665182573979356 z 40 62 + 3695295154959080 z - 476 z ) And in Maple-input format, it is: -(-1-1934571642980131*z^28+1194432989850953*z^26+321*z^2-577140529288741*z^24+ 217198315991819*z^22-38659*z^4+2342775*z^6-82293145*z^8+1843261341*z^10-\ 28059796111*z^12+303130605135*z^14+14130758961485*z^18-2394677786989*z^16+ 28059796111*z^50-303130605135*z^48-63249655254487*z^20-1194432989850953*z^36+ 1934571642980131*z^34+2460058392107011*z^30+63249655254487*z^42-14130758961485* z^44+2394677786989*z^46+38659*z^58-2342775*z^56+82293145*z^54-1843261341*z^52-\ 321*z^60-2460058392107011*z^32+577140529288741*z^38-217198315991819*z^40+z^62)/ (1+15885537192804364*z^28-8665182573979356*z^26-476*z^2+3695295154959080*z^24-\ 1227275333105260*z^22+71876*z^4-5092356*z^6+203338840*z^8-5121565364*z^10+ 87382089756*z^12-1058344358172*z^14-62443867196964*z^18+9393067588012*z^16-\ 1058344358172*z^50+9393067588012*z^48+315730448341012*z^20+15885537192804364*z^ 36-22828470282127588*z^34+z^64-22828470282127588*z^30-1227275333105260*z^42+ 315730448341012*z^44-62443867196964*z^46-5092356*z^58+203338840*z^56-5121565364 *z^54+87382089756*z^52+71876*z^60+25756784156796966*z^32-8665182573979356*z^38+ 3695295154959080*z^40-476*z^62) The first , 40, terms are: [0, 155, 0, 40563, 0, 10916789, 0, 2949154861, 0, 797464607923, 0, 215698361517851, 0, 58347150208682073, 0, 15783527633111282825, 0, 4269648581673975108763, 0, 1154998244484000117479539, 0, 312443053018614941774005469, 0, 84520203318717757818718510149, 0, 22863895597221063048144612552371, 0, 6185003283854203537593735748362139, 0, 1673129837451540521054411391926810257, 0, 452605007862446649561561220554488094449, 0, 122435981186967543273136407491486531165595, 0, 33120644342116540127473017283145405351357747, 0, 8959597261247427707910659467155206054456658853, 0, 2423696298190678952301028907255967150998511102973] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {3, 7}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 252293977293 z + 511071422130 z + 246 z - 726197232695 z 22 4 6 8 10 + 726197232695 z - 21595 z + 939307 z - 23131590 z + 345766781 z 12 14 18 16 - 3283707883 z + 20529468642 z + 252293977293 z - 86747098093 z 20 36 34 30 - 511071422130 z - 345766781 z + 3283707883 z + 86747098093 z 42 44 46 32 38 40 + 21595 z - 246 z + z - 20529468642 z + 23131590 z - 939307 z ) / 28 26 2 / (1 + 3465077483004 z - 5814190436450 z - 358 z / 24 22 4 6 + 6906858051622 z - 5814190436450 z + 39556 z - 2087420 z 8 10 12 14 + 61307332 z - 1080107222 z + 11983973460 z - 87182432530 z 18 16 48 20 - 1456863228148 z + 428829138172 z + z + 3465077483004 z 36 34 30 42 + 11983973460 z - 87182432530 z - 1456863228148 z - 2087420 z 44 46 32 38 40 + 39556 z - 358 z + 428829138172 z - 1080107222 z + 61307332 z ) And in Maple-input format, it is: -(-1-252293977293*z^28+511071422130*z^26+246*z^2-726197232695*z^24+726197232695 *z^22-21595*z^4+939307*z^6-23131590*z^8+345766781*z^10-3283707883*z^12+ 20529468642*z^14+252293977293*z^18-86747098093*z^16-511071422130*z^20-345766781 *z^36+3283707883*z^34+86747098093*z^30+21595*z^42-246*z^44+z^46-20529468642*z^ 32+23131590*z^38-939307*z^40)/(1+3465077483004*z^28-5814190436450*z^26-358*z^2+ 6906858051622*z^24-5814190436450*z^22+39556*z^4-2087420*z^6+61307332*z^8-\ 1080107222*z^10+11983973460*z^12-87182432530*z^14-1456863228148*z^18+ 428829138172*z^16+z^48+3465077483004*z^20+11983973460*z^36-87182432530*z^34-\ 1456863228148*z^30-2087420*z^42+39556*z^44-358*z^46+428829138172*z^32-\ 1080107222*z^38+61307332*z^40) The first , 40, terms are: [0, 112, 0, 22135, 0, 4642171, 0, 981940456, 0, 207981928129, 0, 44061288130933, 0, 9334763651064376, 0, 1977660760169399791, 0, 418987090394498707867, 0, 88766592270779105959744, 0, 18806087969727619146113029, 0, 3984257331024050099348230717, 0, 844104659865443715463081248352, 0, 178831992433217712145326215287891, 0, 37887341509545415892480548668013447, 0, 8026811238489641209877582453477702872, 0, 1700560031168631441654378643708498683085, 0, 360280606293738471353188154785082396263273, 0, 76329040370415273739987266189186863969048008, 0, 16171068611776536674404773204929393494559743843] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 5262341602498650 z + 3127698340699252 z + 324 z 24 22 4 6 - 1430612718912406 z + 502398731118696 z - 40449 z + 2610921 z 8 10 12 14 - 99464609 z + 2440590397 z - 40916332032 z + 487813102209 z 18 16 50 + 27579214633812 z - 4250484157904 z + 40916332032 z 48 20 36 - 487813102209 z - 134963338852996 z - 3127698340699252 z 34 30 42 + 5262341602498650 z + 6823495651050606 z + 134963338852996 z 44 46 58 56 - 27579214633812 z + 4250484157904 z + 40449 z - 2610921 z 54 52 60 32 + 99464609 z - 2440590397 z - 324 z - 6823495651050606 z 38 40 62 / + 1430612718912406 z - 502398731118696 z + z ) / (1 / 28 26 2 + 39588356031607464 z - 20784351095181960 z - 440 z 24 22 4 6 + 8423588084048606 z - 2630430020188888 z + 69070 z - 5327344 z 8 10 12 14 + 235116067 z - 6555277384 z + 123452149794 z - 1643441667424 z 18 16 50 - 115375618733672 z + 15954055343993 z - 1643441667424 z 48 20 36 + 15954055343993 z + 630633263418428 z + 39588356031607464 z 34 64 30 - 58260154643771080 z + z - 58260154643771080 z 42 44 46 - 2630430020188888 z + 630633263418428 z - 115375618733672 z 58 56 54 52 - 5327344 z + 235116067 z - 6555277384 z + 123452149794 z 60 32 38 + 69070 z + 66266621056767282 z - 20784351095181960 z 40 62 + 8423588084048606 z - 440 z ) And in Maple-input format, it is: -(-1-5262341602498650*z^28+3127698340699252*z^26+324*z^2-1430612718912406*z^24+ 502398731118696*z^22-40449*z^4+2610921*z^6-99464609*z^8+2440590397*z^10-\ 40916332032*z^12+487813102209*z^14+27579214633812*z^18-4250484157904*z^16+ 40916332032*z^50-487813102209*z^48-134963338852996*z^20-3127698340699252*z^36+ 5262341602498650*z^34+6823495651050606*z^30+134963338852996*z^42-27579214633812 *z^44+4250484157904*z^46+40449*z^58-2610921*z^56+99464609*z^54-2440590397*z^52-\ 324*z^60-6823495651050606*z^32+1430612718912406*z^38-502398731118696*z^40+z^62) /(1+39588356031607464*z^28-20784351095181960*z^26-440*z^2+8423588084048606*z^24 -2630430020188888*z^22+69070*z^4-5327344*z^6+235116067*z^8-6555277384*z^10+ 123452149794*z^12-1643441667424*z^14-115375618733672*z^18+15954055343993*z^16-\ 1643441667424*z^50+15954055343993*z^48+630633263418428*z^20+39588356031607464*z ^36-58260154643771080*z^34+z^64-58260154643771080*z^30-2630430020188888*z^42+ 630633263418428*z^44-115375618733672*z^46-5327344*z^58+235116067*z^56-\ 6555277384*z^54+123452149794*z^52+69070*z^60+66266621056767282*z^32-\ 20784351095181960*z^38+8423588084048606*z^40-440*z^62) The first , 40, terms are: [0, 116, 0, 22419, 0, 4568663, 0, 944051836, 0, 196100202781, 0, 40824077584901, 0, 8506873882328940, 0, 1773397068550337183, 0, 369762119296920472267, 0, 77103553481680861777348, 0, 16078372737090627155330489, 0, 3352870257906477387419813961, 0, 699188833654230328656055133348, 0, 145805391239230816165100933473243, 0, 30405579490500287677690360016985871, 0, 6340642312149666772645913289624960140, 0, 1322249289692426725264121251578924278709, 0, 275736007319793881328873732090805013392493, 0, 57500768112515682142392030289232338828739164, 0, 11990956291341961151979817069983940864297620967] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {3, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 10901566767725218145 z - 2426260187589388993 z - 345 z 24 22 4 6 + 446774750218826046 z - 67688714575741592 z + 49430 z - 4021396 z 8 10 12 14 + 211988605 z - 7797160062 z + 209856738499 z - 4272495507072 z 18 16 50 - 839862951586898 z + 67418051722719 z - 1320218283944754496286 z 48 20 + 2032958461930820195668 z + 8378566320966697 z 36 34 + 720569877054417321668 z - 330141171607781849224 z 66 80 88 84 86 - 67688714575741592 z + 211988605 z + z + 49430 z - 345 z 82 64 30 - 4021396 z + 446774750218826046 z - 40707155179610673590 z 42 44 - 2633177078817961972854 z + 2870171576919027585512 z 46 58 - 2633177078817961972854 z - 40707155179610673590 z 56 54 + 126767567440112826046 z - 330141171607781849224 z 52 60 + 720569877054417321668 z + 10901566767725218145 z 70 68 78 - 839862951586898 z + 8378566320966697 z - 7797160062 z 32 38 + 126767567440112826046 z - 1320218283944754496286 z 40 62 76 + 2032958461930820195668 z - 2426260187589388993 z + 209856738499 z 74 72 / - 4272495507072 z + 67418051722719 z ) / (-1 / 28 26 2 - 51923268038905751102 z + 10653359575340648049 z + 465 z 24 22 4 6 - 1809008848158576234 z + 252762685748928261 z - 79542 z + 7373601 z 8 10 12 14 - 433234286 z + 17555421489 z - 517152196694 z + 11479055375197 z 18 16 50 + 2665303689074537 z - 197008327313834 z + 15945168519000539887787 z 48 20 - 22485276168204558408239 z - 28849764856376738 z 36 34 - 4775448706944287557811 z + 2012831417806886790743 z 66 80 90 88 84 + 1809008848158576234 z - 17555421489 z + z - 465 z - 7373601 z 86 82 64 + 79542 z + 433234286 z - 10653359575340648049 z 30 42 + 210404883196113610894 z + 22485276168204558408239 z 44 46 - 26698062510692096350679 z + 26698062510692096350679 z 58 56 + 711429762861143145519 z - 2012831417806886790743 z 54 52 + 4775448706944287557811 z - 9515942151316368001211 z 60 70 - 210404883196113610894 z + 28849764856376738 z 68 78 32 - 252762685748928261 z + 517152196694 z - 711429762861143145519 z 38 40 + 9515942151316368001211 z - 15945168519000539887787 z 62 76 74 + 51923268038905751102 z - 11479055375197 z + 197008327313834 z 72 - 2665303689074537 z ) And in Maple-input format, it is: -(1+10901566767725218145*z^28-2426260187589388993*z^26-345*z^2+ 446774750218826046*z^24-67688714575741592*z^22+49430*z^4-4021396*z^6+211988605* z^8-7797160062*z^10+209856738499*z^12-4272495507072*z^14-839862951586898*z^18+ 67418051722719*z^16-1320218283944754496286*z^50+2032958461930820195668*z^48+ 8378566320966697*z^20+720569877054417321668*z^36-330141171607781849224*z^34-\ 67688714575741592*z^66+211988605*z^80+z^88+49430*z^84-345*z^86-4021396*z^82+ 446774750218826046*z^64-40707155179610673590*z^30-2633177078817961972854*z^42+ 2870171576919027585512*z^44-2633177078817961972854*z^46-40707155179610673590*z^ 58+126767567440112826046*z^56-330141171607781849224*z^54+720569877054417321668* z^52+10901566767725218145*z^60-839862951586898*z^70+8378566320966697*z^68-\ 7797160062*z^78+126767567440112826046*z^32-1320218283944754496286*z^38+ 2032958461930820195668*z^40-2426260187589388993*z^62+209856738499*z^76-\ 4272495507072*z^74+67418051722719*z^72)/(-1-51923268038905751102*z^28+ 10653359575340648049*z^26+465*z^2-1809008848158576234*z^24+252762685748928261*z ^22-79542*z^4+7373601*z^6-433234286*z^8+17555421489*z^10-517152196694*z^12+ 11479055375197*z^14+2665303689074537*z^18-197008327313834*z^16+ 15945168519000539887787*z^50-22485276168204558408239*z^48-28849764856376738*z^ 20-4775448706944287557811*z^36+2012831417806886790743*z^34+1809008848158576234* z^66-17555421489*z^80+z^90-465*z^88-7373601*z^84+79542*z^86+433234286*z^82-\ 10653359575340648049*z^64+210404883196113610894*z^30+22485276168204558408239*z^ 42-26698062510692096350679*z^44+26698062510692096350679*z^46+ 711429762861143145519*z^58-2012831417806886790743*z^56+4775448706944287557811*z ^54-9515942151316368001211*z^52-210404883196113610894*z^60+28849764856376738*z^ 70-252762685748928261*z^68+517152196694*z^78-711429762861143145519*z^32+ 9515942151316368001211*z^38-15945168519000539887787*z^40+51923268038905751102*z ^62-11479055375197*z^76+197008327313834*z^74-2665303689074537*z^72) The first , 40, terms are: [0, 120, 0, 25688, 0, 5752085, 0, 1295031068, 0, 291840311145, 0, 65780395961371, 0, 14827475996465760, 0, 3342277800159803633, 0, 753388417242375558283, 0, 169822640820885767769228, 0, 38280032883387784465197567, 0, 8628772697584222015857501176, 0, 1945027557478417548848071535200, 0, 438432247448491970399720030833355, 0, 98827821205142372161436658308613091, 0, 22276961382394067914663627304425939680, 0, 5021490936396064970897645978993243582408, 0, 1131903529912744582601619323842944773621991, 0, 255144461527260732988315484916342695657510732, 0, 57512583473502077237009838027908096594631889811] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {3, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 9218834240948816 z - 4331444432361966 z - 297 z 24 22 4 6 + 1629323436268302 z - 488157019725408 z + 33472 z - 1979261 z 8 10 12 14 + 71095693 z - 1690747068 z + 28162307421 z - 341576727869 z 18 16 50 - 21512488687785 z + 3101482714032 z - 21512488687785 z 48 20 36 + 115711891805489 z + 115711891805489 z + 21738753784012374 z 34 66 64 30 - 24189638815504200 z - 297 z + 33472 z - 15769476451536646 z 42 44 46 - 4331444432361966 z + 1629323436268302 z - 488157019725408 z 58 56 54 52 - 1690747068 z + 28162307421 z - 341576727869 z + 3101482714032 z 60 68 32 38 + 71095693 z + z + 21738753784012374 z - 15769476451536646 z 40 62 / 28 + 9218834240948816 z - 1979261 z ) / (-1 - 62960446128390770 z / 26 2 24 + 26605144704884746 z + 425 z - 9005161451259834 z 22 4 6 8 + 2429607379172005 z - 59837 z + 4151255 z - 169715387 z 10 12 14 + 4524773699 z - 83858221711 z + 1127691325607 z 18 16 50 + 87090377926705 z - 11338335989203 z + 519138022438061 z 48 20 36 - 2429607379172005 z - 519138022438061 z - 227287139733134990 z 34 66 64 + 227287139733134990 z + 59837 z - 4151255 z 30 42 44 + 119773216693463222 z + 62960446128390770 z - 26605144704884746 z 46 58 56 + 9005161451259834 z + 83858221711 z - 1127691325607 z 54 52 60 70 68 + 11338335989203 z - 87090377926705 z - 4524773699 z + z - 425 z 32 38 40 - 183634807191851262 z + 183634807191851262 z - 119773216693463222 z 62 + 169715387 z ) And in Maple-input format, it is: -(1+9218834240948816*z^28-4331444432361966*z^26-297*z^2+1629323436268302*z^24-\ 488157019725408*z^22+33472*z^4-1979261*z^6+71095693*z^8-1690747068*z^10+ 28162307421*z^12-341576727869*z^14-21512488687785*z^18+3101482714032*z^16-\ 21512488687785*z^50+115711891805489*z^48+115711891805489*z^20+21738753784012374 *z^36-24189638815504200*z^34-297*z^66+33472*z^64-15769476451536646*z^30-\ 4331444432361966*z^42+1629323436268302*z^44-488157019725408*z^46-1690747068*z^ 58+28162307421*z^56-341576727869*z^54+3101482714032*z^52+71095693*z^60+z^68+ 21738753784012374*z^32-15769476451536646*z^38+9218834240948816*z^40-1979261*z^ 62)/(-1-62960446128390770*z^28+26605144704884746*z^26+425*z^2-9005161451259834* z^24+2429607379172005*z^22-59837*z^4+4151255*z^6-169715387*z^8+4524773699*z^10-\ 83858221711*z^12+1127691325607*z^14+87090377926705*z^18-11338335989203*z^16+ 519138022438061*z^50-2429607379172005*z^48-519138022438061*z^20-\ 227287139733134990*z^36+227287139733134990*z^34+59837*z^66-4151255*z^64+ 119773216693463222*z^30+62960446128390770*z^42-26605144704884746*z^44+ 9005161451259834*z^46+83858221711*z^58-1127691325607*z^56+11338335989203*z^54-\ 87090377926705*z^52-4524773699*z^60+z^70-425*z^68-183634807191851262*z^32+ 183634807191851262*z^38-119773216693463222*z^40+169715387*z^62) The first , 40, terms are: [0, 128, 0, 28035, 0, 6427733, 0, 1486997176, 0, 344848431299, 0, 80031796281315, 0, 18577741397019016, 0, 4312734152740748285, 0, 1001201548989563739387, 0, 232430504953764762536464, 0, 53959211924695851783015857, 0, 12526747853165344398153769969, 0, 2908112178243771558622732847984, 0, 675124703689690703603442965502827, 0, 156731702824661582194698680203420749, 0, 36385613904956660252211873302177282408, 0, 8447001325444975487257726517946879278723, 0, 1960990176392334484712938732366059576389603, 0, 455248238324281594923902358109853203341884888, 0, 105686892776295381542975371873678636185185201221] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 12759988610434152 z - 5770337154385290 z - 275 z 24 22 4 6 + 2072414934405710 z - 589127241884528 z + 29084 z - 1671283 z 8 10 12 14 + 60214313 z - 1470382192 z + 25548074345 z - 326414986899 z 18 16 50 - 23111791228675 z + 3139073953436 z - 23111791228675 z 48 20 36 + 131917329408593 z + 131917329408593 z + 31515484017173326 z 34 66 64 30 - 35279591888617344 z - 275 z + 29084 z - 22460269647857626 z 42 44 46 - 5770337154385290 z + 2072414934405710 z - 589127241884528 z 58 56 54 52 - 1470382192 z + 25548074345 z - 326414986899 z + 3139073953436 z 60 68 32 38 + 60214313 z + z + 31515484017173326 z - 22460269647857626 z 40 62 / 2 + 12759988610434152 z - 1671283 z ) / ((-1 + z ) (1 / 28 26 2 + 59812703488321350 z - 25835521748190332 z - 398 z 24 22 4 6 + 8780269030897922 z - 2344254961739856 z + 50851 z - 3345926 z 8 10 12 14 + 134797313 z - 3633375408 z + 69130181517 z - 961909244286 z 18 16 50 - 79797772232502 z + 10031964962823 z - 79797772232502 z 48 20 36 + 490117896808205 z + 490117896808205 z + 156268023113280642 z 34 66 64 30 - 176221469249791296 z - 398 z + 50851 z - 108989954245992380 z 42 44 46 - 25835521748190332 z + 8780269030897922 z - 2344254961739856 z 58 56 54 52 - 3633375408 z + 69130181517 z - 961909244286 z + 10031964962823 z 60 68 32 38 + 134797313 z + z + 156268023113280642 z - 108989954245992380 z 40 62 + 59812703488321350 z - 3345926 z )) And in Maple-input format, it is: -(1+12759988610434152*z^28-5770337154385290*z^26-275*z^2+2072414934405710*z^24-\ 589127241884528*z^22+29084*z^4-1671283*z^6+60214313*z^8-1470382192*z^10+ 25548074345*z^12-326414986899*z^14-23111791228675*z^18+3139073953436*z^16-\ 23111791228675*z^50+131917329408593*z^48+131917329408593*z^20+31515484017173326 *z^36-35279591888617344*z^34-275*z^66+29084*z^64-22460269647857626*z^30-\ 5770337154385290*z^42+2072414934405710*z^44-589127241884528*z^46-1470382192*z^ 58+25548074345*z^56-326414986899*z^54+3139073953436*z^52+60214313*z^60+z^68+ 31515484017173326*z^32-22460269647857626*z^38+12759988610434152*z^40-1671283*z^ 62)/(-1+z^2)/(1+59812703488321350*z^28-25835521748190332*z^26-398*z^2+ 8780269030897922*z^24-2344254961739856*z^22+50851*z^4-3345926*z^6+134797313*z^8 -3633375408*z^10+69130181517*z^12-961909244286*z^14-79797772232502*z^18+ 10031964962823*z^16-79797772232502*z^50+490117896808205*z^48+490117896808205*z^ 20+156268023113280642*z^36-176221469249791296*z^34-398*z^66+50851*z^64-\ 108989954245992380*z^30-25835521748190332*z^42+8780269030897922*z^44-\ 2344254961739856*z^46-3633375408*z^58+69130181517*z^56-961909244286*z^54+ 10031964962823*z^52+134797313*z^60+z^68+156268023113280642*z^32-\ 108989954245992380*z^38+59812703488321350*z^40-3345926*z^62) The first , 40, terms are: [0, 124, 0, 27311, 0, 6267707, 0, 1444425076, 0, 333049294821, 0, 76798538828489, 0, 17709316721146196, 0, 4083676232464818167, 0, 941674690813550786251, 0, 217145337678355712752636, 0, 50072597679145354776110229, 0, 11546483417012912089773361485, 0, 2662559673320267990962434521276, 0, 613972562738669150500447361938099, 0, 141578914296155850011972543366520607, 0, 32647369263345399135699159344721680468, 0, 7528315392981452261561119799623908860577, 0, 1735990799106579989872229085453619966265293, 0, 400310547216487606969989915908369949335502324, 0, 92309552732212086215267343011342541538047765331] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 11373181949589560704 z - 2552934413415886624 z - 362 z 24 22 4 6 + 474237694193900053 z - 72488531110532498 z + 53792 z - 4463898 z 8 10 12 14 + 237162651 z - 8726061216 z + 233917255840 z - 4731845374992 z 18 16 50 - 915323997510674 z + 74092991584583 z - 1332068156119594664180 z 48 20 + 2043913239296739702670 z + 9052325071461216 z 36 34 + 730553902058246613760 z - 336708807100540553444 z 66 80 88 84 86 - 72488531110532498 z + 237162651 z + z + 53792 z - 362 z 82 64 30 - 4463898 z + 474237694193900053 z - 42122796236045856704 z 42 44 - 2641590431384707401840 z + 2877223766610332485824 z 46 58 - 2641590431384707401840 z - 42122796236045856704 z 56 54 + 130180962895823456634 z - 336708807100540553444 z 52 60 + 730553902058246613760 z + 11373181949589560704 z 70 68 78 - 915323997510674 z + 9052325071461216 z - 8726061216 z 32 38 + 130180962895823456634 z - 1332068156119594664180 z 40 62 76 + 2043913239296739702670 z - 2552934413415886624 z + 233917255840 z 74 72 / - 4731845374992 z + 74092991584583 z ) / (-1 / 28 26 2 - 54687241802200227624 z + 11335675555715915677 z + 477 z 24 22 4 6 - 1945089857152911561 z + 274666569623567686 z - 86022 z + 8243934 z 8 10 12 14 - 492121299 z + 20029956059 z - 588550800072 z + 12978430444144 z 18 16 50 + 2957727720002311 z - 220797072983311 z + 16019789551490225412534 z 48 20 - 22510681064501547487430 z - 31683487728273470 z 36 34 - 4854060514834524478324 z + 2061778238000847407818 z 66 80 90 88 84 + 1945089857152911561 z - 20029956059 z + z - 477 z - 8243934 z 86 82 64 + 86022 z + 492121299 z - 11335675555715915677 z 30 42 + 219431006415205164504 z + 22510681064501547487430 z 44 46 - 26679968730485358296520 z + 26679968730485358296520 z 58 56 + 735048669853505846194 z - 2061778238000847407818 z 54 52 + 4854060514834524478324 z - 9609654000030760556068 z 60 70 - 219431006415205164504 z + 31683487728273470 z 68 78 32 - 274666569623567686 z + 588550800072 z - 735048669853505846194 z 38 40 + 9609654000030760556068 z - 16019789551490225412534 z 62 76 74 + 54687241802200227624 z - 12978430444144 z + 220797072983311 z 72 - 2957727720002311 z ) And in Maple-input format, it is: -(1+11373181949589560704*z^28-2552934413415886624*z^26-362*z^2+ 474237694193900053*z^24-72488531110532498*z^22+53792*z^4-4463898*z^6+237162651* z^8-8726061216*z^10+233917255840*z^12-4731845374992*z^14-915323997510674*z^18+ 74092991584583*z^16-1332068156119594664180*z^50+2043913239296739702670*z^48+ 9052325071461216*z^20+730553902058246613760*z^36-336708807100540553444*z^34-\ 72488531110532498*z^66+237162651*z^80+z^88+53792*z^84-362*z^86-4463898*z^82+ 474237694193900053*z^64-42122796236045856704*z^30-2641590431384707401840*z^42+ 2877223766610332485824*z^44-2641590431384707401840*z^46-42122796236045856704*z^ 58+130180962895823456634*z^56-336708807100540553444*z^54+730553902058246613760* z^52+11373181949589560704*z^60-915323997510674*z^70+9052325071461216*z^68-\ 8726061216*z^78+130180962895823456634*z^32-1332068156119594664180*z^38+ 2043913239296739702670*z^40-2552934413415886624*z^62+233917255840*z^76-\ 4731845374992*z^74+74092991584583*z^72)/(-1-54687241802200227624*z^28+ 11335675555715915677*z^26+477*z^2-1945089857152911561*z^24+274666569623567686*z ^22-86022*z^4+8243934*z^6-492121299*z^8+20029956059*z^10-588550800072*z^12+ 12978430444144*z^14+2957727720002311*z^18-220797072983311*z^16+ 16019789551490225412534*z^50-22510681064501547487430*z^48-31683487728273470*z^ 20-4854060514834524478324*z^36+2061778238000847407818*z^34+1945089857152911561* z^66-20029956059*z^80+z^90-477*z^88-8243934*z^84+86022*z^86+492121299*z^82-\ 11335675555715915677*z^64+219431006415205164504*z^30+22510681064501547487430*z^ 42-26679968730485358296520*z^44+26679968730485358296520*z^46+ 735048669853505846194*z^58-2061778238000847407818*z^56+4854060514834524478324*z ^54-9609654000030760556068*z^52-219431006415205164504*z^60+31683487728273470*z^ 70-274666569623567686*z^68+588550800072*z^78-735048669853505846194*z^32+ 9609654000030760556068*z^38-16019789551490225412534*z^40+54687241802200227624*z ^62-12978430444144*z^76+220797072983311*z^74-2957727720002311*z^72) The first , 40, terms are: [0, 115, 0, 22625, 0, 4679631, 0, 979029999, 0, 205675043849, 0, 43282013463027, 0, 9114795407996345, 0, 1920084676438544297, 0, 404530467720427477827, 0, 85232789585021139060265, 0, 17958610245758018103888047, 0, 3783931587761609862086283055, 0, 797288994675702191884870685377, 0, 167992191758716530588018333946659, 0, 35396700461539146665620685657612785, 0, 7458244534881829943600371240897133201, 0, 1571486021162970586283335197127706674307, 0, 331119268575062859131092164703321080764609, 0, 69768341304203017277852484663885729898835183, 0, 14700508150804875867845273104445970950040489775] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 29664168628439088 z - 11925794490635696 z - 292 z 24 22 4 6 + 3880604888339628 z - 1016099463106548 z + 32958 z - 1996512 z 8 10 12 14 + 75083505 z - 1903304632 z + 34272561296 z - 454465496928 z 18 16 50 - 35177408262508 z + 4554693827532 z - 1016099463106548 z 48 20 36 + 3880604888339628 z + 212519106972376 z + 147548833554603860 z 34 66 64 - 133549570050589660 z - 1996512 z + 75083505 z 30 42 44 - 59993065247656952 z - 59993065247656952 z + 29664168628439088 z 46 58 56 - 11925794490635696 z - 454465496928 z + 4554693827532 z 54 52 60 70 - 35177408262508 z + 212519106972376 z + 34272561296 z - 292 z 68 32 38 + 32958 z + 98983864600845974 z - 133549570050589660 z 40 62 72 / + 98983864600845974 z - 1903304632 z + z ) / (-1 / 28 26 2 - 184763004412062588 z + 67572219138422580 z + 415 z 24 22 4 6 - 20001324543026304 z + 4761814724834440 z - 56900 z + 3985572 z 8 10 12 14 - 169941123 z + 4837648785 z - 97253312004 z + 1433607258788 z 18 16 50 + 135824808481640 z - 15917534419700 z + 20001324543026304 z 48 20 36 - 67572219138422580 z - 904688863185256 z - 1350779471694062056 z 34 66 64 + 1108540795023407030 z + 169941123 z - 4837648785 z 30 42 44 + 410890024992512796 z + 745961564485651590 z - 410890024992512796 z 46 58 56 + 184763004412062588 z + 15917534419700 z - 135824808481640 z 54 52 60 + 904688863185256 z - 4761814724834440 z - 1433607258788 z 70 68 32 + 56900 z - 3985572 z - 745961564485651590 z 38 40 62 + 1350779471694062056 z - 1108540795023407030 z + 97253312004 z 74 72 + z - 415 z ) And in Maple-input format, it is: -(1+29664168628439088*z^28-11925794490635696*z^26-292*z^2+3880604888339628*z^24 -1016099463106548*z^22+32958*z^4-1996512*z^6+75083505*z^8-1903304632*z^10+ 34272561296*z^12-454465496928*z^14-35177408262508*z^18+4554693827532*z^16-\ 1016099463106548*z^50+3880604888339628*z^48+212519106972376*z^20+ 147548833554603860*z^36-133549570050589660*z^34-1996512*z^66+75083505*z^64-\ 59993065247656952*z^30-59993065247656952*z^42+29664168628439088*z^44-\ 11925794490635696*z^46-454465496928*z^58+4554693827532*z^56-35177408262508*z^54 +212519106972376*z^52+34272561296*z^60-292*z^70+32958*z^68+98983864600845974*z^ 32-133549570050589660*z^38+98983864600845974*z^40-1903304632*z^62+z^72)/(-1-\ 184763004412062588*z^28+67572219138422580*z^26+415*z^2-20001324543026304*z^24+ 4761814724834440*z^22-56900*z^4+3985572*z^6-169941123*z^8+4837648785*z^10-\ 97253312004*z^12+1433607258788*z^14+135824808481640*z^18-15917534419700*z^16+ 20001324543026304*z^50-67572219138422580*z^48-904688863185256*z^20-\ 1350779471694062056*z^36+1108540795023407030*z^34+169941123*z^66-4837648785*z^ 64+410890024992512796*z^30+745961564485651590*z^42-410890024992512796*z^44+ 184763004412062588*z^46+15917534419700*z^58-135824808481640*z^56+ 904688863185256*z^54-4761814724834440*z^52-1433607258788*z^60+56900*z^70-\ 3985572*z^68-745961564485651590*z^32+1350779471694062056*z^38-\ 1108540795023407030*z^40+97253312004*z^62+z^74-415*z^72) The first , 40, terms are: [0, 123, 0, 27103, 0, 6238105, 0, 1442020613, 0, 333542923835, 0, 77157892926063, 0, 17849231387013589, 0, 4129154529201141725, 0, 955219853020039218087, 0, 220976293990803771671267, 0, 51119672820628963620823741, 0, 11825797926173554278165507073, 0, 2735727549129472101717878558583, 0, 632871056835227630897048501426883, 0, 146405578587675754160969489184051225, 0, 33868816106380444427073756570395757353, 0, 7835061447328423027816882346474593368883, 0, 1812528306003723905093487708766151870192455, 0, 419302245700702614413080298112635015357172721, 0, 96999518665354875962970426061356189804166946317] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 32072541327827826 z - 12435212615933550 z - 279 z 24 22 4 6 + 3889793233960807 z - 978234542257331 z + 29873 z - 1739198 z 8 10 12 14 + 63776677 z - 1597962183 z + 28795446142 z - 386361523495 z 18 16 50 - 31453892539282 z + 3955581841205 z - 978234542257331 z 48 20 36 + 3889793233960807 z + 196804340851333 z + 171780579039244212 z 34 66 64 - 154705851990116506 z - 1739198 z + 63776677 z 30 42 44 - 66892977843498224 z - 66892977843498224 z + 32072541327827826 z 46 58 56 - 12435212615933550 z - 386361523495 z + 3955581841205 z 54 52 60 70 - 31453892539282 z + 196804340851333 z + 28795446142 z - 279 z 68 32 38 + 29873 z + 112990706044442558 z - 154705851990116506 z 40 62 72 / + 112990706044442558 z - 1597962183 z + z ) / (-1 / 28 26 2 - 193884656865267288 z + 68192791296289127 z + 387 z 24 22 4 6 - 19369043134738117 z + 4425890702997050 z - 50622 z + 3423944 z 8 10 12 14 - 142389320 z + 3994289646 z - 79993434096 z + 1187542458644 z 18 16 50 + 117509858501536 z - 13415186752530 z + 19369043134738117 z 48 20 36 - 68192791296289127 z - 809226987960952 z - 1554281050618901836 z 34 66 64 + 1262709102105215152 z + 142389320 z - 3994289646 z 30 42 44 + 446315167361758880 z + 833204312573830968 z - 446315167361758880 z 46 58 56 + 193884656865267288 z + 13415186752530 z - 117509858501536 z 54 52 60 + 809226987960952 z - 4425890702997050 z - 1187542458644 z 70 68 32 + 50622 z - 3423944 z - 833204312573830968 z 38 40 62 + 1554281050618901836 z - 1262709102105215152 z + 79993434096 z 74 72 + z - 387 z ) And in Maple-input format, it is: -(1+32072541327827826*z^28-12435212615933550*z^26-279*z^2+3889793233960807*z^24 -978234542257331*z^22+29873*z^4-1739198*z^6+63776677*z^8-1597962183*z^10+ 28795446142*z^12-386361523495*z^14-31453892539282*z^18+3955581841205*z^16-\ 978234542257331*z^50+3889793233960807*z^48+196804340851333*z^20+ 171780579039244212*z^36-154705851990116506*z^34-1739198*z^66+63776677*z^64-\ 66892977843498224*z^30-66892977843498224*z^42+32072541327827826*z^44-\ 12435212615933550*z^46-386361523495*z^58+3955581841205*z^56-31453892539282*z^54 +196804340851333*z^52+28795446142*z^60-279*z^70+29873*z^68+112990706044442558*z ^32-154705851990116506*z^38+112990706044442558*z^40-1597962183*z^62+z^72)/(-1-\ 193884656865267288*z^28+68192791296289127*z^26+387*z^2-19369043134738117*z^24+ 4425890702997050*z^22-50622*z^4+3423944*z^6-142389320*z^8+3994289646*z^10-\ 79993434096*z^12+1187542458644*z^14+117509858501536*z^18-13415186752530*z^16+ 19369043134738117*z^50-68192791296289127*z^48-809226987960952*z^20-\ 1554281050618901836*z^36+1262709102105215152*z^34+142389320*z^66-3994289646*z^ 64+446315167361758880*z^30+833204312573830968*z^42-446315167361758880*z^44+ 193884656865267288*z^46+13415186752530*z^58-117509858501536*z^56+ 809226987960952*z^54-4425890702997050*z^52-1187542458644*z^60+50622*z^70-\ 3423944*z^68-833204312573830968*z^32+1554281050618901836*z^38-\ 1262709102105215152*z^40+79993434096*z^62+z^74-387*z^72) The first , 40, terms are: [0, 108, 0, 21047, 0, 4362759, 0, 914119808, 0, 191994809869, 0, 40348578275997, 0, 8480652976403336, 0, 1782566848077920703, 0, 374684915942198156543, 0, 78756707726466754418980, 0, 16554235731333224952875441, 0, 3479611656011501476963917649, 0, 731395751901406162908389927412, 0, 153735474912315376657201213745007, 0, 32314374584659209367053976689449807, 0, 6792308710427251798291796650453897464, 0, 1427706963752027819413661867296658679229, 0, 300096368011599809139469858935972770591853, 0, 63078651558731085975634050582517442103180688, 0, 13258795195822205416835619175491639434433897591] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 41441010754848990 z - 16187406009172305 z - 291 z 24 22 4 6 + 5101045489225964 z - 1291178064534240 z + 33070 z - 2035903 z 8 10 12 14 + 78097544 z - 2023408834 z + 37316526360 z - 508050153218 z 18 16 50 - 41778879876490 z + 5241705480897 z - 1291178064534240 z 48 20 36 + 5101045489225964 z + 260962129557970 z + 218321646028373982 z 34 66 64 - 196849501072274676 z - 2035903 z + 78097544 z 30 42 44 - 85856338853334155 z - 85856338853334155 z + 41441010754848990 z 46 58 56 - 16187406009172305 z - 508050153218 z + 5241705480897 z 54 52 60 70 - 41778879876490 z + 260962129557970 z + 37316526360 z - 291 z 68 32 38 + 33070 z + 144257592712716325 z - 196849501072274676 z 40 62 72 / + 144257592712716325 z - 2023408834 z + z ) / (-1 / 28 26 2 - 254811386900983129 z + 89881389040236413 z + 409 z 24 22 4 6 - 25591212860429251 z + 5855789243828423 z - 56240 z + 3985000 z 8 10 12 14 - 172306843 z + 4983685669 z - 102109604832 z + 1540435252464 z 18 16 50 + 154994648756781 z - 17586092430877 z + 25591212860429251 z 48 20 36 - 89881389040236413 z - 1070336254155823 z - 2026026879962694415 z 34 66 64 + 1647548778851068777 z + 172306843 z - 4983685669 z 30 42 + 584873163134517937 z + 1089137677222879017 z 44 46 58 - 584873163134517937 z + 254811386900983129 z + 17586092430877 z 56 54 52 - 154994648756781 z + 1070336254155823 z - 5855789243828423 z 60 70 68 32 - 1540435252464 z + 56240 z - 3985000 z - 1089137677222879017 z 38 40 62 + 2026026879962694415 z - 1647548778851068777 z + 102109604832 z 74 72 + z - 409 z ) And in Maple-input format, it is: -(1+41441010754848990*z^28-16187406009172305*z^26-291*z^2+5101045489225964*z^24 -1291178064534240*z^22+33070*z^4-2035903*z^6+78097544*z^8-2023408834*z^10+ 37316526360*z^12-508050153218*z^14-41778879876490*z^18+5241705480897*z^16-\ 1291178064534240*z^50+5101045489225964*z^48+260962129557970*z^20+ 218321646028373982*z^36-196849501072274676*z^34-2035903*z^66+78097544*z^64-\ 85856338853334155*z^30-85856338853334155*z^42+41441010754848990*z^44-\ 16187406009172305*z^46-508050153218*z^58+5241705480897*z^56-41778879876490*z^54 +260962129557970*z^52+37316526360*z^60-291*z^70+33070*z^68+144257592712716325*z ^32-196849501072274676*z^38+144257592712716325*z^40-2023408834*z^62+z^72)/(-1-\ 254811386900983129*z^28+89881389040236413*z^26+409*z^2-25591212860429251*z^24+ 5855789243828423*z^22-56240*z^4+3985000*z^6-172306843*z^8+4983685669*z^10-\ 102109604832*z^12+1540435252464*z^14+154994648756781*z^18-17586092430877*z^16+ 25591212860429251*z^50-89881389040236413*z^48-1070336254155823*z^20-\ 2026026879962694415*z^36+1647548778851068777*z^34+172306843*z^66-4983685669*z^ 64+584873163134517937*z^30+1089137677222879017*z^42-584873163134517937*z^44+ 254811386900983129*z^46+17586092430877*z^58-154994648756781*z^56+ 1070336254155823*z^54-5855789243828423*z^52-1540435252464*z^60+56240*z^70-\ 3985000*z^68-1089137677222879017*z^32+2026026879962694415*z^38-\ 1647548778851068777*z^40+102109604832*z^62+z^74-409*z^72) The first , 40, terms are: [0, 118, 0, 25092, 0, 5575405, 0, 1245187266, 0, 278340503955, 0, 62229681728669, 0, 13913494798145624, 0, 3110848052299396591, 0, 695540224305648210381, 0, 155512714811902217462146, 0, 34770393050079522226453403, 0, 7774157005313515529698091476, 0, 1738189078781024944673174292710, 0, 388633941295821630746365840656827, 0, 86892929090399063034771643122511715, 0, 19428002354088798294926507886743686526, 0, 4343820370965530927383004063994272881972, 0, 971215417384562351194352996025329425062531, 0, 217149722228609984212402447557841018736889674, 0, 48551537609394222032226061110820182611758670221] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 658181043995665787621 z - 90211561854776900270 z - 380 z 24 22 4 + 10552887938678713789 z - 1046837031414476328 z + 62027 z 6 102 8 10 - 5932206 z - 17408581178 z + 379024582 z - 17408581178 z 12 14 18 + 602410912374 z - 16227181341128 z - 6079681377981190 z 16 50 + 348535605293547 z - 167624813963071497236976618 z 48 20 + 106143735326309312581038703 z + 87380486547515151 z 36 34 + 420420658939733285303751 z - 103784961169345444805496 z 66 80 - 58938893919006369142698280 z + 22233277618683752956815 z 100 90 88 + 602410912374 z - 1046837031414476328 z + 10552887938678713789 z 84 94 + 658181043995665787621 z - 6079681377981190 z 86 96 98 - 90211561854776900270 z + 348535605293547 z - 16227181341128 z 92 82 + 87380486547515151 z - 4120463628337611701962 z 64 112 110 106 + 106143735326309312581038703 z + z - 380 z - 5932206 z 108 30 42 + 62027 z - 4120463628337611701962 z - 12218969685508696705188086 z 44 46 + 28679682636819124680121286 z - 58938893919006369142698280 z 58 56 - 282391398533271005826453122 z + 301401365824788993247499108 z 54 52 - 282391398533271005826453122 z + 232243341146755644000902481 z 60 70 + 232243341146755644000902481 z - 12218969685508696705188086 z 68 78 + 28679682636819124680121286 z - 103784961169345444805496 z 32 38 + 22233277618683752956815 z - 1481722408088266465462624 z 40 62 + 4553030417741383880600034 z - 167624813963071497236976618 z 76 74 + 420420658939733285303751 z - 1481722408088266465462624 z 72 104 / 2 + 4553030417741383880600034 z + 379024582 z ) / ((-1 + z ) (1 / 28 26 2 + 2252987676667336454286 z - 295392937026164790950 z - 495 z 24 22 4 + 32969930015146218669 z - 3112210832624720818 z + 94768 z 6 102 8 10 - 10185319 z - 35685231579 z + 715606963 z - 35685231579 z 12 14 18 + 1329519968995 z - 38324984572358 z - 16222947479356440 z 16 50 + 876703623213708 z - 783933910044791629610874901 z 48 20 + 489961169419588729795901983 z + 246494580278476213 z 36 34 + 1676310375041057084568451 z - 399833867164690363603592 z 66 80 - 267593719786558347980590569 z + 82550286509597063421526 z 100 90 88 + 1329519968995 z - 3112210832624720818 z + 32969930015146218669 z 84 94 + 2252987676667336454286 z - 16222947479356440 z 86 96 98 - 295392937026164790950 z + 876703623213708 z - 38324984572358 z 92 82 + 246494580278476213 z - 14707824462765608975174 z 64 112 110 106 + 489961169419588729795901983 z + z - 495 z - 10185319 z 108 30 + 94768 z - 14707824462765608975174 z 42 44 - 53140975304570252719654517 z + 127644141983122812380423508 z 46 58 - 267593719786558347980590569 z - 1340810224071819777859328244 z 56 54 + 1433804628162007951178307827 z - 1340810224071819777859328244 z 52 60 + 1096432607129832338176426934 z + 1096432607129832338176426934 z 70 68 - 53140975304570252719654517 z + 127644141983122812380423508 z 78 32 - 399833867164690363603592 z + 82550286509597063421526 z 38 40 - 6098476686844819847138066 z + 19290766627683564284190412 z 62 76 - 783933910044791629610874901 z + 1676310375041057084568451 z 74 72 - 6098476686844819847138066 z + 19290766627683564284190412 z 104 + 715606963 z )) And in Maple-input format, it is: -(1+658181043995665787621*z^28-90211561854776900270*z^26-380*z^2+ 10552887938678713789*z^24-1046837031414476328*z^22+62027*z^4-5932206*z^6-\ 17408581178*z^102+379024582*z^8-17408581178*z^10+602410912374*z^12-\ 16227181341128*z^14-6079681377981190*z^18+348535605293547*z^16-\ 167624813963071497236976618*z^50+106143735326309312581038703*z^48+ 87380486547515151*z^20+420420658939733285303751*z^36-103784961169345444805496*z ^34-58938893919006369142698280*z^66+22233277618683752956815*z^80+602410912374*z ^100-1046837031414476328*z^90+10552887938678713789*z^88+658181043995665787621*z ^84-6079681377981190*z^94-90211561854776900270*z^86+348535605293547*z^96-\ 16227181341128*z^98+87380486547515151*z^92-4120463628337611701962*z^82+ 106143735326309312581038703*z^64+z^112-380*z^110-5932206*z^106+62027*z^108-\ 4120463628337611701962*z^30-12218969685508696705188086*z^42+ 28679682636819124680121286*z^44-58938893919006369142698280*z^46-\ 282391398533271005826453122*z^58+301401365824788993247499108*z^56-\ 282391398533271005826453122*z^54+232243341146755644000902481*z^52+ 232243341146755644000902481*z^60-12218969685508696705188086*z^70+ 28679682636819124680121286*z^68-103784961169345444805496*z^78+ 22233277618683752956815*z^32-1481722408088266465462624*z^38+ 4553030417741383880600034*z^40-167624813963071497236976618*z^62+ 420420658939733285303751*z^76-1481722408088266465462624*z^74+ 4553030417741383880600034*z^72+379024582*z^104)/(-1+z^2)/(1+ 2252987676667336454286*z^28-295392937026164790950*z^26-495*z^2+ 32969930015146218669*z^24-3112210832624720818*z^22+94768*z^4-10185319*z^6-\ 35685231579*z^102+715606963*z^8-35685231579*z^10+1329519968995*z^12-\ 38324984572358*z^14-16222947479356440*z^18+876703623213708*z^16-\ 783933910044791629610874901*z^50+489961169419588729795901983*z^48+ 246494580278476213*z^20+1676310375041057084568451*z^36-399833867164690363603592 *z^34-267593719786558347980590569*z^66+82550286509597063421526*z^80+ 1329519968995*z^100-3112210832624720818*z^90+32969930015146218669*z^88+ 2252987676667336454286*z^84-16222947479356440*z^94-295392937026164790950*z^86+ 876703623213708*z^96-38324984572358*z^98+246494580278476213*z^92-\ 14707824462765608975174*z^82+489961169419588729795901983*z^64+z^112-495*z^110-\ 10185319*z^106+94768*z^108-14707824462765608975174*z^30-\ 53140975304570252719654517*z^42+127644141983122812380423508*z^44-\ 267593719786558347980590569*z^46-1340810224071819777859328244*z^58+ 1433804628162007951178307827*z^56-1340810224071819777859328244*z^54+ 1096432607129832338176426934*z^52+1096432607129832338176426934*z^60-\ 53140975304570252719654517*z^70+127644141983122812380423508*z^68-\ 399833867164690363603592*z^78+82550286509597063421526*z^32-\ 6098476686844819847138066*z^38+19290766627683564284190412*z^40-\ 783933910044791629610874901*z^62+1676310375041057084568451*z^76-\ 6098476686844819847138066*z^74+19290766627683564284190412*z^72+715606963*z^104) The first , 40, terms are: [0, 116, 0, 24300, 0, 5350173, 0, 1184517300, 0, 262453517053, 0, 58159467343511, 0, 12888416930106072, 0, 2856150661008192325, 0, 632940933047009297275, 0, 140263727590044633144548, 0, 31083334242535179264742831, 0, 6888264694041552462730895548, 0, 1526483301246455484507292441676, 0, 338278416188560484249408141663519, 0, 74964650310168219069826804623057999, 0, 16612643690699776614783982379950669036, 0, 3681467588461910824716308933116351786364, 0, 815836651725647657199915029738848949613479, 0, 180794595173229037263131949796270600722305684, 0, 40065233125673083902925273920284762044375432603] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 626019341313524805773 z - 85546937763440688298 z - 380 z 24 22 4 6 + 9989120424734166973 z - 990472874895558520 z + 61795 z - 5873354 z 102 8 10 12 - 16960461698 z + 372362134 z - 16960461698 z + 582177087602 z 14 18 16 - 15569706529676 z - 5774065079541018 z + 332452625035731 z 50 48 - 168230148025039583207216954 z + 106186018924078411568396479 z 20 36 + 82764727495679159 z + 407816351738665778113351 z 34 66 - 100106616580218215923256 z - 58728678992841715955924096 z 80 100 90 + 21331966644453244564223 z + 582177087602 z - 990472874895558520 z 88 84 + 9989120424734166973 z + 626019341313524805773 z 94 86 96 - 5774065079541018 z - 85546937763440688298 z + 332452625035731 z 98 92 82 - 15569706529676 z + 82764727495679159 z - 3934722525440351540318 z 64 112 110 106 + 106186018924078411568396479 z + z - 380 z - 5873354 z 108 30 42 + 61795 z - 3934722525440351540318 z - 12056870250864055662980858 z 44 46 + 28445461032478101313072486 z - 58728678992841715955924096 z 58 56 - 284488202089551796152576042 z + 303786284930846328654511540 z 54 52 - 284488202089551796152576042 z + 233631444496942226604717797 z 60 70 + 233631444496942226604717797 z - 12056870250864055662980858 z 68 78 + 28445461032478101313072486 z - 100106616580218215923256 z 32 38 + 21331966644453244564223 z - 1445653063486104998575712 z 40 62 + 4467828975228088941639610 z - 168230148025039583207216954 z 76 74 + 407816351738665778113351 z - 1445653063486104998575712 z 72 104 / + 4467828975228088941639610 z + 372362134 z ) / (-1 / 28 26 2 - 2413872436233386010464 z + 309858694510361148687 z + 500 z 24 22 4 - 33974080713508483367 z + 3161347714678211135 z - 95939 z 6 102 8 10 + 10288399 z + 1323982629198 z - 719499722 z + 35691472358 z 12 14 18 - 1323982629198 z + 38076746587489 z + 16185293003715584 z 16 50 - 871426975280322 z + 1294111487069565321092829064 z 48 20 - 766178522875177847617725392 z - 247701072531129729 z 36 34 - 2016858924359058448214431 z + 465212811982078145119354 z 66 80 + 766178522875177847617725392 z - 465212811982078145119354 z 100 90 - 38076746587489 z + 33974080713508483367 z 88 84 - 309858694510361148687 z - 16147342304413987247956 z 94 86 + 247701072531129729 z + 2413872436233386010464 z 96 98 92 - 16185293003715584 z + 871426975280322 z - 3161347714678211135 z 82 64 + 93158824533624181311108 z - 1294111487069565321092829064 z 112 114 110 106 108 - 500 z + z + 95939 z + 719499722 z - 10288399 z 30 42 + 16147342304413987247956 z + 71936232113911017761108121 z 44 46 - 180762205829336746112741905 z + 397610629101512802794755717 z 58 56 + 2838182906122321522771821215 z - 2838182906122321522771821215 z 54 52 + 2490157184391525226099227418 z - 1916738211788042326738637663 z 60 70 - 2490157184391525226099227418 z + 180762205829336746112741905 z 68 78 - 397610629101512802794755717 z + 2016858924359058448214431 z 32 38 - 93158824533624181311108 z + 7609457317999167575912285 z 40 62 - 25034950350308307699412166 z + 1916738211788042326738637663 z 76 74 - 7609457317999167575912285 z + 25034950350308307699412166 z 72 104 - 71936232113911017761108121 z - 35691472358 z ) And in Maple-input format, it is: -(1+626019341313524805773*z^28-85546937763440688298*z^26-380*z^2+ 9989120424734166973*z^24-990472874895558520*z^22+61795*z^4-5873354*z^6-\ 16960461698*z^102+372362134*z^8-16960461698*z^10+582177087602*z^12-\ 15569706529676*z^14-5774065079541018*z^18+332452625035731*z^16-\ 168230148025039583207216954*z^50+106186018924078411568396479*z^48+ 82764727495679159*z^20+407816351738665778113351*z^36-100106616580218215923256*z ^34-58728678992841715955924096*z^66+21331966644453244564223*z^80+582177087602*z ^100-990472874895558520*z^90+9989120424734166973*z^88+626019341313524805773*z^ 84-5774065079541018*z^94-85546937763440688298*z^86+332452625035731*z^96-\ 15569706529676*z^98+82764727495679159*z^92-3934722525440351540318*z^82+ 106186018924078411568396479*z^64+z^112-380*z^110-5873354*z^106+61795*z^108-\ 3934722525440351540318*z^30-12056870250864055662980858*z^42+ 28445461032478101313072486*z^44-58728678992841715955924096*z^46-\ 284488202089551796152576042*z^58+303786284930846328654511540*z^56-\ 284488202089551796152576042*z^54+233631444496942226604717797*z^52+ 233631444496942226604717797*z^60-12056870250864055662980858*z^70+ 28445461032478101313072486*z^68-100106616580218215923256*z^78+ 21331966644453244564223*z^32-1445653063486104998575712*z^38+ 4467828975228088941639610*z^40-168230148025039583207216954*z^62+ 407816351738665778113351*z^76-1445653063486104998575712*z^74+ 4467828975228088941639610*z^72+372362134*z^104)/(-1-2413872436233386010464*z^28 +309858694510361148687*z^26+500*z^2-33974080713508483367*z^24+ 3161347714678211135*z^22-95939*z^4+10288399*z^6+1323982629198*z^102-719499722*z ^8+35691472358*z^10-1323982629198*z^12+38076746587489*z^14+16185293003715584*z^ 18-871426975280322*z^16+1294111487069565321092829064*z^50-\ 766178522875177847617725392*z^48-247701072531129729*z^20-\ 2016858924359058448214431*z^36+465212811982078145119354*z^34+ 766178522875177847617725392*z^66-465212811982078145119354*z^80-38076746587489*z ^100+33974080713508483367*z^90-309858694510361148687*z^88-\ 16147342304413987247956*z^84+247701072531129729*z^94+2413872436233386010464*z^ 86-16185293003715584*z^96+871426975280322*z^98-3161347714678211135*z^92+ 93158824533624181311108*z^82-1294111487069565321092829064*z^64-500*z^112+z^114+ 95939*z^110+719499722*z^106-10288399*z^108+16147342304413987247956*z^30+ 71936232113911017761108121*z^42-180762205829336746112741905*z^44+ 397610629101512802794755717*z^46+2838182906122321522771821215*z^58-\ 2838182906122321522771821215*z^56+2490157184391525226099227418*z^54-\ 1916738211788042326738637663*z^52-2490157184391525226099227418*z^60+ 180762205829336746112741905*z^70-397610629101512802794755717*z^68+ 2016858924359058448214431*z^78-93158824533624181311108*z^32+ 7609457317999167575912285*z^38-25034950350308307699412166*z^40+ 1916738211788042326738637663*z^62-7609457317999167575912285*z^76+ 25034950350308307699412166*z^74-71936232113911017761108121*z^72-35691472358*z^ 104) The first , 40, terms are: [0, 120, 0, 25856, 0, 5830365, 0, 1322054008, 0, 300075504829, 0, 68124120705955, 0, 15466457466225232, 0, 3511439400900453521, 0, 797224235092683551263, 0, 180998939541001992727900, 0, 41093356780486107211794587, 0, 9329690109055175011445743888, 0, 2118179795234044824825348718156, 0, 480904038458636107501356490212691, 0, 109182749640614778242929422862037239, 0, 24788464780050271345664185371731764228, 0, 5627885249126717348552561008977052694176, 0, 1277735134405834415287339449315271523495955, 0, 290092459498842787789237353746128902174717956, 0, 65861564570059352676856028386019213909225243803] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 8420357750207355 z - 3979216146890164 z - 304 z 24 22 4 6 + 1508368768475625 z - 456277891082988 z + 34915 z - 2066912 z 8 10 12 14 + 73517555 z - 1723123323 z + 28245157274 z - 337113685535 z 18 16 50 - 20606234766307 z + 3013991318198 z - 20606234766307 z 48 20 36 + 109398744658177 z + 109398744658177 z + 19731173687082928 z 34 66 64 30 - 21939057813667930 z - 304 z + 34915 z - 14346373242199906 z 42 44 46 - 3979216146890164 z + 1508368768475625 z - 456277891082988 z 58 56 54 52 - 1723123323 z + 28245157274 z - 337113685535 z + 3013991318198 z 60 68 32 38 + 73517555 z + z + 19731173687082928 z - 14346373242199906 z 40 62 / 2 + 8420357750207355 z - 2066912 z ) / ((-1 + z ) (1 / 28 26 2 + 38723533955874782 z - 17675525232017702 z - 415 z 24 22 4 6 + 6415075446838151 z - 1842975355564781 z + 59620 z - 4145121 z 8 10 12 14 + 167612638 z - 4382824873 z + 79154309603 z - 1031206298394 z 18 16 50 - 73437863110912 z + 9986212235922 z - 73437863110912 z 48 20 36 + 416526588178569 z + 416526588178569 z + 94509612227794825 z 34 66 64 30 - 105630034431508487 z - 415 z + 59620 z - 67666822542181282 z 42 44 46 - 17675525232017702 z + 6415075446838151 z - 1842975355564781 z 58 56 54 52 - 4382824873 z + 79154309603 z - 1031206298394 z + 9986212235922 z 60 68 32 38 + 167612638 z + z + 94509612227794825 z - 67666822542181282 z 40 62 + 38723533955874782 z - 4145121 z )) And in Maple-input format, it is: -(1+8420357750207355*z^28-3979216146890164*z^26-304*z^2+1508368768475625*z^24-\ 456277891082988*z^22+34915*z^4-2066912*z^6+73517555*z^8-1723123323*z^10+ 28245157274*z^12-337113685535*z^14-20606234766307*z^18+3013991318198*z^16-\ 20606234766307*z^50+109398744658177*z^48+109398744658177*z^20+19731173687082928 *z^36-21939057813667930*z^34-304*z^66+34915*z^64-14346373242199906*z^30-\ 3979216146890164*z^42+1508368768475625*z^44-456277891082988*z^46-1723123323*z^ 58+28245157274*z^56-337113685535*z^54+3013991318198*z^52+73517555*z^60+z^68+ 19731173687082928*z^32-14346373242199906*z^38+8420357750207355*z^40-2066912*z^ 62)/(-1+z^2)/(1+38723533955874782*z^28-17675525232017702*z^26-415*z^2+ 6415075446838151*z^24-1842975355564781*z^22+59620*z^4-4145121*z^6+167612638*z^8 -4382824873*z^10+79154309603*z^12-1031206298394*z^14-73437863110912*z^18+ 9986212235922*z^16-73437863110912*z^50+416526588178569*z^48+416526588178569*z^ 20+94509612227794825*z^36-105630034431508487*z^34-415*z^66+59620*z^64-\ 67666822542181282*z^30-17675525232017702*z^42+6415075446838151*z^44-\ 1842975355564781*z^46-4382824873*z^58+79154309603*z^56-1031206298394*z^54+ 9986212235922*z^52+167612638*z^60+z^68+94509612227794825*z^32-67666822542181282 *z^38+38723533955874782*z^40-4145121*z^62) The first , 40, terms are: [0, 112, 0, 21472, 0, 4346261, 0, 891663844, 0, 183879023901, 0, 38003911314459, 0, 7862285462711056, 0, 1627259460630409521, 0, 336858639803573472803, 0, 69738921116826853166764, 0, 14438396402952983052288079, 0, 2989302471437869045271677968, 0, 618904910080140857901762066912, 0, 128138429022057099269093021877623, 0, 26529891952020580841750775907551631, 0, 5492775397765556580525673930807610272, 0, 1137230035784891314728595103458634315216, 0, 235453340049309347639669625833038656654383, 0, 48748517550395612706227264637425897108586908, 0, 10092946738638389434217677354702226103818802491] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 565203264355250486028 z - 79445019612159080672 z - 389 z 24 22 4 6 + 9530748878589361212 z - 969296498043753032 z + 64461 z - 6209198 z 102 8 10 12 - 18098580470 z + 396644902 z - 18098580470 z + 618926235549 z 14 18 16 - 16407807019361 z - 5902590104979000 z + 345732022237121 z 50 48 - 117075781869656368161015634 z + 74836835398530459746579958 z 20 36 + 82894391005832696 z + 328160074303106744596828 z 34 66 - 82859605415416946927744 z - 42051569852430152397664184 z 80 100 90 + 18174319522807683443400 z + 618926235549 z - 969296498043753032 z 88 84 + 9530748878589361212 z + 565203264355250486028 z 94 86 96 - 5902590104979000 z - 79445019612159080672 z + 345732022237121 z 98 92 82 - 16407807019361 z + 82894391005832696 z - 3451323182761996036544 z 64 112 110 106 + 74836835398530459746579958 z + z - 389 z - 6209198 z 108 30 42 + 64461 z - 3451323182761996036544 z - 8987530448324056120176648 z 44 46 + 20754380345140476218596936 z - 42051569852430152397664184 z 58 56 - 195081013359688097027420636 z + 207924812663167370917623204 z 54 52 - 195081013359688097027420636 z + 161103510625552686042380782 z 60 70 + 161103510625552686042380782 z - 8987530448324056120176648 z 68 78 + 20754380345140476218596936 z - 82859605415416946927744 z 32 38 + 18174319522807683443400 z - 1132141388917208188599784 z 40 62 + 3410386355962176733233228 z - 117075781869656368161015634 z 76 74 + 328160074303106744596828 z - 1132141388917208188599784 z 72 104 / + 3410386355962176733233228 z + 396644902 z ) / (-1 / 28 26 2 - 2232825382055362505832 z + 295649770358664536172 z + 501 z 24 22 4 - 33379087157634842860 z + 3190521707328567328 z - 98322 z 6 102 8 10 + 10791814 z + 1435273813750 z - 768496955 z + 38547305947 z 12 14 18 - 1435273813750 z + 41142879031618 z + 17061684893767473 z 16 50 - 932609474730101 z + 886216394957514108437001470 z 48 20 - 533049503819612591801545238 z - 255987889495484768 z 36 34 - 1638723080114992548209512 z + 390336453742580982339236 z 66 80 + 533049503819612591801545238 z - 390336453742580982339236 z 100 90 - 41142879031618 z + 33379087157634842860 z 88 84 - 295649770358664536172 z - 14463669040995051506312 z 94 86 + 255987889495484768 z + 2232825382055362505832 z 96 98 92 - 17061684893767473 z + 932609474730101 z - 3190521707328567328 z 82 64 112 + 80761335043666903604452 z - 886216394957514108437001470 z - 501 z 114 110 106 108 + z + 98322 z + 768496955 z - 10791814 z 30 42 + 14463669040995051506312 z + 53510544622872004910365516 z 44 46 - 131114792142130198783003488 z + 282010913509488631514783456 z 58 56 + 1896733908389401399672967586 z - 1896733908389401399672967586 z 54 52 + 1671033313111024962159847188 z - 1296806600046591645024747324 z 60 70 - 1671033313111024962159847188 z + 131114792142130198783003488 z 68 78 - 282010913509488631514783456 z + 1638723080114992548209512 z 32 38 - 80761335043666903604452 z + 5993477257414926102322120 z 40 62 - 19143747234804219266135660 z + 1296806600046591645024747324 z 76 74 - 5993477257414926102322120 z + 19143747234804219266135660 z 72 104 - 53510544622872004910365516 z - 38547305947 z ) And in Maple-input format, it is: -(1+565203264355250486028*z^28-79445019612159080672*z^26-389*z^2+ 9530748878589361212*z^24-969296498043753032*z^22+64461*z^4-6209198*z^6-\ 18098580470*z^102+396644902*z^8-18098580470*z^10+618926235549*z^12-\ 16407807019361*z^14-5902590104979000*z^18+345732022237121*z^16-\ 117075781869656368161015634*z^50+74836835398530459746579958*z^48+ 82894391005832696*z^20+328160074303106744596828*z^36-82859605415416946927744*z^ 34-42051569852430152397664184*z^66+18174319522807683443400*z^80+618926235549*z^ 100-969296498043753032*z^90+9530748878589361212*z^88+565203264355250486028*z^84 -5902590104979000*z^94-79445019612159080672*z^86+345732022237121*z^96-\ 16407807019361*z^98+82894391005832696*z^92-3451323182761996036544*z^82+ 74836835398530459746579958*z^64+z^112-389*z^110-6209198*z^106+64461*z^108-\ 3451323182761996036544*z^30-8987530448324056120176648*z^42+ 20754380345140476218596936*z^44-42051569852430152397664184*z^46-\ 195081013359688097027420636*z^58+207924812663167370917623204*z^56-\ 195081013359688097027420636*z^54+161103510625552686042380782*z^52+ 161103510625552686042380782*z^60-8987530448324056120176648*z^70+ 20754380345140476218596936*z^68-82859605415416946927744*z^78+ 18174319522807683443400*z^32-1132141388917208188599784*z^38+ 3410386355962176733233228*z^40-117075781869656368161015634*z^62+ 328160074303106744596828*z^76-1132141388917208188599784*z^74+ 3410386355962176733233228*z^72+396644902*z^104)/(-1-2232825382055362505832*z^28 +295649770358664536172*z^26+501*z^2-33379087157634842860*z^24+ 3190521707328567328*z^22-98322*z^4+10791814*z^6+1435273813750*z^102-768496955*z ^8+38547305947*z^10-1435273813750*z^12+41142879031618*z^14+17061684893767473*z^ 18-932609474730101*z^16+886216394957514108437001470*z^50-\ 533049503819612591801545238*z^48-255987889495484768*z^20-\ 1638723080114992548209512*z^36+390336453742580982339236*z^34+ 533049503819612591801545238*z^66-390336453742580982339236*z^80-41142879031618*z ^100+33379087157634842860*z^90-295649770358664536172*z^88-\ 14463669040995051506312*z^84+255987889495484768*z^94+2232825382055362505832*z^ 86-17061684893767473*z^96+932609474730101*z^98-3190521707328567328*z^92+ 80761335043666903604452*z^82-886216394957514108437001470*z^64-501*z^112+z^114+ 98322*z^110+768496955*z^106-10791814*z^108+14463669040995051506312*z^30+ 53510544622872004910365516*z^42-131114792142130198783003488*z^44+ 282010913509488631514783456*z^46+1896733908389401399672967586*z^58-\ 1896733908389401399672967586*z^56+1671033313111024962159847188*z^54-\ 1296806600046591645024747324*z^52-1671033313111024962159847188*z^60+ 131114792142130198783003488*z^70-282010913509488631514783456*z^68+ 1638723080114992548209512*z^78-80761335043666903604452*z^32+ 5993477257414926102322120*z^38-19143747234804219266135660*z^40+ 1296806600046591645024747324*z^62-5993477257414926102322120*z^76+ 19143747234804219266135660*z^74-53510544622872004910365516*z^72-38547305947*z^ 104) The first , 40, terms are: [0, 112, 0, 22251, 0, 4718303, 0, 1012938096, 0, 218074718361, 0, 46981507737749, 0, 10123331464699740, 0, 2181417932959356663, 0, 470066279048463887755, 0, 101293256643648311724508, 0, 21827413739063008225968661, 0, 4703532028835744924743660061, 0, 1013551803764272904617547621700, 0, 218407627432289278311279656689387, 0, 47064088570409600673859194199177399, 0, 10141717396447918905613311260054083236, 0, 2185412166549029913207836355427074707069, 0, 470928754099236238057113580456328606109681, 0, 101479205998217640264047561112958043420287656, 0, 21867488787680882961681752804024986932939838527] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 5}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 40660045714480964 z - 18325453373234560 z - 324 z 24 22 4 6 + 6530698067358260 z - 1830177651468644 z + 41313 z - 2817220 z 8 10 12 14 + 117525596 z - 3241126784 z + 62172955180 z - 859679119212 z 18 16 50 - 67837795154736 z + 8793565970308 z - 67837795154736 z 48 20 36 + 400509486756868 z + 400509486756868 z + 100501808228551446 z 34 66 64 30 - 112488798837889304 z - 324 z + 41313 z - 71633186278477932 z 42 44 46 - 18325453373234560 z + 6530698067358260 z - 1830177651468644 z 58 56 54 52 - 3241126784 z + 62172955180 z - 859679119212 z + 8793565970308 z 60 68 32 38 + 117525596 z + z + 100501808228551446 z - 71633186278477932 z 40 62 / 2 + 40660045714480964 z - 2817220 z ) / ((-1 + z ) (1 / 28 26 2 + 189394153180143224 z - 82067894438373576 z - 454 z 24 22 4 6 + 27851412122814544 z - 7369341092870592 z + 70477 z - 5559968 z 8 10 12 14 + 261763000 z - 8030640696 z + 169747892608 z - 2567946255488 z 18 16 50 - 238213698155032 z + 28563396109576 z - 238213698155032 z 48 20 36 + 1511105180932144 z + 1511105180932144 z + 490417487702364026 z 34 66 64 30 - 552178770759884356 z - 454 z + 70477 z - 343435486906314016 z 42 44 46 - 82067894438373576 z + 27851412122814544 z - 7369341092870592 z 58 56 54 - 8030640696 z + 169747892608 z - 2567946255488 z 52 60 68 32 + 28563396109576 z + 261763000 z + z + 490417487702364026 z 38 40 62 - 343435486906314016 z + 189394153180143224 z - 5559968 z )) And in Maple-input format, it is: -(1+40660045714480964*z^28-18325453373234560*z^26-324*z^2+6530698067358260*z^24 -1830177651468644*z^22+41313*z^4-2817220*z^6+117525596*z^8-3241126784*z^10+ 62172955180*z^12-859679119212*z^14-67837795154736*z^18+8793565970308*z^16-\ 67837795154736*z^50+400509486756868*z^48+400509486756868*z^20+ 100501808228551446*z^36-112488798837889304*z^34-324*z^66+41313*z^64-\ 71633186278477932*z^30-18325453373234560*z^42+6530698067358260*z^44-\ 1830177651468644*z^46-3241126784*z^58+62172955180*z^56-859679119212*z^54+ 8793565970308*z^52+117525596*z^60+z^68+100501808228551446*z^32-\ 71633186278477932*z^38+40660045714480964*z^40-2817220*z^62)/(-1+z^2)/(1+ 189394153180143224*z^28-82067894438373576*z^26-454*z^2+27851412122814544*z^24-\ 7369341092870592*z^22+70477*z^4-5559968*z^6+261763000*z^8-8030640696*z^10+ 169747892608*z^12-2567946255488*z^14-238213698155032*z^18+28563396109576*z^16-\ 238213698155032*z^50+1511105180932144*z^48+1511105180932144*z^20+ 490417487702364026*z^36-552178770759884356*z^34-454*z^66+70477*z^64-\ 343435486906314016*z^30-82067894438373576*z^42+27851412122814544*z^44-\ 7369341092870592*z^46-8030640696*z^58+169747892608*z^56-2567946255488*z^54+ 28563396109576*z^52+261763000*z^60+z^68+490417487702364026*z^32-\ 343435486906314016*z^38+189394153180143224*z^40-5559968*z^62) The first , 40, terms are: [0, 131, 0, 29987, 0, 7165349, 0, 1721016821, 0, 413689405955, 0, 99453824494115, 0, 23909968421105745, 0, 5748287750965246833, 0, 1381969302614831095715, 0, 332244941140248255899779, 0, 79876380412370844693311637, 0, 19203411168352007206772704133, 0, 4616771560305287524177447460131, 0, 1109937159950658023718258007798403, 0, 266844586774279858963669968128301665, 0, 64153211607901154639630039163382600865, 0, 15423339140606820798392890683325886439811, 0, 3707988801882450497579862738118215426219555, 0, 891452935680944058320081542267158054787712197, 0, 214317890100127278342402731355181704444363843669] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 6}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 37963253668194800 z - 15008468216398049 z - 293 z 24 22 4 6 + 4792176686614012 z - 1229507837075188 z + 33320 z - 2048523 z 8 10 12 14 + 78490628 z - 2030167642 z + 37317255686 z - 505225585900 z 18 16 50 - 40794696291824 z + 5170898596199 z - 1229507837075188 z 48 20 36 + 4792176686614012 z + 251778187695616 z + 195168166626530474 z 34 66 64 - 176263731432091708 z - 2048523 z + 78490628 z 30 42 44 - 77853115892820445 z - 77853115892820445 z + 37963253668194800 z 46 58 56 - 15008468216398049 z - 505225585900 z + 5170898596199 z 54 52 60 70 - 40794696291824 z + 251778187695616 z + 37317255686 z - 293 z 68 32 38 + 33320 z + 129799625261245433 z - 176263731432091708 z 40 62 72 / 2 + 129799625261245433 z - 2030167642 z + z ) / ((-1 + z ) (1 / 28 26 2 + 172416596076744669 z - 64996540424663742 z - 418 z 24 22 4 6 + 19635191961188827 z - 4734902881898322 z + 57668 z - 4053048 z 8 10 12 14 + 172884373 z - 4913560120 z + 98544426750 z - 1449213735046 z 18 16 50 - 136644669228132 z + 16055423885405 z - 4734902881898322 z 48 20 36 + 19635191961188827 z + 906333100509529 z + 970292225110336527 z 34 66 64 - 871154200745626554 z - 4053048 z + 172884373 z 30 42 44 - 367489817181917176 z - 367489817181917176 z + 172416596076744669 z 46 58 56 - 64996540424663742 z - 1449213735046 z + 16055423885405 z 54 52 60 70 - 136644669228132 z + 906333100509529 z + 98544426750 z - 418 z 68 32 38 + 57668 z + 630393183959236569 z - 871154200745626554 z 40 62 72 + 630393183959236569 z - 4913560120 z + z )) And in Maple-input format, it is: -(1+37963253668194800*z^28-15008468216398049*z^26-293*z^2+4792176686614012*z^24 -1229507837075188*z^22+33320*z^4-2048523*z^6+78490628*z^8-2030167642*z^10+ 37317255686*z^12-505225585900*z^14-40794696291824*z^18+5170898596199*z^16-\ 1229507837075188*z^50+4792176686614012*z^48+251778187695616*z^20+ 195168166626530474*z^36-176263731432091708*z^34-2048523*z^66+78490628*z^64-\ 77853115892820445*z^30-77853115892820445*z^42+37963253668194800*z^44-\ 15008468216398049*z^46-505225585900*z^58+5170898596199*z^56-40794696291824*z^54 +251778187695616*z^52+37317255686*z^60-293*z^70+33320*z^68+129799625261245433*z ^32-176263731432091708*z^38+129799625261245433*z^40-2030167642*z^62+z^72)/(-1+z ^2)/(1+172416596076744669*z^28-64996540424663742*z^26-418*z^2+19635191961188827 *z^24-4734902881898322*z^22+57668*z^4-4053048*z^6+172884373*z^8-4913560120*z^10 +98544426750*z^12-1449213735046*z^14-136644669228132*z^18+16055423885405*z^16-\ 4734902881898322*z^50+19635191961188827*z^48+906333100509529*z^20+ 970292225110336527*z^36-871154200745626554*z^34-4053048*z^66+172884373*z^64-\ 367489817181917176*z^30-367489817181917176*z^42+172416596076744669*z^44-\ 64996540424663742*z^46-1449213735046*z^58+16055423885405*z^56-136644669228132*z ^54+906333100509529*z^52+98544426750*z^60-418*z^70+57668*z^68+ 630393183959236569*z^32-871154200745626554*z^38+630393183959236569*z^40-\ 4913560120*z^62+z^72) The first , 40, terms are: [0, 126, 0, 28028, 0, 6487089, 0, 1509559306, 0, 351673607413, 0, 81949109442601, 0, 19097494183404008, 0, 4450566067536526109, 0, 1037183864739717085029, 0, 241711130139494659566706, 0, 56329726520123999445153829, 0, 13127398230072463941770698700, 0, 3059283210010563408363270444270, 0, 712952683016818470250520801899217, 0, 166150530606582110402498416065170313, 0, 38720660548329344112307370734976370702, 0, 9023682006280634049840898547101474434588, 0, 2102929955178857665111419556540445723444589, 0, 490078705491032293224171787817427202964867106, 0, 114210716807017766650402661823190527639061785661] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 28320598705941574 z - 11323172955137778 z - 295 z 24 22 4 6 + 3665612092996439 z - 955992143038991 z + 33397 z - 2005238 z 8 10 12 14 + 74329493 z - 1855137375 z + 32940270634 z - 431978463435 z 18 16 50 - 33052979464866 z + 4296676790537 z - 955992143038991 z 48 20 36 + 3665612092996439 z + 199545908047677 z + 142687169070206588 z 34 66 64 - 129030118832413294 z - 2005238 z + 74329493 z 30 42 44 - 57570089479734440 z - 57570089479734440 z + 28320598705941574 z 46 58 56 - 11323172955137778 z - 431978463435 z + 4296676790537 z 54 52 60 70 - 33052979464866 z + 199545908047677 z + 32940270634 z - 295 z 68 32 38 + 33397 z + 95379189310875818 z - 129030118832413294 z 40 62 72 / + 95379189310875818 z - 1855137375 z + z ) / (-1 / 28 26 2 - 179944288132783264 z + 65120999842462367 z + 431 z 24 22 4 6 - 19083835491392529 z + 4505979290322394 z - 59350 z + 4101228 z 8 10 12 14 - 171275088 z + 4769862394 z - 94044865396 z + 1365836042968 z 18 16 50 + 127699199962544 z - 15023092586166 z + 19083835491392529 z 48 20 36 - 65120999842462367 z - 851565761780516 z - 1352049292415500628 z 34 66 64 + 1106116032718251776 z + 171275088 z - 4769862394 z 30 42 44 + 404159913960144000 z + 739882761973822448 z - 404159913960144000 z 46 58 56 + 179944288132783264 z + 15023092586166 z - 127699199962544 z 54 52 60 + 851565761780516 z - 4505979290322394 z - 1365836042968 z 70 68 32 + 59350 z - 4101228 z - 739882761973822448 z 38 40 62 + 1352049292415500628 z - 1106116032718251776 z + 94044865396 z 74 72 + z - 431 z ) And in Maple-input format, it is: -(1+28320598705941574*z^28-11323172955137778*z^26-295*z^2+3665612092996439*z^24 -955992143038991*z^22+33397*z^4-2005238*z^6+74329493*z^8-1855137375*z^10+ 32940270634*z^12-431978463435*z^14-33052979464866*z^18+4296676790537*z^16-\ 955992143038991*z^50+3665612092996439*z^48+199545908047677*z^20+ 142687169070206588*z^36-129030118832413294*z^34-2005238*z^66+74329493*z^64-\ 57570089479734440*z^30-57570089479734440*z^42+28320598705941574*z^44-\ 11323172955137778*z^46-431978463435*z^58+4296676790537*z^56-33052979464866*z^54 +199545908047677*z^52+32940270634*z^60-295*z^70+33397*z^68+95379189310875818*z^ 32-129030118832413294*z^38+95379189310875818*z^40-1855137375*z^62+z^72)/(-1-\ 179944288132783264*z^28+65120999842462367*z^26+431*z^2-19083835491392529*z^24+ 4505979290322394*z^22-59350*z^4+4101228*z^6-171275088*z^8+4769862394*z^10-\ 94044865396*z^12+1365836042968*z^14+127699199962544*z^18-15023092586166*z^16+ 19083835491392529*z^50-65120999842462367*z^48-851565761780516*z^20-\ 1352049292415500628*z^36+1106116032718251776*z^34+171275088*z^66-4769862394*z^ 64+404159913960144000*z^30+739882761973822448*z^42-404159913960144000*z^44+ 179944288132783264*z^46+15023092586166*z^58-127699199962544*z^56+ 851565761780516*z^54-4505979290322394*z^52-1365836042968*z^60+59350*z^70-\ 4101228*z^68-739882761973822448*z^32+1352049292415500628*z^38-\ 1106116032718251776*z^40+94044865396*z^62+z^74-431*z^72) The first , 40, terms are: [0, 136, 0, 32663, 0, 8102143, 0, 2014295996, 0, 500879110441, 0, 124552403460553, 0, 30972244341855396, 0, 7701822221392757839, 0, 1915200980134885014967, 0, 476250267904570523143600, 0, 118428468620485696895206129, 0, 29449436876652956055081725009, 0, 7323149094045739151083934857184, 0, 1821036948288235914929784641172855, 0, 452834637738451941751356096496380719, 0, 112605737807578968405844439939747327572, 0, 28001506798448995569235704808244221154377, 0, 6963094405754800038113113299388684298432169, 0, 1731502667068733273577311064558936431199379724, 0, 430570276856845377868859229609500953904158148767] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 622589263412822063284 z - 85301190842214854320 z - 380 z 24 22 4 6 + 9984905353216345452 z - 992219084750948432 z + 61782 z - 5876304 z 102 8 10 12 - 17020177472 z + 373080822 z - 17020177472 z + 584976850590 z 14 18 16 - 15655595871052 z - 5802095523836328 z + 334291216838963 z 50 48 - 163708728307732461100115664 z + 103424606968531946160996779 z 20 36 + 83059567709503636 z + 401380474123883721897906 z 34 66 - 98769750634128774420148 z - 57267070989364837231126916 z 80 100 90 + 21101742296478299174177 z + 584976850590 z - 992219084750948432 z 88 84 + 9984905353216345452 z + 622589263412822063284 z 94 86 96 - 5802095523836328 z - 85301190842214854320 z + 334291216838963 z 98 92 82 - 15655595871052 z + 83059567709503636 z - 3902670961006562368136 z 64 112 110 106 + 103424606968531946160996779 z + z - 380 z - 5876304 z 108 30 42 + 61782 z - 3902670961006562368136 z - 11792548747030925759009072 z 44 46 + 27776279218013749183306650 z - 57267070989364837231126916 z 58 56 - 276558700002624098522750336 z + 295281032520131199044047592 z 54 52 - 276558700002624098522750336 z + 227206793993527831030750072 z 60 70 + 227206793993527831030750072 z - 11792548747030925759009072 z 68 78 + 27776279218013749183306650 z - 98769750634128774420148 z 32 38 + 21101742296478299174177 z - 1419580374681351823147968 z 40 62 + 4378075828136174407007786 z - 163708728307732461100115664 z 76 74 + 401380474123883721897906 z - 1419580374681351823147968 z 72 104 / + 4378075828136174407007786 z + 373080822 z ) / (-1 / 28 26 2 - 2405087831575149680828 z + 309656186212137746928 z + 499 z 24 22 4 - 34051647612311133552 z + 3177260806347033052 z - 95906 z 6 102 8 10 + 10314422 z + 1337259316006 z - 723552980 z + 35986583916 z 12 14 18 - 1337259316006 z + 38486363108266 z + 16336047748467839 z 16 50 - 880558357015837 z + 1264637787986898977964934867 z 48 20 - 749136388728811408871832489 z - 249543262467485404 z 36 34 - 1988225968797006481432134 z + 459679526872296325330183 z 66 80 + 749136388728811408871832489 z - 459679526872296325330183 z 100 90 - 38486363108266 z + 34051647612311133552 z 88 84 - 309656186212137746928 z - 16041383104554139782620 z 94 86 + 249543262467485404 z + 2405087831575149680828 z 96 98 92 - 16336047748467839 z + 880558357015837 z - 3177260806347033052 z 82 64 + 92289194417417978621165 z - 1264637787986898977964934867 z 112 114 110 106 108 - 499 z + z + 95906 z + 723552980 z - 10314422 z 30 42 + 16041383104554139782620 z + 70540485003441099654618044 z 44 46 - 177041236569246570360222186 z + 389053038516307397543680046 z 58 56 + 2771507751730813890196902688 z - 2771507751730813890196902688 z 54 52 + 2431937591330740555554956712 z - 1872372466226920386800795272 z 60 70 - 2431937591330740555554956712 z + 177041236569246570360222186 z 68 78 - 389053038516307397543680046 z + 1988225968797006481432134 z 32 38 - 92289194417417978621165 z + 7486039880732111162729466 z 40 62 - 24585511262454335668669828 z + 1872372466226920386800795272 z 76 74 - 7486039880732111162729466 z + 24585511262454335668669828 z 72 104 - 70540485003441099654618044 z - 35986583916 z ) And in Maple-input format, it is: -(1+622589263412822063284*z^28-85301190842214854320*z^26-380*z^2+ 9984905353216345452*z^24-992219084750948432*z^22+61782*z^4-5876304*z^6-\ 17020177472*z^102+373080822*z^8-17020177472*z^10+584976850590*z^12-\ 15655595871052*z^14-5802095523836328*z^18+334291216838963*z^16-\ 163708728307732461100115664*z^50+103424606968531946160996779*z^48+ 83059567709503636*z^20+401380474123883721897906*z^36-98769750634128774420148*z^ 34-57267070989364837231126916*z^66+21101742296478299174177*z^80+584976850590*z^ 100-992219084750948432*z^90+9984905353216345452*z^88+622589263412822063284*z^84 -5802095523836328*z^94-85301190842214854320*z^86+334291216838963*z^96-\ 15655595871052*z^98+83059567709503636*z^92-3902670961006562368136*z^82+ 103424606968531946160996779*z^64+z^112-380*z^110-5876304*z^106+61782*z^108-\ 3902670961006562368136*z^30-11792548747030925759009072*z^42+ 27776279218013749183306650*z^44-57267070989364837231126916*z^46-\ 276558700002624098522750336*z^58+295281032520131199044047592*z^56-\ 276558700002624098522750336*z^54+227206793993527831030750072*z^52+ 227206793993527831030750072*z^60-11792548747030925759009072*z^70+ 27776279218013749183306650*z^68-98769750634128774420148*z^78+ 21101742296478299174177*z^32-1419580374681351823147968*z^38+ 4378075828136174407007786*z^40-163708728307732461100115664*z^62+ 401380474123883721897906*z^76-1419580374681351823147968*z^74+ 4378075828136174407007786*z^72+373080822*z^104)/(-1-2405087831575149680828*z^28 +309656186212137746928*z^26+499*z^2-34051647612311133552*z^24+ 3177260806347033052*z^22-95906*z^4+10314422*z^6+1337259316006*z^102-723552980*z ^8+35986583916*z^10-1337259316006*z^12+38486363108266*z^14+16336047748467839*z^ 18-880558357015837*z^16+1264637787986898977964934867*z^50-\ 749136388728811408871832489*z^48-249543262467485404*z^20-\ 1988225968797006481432134*z^36+459679526872296325330183*z^34+ 749136388728811408871832489*z^66-459679526872296325330183*z^80-38486363108266*z ^100+34051647612311133552*z^90-309656186212137746928*z^88-\ 16041383104554139782620*z^84+249543262467485404*z^94+2405087831575149680828*z^ 86-16336047748467839*z^96+880558357015837*z^98-3177260806347033052*z^92+ 92289194417417978621165*z^82-1264637787986898977964934867*z^64-499*z^112+z^114+ 95906*z^110+723552980*z^106-10314422*z^108+16041383104554139782620*z^30+ 70540485003441099654618044*z^42-177041236569246570360222186*z^44+ 389053038516307397543680046*z^46+2771507751730813890196902688*z^58-\ 2771507751730813890196902688*z^56+2431937591330740555554956712*z^54-\ 1872372466226920386800795272*z^52-2431937591330740555554956712*z^60+ 177041236569246570360222186*z^70-389053038516307397543680046*z^68+ 1988225968797006481432134*z^78-92289194417417978621165*z^32+ 7486039880732111162729466*z^38-24585511262454335668669828*z^40+ 1872372466226920386800795272*z^62-7486039880732111162729466*z^76+ 24585511262454335668669828*z^74-70540485003441099654618044*z^72-35986583916*z^ 104) The first , 40, terms are: [0, 119, 0, 25257, 0, 5628547, 0, 1263291171, 0, 283945824025, 0, 63842315552111, 0, 14355383611808753, 0, 3227964900164652441, 0, 725846305842793806807, 0, 163215343660416009549473, 0, 36700958223668014250879387, 0, 8252658013048050830852242747, 0, 1855710818882749587087780325681, 0, 417279214706891362943418859920127, 0, 93830321726455949539859273316793257, 0, 21098892460239162673050541654216481145, 0, 4744343351684966815925751079127224798287, 0, 1066823478120717456476731982425823668930881, 0, 239888273067000813569529999284391680447201707, 0, 53941804558404099446637423636625946592105649547] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 703212459330203155373 z - 95038165805510094750 z - 377 z 24 22 4 + 10964427131371839591 z - 1073231334129993370 z + 61034 z 6 102 8 10 - 5802958 z - 16980793944 z + 369681579 z - 16980793944 z 12 14 18 + 589330120477 z - 15962660455426 z - 6086321185623110 z 16 50 + 345537651424463 z - 199773708843660367569388328 z 48 20 + 125942326484652625109426558 z + 88470155511680271 z 36 34 + 473619270946872126151048 z - 115486294747233708418746 z 66 80 - 69536227541392586592185220 z + 24418940260611371995379 z 100 90 88 + 589330120477 z - 1073231334129993370 z + 10964427131371839591 z 84 94 + 703212459330203155373 z - 6086321185623110 z 86 96 98 - 95038165805510094750 z + 345537651424463 z - 15962660455426 z 92 82 + 88470155511680271 z - 4464295494700040795500 z 64 112 110 106 + 125942326484652625109426558 z + z - 377 z - 5802958 z 108 30 42 + 61034 z - 4464295494700040795500 z - 14200472711221514161062848 z 44 46 + 33602874772262239248677226 z - 69536227541392586592185220 z 58 56 - 338259557208623787444912264 z + 361258973881811920889948074 z 54 52 - 338259557208623787444912264 z + 277662597221516679535145746 z 60 70 + 277662597221516679535145746 z - 14200472711221514161062848 z 68 78 + 33602874772262239248677226 z - 115486294747233708418746 z 32 38 + 24418940260611371995379 z - 1688390586898446575050347 z 40 62 + 5242365703292948722170805 z - 199773708843660367569388328 z 76 74 + 473619270946872126151048 z - 1688390586898446575050347 z 72 104 / + 5242365703292948722170805 z + 369681579 z ) / (-1 / 28 26 2 - 2697059441345925397876 z + 342102558130507221022 z + 502 z 24 22 4 - 37030914262325775184 z + 3399318060097447107 z - 95561 z 6 102 8 10 + 10197788 z + 1331537684550 z - 713459098 z + 35576944688 z 12 14 18 - 1331537684550 z + 38736118625700 z + 16917513315322873 z 16 50 - 898211940893955 z + 1544833801274916968383462057 z 48 20 - 912852520178457187521416977 z - 262622846883320545 z 36 34 - 2339107087685061458308764 z + 535394806860594446549479 z 66 80 + 912852520178457187521416977 z - 535394806860594446549479 z 100 90 - 38736118625700 z + 37030914262325775184 z 88 84 - 342102558130507221022 z - 18239941071688123004522 z 94 86 + 262622846883320545 z + 2697059441345925397876 z 96 98 92 - 16917513315322873 z + 898211940893955 z - 3399318060097447107 z 82 64 + 106274967503500316489689 z - 1544833801274916968383462057 z 112 114 110 106 108 - 502 z + z + 95561 z + 713459098 z - 10197788 z 30 42 + 18239941071688123004522 z + 84877850909150444368360776 z 44 46 - 214130710160450879438618752 z + 472526448163686342952160312 z 58 56 + 3397305142549576385906604445 z - 3397305142549576385906604445 z 54 52 + 2979408483678917009166840376 z - 2291271074355142367669381728 z 60 70 - 2979408483678917009166840376 z + 214130710160450879438618752 z 68 78 - 472526448163686342952160312 z + 2339107087685061458308764 z 32 38 - 106274967503500316489689 z + 8884498708869017483965859 z 40 62 - 29397265594572471208720095 z + 2291271074355142367669381728 z 76 74 - 8884498708869017483965859 z + 29397265594572471208720095 z 72 104 - 84877850909150444368360776 z - 35576944688 z ) And in Maple-input format, it is: -(1+703212459330203155373*z^28-95038165805510094750*z^26-377*z^2+ 10964427131371839591*z^24-1073231334129993370*z^22+61034*z^4-5802958*z^6-\ 16980793944*z^102+369681579*z^8-16980793944*z^10+589330120477*z^12-\ 15962660455426*z^14-6086321185623110*z^18+345537651424463*z^16-\ 199773708843660367569388328*z^50+125942326484652625109426558*z^48+ 88470155511680271*z^20+473619270946872126151048*z^36-115486294747233708418746*z ^34-69536227541392586592185220*z^66+24418940260611371995379*z^80+589330120477*z ^100-1073231334129993370*z^90+10964427131371839591*z^88+703212459330203155373*z ^84-6086321185623110*z^94-95038165805510094750*z^86+345537651424463*z^96-\ 15962660455426*z^98+88470155511680271*z^92-4464295494700040795500*z^82+ 125942326484652625109426558*z^64+z^112-377*z^110-5802958*z^106+61034*z^108-\ 4464295494700040795500*z^30-14200472711221514161062848*z^42+ 33602874772262239248677226*z^44-69536227541392586592185220*z^46-\ 338259557208623787444912264*z^58+361258973881811920889948074*z^56-\ 338259557208623787444912264*z^54+277662597221516679535145746*z^52+ 277662597221516679535145746*z^60-14200472711221514161062848*z^70+ 33602874772262239248677226*z^68-115486294747233708418746*z^78+ 24418940260611371995379*z^32-1688390586898446575050347*z^38+ 5242365703292948722170805*z^40-199773708843660367569388328*z^62+ 473619270946872126151048*z^76-1688390586898446575050347*z^74+ 5242365703292948722170805*z^72+369681579*z^104)/(-1-2697059441345925397876*z^28 +342102558130507221022*z^26+502*z^2-37030914262325775184*z^24+ 3399318060097447107*z^22-95561*z^4+10197788*z^6+1331537684550*z^102-713459098*z ^8+35576944688*z^10-1331537684550*z^12+38736118625700*z^14+16917513315322873*z^ 18-898211940893955*z^16+1544833801274916968383462057*z^50-\ 912852520178457187521416977*z^48-262622846883320545*z^20-\ 2339107087685061458308764*z^36+535394806860594446549479*z^34+ 912852520178457187521416977*z^66-535394806860594446549479*z^80-38736118625700*z ^100+37030914262325775184*z^90-342102558130507221022*z^88-\ 18239941071688123004522*z^84+262622846883320545*z^94+2697059441345925397876*z^ 86-16917513315322873*z^96+898211940893955*z^98-3399318060097447107*z^92+ 106274967503500316489689*z^82-1544833801274916968383462057*z^64-502*z^112+z^114 +95561*z^110+713459098*z^106-10197788*z^108+18239941071688123004522*z^30+ 84877850909150444368360776*z^42-214130710160450879438618752*z^44+ 472526448163686342952160312*z^46+3397305142549576385906604445*z^58-\ 3397305142549576385906604445*z^56+2979408483678917009166840376*z^54-\ 2291271074355142367669381728*z^52-2979408483678917009166840376*z^60+ 214130710160450879438618752*z^70-472526448163686342952160312*z^68+ 2339107087685061458308764*z^78-106274967503500316489689*z^32+ 8884498708869017483965859*z^38-29397265594572471208720095*z^40+ 2291271074355142367669381728*z^62-8884498708869017483965859*z^76+ 29397265594572471208720095*z^74-84877850909150444368360776*z^72-35576944688*z^ 104) The first , 40, terms are: [0, 125, 0, 28223, 0, 6617651, 0, 1555988680, 0, 365942904367, 0, 86065860097815, 0, 20241830618120233, 0, 4760678348256163553, 0, 1119664531661708296425, 0, 263334043036490430106933, 0, 61933567058043165820035327, 0, 14566163514713396344394021571, 0, 3425817850150481264654751452056, 0, 805718536023242795366113190338711, 0, 189497044994492409322896871949305031, 0, 44567834120951796060138791979464776369, 0, 10481914577035047224037597272112073539009, 0, 2465242822931129246792404141083039428114961, 0, 579800773164909999037275052588801560985068761, 0, 136363417605625074316830115808206052776413467495] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 587338867993369993316 z - 82548774613091164400 z - 392 z 24 22 4 + 9900188704772779988 z - 1006334960795844592 z + 65368 z 6 102 8 10 - 6328456 z - 18589350032 z + 405964010 z - 18589350032 z 12 14 18 + 637585891988 z - 16943351744660 z - 6116255260531580 z 16 50 + 357707758965747 z - 121147963397904144825390728 z 48 20 + 77465851800697672709225323 z + 85992359215688964 z 36 34 + 340660599214173761830724 z - 86051394198403704502724 z 66 80 - 43546600135573866180324064 z + 18880578951415881184545 z 100 90 88 + 637585891988 z - 1006334960795844592 z + 9900188704772779988 z 84 94 + 587338867993369993316 z - 6116255260531580 z 86 96 98 - 82548774613091164400 z + 357707758965747 z - 16943351744660 z 92 82 + 85992359215688964 z - 3586216336084903497788 z 64 112 110 106 + 77465851800697672709225323 z + z - 392 z - 6328456 z 108 30 42 + 65368 z - 3586216336084903497788 z - 9316017676220337743740328 z 44 46 + 21502204073871583730626768 z - 43546600135573866180324064 z 58 56 - 201784836805962060658590816 z + 215058804302757204533025176 z 54 52 - 201784836805962060658590816 z + 166665409433067443753600152 z 60 70 + 166665409433067443753600152 z - 9316017676220337743740328 z 68 78 + 21502204073871583730626768 z - 86051394198403704502724 z 32 38 + 18880578951415881184545 z - 1174714768493061884926720 z 40 62 + 3536847993342597751903222 z - 121147963397904144825390728 z 76 74 + 340660599214173761830724 z - 1174714768493061884926720 z 72 104 / + 3536847993342597751903222 z + 405964010 z ) / (-1 / 28 26 2 - 2330877751898270876700 z + 308223320827289062648 z + 517 z 24 22 4 - 34758735295338154504 z + 3319367161097782316 z - 102504 z 6 102 8 10 + 11276088 z + 1494572008564 z - 802489658 z + 40195489566 z 12 14 18 - 1494572008564 z + 42798275612952 z + 17735081813809515 z 16 50 - 969558821340511 z + 938721811480508906296906283 z 48 20 - 564194203722578400838815143 z - 266161373977858488 z 36 34 - 1721223127883854680059032 z + 409343540356430628116661 z 66 80 + 564194203722578400838815143 z - 409343540356430628116661 z 100 90 - 42798275612952 z + 34758735295338154504 z 88 84 - 308223320827289062648 z - 15120836614034898384808 z 94 86 + 266161373977858488 z + 2330877751898270876700 z 96 98 92 - 17735081813809515 z + 969558821340511 z - 3319367161097782316 z 82 64 112 + 84560689317200905725865 z - 938721811480508906296906283 z - 517 z 114 110 106 108 + z + 102504 z + 802489658 z - 11276088 z 30 42 + 15120836614034898384808 z + 56450485864833079946793486 z 44 46 - 138491003799264849659483584 z + 298205650922144177815849384 z 58 56 + 2011497034394677163077353472 z - 2011497034394677163077353472 z 54 52 + 1771785507797560380869062392 z - 1374448632591940530536589024 z 60 70 - 1771785507797560380869062392 z + 138491003799264849659483584 z 68 78 - 298205650922144177815849384 z + 1721223127883854680059032 z 32 38 - 84560689317200905725865 z + 6304879816533517769228628 z 40 62 - 20167921685305966746471322 z + 1374448632591940530536589024 z 76 74 - 6304879816533517769228628 z + 20167921685305966746471322 z 72 104 - 56450485864833079946793486 z - 40195489566 z ) And in Maple-input format, it is: -(1+587338867993369993316*z^28-82548774613091164400*z^26-392*z^2+ 9900188704772779988*z^24-1006334960795844592*z^22+65368*z^4-6328456*z^6-\ 18589350032*z^102+405964010*z^8-18589350032*z^10+637585891988*z^12-\ 16943351744660*z^14-6116255260531580*z^18+357707758965747*z^16-\ 121147963397904144825390728*z^50+77465851800697672709225323*z^48+ 85992359215688964*z^20+340660599214173761830724*z^36-86051394198403704502724*z^ 34-43546600135573866180324064*z^66+18880578951415881184545*z^80+637585891988*z^ 100-1006334960795844592*z^90+9900188704772779988*z^88+587338867993369993316*z^ 84-6116255260531580*z^94-82548774613091164400*z^86+357707758965747*z^96-\ 16943351744660*z^98+85992359215688964*z^92-3586216336084903497788*z^82+ 77465851800697672709225323*z^64+z^112-392*z^110-6328456*z^106+65368*z^108-\ 3586216336084903497788*z^30-9316017676220337743740328*z^42+ 21502204073871583730626768*z^44-43546600135573866180324064*z^46-\ 201784836805962060658590816*z^58+215058804302757204533025176*z^56-\ 201784836805962060658590816*z^54+166665409433067443753600152*z^52+ 166665409433067443753600152*z^60-9316017676220337743740328*z^70+ 21502204073871583730626768*z^68-86051394198403704502724*z^78+ 18880578951415881184545*z^32-1174714768493061884926720*z^38+ 3536847993342597751903222*z^40-121147963397904144825390728*z^62+ 340660599214173761830724*z^76-1174714768493061884926720*z^74+ 3536847993342597751903222*z^72+405964010*z^104)/(-1-2330877751898270876700*z^28 +308223320827289062648*z^26+517*z^2-34758735295338154504*z^24+ 3319367161097782316*z^22-102504*z^4+11276088*z^6+1494572008564*z^102-802489658* z^8+40195489566*z^10-1494572008564*z^12+42798275612952*z^14+17735081813809515*z ^18-969558821340511*z^16+938721811480508906296906283*z^50-\ 564194203722578400838815143*z^48-266161373977858488*z^20-\ 1721223127883854680059032*z^36+409343540356430628116661*z^34+ 564194203722578400838815143*z^66-409343540356430628116661*z^80-42798275612952*z ^100+34758735295338154504*z^90-308223320827289062648*z^88-\ 15120836614034898384808*z^84+266161373977858488*z^94+2330877751898270876700*z^ 86-17735081813809515*z^96+969558821340511*z^98-3319367161097782316*z^92+ 84560689317200905725865*z^82-938721811480508906296906283*z^64-517*z^112+z^114+ 102504*z^110+802489658*z^106-11276088*z^108+15120836614034898384808*z^30+ 56450485864833079946793486*z^42-138491003799264849659483584*z^44+ 298205650922144177815849384*z^46+2011497034394677163077353472*z^58-\ 2011497034394677163077353472*z^56+1771785507797560380869062392*z^54-\ 1374448632591940530536589024*z^52-1771785507797560380869062392*z^60+ 138491003799264849659483584*z^70-298205650922144177815849384*z^68+ 1721223127883854680059032*z^78-84560689317200905725865*z^32+ 6304879816533517769228628*z^38-20167921685305966746471322*z^40+ 1374448632591940530536589024*z^62-6304879816533517769228628*z^76+ 20167921685305966746471322*z^74-56450485864833079946793486*z^72-40195489566*z^ 104) The first , 40, terms are: [0, 125, 0, 27489, 0, 6346445, 0, 1476364961, 0, 344008001873, 0, 80189707183569, 0, 18694514368179505, 0, 4358347179616808105, 0, 1016091221296649571937, 0, 236888758271736573143273, 0, 55227634362183556747002561, 0, 12875630197490848635649279277, 0, 3001791794542401102013371160665, 0, 699830139437182093538230276164461, 0, 163156627456136828090960502778938169, 0, 38037923210664347016800057249744248873, 0, 8868065153455869746714247627208615045229, 0, 2067478267283606915893142678166784279108681, 0, 482006651033000107934089970157309962248276493, 0, 112373810800115545571349098073918002856353425377] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 7}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 6457441672946300160 z - 1490994688750118016 z - 342 z 24 22 4 6 + 285630739338108725 z - 45124823424371646 z + 47948 z - 3784422 z 8 10 12 14 + 192394007 z - 6795998464 z + 175170767840 z - 3409705331312 z 18 16 50 - 611784273162846 z + 51402456661011 z - 687991977403729635868 z 48 20 + 1046416588294267163622 z + 5834965645501308 z 36 34 + 381975200765991812024 z - 178835843943915091340 z 66 80 88 84 86 - 45124823424371646 z + 192394007 z + z + 47948 z - 342 z 82 64 30 - 3784422 z + 285630739338108725 z - 23317195486660380000 z 42 44 - 1345287859949765349552 z + 1462711621036317976896 z 46 58 - 1345287859949765349552 z - 23317195486660380000 z 56 54 + 70473052963863130042 z - 178835843943915091340 z 52 60 + 381975200765991812024 z + 6457441672946300160 z 70 68 78 - 611784273162846 z + 5834965645501308 z - 6795998464 z 32 38 + 70473052963863130042 z - 687991977403729635868 z 40 62 76 + 1046416588294267163622 z - 1490994688750118016 z + 175170767840 z 74 72 / - 3409705331312 z + 51402456661011 z ) / (-1 / 28 26 2 - 31073444700092195832 z + 6654333463715889373 z + 455 z 24 22 4 6 - 1181927198077195515 z + 172986222010938712 z - 76288 z + 6916140 z 8 10 12 14 - 395837453 z + 15550840903 z - 442023040648 z + 9427601613856 z 18 16 50 + 2003278620838249 z - 154960823391275 z + 7994474726782424142042 z 48 20 - 11141201106918344005790 z - 20694323731812660 z 36 34 - 2492757952570570081032 z + 1080359318822508515310 z 66 80 90 88 84 + 1181927198077195515 z - 15550840903 z + z - 455 z - 6916140 z 86 82 64 + 76288 z + 395837453 z - 6654333463715889373 z 30 42 + 120985202229483115656 z + 11141201106918344005790 z 44 46 - 13150090124994859971032 z + 13150090124994859971032 z 58 56 + 394430279819497168146 z - 1080359318822508515310 z 54 52 + 2492757952570570081032 z - 4855095800802144755520 z 60 70 - 120985202229483115656 z + 20694323731812660 z 68 78 32 - 172986222010938712 z + 442023040648 z - 394430279819497168146 z 38 40 + 4855095800802144755520 z - 7994474726782424142042 z 62 76 74 + 31073444700092195832 z - 9427601613856 z + 154960823391275 z 72 - 2003278620838249 z ) And in Maple-input format, it is: -(1+6457441672946300160*z^28-1490994688750118016*z^26-342*z^2+ 285630739338108725*z^24-45124823424371646*z^22+47948*z^4-3784422*z^6+192394007* z^8-6795998464*z^10+175170767840*z^12-3409705331312*z^14-611784273162846*z^18+ 51402456661011*z^16-687991977403729635868*z^50+1046416588294267163622*z^48+ 5834965645501308*z^20+381975200765991812024*z^36-178835843943915091340*z^34-\ 45124823424371646*z^66+192394007*z^80+z^88+47948*z^84-342*z^86-3784422*z^82+ 285630739338108725*z^64-23317195486660380000*z^30-1345287859949765349552*z^42+ 1462711621036317976896*z^44-1345287859949765349552*z^46-23317195486660380000*z^ 58+70473052963863130042*z^56-178835843943915091340*z^54+381975200765991812024*z ^52+6457441672946300160*z^60-611784273162846*z^70+5834965645501308*z^68-\ 6795998464*z^78+70473052963863130042*z^32-687991977403729635868*z^38+ 1046416588294267163622*z^40-1490994688750118016*z^62+175170767840*z^76-\ 3409705331312*z^74+51402456661011*z^72)/(-1-31073444700092195832*z^28+ 6654333463715889373*z^26+455*z^2-1181927198077195515*z^24+172986222010938712*z^ 22-76288*z^4+6916140*z^6-395837453*z^8+15550840903*z^10-442023040648*z^12+ 9427601613856*z^14+2003278620838249*z^18-154960823391275*z^16+ 7994474726782424142042*z^50-11141201106918344005790*z^48-20694323731812660*z^20 -2492757952570570081032*z^36+1080359318822508515310*z^34+1181927198077195515*z^ 66-15550840903*z^80+z^90-455*z^88-6916140*z^84+76288*z^86+395837453*z^82-\ 6654333463715889373*z^64+120985202229483115656*z^30+11141201106918344005790*z^ 42-13150090124994859971032*z^44+13150090124994859971032*z^46+ 394430279819497168146*z^58-1080359318822508515310*z^56+2492757952570570081032*z ^54-4855095800802144755520*z^52-120985202229483115656*z^60+20694323731812660*z^ 70-172986222010938712*z^68+442023040648*z^78-394430279819497168146*z^32+ 4855095800802144755520*z^38-7994474726782424142042*z^40+31073444700092195832*z^ 62-9427601613856*z^76+154960823391275*z^74-2003278620838249*z^72) The first , 40, terms are: [0, 113, 0, 23075, 0, 5010299, 0, 1097420819, 0, 240715923283, 0, 52814078501009, 0, 11588226337261849, 0, 2542663768139663177, 0, 557907158391514590417, 0, 122415149539681721882483, 0, 26860151345584120316766595, 0, 5893614918077915619293435275, 0, 1293168329481801541538355232195, 0, 283745096076466656016005592760817, 0, 62258932378903792873694377873835729, 0, 13660763533658264445775749870708079025, 0, 2997424677814115983034515327717836023793, 0, 657690521988987352519617695668825569613187, 0, 144309488714219303248287716067509606622614763, 0, 31664176138629228366312068634923128931798417187] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 15439671885420 z - 17929169965688 z - 282 z 24 22 4 6 + 15439671885420 z - 9844389440960 z + 28289 z - 1414880 z 8 10 12 14 + 40696250 z - 731359268 z + 8656282042 z - 69897763680 z 18 16 50 48 - 1589123239302 z + 394783287695 z - 282 z + 28289 z 20 36 34 + 4625575759375 z + 394783287695 z - 1589123239302 z 30 42 44 46 52 - 9844389440960 z - 731359268 z + 40696250 z - 1414880 z + z 32 38 40 / 2 + 4625575759375 z - 69897763680 z + 8656282042 z ) / ((-1 + z ) (1 / 28 26 2 24 + 71885553914124 z - 84277855682952 z - 390 z + 71885553914124 z 22 4 6 8 - 44572364788992 z + 50225 z - 3050112 z + 102728522 z 10 12 14 18 - 2101083868 z + 27710424202 z - 245535662208 z - 6493023370970 z 16 50 48 20 + 1503792503039 z - 390 z + 50225 z + 20034651343343 z 36 34 30 + 1503792503039 z - 6493023370970 z - 44572364788992 z 42 44 46 52 32 - 2101083868 z + 102728522 z - 3050112 z + z + 20034651343343 z 38 40 - 245535662208 z + 27710424202 z )) And in Maple-input format, it is: -(1+15439671885420*z^28-17929169965688*z^26-282*z^2+15439671885420*z^24-\ 9844389440960*z^22+28289*z^4-1414880*z^6+40696250*z^8-731359268*z^10+8656282042 *z^12-69897763680*z^14-1589123239302*z^18+394783287695*z^16-282*z^50+28289*z^48 +4625575759375*z^20+394783287695*z^36-1589123239302*z^34-9844389440960*z^30-\ 731359268*z^42+40696250*z^44-1414880*z^46+z^52+4625575759375*z^32-69897763680*z ^38+8656282042*z^40)/(-1+z^2)/(1+71885553914124*z^28-84277855682952*z^26-390*z^ 2+71885553914124*z^24-44572364788992*z^22+50225*z^4-3050112*z^6+102728522*z^8-\ 2101083868*z^10+27710424202*z^12-245535662208*z^14-6493023370970*z^18+ 1503792503039*z^16-390*z^50+50225*z^48+20034651343343*z^20+1503792503039*z^36-\ 6493023370970*z^34-44572364788992*z^30-2101083868*z^42+102728522*z^44-3050112*z ^46+z^52+20034651343343*z^32-245535662208*z^38+27710424202*z^40) The first , 40, terms are: [0, 109, 0, 20293, 0, 4102985, 0, 849991289, 0, 177531728981, 0, 37185727649501, 0, 7796778201696849, 0, 1635347615979682993, 0, 343052478566978450685, 0, 71966577578334096264501, 0, 15097609921890698736299801, 0, 3167291739640261486311084201, 0, 664459987263944463788299586661, 0, 139395873911351656970040919468109, 0, 29243619089497003187450643451653025, 0, 6134968801764449591960289481625006177, 0, 1287044648615247480682423523707897971853, 0, 270006906436193344289836197921279316991525, 0, 56644289622017670632570090201658796617490921, 0, 11883309186373537695812158385820295768925518297] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4920558460893749413 z - 1135094160404791153 z - 329 z 24 22 4 6 + 217472084217449952 z - 34409917436711730 z + 44056 z - 3329490 z 8 10 12 14 + 163030711 z - 5584496108 z + 140506111805 z - 2685268718948 z 18 16 50 - 470990295526112 z + 39943161654361 z - 529071732301334803150 z 48 20 + 805741028601630771872 z + 4465026664230471 z 36 34 + 293249810634887898132 z - 137029694111978636420 z 66 80 88 84 86 - 34409917436711730 z + 163030711 z + z + 44056 z - 329 z 82 64 30 - 3329490 z + 217472084217449952 z - 17795634003917405074 z 42 44 - 1036710734184614618658 z + 1127511222131199837456 z 46 58 - 1036710734184614618658 z - 17795634003917405074 z 56 54 + 53888786404754260166 z - 137029694111978636420 z 52 60 + 293249810634887898132 z + 4920558460893749413 z 70 68 78 - 470990295526112 z + 4465026664230471 z - 5584496108 z 32 38 + 53888786404754260166 z - 529071732301334803150 z 40 62 76 + 805741028601630771872 z - 1135094160404791153 z + 140506111805 z 74 72 / - 2685268718948 z + 39943161654361 z ) / (-1 / 28 26 2 - 23579539788271530520 z + 5037313849622334477 z + 439 z 24 22 4 6 - 893587258523175560 z + 130842087898709853 z - 70362 z + 6101775 z 8 10 12 14 - 335564634 z + 12746790309 z - 352710618800 z + 7370777366703 z 18 16 50 + 1528379584186663 z - 119390405687972 z + 6179695866280531462003 z 48 20 - 8625731185156428569703 z - 15696158902333374 z 36 34 - 1917034452047378243807 z + 828125218040503863247 z 66 80 90 88 84 + 893587258523175560 z - 12746790309 z + z - 439 z - 6101775 z 86 82 64 + 70362 z + 335564634 z - 5037313849622334477 z 30 42 + 92095271141438248966 z + 8625731185156428569703 z 44 46 - 10189300861220124185243 z + 10189300861220124185243 z 58 56 + 301286936143278742351 z - 828125218040503863247 z 54 52 + 1917034452047378243807 z - 3744439138517707910631 z 60 70 - 92095271141438248966 z + 15696158902333374 z 68 78 32 - 130842087898709853 z + 352710618800 z - 301286936143278742351 z 38 40 + 3744439138517707910631 z - 6179695866280531462003 z 62 76 74 + 23579539788271530520 z - 7370777366703 z + 119390405687972 z 72 - 1528379584186663 z ) And in Maple-input format, it is: -(1+4920558460893749413*z^28-1135094160404791153*z^26-329*z^2+ 217472084217449952*z^24-34409917436711730*z^22+44056*z^4-3329490*z^6+163030711* z^8-5584496108*z^10+140506111805*z^12-2685268718948*z^14-470990295526112*z^18+ 39943161654361*z^16-529071732301334803150*z^50+805741028601630771872*z^48+ 4465026664230471*z^20+293249810634887898132*z^36-137029694111978636420*z^34-\ 34409917436711730*z^66+163030711*z^80+z^88+44056*z^84-329*z^86-3329490*z^82+ 217472084217449952*z^64-17795634003917405074*z^30-1036710734184614618658*z^42+ 1127511222131199837456*z^44-1036710734184614618658*z^46-17795634003917405074*z^ 58+53888786404754260166*z^56-137029694111978636420*z^54+293249810634887898132*z ^52+4920558460893749413*z^60-470990295526112*z^70+4465026664230471*z^68-\ 5584496108*z^78+53888786404754260166*z^32-529071732301334803150*z^38+ 805741028601630771872*z^40-1135094160404791153*z^62+140506111805*z^76-\ 2685268718948*z^74+39943161654361*z^72)/(-1-23579539788271530520*z^28+ 5037313849622334477*z^26+439*z^2-893587258523175560*z^24+130842087898709853*z^ 22-70362*z^4+6101775*z^6-335564634*z^8+12746790309*z^10-352710618800*z^12+ 7370777366703*z^14+1528379584186663*z^18-119390405687972*z^16+ 6179695866280531462003*z^50-8625731185156428569703*z^48-15696158902333374*z^20-\ 1917034452047378243807*z^36+828125218040503863247*z^34+893587258523175560*z^66-\ 12746790309*z^80+z^90-439*z^88-6101775*z^84+70362*z^86+335564634*z^82-\ 5037313849622334477*z^64+92095271141438248966*z^30+8625731185156428569703*z^42-\ 10189300861220124185243*z^44+10189300861220124185243*z^46+301286936143278742351 *z^58-828125218040503863247*z^56+1917034452047378243807*z^54-\ 3744439138517707910631*z^52-92095271141438248966*z^60+15696158902333374*z^70-\ 130842087898709853*z^68+352710618800*z^78-301286936143278742351*z^32+ 3744439138517707910631*z^38-6179695866280531462003*z^40+23579539788271530520*z^ 62-7370777366703*z^76+119390405687972*z^74-1528379584186663*z^72) The first , 40, terms are: [0, 110, 0, 21984, 0, 4683441, 0, 1007853718, 0, 217303112621, 0, 46871655855617, 0, 10110987544233728, 0, 2181150906986654089, 0, 470521965281009660441, 0, 101501992935922670751494, 0, 21896230149098281269635977, 0, 4723502566227505651227600032, 0, 1018964299557766594221655165046, 0, 219813206998305644502700162846405, 0, 47418585745704373814179777484417693, 0, 10229241023617062491962985514895123478, 0, 2206674245521318044106390378903440769808, 0, 476028594365728771753063738773727204865793, 0, 102689929478408894177600998502071304264435686, 0, 22152496175860740949272891112254849878309534337] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {3, 7}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 11595213877743408 z - 5398581888876634 z - 295 z 24 22 4 6 + 2007195176832974 z - 592808931027040 z + 33240 z - 1985951 z 8 10 12 14 + 72764645 z - 1776096336 z + 30445175157 z - 379953093863 z 18 16 50 - 25159995248287 z + 3542726737656 z - 25159995248287 z 48 20 36 + 138116974016977 z + 138116974016977 z + 27632631297962374 z 34 66 64 30 - 30789145876739232 z - 295 z + 33240 z - 19965225094296538 z 42 44 46 - 5398581888876634 z + 2007195176832974 z - 592808931027040 z 58 56 54 52 - 1776096336 z + 30445175157 z - 379953093863 z + 3542726737656 z 60 68 32 38 + 72764645 z + z + 27632631297962374 z - 19965225094296538 z 40 62 / 28 + 11595213877743408 z - 1985951 z ) / (-1 - 77635796257520778 z / 26 2 24 + 32571116323624586 z + 421 z - 10919561676953522 z 22 4 6 8 + 2909656597849677 z - 57917 z + 4026723 z - 168188103 z 10 12 14 + 4627171847 z - 88760435035 z + 1233772975787 z 18 16 50 + 100593233341433 z - 12775270955207 z + 611849029513201 z 48 20 36 - 2909656597849677 z - 611849029513201 z - 283159478700134558 z 34 66 64 + 283159478700134558 z + 57917 z - 4026723 z 30 42 44 + 148460457772931478 z + 77635796257520778 z - 32571116323624586 z 46 58 56 + 10919561676953522 z + 88760435035 z - 1233772975787 z 54 52 60 70 + 12775270955207 z - 100593233341433 z - 4627171847 z + z 68 32 38 - 421 z - 228391731320288686 z + 228391731320288686 z 40 62 - 148460457772931478 z + 168188103 z ) And in Maple-input format, it is: -(1+11595213877743408*z^28-5398581888876634*z^26-295*z^2+2007195176832974*z^24-\ 592808931027040*z^22+33240*z^4-1985951*z^6+72764645*z^8-1776096336*z^10+ 30445175157*z^12-379953093863*z^14-25159995248287*z^18+3542726737656*z^16-\ 25159995248287*z^50+138116974016977*z^48+138116974016977*z^20+27632631297962374 *z^36-30789145876739232*z^34-295*z^66+33240*z^64-19965225094296538*z^30-\ 5398581888876634*z^42+2007195176832974*z^44-592808931027040*z^46-1776096336*z^ 58+30445175157*z^56-379953093863*z^54+3542726737656*z^52+72764645*z^60+z^68+ 27632631297962374*z^32-19965225094296538*z^38+11595213877743408*z^40-1985951*z^ 62)/(-1-77635796257520778*z^28+32571116323624586*z^26+421*z^2-10919561676953522 *z^24+2909656597849677*z^22-57917*z^4+4026723*z^6-168188103*z^8+4627171847*z^10 -88760435035*z^12+1233772975787*z^14+100593233341433*z^18-12775270955207*z^16+ 611849029513201*z^50-2909656597849677*z^48-611849029513201*z^20-\ 283159478700134558*z^36+283159478700134558*z^34+57917*z^66-4026723*z^64+ 148460457772931478*z^30+77635796257520778*z^42-32571116323624586*z^44+ 10919561676953522*z^46+88760435035*z^58-1233772975787*z^56+12775270955207*z^54-\ 100593233341433*z^52-4627171847*z^60+z^70-421*z^68-228391731320288686*z^32+ 228391731320288686*z^38-148460457772931478*z^40+168188103*z^62) The first , 40, terms are: [0, 126, 0, 28369, 0, 6686579, 0, 1583946026, 0, 375468160323, 0, 89013075057595, 0, 21102962257098522, 0, 5003050369871496107, 0, 1186114813137430309513, 0, 281202174774043764012270, 0, 66666958415776078240774201, 0, 15805295240765503634551176329, 0, 3747093976500062184312741741198, 0, 888355014166525878661252258772057, 0, 210609778207021009167590780681171707, 0, 49931027540355611589273735889796938682, 0, 11837567716375210765269981998553464371211, 0, 2806431518493939847031046388012064844547443, 0, 665344271450688069310545082015899198803911050, 0, 157738749951703606663091704225759816643843138979] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 7}, {3, 7}, {4, 5}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 475425 z + 7354052 z + 228 z - 62252159 z + 298290672 z 4 6 8 10 12 - 15743 z + 475425 z - 7354052 z + 62252159 z - 298290672 z 14 18 16 20 34 + 833309624 z + 1383993168 z - 1383993168 z - 833309624 z + z 30 32 / 12 6 4 + 15743 z - 228 z ) / (1 + 1414096579 z - 1218746 z + 30780 z / 32 34 36 24 26 + 30780 z - 328 z + z + 1414096579 z - 243528424 z 28 30 22 18 + 23619108 z - 1218746 z - 4828986576 z - 12684266772 z 20 14 16 10 + 9977346592 z - 4828986576 z + 9977346592 z - 243528424 z 2 8 - 328 z + 23619108 z ) And in Maple-input format, it is: -(-1-475425*z^28+7354052*z^26+228*z^2-62252159*z^24+298290672*z^22-15743*z^4+ 475425*z^6-7354052*z^8+62252159*z^10-298290672*z^12+833309624*z^14+1383993168*z ^18-1383993168*z^16-833309624*z^20+z^34+15743*z^30-228*z^32)/(1+1414096579*z^12 -1218746*z^6+30780*z^4+30780*z^32-328*z^34+z^36+1414096579*z^24-243528424*z^26+ 23619108*z^28-1218746*z^30-4828986576*z^22-12684266772*z^18+9977346592*z^20-\ 4828986576*z^14+9977346592*z^16-243528424*z^10-328*z^2+23619108*z^8) The first , 40, terms are: [0, 100, 0, 17763, 0, 3491585, 0, 704104284, 0, 142943169515, 0, 29072075812899, 0, 5915694646875420, 0, 1203916107500859497, 0, 245021304738551344043, 0, 49867354593243225117092, 0, 10149162163179658069891033, 0, 2065591538002722512035537641, 0, 420396228764491091173331427876, 0, 85560479328038091894218955337563, 0, 17413561955081920314369261445435481, 0, 3544067823630128954867834109611325084, 0, 721300833918126893075860930494103250419, 0, 146801618691376777087470406512428874484891, 0, 29877568747968504527190631312458356360589276, 0, 6080785227586721807326670246283557710741088913] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 991491319941 z + 2073943508731 z + 305 z 24 22 4 6 - 2991474000035 z + 2991474000035 z - 34407 z + 1874315 z 8 10 12 14 - 55346297 z + 958528625 z - 10287905947 z + 71020927435 z 18 16 20 36 + 991491319941 z - 323536562617 z - 2073943508731 z - 958528625 z 34 30 42 44 46 + 10287905947 z + 323536562617 z + 34407 z - 305 z + z 32 38 40 / - 71020927435 z + 55346297 z - 1874315 z ) / (1 / 28 26 2 24 + 14719676546248 z - 25505637642051 z - 441 z + 30622121914162 z 22 4 6 8 - 25505637642051 z + 63644 z - 4164185 z + 144255094 z 10 12 14 18 - 2912424145 z + 36505320738 z - 296305508651 z - 5863571049859 z 16 48 20 36 + 1601099657054 z + z + 14719676546248 z + 36505320738 z 34 30 42 44 46 - 296305508651 z - 5863571049859 z - 4164185 z + 63644 z - 441 z 32 38 40 + 1601099657054 z - 2912424145 z + 144255094 z ) And in Maple-input format, it is: -(-1-991491319941*z^28+2073943508731*z^26+305*z^2-2991474000035*z^24+ 2991474000035*z^22-34407*z^4+1874315*z^6-55346297*z^8+958528625*z^10-\ 10287905947*z^12+71020927435*z^14+991491319941*z^18-323536562617*z^16-\ 2073943508731*z^20-958528625*z^36+10287905947*z^34+323536562617*z^30+34407*z^42 -305*z^44+z^46-71020927435*z^32+55346297*z^38-1874315*z^40)/(1+14719676546248*z ^28-25505637642051*z^26-441*z^2+30622121914162*z^24-25505637642051*z^22+63644*z ^4-4164185*z^6+144255094*z^8-2912424145*z^10+36505320738*z^12-296305508651*z^14 -5863571049859*z^18+1601099657054*z^16+z^48+14719676546248*z^20+36505320738*z^ 36-296305508651*z^34-5863571049859*z^30-4164185*z^42+63644*z^44-441*z^46+ 1601099657054*z^32-2912424145*z^38+144255094*z^40) The first , 40, terms are: [0, 136, 0, 30739, 0, 7190185, 0, 1691939032, 0, 398871064423, 0, 94097247116623, 0, 22204247845963864, 0, 5240107935213504601, 0, 1236693670949377159891, 0, 291871067050198779667720, 0, 68884691752468044425230777, 0, 16257564500696872539807898921, 0, 3836972476905320553245395255048, 0, 905570054727272573304323862186355, 0, 213725080472328432303580982200019545, 0, 50441611434314579107089768454677387352, 0, 11904808832925782728742800320090119792287, 0, 2809673812505905681601992397623258308170071, 0, 663115810878726663056208250798689735945022168, 0, 156503070690568175477470487456318945723740553065] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 102613451 z + 644342449 z + 222 z - 2490995544 z 22 4 6 8 10 + 6054858075 z - 16122 z + 542879 z - 9820140 z + 102613451 z 12 14 18 16 - 644342449 z + 2490995544 z + 9400624077 z - 6054858075 z 20 36 34 30 32 38 - 9400624077 z - 222 z + 16122 z + 9820140 z - 542879 z + z ) / 40 38 36 34 32 / (z - 350 z + 32194 z - 1324680 z + 28943442 z / 30 28 26 24 - 363449910 z + 2755552272 z - 13014028506 z + 39055368074 z 22 20 18 16 - 75297474736 z + 93688804423 z - 75297474736 z + 39055368074 z 14 12 10 8 - 13014028506 z + 2755552272 z - 363449910 z + 28943442 z 6 4 2 - 1324680 z + 32194 z - 350 z + 1) And in Maple-input format, it is: -(-1-102613451*z^28+644342449*z^26+222*z^2-2490995544*z^24+6054858075*z^22-\ 16122*z^4+542879*z^6-9820140*z^8+102613451*z^10-644342449*z^12+2490995544*z^14+ 9400624077*z^18-6054858075*z^16-9400624077*z^20-222*z^36+16122*z^34+9820140*z^ 30-542879*z^32+z^38)/(z^40-350*z^38+32194*z^36-1324680*z^34+28943442*z^32-\ 363449910*z^30+2755552272*z^28-13014028506*z^26+39055368074*z^24-75297474736*z^ 22+93688804423*z^20-75297474736*z^18+39055368074*z^16-13014028506*z^14+ 2755552272*z^12-363449910*z^10+28943442*z^8-1324680*z^6+32194*z^4-350*z^2+1) The first , 40, terms are: [0, 128, 0, 28728, 0, 6715769, 0, 1576085656, 0, 370033995337, 0, 86880564814487, 0, 20398853916800880, 0, 4789488573342362273, 0, 1124533878259849145383, 0, 264031625601307930267576, 0, 61992529298422007742867839, 0, 14555353665224008634402918712, 0, 3417481472729148623359560958304, 0, 802397515382811673820667616992319, 0, 188396565667050818041671854039542111, 0, 44234017771366454450164007940092684576, 0, 10385796159657955149339592201385338592856, 0, 2438502476250061632934928953240758638438447, 0, 572541019991819394287455886931766927744218904, 0, 134428085583644766600671466990255450899043139111] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 6}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 22378422963626768 z + 11028981785144024 z + 334 z 24 22 4 6 - 4271390205659626 z + 1293759442085892 z - 43578 z + 2983857 z 8 10 12 14 - 122481420 z + 3272201160 z - 60122543957 z + 789665425446 z 18 16 50 + 55205922166629 z - 7623229108938 z + 7623229108938 z 48 20 36 - 55205922166629 z - 304489954274988 z - 35802770218159690 z 34 66 64 30 + 45261849795826268 z + z - 334 z + 35802770218159690 z 42 44 46 + 4271390205659626 z - 1293759442085892 z + 304489954274988 z 58 56 54 52 + 122481420 z - 3272201160 z + 60122543957 z - 789665425446 z 60 32 38 - 2983857 z - 45261849795826268 z + 22378422963626768 z 40 62 / 28 - 11028981785144024 z + 43578 z ) / (1 + 161382131454165422 z / 26 2 24 - 70609879319392380 z - 470 z + 24282537559006746 z 22 4 6 8 - 6536761643420416 z + 76731 z - 6143078 z + 285503161 z 10 12 14 - 8520491472 z + 174127342605 z - 2543921348286 z 18 16 50 - 221555355386446 z + 27372791072967 z - 221555355386446 z 48 20 36 + 1369544905514357 z + 1369544905514357 z + 413599022711595722 z 34 66 64 30 - 465114674378367072 z - 470 z + 76731 z - 290732698875826300 z 42 44 46 - 70609879319392380 z + 24282537559006746 z - 6536761643420416 z 58 56 54 - 8520491472 z + 174127342605 z - 2543921348286 z 52 60 68 32 + 27372791072967 z + 285503161 z + z + 413599022711595722 z 38 40 62 - 290732698875826300 z + 161382131454165422 z - 6143078 z ) And in Maple-input format, it is: -(-1-22378422963626768*z^28+11028981785144024*z^26+334*z^2-4271390205659626*z^ 24+1293759442085892*z^22-43578*z^4+2983857*z^6-122481420*z^8+3272201160*z^10-\ 60122543957*z^12+789665425446*z^14+55205922166629*z^18-7623229108938*z^16+ 7623229108938*z^50-55205922166629*z^48-304489954274988*z^20-35802770218159690*z ^36+45261849795826268*z^34+z^66-334*z^64+35802770218159690*z^30+ 4271390205659626*z^42-1293759442085892*z^44+304489954274988*z^46+122481420*z^58 -3272201160*z^56+60122543957*z^54-789665425446*z^52-2983857*z^60-\ 45261849795826268*z^32+22378422963626768*z^38-11028981785144024*z^40+43578*z^62 )/(1+161382131454165422*z^28-70609879319392380*z^26-470*z^2+24282537559006746*z ^24-6536761643420416*z^22+76731*z^4-6143078*z^6+285503161*z^8-8520491472*z^10+ 174127342605*z^12-2543921348286*z^14-221555355386446*z^18+27372791072967*z^16-\ 221555355386446*z^50+1369544905514357*z^48+1369544905514357*z^20+ 413599022711595722*z^36-465114674378367072*z^34-470*z^66+76731*z^64-\ 290732698875826300*z^30-70609879319392380*z^42+24282537559006746*z^44-\ 6536761643420416*z^46-8520491472*z^58+174127342605*z^56-2543921348286*z^54+ 27372791072967*z^52+285503161*z^60+z^68+413599022711595722*z^32-\ 290732698875826300*z^38+161382131454165422*z^40-6143078*z^62) The first , 40, terms are: [0, 136, 0, 30767, 0, 7184295, 0, 1688272840, 0, 397654036397, 0, 93748924667617, 0, 22109878434794392, 0, 5215189227904914011, 0, 1230209971825402690499, 0, 290200792106010587205880, 0, 68457654258207653817210645, 0, 16149053043622652718739577229, 0, 3809541937893840260146252818328, 0, 898666840044867758317783442071483, 0, 211994590412334302388754130973012307, 0, 50009312310357897456956093732377707896, 0, 11797147316925587544128441854811751584473, 0, 2782935427775182120583987263643337895118581, 0, 656491729271165970636590351245790360652825064, 0, 154865753389167309383742421855914912117644503023] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 824260041 z - 3461207222 z - 234 z + 9472326567 z 22 4 6 8 10 - 17197516984 z + 18191 z - 634052 z + 11671552 z - 125009860 z 12 14 18 16 + 824260041 z - 3461207222 z - 17197516984 z + 9472326567 z 20 36 34 30 32 + 20952225472 z + 18191 z - 634052 z - 125009860 z + 11671552 z 38 40 / 2 40 38 36 34 - 234 z + z ) / ((-1 + z ) (z - 365 z + 35335 z - 1465768 z / 32 30 28 26 + 31799114 z - 397533608 z + 3027754969 z - 14480014887 z 24 22 20 18 + 44098790527 z - 85986228816 z + 107447149940 z - 85986228816 z 16 14 12 10 + 44098790527 z - 14480014887 z + 3027754969 z - 397533608 z 8 6 4 2 + 31799114 z - 1465768 z + 35335 z - 365 z + 1)) And in Maple-input format, it is: -(1+824260041*z^28-3461207222*z^26-234*z^2+9472326567*z^24-17197516984*z^22+ 18191*z^4-634052*z^6+11671552*z^8-125009860*z^10+824260041*z^12-3461207222*z^14 -17197516984*z^18+9472326567*z^16+20952225472*z^20+18191*z^36-634052*z^34-\ 125009860*z^30+11671552*z^32-234*z^38+z^40)/(-1+z^2)/(z^40-365*z^38+35335*z^36-\ 1465768*z^34+31799114*z^32-397533608*z^30+3027754969*z^28-14480014887*z^26+ 44098790527*z^24-85986228816*z^22+107447149940*z^20-85986228816*z^18+ 44098790527*z^16-14480014887*z^14+3027754969*z^12-397533608*z^10+31799114*z^8-\ 1465768*z^6+35335*z^4-365*z^2+1) The first , 40, terms are: [0, 132, 0, 30803, 0, 7428549, 0, 1795734100, 0, 434191315447, 0, 104986744003671, 0, 25385777026988004, 0, 6138284706115901797, 0, 1484238610242749529267, 0, 358889251643956699977940, 0, 86779508448095076672087969, 0, 20983306354779601322870781409, 0, 5073768610083998509927405108404, 0, 1226838491448694429664084696634995, 0, 296649847448686903002984754184567717, 0, 71730005706280819985899469851077747268, 0, 17344332932873977017223163327880030580247, 0, 4193863947514171788186183406760884592667959, 0, 1014077328792786897742795557236631459703684852, 0, 245204146257792372728027719366874987196319222405] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 19350855678908 z + 19350855678908 z + 262 z 24 22 4 6 - 14666005650097 z + 8401810171474 z - 24431 z + 1126357 z 8 10 12 14 - 30185499 z + 515556492 z - 5938156592 z + 47837594204 z 18 16 50 48 + 1163305703753 z - 276236760619 z + 24431 z - 1126357 z 20 36 34 - 3619152269135 z - 1163305703753 z + 3619152269135 z 30 42 44 46 54 + 14666005650097 z + 5938156592 z - 515556492 z + 30185499 z + z 52 32 38 40 / - 262 z - 8401810171474 z + 276236760619 z - 47837594204 z ) / (1 / 28 26 2 24 + 198315726539767 z - 172574326223710 z - 398 z + 113613671785833 z 22 4 6 8 - 56433750740928 z + 46823 z - 2536398 z + 77940937 z 10 12 14 18 - 1515443124 z + 19869933965 z - 182785047482 z - 5866264340222 z 16 50 48 20 + 1210261163403 z - 2536398 z + 77940937 z + 21056900282529 z 36 34 30 + 21056900282529 z - 56433750740928 z - 172574326223710 z 42 44 46 56 54 - 182785047482 z + 19869933965 z - 1515443124 z + z - 398 z 52 32 38 40 + 46823 z + 113613671785833 z - 5866264340222 z + 1210261163403 z ) And in Maple-input format, it is: -(-1-19350855678908*z^28+19350855678908*z^26+262*z^2-14666005650097*z^24+ 8401810171474*z^22-24431*z^4+1126357*z^6-30185499*z^8+515556492*z^10-5938156592 *z^12+47837594204*z^14+1163305703753*z^18-276236760619*z^16+24431*z^50-1126357* z^48-3619152269135*z^20-1163305703753*z^36+3619152269135*z^34+14666005650097*z^ 30+5938156592*z^42-515556492*z^44+30185499*z^46+z^54-262*z^52-8401810171474*z^ 32+276236760619*z^38-47837594204*z^40)/(1+198315726539767*z^28-172574326223710* z^26-398*z^2+113613671785833*z^24-56433750740928*z^22+46823*z^4-2536398*z^6+ 77940937*z^8-1515443124*z^10+19869933965*z^12-182785047482*z^14-5866264340222*z ^18+1210261163403*z^16-2536398*z^50+77940937*z^48+21056900282529*z^20+ 21056900282529*z^36-56433750740928*z^34-172574326223710*z^30-182785047482*z^42+ 19869933965*z^44-1515443124*z^46+z^56-398*z^54+46823*z^52+113613671785833*z^32-\ 5866264340222*z^38+1210261163403*z^40) The first , 40, terms are: [0, 136, 0, 31736, 0, 7673041, 0, 1865090280, 0, 453926178825, 0, 110514017749087, 0, 26908487583262976, 0, 6551972162921105449, 0, 1595356525011968152623, 0, 388458177339013992552408, 0, 94586902397581717818699847, 0, 23031265300717822739451473672, 0, 5607956153415183329334731118296, 0, 1365499121788343322391984096315591, 0, 332489735265919856718403571141579351, 0, 80958985895915264972421297034583950904, 0, 19712961645027646528872779853516560160648, 0, 4799971893670366949517139089618828013035383, 0, 1168760463052730537467901044369142866867895224, 0, 284585212218083874436878466397406182624668513631] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 10542881053848174983 z - 2362180693507231147 z - 351 z 24 22 4 6 + 437971329191819913 z - 66818787790991624 z + 50895 z - 4159872 z 8 10 12 14 + 219082342 z - 8023058008 z + 214615664869 z - 4339295509369 z 18 16 50 - 840872553599448 z + 67984218256523 z - 1243837397627561354685 z 48 20 + 1910060123168771717611 z + 8329277848543892 z 36 34 + 681435108787803833157 z - 313661683785349019648 z 66 80 88 84 86 - 66818787790991624 z + 219082342 z + z + 50895 z - 351 z 82 64 30 - 4159872 z + 437971329191819913 z - 39116609315567239080 z 42 44 - 2469819579783713093920 z + 2690582358228705877400 z 46 58 - 2469819579783713093920 z - 39116609315567239080 z 56 54 + 121089454286268511866 z - 313661683785349019648 z 52 60 + 681435108787803833157 z + 10542881053848174983 z 70 68 78 - 840872553599448 z + 8329277848543892 z - 8023058008 z 32 38 + 121089454286268511866 z - 1243837397627561354685 z 40 62 76 + 1910060123168771717611 z - 2362180693507231147 z + 214615664869 z 74 72 / - 4339295509369 z + 67984218256523 z ) / (-1 / 28 26 2 - 51280830948013227082 z + 10559059007177880061 z + 491 z 24 22 4 6 - 1801231842143114547 z + 253128858565616922 z - 85696 z + 7969522 z 8 10 12 14 - 465532763 z + 18672547219 z - 543574651872 z + 11922628339338 z 18 16 50 + 2710903972752389 z - 202363019819043 z + 15625260855587001951061 z 48 20 - 22030579167003406257575 z - 29096286187888794 z 36 34 - 4683788498767289073810 z + 1975996142715799577893 z 66 80 90 88 84 + 1801231842143114547 z - 18672547219 z + z - 491 z - 7969522 z 86 82 64 + 85696 z + 465532763 z - 10559059007177880061 z 30 42 + 207244046831212959356 z + 22030579167003406257575 z 44 46 - 26156479150479301824812 z + 26156479150479301824812 z 58 56 + 699369556519548906701 z - 1975996142715799577893 z 54 52 + 4683788498767289073810 z - 9328020572699538591436 z 60 70 - 207244046831212959356 z + 29096286187888794 z 68 78 32 - 253128858565616922 z + 543574651872 z - 699369556519548906701 z 38 40 + 9328020572699538591436 z - 15625260855587001951061 z 62 76 74 + 51280830948013227082 z - 11922628339338 z + 202363019819043 z 72 - 2710903972752389 z ) And in Maple-input format, it is: -(1+10542881053848174983*z^28-2362180693507231147*z^26-351*z^2+ 437971329191819913*z^24-66818787790991624*z^22+50895*z^4-4159872*z^6+219082342* z^8-8023058008*z^10+214615664869*z^12-4339295509369*z^14-840872553599448*z^18+ 67984218256523*z^16-1243837397627561354685*z^50+1910060123168771717611*z^48+ 8329277848543892*z^20+681435108787803833157*z^36-313661683785349019648*z^34-\ 66818787790991624*z^66+219082342*z^80+z^88+50895*z^84-351*z^86-4159872*z^82+ 437971329191819913*z^64-39116609315567239080*z^30-2469819579783713093920*z^42+ 2690582358228705877400*z^44-2469819579783713093920*z^46-39116609315567239080*z^ 58+121089454286268511866*z^56-313661683785349019648*z^54+681435108787803833157* z^52+10542881053848174983*z^60-840872553599448*z^70+8329277848543892*z^68-\ 8023058008*z^78+121089454286268511866*z^32-1243837397627561354685*z^38+ 1910060123168771717611*z^40-2362180693507231147*z^62+214615664869*z^76-\ 4339295509369*z^74+67984218256523*z^72)/(-1-51280830948013227082*z^28+ 10559059007177880061*z^26+491*z^2-1801231842143114547*z^24+253128858565616922*z ^22-85696*z^4+7969522*z^6-465532763*z^8+18672547219*z^10-543574651872*z^12+ 11922628339338*z^14+2710903972752389*z^18-202363019819043*z^16+ 15625260855587001951061*z^50-22030579167003406257575*z^48-29096286187888794*z^ 20-4683788498767289073810*z^36+1975996142715799577893*z^34+1801231842143114547* z^66-18672547219*z^80+z^90-491*z^88-7969522*z^84+85696*z^86+465532763*z^82-\ 10559059007177880061*z^64+207244046831212959356*z^30+22030579167003406257575*z^ 42-26156479150479301824812*z^44+26156479150479301824812*z^46+ 699369556519548906701*z^58-1975996142715799577893*z^56+4683788498767289073810*z ^54-9328020572699538591436*z^52-207244046831212959356*z^60+29096286187888794*z^ 70-253128858565616922*z^68+543574651872*z^78-699369556519548906701*z^32+ 9328020572699538591436*z^38-15625260855587001951061*z^40+51280830948013227082*z ^62-11922628339338*z^76+202363019819043*z^74-2710903972752389*z^72) The first , 40, terms are: [0, 140, 0, 33939, 0, 8476259, 0, 2122689284, 0, 531811456729, 0, 133250658130673, 0, 33388011683774220, 0, 8365926789923825171, 0, 2096226116884160311571, 0, 525245492295343542823412, 0, 131609298527672050943655241, 0, 32976975617299501258961392313, 0, 8262949033063867836291877060996, 0, 2070424152742395701729342714165683, 0, 518780420444607433379451448355522515, 0, 129989366816035129859783516331066681564, 0, 32571074041392189708590959118163638833217, 0, 8161243417053825739083811045361721386632201, 0, 2044940060245771146243582281981720827887916532, 0, 512394942327140815330455736456754498853732568163] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 11121925254174101569 z - 2488479938923174411 z - 347 z 24 22 4 6 + 460555488328321869 z - 70089947283011936 z + 50029 z - 4098546 z 8 10 12 14 + 217587678 z - 8053802270 z + 217822682347 z - 4448194811857 z 18 16 50 - 874757886295424 z + 70268839239199 z - 1318038678819924873737 z 48 20 + 2025049877244066561427 z + 8706720200986692 z 36 34 + 721600678407775140583 z - 331886231540254666134 z 66 80 88 84 86 - 70089947283011936 z + 217587678 z + z + 50029 z - 347 z 82 64 30 - 4098546 z + 460555488328321869 z - 41311727015400466202 z 42 44 - 2619360781327734056672 z + 2853807932120379860088 z 46 58 - 2619360781327734056672 z - 41311727015400466202 z 56 54 + 128011088145141472994 z - 331886231540254666134 z 52 60 + 721600678407775140583 z + 11121925254174101569 z 70 68 78 - 874757886295424 z + 8706720200986692 z - 8053802270 z 32 38 + 128011088145141472994 z - 1318038678819924873737 z 40 62 76 + 2025049877244066561427 z - 2488479938923174411 z + 217822682347 z 74 72 / 2 - 4448194811857 z + 70268839239199 z ) / ((-1 + z ) (1 / 28 26 2 + 44079034883505874344 z - 9369524568546519160 z - 472 z 24 22 4 6 + 1642181383708343043 z - 236003747670715392 z + 81254 z - 7560140 z 8 10 12 14 + 444742435 z - 17986348004 z + 526805955796 z - 11579516114616 z 18 16 50 - 2605584712482320 z + 196013684203893 z - 6309044174581889379128 z 48 20 + 9883498031329434007271 z + 27610941829426346 z 36 34 + 3363903250781607287806 z - 1497204394571622156500 z 66 80 88 84 86 - 236003747670715392 z + 444742435 z + z + 81254 z - 472 z 82 64 30 - 7560140 z + 1642181383708343043 z - 171716426528761364156 z 42 44 - 12936981044327927966512 z + 14151430589005841431564 z 46 58 - 12936981044327927966512 z - 171716426528761364156 z 56 54 + 555688586115094277197 z - 1497204394571622156500 z 52 60 + 3363903250781607287806 z + 44079034883505874344 z 70 68 78 - 2605584712482320 z + 27610941829426346 z - 17986348004 z 32 38 + 555688586115094277197 z - 6309044174581889379128 z 40 62 76 + 9883498031329434007271 z - 9369524568546519160 z + 526805955796 z 74 72 - 11579516114616 z + 196013684203893 z )) And in Maple-input format, it is: -(1+11121925254174101569*z^28-2488479938923174411*z^26-347*z^2+ 460555488328321869*z^24-70089947283011936*z^22+50029*z^4-4098546*z^6+217587678* z^8-8053802270*z^10+217822682347*z^12-4448194811857*z^14-874757886295424*z^18+ 70268839239199*z^16-1318038678819924873737*z^50+2025049877244066561427*z^48+ 8706720200986692*z^20+721600678407775140583*z^36-331886231540254666134*z^34-\ 70089947283011936*z^66+217587678*z^80+z^88+50029*z^84-347*z^86-4098546*z^82+ 460555488328321869*z^64-41311727015400466202*z^30-2619360781327734056672*z^42+ 2853807932120379860088*z^44-2619360781327734056672*z^46-41311727015400466202*z^ 58+128011088145141472994*z^56-331886231540254666134*z^54+721600678407775140583* z^52+11121925254174101569*z^60-874757886295424*z^70+8706720200986692*z^68-\ 8053802270*z^78+128011088145141472994*z^32-1318038678819924873737*z^38+ 2025049877244066561427*z^40-2488479938923174411*z^62+217822682347*z^76-\ 4448194811857*z^74+70268839239199*z^72)/(-1+z^2)/(1+44079034883505874344*z^28-\ 9369524568546519160*z^26-472*z^2+1642181383708343043*z^24-236003747670715392*z^ 22+81254*z^4-7560140*z^6+444742435*z^8-17986348004*z^10+526805955796*z^12-\ 11579516114616*z^14-2605584712482320*z^18+196013684203893*z^16-\ 6309044174581889379128*z^50+9883498031329434007271*z^48+27610941829426346*z^20+ 3363903250781607287806*z^36-1497204394571622156500*z^34-236003747670715392*z^66 +444742435*z^80+z^88+81254*z^84-472*z^86-7560140*z^82+1642181383708343043*z^64-\ 171716426528761364156*z^30-12936981044327927966512*z^42+14151430589005841431564 *z^44-12936981044327927966512*z^46-171716426528761364156*z^58+ 555688586115094277197*z^56-1497204394571622156500*z^54+3363903250781607287806*z ^52+44079034883505874344*z^60-2605584712482320*z^70+27610941829426346*z^68-\ 17986348004*z^78+555688586115094277197*z^32-6309044174581889379128*z^38+ 9883498031329434007271*z^40-9369524568546519160*z^62+526805955796*z^76-\ 11579516114616*z^74+196013684203893*z^72) The first , 40, terms are: [0, 126, 0, 27901, 0, 6442545, 0, 1495187406, 0, 347289908081, 0, 80678118916073, 0, 18742731750823606, 0, 4354245029337714161, 0, 1011564211888657355485, 0, 235003420039435972075862, 0, 54595260088597838980961513, 0, 12683400392747560497346571145, 0, 2946567995380311489119690958054, 0, 684537481215265193434426052255149, 0, 159029611400204532579132741159888577, 0, 36945263038312975927386817143151605926, 0, 8583008214407812468834184966531546731945, 0, 1993977683481128695805436679676853288704593, 0, 463234672844460209238171488795521800501217374, 0, 107617233584433027050646898959748691640307140161] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 46436211410804032 z - 21275049520068278 z - 349 z 24 22 4 6 + 7727538757524294 z - 2209784638314016 z + 47680 z - 3417413 z 8 10 12 14 + 147170485 z - 4131656188 z + 79812050861 z - 1101817741829 z 18 16 50 - 85027043092013 z + 11175205678976 z - 85027043092013 z 48 20 36 + 493228845873081 z + 493228845873081 z + 112448324949549022 z 34 66 64 30 - 125521066334708424 z - 349 z + 47680 z - 80785058813398070 z 42 44 46 - 21275049520068278 z + 7727538757524294 z - 2209784638314016 z 58 56 54 - 4131656188 z + 79812050861 z - 1101817741829 z 52 60 68 32 + 11175205678976 z + 147170485 z + z + 112448324949549022 z 38 40 62 / - 80785058813398070 z + 46436211410804032 z - 3417413 z ) / (-1 / 28 26 2 - 330688852339335402 z + 135059842433890502 z + 511 z 24 22 4 6 - 43701236948629754 z + 11133884458897221 z - 86249 z + 7146103 z 8 10 12 14 - 346270077 z + 10824246583 z - 232133695347 z + 3559853090001 z 18 16 50 + 341401122555437 z - 40195989106227 z + 2216342879170267 z 48 20 36 - 11133884458897221 z - 2216342879170267 z - 1251883749911599894 z 34 66 64 + 1251883749911599894 z + 86249 z - 7146103 z 30 42 44 + 644570560267207894 z + 330688852339335402 z - 135059842433890502 z 46 58 56 + 43701236948629754 z + 232133695347 z - 3559853090001 z 54 52 60 70 + 40195989106227 z - 341401122555437 z - 10824246583 z + z 68 32 38 - 511 z - 1003781562244263250 z + 1003781562244263250 z 40 62 - 644570560267207894 z + 346270077 z ) And in Maple-input format, it is: -(1+46436211410804032*z^28-21275049520068278*z^26-349*z^2+7727538757524294*z^24 -2209784638314016*z^22+47680*z^4-3417413*z^6+147170485*z^8-4131656188*z^10+ 79812050861*z^12-1101817741829*z^14-85027043092013*z^18+11175205678976*z^16-\ 85027043092013*z^50+493228845873081*z^48+493228845873081*z^20+ 112448324949549022*z^36-125521066334708424*z^34-349*z^66+47680*z^64-\ 80785058813398070*z^30-21275049520068278*z^42+7727538757524294*z^44-\ 2209784638314016*z^46-4131656188*z^58+79812050861*z^56-1101817741829*z^54+ 11175205678976*z^52+147170485*z^60+z^68+112448324949549022*z^32-\ 80785058813398070*z^38+46436211410804032*z^40-3417413*z^62)/(-1-\ 330688852339335402*z^28+135059842433890502*z^26+511*z^2-43701236948629754*z^24+ 11133884458897221*z^22-86249*z^4+7146103*z^6-346270077*z^8+10824246583*z^10-\ 232133695347*z^12+3559853090001*z^14+341401122555437*z^18-40195989106227*z^16+ 2216342879170267*z^50-11133884458897221*z^48-2216342879170267*z^20-\ 1251883749911599894*z^36+1251883749911599894*z^34+86249*z^66-7146103*z^64+ 644570560267207894*z^30+330688852339335402*z^42-135059842433890502*z^44+ 43701236948629754*z^46+232133695347*z^58-3559853090001*z^56+40195989106227*z^54 -341401122555437*z^52-10824246583*z^60+z^70-511*z^68-1003781562244263250*z^32+ 1003781562244263250*z^38-644570560267207894*z^40+346270077*z^62) The first , 40, terms are: [0, 162, 0, 44213, 0, 12349195, 0, 3455680702, 0, 967294609027, 0, 270778727170643, 0, 75801655169171278, 0, 21219994516514779363, 0, 5940357349243212281885, 0, 1662953456264239596823570, 0, 465530010750447064214593857, 0, 130321266765865650849812900513, 0, 36482358644904727256245550260850, 0, 10212934013569609478496712473091117, 0, 2859026257542952313694178734532272979, 0, 800360712576964985223074186963373465838, 0, 224054350191819300077164767149783519600979, 0, 62722158962700426399214047455488847113219267, 0, 17558548725427472551125131522517509367724762782, 0, 4915370236659459558678696131781097372952628244443] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 21575400112296 z - 25156234750032 z - 280 z 24 22 4 6 + 21575400112296 z - 13593118237042 z + 28951 z - 1505862 z 8 10 12 14 + 45212976 z - 846520362 z + 10400263414 z - 86904603328 z 18 16 50 48 - 2099825634334 z + 506715044170 z - 280 z + 28951 z 20 36 34 + 6266252165840 z + 506715044170 z - 2099825634334 z 30 42 44 46 52 - 13593118237042 z - 846520362 z + 45212976 z - 1505862 z + z 32 38 40 / 2 + 6266252165840 z - 86904603328 z + 10400263414 z ) / ((-1 + z ) (1 / 28 26 2 24 + 107595775147822 z - 127192627790554 z - 407 z + 107595775147822 z 22 4 6 8 - 65136024700956 z + 51945 z - 3171588 z + 109171828 z 10 12 14 18 - 2312461556 z + 31882558300 z - 297099869948 z - 8726205356524 z 16 50 48 20 + 1918713034380 z - 407 z + 51945 z + 28210899102092 z 36 34 30 + 1918713034380 z - 8726205356524 z - 65136024700956 z 42 44 46 52 32 - 2312461556 z + 109171828 z - 3171588 z + z + 28210899102092 z 38 40 - 297099869948 z + 31882558300 z )) And in Maple-input format, it is: -(1+21575400112296*z^28-25156234750032*z^26-280*z^2+21575400112296*z^24-\ 13593118237042*z^22+28951*z^4-1505862*z^6+45212976*z^8-846520362*z^10+ 10400263414*z^12-86904603328*z^14-2099825634334*z^18+506715044170*z^16-280*z^50 +28951*z^48+6266252165840*z^20+506715044170*z^36-2099825634334*z^34-\ 13593118237042*z^30-846520362*z^42+45212976*z^44-1505862*z^46+z^52+ 6266252165840*z^32-86904603328*z^38+10400263414*z^40)/(-1+z^2)/(1+ 107595775147822*z^28-127192627790554*z^26-407*z^2+107595775147822*z^24-\ 65136024700956*z^22+51945*z^4-3171588*z^6+109171828*z^8-2312461556*z^10+ 31882558300*z^12-297099869948*z^14-8726205356524*z^18+1918713034380*z^16-407*z^ 50+51945*z^48+28210899102092*z^20+1918713034380*z^36-8726205356524*z^34-\ 65136024700956*z^30-2312461556*z^42+109171828*z^44-3171588*z^46+z^52+ 28210899102092*z^32-297099869948*z^38+31882558300*z^40) The first , 40, terms are: [0, 128, 0, 28823, 0, 6776399, 0, 1601310880, 0, 378683846025, 0, 89563227823449, 0, 21183198752276832, 0, 5010198879444435967, 0, 1185000981331456303655, 0, 280273813276289332153792, 0, 66289745247797801948295409, 0, 15678704727991273002973499025, 0, 3708292761132362065802105018176, 0, 877077248894915475446790773542791, 0, 207444382139099746542138710591761695, 0, 49064289076735574634223067897768276960, 0, 11604577756220648651837117916717923551545, 0, 2744689211532704947013173186626525591343273, 0, 649167856527098713078387701586622413471082016, 0, 153539753855304584267100160145697626968961338927] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 32159141672465736 z - 14671801020327854 z - 325 z 24 22 4 6 + 5311339325336022 z - 1517140610274496 z + 41468 z - 2816057 z 8 10 12 14 + 115936669 z - 3132234112 z + 58597343797 z - 788520627505 z 18 16 50 - 58939414799245 z + 7846504316220 z - 58939414799245 z 48 20 36 + 339454276998313 z + 339454276998313 z + 78397120905041838 z 34 66 64 30 - 87596823745534080 z - 325 z + 41468 z - 56167876831061526 z 42 44 46 - 14671801020327854 z + 5311339325336022 z - 1517140610274496 z 58 56 54 52 - 3132234112 z + 58597343797 z - 788520627505 z + 7846504316220 z 60 68 32 38 + 115936669 z + z + 78397120905041838 z - 56167876831061526 z 40 62 / 28 + 32159141672465736 z - 2816057 z ) / (-1 - 220956993631951274 z / 26 2 24 + 90106266630575306 z + 461 z - 29161006087549106 z 22 4 6 8 + 7450652552732909 z - 72125 z + 5689103 z - 265357567 z 10 12 14 + 8029564299 z - 167416414255 z + 2506839782931 z 18 16 50 + 232321395555641 z - 27762250333283 z + 1492577907739529 z 48 20 36 - 7450652552732909 z - 1492577907739529 z - 841061873646620646 z 34 66 64 + 841061873646620646 z + 72125 z - 5689103 z 30 42 44 + 431643787843607342 z + 220956993631951274 z - 90106266630575306 z 46 58 56 + 29161006087549106 z + 167416414255 z - 2506839782931 z 54 52 60 70 + 27762250333283 z - 232321395555641 z - 8029564299 z + z 68 32 38 - 461 z - 673569319289567502 z + 673569319289567502 z 40 62 - 431643787843607342 z + 265357567 z ) And in Maple-input format, it is: -(1+32159141672465736*z^28-14671801020327854*z^26-325*z^2+5311339325336022*z^24 -1517140610274496*z^22+41468*z^4-2816057*z^6+115936669*z^8-3132234112*z^10+ 58597343797*z^12-788520627505*z^14-58939414799245*z^18+7846504316220*z^16-\ 58939414799245*z^50+339454276998313*z^48+339454276998313*z^20+78397120905041838 *z^36-87596823745534080*z^34-325*z^66+41468*z^64-56167876831061526*z^30-\ 14671801020327854*z^42+5311339325336022*z^44-1517140610274496*z^46-3132234112*z ^58+58597343797*z^56-788520627505*z^54+7846504316220*z^52+115936669*z^60+z^68+ 78397120905041838*z^32-56167876831061526*z^38+32159141672465736*z^40-2816057*z^ 62)/(-1-220956993631951274*z^28+90106266630575306*z^26+461*z^2-\ 29161006087549106*z^24+7450652552732909*z^22-72125*z^4+5689103*z^6-265357567*z^ 8+8029564299*z^10-167416414255*z^12+2506839782931*z^14+232321395555641*z^18-\ 27762250333283*z^16+1492577907739529*z^50-7450652552732909*z^48-\ 1492577907739529*z^20-841061873646620646*z^36+841061873646620646*z^34+72125*z^ 66-5689103*z^64+431643787843607342*z^30+220956993631951274*z^42-\ 90106266630575306*z^44+29161006087549106*z^46+167416414255*z^58-2506839782931*z ^56+27762250333283*z^54-232321395555641*z^52-8029564299*z^60+z^70-461*z^68-\ 673569319289567502*z^32+673569319289567502*z^38-431643787843607342*z^40+ 265357567*z^62) The first , 40, terms are: [0, 136, 0, 32039, 0, 7834025, 0, 1924969760, 0, 473463878327, 0, 116478389683415, 0, 28656697739819952, 0, 7050374248085906841, 0, 1734600375693544920119, 0, 426763230271652444917656, 0, 104996451195268540816460033, 0, 25832251694457472037895588737, 0, 6355502742671990012149230407672, 0, 1563642832025178962402524563176343, 0, 384702675202214079423133034980307385, 0, 94648307972924081788448593081212874512, 0, 23286300771572287116958538397974851603607, 0, 5729123058204357302565197771960876879090679, 0, 1409534787774019542610062890491987509838232768, 0, 346787509669689627251698957753931453849502534793] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 51478350219780 z - 45099482286735 z - 263 z 24 22 4 6 + 30292479266301 z - 15547997491763 z + 24933 z - 1188212 z 8 10 12 14 + 33313744 z - 600035181 z + 7331800188 z - 63020199800 z 18 16 50 48 - 1782856886638 z + 390903249561 z - 1188212 z + 33313744 z 20 36 34 + 6064035353222 z + 6064035353222 z - 15547997491763 z 30 42 44 46 - 45099482286735 z - 63020199800 z + 7331800188 z - 600035181 z 56 54 52 32 38 + z - 263 z + 24933 z + 30292479266301 z - 1782856886638 z 40 / 2 28 + 390903249561 z ) / ((-1 + z ) (1 + 263846337073545 z / 26 2 24 22 - 228745279058288 z - 402 z + 148984567469067 z - 72779471680158 z 4 6 8 10 12 + 46419 z - 2546058 z + 80624877 z - 1627351096 z + 22175267903 z 14 18 16 50 - 211536038454 z - 7220719911392 z + 1447225915413 z - 2546058 z 48 20 36 + 80624877 z + 26581553822459 z + 26581553822459 z 34 30 42 - 72779471680158 z - 228745279058288 z - 211536038454 z 44 46 56 54 52 + 22175267903 z - 1627351096 z + z - 402 z + 46419 z 32 38 40 + 148984567469067 z - 7220719911392 z + 1447225915413 z )) And in Maple-input format, it is: -(1+51478350219780*z^28-45099482286735*z^26-263*z^2+30292479266301*z^24-\ 15547997491763*z^22+24933*z^4-1188212*z^6+33313744*z^8-600035181*z^10+ 7331800188*z^12-63020199800*z^14-1782856886638*z^18+390903249561*z^16-1188212*z ^50+33313744*z^48+6064035353222*z^20+6064035353222*z^36-15547997491763*z^34-\ 45099482286735*z^30-63020199800*z^42+7331800188*z^44-600035181*z^46+z^56-263*z^ 54+24933*z^52+30292479266301*z^32-1782856886638*z^38+390903249561*z^40)/(-1+z^2 )/(1+263846337073545*z^28-228745279058288*z^26-402*z^2+148984567469067*z^24-\ 72779471680158*z^22+46419*z^4-2546058*z^6+80624877*z^8-1627351096*z^10+ 22175267903*z^12-211536038454*z^14-7220719911392*z^18+1447225915413*z^16-\ 2546058*z^50+80624877*z^48+26581553822459*z^20+26581553822459*z^36-\ 72779471680158*z^34-228745279058288*z^30-211536038454*z^42+22175267903*z^44-\ 1627351096*z^46+z^56-402*z^54+46419*z^52+148984567469067*z^32-7220719911392*z^ 38+1447225915413*z^40) The first , 40, terms are: [0, 140, 0, 34532, 0, 8765721, 0, 2228852380, 0, 566795111855, 0, 144136849964491, 0, 36654250074299376, 0, 9321240023069088691, 0, 2370407699919491309731, 0, 602798840099973676863068, 0, 153292803494136862503572425, 0, 38982629098431962834142507292, 0, 9913351030230213284414765995116, 0, 2520982574076359451223751952110349, 0, 641090295241228718460359964811678133, 0, 163030387785649705160487669114572223084, 0, 41458913882854678059898977878543383303788, 0, 10543074599110201058962227631808475124893409, 0, 2681122383391031294957695674383608557303092668, 0, 681814129943373764902956039462738211264560024235] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 4}, {3, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 195 z + 10757 z + 195 z - 248911 z + 2684765 z 4 6 8 10 12 - 10757 z + 248911 z - 2684765 z + 15001839 z - 45931545 z 14 18 16 20 30 / + 79759619 z + 45931545 z - 79759619 z - 15001839 z + z ) / ( / 32 30 28 26 24 22 z - 332 z + 23512 z - 675036 z + 9218332 z - 67298244 z 20 18 16 14 + 277021688 z - 651320148 z + 869208998 z - 651320148 z 12 10 8 6 4 + 277021688 z - 67298244 z + 9218332 z - 675036 z + 23512 z 2 - 332 z + 1) And in Maple-input format, it is: -(-1-195*z^28+10757*z^26+195*z^2-248911*z^24+2684765*z^22-10757*z^4+248911*z^6-\ 2684765*z^8+15001839*z^10-45931545*z^12+79759619*z^14+45931545*z^18-79759619*z^ 16-15001839*z^20+z^30)/(z^32-332*z^30+23512*z^28-675036*z^26+9218332*z^24-\ 67298244*z^22+277021688*z^20-651320148*z^18+869208998*z^16-651320148*z^14+ 277021688*z^12-67298244*z^10+9218332*z^8-675036*z^6+23512*z^4-332*z^2+1) The first , 40, terms are: [0, 137, 0, 32729, 0, 8071009, 0, 1995997105, 0, 493788113417, 0, 122163273334265, 0, 30223394509366417, 0, 7477323269949633457, 0, 1849903690984631879513, 0, 457669621774969839738089, 0, 113228317829757736377600337, 0, 28012896971266723385411227201, 0, 6930442947389872933541219472761, 0, 1714604508648206707239068576348585, 0, 424196352728888480739207281194121569, 0, 104946968680475365244344445022641485729, 0, 25964075750222086974545470469816065169513, 0, 6423560756821451767721711199611245798176569, 0, 1589200909499874986105542180226385154919295361, 0, 393171268454685520275476271978431424621828452113] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 11850469308343353963 z - 2659561620385180332 z - 364 z 24 22 4 6 + 493791552726010397 z - 75407385241588628 z + 54343 z - 4527372 z 8 10 12 14 + 241498180 z - 8922642640 z + 240169391481 z - 4876661350980 z 18 16 50 - 948905147752628 z + 76609038946719 z - 1386459527726463304100 z 48 20 + 2126895451707446753347 z + 9403345649352728 z 36 34 + 760596418532397633493 z - 350660527052699372684 z 66 80 88 84 86 - 75407385241588628 z + 241498180 z + z + 54343 z - 364 z 82 64 30 - 4527372 z + 493791552726010397 z - 43888726231967846296 z 42 44 - 2748443435803316031928 z + 2993460204241684039376 z 46 58 - 2748443435803316031928 z - 43888726231967846296 z 56 54 + 135612426195776943164 z - 350660527052699372684 z 52 60 + 760596418532397633493 z + 11850469308343353963 z 70 68 78 - 948905147752628 z + 9403345649352728 z - 8922642640 z 32 38 + 135612426195776943164 z - 1386459527726463304100 z 40 62 76 + 2126895451707446753347 z - 2659561620385180332 z + 240169391481 z 74 72 / - 4876661350980 z + 76609038946719 z ) / (-1 / 28 26 2 - 57364444219466162986 z + 11873800579940556778 z + 486 z 24 22 4 6 - 2034132420728277525 z + 286719069268010692 z - 88206 z + 8469239 z 8 10 12 14 - 506223844 z + 20637856200 z - 607648036497 z + 13429842247766 z 18 16 50 + 3074702494974279 z - 229007851113574 z + 16882527258192470291594 z 48 20 - 23728955928913565189219 z - 33007695509374132 z 36 34 - 5110829204740756229925 z + 2169374636447417493412 z 66 80 90 88 84 + 2034132420728277525 z - 20637856200 z + z - 486 z - 8469239 z 86 82 64 + 88206 z + 506223844 z - 11873800579940556778 z 30 42 + 230452707931617308563 z + 23728955928913565189219 z 44 46 - 28127391614975650206664 z + 28127391614975650206664 z 58 56 + 772757695091237894768 z - 2169374636447417493412 z 54 52 + 5110829204740756229925 z - 10123289340010017059122 z 60 70 - 230452707931617308563 z + 33007695509374132 z 68 78 32 - 286719069268010692 z + 607648036497 z - 772757695091237894768 z 38 40 + 10123289340010017059122 z - 16882527258192470291594 z 62 76 74 + 57364444219466162986 z - 13429842247766 z + 229007851113574 z 72 - 3074702494974279 z ) And in Maple-input format, it is: -(1+11850469308343353963*z^28-2659561620385180332*z^26-364*z^2+ 493791552726010397*z^24-75407385241588628*z^22+54343*z^4-4527372*z^6+241498180* z^8-8922642640*z^10+240169391481*z^12-4876661350980*z^14-948905147752628*z^18+ 76609038946719*z^16-1386459527726463304100*z^50+2126895451707446753347*z^48+ 9403345649352728*z^20+760596418532397633493*z^36-350660527052699372684*z^34-\ 75407385241588628*z^66+241498180*z^80+z^88+54343*z^84-364*z^86-4527372*z^82+ 493791552726010397*z^64-43888726231967846296*z^30-2748443435803316031928*z^42+ 2993460204241684039376*z^44-2748443435803316031928*z^46-43888726231967846296*z^ 58+135612426195776943164*z^56-350660527052699372684*z^54+760596418532397633493* z^52+11850469308343353963*z^60-948905147752628*z^70+9403345649352728*z^68-\ 8922642640*z^78+135612426195776943164*z^32-1386459527726463304100*z^38+ 2126895451707446753347*z^40-2659561620385180332*z^62+240169391481*z^76-\ 4876661350980*z^74+76609038946719*z^72)/(-1-57364444219466162986*z^28+ 11873800579940556778*z^26+486*z^2-2034132420728277525*z^24+286719069268010692*z ^22-88206*z^4+8469239*z^6-506223844*z^8+20637856200*z^10-607648036497*z^12+ 13429842247766*z^14+3074702494974279*z^18-229007851113574*z^16+ 16882527258192470291594*z^50-23728955928913565189219*z^48-33007695509374132*z^ 20-5110829204740756229925*z^36+2169374636447417493412*z^34+2034132420728277525* z^66-20637856200*z^80+z^90-486*z^88-8469239*z^84+88206*z^86+506223844*z^82-\ 11873800579940556778*z^64+230452707931617308563*z^30+23728955928913565189219*z^ 42-28127391614975650206664*z^44+28127391614975650206664*z^46+ 772757695091237894768*z^58-2169374636447417493412*z^56+5110829204740756229925*z ^54-10123289340010017059122*z^52-230452707931617308563*z^60+33007695509374132*z ^70-286719069268010692*z^68+607648036497*z^78-772757695091237894768*z^32+ 10123289340010017059122*z^38-16882527258192470291594*z^40+57364444219466162986* z^62-13429842247766*z^76+229007851113574*z^74-3074702494974279*z^72) The first , 40, terms are: [0, 122, 0, 25429, 0, 5539229, 0, 1217596414, 0, 268478807153, 0, 59272018942113, 0, 13092004946416022, 0, 2892353519810986749, 0, 639047315342563962581, 0, 141198346282811483493602, 0, 31198401713886286240479217, 0, 6893465433646955673118354513, 0, 1523154274773570757444649936274, 0, 336550786716478685745575847380373, 0, 74363102556601747781830690936997373, 0, 16431017701510305724386883538882754278, 0, 3630541918097247794739023978457804937345, 0, 802192245484419503669700260258018610100497, 0, 177249683540156429854246848133047496843721134, 0, 39164490356520790499639823511828027739249136157] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 2363665436 z + 7114953294 z + 213 z - 14721629070 z 22 4 6 8 10 + 21128532576 z - 14230 z + 442730 z - 7612365 z + 79468929 z 12 14 18 16 - 532146308 z + 2363665436 z + 14721629070 z - 7114953294 z 20 36 34 30 42 - 21128532576 z - 442730 z + 7612365 z + 532146308 z + z 32 38 40 / 44 42 40 - 79468929 z + 14230 z - 213 z ) / (z - 326 z + 27983 z / 38 36 34 32 - 1066328 z + 22075211 z - 274810158 z + 2182972869 z 30 28 26 24 - 11485086560 z + 40992184778 z - 100787481740 z + 172326194134 z 22 20 18 16 - 205950268560 z + 172326194134 z - 100787481740 z + 40992184778 z 14 12 10 8 - 11485086560 z + 2182972869 z - 274810158 z + 22075211 z 6 4 2 - 1066328 z + 27983 z - 326 z + 1) And in Maple-input format, it is: -(-1-2363665436*z^28+7114953294*z^26+213*z^2-14721629070*z^24+21128532576*z^22-\ 14230*z^4+442730*z^6-7612365*z^8+79468929*z^10-532146308*z^12+2363665436*z^14+ 14721629070*z^18-7114953294*z^16-21128532576*z^20-442730*z^36+7612365*z^34+ 532146308*z^30+z^42-79468929*z^32+14230*z^38-213*z^40)/(z^44-326*z^42+27983*z^ 40-1066328*z^38+22075211*z^36-274810158*z^34+2182972869*z^32-11485086560*z^30+ 40992184778*z^28-100787481740*z^26+172326194134*z^24-205950268560*z^22+ 172326194134*z^20-100787481740*z^18+40992184778*z^16-11485086560*z^14+ 2182972869*z^12-274810158*z^10+22075211*z^8-1066328*z^6+27983*z^4-326*z^2+1) The first , 40, terms are: [0, 113, 0, 23085, 0, 4987229, 0, 1085881317, 0, 236756704501, 0, 51634287174185, 0, 11261532715230841, 0, 2456188062371785865, 0, 535706255147115424697, 0, 116840127543179365034597, 0, 25483399897060373552607701, 0, 5558053529800944402685600109, 0, 1212238527538927427845331369661, 0, 264395123405389561352579687464161, 0, 57665863363387114939960457749041137, 0, 12577205489783081582492564228133239825, 0, 2743149737247437623026508702746483038209, 0, 598294310058393282840116430161544496103837, 0, 130490901239517594840616529332362974864891213, 0, 28460700728778037277940652880611002340988285429] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 681127636182561341559 z - 92818145323181294682 z - 383 z 24 22 4 + 10800170203030289909 z - 1066347969555319030 z + 62736 z 6 102 8 10 - 6002978 z - 17577012580 z + 383180601 z - 17577012580 z 12 14 18 + 607701422759 z - 16368343749798 z - 6150355224454826 z 16 50 + 351890895970989 z - 183312224702567613790189048 z 48 20 + 115779084032805295550427674 z + 88661407517692521 z 36 34 + 445928506261286716651262 z - 109413311754448284371878 z 66 80 - 64080085866153524638540612 z + 23293503389668428043745 z 100 90 88 + 607701422759 z - 1066347969555319030 z + 10800170203030289909 z 84 94 + 681127636182561341559 z - 6150355224454826 z 86 96 98 - 92818145323181294682 z + 351890895970989 z - 16368343749798 z 92 82 + 88661407517692521 z - 4290145798563271049808 z 64 112 110 106 + 115779084032805295550427674 z + z - 383 z - 6002978 z 108 30 42 + 62736 z - 4290145798563271049808 z - 13173751807572097443045136 z 44 46 + 31059970258505879348133662 z - 64080085866153524638540612 z 58 56 - 309741498432046204246755800 z + 330716838125183045755466662 z 54 52 - 309741498432046204246755800 z + 254450585369077996704956982 z 60 70 + 254450585369077996704956982 z - 13173751807572097443045136 z 68 78 + 31059970258505879348133662 z - 109413311754448284371878 z 32 38 + 23293503389668428043745 z - 1580820316293671067573629 z 40 62 + 4884205723395759913237289 z - 183312224702567613790189048 z 76 74 + 445928506261286716651262 z - 1580820316293671067573629 z 72 104 / + 4884205723395759913237289 z + 383180601 z ) / (-1 / 28 26 2 - 2633844728784834233928 z + 337103730679756634546 z + 512 z 24 22 4 - 36835690097480699200 z + 3414351253429272163 z - 99155 z 6 102 8 10 + 10687564 z + 1394927072750 z - 750698490 z + 37413629116 z 12 14 18 - 1394927072750 z + 40330120008780 z + 17322169422135997 z 16 50 - 927896225240947 z + 1427681472537365246214611773 z 48 20 - 845181918607301104129505181 z - 266365373095265277 z 36 34 - 2217512569261862131184712 z + 510800585557425218499883 z 66 80 + 845181918607301104129505181 z - 510800585557425218499883 z 100 90 - 40330120008780 z + 36835690097480699200 z 88 84 - 337103730679756634546 z - 17662803446728836238022 z 94 86 + 266365373095265277 z + 2633844728784834233928 z 96 98 92 - 17322169422135997 z + 927896225240947 z - 3414351253429272163 z 82 64 + 102114015065086587187457 z - 1427681472537365246214611773 z 112 114 110 106 108 - 512 z + z + 99155 z + 750698490 z - 10687564 z 30 42 + 17662803446728836238022 z + 79287086952004337683310780 z 44 46 - 199314913489407382785432108 z + 438537663011608342316454036 z 58 56 + 3131328214034971050097952113 z - 3131328214034971050097952113 z 54 52 + 2747342551582097985587738324 z - 2114664058824915933988291372 z 60 70 - 2747342551582097985587738324 z + 199314913489407382785432108 z 68 78 - 438537663011608342316454036 z + 2217512569261862131184712 z 32 38 - 102114015065086587187457 z + 8375392519389021595565501 z 40 62 - 27577001473739548819931201 z + 2114664058824915933988291372 z 76 74 - 8375392519389021595565501 z + 27577001473739548819931201 z 72 104 - 79287086952004337683310780 z - 37413629116 z ) And in Maple-input format, it is: -(1+681127636182561341559*z^28-92818145323181294682*z^26-383*z^2+ 10800170203030289909*z^24-1066347969555319030*z^22+62736*z^4-6002978*z^6-\ 17577012580*z^102+383180601*z^8-17577012580*z^10+607701422759*z^12-\ 16368343749798*z^14-6150355224454826*z^18+351890895970989*z^16-\ 183312224702567613790189048*z^50+115779084032805295550427674*z^48+ 88661407517692521*z^20+445928506261286716651262*z^36-109413311754448284371878*z ^34-64080085866153524638540612*z^66+23293503389668428043745*z^80+607701422759*z ^100-1066347969555319030*z^90+10800170203030289909*z^88+681127636182561341559*z ^84-6150355224454826*z^94-92818145323181294682*z^86+351890895970989*z^96-\ 16368343749798*z^98+88661407517692521*z^92-4290145798563271049808*z^82+ 115779084032805295550427674*z^64+z^112-383*z^110-6002978*z^106+62736*z^108-\ 4290145798563271049808*z^30-13173751807572097443045136*z^42+ 31059970258505879348133662*z^44-64080085866153524638540612*z^46-\ 309741498432046204246755800*z^58+330716838125183045755466662*z^56-\ 309741498432046204246755800*z^54+254450585369077996704956982*z^52+ 254450585369077996704956982*z^60-13173751807572097443045136*z^70+ 31059970258505879348133662*z^68-109413311754448284371878*z^78+ 23293503389668428043745*z^32-1580820316293671067573629*z^38+ 4884205723395759913237289*z^40-183312224702567613790189048*z^62+ 445928506261286716651262*z^76-1580820316293671067573629*z^74+ 4884205723395759913237289*z^72+383180601*z^104)/(-1-2633844728784834233928*z^28 +337103730679756634546*z^26+512*z^2-36835690097480699200*z^24+ 3414351253429272163*z^22-99155*z^4+10687564*z^6+1394927072750*z^102-750698490*z ^8+37413629116*z^10-1394927072750*z^12+40330120008780*z^14+17322169422135997*z^ 18-927896225240947*z^16+1427681472537365246214611773*z^50-\ 845181918607301104129505181*z^48-266365373095265277*z^20-\ 2217512569261862131184712*z^36+510800585557425218499883*z^34+ 845181918607301104129505181*z^66-510800585557425218499883*z^80-40330120008780*z ^100+36835690097480699200*z^90-337103730679756634546*z^88-\ 17662803446728836238022*z^84+266365373095265277*z^94+2633844728784834233928*z^ 86-17322169422135997*z^96+927896225240947*z^98-3414351253429272163*z^92+ 102114015065086587187457*z^82-1427681472537365246214611773*z^64-512*z^112+z^114 +99155*z^110+750698490*z^106-10687564*z^108+17662803446728836238022*z^30+ 79287086952004337683310780*z^42-199314913489407382785432108*z^44+ 438537663011608342316454036*z^46+3131328214034971050097952113*z^58-\ 3131328214034971050097952113*z^56+2747342551582097985587738324*z^54-\ 2114664058824915933988291372*z^52-2747342551582097985587738324*z^60+ 199314913489407382785432108*z^70-438537663011608342316454036*z^68+ 2217512569261862131184712*z^78-102114015065086587187457*z^32+ 8375392519389021595565501*z^38-27577001473739548819931201*z^40+ 2114664058824915933988291372*z^62-8375392519389021595565501*z^76+ 27577001473739548819931201*z^74-79287086952004337683310780*z^72-37413629116*z^ 104) The first , 40, terms are: [0, 129, 0, 29629, 0, 7063639, 0, 1689897540, 0, 404490760517, 0, 96827259637163, 0, 23179045157103967, 0, 5548754205684395863, 0, 1328299214794749434591, 0, 317977550778571029777931, 0, 76119693590011564387456707, 0, 18222065760433078831548849521, 0, 4362125832806356064567946049028, 0, 1044236259936564812046513407410635, 0, 249976595951517241094373695873495789, 0, 59841149866299093590331745909847715053, 0, 14325193939592632573307122658645925686409, 0, 3429265344433401537216325806582308856858761, 0, 820921577196787812018870750855242194515438309, 0, 196517961784821574616872846008101912412833178829] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 5266790 z - 40607460 z - 217 z + 189608751 z 22 4 6 8 10 - 557752799 z + 14329 z - 386654 z + 5266790 z - 40607460 z 12 14 18 16 + 189608751 z - 557752799 z - 1306629028 z + 1057140669 z 20 36 34 30 32 / 2 + 1057140669 z + z - 217 z - 386654 z + 14329 z ) / ((-1 + z ) / 36 34 32 30 28 26 (z - 356 z + 29313 z - 945622 z + 15376525 z - 141150173 z 24 22 20 18 + 774079371 z - 2602484037 z + 5400920337 z - 6898017147 z 16 14 12 10 + 5400920337 z - 2602484037 z + 774079371 z - 141150173 z 8 6 4 2 + 15376525 z - 945622 z + 29313 z - 356 z + 1)) And in Maple-input format, it is: -(1+5266790*z^28-40607460*z^26-217*z^2+189608751*z^24-557752799*z^22+14329*z^4-\ 386654*z^6+5266790*z^8-40607460*z^10+189608751*z^12-557752799*z^14-1306629028*z ^18+1057140669*z^16+1057140669*z^20+z^36-217*z^34-386654*z^30+14329*z^32)/(-1+z ^2)/(z^36-356*z^34+29313*z^32-945622*z^30+15376525*z^28-141150173*z^26+ 774079371*z^24-2602484037*z^22+5400920337*z^20-6898017147*z^18+5400920337*z^16-\ 2602484037*z^14+774079371*z^12-141150173*z^10+15376525*z^8-945622*z^6+29313*z^4 -356*z^2+1) The first , 40, terms are: [0, 140, 0, 34640, 0, 8801101, 0, 2239694440, 0, 570053616569, 0, 145095937123055, 0, 36931529209015696, 0, 9400259539399123765, 0, 2392668356796984652655, 0, 609011074534369126566320, 0, 155012913466795158513559183, 0, 39455774124914947108755682528, 0, 10042764036407675072658806434364, 0, 2556206581622429484630859544535551, 0, 650636823127543776928792434048956335, 0, 165608006275725788013668507543650703212, 0, 42152566174801391410492689078500472679136, 0, 10729184385947164402970537455955180030744567, 0, 2730922646803709799135956182350651709914060160, 0, 695107683356964018038506719537899465339404817167] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 5}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 80720436 z + 471278484 z + 223 z - 1723923276 z 22 4 6 8 10 + 4035661814 z - 15663 z + 500689 z - 8387820 z + 80720436 z 12 14 18 16 - 471278484 z + 1723923276 z + 6146408490 z - 4035661814 z 20 36 34 30 32 38 - 6146408490 z - 223 z + 15663 z + 8387820 z - 500689 z + z ) / 26 32 28 20 / (-9418181296 z + 25861757 z + 2119569768 z + 61810596004 z / 14 30 34 36 2 - 9418181296 z - 300392624 z - 1272420 z + 32326 z - 356 z 16 10 8 22 + 26927220098 z - 300392624 z + 25861757 z - 50258045144 z 6 12 40 38 24 - 1272420 z + 2119569768 z + z - 356 z + 26927220098 z + 1 18 4 - 50258045144 z + 32326 z ) And in Maple-input format, it is: -(-1-80720436*z^28+471278484*z^26+223*z^2-1723923276*z^24+4035661814*z^22-15663 *z^4+500689*z^6-8387820*z^8+80720436*z^10-471278484*z^12+1723923276*z^14+ 6146408490*z^18-4035661814*z^16-6146408490*z^20-223*z^36+15663*z^34+8387820*z^ 30-500689*z^32+z^38)/(-9418181296*z^26+25861757*z^32+2119569768*z^28+ 61810596004*z^20-9418181296*z^14-300392624*z^30-1272420*z^34+32326*z^36-356*z^2 +26927220098*z^16-300392624*z^10+25861757*z^8-50258045144*z^22-1272420*z^6+ 2119569768*z^12+z^40-356*z^38+26927220098*z^24+1-50258045144*z^18+32326*z^4) The first , 40, terms are: [0, 133, 0, 30685, 0, 7396233, 0, 1792893561, 0, 435003745965, 0, 105560107018677, 0, 25616444390056977, 0, 6216416236780678129, 0, 1508557009770570044757, 0, 366086271135477341240077, 0, 88839308859290038529581209, 0, 21558915115400275843488800105, 0, 5231769893512544934874869841789, 0, 1269610092973959650341739822815461, 0, 308100283661013307964689089188900897, 0, 74767667111312986127547538810446483425, 0, 18144105480362807230589563613673847382053, 0, 4403087275579794960252525878834325456779325, 0, 1068511069744170808687231965230520564999096745, 0, 259298949738737832929556381912605111945781949529] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 13110281007957356863 z - 2984240152986715850 z - 370 z 24 22 4 6 + 561036365410344557 z - 86538647904738684 z + 56491 z - 4827732 z 8 10 12 14 + 263964244 z - 9964467160 z + 272785165221 z - 5604529620014 z 18 16 50 - 1099531229773644 z + 88624658140831 z - 1433715458723895712406 z 48 20 + 2181982650743899027187 z + 10864840764151224 z 36 34 + 794892762052407469681 z - 371162958679112870420 z 66 80 88 84 86 - 86538647904738684 z + 263964244 z + z + 56491 z - 370 z 82 64 30 - 4827732 z + 561036365410344557 z - 47832133466007774656 z 42 44 - 2805782848686147394872 z + 3050821099013758538128 z 46 58 - 2805782848686147394872 z - 47832133466007774656 z 56 54 + 145597050087023146028 z - 371162958679112870420 z 52 60 + 794892762052407469681 z + 13110281007957356863 z 70 68 78 - 1099531229773644 z + 10864840764151224 z - 9964467160 z 32 38 + 145597050087023146028 z - 1433715458723895712406 z 40 62 76 + 2181982650743899027187 z - 2984240152986715850 z + 272785165221 z 74 72 / 2 - 5604529620014 z + 88624658140831 z ) / ((-1 + z ) (1 / 28 26 2 + 53224138466482754195 z - 11520668851434236227 z - 511 z 24 22 4 6 + 2052332650228098217 z - 299028905951856992 z + 93843 z - 9130392 z 8 10 12 14 + 553517418 z - 22831421680 z + 676844000969 z - 14968374623537 z 18 16 50 - 3360869543333568 z + 253658328402747 z - 6968344842447943718861 z 48 20 + 10796746148606792734939 z + 35360051963086572 z 36 34 + 3769611190159213580497 z - 1706717584232152098680 z 66 80 88 84 86 - 299028905951856992 z + 553517418 z + z + 93843 z - 511 z 82 64 30 - 9130392 z + 2052332650228098217 z - 203386379539590666816 z 42 44 - 14035289002648973853728 z + 15316854621395528685800 z 46 58 - 14035289002648973853728 z - 203386379539590666816 z 56 54 + 645464147582729720310 z - 1706717584232152098680 z 52 60 + 3769611190159213580497 z + 53224138466482754195 z 70 68 78 - 3360869543333568 z + 35360051963086572 z - 22831421680 z 32 38 + 645464147582729720310 z - 6968344842447943718861 z 40 62 + 10796746148606792734939 z - 11520668851434236227 z 76 74 72 + 676844000969 z - 14968374623537 z + 253658328402747 z )) And in Maple-input format, it is: -(1+13110281007957356863*z^28-2984240152986715850*z^26-370*z^2+ 561036365410344557*z^24-86538647904738684*z^22+56491*z^4-4827732*z^6+263964244* z^8-9964467160*z^10+272785165221*z^12-5604529620014*z^14-1099531229773644*z^18+ 88624658140831*z^16-1433715458723895712406*z^50+2181982650743899027187*z^48+ 10864840764151224*z^20+794892762052407469681*z^36-371162958679112870420*z^34-\ 86538647904738684*z^66+263964244*z^80+z^88+56491*z^84-370*z^86-4827732*z^82+ 561036365410344557*z^64-47832133466007774656*z^30-2805782848686147394872*z^42+ 3050821099013758538128*z^44-2805782848686147394872*z^46-47832133466007774656*z^ 58+145597050087023146028*z^56-371162958679112870420*z^54+794892762052407469681* z^52+13110281007957356863*z^60-1099531229773644*z^70+10864840764151224*z^68-\ 9964467160*z^78+145597050087023146028*z^32-1433715458723895712406*z^38+ 2181982650743899027187*z^40-2984240152986715850*z^62+272785165221*z^76-\ 5604529620014*z^74+88624658140831*z^72)/(-1+z^2)/(1+53224138466482754195*z^28-\ 11520668851434236227*z^26-511*z^2+2052332650228098217*z^24-299028905951856992*z ^22+93843*z^4-9130392*z^6+553517418*z^8-22831421680*z^10+676844000969*z^12-\ 14968374623537*z^14-3360869543333568*z^18+253658328402747*z^16-\ 6968344842447943718861*z^50+10796746148606792734939*z^48+35360051963086572*z^20 +3769611190159213580497*z^36-1706717584232152098680*z^34-299028905951856992*z^ 66+553517418*z^80+z^88+93843*z^84-511*z^86-9130392*z^82+2052332650228098217*z^ 64-203386379539590666816*z^30-14035289002648973853728*z^42+ 15316854621395528685800*z^44-14035289002648973853728*z^46-203386379539590666816 *z^58+645464147582729720310*z^56-1706717584232152098680*z^54+ 3769611190159213580497*z^52+53224138466482754195*z^60-3360869543333568*z^70+ 35360051963086572*z^68-22831421680*z^78+645464147582729720310*z^32-\ 6968344842447943718861*z^38+10796746148606792734939*z^40-11520668851434236227*z ^62+676844000969*z^76-14968374623537*z^74+253658328402747*z^72) The first , 40, terms are: [0, 142, 0, 34841, 0, 8836827, 0, 2248225514, 0, 572207542963, 0, 145644719497723, 0, 37071606431437946, 0, 9436031201446999011, 0, 2401804591275700888161, 0, 611344580520632167149438, 0, 155608920115145668077488585, 0, 39608000476199615594740424377, 0, 10081643830396908754532907075294, 0, 2566136670466392317314245236883281, 0, 653172986959339142564422423426852691, 0, 166255739942703211671147268814023835674, 0, 42317994798715221985724967463891526368235, 0, 10771433722638095941605084431402430903287747, 0, 2741712715674823129274381540373655087381248714, 0, 697863330811805388796328297859125322256918790763] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1172002075677551 z - 664512989998502 z - 270 z 24 22 4 6 + 298961306485960 z - 106236591116552 z + 26800 z - 1379384 z 8 10 12 14 + 43106094 z - 893269315 z + 12968575138 z - 136837923932 z 18 16 50 - 6435531472013 z + 1076650050582 z - 136837923932 z 48 20 36 + 1076650050582 z + 29633292666222 z + 1172002075677551 z 34 64 30 42 - 1645331728775576 z + z - 1645331728775576 z - 106236591116552 z 44 46 58 56 + 29633292666222 z - 6435531472013 z - 1379384 z + 43106094 z 54 52 60 32 - 893269315 z + 12968575138 z + 26800 z + 1841936479597816 z 38 40 62 / - 664512989998502 z + 298961306485960 z - 270 z ) / (-1 / 28 26 2 - 8826345244081446 z + 4474228890384413 z + 403 z 24 22 4 6 - 1799623910538107 z + 571730040305952 z - 49228 z + 2942286 z 8 10 12 14 - 104542761 z + 2441657681 z - 39789159040 z + 470282062044 z 18 16 50 + 27684963340137 z - 4140662301525 z + 4140662301525 z 48 20 36 - 27684963340137 z - 142581109497384 z - 13859901234418376 z 34 66 64 30 + 17359477858942641 z + z - 403 z + 13859901234418376 z 42 44 46 + 1799623910538107 z - 571730040305952 z + 142581109497384 z 58 56 54 52 + 104542761 z - 2441657681 z + 39789159040 z - 470282062044 z 60 32 38 - 2942286 z - 17359477858942641 z + 8826345244081446 z 40 62 - 4474228890384413 z + 49228 z ) And in Maple-input format, it is: -(1+1172002075677551*z^28-664512989998502*z^26-270*z^2+298961306485960*z^24-\ 106236591116552*z^22+26800*z^4-1379384*z^6+43106094*z^8-893269315*z^10+ 12968575138*z^12-136837923932*z^14-6435531472013*z^18+1076650050582*z^16-\ 136837923932*z^50+1076650050582*z^48+29633292666222*z^20+1172002075677551*z^36-\ 1645331728775576*z^34+z^64-1645331728775576*z^30-106236591116552*z^42+ 29633292666222*z^44-6435531472013*z^46-1379384*z^58+43106094*z^56-893269315*z^ 54+12968575138*z^52+26800*z^60+1841936479597816*z^32-664512989998502*z^38+ 298961306485960*z^40-270*z^62)/(-1-8826345244081446*z^28+4474228890384413*z^26+ 403*z^2-1799623910538107*z^24+571730040305952*z^22-49228*z^4+2942286*z^6-\ 104542761*z^8+2441657681*z^10-39789159040*z^12+470282062044*z^14+27684963340137 *z^18-4140662301525*z^16+4140662301525*z^50-27684963340137*z^48-142581109497384 *z^20-13859901234418376*z^36+17359477858942641*z^34+z^66-403*z^64+ 13859901234418376*z^30+1799623910538107*z^42-571730040305952*z^44+ 142581109497384*z^46+104542761*z^58-2441657681*z^56+39789159040*z^54-\ 470282062044*z^52-2942286*z^60-17359477858942641*z^32+8826345244081446*z^38-\ 4474228890384413*z^40+49228*z^62) The first , 40, terms are: [0, 133, 0, 31171, 0, 7577491, 0, 1849130256, 0, 451532964279, 0, 110273163531035, 0, 26931668768853301, 0, 6577482011196237741, 0, 1606411313625517786205, 0, 392332237590685837310765, 0, 95818920108495416390225339, 0, 23401761936165375774602967455, 0, 5715389648155246851891059799504, 0, 1395864078599345530834163285261083, 0, 340910532144687037361188488955513971, 0, 83260249128158500390566055139902627245, 0, 20334570015641718373674433378474824727121, 0, 4966292343008664170439649308813256165916593, 0, 1212912769597288147292831986362741327193953341, 0, 296228511139383262124273527207518828762230632979] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 671250191172469832584 z - 91887600605418449956 z - 383 z 24 22 4 + 10737309964360978088 z - 1064212307921996304 z + 62797 z 6 102 8 10 - 6020298 z - 17690068442 z + 385058654 z - 17690068442 z 12 14 18 + 612058604897 z - 16482887720367 z - 6174856919193216 z 16 50 + 353972105099253 z - 173099070399493184785807018 z 48 20 + 109550328707141425182113858 z + 88777192773881864 z 36 34 + 431200952909813127698496 z - 106295917298316879186792 z 66 80 - 60787904134919230126644032 z + 22738335281389536767480 z 100 90 88 + 612058604897 z - 1064212307921996304 z + 10737309964360978088 z 84 94 + 671250191172469832584 z - 6174856919193216 z 86 96 98 - 91887600605418449956 z + 353972105099253 z - 16482887720367 z 92 82 + 88777192773881864 z - 4208038587050616689256 z 64 112 110 106 + 109550328707141425182113858 z + z - 383 z - 6020298 z 108 30 42 + 62797 z - 4208038587050616689256 z - 12579041748608796247569968 z 44 46 + 29554242545854999196335736 z - 60787904134919230126644032 z 58 56 - 291795792409160712353436300 z + 311463139919139295087774804 z 54 52 - 291795792409160712353436300 z + 239921498766377272624920330 z 60 70 + 239921498766377272624920330 z - 12579041748608796247569968 z 68 78 + 29554242545854999196335736 z - 106295917298316879186792 z 32 38 + 22738335281389536767480 z - 1521761941570558404506884 z 40 62 + 4681906203876009076405664 z - 173099070399493184785807018 z 76 74 + 431200952909813127698496 z - 1521761941570558404506884 z 72 104 / 2 + 4681906203876009076405664 z + 385058654 z ) / ((-1 + z ) (1 / 28 26 2 + 2307991234511721097536 z - 301935392865723123416 z - 508 z 24 22 4 + 33653788104882746172 z - 3175386550084861432 z + 98246 z 6 102 8 10 - 10592328 z - 36981268208 z + 743672247 z - 36981268208 z 12 14 18 + 1372722947230 z - 39420706555428 z - 16588460812368280 z 16 50 + 898758458357249 z - 835840614382708972013293376 z 48 20 + 521134962289718993727598982 z + 251643926312532952 z 36 34 + 1742293943551715576829904 z - 413841112548438617434456 z 66 80 - 283766514208604400973945064 z + 85109274771059641267716 z 100 90 88 + 1372722947230 z - 3175386550084861432 z + 33653788104882746172 z 84 94 + 2307991234511721097536 z - 16588460812368280 z 86 96 98 - 301935392865723123416 z + 898758458357249 z - 39420706555428 z 92 82 + 251643926312532952 z - 15110974489339526327144 z 64 112 110 106 + 521134962289718993727598982 z + z - 508 z - 10592328 z 108 30 + 98246 z - 15110974489339526327144 z 42 44 - 55939880923504399551287432 z + 134887800752601758271105320 z 46 58 - 283766514208604400973945064 z - 1433718049120954302098567432 z 56 54 + 1533721514243723240046902922 z - 1433718049120954302098567432 z 52 60 + 1171123940590647762298914572 z + 1171123940590647762298914572 z 70 68 - 55939880923504399551287432 z + 134887800752601758271105320 z 78 32 - 413841112548438617434456 z + 85109274771059641267716 z 38 40 - 6365837105640306455657672 z + 20222947085484221698758076 z 62 76 - 835840614382708972013293376 z + 1742293943551715576829904 z 74 72 - 6365837105640306455657672 z + 20222947085484221698758076 z 104 + 743672247 z )) And in Maple-input format, it is: -(1+671250191172469832584*z^28-91887600605418449956*z^26-383*z^2+ 10737309964360978088*z^24-1064212307921996304*z^22+62797*z^4-6020298*z^6-\ 17690068442*z^102+385058654*z^8-17690068442*z^10+612058604897*z^12-\ 16482887720367*z^14-6174856919193216*z^18+353972105099253*z^16-\ 173099070399493184785807018*z^50+109550328707141425182113858*z^48+ 88777192773881864*z^20+431200952909813127698496*z^36-106295917298316879186792*z ^34-60787904134919230126644032*z^66+22738335281389536767480*z^80+612058604897*z ^100-1064212307921996304*z^90+10737309964360978088*z^88+671250191172469832584*z ^84-6174856919193216*z^94-91887600605418449956*z^86+353972105099253*z^96-\ 16482887720367*z^98+88777192773881864*z^92-4208038587050616689256*z^82+ 109550328707141425182113858*z^64+z^112-383*z^110-6020298*z^106+62797*z^108-\ 4208038587050616689256*z^30-12579041748608796247569968*z^42+ 29554242545854999196335736*z^44-60787904134919230126644032*z^46-\ 291795792409160712353436300*z^58+311463139919139295087774804*z^56-\ 291795792409160712353436300*z^54+239921498766377272624920330*z^52+ 239921498766377272624920330*z^60-12579041748608796247569968*z^70+ 29554242545854999196335736*z^68-106295917298316879186792*z^78+ 22738335281389536767480*z^32-1521761941570558404506884*z^38+ 4681906203876009076405664*z^40-173099070399493184785807018*z^62+ 431200952909813127698496*z^76-1521761941570558404506884*z^74+ 4681906203876009076405664*z^72+385058654*z^104)/(-1+z^2)/(1+ 2307991234511721097536*z^28-301935392865723123416*z^26-508*z^2+ 33653788104882746172*z^24-3175386550084861432*z^22+98246*z^4-10592328*z^6-\ 36981268208*z^102+743672247*z^8-36981268208*z^10+1372722947230*z^12-\ 39420706555428*z^14-16588460812368280*z^18+898758458357249*z^16-\ 835840614382708972013293376*z^50+521134962289718993727598982*z^48+ 251643926312532952*z^20+1742293943551715576829904*z^36-413841112548438617434456 *z^34-283766514208604400973945064*z^66+85109274771059641267716*z^80+ 1372722947230*z^100-3175386550084861432*z^90+33653788104882746172*z^88+ 2307991234511721097536*z^84-16588460812368280*z^94-301935392865723123416*z^86+ 898758458357249*z^96-39420706555428*z^98+251643926312532952*z^92-\ 15110974489339526327144*z^82+521134962289718993727598982*z^64+z^112-508*z^110-\ 10592328*z^106+98246*z^108-15110974489339526327144*z^30-\ 55939880923504399551287432*z^42+134887800752601758271105320*z^44-\ 283766514208604400973945064*z^46-1433718049120954302098567432*z^58+ 1533721514243723240046902922*z^56-1433718049120954302098567432*z^54+ 1171123940590647762298914572*z^52+1171123940590647762298914572*z^60-\ 55939880923504399551287432*z^70+134887800752601758271105320*z^68-\ 413841112548438617434456*z^78+85109274771059641267716*z^32-\ 6365837105640306455657672*z^38+20222947085484221698758076*z^40-\ 835840614382708972013293376*z^62+1742293943551715576829904*z^76-\ 6365837105640306455657672*z^74+20222947085484221698758076*z^72+743672247*z^104) The first , 40, terms are: [0, 126, 0, 28177, 0, 6569365, 0, 1539021730, 0, 360836828521, 0, 84614460415293, 0, 19842330037887086, 0, 4653116244398814189, 0, 1091178601109134895721, 0, 255886832181251395902314, 0, 60006745233174864509185061, 0, 14071882913219652525839334765, 0, 3299927179960110397034651663442, 0, 773849489147727991254355928855353, 0, 181471589926394070063977794632578189, 0, 42555998826559584867016894240992043638, 0, 9979595356396968212399400381758479017557, 0, 2340265208756685023341962669119486102497361, 0, 548803939611605619074450613737212200910721434, 0, 128697278841004762675190948116737944972148649685] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 627354393363944479440 z - 86644118882660977512 z - 389 z 24 22 4 + 10222749540748808568 z - 1023731550618171100 z + 64405 z 6 102 8 10 - 6197206 z - 18092622114 z + 395847022 z - 18092622114 z 12 14 18 + 621098483809 z - 16566091627865 z - 6072178851658484 z 16 50 + 351983076350901 z - 154199890109498685993241466 z 48 20 + 97722888280520106943444018 z + 86332440745299992 z 36 34 + 392179165478497448720232 z - 97217792111389483173588 z 66 80 - 54323735996961239214642508 z + 20929537140925044180968 z 100 90 88 + 621098483809 z - 1023731550618171100 z + 10222749540748808568 z 84 94 + 627354393363944479440 z - 6072178851658484 z 86 96 98 - 86644118882660977512 z + 351983076350901 z - 16566091627865 z 92 82 + 86332440745299992 z - 3901326601561637068316 z 64 112 110 106 + 97722888280520106943444018 z + z - 389 z - 6197206 z 108 30 42 + 64405 z - 3901326601561637068316 z - 11299720078534488493618980 z 44 46 + 26472661855662622034913192 z - 54323735996961239214642508 z 58 56 - 259545527928812664816975832 z + 276988098250435818768379348 z 54 52 - 259545527928812664816975832 z + 213523802809011312960387218 z 60 70 + 213523802809011312960387218 z - 11299720078534488493618980 z 68 78 + 26472661855662622034913192 z - 97217792111389483173588 z 32 38 + 20929537140925044180968 z - 1377388962624900146202176 z 40 62 + 4220321432226313242086368 z - 154199890109498685993241466 z 76 74 + 392179165478497448720232 z - 1377388962624900146202176 z 72 104 / 2 + 4220321432226313242086368 z + 395847022 z ) / ((-1 + z ) (1 / 28 26 2 + 2170706293002802610360 z - 286424363390342929456 z - 524 z 24 22 4 + 32236359892327756780 z - 3074706810745748000 z + 103034 z 6 102 8 10 - 11165304 z - 38573965448 z + 781770695 z - 38573965448 z 12 14 18 + 1416328974426 z - 40164783440412 z - 16460747459469184 z 16 50 + 903681189471185 z - 755574078964859081440367960 z 48 20 + 471525847756263754939777398 z + 246564274324104464 z 36 34 + 1599646161126596504069048 z - 381725032695060493011968 z 66 80 - 257076192664529273778766528 z + 78937683107863691636676 z 100 90 88 + 1416328974426 z - 3074706810745748000 z + 32236359892327756780 z 84 94 + 2170706293002802610360 z - 16460747459469184 z 86 96 98 - 286424363390342929456 z + 903681189471185 z - 40164783440412 z 92 82 + 246564274324104464 z - 14106042192965096277792 z 64 112 110 106 + 471525847756263754939777398 z + z - 524 z - 11165304 z 108 30 + 103034 z - 14106042192965096277792 z 42 44 - 50866985291480751460884576 z + 122400807358254492151174576 z 46 58 - 257076192664529273778766528 z - 1294739053447604208696326432 z 56 54 + 1384880861691012363774802794 z - 1294739053447604208696326432 z 52 60 + 1057989889313989766947540860 z + 1057989889313989766947540860 z 70 68 - 50866985291480751460884576 z + 122400807358254492151174576 z 78 32 - 381725032695060493011968 z + 78937683107863691636676 z 38 40 - 5822091967468480876360272 z + 18436760618281396409014284 z 62 76 - 755574078964859081440367960 z + 1599646161126596504069048 z 74 72 - 5822091967468480876360272 z + 18436760618281396409014284 z 104 + 781770695 z )) And in Maple-input format, it is: -(1+627354393363944479440*z^28-86644118882660977512*z^26-389*z^2+ 10222749540748808568*z^24-1023731550618171100*z^22+64405*z^4-6197206*z^6-\ 18092622114*z^102+395847022*z^8-18092622114*z^10+621098483809*z^12-\ 16566091627865*z^14-6072178851658484*z^18+351983076350901*z^16-\ 154199890109498685993241466*z^50+97722888280520106943444018*z^48+ 86332440745299992*z^20+392179165478497448720232*z^36-97217792111389483173588*z^ 34-54323735996961239214642508*z^66+20929537140925044180968*z^80+621098483809*z^ 100-1023731550618171100*z^90+10222749540748808568*z^88+627354393363944479440*z^ 84-6072178851658484*z^94-86644118882660977512*z^86+351983076350901*z^96-\ 16566091627865*z^98+86332440745299992*z^92-3901326601561637068316*z^82+ 97722888280520106943444018*z^64+z^112-389*z^110-6197206*z^106+64405*z^108-\ 3901326601561637068316*z^30-11299720078534488493618980*z^42+ 26472661855662622034913192*z^44-54323735996961239214642508*z^46-\ 259545527928812664816975832*z^58+276988098250435818768379348*z^56-\ 259545527928812664816975832*z^54+213523802809011312960387218*z^52+ 213523802809011312960387218*z^60-11299720078534488493618980*z^70+ 26472661855662622034913192*z^68-97217792111389483173588*z^78+ 20929537140925044180968*z^32-1377388962624900146202176*z^38+ 4220321432226313242086368*z^40-154199890109498685993241466*z^62+ 392179165478497448720232*z^76-1377388962624900146202176*z^74+ 4220321432226313242086368*z^72+395847022*z^104)/(-1+z^2)/(1+ 2170706293002802610360*z^28-286424363390342929456*z^26-524*z^2+ 32236359892327756780*z^24-3074706810745748000*z^22+103034*z^4-11165304*z^6-\ 38573965448*z^102+781770695*z^8-38573965448*z^10+1416328974426*z^12-\ 40164783440412*z^14-16460747459469184*z^18+903681189471185*z^16-\ 755574078964859081440367960*z^50+471525847756263754939777398*z^48+ 246564274324104464*z^20+1599646161126596504069048*z^36-381725032695060493011968 *z^34-257076192664529273778766528*z^66+78937683107863691636676*z^80+ 1416328974426*z^100-3074706810745748000*z^90+32236359892327756780*z^88+ 2170706293002802610360*z^84-16460747459469184*z^94-286424363390342929456*z^86+ 903681189471185*z^96-40164783440412*z^98+246564274324104464*z^92-\ 14106042192965096277792*z^82+471525847756263754939777398*z^64+z^112-524*z^110-\ 11165304*z^106+103034*z^108-14106042192965096277792*z^30-\ 50866985291480751460884576*z^42+122400807358254492151174576*z^44-\ 257076192664529273778766528*z^46-1294739053447604208696326432*z^58+ 1384880861691012363774802794*z^56-1294739053447604208696326432*z^54+ 1057989889313989766947540860*z^52+1057989889313989766947540860*z^60-\ 50866985291480751460884576*z^70+122400807358254492151174576*z^68-\ 381725032695060493011968*z^78+78937683107863691636676*z^32-\ 5822091967468480876360272*z^38+18436760618281396409014284*z^40-\ 755574078964859081440367960*z^62+1599646161126596504069048*z^76-\ 5822091967468480876360272*z^74+18436760618281396409014284*z^72+781770695*z^104) The first , 40, terms are: [0, 136, 0, 32247, 0, 7916919, 0, 1952352640, 0, 481918751849, 0, 118984897737857, 0, 29378941219208852, 0, 7254161882040865507, 0, 1791183632923619980267, 0, 442276068703080772383340, 0, 109206097650025982483463313, 0, 26964996437012213604956827505, 0, 6658154279107178733206812052724, 0, 1644020933674434844135126119553891, 0, 405939051644281385782029509523423643, 0, 100233829343961143443110825806012084572, 0, 24749578797888487322423333382388613556065, 0, 6111126898928279511684364124750865948412521, 0, 1508949799913616105195175436820080362711776024, 0, 372587500852267900055197490034302120533083738975] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 728241291225660 z - 408504583629708 z - 244 z 24 22 4 6 + 181425141298072 z - 63623898435620 z + 21364 z - 992484 z 8 10 12 14 + 28706664 z - 562816364 z + 7873853340 z - 81278457372 z 18 16 50 - 3785119466804 z + 633330335948 z - 81278457372 z 48 20 36 + 633330335948 z + 17548554599604 z + 728241291225660 z 34 64 30 42 - 1029579368469948 z + z - 1029579368469948 z - 63623898435620 z 44 46 58 56 + 17548554599604 z - 3785119466804 z - 992484 z + 28706664 z 54 52 60 32 - 562816364 z + 7873853340 z + 21364 z + 1155437207172198 z 38 40 62 / 2 - 408504583629708 z + 181425141298072 z - 244 z ) / ((-1 + z ) (1 / 28 26 2 + 3563457471613848 z - 1924873910649822 z - 374 z 24 22 4 6 + 813068981775104 z - 268390328889762 z + 39368 z - 2079434 z 8 10 12 14 + 67007488 z - 1449870774 z + 22264485464 z - 251254017930 z 18 16 50 - 13817471774366 z + 2132133139516 z - 251254017930 z 48 20 36 + 2132133139516 z + 69094791416392 z + 3563457471613848 z 34 64 30 42 - 5157876307155554 z + z - 5157876307155554 z - 268390328889762 z 44 46 58 56 + 69094791416392 z - 13817471774366 z - 2079434 z + 67007488 z 54 52 60 32 - 1449870774 z + 22264485464 z + 39368 z + 5834873067285894 z 38 40 62 - 1924873910649822 z + 813068981775104 z - 374 z )) And in Maple-input format, it is: -(1+728241291225660*z^28-408504583629708*z^26-244*z^2+181425141298072*z^24-\ 63623898435620*z^22+21364*z^4-992484*z^6+28706664*z^8-562816364*z^10+7873853340 *z^12-81278457372*z^14-3785119466804*z^18+633330335948*z^16-81278457372*z^50+ 633330335948*z^48+17548554599604*z^20+728241291225660*z^36-1029579368469948*z^ 34+z^64-1029579368469948*z^30-63623898435620*z^42+17548554599604*z^44-\ 3785119466804*z^46-992484*z^58+28706664*z^56-562816364*z^54+7873853340*z^52+ 21364*z^60+1155437207172198*z^32-408504583629708*z^38+181425141298072*z^40-244* z^62)/(-1+z^2)/(1+3563457471613848*z^28-1924873910649822*z^26-374*z^2+ 813068981775104*z^24-268390328889762*z^22+39368*z^4-2079434*z^6+67007488*z^8-\ 1449870774*z^10+22264485464*z^12-251254017930*z^14-13817471774366*z^18+ 2132133139516*z^16-251254017930*z^50+2132133139516*z^48+69094791416392*z^20+ 3563457471613848*z^36-5157876307155554*z^34+z^64-5157876307155554*z^30-\ 268390328889762*z^42+69094791416392*z^44-13817471774366*z^46-2079434*z^58+ 67007488*z^56-1449870774*z^54+22264485464*z^52+39368*z^60+5834873067285894*z^32 -1924873910649822*z^38+813068981775104*z^40-374*z^62) The first , 40, terms are: [0, 131, 0, 30747, 0, 7450241, 0, 1809075905, 0, 439366466763, 0, 106710370909587, 0, 25917171511118145, 0, 6294608862725495809, 0, 1528797343872168591795, 0, 371305253182914646936619, 0, 90180423032127592999284737, 0, 21902487589104128070571597761, 0, 5319546598616680730092129639739, 0, 1291980004546576997268327801809379, 0, 313788459449358724391461646863185089, 0, 76211084488234607327325125850615941185, 0, 18509697294365436473434160014629013424163, 0, 4495525765440738428239825277034462771473915, 0, 1091846699940015825053838412357882908974271553, 0, 265181266523789440538810691470326562373640272001] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 324261074115073 z + 203279883252663 z + 251 z 24 22 4 6 - 100639700542549 z + 39214162300387 z - 22413 z + 1044599 z 8 10 12 14 - 29716817 z + 562110795 z - 7459784157 z + 72014926439 z 18 16 50 48 + 2839061022503 z - 518716787685 z + 7459784157 z - 72014926439 z 20 36 34 - 11965625400177 z - 203279883252663 z + 324261074115073 z 30 42 44 + 409335459208627 z + 11965625400177 z - 2839061022503 z 46 58 56 54 + 518716787685 z + 22413 z - 1044599 z + 29716817 z 52 60 32 38 - 562110795 z - 251 z - 409335459208627 z + 100639700542549 z 40 62 / 28 - 39214162300387 z + z ) / (1 + 2652144954364920 z / 26 2 24 - 1477299582309066 z - 402 z + 649556255069008 z 22 4 6 8 - 224769304779894 z + 43752 z - 2347438 z + 75496624 z 10 12 14 18 - 1604436754 z + 23878863352 z - 258526629230 z - 12853408331530 z 16 50 48 + 2090064773148 z - 258526629230 z + 2090064773148 z 20 36 34 64 + 60931024185128 z + 2652144954364920 z - 3765282682373622 z + z 30 42 44 - 3765282682373622 z - 224769304779894 z + 60931024185128 z 46 58 56 54 - 12853408331530 z - 2347438 z + 75496624 z - 1604436754 z 52 60 32 + 23878863352 z + 43752 z + 4231437853704134 z 38 40 62 - 1477299582309066 z + 649556255069008 z - 402 z ) And in Maple-input format, it is: -(-1-324261074115073*z^28+203279883252663*z^26+251*z^2-100639700542549*z^24+ 39214162300387*z^22-22413*z^4+1044599*z^6-29716817*z^8+562110795*z^10-\ 7459784157*z^12+72014926439*z^14+2839061022503*z^18-518716787685*z^16+ 7459784157*z^50-72014926439*z^48-11965625400177*z^20-203279883252663*z^36+ 324261074115073*z^34+409335459208627*z^30+11965625400177*z^42-2839061022503*z^ 44+518716787685*z^46+22413*z^58-1044599*z^56+29716817*z^54-562110795*z^52-251*z ^60-409335459208627*z^32+100639700542549*z^38-39214162300387*z^40+z^62)/(1+ 2652144954364920*z^28-1477299582309066*z^26-402*z^2+649556255069008*z^24-\ 224769304779894*z^22+43752*z^4-2347438*z^6+75496624*z^8-1604436754*z^10+ 23878863352*z^12-258526629230*z^14-12853408331530*z^18+2090064773148*z^16-\ 258526629230*z^50+2090064773148*z^48+60931024185128*z^20+2652144954364920*z^36-\ 3765282682373622*z^34+z^64-3765282682373622*z^30-224769304779894*z^42+ 60931024185128*z^44-12853408331530*z^46-2347438*z^58+75496624*z^56-1604436754*z ^54+23878863352*z^52+43752*z^60+4231437853704134*z^32-1477299582309066*z^38+ 649556255069008*z^40-402*z^62) The first , 40, terms are: [0, 151, 0, 39363, 0, 10520213, 0, 2815598981, 0, 753634968915, 0, 201722795911559, 0, 53994466519261745, 0, 14452519459449253841, 0, 3868457894814724327591, 0, 1035457280463365280257587, 0, 277157412308065411525000101, 0, 74185804327511566182416696437, 0, 19857067930796089557909098868451, 0, 5315075443106911631079493011433847, 0, 1422668596611176252915518462201529953, 0, 380800979675370616195378719763497666977, 0, 101927733884853558998035640719055310971831, 0, 27282658105969887708967675720553555926621347, 0, 7302658510667170916652903820817917048677400757, 0, 1954678357082459372311348604226745589586758262629] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 6}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1405260539834367 z - 787351171577454 z - 270 z 24 22 4 6 + 348935347360414 z - 121927802865326 z + 27030 z - 1415134 z 8 10 12 14 + 44950330 z - 943982295 z + 13861625694 z - 147906196988 z 18 16 50 - 7144613676353 z + 1178272429774 z - 147906196988 z 48 20 36 + 1178272429774 z + 33433203259690 z + 1405260539834367 z 34 64 30 42 - 1987869210358932 z + z - 1987869210358932 z - 121927802865326 z 44 46 58 56 + 33433203259690 z - 7144613676353 z - 1415134 z + 44950330 z 54 52 60 32 - 943982295 z + 13861625694 z + 27030 z + 2231240769354328 z 38 40 62 / 2 - 787351171577454 z + 348935347360414 z - 270 z ) / ((-1 + z ) (1 / 28 26 2 + 6694512448074472 z - 3628946982823588 z - 388 z 24 22 4 6 + 1538252683323369 z - 508885315551406 z + 47328 z - 2861570 z 8 10 12 14 + 102576287 z - 2401440424 z + 38984230800 z - 456615424000 z 18 16 50 - 26034385216136 z + 3965827939585 z - 456615424000 z 48 20 36 + 3965827939585 z + 130914922971410 z + 6694512448074472 z 34 64 30 42 - 9665185625231944 z + z - 9665185625231944 z - 508885315551406 z 44 46 58 56 + 130914922971410 z - 26034385216136 z - 2861570 z + 102576287 z 54 52 60 32 - 2401440424 z + 38984230800 z + 47328 z + 10923655001957273 z 38 40 62 - 3628946982823588 z + 1538252683323369 z - 388 z )) And in Maple-input format, it is: -(1+1405260539834367*z^28-787351171577454*z^26-270*z^2+348935347360414*z^24-\ 121927802865326*z^22+27030*z^4-1415134*z^6+44950330*z^8-943982295*z^10+ 13861625694*z^12-147906196988*z^14-7144613676353*z^18+1178272429774*z^16-\ 147906196988*z^50+1178272429774*z^48+33433203259690*z^20+1405260539834367*z^36-\ 1987869210358932*z^34+z^64-1987869210358932*z^30-121927802865326*z^42+ 33433203259690*z^44-7144613676353*z^46-1415134*z^58+44950330*z^56-943982295*z^ 54+13861625694*z^52+27030*z^60+2231240769354328*z^32-787351171577454*z^38+ 348935347360414*z^40-270*z^62)/(-1+z^2)/(1+6694512448074472*z^28-\ 3628946982823588*z^26-388*z^2+1538252683323369*z^24-508885315551406*z^22+47328* z^4-2861570*z^6+102576287*z^8-2401440424*z^10+38984230800*z^12-456615424000*z^ 14-26034385216136*z^18+3965827939585*z^16-456615424000*z^50+3965827939585*z^48+ 130914922971410*z^20+6694512448074472*z^36-9665185625231944*z^34+z^64-\ 9665185625231944*z^30-508885315551406*z^42+130914922971410*z^44-26034385216136* z^46-2861570*z^58+102576287*z^56-2401440424*z^54+38984230800*z^52+47328*z^60+ 10923655001957273*z^32-3628946982823588*z^38+1538252683323369*z^40-388*z^62) The first , 40, terms are: [0, 119, 0, 25605, 0, 5775905, 0, 1310730200, 0, 297766227543, 0, 67660496408311, 0, 15375039347742517, 0, 3493832074744512975, 0, 793942186353830043263, 0, 180416381583841335410237, 0, 40998041568802984107009151, 0, 9316445903874607827970386887, 0, 2117080755628926600706415116952, 0, 481088064958981324532057636684081, 0, 109323050480911663662764947343587053, 0, 24842706019595356239639069348216896095, 0, 5645287427227615571644432296676084003345, 0, 1282842139298078858238260862501289639011441, 0, 291514643952933866047526073097069683840798479, 0, 66244150418632046023272723145373938031860153565] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 723756146731529618204 z - 97747781971517285908 z - 380 z 24 22 4 + 11269520376615349548 z - 1102334843527925028 z + 61800 z 6 102 8 10 - 5893792 z - 17320785796 z + 376350446 z - 17320785796 z 12 14 18 + 602129372364 z - 16332543685772 z - 6241231479532944 z 16 50 + 353971307287187 z - 207883732865674808589969296 z 48 20 + 130943791795085040069596811 z + 90800601886923244 z 36 34 + 489135716677687064013836 z - 119151133072399441941316 z 66 80 - 72224771050317037696982644 z + 25171161857818326688097 z 100 90 88 + 602129372364 z - 1102334843527925028 z + 11269520376615349548 z 84 94 + 723756146731529618204 z - 6241231479532944 z 86 96 98 - 97747781971517285908 z + 353971307287187 z - 16332543685772 z 92 82 + 90800601886923244 z - 4598090754644580629792 z 64 112 110 106 + 130943791795085040069596811 z + z - 380 z - 5893792 z 108 30 42 + 61800 z - 4598090754644580629792 z - 14715781336262883339722504 z 44 46 + 34863132238433575180802560 z - 72224771050317037696982644 z 58 56 - 352360121515721453172536456 z + 376369769282661144765500648 z 54 52 - 352360121515721453172536456 z + 289120804611082314444560312 z 60 70 + 289120804611082314444560312 z - 14715781336262883339722504 z 68 78 + 34863132238433575180802560 z - 119151133072399441941316 z 32 38 + 25171161857818326688097 z - 1745591306042979664513300 z 40 62 + 5426190872936190909800946 z - 207883732865674808589969296 z 76 74 + 489135716677687064013836 z - 1745591306042979664513300 z 72 104 / + 5426190872936190909800946 z + 376350446 z ) / (-1 / 28 26 2 - 2784998224730095583196 z + 352914473973671297640 z + 513 z 24 22 4 - 38178834963899888088 z + 3503940402240581820 z - 98404 z 6 102 8 10 + 10533340 z + 1376121824080 z - 737695134 z + 36786214602 z 12 14 18 - 1376121824080 z + 40004685639324 z + 17447653290671419 z 16 50 - 926942036577571 z + 1634571069690025436338073099 z 48 20 - 964017964971934998453083547 z - 270739975270575496 z 36 34 - 2433175435537690641815476 z + 555656939643260637452481 z 66 80 + 964017964971934998453083547 z - 555656939643260637452481 z 100 90 - 40004685639324 z + 38178834963899888088 z 88 84 - 352914473973671297640 z - 18860034560753655747688 z 94 86 + 270739975270575496 z + 2784998224730095583196 z 96 98 92 - 17447653290671419 z + 926942036577571 z - 3503940402240581820 z 82 64 + 110074865950209646991929 z - 1634571069690025436338073099 z 112 114 110 106 108 - 513 z + z + 98404 z + 737695134 z - 10533340 z 30 42 + 18860034560753655747688 z + 88978949251571345768864826 z 44 46 - 225065434066460114333628252 z + 497890110395095396531577868 z 58 56 + 3605897977383176341258590208 z - 3605897977383176341258590208 z 54 52 + 3160632548053622575088933544 z - 2428070517454217848989620816 z 60 70 - 3160632548053622575088933544 z + 225065434066460114333628252 z 68 78 - 497890110395095396531577868 z + 2433175435537690641815476 z 32 38 - 110074865950209646991929 z + 9264780327483947986674488 z 40 62 - 30735783681331662795200030 z + 2428070517454217848989620816 z 76 74 - 9264780327483947986674488 z + 30735783681331662795200030 z 72 104 - 88978949251571345768864826 z - 36786214602 z ) And in Maple-input format, it is: -(1+723756146731529618204*z^28-97747781971517285908*z^26-380*z^2+ 11269520376615349548*z^24-1102334843527925028*z^22+61800*z^4-5893792*z^6-\ 17320785796*z^102+376350446*z^8-17320785796*z^10+602129372364*z^12-\ 16332543685772*z^14-6241231479532944*z^18+353971307287187*z^16-\ 207883732865674808589969296*z^50+130943791795085040069596811*z^48+ 90800601886923244*z^20+489135716677687064013836*z^36-119151133072399441941316*z ^34-72224771050317037696982644*z^66+25171161857818326688097*z^80+602129372364*z ^100-1102334843527925028*z^90+11269520376615349548*z^88+723756146731529618204*z ^84-6241231479532944*z^94-97747781971517285908*z^86+353971307287187*z^96-\ 16332543685772*z^98+90800601886923244*z^92-4598090754644580629792*z^82+ 130943791795085040069596811*z^64+z^112-380*z^110-5893792*z^106+61800*z^108-\ 4598090754644580629792*z^30-14715781336262883339722504*z^42+ 34863132238433575180802560*z^44-72224771050317037696982644*z^46-\ 352360121515721453172536456*z^58+376369769282661144765500648*z^56-\ 352360121515721453172536456*z^54+289120804611082314444560312*z^52+ 289120804611082314444560312*z^60-14715781336262883339722504*z^70+ 34863132238433575180802560*z^68-119151133072399441941316*z^78+ 25171161857818326688097*z^32-1745591306042979664513300*z^38+ 5426190872936190909800946*z^40-207883732865674808589969296*z^62+ 489135716677687064013836*z^76-1745591306042979664513300*z^74+ 5426190872936190909800946*z^72+376350446*z^104)/(-1-2784998224730095583196*z^28 +352914473973671297640*z^26+513*z^2-38178834963899888088*z^24+ 3503940402240581820*z^22-98404*z^4+10533340*z^6+1376121824080*z^102-737695134*z ^8+36786214602*z^10-1376121824080*z^12+40004685639324*z^14+17447653290671419*z^ 18-926942036577571*z^16+1634571069690025436338073099*z^50-\ 964017964971934998453083547*z^48-270739975270575496*z^20-\ 2433175435537690641815476*z^36+555656939643260637452481*z^34+ 964017964971934998453083547*z^66-555656939643260637452481*z^80-40004685639324*z ^100+38178834963899888088*z^90-352914473973671297640*z^88-\ 18860034560753655747688*z^84+270739975270575496*z^94+2784998224730095583196*z^ 86-17447653290671419*z^96+926942036577571*z^98-3503940402240581820*z^92+ 110074865950209646991929*z^82-1634571069690025436338073099*z^64-513*z^112+z^114 +98404*z^110+737695134*z^106-10533340*z^108+18860034560753655747688*z^30+ 88978949251571345768864826*z^42-225065434066460114333628252*z^44+ 497890110395095396531577868*z^46+3605897977383176341258590208*z^58-\ 3605897977383176341258590208*z^56+3160632548053622575088933544*z^54-\ 2428070517454217848989620816*z^52-3160632548053622575088933544*z^60+ 225065434066460114333628252*z^70-497890110395095396531577868*z^68+ 2433175435537690641815476*z^78-110074865950209646991929*z^32+ 9264780327483947986674488*z^38-30735783681331662795200030*z^40+ 2428070517454217848989620816*z^62-9264780327483947986674488*z^76+ 30735783681331662795200030*z^74-88978949251571345768864826*z^72-36786214602*z^ 104) The first , 40, terms are: [0, 133, 0, 31625, 0, 7775441, 0, 1916364265, 0, 472429225265, 0, 116468612608425, 0, 28713296137975133, 0, 7078765467078037309, 0, 1745147151115412702057, 0, 430235845763293333154489, 0, 106067206773805381335020465, 0, 26149035399652640904403130873, 0, 6446592431557315004962993131201, 0, 1589295870554299417693272549727349, 0, 391813410106094093574814859073060289, 0, 96594819871743432312209085673550975777, 0, 23813782237646116013431559074105898990261, 0, 5870876152727704179714496166917544926441137, 0, 1447362978997072054813059899541478860725920137, 0, 356822310414089614312536626557339812592792887489] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 7}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 20729026061884 z + 20729026061884 z + 265 z 24 22 4 6 - 15705630855084 z + 8994847645551 z - 25025 z + 1175649 z 8 10 12 14 - 31985994 z + 551523010 z - 6380562018 z + 51437535338 z 18 16 50 48 + 1247088895775 z - 296637387471 z + 25025 z - 1175649 z 20 36 34 - 3875560594103 z - 1247088895775 z + 3875560594103 z 30 42 44 46 54 + 15705630855084 z + 6380562018 z - 551523010 z + 31985994 z + z 52 32 38 40 / - 265 z - 8994847645551 z + 296637387471 z - 51437535338 z ) / (1 / 28 26 2 24 + 211011135399656 z - 183693806807256 z - 406 z + 121094190063163 z 22 4 6 8 - 60302669423610 z + 47438 z - 2608734 z + 81776315 z 10 12 14 18 - 1616710036 z + 21420020780 z - 197847002580 z - 6320671127506 z 16 50 48 20 + 1308846606089 z - 2608734 z + 81776315 z + 22586529861394 z 36 34 30 + 22586529861394 z - 60302669423610 z - 183693806807256 z 42 44 46 56 54 - 197847002580 z + 21420020780 z - 1616710036 z + z - 406 z 52 32 38 40 + 47438 z + 121094190063163 z - 6320671127506 z + 1308846606089 z ) And in Maple-input format, it is: -(-1-20729026061884*z^28+20729026061884*z^26+265*z^2-15705630855084*z^24+ 8994847645551*z^22-25025*z^4+1175649*z^6-31985994*z^8+551523010*z^10-6380562018 *z^12+51437535338*z^14+1247088895775*z^18-296637387471*z^16+25025*z^50-1175649* z^48-3875560594103*z^20-1247088895775*z^36+3875560594103*z^34+15705630855084*z^ 30+6380562018*z^42-551523010*z^44+31985994*z^46+z^54-265*z^52-8994847645551*z^ 32+296637387471*z^38-51437535338*z^40)/(1+211011135399656*z^28-183693806807256* z^26-406*z^2+121094190063163*z^24-60302669423610*z^22+47438*z^4-2608734*z^6+ 81776315*z^8-1616710036*z^10+21420020780*z^12-197847002580*z^14-6320671127506*z ^18+1308846606089*z^16-2608734*z^50+81776315*z^48+22586529861394*z^20+ 22586529861394*z^36-60302669423610*z^34-183693806807256*z^30-197847002580*z^42+ 21420020780*z^44-1616710036*z^46+z^56-406*z^54+47438*z^52+121094190063163*z^32-\ 6320671127506*z^38+1308846606089*z^40) The first , 40, terms are: [0, 141, 0, 34833, 0, 8886525, 0, 2273562469, 0, 581912147497, 0, 148950057664629, 0, 38126828452446809, 0, 9759377654537031337, 0, 2498123342112295025317, 0, 639448683490261132203257, 0, 163680722052929516014289301, 0, 41897621585648601437404879565, 0, 10724602599771306412873507166017, 0, 2745194037483147799821875772249693, 0, 702691799893903985667114413910623153, 0, 179869167317071732632514150136971688017, 0, 46041404434107990370671142328409155234493, 0, 11785293465721506022614598600005312918625505, 0, 3016700810505824801374798622037030932696623917, 0, 772189831893140101858001454677037940384938805557] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 5}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 56224903885956 z - 48928331651995 z - 245 z 24 22 4 6 + 32226746605643 z - 16038611610121 z + 21395 z - 975196 z 8 10 12 14 + 26979522 z - 490338763 z + 6141947088 z - 54717943244 z 18 16 50 48 - 1691047276516 z + 354224205247 z - 975196 z + 26979522 z 20 36 34 + 6013613932542 z + 6013613932542 z - 16038611610121 z 30 42 44 46 - 48928331651995 z - 54717943244 z + 6141947088 z - 490338763 z 56 54 52 32 38 + z - 245 z + 21395 z + 32226746605643 z - 1691047276516 z 40 / 2 28 + 354224205247 z ) / ((-1 + z ) (1 + 281486426382273 z / 26 2 24 22 - 242174053096892 z - 370 z + 154290400480811 z - 72866248852546 z 4 6 8 10 12 + 39111 z - 2047120 z + 63822421 z - 1296354814 z + 18056840129 z 14 18 16 50 - 178067092604 z - 6624716305376 z + 1269151696779 z - 2047120 z 48 20 36 + 63822421 z + 25524041441645 z + 25524041441645 z 34 30 42 - 72866248852546 z - 242174053096892 z - 178067092604 z 44 46 56 54 52 + 18056840129 z - 1296354814 z + z - 370 z + 39111 z 32 38 40 + 154290400480811 z - 6624716305376 z + 1269151696779 z )) And in Maple-input format, it is: -(1+56224903885956*z^28-48928331651995*z^26-245*z^2+32226746605643*z^24-\ 16038611610121*z^22+21395*z^4-975196*z^6+26979522*z^8-490338763*z^10+6141947088 *z^12-54717943244*z^14-1691047276516*z^18+354224205247*z^16-975196*z^50+ 26979522*z^48+6013613932542*z^20+6013613932542*z^36-16038611610121*z^34-\ 48928331651995*z^30-54717943244*z^42+6141947088*z^44-490338763*z^46+z^56-245*z^ 54+21395*z^52+32226746605643*z^32-1691047276516*z^38+354224205247*z^40)/(-1+z^2 )/(1+281486426382273*z^28-242174053096892*z^26-370*z^2+154290400480811*z^24-\ 72866248852546*z^22+39111*z^4-2047120*z^6+63822421*z^8-1296354814*z^10+ 18056840129*z^12-178067092604*z^14-6624716305376*z^18+1269151696779*z^16-\ 2047120*z^50+63822421*z^48+25524041441645*z^20+25524041441645*z^36-\ 72866248852546*z^34-242174053096892*z^30-178067092604*z^42+18056840129*z^44-\ 1296354814*z^46+z^56-370*z^54+39111*z^52+154290400480811*z^32-6624716305376*z^ 38+1269151696779*z^40) The first , 40, terms are: [0, 126, 0, 28660, 0, 6769289, 0, 1603855846, 0, 380133876623, 0, 90100486169661, 0, 21356024195103416, 0, 5061905899922682379, 0, 1199797046620421344605, 0, 284381616327200274380386, 0, 67405486761231057735364695, 0, 15976769898771611180490562260, 0, 3786890187723898439225602047554, 0, 897586770347152799284729764185483, 0, 212750296513781792336281305872848603, 0, 50427089794567547135218159825129457906, 0, 11952469288261287843018009397474101248340, 0, 2833031266900916468152359459619250514641663, 0, 671498580391312614381648493301344247026901346, 0, 159161795612938602314783261228235271289518233757] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 23768465663982950 z + 11855326556031462 z + 347 z 24 22 4 6 - 4655529676765290 z + 1430316105950410 z - 46754 z + 3273282 z 8 10 12 14 - 136652930 z + 3696337966 z - 68442587790 z + 901499787518 z 18 16 50 + 62479772709934 z - 8684918423074 z + 8684918423074 z 48 20 36 - 62479772709934 z - 341043802262266 z - 37693445476590182 z 34 66 64 30 + 47430760546609492 z + z - 347 z + 37693445476590182 z 42 44 46 + 4655529676765290 z - 1430316105950410 z + 341043802262266 z 58 56 54 52 + 136652930 z - 3696337966 z + 68442587790 z - 901499787518 z 60 32 38 - 3273282 z - 47430760546609492 z + 23768465663982950 z 40 62 / 28 - 11855326556031462 z + 46754 z ) / (1 + 178760074320876224 z / 26 2 24 - 79048661450720464 z - 516 z + 27498055590281968 z 22 4 6 8 - 7484587371591172 z + 86017 z - 6954468 z + 326369184 z 10 12 14 - 9830656560 z + 202371861168 z - 2968921959628 z 18 16 50 - 257821247682736 z + 31960239843724 z - 257821247682736 z 48 20 36 + 1582975680535564 z + 1582975680535564 z + 451680957380569702 z 34 66 64 30 - 506955594743592088 z - 516 z + 86017 z - 319282379794660172 z 42 44 46 - 79048661450720464 z + 27498055590281968 z - 7484587371591172 z 58 56 54 - 9830656560 z + 202371861168 z - 2968921959628 z 52 60 68 32 + 31960239843724 z + 326369184 z + z + 451680957380569702 z 38 40 62 - 319282379794660172 z + 178760074320876224 z - 6954468 z ) And in Maple-input format, it is: -(-1-23768465663982950*z^28+11855326556031462*z^26+347*z^2-4655529676765290*z^ 24+1430316105950410*z^22-46754*z^4+3273282*z^6-136652930*z^8+3696337966*z^10-\ 68442587790*z^12+901499787518*z^14+62479772709934*z^18-8684918423074*z^16+ 8684918423074*z^50-62479772709934*z^48-341043802262266*z^20-37693445476590182*z ^36+47430760546609492*z^34+z^66-347*z^64+37693445476590182*z^30+ 4655529676765290*z^42-1430316105950410*z^44+341043802262266*z^46+136652930*z^58 -3696337966*z^56+68442587790*z^54-901499787518*z^52-3273282*z^60-\ 47430760546609492*z^32+23768465663982950*z^38-11855326556031462*z^40+46754*z^62 )/(1+178760074320876224*z^28-79048661450720464*z^26-516*z^2+27498055590281968*z ^24-7484587371591172*z^22+86017*z^4-6954468*z^6+326369184*z^8-9830656560*z^10+ 202371861168*z^12-2968921959628*z^14-257821247682736*z^18+31960239843724*z^16-\ 257821247682736*z^50+1582975680535564*z^48+1582975680535564*z^20+ 451680957380569702*z^36-506955594743592088*z^34-516*z^66+86017*z^64-\ 319282379794660172*z^30-79048661450720464*z^42+27498055590281968*z^44-\ 7484587371591172*z^46-9830656560*z^58+202371861168*z^56-2968921959628*z^54+ 31960239843724*z^52+326369184*z^60+z^68+451680957380569702*z^32-\ 319282379794660172*z^38+178760074320876224*z^40-6954468*z^62) The first , 40, terms are: [0, 169, 0, 47941, 0, 13881869, 0, 4024892245, 0, 1167149749533, 0, 338462114896673, 0, 98151259428256649, 0, 28463103364203406841, 0, 8254081850423141770417, 0, 2393620709056267512091629, 0, 694131746988953267960544709, 0, 201292913353110353670623941533, 0, 58373409998594054049671061979765, 0, 16927843809412227889437913884794137, 0, 4908945632954403705179793791874385553, 0, 1423556803810125987085476396232510010225, 0, 412820618770978981522739676716966636653113, 0, 119714831770111664465771199547323229290568213, 0, 34716388412078225958334006592564963162524653693, 0, 10067487934103384596399726645935831160750208784613] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 98810269226064 z - 86309740487554 z - 282 z 24 22 4 6 + 57463411143964 z - 29070046102992 z + 29272 z - 1532604 z 8 10 12 14 + 46698579 z - 901764144 z + 11666228490 z - 105061510758 z 18 16 50 48 - 3185766035522 z + 677007792512 z - 1532604 z + 46698579 z 20 36 34 + 11114252468102 z + 11114252468102 z - 29070046102992 z 30 42 44 46 - 86309740487554 z - 105061510758 z + 11666228490 z - 901764144 z 56 54 52 32 38 + z - 282 z + 29272 z + 57463411143964 z - 3185766035522 z 40 / 2 28 + 677007792512 z ) / ((-1 + z ) (1 + 491472415906864 z / 26 2 24 22 - 425133240578546 z - 409 z + 275047685123362 z - 132876366862064 z 4 6 8 10 12 + 52264 z - 3200621 z + 111320213 z - 2421880080 z + 35000339224 z 14 18 16 50 - 349521299000 z - 12706749339600 z + 2477768083904 z - 3200621 z 48 20 36 + 111320213 z + 47772103570224 z + 47772103570224 z 34 30 42 - 132876366862064 z - 425133240578546 z - 349521299000 z 44 46 56 54 52 + 35000339224 z - 2421880080 z + z - 409 z + 52264 z 32 38 40 + 275047685123362 z - 12706749339600 z + 2477768083904 z )) And in Maple-input format, it is: -(1+98810269226064*z^28-86309740487554*z^26-282*z^2+57463411143964*z^24-\ 29070046102992*z^22+29272*z^4-1532604*z^6+46698579*z^8-901764144*z^10+ 11666228490*z^12-105061510758*z^14-3185766035522*z^18+677007792512*z^16-1532604 *z^50+46698579*z^48+11114252468102*z^20+11114252468102*z^36-29070046102992*z^34 -86309740487554*z^30-105061510758*z^42+11666228490*z^44-901764144*z^46+z^56-282 *z^54+29272*z^52+57463411143964*z^32-3185766035522*z^38+677007792512*z^40)/(-1+ z^2)/(1+491472415906864*z^28-425133240578546*z^26-409*z^2+275047685123362*z^24-\ 132876366862064*z^22+52264*z^4-3200621*z^6+111320213*z^8-2421880080*z^10+ 35000339224*z^12-349521299000*z^14-12706749339600*z^18+2477768083904*z^16-\ 3200621*z^50+111320213*z^48+47772103570224*z^20+47772103570224*z^36-\ 132876366862064*z^34-425133240578546*z^30-349521299000*z^42+35000339224*z^44-\ 2421880080*z^46+z^56-409*z^54+52264*z^52+275047685123362*z^32-12706749339600*z^ 38+2477768083904*z^40) The first , 40, terms are: [0, 128, 0, 29079, 0, 6900527, 0, 1646084928, 0, 392986774121, 0, 93835309062265, 0, 22406107573567680, 0, 5350185919664479263, 0, 1277532080433239039463, 0, 305052690554759377200384, 0, 72841340328673309226955441, 0, 17393260546811449703837242577, 0, 4153211788532966042851315320832, 0, 991715620327358967089394770737607, 0, 236804651881005045540070404571382463, 0, 56544882429454705722263559441213694400, 0, 13501946450785733042238860114511534010329, 0, 3224032841306142836831994241245894213775881, 0, 769843651781208571521630581534908532135541312, 0, 183825437692445656758802634770136136484796980623] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {3, 6}, {3, 7}, {4, 6}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1188638479918040 z - 672843305321924 z - 272 z 24 22 4 6 + 302213821263792 z - 107270638826540 z + 27256 z - 1410700 z 8 10 12 14 + 44151248 z - 913719428 z + 13226318912 z - 139063824464 z 18 16 50 - 6503915965104 z + 1090613739100 z - 139063824464 z 48 20 36 + 1090613739100 z + 29916381410416 z + 1188638479918040 z 34 64 30 42 - 1670671010617072 z + z - 1670671010617072 z - 107270638826540 z 44 46 58 56 + 29916381410416 z - 6503915965104 z - 1410700 z + 44151248 z 54 52 60 32 - 913719428 z + 13226318912 z + 27256 z + 1871112572357190 z 38 40 62 / - 672843305321924 z + 302213821263792 z - 272 z ) / (-1 / 28 26 2 - 8998061243018030 z + 4545427012758154 z + 407 z 24 22 4 6 - 1821949110722398 z + 577248735070418 z - 49874 z + 2995354 z 8 10 12 14 - 106688702 z + 2490346842 z - 40473608550 z + 476696242526 z 18 16 50 + 27920750169922 z - 4183967622190 z + 4183967622190 z 48 20 36 - 27920750169922 z - 143756504936346 z - 14169158303755830 z 34 66 64 30 + 17774465985892088 z + z - 407 z + 14169158303755830 z 42 44 46 + 1821949110722398 z - 577248735070418 z + 143756504936346 z 58 56 54 52 + 106688702 z - 2490346842 z + 40473608550 z - 476696242526 z 60 32 38 - 2995354 z - 17774465985892088 z + 8998061243018030 z 40 62 - 4545427012758154 z + 49874 z ) And in Maple-input format, it is: -(1+1188638479918040*z^28-672843305321924*z^26-272*z^2+302213821263792*z^24-\ 107270638826540*z^22+27256*z^4-1410700*z^6+44151248*z^8-913719428*z^10+ 13226318912*z^12-139063824464*z^14-6503915965104*z^18+1090613739100*z^16-\ 139063824464*z^50+1090613739100*z^48+29916381410416*z^20+1188638479918040*z^36-\ 1670671010617072*z^34+z^64-1670671010617072*z^30-107270638826540*z^42+ 29916381410416*z^44-6503915965104*z^46-1410700*z^58+44151248*z^56-913719428*z^ 54+13226318912*z^52+27256*z^60+1871112572357190*z^32-672843305321924*z^38+ 302213821263792*z^40-272*z^62)/(-1-8998061243018030*z^28+4545427012758154*z^26+ 407*z^2-1821949110722398*z^24+577248735070418*z^22-49874*z^4+2995354*z^6-\ 106688702*z^8+2490346842*z^10-40473608550*z^12+476696242526*z^14+27920750169922 *z^18-4183967622190*z^16+4183967622190*z^50-27920750169922*z^48-143756504936346 *z^20-14169158303755830*z^36+17774465985892088*z^34+z^66-407*z^64+ 14169158303755830*z^30+1821949110722398*z^42-577248735070418*z^44+ 143756504936346*z^46+106688702*z^58-2490346842*z^56+40473608550*z^54-\ 476696242526*z^52-2995354*z^60-17774465985892088*z^32+8998061243018030*z^38-\ 4545427012758154*z^40+49874*z^62) The first , 40, terms are: [0, 135, 0, 32327, 0, 8008753, 0, 1989121009, 0, 494148164943, 0, 122761956126975, 0, 30498027811520369, 0, 7576696906332743633, 0, 1882296816654828987887, 0, 467623478275028309387487, 0, 116172814080663546444646641, 0, 28861088818997251759128081585, 0, 7170029016912684863481310640247, 0, 1781267381397867470846498975332567, 0, 442524496981537035617990423197501857, 0, 109937414491558743105185086841664451937, 0, 27312013656944745264924551253065312017847, 0, 6785188586133784674539948996104718233945239, 0, 1685660556840499662035510109061411280446608305, 0, 418772665905649785370631459523193395804825512177] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 6}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 6 4 2 z - 153 z + 153 z - 1 f(z) = - ---------------------------------- 8 6 4 2 z - 382 z + 2706 z - 382 z + 1 And in Maple-input format, it is: -(z^6-153*z^4+153*z^2-1)/(z^8-382*z^6+2706*z^4-382*z^2+1) The first , 40, terms are: [0, 229, 0, 84925, 0, 31822057, 0, 11926306201, 0, 4469770923661, 0, 1675192064199445, 0, 627832724221908049, 0, 235300738367611708849, 0, 88186606623581942864053, 0, 33050799762611005594598125, 0, 12386862436048215103197228601, 0, 4642379673461222258216921158729, 0, 1739882810827563143350167375659869, 0, 652077686088142823984587364593276741, 0, 244387326576219572647061244116781515425, 0, 91592101164152718725134639349741712942433, 0, 34327119630927291249149683474044430819324549, 0, 12865204828570702430093573106415236743705369629, 0, 4821654046730976457937879311955026563515921462665, 0, 1807071714453227599217541144794812962257446537557241] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 23226357193600 z - 27058678259956 z - 310 z 24 22 4 6 + 23226357193600 z - 14668234498122 z + 32959 z - 1719894 z 8 10 12 14 + 51076768 z - 944097530 z + 11487666154 z - 95380270128 z 18 16 50 48 - 2283682543014 z + 553497048086 z - 310 z + 32959 z 20 36 34 + 6786189237312 z + 553497048086 z - 2283682543014 z 30 42 44 46 52 - 14668234498122 z - 944097530 z + 51076768 z - 1719894 z + z 32 38 40 / + 6786189237312 z - 95380270128 z + 11487666154 z ) / (-1 / 28 26 2 24 - 246034573985228 z + 246034573985228 z + 430 z - 182891267824008 z 22 4 6 8 + 100793539769740 z - 59862 z + 3894375 z - 139318020 z 10 12 14 18 + 3017763764 z - 42207055554 z + 398133345680 z + 12167581053614 z 16 50 48 20 - 2610898846912 z + 59862 z - 3894375 z - 40958210888412 z 36 34 30 - 12167581053614 z + 40958210888412 z + 182891267824008 z 42 44 46 54 52 + 42207055554 z - 3017763764 z + 139318020 z + z - 430 z 32 38 40 - 100793539769740 z + 2610898846912 z - 398133345680 z ) And in Maple-input format, it is: -(1+23226357193600*z^28-27058678259956*z^26-310*z^2+23226357193600*z^24-\ 14668234498122*z^22+32959*z^4-1719894*z^6+51076768*z^8-944097530*z^10+ 11487666154*z^12-95380270128*z^14-2283682543014*z^18+553497048086*z^16-310*z^50 +32959*z^48+6786189237312*z^20+553497048086*z^36-2283682543014*z^34-\ 14668234498122*z^30-944097530*z^42+51076768*z^44-1719894*z^46+z^52+ 6786189237312*z^32-95380270128*z^38+11487666154*z^40)/(-1-246034573985228*z^28+ 246034573985228*z^26+430*z^2-182891267824008*z^24+100793539769740*z^22-59862*z^ 4+3894375*z^6-139318020*z^8+3017763764*z^10-42207055554*z^12+398133345680*z^14+ 12167581053614*z^18-2610898846912*z^16+59862*z^50-3894375*z^48-40958210888412*z ^20-12167581053614*z^36+40958210888412*z^34+182891267824008*z^30+42207055554*z^ 42-3017763764*z^44+139318020*z^46+z^54-430*z^52-100793539769740*z^32+ 2610898846912*z^38-398133345680*z^40) The first , 40, terms are: [0, 120, 0, 24697, 0, 5610751, 0, 1313294864, 0, 310380898367, 0, 73588372697007, 0, 17465530505713216, 0, 4146740174023078591, 0, 984651684060659207385, 0, 233816599523214117456296, 0, 55523097394453571809577489, 0, 13184810978490512377472232753, 0, 3130940078263094474539812590664, 0, 743491165266628869200310319180377, 0, 176553746611510864227559303466675967, 0, 41925484202103934430747261566778228448, 0, 9955870608996115345492604250021046587791, 0, 2364179258893517776921886067014241503457247, 0, 561411834154918270633415268845432265191872368, 0, 133316137746612553101934285238567913226703432767] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 575507 z + 8694064 z + 256 z - 70880457 z + 326918070 z 4 6 8 10 12 - 18923 z + 575507 z - 8694064 z + 70880457 z - 326918070 z 14 18 16 20 34 + 883106000 z + 1439020930 z - 1439020930 z - 883106000 z + z 30 32 / 8 36 28 + 18923 z - 256 z ) / (30318464 z + z + 30318464 z / 18 22 10 20 - 14208612932 z - 5585702336 z - 303011784 z + 1 + 11270559568 z 6 34 30 12 32 - 1572062 z - 376 z - 1572062 z + 1693446143 z + 38448 z 26 16 4 14 - 303011784 z + 11270559568 z + 38448 z - 5585702336 z 24 2 + 1693446143 z - 376 z ) And in Maple-input format, it is: -(-1-575507*z^28+8694064*z^26+256*z^2-70880457*z^24+326918070*z^22-18923*z^4+ 575507*z^6-8694064*z^8+70880457*z^10-326918070*z^12+883106000*z^14+1439020930*z ^18-1439020930*z^16-883106000*z^20+z^34+18923*z^30-256*z^32)/(30318464*z^8+z^36 +30318464*z^28-14208612932*z^18-5585702336*z^22-303011784*z^10+1+11270559568*z^ 20-1572062*z^6-376*z^34-1572062*z^30+1693446143*z^12+38448*z^32-303011784*z^26+ 11270559568*z^16+38448*z^4-5585702336*z^14+1693446143*z^24-376*z^2) The first , 40, terms are: [0, 120, 0, 25595, 0, 6006515, 0, 1441396120, 0, 347857294937, 0, 84077152658409, 0, 20330004957901144, 0, 4916413178704095683, 0, 1188978052386621735083, 0, 287543429127232078293176, 0, 69539928279083850784443633, 0, 16817652166683219798929370129, 0, 4067209931290707000099233329208, 0, 983621108581053606265072297433675, 0, 237880640287400570141032912832491299, 0, 57529468227133638883535454480226267608, 0, 13913026782411711620043593719407237716425, 0, 3364750627705696019858869372450498326024825, 0, 813737151883047470808602432708439407798986008, 0, 196795610029710016597941083998223749294078035859] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 564557935935 z + 1135853325928 z + 336 z 24 22 4 6 - 1607899141327 z + 1607899141327 z - 38713 z + 1988589 z 8 10 12 14 - 52419952 z + 797118341 z - 7551085547 z + 46801384360 z 18 16 20 36 + 564557935935 z - 195841117451 z - 1135853325928 z - 797118341 z 34 30 42 44 46 + 7551085547 z + 195841117451 z + 38713 z - 336 z + z 32 38 40 / 28 - 46801384360 z + 52419952 z - 1988589 z ) / (1 + 8310354059852 z / 26 2 24 22 - 13802868821956 z - 468 z + 16337762991470 z - 13802868821956 z 4 6 8 10 12 + 71340 z - 4670644 z + 152637208 z - 2776503956 z + 30746032888 z 14 18 16 48 - 220348884756 z - 3547757154764 z + 1063622034432 z + z 20 36 34 + 8310354059852 z + 30746032888 z - 220348884756 z 30 42 44 46 - 3547757154764 z - 4670644 z + 71340 z - 468 z 32 38 40 + 1063622034432 z - 2776503956 z + 152637208 z ) And in Maple-input format, it is: -(-1-564557935935*z^28+1135853325928*z^26+336*z^2-1607899141327*z^24+ 1607899141327*z^22-38713*z^4+1988589*z^6-52419952*z^8+797118341*z^10-7551085547 *z^12+46801384360*z^14+564557935935*z^18-195841117451*z^16-1135853325928*z^20-\ 797118341*z^36+7551085547*z^34+195841117451*z^30+38713*z^42-336*z^44+z^46-\ 46801384360*z^32+52419952*z^38-1988589*z^40)/(1+8310354059852*z^28-\ 13802868821956*z^26-468*z^2+16337762991470*z^24-13802868821956*z^22+71340*z^4-\ 4670644*z^6+152637208*z^8-2776503956*z^10+30746032888*z^12-220348884756*z^14-\ 3547757154764*z^18+1063622034432*z^16+z^48+8310354059852*z^20+30746032888*z^36-\ 220348884756*z^34-3547757154764*z^30-4670644*z^42+71340*z^44-468*z^46+ 1063622034432*z^32-2776503956*z^38+152637208*z^40) The first , 40, terms are: [0, 132, 0, 29149, 0, 6906907, 0, 1669250568, 0, 406446396559, 0, 99286363405459, 0, 24289403677738512, 0, 5946237489862600999, 0, 1456154107585495215817, 0, 356646572653489779710220, 0, 87357386599918530288996037, 0, 21398125565201538275514848029, 0, 5241537082069951510050785419260, 0, 1283940235468129071856552631699089, 0, 314508558376509048182329230223575535, 0, 77040809787317753862367162952104313088, 0, 18871635743170348036024379837042700747115, 0, 4622728731779725703171694968304831067442743, 0, 1132367372363395283324885443178493480427813656, 0, 277380751389066067924916954769178959897308813011] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 41371596156820 z - 36393631701300 z - 296 z 24 22 4 6 + 24748590863796 z - 12969605086580 z + 29362 z - 1414433 z 8 10 12 14 + 38674402 z - 664484707 z + 7670254986 z - 62212694724 z 18 16 50 48 - 1591581587676 z + 365665091675 z - 1414433 z + 38674402 z 20 36 34 + 5209934060284 z + 5209934060284 z - 12969605086580 z 30 42 44 46 - 36393631701300 z - 62212694724 z + 7670254986 z - 664484707 z 56 54 52 32 38 + z - 296 z + 29362 z + 24748590863796 z - 1591581587676 z 40 / 28 26 + 365665091675 z ) / (-1 - 391476274945674 z + 304226431062018 z / 2 24 22 4 + 425 z - 183403863930654 z + 85454039821674 z - 55247 z 6 8 10 12 + 3294037 z - 107478697 z + 2145739857 z - 28275860509 z 14 18 16 50 + 259033411871 z + 8339542934273 z - 1709879142513 z + 107478697 z 48 20 36 - 2145739857 z - 30590216689938 z - 85454039821674 z 34 30 42 + 183403863930654 z + 391476274945674 z + 1709879142513 z 44 46 58 56 54 - 259033411871 z + 28275860509 z + z - 425 z + 55247 z 52 32 38 - 3294037 z - 304226431062018 z + 30590216689938 z 40 - 8339542934273 z ) And in Maple-input format, it is: -(1+41371596156820*z^28-36393631701300*z^26-296*z^2+24748590863796*z^24-\ 12969605086580*z^22+29362*z^4-1414433*z^6+38674402*z^8-664484707*z^10+ 7670254986*z^12-62212694724*z^14-1591581587676*z^18+365665091675*z^16-1414433*z ^50+38674402*z^48+5209934060284*z^20+5209934060284*z^36-12969605086580*z^34-\ 36393631701300*z^30-62212694724*z^42+7670254986*z^44-664484707*z^46+z^56-296*z^ 54+29362*z^52+24748590863796*z^32-1591581587676*z^38+365665091675*z^40)/(-1-\ 391476274945674*z^28+304226431062018*z^26+425*z^2-183403863930654*z^24+ 85454039821674*z^22-55247*z^4+3294037*z^6-107478697*z^8+2145739857*z^10-\ 28275860509*z^12+259033411871*z^14+8339542934273*z^18-1709879142513*z^16+ 107478697*z^50-2145739857*z^48-30590216689938*z^20-85454039821674*z^36+ 183403863930654*z^34+391476274945674*z^30+1709879142513*z^42-259033411871*z^44+ 28275860509*z^46+z^58-425*z^56+55247*z^54-3294037*z^52-304226431062018*z^32+ 30590216689938*z^38-8339542934273*z^40) The first , 40, terms are: [0, 129, 0, 28940, 0, 7052241, 0, 1754480723, 0, 438985082765, 0, 110014967803311, 0, 27583758138801060, 0, 6916910128105061655, 0, 1734551281983583775959, 0, 434977532182961737157095, 0, 109080681474980022231864871, 0, 27354529090923516164938344356, 0, 6859788924317397389213611300719, 0, 1720252881373514381299829438672653, 0, 431393745459997216178580916019289395, 0, 108182097376615977977810693473019714353, 0, 27129197712850460502163473565492112753740, 0, 6803282490803679536087454180321694248287089, 0, 1706082617924095271153316012798091654765607345, 0, 427840223191283670365498762257797461380725238929] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 5186333721275935 z + 3145290045080870 z + 376 z 24 22 4 6 - 1480991042375449 z + 539410070487009 z - 52397 z + 3599681 z 8 10 12 14 - 139465880 z + 3378457489 z - 54919764415 z + 628380016768 z 18 16 50 + 32345759121823 z - 5227990837593 z + 54919764415 z 48 20 36 - 628380016768 z - 151161161880558 z - 3145290045080870 z 34 30 42 + 5186333721275935 z + 6655788600780449 z + 151161161880558 z 44 46 58 56 - 32345759121823 z + 5227990837593 z + 52397 z - 3599681 z 54 52 60 32 + 139465880 z - 3378457489 z - 376 z - 6655788600780449 z 38 40 62 / + 1480991042375449 z - 539410070487009 z + z ) / (1 / 28 26 2 + 41189544288901188 z - 22287567538115756 z - 508 z 24 22 4 6 + 9398307166153792 z - 3075763904743980 z + 90532 z - 7645700 z 8 10 12 14 + 353377292 z - 9931260088 z + 183532251904 z - 2356220043488 z 18 16 50 - 149855264036844 z + 21829362333834 z - 2356220043488 z 48 20 36 + 21829362333834 z + 776454870948820 z + 41189544288901188 z 34 64 30 - 59481790932170428 z + z - 59481790932170428 z 42 44 46 - 3075763904743980 z + 776454870948820 z - 149855264036844 z 58 56 54 52 - 7645700 z + 353377292 z - 9931260088 z + 183532251904 z 60 32 38 + 90532 z + 67222048498087747 z - 22287567538115756 z 40 62 + 9398307166153792 z - 508 z ) And in Maple-input format, it is: -(-1-5186333721275935*z^28+3145290045080870*z^26+376*z^2-1480991042375449*z^24+ 539410070487009*z^22-52397*z^4+3599681*z^6-139465880*z^8+3378457489*z^10-\ 54919764415*z^12+628380016768*z^14+32345759121823*z^18-5227990837593*z^16+ 54919764415*z^50-628380016768*z^48-151161161880558*z^20-3145290045080870*z^36+ 5186333721275935*z^34+6655788600780449*z^30+151161161880558*z^42-32345759121823 *z^44+5227990837593*z^46+52397*z^58-3599681*z^56+139465880*z^54-3378457489*z^52 -376*z^60-6655788600780449*z^32+1480991042375449*z^38-539410070487009*z^40+z^62 )/(1+41189544288901188*z^28-22287567538115756*z^26-508*z^2+9398307166153792*z^ 24-3075763904743980*z^22+90532*z^4-7645700*z^6+353377292*z^8-9931260088*z^10+ 183532251904*z^12-2356220043488*z^14-149855264036844*z^18+21829362333834*z^16-\ 2356220043488*z^50+21829362333834*z^48+776454870948820*z^20+41189544288901188*z ^36-59481790932170428*z^34+z^64-59481790932170428*z^30-3075763904743980*z^42+ 776454870948820*z^44-149855264036844*z^46-7645700*z^58+353377292*z^56-\ 9931260088*z^54+183532251904*z^52+90532*z^60+67222048498087747*z^32-\ 22287567538115756*z^38+9398307166153792*z^40-508*z^62) The first , 40, terms are: [0, 132, 0, 28921, 0, 6787663, 0, 1625177820, 0, 392117915599, 0, 94924026905347, 0, 23013926177113452, 0, 5583524034483592807, 0, 1355088199020050442673, 0, 328922120527277854264644, 0, 79845377449811499759079345, 0, 19383006669752725889978613997, 0, 4705431107939829294593557498740, 0, 1142302025014440223341476120168845, 0, 277309026772530227512527539235387499, 0, 67320570656286323220892327186380252204, 0, 16343004901577929152468015225549544795903, 0, 3967493222312262168728013308457689186603899, 0, 963164687376409461672676218415683429588119244, 0, 233821770839433259036103723298475723465607015299] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 10304786263653845 z - 4911756107694792 z - 350 z 24 22 4 6 + 1881638637979530 z - 576188290298877 z + 44304 z - 2752529 z 8 10 12 14 + 99735579 z - 2342744927 z + 38146327178 z - 450141356273 z 18 16 50 - 26753959362943 z + 3970039454204 z - 26753959362943 z 48 20 36 + 140024204421843 z + 140024204421843 z + 23901027271631342 z 34 66 64 30 - 26540677846559058 z - 350 z + 44304 z - 17445994896501894 z 42 44 46 - 4911756107694792 z + 1881638637979530 z - 576188290298877 z 58 56 54 52 - 2342744927 z + 38146327178 z - 450141356273 z + 3970039454204 z 60 68 32 38 + 99735579 z + z + 23901027271631342 z - 17445994896501894 z 40 62 / 28 + 10304786263653845 z - 2752529 z ) / (-1 - 75149756379319266 z / 26 2 24 + 32441485883798354 z + 491 z - 11264694527208950 z 22 4 6 8 + 3128024969046453 z - 80107 z + 5993231 z - 252752640 z 10 12 14 + 6773038470 z - 124184171474 z + 1635784894450 z 18 16 50 + 119328371839123 z - 16011713793803 z + 689308260169047 z 48 20 36 - 3128024969046453 z - 689308260169047 z - 262385736098917189 z 34 66 64 + 262385736098917189 z + 80107 z - 5993231 z 30 42 44 + 140621674781032644 z + 75149756379319266 z - 32441485883798354 z 46 58 56 + 11264694527208950 z + 124184171474 z - 1635784894450 z 54 52 60 70 + 16011713793803 z - 119328371839123 z - 6773038470 z + z 68 32 38 - 491 z - 213196088842227037 z + 213196088842227037 z 40 62 - 140621674781032644 z + 252752640 z ) And in Maple-input format, it is: -(1+10304786263653845*z^28-4911756107694792*z^26-350*z^2+1881638637979530*z^24-\ 576188290298877*z^22+44304*z^4-2752529*z^6+99735579*z^8-2342744927*z^10+ 38146327178*z^12-450141356273*z^14-26753959362943*z^18+3970039454204*z^16-\ 26753959362943*z^50+140024204421843*z^48+140024204421843*z^20+23901027271631342 *z^36-26540677846559058*z^34-350*z^66+44304*z^64-17445994896501894*z^30-\ 4911756107694792*z^42+1881638637979530*z^44-576188290298877*z^46-2342744927*z^ 58+38146327178*z^56-450141356273*z^54+3970039454204*z^52+99735579*z^60+z^68+ 23901027271631342*z^32-17445994896501894*z^38+10304786263653845*z^40-2752529*z^ 62)/(-1-75149756379319266*z^28+32441485883798354*z^26+491*z^2-11264694527208950 *z^24+3128024969046453*z^22-80107*z^4+5993231*z^6-252752640*z^8+6773038470*z^10 -124184171474*z^12+1635784894450*z^14+119328371839123*z^18-16011713793803*z^16+ 689308260169047*z^50-3128024969046453*z^48-689308260169047*z^20-\ 262385736098917189*z^36+262385736098917189*z^34+80107*z^66-5993231*z^64+ 140621674781032644*z^30+75149756379319266*z^42-32441485883798354*z^44+ 11264694527208950*z^46+124184171474*z^58-1635784894450*z^56+16011713793803*z^54 -119328371839123*z^52-6773038470*z^60+z^70-491*z^68-213196088842227037*z^32+ 213196088842227037*z^38-140621674781032644*z^40+252752640*z^62) The first , 40, terms are: [0, 141, 0, 33428, 0, 8358763, 0, 2118364347, 0, 539655363907, 0, 137790913796515, 0, 35218398701277156, 0, 9005755704799601517, 0, 2303361497557565761625, 0, 589176598086826578469609, 0, 150711942946158460340210917, 0, 38553017180472641553803552484, 0, 9862180339557656794576570413995, 0, 2522837397529439283880496359301955, 0, 645366436350800729908531046917660539, 0, 165091172006027060386883655175342679539, 0, 42231984486996150085768840198018028693108, 0, 10803368807100558942727769480414111154742629, 0, 2763611230409176525843208125230946997317634705, 0, 706959786073883090134410382433685541354536844977] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1203715834280826969730 z - 165789049564646796853 z - 443 z 24 22 4 + 19451465308925658168 z - 1930469597765239139 z + 81714 z 6 102 8 10 - 8603535 z - 28729919445 z + 591827544 z - 28729919445 z 12 14 18 + 1035242217650 z - 28698981550897 z - 11102389475684235 z 16 50 + 628588509455924 z - 282802737022210858835788519 z 48 20 + 179935957586184899988045320 z + 160690471251031594 z 36 34 + 746041471444370107551330 z - 185733652581257898038931 z 66 80 - 100505905599866112637359657 z + 40110340762152578311048 z 100 90 88 + 1035242217650 z - 1930469597765239139 z + 19451465308925658168 z 84 94 + 1203715834280826969730 z - 11102389475684235 z 86 96 98 - 165789049564646796853 z + 628588509455924 z - 28698981550897 z 92 82 + 160690471251031594 z - 7488380119217250771865 z 64 112 110 106 + 179935957586184899988045320 z + z - 443 z - 8603535 z 108 30 42 + 81714 z - 7488380119217250771865 z - 21140627222719076327435525 z 44 46 + 49243031802597494531994778 z - 100505905599866112637359657 z 58 56 - 473728603998471651845498211 z + 505251810486646228206778854 z 54 52 - 473728603998471651845498211 z + 390443618342620778123015674 z 60 70 + 390443618342620778123015674 z - 21140627222719076327435525 z 68 78 + 49243031802597494531994778 z - 185733652581257898038931 z 32 38 + 40110340762152578311048 z - 2606775512144485739469283 z 40 62 + 7942151003210658507114732 z - 282802737022210858835788519 z 76 74 + 746041471444370107551330 z - 2606775512144485739469283 z 72 104 / 2 + 7942151003210658507114732 z + 591827544 z ) / ((-1 + z ) (1 / 28 26 2 + 4535143797159559046364 z - 598992240525753440784 z - 577 z 24 22 4 + 67175859894147432126 z - 6350057463719269026 z + 127048 z 6 102 8 10 - 15341740 z - 63224862430 z + 1181908494 z - 63224862430 z 12 14 18 + 2480815649460 z - 74201726467145 z - 32721354248533243 z 16 50 + 1740540895243016 z - 1406724475779994901031207731 z 48 20 + 885140736639572081200586176 z + 501458278664685868 z 36 34 + 3230435404341609374824340 z - 779899793947448202567693 z 66 80 - 487455774823511352434007409 z + 162893527856925247657840 z 100 90 88 + 2480815649460 z - 6350057463719269026 z + 67175859894147432126 z 84 94 + 4535143797159559046364 z - 32721354248533243 z 86 96 98 - 598992240525753440784 z + 1740540895243016 z - 74201726467145 z 92 82 + 501458278664685868 z - 29333083831024954602639 z 64 112 110 106 + 885140736639572081200586176 z + z - 577 z - 15341740 z 108 30 + 127048 z - 29333083831024954602639 z 42 44 - 98799568697346578843862270 z + 234777374407404398682508140 z 46 58 - 487455774823511352434007409 z - 2386879605624507482170088878 z 56 54 + 2549822678622613935372614356 z - 2386879605624507482170088878 z 52 60 + 1957768837897100545453030424 z + 1957768837897100545453030424 z 70 68 - 98799568697346578843862270 z + 234777374407404398682508140 z 78 32 - 779899793947448202567693 z + 162893527856925247657840 z 38 40 - 11609348537976228583621444 z + 36282391705779552803750758 z 62 76 - 1406724475779994901031207731 z + 3230435404341609374824340 z 74 72 - 11609348537976228583621444 z + 36282391705779552803750758 z 104 + 1181908494 z )) And in Maple-input format, it is: -(1+1203715834280826969730*z^28-165789049564646796853*z^26-443*z^2+ 19451465308925658168*z^24-1930469597765239139*z^22+81714*z^4-8603535*z^6-\ 28729919445*z^102+591827544*z^8-28729919445*z^10+1035242217650*z^12-\ 28698981550897*z^14-11102389475684235*z^18+628588509455924*z^16-\ 282802737022210858835788519*z^50+179935957586184899988045320*z^48+ 160690471251031594*z^20+746041471444370107551330*z^36-185733652581257898038931* z^34-100505905599866112637359657*z^66+40110340762152578311048*z^80+ 1035242217650*z^100-1930469597765239139*z^90+19451465308925658168*z^88+ 1203715834280826969730*z^84-11102389475684235*z^94-165789049564646796853*z^86+ 628588509455924*z^96-28698981550897*z^98+160690471251031594*z^92-\ 7488380119217250771865*z^82+179935957586184899988045320*z^64+z^112-443*z^110-\ 8603535*z^106+81714*z^108-7488380119217250771865*z^30-\ 21140627222719076327435525*z^42+49243031802597494531994778*z^44-\ 100505905599866112637359657*z^46-473728603998471651845498211*z^58+ 505251810486646228206778854*z^56-473728603998471651845498211*z^54+ 390443618342620778123015674*z^52+390443618342620778123015674*z^60-\ 21140627222719076327435525*z^70+49243031802597494531994778*z^68-\ 185733652581257898038931*z^78+40110340762152578311048*z^32-\ 2606775512144485739469283*z^38+7942151003210658507114732*z^40-\ 282802737022210858835788519*z^62+746041471444370107551330*z^76-\ 2606775512144485739469283*z^74+7942151003210658507114732*z^72+591827544*z^104)/ (-1+z^2)/(1+4535143797159559046364*z^28-598992240525753440784*z^26-577*z^2+ 67175859894147432126*z^24-6350057463719269026*z^22+127048*z^4-15341740*z^6-\ 63224862430*z^102+1181908494*z^8-63224862430*z^10+2480815649460*z^12-\ 74201726467145*z^14-32721354248533243*z^18+1740540895243016*z^16-\ 1406724475779994901031207731*z^50+885140736639572081200586176*z^48+ 501458278664685868*z^20+3230435404341609374824340*z^36-779899793947448202567693 *z^34-487455774823511352434007409*z^66+162893527856925247657840*z^80+ 2480815649460*z^100-6350057463719269026*z^90+67175859894147432126*z^88+ 4535143797159559046364*z^84-32721354248533243*z^94-598992240525753440784*z^86+ 1740540895243016*z^96-74201726467145*z^98+501458278664685868*z^92-\ 29333083831024954602639*z^82+885140736639572081200586176*z^64+z^112-577*z^110-\ 15341740*z^106+127048*z^108-29333083831024954602639*z^30-\ 98799568697346578843862270*z^42+234777374407404398682508140*z^44-\ 487455774823511352434007409*z^46-2386879605624507482170088878*z^58+ 2549822678622613935372614356*z^56-2386879605624507482170088878*z^54+ 1957768837897100545453030424*z^52+1957768837897100545453030424*z^60-\ 98799568697346578843862270*z^70+234777374407404398682508140*z^68-\ 779899793947448202567693*z^78+162893527856925247657840*z^32-\ 11609348537976228583621444*z^38+36282391705779552803750758*z^40-\ 1406724475779994901031207731*z^62+3230435404341609374824340*z^76-\ 11609348537976228583621444*z^74+36282391705779552803750758*z^72+1181908494*z^ 104) The first , 40, terms are: [0, 135, 0, 32119, 0, 8200660, 0, 2123657795, 0, 551755044671, 0, 143468496247697, 0, 37312442533249025, 0, 9704488954469863911, 0, 2524045499850336264619, 0, 656482396612772235198052, 0, 170745530520612703666064343, 0, 44409480524364407385503721519, 0, 11550534046519267035308088552473, 0, 3004197233410890820496377859096137, 0, 781366559435738682129773641170127439, 0, 203226903325448798765704291823500655975, 0, 52857616883435981742538995828778673484900, 0, 13747823821707055056006044616785850407009771, 0, 3575693929828950232771365395370011016229399175, 0, 930008068598683219052995821657898496143101868641] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 10399649333049092452 z - 2410235905260318876 z - 389 z 24 22 4 6 + 462960502187657728 z - 73217102066107644 z + 60304 z - 5140693 z 8 10 12 14 + 276863465 z - 10200412152 z + 270919763408 z - 5383587008128 z 18 16 50 - 987297824417480 z + 82271144598208 z - 1084766744648346392968 z 48 20 + 1645520295348306096606 z + 9456273984751828 z 36 34 + 604380948746142959056 z - 284139496194645727616 z 66 80 88 84 86 - 73217102066107644 z + 276863465 z + z + 60304 z - 389 z 82 64 30 - 5140693 z + 462960502187657728 z - 37388069976896170936 z 42 44 - 2112000356045617769486 z + 2295054080199383404768 z 46 58 - 2112000356045617769486 z - 37388069976896170936 z 56 54 + 112478679719108662688 z - 284139496194645727616 z 52 60 + 604380948746142959056 z + 10399649333049092452 z 70 68 78 - 987297824417480 z + 9456273984751828 z - 10200412152 z 32 38 + 112478679719108662688 z - 1084766744648346392968 z 40 62 76 + 1645520295348306096606 z - 2410235905260318876 z + 270919763408 z 74 72 / - 5383587008128 z + 82271144598208 z ) / (-1 / 28 26 2 - 54377771557922382900 z + 11700224893484448164 z + 530 z 24 22 4 6 - 2085110476869848700 z + 305558538439171084 z - 99450 z + 9839142 z 8 10 12 14 - 602336222 z + 24892955041 z - 734246383940 z + 16073800165572 z 18 16 50 + 3513101618083932 z - 268851729904940 z + 13559376463359834639590 z 48 20 - 18847158255546903128528 z - 36493641870692956 z 36 34 - 4263984560734252474092 z + 1858087076501598053412 z 66 80 90 88 84 + 2085110476869848700 z - 24892955041 z + z - 530 z - 9839142 z 86 82 64 + 99450 z + 602336222 z - 11700224893484448164 z 30 42 + 210544993515063555868 z + 18847158255546903128528 z 44 46 - 22215729079846487978048 z + 22215729079846487978048 z 58 56 + 682357040986604865044 z - 1858087076501598053412 z 54 52 + 4263984560734252474092 z - 8265744084476360884300 z 60 70 - 210544993515063555868 z + 36493641870692956 z 68 78 32 - 305558538439171084 z + 734246383940 z - 682357040986604865044 z 38 40 + 8265744084476360884300 z - 13559376463359834639590 z 62 76 74 + 54377771557922382900 z - 16073800165572 z + 268851729904940 z 72 - 3513101618083932 z ) And in Maple-input format, it is: -(1+10399649333049092452*z^28-2410235905260318876*z^26-389*z^2+ 462960502187657728*z^24-73217102066107644*z^22+60304*z^4-5140693*z^6+276863465* z^8-10200412152*z^10+270919763408*z^12-5383587008128*z^14-987297824417480*z^18+ 82271144598208*z^16-1084766744648346392968*z^50+1645520295348306096606*z^48+ 9456273984751828*z^20+604380948746142959056*z^36-284139496194645727616*z^34-\ 73217102066107644*z^66+276863465*z^80+z^88+60304*z^84-389*z^86-5140693*z^82+ 462960502187657728*z^64-37388069976896170936*z^30-2112000356045617769486*z^42+ 2295054080199383404768*z^44-2112000356045617769486*z^46-37388069976896170936*z^ 58+112478679719108662688*z^56-284139496194645727616*z^54+604380948746142959056* z^52+10399649333049092452*z^60-987297824417480*z^70+9456273984751828*z^68-\ 10200412152*z^78+112478679719108662688*z^32-1084766744648346392968*z^38+ 1645520295348306096606*z^40-2410235905260318876*z^62+270919763408*z^76-\ 5383587008128*z^74+82271144598208*z^72)/(-1-54377771557922382900*z^28+ 11700224893484448164*z^26+530*z^2-2085110476869848700*z^24+305558538439171084*z ^22-99450*z^4+9839142*z^6-602336222*z^8+24892955041*z^10-734246383940*z^12+ 16073800165572*z^14+3513101618083932*z^18-268851729904940*z^16+ 13559376463359834639590*z^50-18847158255546903128528*z^48-36493641870692956*z^ 20-4263984560734252474092*z^36+1858087076501598053412*z^34+2085110476869848700* z^66-24892955041*z^80+z^90-530*z^88-9839142*z^84+99450*z^86+602336222*z^82-\ 11700224893484448164*z^64+210544993515063555868*z^30+18847158255546903128528*z^ 42-22215729079846487978048*z^44+22215729079846487978048*z^46+ 682357040986604865044*z^58-1858087076501598053412*z^56+4263984560734252474092*z ^54-8265744084476360884300*z^52-210544993515063555868*z^60+36493641870692956*z^ 70-305558538439171084*z^68+734246383940*z^78-682357040986604865044*z^32+ 8265744084476360884300*z^38-13559376463359834639590*z^40+54377771557922382900*z ^62-16073800165572*z^76+268851729904940*z^74-3513101618083932*z^72) The first , 40, terms are: [0, 141, 0, 35584, 0, 9535519, 0, 2576842535, 0, 697298343515, 0, 188735505358499, 0, 51086781086874320, 0, 13828259436800244649, 0, 3743064903080546895533, 0, 1013181812343537729153509, 0, 274250517198632553276963489, 0, 74234798698567759359623305744, 0, 20094056424298888538295177212459, 0, 5439108223736611305323147812076675, 0, 1472271086219929563042665219520259743, 0, 398517930203226613194592942935743779959, 0, 107871805799009750895621279092547424852352, 0, 29199003619370329592472117660440281328849077, 0, 7903657550272445255533748338602822930566772969, 0, 2139381311989477689715754218687839742082115017049] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 13797261365138620 z - 6417156222206490 z - 327 z 24 22 4 6 + 2382550726459430 z - 702591572561704 z + 38792 z - 2373345 z 8 10 12 14 + 87560863 z - 2133480644 z + 36390014915 z - 451898388077 z 18 16 50 - 29779132658779 z + 4199246891972 z - 29779132658779 z 48 20 36 + 163502565716813 z + 163502565716813 z + 32911582939569210 z 34 66 64 30 - 36674440803665336 z - 327 z + 38792 z - 23771931958078662 z 42 44 46 - 6417156222206490 z + 2382550726459430 z - 702591572561704 z 58 56 54 52 - 2133480644 z + 36390014915 z - 451898388077 z + 4199246891972 z 60 68 32 38 + 87560863 z + z + 32911582939569210 z - 23771931958078662 z 40 62 / 2 + 13797261365138620 z - 2373345 z ) / ((-1 + z ) (1 / 28 26 2 + 68331806412012466 z - 30673932233975112 z - 466 z 24 22 4 6 + 10897623049636334 z - 3051261434992692 z + 69747 z - 5054068 z 8 10 12 14 + 213330253 z - 5822922262 z + 109672001761 z - 1487833493052 z 18 16 50 - 114173658862406 z + 14974546303711 z - 114173658862406 z 48 20 36 + 669425820481773 z + 669425820481773 z + 169983559086168942 z 34 66 64 30 - 190436562934350812 z - 466 z + 69747 z - 120838356395769776 z 42 44 46 - 30673932233975112 z + 10897623049636334 z - 3051261434992692 z 58 56 54 - 5822922262 z + 109672001761 z - 1487833493052 z 52 60 68 32 + 14974546303711 z + 213330253 z + z + 169983559086168942 z 38 40 62 - 120838356395769776 z + 68331806412012466 z - 5054068 z )) And in Maple-input format, it is: -(1+13797261365138620*z^28-6417156222206490*z^26-327*z^2+2382550726459430*z^24-\ 702591572561704*z^22+38792*z^4-2373345*z^6+87560863*z^8-2133480644*z^10+ 36390014915*z^12-451898388077*z^14-29779132658779*z^18+4199246891972*z^16-\ 29779132658779*z^50+163502565716813*z^48+163502565716813*z^20+32911582939569210 *z^36-36674440803665336*z^34-327*z^66+38792*z^64-23771931958078662*z^30-\ 6417156222206490*z^42+2382550726459430*z^44-702591572561704*z^46-2133480644*z^ 58+36390014915*z^56-451898388077*z^54+4199246891972*z^52+87560863*z^60+z^68+ 32911582939569210*z^32-23771931958078662*z^38+13797261365138620*z^40-2373345*z^ 62)/(-1+z^2)/(1+68331806412012466*z^28-30673932233975112*z^26-466*z^2+ 10897623049636334*z^24-3051261434992692*z^22+69747*z^4-5054068*z^6+213330253*z^ 8-5822922262*z^10+109672001761*z^12-1487833493052*z^14-114173658862406*z^18+ 14974546303711*z^16-114173658862406*z^50+669425820481773*z^48+669425820481773*z ^20+169983559086168942*z^36-190436562934350812*z^34-466*z^66+69747*z^64-\ 120838356395769776*z^30-30673932233975112*z^42+10897623049636334*z^44-\ 3051261434992692*z^46-5822922262*z^58+109672001761*z^56-1487833493052*z^54+ 14974546303711*z^52+213330253*z^60+z^68+169983559086168942*z^32-\ 120838356395769776*z^38+68331806412012466*z^40-5054068*z^62) The first , 40, terms are: [0, 140, 0, 33959, 0, 8779503, 0, 2302175276, 0, 606009210269, 0, 159698074991933, 0, 42097852167941292, 0, 11098420712837872291, 0, 2926001028430854250503, 0, 771420736280473834796828, 0, 203380441701573002366137781, 0, 53620069266620396306638765621, 0, 14136622000812824844419091569948, 0, 3727039084004350161271594931550551, 0, 982612436239194804514839340148329091, 0, 259060123390483455435660691752660237964, 0, 68299713255186685799382004481439149087757, 0, 18006827032421806546711475740180079488993517, 0, 4747396502355432234912433655303363922553777996, 0, 1251623815283900434564596065464794133263518677071] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 8070381887407545071 z - 1850356881512718331 z - 371 z 24 22 4 6 + 351853825552392365 z - 55168249132066212 z + 54251 z - 4376906 z 8 10 12 14 + 225130748 z - 8007928346 z + 207498280661 z - 4059680248817 z 18 16 50 - 737224536144996 z + 61548114546207 z - 880893376877342659233 z 48 20 + 1342785612838528590451 z + 7080870402509496 z 36 34 + 487553120503985293585 z - 227351321157921983414 z 66 80 88 84 86 - 55168249132066212 z + 225130748 z + z + 54251 z - 371 z 82 64 30 - 4376906 z + 351853825552392365 z - 29329885522505770150 z 42 44 - 1728589655820064842296 z + 1880296828534732400400 z 46 58 - 1728589655820064842296 z - 29329885522505770150 z 56 54 + 89153979851818397892 z - 227351321157921983414 z 52 60 + 487553120503985293585 z + 8070381887407545071 z 70 68 78 - 737224536144996 z + 7080870402509496 z - 8007928346 z 32 38 + 89153979851818397892 z - 880893376877342659233 z 40 62 76 + 1342785612838528590451 z - 1850356881512718331 z + 207498280661 z 74 72 / 2 - 4059680248817 z + 61548114546207 z ) / ((-1 + z ) (1 / 28 26 2 + 34457994432962866850 z - 7552136234128906410 z - 510 z 24 22 4 6 + 1366909244823048431 z - 203122074229805244 z + 90240 z - 8390060 z 8 10 12 14 + 484964909 z - 19084161872 z + 541037062946 z - 11482447745958 z 18 16 50 - 2404712559517404 z + 187510624515297 z - 4394276401397337269682 z 48 20 + 6800877246145539549519 z + 24597834247813274 z 36 34 + 2381814620422355045020 z - 1081814248195334842156 z 66 80 88 84 86 - 203122074229805244 z + 484964909 z + z + 90240 z - 510 z 82 64 30 - 8390060 z + 1366909244823048431 z - 130436176938622842840 z 42 44 - 8835957618165989146728 z + 9641192269729433139500 z 46 58 - 8835957618165989146728 z - 130436176938622842840 z 56 54 + 411104428402684361379 z - 1081814248195334842156 z 52 60 + 2381814620422355045020 z + 34457994432962866850 z 70 68 78 - 2404712559517404 z + 24597834247813274 z - 19084161872 z 32 38 + 411104428402684361379 z - 4394276401397337269682 z 40 62 76 + 6800877246145539549519 z - 7552136234128906410 z + 541037062946 z 74 72 - 11482447745958 z + 187510624515297 z )) And in Maple-input format, it is: -(1+8070381887407545071*z^28-1850356881512718331*z^26-371*z^2+ 351853825552392365*z^24-55168249132066212*z^22+54251*z^4-4376906*z^6+225130748* z^8-8007928346*z^10+207498280661*z^12-4059680248817*z^14-737224536144996*z^18+ 61548114546207*z^16-880893376877342659233*z^50+1342785612838528590451*z^48+ 7080870402509496*z^20+487553120503985293585*z^36-227351321157921983414*z^34-\ 55168249132066212*z^66+225130748*z^80+z^88+54251*z^84-371*z^86-4376906*z^82+ 351853825552392365*z^64-29329885522505770150*z^30-1728589655820064842296*z^42+ 1880296828534732400400*z^44-1728589655820064842296*z^46-29329885522505770150*z^ 58+89153979851818397892*z^56-227351321157921983414*z^54+487553120503985293585*z ^52+8070381887407545071*z^60-737224536144996*z^70+7080870402509496*z^68-\ 8007928346*z^78+89153979851818397892*z^32-880893376877342659233*z^38+ 1342785612838528590451*z^40-1850356881512718331*z^62+207498280661*z^76-\ 4059680248817*z^74+61548114546207*z^72)/(-1+z^2)/(1+34457994432962866850*z^28-\ 7552136234128906410*z^26-510*z^2+1366909244823048431*z^24-203122074229805244*z^ 22+90240*z^4-8390060*z^6+484964909*z^8-19084161872*z^10+541037062946*z^12-\ 11482447745958*z^14-2404712559517404*z^18+187510624515297*z^16-\ 4394276401397337269682*z^50+6800877246145539549519*z^48+24597834247813274*z^20+ 2381814620422355045020*z^36-1081814248195334842156*z^34-203122074229805244*z^66 +484964909*z^80+z^88+90240*z^84-510*z^86-8390060*z^82+1366909244823048431*z^64-\ 130436176938622842840*z^30-8835957618165989146728*z^42+9641192269729433139500*z ^44-8835957618165989146728*z^46-130436176938622842840*z^58+ 411104428402684361379*z^56-1081814248195334842156*z^54+2381814620422355045020*z ^52+34457994432962866850*z^60-2404712559517404*z^70+24597834247813274*z^68-\ 19084161872*z^78+411104428402684361379*z^32-4394276401397337269682*z^38+ 6800877246145539549519*z^40-7552136234128906410*z^62+541037062946*z^76-\ 11482447745958*z^74+187510624515297*z^72) The first , 40, terms are: [0, 140, 0, 35041, 0, 9304345, 0, 2493567324, 0, 669493288889, 0, 179822876209233, 0, 48304096519536796, 0, 12975753955425209169, 0, 3485648293288441405721, 0, 936343266402187765124364, 0, 251528243099428682070755449, 0, 67567594351284035566988941353, 0, 18150565666652563849028303730444, 0, 4875754993506648102723373457362249, 0, 1309765613949402852610725161894342529, 0, 351840067076841394555330022748713839900, 0, 94514187493347904209164998512222971684161, 0, 25389182397260921123684561800713981293992457, 0, 6820252069028472204600945885668821751441920220, 0, 1832112494105074294301422321091484655563800457257] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1265101785293885918380 z - 172992860499685140876 z - 440 z 24 22 4 + 20149156042242807468 z - 1985113759930379476 z + 80694 z 6 102 8 10 - 8475680 z - 28418485260 z + 583512631 z - 28418485260 z 12 14 18 + 1029032038890 z - 28696306862892 z - 11252549544620428 z 16 50 + 632672251525625 z - 315014123087837794707886444 z 48 20 + 199896650826078220768311198 z + 164037875304356380 z 36 34 + 805457930411084069265860 z - 199260789566491310815132 z 66 80 - 111281199605884989959039332 z + 42747399713427585014988 z 100 90 88 + 1029032038890 z - 1985113759930379476 z + 20149156042242807468 z 84 94 + 1265101785293885918380 z - 11252549544620428 z 86 96 98 - 172992860499685140876 z + 632672251525625 z - 28696306862892 z 92 82 + 164037875304356380 z - 7926056195158576133364 z 64 112 110 106 + 199896650826078220768311198 z + z - 440 z - 8475680 z 108 30 42 + 80694 z - 7926056195158576133364 z - 23207409254403910655524924 z 44 46 + 54305124173517252457472612 z - 111281199605884989959039332 z 58 56 - 529333199557799153155748676 z + 564779306648996412174235898 z 54 52 - 529333199557799153155748676 z + 435759386097775423408324976 z 60 70 + 435759386097775423408324976 z - 23207409254403910655524924 z 68 78 + 54305124173517252457472612 z - 199260789566491310815132 z 32 38 + 42747399713427585014988 z - 2831255877779778834743332 z 40 62 + 8674244051118158973829820 z - 315014123087837794707886444 z 76 74 + 805457930411084069265860 z - 2831255877779778834743332 z 72 104 / 2 + 8674244051118158973829820 z + 583512631 z ) / ((-1 + z ) (1 / 28 26 2 + 4738169392420823598038 z - 619885315716851098119 z - 587 z 24 22 4 + 68893606092272833364 z - 6458087911244781597 z + 128712 z 6 102 8 10 - 15417103 z - 62863194314 z + 1179776683 z - 62863194314 z 12 14 18 + 2463852154726 z - 73792718793588 z - 32817139263782447 z 16 50 + 1736816561535741 z - 1609253193546866639502957621 z 48 20 + 1007983111480025503467018962 z + 506168910342542106 z 36 34 + 3510833932471989133770298 z - 839363856879671669950719 z 66 80 - 551962897343958265580707141 z + 173581416497702195487848 z 100 90 88 + 2463852154726 z - 6458087911244781597 z + 68893606092272833364 z 84 94 + 4738169392420823598038 z - 32817139263782447 z 86 96 98 - 619885315716851098119 z + 1736816561535741 z - 73792718793588 z 92 82 + 506168910342542106 z - 30948325290046903507397 z 64 112 110 106 + 1007983111480025503467018962 z + z - 587 z - 15417103 z 108 30 + 128712 z - 30948325290046903507397 z 42 44 - 110286677588050042337640047 z + 264071576497118915932981446 z 46 58 - 551962897343958265580707141 z - 2745145803539726372424286051 z 56 54 + 2934537853585911870658799130 z - 2745145803539726372424286051 z 52 60 + 2247079288744446965910012802 z + 2247079288744446965910012802 z 70 68 - 110286677588050042337640047 z + 264071576497118915932981446 z 78 32 - 839363856879671669950719 z + 173581416497702195487848 z 38 40 - 12736650034785177687214597 z + 40164034177573606513663772 z 62 76 - 1609253193546866639502957621 z + 3510833932471989133770298 z 74 72 - 12736650034785177687214597 z + 40164034177573606513663772 z 104 + 1179776683 z )) And in Maple-input format, it is: -(1+1265101785293885918380*z^28-172992860499685140876*z^26-440*z^2+ 20149156042242807468*z^24-1985113759930379476*z^22+80694*z^4-8475680*z^6-\ 28418485260*z^102+583512631*z^8-28418485260*z^10+1029032038890*z^12-\ 28696306862892*z^14-11252549544620428*z^18+632672251525625*z^16-\ 315014123087837794707886444*z^50+199896650826078220768311198*z^48+ 164037875304356380*z^20+805457930411084069265860*z^36-199260789566491310815132* z^34-111281199605884989959039332*z^66+42747399713427585014988*z^80+ 1029032038890*z^100-1985113759930379476*z^90+20149156042242807468*z^88+ 1265101785293885918380*z^84-11252549544620428*z^94-172992860499685140876*z^86+ 632672251525625*z^96-28696306862892*z^98+164037875304356380*z^92-\ 7926056195158576133364*z^82+199896650826078220768311198*z^64+z^112-440*z^110-\ 8475680*z^106+80694*z^108-7926056195158576133364*z^30-\ 23207409254403910655524924*z^42+54305124173517252457472612*z^44-\ 111281199605884989959039332*z^46-529333199557799153155748676*z^58+ 564779306648996412174235898*z^56-529333199557799153155748676*z^54+ 435759386097775423408324976*z^52+435759386097775423408324976*z^60-\ 23207409254403910655524924*z^70+54305124173517252457472612*z^68-\ 199260789566491310815132*z^78+42747399713427585014988*z^32-\ 2831255877779778834743332*z^38+8674244051118158973829820*z^40-\ 315014123087837794707886444*z^62+805457930411084069265860*z^76-\ 2831255877779778834743332*z^74+8674244051118158973829820*z^72+583512631*z^104)/ (-1+z^2)/(1+4738169392420823598038*z^28-619885315716851098119*z^26-587*z^2+ 68893606092272833364*z^24-6458087911244781597*z^22+128712*z^4-15417103*z^6-\ 62863194314*z^102+1179776683*z^8-62863194314*z^10+2463852154726*z^12-\ 73792718793588*z^14-32817139263782447*z^18+1736816561535741*z^16-\ 1609253193546866639502957621*z^50+1007983111480025503467018962*z^48+ 506168910342542106*z^20+3510833932471989133770298*z^36-839363856879671669950719 *z^34-551962897343958265580707141*z^66+173581416497702195487848*z^80+ 2463852154726*z^100-6458087911244781597*z^90+68893606092272833364*z^88+ 4738169392420823598038*z^84-32817139263782447*z^94-619885315716851098119*z^86+ 1736816561535741*z^96-73792718793588*z^98+506168910342542106*z^92-\ 30948325290046903507397*z^82+1007983111480025503467018962*z^64+z^112-587*z^110-\ 15417103*z^106+128712*z^108-30948325290046903507397*z^30-\ 110286677588050042337640047*z^42+264071576497118915932981446*z^44-\ 551962897343958265580707141*z^46-2745145803539726372424286051*z^58+ 2934537853585911870658799130*z^56-2745145803539726372424286051*z^54+ 2247079288744446965910012802*z^52+2247079288744446965910012802*z^60-\ 110286677588050042337640047*z^70+264071576497118915932981446*z^68-\ 839363856879671669950719*z^78+173581416497702195487848*z^32-\ 12736650034785177687214597*z^38+40164034177573606513663772*z^40-\ 1609253193546866639502957621*z^62+3510833932471989133770298*z^76-\ 12736650034785177687214597*z^74+40164034177573606513663772*z^72+1179776683*z^ 104) The first , 40, terms are: [0, 148, 0, 38419, 0, 10524255, 0, 2909823124, 0, 806190821561, 0, 223473630543689, 0, 61953948807981268, 0, 17176134510152762567, 0, 4761955876320327211915, 0, 1320219652866089697986580, 0, 366022048983185696789967601, 0, 101477173175517954470658737809, 0, 28133870783427334223414552275092, 0, 7799928471247011147249315392906395, 0, 2162478272621778070708926957224036855, 0, 599532713601711406043855986751694304596, 0, 166216456036858967509524039748985896780969, 0, 46082406567152464837219966833308332166223449, 0, 12776040626026161605970781331181623957690519316, 0, 3542072262229167810352815572448945122253687245103] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 16557361222060152 z - 7682870167793006 z - 325 z 24 22 4 6 + 2842173450850614 z - 833523580563456 z + 38748 z - 2406937 z 8 10 12 14 + 90878585 z - 2274431856 z + 39845711553 z - 506927762401 z 18 16 50 - 34633448726733 z + 4806963263468 z - 34633448726733 z 48 20 36 + 192390888452969 z + 192390888452969 z + 39585678291925350 z 34 66 64 30 - 44123185870679584 z - 325 z + 38748 z - 28569321423618310 z 42 44 46 - 7682870167793006 z + 2842173450850614 z - 833523580563456 z 58 56 54 52 - 2274431856 z + 39845711553 z - 506927762401 z + 4806963263468 z 60 68 32 38 + 90878585 z + z + 39585678291925350 z - 28569321423618310 z 40 62 / 2 + 16557361222060152 z - 2406937 z ) / ((-1 + z ) (1 / 28 26 2 + 81938134706586570 z - 36717524778294576 z - 476 z 24 22 4 6 + 13001720698912150 z - 3619386291880688 z + 69415 z - 5017924 z 8 10 12 14 + 215758009 z - 6063719860 z + 117926479201 z - 1648450553644 z 18 16 50 - 132294594486740 z + 17015031924015 z - 132294594486740 z 48 20 36 + 786658268388905 z + 786658268388905 z + 204084113776169830 z 34 66 64 30 - 228664714055774968 z - 476 z + 69415 z - 145024544342122800 z 42 44 46 - 36717524778294576 z + 13001720698912150 z - 3619386291880688 z 58 56 54 - 6063719860 z + 117926479201 z - 1648450553644 z 52 60 68 32 + 17015031924015 z + 215758009 z + z + 204084113776169830 z 38 40 62 - 145024544342122800 z + 81938134706586570 z - 5017924 z )) And in Maple-input format, it is: -(1+16557361222060152*z^28-7682870167793006*z^26-325*z^2+2842173450850614*z^24-\ 833523580563456*z^22+38748*z^4-2406937*z^6+90878585*z^8-2274431856*z^10+ 39845711553*z^12-506927762401*z^14-34633448726733*z^18+4806963263468*z^16-\ 34633448726733*z^50+192390888452969*z^48+192390888452969*z^20+39585678291925350 *z^36-44123185870679584*z^34-325*z^66+38748*z^64-28569321423618310*z^30-\ 7682870167793006*z^42+2842173450850614*z^44-833523580563456*z^46-2274431856*z^ 58+39845711553*z^56-506927762401*z^54+4806963263468*z^52+90878585*z^60+z^68+ 39585678291925350*z^32-28569321423618310*z^38+16557361222060152*z^40-2406937*z^ 62)/(-1+z^2)/(1+81938134706586570*z^28-36717524778294576*z^26-476*z^2+ 13001720698912150*z^24-3619386291880688*z^22+69415*z^4-5017924*z^6+215758009*z^ 8-6063719860*z^10+117926479201*z^12-1648450553644*z^14-132294594486740*z^18+ 17015031924015*z^16-132294594486740*z^50+786658268388905*z^48+786658268388905*z ^20+204084113776169830*z^36-228664714055774968*z^34-476*z^66+69415*z^64-\ 145024544342122800*z^30-36717524778294576*z^42+13001720698912150*z^44-\ 3619386291880688*z^46-6063719860*z^58+117926479201*z^56-1648450553644*z^54+ 17015031924015*z^52+215758009*z^60+z^68+204084113776169830*z^32-\ 145024544342122800*z^38+81938134706586570*z^40-5017924*z^62) The first , 40, terms are: [0, 152, 0, 41361, 0, 11786167, 0, 3374618188, 0, 966810410455, 0, 277012175832907, 0, 79371324600342036, 0, 22742058929967962227, 0, 6516227045122173703245, 0, 1867078987235977911307808, 0, 534969701322533017599592373, 0, 153283596908287917627889527949, 0, 43919984718956747174727677809936, 0, 12584288841472600760550616828461397, 0, 3605746374149049162055666297776131195, 0, 1033145939241935770906837611733241392388, 0, 296024850618101580238958677698184866127715, 0, 84819296921211254229853309413680906413864191, 0, 24303071567091210385811913367770420213627989596, 0, 6963501337953945641647852797894363434222234854127] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 2484988842188126 z + 1531215256893216 z + 349 z 24 22 4 6 - 737815027241458 z + 276727633626523 z - 44227 z + 2784760 z 8 10 12 14 - 100097829 z + 2272980466 z - 34906143142 z + 379478012365 z 18 16 50 + 17860800294811 z - 3013165118824 z + 34906143142 z 48 20 36 - 379478012365 z - 80277979542535 z - 1531215256893216 z 34 30 42 + 2484988842188126 z + 3163278186391112 z + 80277979542535 z 44 46 58 56 - 17860800294811 z + 3013165118824 z + 44227 z - 2784760 z 54 52 60 32 + 100097829 z - 2272980466 z - 349 z - 3163278186391112 z 38 40 62 / + 737815027241458 z - 276727633626523 z + z ) / (1 / 28 26 2 + 21300917094589904 z - 11629013864580040 z - 514 z 24 22 4 6 + 4964757321914983 z - 1650780688728510 z + 82725 z - 6203980 z 8 10 12 14 + 257737912 z - 6662471602 z + 115556118580 z - 1413441967842 z 18 16 50 - 84028237763948 z + 12611484315064 z - 1413441967842 z 48 20 36 + 12611484315064 z + 424998794765875 z + 21300917094589904 z 34 64 30 - 30595156517810732 z + z - 30595156517810732 z 42 44 46 - 1650780688728510 z + 424998794765875 z - 84028237763948 z 58 56 54 52 - 6203980 z + 257737912 z - 6662471602 z + 115556118580 z 60 32 38 + 82725 z + 34513997828298648 z - 11629013864580040 z 40 62 + 4964757321914983 z - 514 z ) And in Maple-input format, it is: -(-1-2484988842188126*z^28+1531215256893216*z^26+349*z^2-737815027241458*z^24+ 276727633626523*z^22-44227*z^4+2784760*z^6-100097829*z^8+2272980466*z^10-\ 34906143142*z^12+379478012365*z^14+17860800294811*z^18-3013165118824*z^16+ 34906143142*z^50-379478012365*z^48-80277979542535*z^20-1531215256893216*z^36+ 2484988842188126*z^34+3163278186391112*z^30+80277979542535*z^42-17860800294811* z^44+3013165118824*z^46+44227*z^58-2784760*z^56+100097829*z^54-2272980466*z^52-\ 349*z^60-3163278186391112*z^32+737815027241458*z^38-276727633626523*z^40+z^62)/ (1+21300917094589904*z^28-11629013864580040*z^26-514*z^2+4964757321914983*z^24-\ 1650780688728510*z^22+82725*z^4-6203980*z^6+257737912*z^8-6662471602*z^10+ 115556118580*z^12-1413441967842*z^14-84028237763948*z^18+12611484315064*z^16-\ 1413441967842*z^50+12611484315064*z^48+424998794765875*z^20+21300917094589904*z ^36-30595156517810732*z^34+z^64-30595156517810732*z^30-1650780688728510*z^42+ 424998794765875*z^44-84028237763948*z^46-6203980*z^58+257737912*z^56-6662471602 *z^54+115556118580*z^52+82725*z^60+34513997828298648*z^32-11629013864580040*z^ 38+4964757321914983*z^40-514*z^62) The first , 40, terms are: [0, 165, 0, 46312, 0, 13573963, 0, 4011873399, 0, 1188378295327, 0, 352239111496891, 0, 104423885866440264, 0, 30958868268002105749, 0, 9178610476603161369481, 0, 2721264289411962400278729, 0, 806798460211353618905305669, 0, 239199109985268371582481089128, 0, 70917612213842622902442973005963, 0, 21025612837625292434598593011035487, 0, 6233661644148111664479906709791276439, 0, 1848152436475700109838958001747019664475, 0, 547939177002533485032534867308437467453448, 0, 162452693746257647931455746959121143454141717, 0, 48163881712997406561403015951182470678897131041, 0, 14279600099022909386337743286647882896682620013857] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 26 2 24 22 4 6 f(z) = - (-1 + z + 249 z - 249 z + 17848 z - 17848 z + 482088 z 8 10 12 14 18 - 5528600 z + 26590664 z - 56442886 z + 56442886 z + 5528600 z 16 20 / 28 26 24 22 - 26590664 z - 482088 z ) / (z - 393 z + 38049 z - 1415104 z / 20 18 16 14 + 21752672 z - 141055808 z + 413029982 z - 588065326 z 12 10 8 6 4 + 413029982 z - 141055808 z + 21752672 z - 1415104 z + 38049 z 2 - 393 z + 1) And in Maple-input format, it is: -(-1+z^26+249*z^2-249*z^24+17848*z^22-17848*z^4+482088*z^6-5528600*z^8+26590664 *z^10-56442886*z^12+56442886*z^14+5528600*z^18-26590664*z^16-482088*z^20)/(z^28 -393*z^26+38049*z^24-1415104*z^22+21752672*z^20-141055808*z^18+413029982*z^16-\ 588065326*z^14+413029982*z^12-141055808*z^10+21752672*z^8-1415104*z^6+38049*z^4 -393*z^2+1) The first , 40, terms are: [0, 144, 0, 36391, 0, 9755623, 0, 2636869584, 0, 713577177025, 0, 193139154861505, 0, 52277097897683280, 0, 14149935869819083495, 0, 3829991216998487246887, 0, 1036671446848014997757712, 0, 280597957298542904484946177, 0, 75950016949113918093515470081, 0, 20557544804076220189109864743440, 0, 5564352259735731984153166262967847, 0, 1506114488208504387874420828909811431, 0, 407663056852245237461934578917415417424, 0, 110342984695689438501788865228022172231617, 0, 29866758998399182059948147190464426952024513, 0, 8084096107501438795575134883381906551471678160, 0, 2188138655380169919541753387691746803305493140455] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 53530857185964932 z - 24450862446226402 z - 383 z 24 22 4 6 + 8855562520120230 z - 2527416649128376 z + 55376 z - 4075225 z 8 10 12 14 + 176219087 z - 4910654332 z + 93789527339 z - 1280611837397 z 18 16 50 - 97454249148715 z + 12877853732820 z - 97454249148715 z 48 20 36 + 564008386645821 z + 564008386645821 z + 130194324007019986 z 34 66 64 30 - 145418732101752040 z - 383 z + 55376 z - 93371338997806134 z 42 44 46 - 24450862446226402 z + 8855562520120230 z - 2527416649128376 z 58 56 54 - 4910654332 z + 93789527339 z - 1280611837397 z 52 60 68 32 + 12877853732820 z + 176219087 z + z + 130194324007019986 z 38 40 62 / - 93371338997806134 z + 53530857185964932 z - 4075225 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 277783541872742706 z - 121443142626808632 z - 542 z 24 22 4 6 + 41669213129452414 z - 11167460530815812 z + 99435 z - 8607936 z 8 10 12 14 + 422041789 z - 13098756918 z + 275917925953 z - 4128737904608 z 18 16 50 - 371792474132570 z + 45272510131367 z - 371792474132570 z 48 20 36 + 2322818884012605 z + 2322818884012605 z + 711718176061564126 z 34 66 64 30 - 800260276411867836 z - 542 z + 99435 z - 500422623180807504 z 42 44 46 - 121443142626808632 z + 41669213129452414 z - 11167460530815812 z 58 56 54 - 13098756918 z + 275917925953 z - 4128737904608 z 52 60 68 32 + 45272510131367 z + 422041789 z + z + 711718176061564126 z 38 40 62 - 500422623180807504 z + 277783541872742706 z - 8607936 z )) And in Maple-input format, it is: -(1+53530857185964932*z^28-24450862446226402*z^26-383*z^2+8855562520120230*z^24 -2527416649128376*z^22+55376*z^4-4075225*z^6+176219087*z^8-4910654332*z^10+ 93789527339*z^12-1280611837397*z^14-97454249148715*z^18+12877853732820*z^16-\ 97454249148715*z^50+564008386645821*z^48+564008386645821*z^20+ 130194324007019986*z^36-145418732101752040*z^34-383*z^66+55376*z^64-\ 93371338997806134*z^30-24450862446226402*z^42+8855562520120230*z^44-\ 2527416649128376*z^46-4910654332*z^58+93789527339*z^56-1280611837397*z^54+ 12877853732820*z^52+176219087*z^60+z^68+130194324007019986*z^32-\ 93371338997806134*z^38+53530857185964932*z^40-4075225*z^62)/(-1+z^2)/(1+ 277783541872742706*z^28-121443142626808632*z^26-542*z^2+41669213129452414*z^24-\ 11167460530815812*z^22+99435*z^4-8607936*z^6+422041789*z^8-13098756918*z^10+ 275917925953*z^12-4128737904608*z^14-371792474132570*z^18+45272510131367*z^16-\ 371792474132570*z^50+2322818884012605*z^48+2322818884012605*z^20+ 711718176061564126*z^36-800260276411867836*z^34-542*z^66+99435*z^64-\ 500422623180807504*z^30-121443142626808632*z^42+41669213129452414*z^44-\ 11167460530815812*z^46-13098756918*z^58+275917925953*z^56-4128737904608*z^54+ 45272510131367*z^52+422041789*z^60+z^68+711718176061564126*z^32-\ 500422623180807504*z^38+277783541872742706*z^40-8607936*z^62) The first , 40, terms are: [0, 160, 0, 42279, 0, 11593323, 0, 3206995528, 0, 890178045017, 0, 247448911939521, 0, 68827794495613064, 0, 19149511165780410199, 0, 5328451819333320437199, 0, 1482742079496279087226528, 0, 412609549864898870015625997, 0, 114819812936778928124066220813, 0, 31951854093831853743728688785696, 0, 8891519992515371622293421147900127, 0, 2474322252801178196273872082028316183, 0, 688551841354706418666172620965084767752, 0, 191609520212966430015784954384535704243153, 0, 53320909293039541254495327410221075668539561, 0, 14838090677231610574674022159684179695034621064, 0, 4129129456186494327608439138326094696599509486187] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 60039895505832 z - 52551342871342 z - 293 z 24 22 4 6 + 35207118866374 z - 18005864950800 z + 29052 z - 1404853 z 8 10 12 14 + 39341409 z - 703294448 z + 8521716652 z - 72781737196 z 18 16 50 48 - 2052394043660 z + 450009093296 z - 1404853 z + 39341409 z 20 36 34 + 6997451814188 z + 6997451814188 z - 18005864950800 z 30 42 44 46 - 52551342871342 z - 72781737196 z + 8521716652 z - 703294448 z 56 54 52 32 38 + z - 293 z + 29052 z + 35207118866374 z - 2052394043660 z 40 / 28 26 + 450009093296 z ) / (-1 - 593779632992770 z + 455839535559986 z / 2 24 22 4 + 441 z - 268275414098554 z + 120688256112120 z - 56029 z 6 8 10 12 + 3247065 z - 106120205 z + 2184966749 z - 30261908784 z 14 18 16 + 294103570264 z + 10678665754984 z - 2064292154848 z 50 48 20 + 106120205 z - 2184966749 z - 41305095462320 z 36 34 30 - 120688256112120 z + 268275414098554 z + 593779632992770 z 42 44 46 58 56 + 2064292154848 z - 294103570264 z + 30261908784 z + z - 441 z 54 52 32 38 + 56029 z - 3247065 z - 455839535559986 z + 41305095462320 z 40 - 10678665754984 z ) And in Maple-input format, it is: -(1+60039895505832*z^28-52551342871342*z^26-293*z^2+35207118866374*z^24-\ 18005864950800*z^22+29052*z^4-1404853*z^6+39341409*z^8-703294448*z^10+ 8521716652*z^12-72781737196*z^14-2052394043660*z^18+450009093296*z^16-1404853*z ^50+39341409*z^48+6997451814188*z^20+6997451814188*z^36-18005864950800*z^34-\ 52551342871342*z^30-72781737196*z^42+8521716652*z^44-703294448*z^46+z^56-293*z^ 54+29052*z^52+35207118866374*z^32-2052394043660*z^38+450009093296*z^40)/(-1-\ 593779632992770*z^28+455839535559986*z^26+441*z^2-268275414098554*z^24+ 120688256112120*z^22-56029*z^4+3247065*z^6-106120205*z^8+2184966749*z^10-\ 30261908784*z^12+294103570264*z^14+10678665754984*z^18-2064292154848*z^16+ 106120205*z^50-2184966749*z^48-41305095462320*z^20-120688256112120*z^36+ 268275414098554*z^34+593779632992770*z^30+2064292154848*z^42-294103570264*z^44+ 30261908784*z^46+z^58-441*z^56+56029*z^54-3247065*z^52-455839535559986*z^32+ 41305095462320*z^38-10678665754984*z^40) The first , 40, terms are: [0, 148, 0, 38291, 0, 10436251, 0, 2870767076, 0, 791384821113, 0, 218279869080009, 0, 60214399443343172, 0, 16611264949100734731, 0, 4582569934146445767587, 0, 1264202208069177351392052, 0, 348758064130196346401414833, 0, 96212620195007861813677681745, 0, 26542378865131367683847642120692, 0, 7322302229996442758865947037153155, 0, 2020019018232698542731675479194300971, 0, 557266923542603564232116271401476550788, 0, 153734406126597044269876126210689715635753, 0, 42411036129668730996166601754219959637268249, 0, 11700022336790725863856277230131639555811908772, 0, 3227709935293944136538824294696048240113322282363] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 17448156668664198341 z - 3900836786325000863 z - 395 z 24 22 4 6 + 721116600558758625 z - 109551709657369592 z + 62753 z - 5518496 z 8 10 12 14 + 308109220 z - 11810047412 z + 327094391371 z - 6786459495105 z 18 16 50 - 1357674132661992 z + 108329212006751 z - 2071776622420123086773 z 48 20 + 3183623359760014424223 z + 13571590101316948 z 36 34 + 1133996239011565155215 z - 521404121652483411384 z 66 80 88 84 86 - 109551709657369592 z + 308109220 z + z + 62753 z - 395 z 82 64 30 - 5518496 z + 721116600558758625 z - 64848229515509830852 z 42 44 - 4118355075873970388448 z + 4487116339004747777544 z 46 58 - 4118355075873970388448 z - 64848229515509830852 z 56 54 + 201035080641643173268 z - 521404121652483411384 z 52 60 + 1133996239011565155215 z + 17448156668664198341 z 70 68 78 - 1357674132661992 z + 13571590101316948 z - 11810047412 z 32 38 + 201035080641643173268 z - 2071776622420123086773 z 40 62 76 + 3183623359760014424223 z - 3900836786325000863 z + 327094391371 z 74 72 / - 6786459495105 z + 108329212006751 z ) / (-1 / 28 26 2 - 90550595577688514444 z + 18702157322791840267 z + 547 z 24 22 4 - 3192871749150215873 z + 447616756957293552 z - 104860 z 6 8 10 12 + 10599692 z - 665612233 z + 28362407475 z - 866804846988 z 14 18 16 + 19739389296064 z + 4703638479890099 z - 344431210569545 z 50 48 + 26789643193863394880353 z - 37669505174860434841363 z 20 36 - 51107726444192860 z - 8099965817433417232144 z 34 66 80 + 3435617366748639354819 z + 3192871749150215873 z - 28362407475 z 90 88 84 86 82 + z - 547 z - 10599692 z + 104860 z + 665612233 z 64 30 - 18702157322791840267 z + 364290837237037303376 z 42 44 + 37669505174860434841363 z - 44661930397834021668652 z 46 58 + 44661930397834021668652 z + 1222792809611714241561 z 56 54 - 3435617366748639354819 z + 8099965817433417232144 z 52 60 - 16054772926790702910432 z - 364290837237037303376 z 70 68 78 + 51107726444192860 z - 447616756957293552 z + 866804846988 z 32 38 - 1222792809611714241561 z + 16054772926790702910432 z 40 62 - 26789643193863394880353 z + 90550595577688514444 z 76 74 72 - 19739389296064 z + 344431210569545 z - 4703638479890099 z ) And in Maple-input format, it is: -(1+17448156668664198341*z^28-3900836786325000863*z^26-395*z^2+ 721116600558758625*z^24-109551709657369592*z^22+62753*z^4-5518496*z^6+308109220 *z^8-11810047412*z^10+327094391371*z^12-6786459495105*z^14-1357674132661992*z^ 18+108329212006751*z^16-2071776622420123086773*z^50+3183623359760014424223*z^48 +13571590101316948*z^20+1133996239011565155215*z^36-521404121652483411384*z^34-\ 109551709657369592*z^66+308109220*z^80+z^88+62753*z^84-395*z^86-5518496*z^82+ 721116600558758625*z^64-64848229515509830852*z^30-4118355075873970388448*z^42+ 4487116339004747777544*z^44-4118355075873970388448*z^46-64848229515509830852*z^ 58+201035080641643173268*z^56-521404121652483411384*z^54+1133996239011565155215 *z^52+17448156668664198341*z^60-1357674132661992*z^70+13571590101316948*z^68-\ 11810047412*z^78+201035080641643173268*z^32-2071776622420123086773*z^38+ 3183623359760014424223*z^40-3900836786325000863*z^62+327094391371*z^76-\ 6786459495105*z^74+108329212006751*z^72)/(-1-90550595577688514444*z^28+ 18702157322791840267*z^26+547*z^2-3192871749150215873*z^24+447616756957293552*z ^22-104860*z^4+10599692*z^6-665612233*z^8+28362407475*z^10-866804846988*z^12+ 19739389296064*z^14+4703638479890099*z^18-344431210569545*z^16+ 26789643193863394880353*z^50-37669505174860434841363*z^48-51107726444192860*z^ 20-8099965817433417232144*z^36+3435617366748639354819*z^34+3192871749150215873* z^66-28362407475*z^80+z^90-547*z^88-10599692*z^84+104860*z^86+665612233*z^82-\ 18702157322791840267*z^64+364290837237037303376*z^30+37669505174860434841363*z^ 42-44661930397834021668652*z^44+44661930397834021668652*z^46+ 1222792809611714241561*z^58-3435617366748639354819*z^56+8099965817433417232144* z^54-16054772926790702910432*z^52-364290837237037303376*z^60+51107726444192860* z^70-447616756957293552*z^68+866804846988*z^78-1222792809611714241561*z^32+ 16054772926790702910432*z^38-26789643193863394880353*z^40+90550595577688514444* z^62-19739389296064*z^76+344431210569545*z^74-4703638479890099*z^72) The first , 40, terms are: [0, 152, 0, 41037, 0, 11589715, 0, 3290084456, 0, 934737543783, 0, 265607236035883, 0, 75475311785756936, 0, 21447335813523629203, 0, 6094563379878185131493, 0, 1731857031276504177549192, 0, 492131906738426993807362665, 0, 139846312634874071952090166901, 0, 39739328034112114070686779783752, 0, 11292497922783928388680371337705425, 0, 3208924651970559906915077081398034759, 0, 911861794734394543853741320426074718968, 0, 259118559297645040557538980655853469075559, 0, 73632241377516872145108555191279491331196707, 0, 20923653577648681111583997977351205060363772616, 0, 5945755158979236739741291448158114926052732337799] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 47297939386018108 z - 18807938325609146 z - 329 z 24 22 4 6 + 6041290813436335 z - 1558736921365421 z + 39948 z - 2540545 z 8 10 12 14 + 99061515 z - 2585814866 z + 47750980244 z - 647622898198 z 18 16 50 - 52159726376665 z + 6625776658239 z - 1558736921365421 z 48 20 36 + 6041290813436335 z + 320745485285800 z + 240080669249193720 z 34 66 64 - 217020023655770024 z - 2540545 z + 99061515 z 30 42 44 - 96496091379452518 z - 96496091379452518 z + 47297939386018108 z 46 58 56 - 18807938325609146 z - 647622898198 z + 6625776658239 z 54 52 60 70 - 52159726376665 z + 320745485285800 z + 47750980244 z - 329 z 68 32 38 + 39948 z + 160228616301461766 z - 217020023655770024 z 40 62 72 / 2 + 160228616301461766 z - 2585814866 z + z ) / ((-1 + z ) (1 / 28 26 2 + 232069779165481562 z - 88544397672821000 z - 484 z 24 22 4 6 + 27074840283626543 z - 6601126076314276 z + 72147 z - 5305232 z 8 10 12 14 + 233190851 z - 6769608556 z + 137777455170 z - 2044395550148 z 18 16 50 - 193220966665048 z + 22731311698619 z - 6601126076314276 z 48 20 36 + 27074840283626543 z + 1274646350938659 z + 1271382045299743964 z 34 66 64 - 1143657452260650208 z - 5305232 z + 233190851 z 30 42 44 - 489303918989243512 z - 489303918989243512 z + 232069779165481562 z 46 58 56 - 88544397672821000 z - 2044395550148 z + 22731311698619 z 54 52 60 70 - 193220966665048 z + 1274646350938659 z + 137777455170 z - 484 z 68 32 38 + 72147 z + 832169011221551698 z - 1143657452260650208 z 40 62 72 + 832169011221551698 z - 6769608556 z + z )) And in Maple-input format, it is: -(1+47297939386018108*z^28-18807938325609146*z^26-329*z^2+6041290813436335*z^24 -1558736921365421*z^22+39948*z^4-2540545*z^6+99061515*z^8-2585814866*z^10+ 47750980244*z^12-647622898198*z^14-52159726376665*z^18+6625776658239*z^16-\ 1558736921365421*z^50+6041290813436335*z^48+320745485285800*z^20+ 240080669249193720*z^36-217020023655770024*z^34-2540545*z^66+99061515*z^64-\ 96496091379452518*z^30-96496091379452518*z^42+47297939386018108*z^44-\ 18807938325609146*z^46-647622898198*z^58+6625776658239*z^56-52159726376665*z^54 +320745485285800*z^52+47750980244*z^60-329*z^70+39948*z^68+160228616301461766*z ^32-217020023655770024*z^38+160228616301461766*z^40-2585814866*z^62+z^72)/(-1+z ^2)/(1+232069779165481562*z^28-88544397672821000*z^26-484*z^2+27074840283626543 *z^24-6601126076314276*z^22+72147*z^4-5305232*z^6+233190851*z^8-6769608556*z^10 +137777455170*z^12-2044395550148*z^14-193220966665048*z^18+22731311698619*z^16-\ 6601126076314276*z^50+27074840283626543*z^48+1274646350938659*z^20+ 1271382045299743964*z^36-1143657452260650208*z^34-5305232*z^66+233190851*z^64-\ 489303918989243512*z^30-489303918989243512*z^42+232069779165481562*z^44-\ 88544397672821000*z^46-2044395550148*z^58+22731311698619*z^56-193220966665048*z ^54+1274646350938659*z^52+137777455170*z^60-484*z^70+72147*z^68+ 832169011221551698*z^32-1143657452260650208*z^38+832169011221551698*z^40-\ 6769608556*z^62+z^72) The first , 40, terms are: [0, 156, 0, 42977, 0, 12350243, 0, 3567841924, 0, 1031708046683, 0, 298400207677227, 0, 86310211760941652, 0, 24964912425236219827, 0, 7221027990235471179713, 0, 2088662501964610129809452, 0, 604139973027702432042954497, 0, 174745856986215623475180941185, 0, 50544767981104429331076615138444, 0, 14619937884799144695027917183117057, 0, 4228777623887863008290159069490630419, 0, 1223162528795500353769043407402772341940, 0, 353796464356809588469672240700834149554699, 0, 102334673639234626398884655539849828432104571, 0, 29600028501979998694518968545901444735023218724, 0, 8561728455860913978821270691192281027052902327235] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1362394589311962055306 z - 183714057341302006201 z - 437 z 24 22 4 + 21107108591143245856 z - 2052323247443069627 z + 79686 z 6 102 8 10 - 8335691 z - 27923277349 z + 572854680 z - 27923277349 z 12 14 18 + 1014641824558 z - 28463266734239 z - 11362326363456153 z 16 50 + 632604218331624 z - 381193827982045334403124381 z 48 20 + 240646277271307341375085964 z + 167510817097511214 z 36 34 + 915701078796175699721342 z - 223631262121070572831853 z 66 80 - 133091220666693795969623947 z + 47334349633910979878628 z 100 90 88 + 1014641824558 z - 2052323247443069627 z + 21107108591143245856 z 84 94 + 1362394589311962055306 z - 11362326363456153 z 86 96 98 - 183714057341302006201 z + 632604218331624 z - 28463266734239 z 92 82 + 167510817097511214 z - 8655993490790622236839 z 64 112 110 106 + 240646277271307341375085964 z + z - 437 z - 8335691 z 108 30 42 + 79686 z - 8655993490790622236839 z - 27288103347303628452609129 z 44 46 + 64438790227796189632355646 z - 133091220666693795969623947 z 58 56 - 644373254537836365058579579 z + 688039593047742963825651130 z 54 52 - 644373254537836365058579579 z + 529272359846494073587396870 z 60 70 + 529272359846494073587396870 z - 27288103347303628452609129 z 68 78 + 64438790227796189632355646 z - 223631262121070572831853 z 32 38 + 47334349633910979878628 z - 3258161307862183483099975 z 40 62 + 10095397907384474453985532 z - 381193827982045334403124381 z 76 74 + 915701078796175699721342 z - 3258161307862183483099975 z 72 104 / 2 + 10095397907384474453985532 z + 572854680 z ) / ((-1 + z ) (1 / 28 26 2 + 5064020205580385931020 z - 651611052730895483140 z - 595 z 24 22 4 + 71292973757044776162 z - 6587745029768480614 z + 129828 z 6 102 8 10 - 15410388 z - 61951479674 z + 1169375482 z - 61951479674 z 12 14 18 + 2422529111708 z - 72648985190107 z - 32714205490123077 z 16 50 + 1717965275080368 z - 2003308928240119600376366237 z 48 20 + 1245857851239263740229112228 z + 509862769262415792 z 36 34 + 4017717023357014192714688 z - 944617234006280305784411 z 66 80 - 676085796367358615125749767 z + 192021810205246788362460 z 100 90 88 + 2422529111708 z - 6587745029768480614 z + 71292973757044776162 z 84 94 + 5064020205580385931020 z - 32714205490123077 z 86 96 98 - 651611052730895483140 z + 1717965275080368 z - 72648985190107 z 92 82 + 509862769262415792 z - 33648315442930067344445 z 64 112 110 106 + 1245857851239263740229112228 z + z - 595 z - 15410388 z 108 30 + 129828 z - 33648315442930067344445 z 42 44 - 131993833476824154694380606 z + 319985200988669296193295172 z 46 58 - 676085796367358615125749767 z - 3445896848620334714644727966 z 56 54 + 3687506621313281106008837364 z - 3445896848620334714644727966 z 52 60 + 2811853187952811393570127148 z + 2811853187952811393570127148 z 70 68 - 131993833476824154694380606 z + 319985200988669296193295172 z 78 32 - 944617234006280305784411 z + 192021810205246788362460 z 38 40 - 14810564901202077352774252 z + 47410469891300409296849022 z 62 76 - 2003308928240119600376366237 z + 4017717023357014192714688 z 74 72 - 14810564901202077352774252 z + 47410469891300409296849022 z 104 + 1169375482 z )) And in Maple-input format, it is: -(1+1362394589311962055306*z^28-183714057341302006201*z^26-437*z^2+ 21107108591143245856*z^24-2052323247443069627*z^22+79686*z^4-8335691*z^6-\ 27923277349*z^102+572854680*z^8-27923277349*z^10+1014641824558*z^12-\ 28463266734239*z^14-11362326363456153*z^18+632604218331624*z^16-\ 381193827982045334403124381*z^50+240646277271307341375085964*z^48+ 167510817097511214*z^20+915701078796175699721342*z^36-223631262121070572831853* z^34-133091220666693795969623947*z^66+47334349633910979878628*z^80+ 1014641824558*z^100-2052323247443069627*z^90+21107108591143245856*z^88+ 1362394589311962055306*z^84-11362326363456153*z^94-183714057341302006201*z^86+ 632604218331624*z^96-28463266734239*z^98+167510817097511214*z^92-\ 8655993490790622236839*z^82+240646277271307341375085964*z^64+z^112-437*z^110-\ 8335691*z^106+79686*z^108-8655993490790622236839*z^30-\ 27288103347303628452609129*z^42+64438790227796189632355646*z^44-\ 133091220666693795969623947*z^46-644373254537836365058579579*z^58+ 688039593047742963825651130*z^56-644373254537836365058579579*z^54+ 529272359846494073587396870*z^52+529272359846494073587396870*z^60-\ 27288103347303628452609129*z^70+64438790227796189632355646*z^68-\ 223631262121070572831853*z^78+47334349633910979878628*z^32-\ 3258161307862183483099975*z^38+10095397907384474453985532*z^40-\ 381193827982045334403124381*z^62+915701078796175699721342*z^76-\ 3258161307862183483099975*z^74+10095397907384474453985532*z^72+572854680*z^104) /(-1+z^2)/(1+5064020205580385931020*z^28-651611052730895483140*z^26-595*z^2+ 71292973757044776162*z^24-6587745029768480614*z^22+129828*z^4-15410388*z^6-\ 61951479674*z^102+1169375482*z^8-61951479674*z^10+2422529111708*z^12-\ 72648985190107*z^14-32714205490123077*z^18+1717965275080368*z^16-\ 2003308928240119600376366237*z^50+1245857851239263740229112228*z^48+ 509862769262415792*z^20+4017717023357014192714688*z^36-944617234006280305784411 *z^34-676085796367358615125749767*z^66+192021810205246788362460*z^80+ 2422529111708*z^100-6587745029768480614*z^90+71292973757044776162*z^88+ 5064020205580385931020*z^84-32714205490123077*z^94-651611052730895483140*z^86+ 1717965275080368*z^96-72648985190107*z^98+509862769262415792*z^92-\ 33648315442930067344445*z^82+1245857851239263740229112228*z^64+z^112-595*z^110-\ 15410388*z^106+129828*z^108-33648315442930067344445*z^30-\ 131993833476824154694380606*z^42+319985200988669296193295172*z^44-\ 676085796367358615125749767*z^46-3445896848620334714644727966*z^58+ 3687506621313281106008837364*z^56-3445896848620334714644727966*z^54+ 2811853187952811393570127148*z^52+2811853187952811393570127148*z^60-\ 131993833476824154694380606*z^70+319985200988669296193295172*z^68-\ 944617234006280305784411*z^78+192021810205246788362460*z^32-\ 14810564901202077352774252*z^38+47410469891300409296849022*z^40-\ 2003308928240119600376366237*z^62+4017717023357014192714688*z^76-\ 14810564901202077352774252*z^74+47410469891300409296849022*z^72+1169375482*z^ 104) The first , 40, terms are: [0, 159, 0, 44027, 0, 12707360, 0, 3690416293, 0, 1073161811657, 0, 312168204459883, 0, 90812230369976495, 0, 26418482411682011093, 0, 7685520949528240708141, 0, 2235832300675291853381648, 0, 650437028312126320641941195, 0, 189221863973343177366840743227, 0, 55047472426123552180214340262317, 0, 16014133720303097940756096113845861, 0, 4658751215233922321422435613744499283, 0, 1355300465807119751992819563942566969451, 0, 394277193149614541492429332303547507270800, 0, 114701137468881223812970808892037969425079869, 0, 33368277864627774127174254242346071000124727533, 0, 9707331524537488532488073543915750021777456620183] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 65531509706574163 z - 25509602452770745 z - 330 z 24 22 4 6 + 7999287609665557 z - 2011198977166056 z + 40661 z - 2650739 z 8 10 12 14 + 106183435 z - 2846271236 z + 53943194401 z - 750802539320 z 18 16 50 - 63766624857872 z + 7885997624514 z - 2011198977166056 z 48 20 36 + 7999287609665557 z + 402885444596419 z + 346855682860248916 z 34 66 64 - 312668623065299168 z - 2650739 z + 106183435 z 30 42 44 - 136076730582981778 z - 136076730582981778 z + 65531509706574163 z 46 58 56 - 25509602452770745 z - 750802539320 z + 7885997624514 z 54 52 60 70 - 63766624857872 z + 402885444596419 z + 53943194401 z - 330 z 68 32 38 + 40661 z + 228961211472915523 z - 312668623065299168 z 40 62 72 / + 228961211472915523 z - 2846271236 z + z ) / (-1 / 28 26 2 - 441179180159865796 z + 155038784480432067 z + 491 z 24 22 4 6 - 43899505165812925 z + 9967629633820952 z - 73492 z + 5522042 z 8 10 12 14 - 250154379 z + 7519304865 z - 159060178734 z + 2463474598344 z 18 16 50 + 257599657914913 z - 28731227470221 z + 43899505165812925 z 48 20 36 - 155038784480432067 z - 1803076341926302 z - 3524566308619310046 z 34 66 64 + 2865308207357416325 z + 250154379 z - 7519304865 z 30 42 + 1015047301120630522 z + 1892803197471355771 z 44 46 58 - 1015047301120630522 z + 441179180159865796 z + 28731227470221 z 56 54 52 - 257599657914913 z + 1803076341926302 z - 9967629633820952 z 60 70 68 32 - 2463474598344 z + 73492 z - 5522042 z - 1892803197471355771 z 38 40 62 + 3524566308619310046 z - 2865308207357416325 z + 159060178734 z 74 72 + z - 491 z ) And in Maple-input format, it is: -(1+65531509706574163*z^28-25509602452770745*z^26-330*z^2+7999287609665557*z^24 -2011198977166056*z^22+40661*z^4-2650739*z^6+106183435*z^8-2846271236*z^10+ 53943194401*z^12-750802539320*z^14-63766624857872*z^18+7885997624514*z^16-\ 2011198977166056*z^50+7999287609665557*z^48+402885444596419*z^20+ 346855682860248916*z^36-312668623065299168*z^34-2650739*z^66+106183435*z^64-\ 136076730582981778*z^30-136076730582981778*z^42+65531509706574163*z^44-\ 25509602452770745*z^46-750802539320*z^58+7885997624514*z^56-63766624857872*z^54 +402885444596419*z^52+53943194401*z^60-330*z^70+40661*z^68+228961211472915523*z ^32-312668623065299168*z^38+228961211472915523*z^40-2846271236*z^62+z^72)/(-1-\ 441179180159865796*z^28+155038784480432067*z^26+491*z^2-43899505165812925*z^24+ 9967629633820952*z^22-73492*z^4+5522042*z^6-250154379*z^8+7519304865*z^10-\ 159060178734*z^12+2463474598344*z^14+257599657914913*z^18-28731227470221*z^16+ 43899505165812925*z^50-155038784480432067*z^48-1803076341926302*z^20-\ 3524566308619310046*z^36+2865308207357416325*z^34+250154379*z^66-7519304865*z^ 64+1015047301120630522*z^30+1892803197471355771*z^42-1015047301120630522*z^44+ 441179180159865796*z^46+28731227470221*z^58-257599657914913*z^56+ 1803076341926302*z^54-9967629633820952*z^52-2463474598344*z^60+73492*z^70-\ 5522042*z^68-1892803197471355771*z^32+3524566308619310046*z^38-\ 2865308207357416325*z^40+159060178734*z^62+z^74-491*z^72) The first , 40, terms are: [0, 161, 0, 46220, 0, 13733111, 0, 4091235079, 0, 1219149590027, 0, 363307571711603, 0, 108266496378444188, 0, 32263691992541870037, 0, 9614663760446108107765, 0, 2865194762816923971281813, 0, 853835481498472230529744661, 0, 254445191452587648894847930316, 0, 75825328021902009403710042189075, 0, 22596144721514676764282564785539083, 0, 6733709824904123570184281865237091767, 0, 2006663019947376883207491905530898387431, 0, 597990792643366716500957426990356661153980, 0, 178202809605584827277058433511306437374287441, 0, 53104900179062494729765656184404181917920523249, 0, 15825398203709420018875073381370407804264443309969] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 238210681 z - 887373732 z - 220 z + 2240098283 z 22 4 6 8 10 - 3885457568 z + 12571 z - 330768 z + 4813120 z - 42356176 z 12 14 18 16 + 238210681 z - 887373732 z - 3885457568 z + 2240098283 z 20 36 34 30 32 + 4664800384 z + 12571 z - 330768 z - 42356176 z + 4813120 z 38 40 / 2 4 18 - 220 z + z ) / (376 z - 27900 z + 32327837859 z / 16 10 22 40 - 15499749920 z - 1 + 169374652 z + 46636908200 z - 376 z 30 24 14 28 + 1142580685 z - 32327837859 z + 5108686060 z - 5108686060 z 36 8 38 12 6 - 907407 z - 16008732 z + 27900 z - 1142580685 z + 907407 z 26 42 32 34 20 + 15499749920 z + z - 169374652 z + 16008732 z - 46636908200 z ) And in Maple-input format, it is: -(1+238210681*z^28-887373732*z^26-220*z^2+2240098283*z^24-3885457568*z^22+12571 *z^4-330768*z^6+4813120*z^8-42356176*z^10+238210681*z^12-887373732*z^14-\ 3885457568*z^18+2240098283*z^16+4664800384*z^20+12571*z^36-330768*z^34-42356176 *z^30+4813120*z^32-220*z^38+z^40)/(376*z^2-27900*z^4+32327837859*z^18-\ 15499749920*z^16-1+169374652*z^10+46636908200*z^22-376*z^40+1142580685*z^30-\ 32327837859*z^24+5108686060*z^14-5108686060*z^28-907407*z^36-16008732*z^8+27900 *z^38-1142580685*z^12+907407*z^6+15499749920*z^26+z^42-169374652*z^32+16008732* z^34-46636908200*z^20) The first , 40, terms are: [0, 156, 0, 43327, 0, 12515191, 0, 3627248396, 0, 1051616447369, 0, 304895833626425, 0, 88398937282834732, 0, 25629654173579743303, 0, 7430849470179432149743, 0, 2154438904154633779477628, 0, 624640158859736264032348625, 0, 181102990353324570436251620657, 0, 52507499957937394990768011552956, 0, 15223589331435847898000320986867855, 0, 4413801310629413743201220302371272615, 0, 1279700968383687191481368015036003825900, 0, 371025892021546114338306313286771870453465, 0, 107572171899075987304507738644483729197801961, 0, 31188583912662248307892335086247726104904616780, 0, 9042559514274619177171224305546398739443588624919] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 51231589139277546 z - 23353823913249982 z - 368 z 24 22 4 6 + 8440923705024820 z - 2404312901911552 z + 51483 z - 3743110 z 8 10 12 14 + 162139490 z - 4550711866 z + 87588036664 z - 1203541448332 z 18 16 50 - 92305809314386 z + 12158620311931 z - 92305809314386 z 48 20 36 + 535465411319723 z + 535465411319723 z + 124961491597329414 z 34 66 64 30 - 139631566925923980 z - 368 z + 51483 z - 89514282051683822 z 42 44 46 - 23353823913249982 z + 8440923705024820 z - 2404312901911552 z 58 56 54 - 4550711866 z + 87588036664 z - 1203541448332 z 52 60 68 32 + 12158620311931 z + 162139490 z + z + 124961491597329414 z 38 40 62 / - 89514282051683822 z + 51231589139277546 z - 3743110 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 262110915555607418 z - 114714713509355248 z - 531 z 24 22 4 6 + 39410343131020360 z - 10573759756172864 z + 91635 z - 7778202 z 8 10 12 14 + 382771790 z - 12017400196 z + 256163176440 z - 3869332806192 z 18 16 50 - 351779981552573 z + 42693360259219 z - 351779981552573 z 48 20 36 + 2200165484298615 z + 2200165484298615 z + 670779048389528826 z 34 66 64 30 - 754128826179875448 z - 531 z + 91635 z - 471834876239673206 z 42 44 46 - 114714713509355248 z + 39410343131020360 z - 10573759756172864 z 58 56 54 - 12017400196 z + 256163176440 z - 3869332806192 z 52 60 68 32 + 42693360259219 z + 382771790 z + z + 670779048389528826 z 38 40 62 - 471834876239673206 z + 262110915555607418 z - 7778202 z )) And in Maple-input format, it is: -(1+51231589139277546*z^28-23353823913249982*z^26-368*z^2+8440923705024820*z^24 -2404312901911552*z^22+51483*z^4-3743110*z^6+162139490*z^8-4550711866*z^10+ 87588036664*z^12-1203541448332*z^14-92305809314386*z^18+12158620311931*z^16-\ 92305809314386*z^50+535465411319723*z^48+535465411319723*z^20+ 124961491597329414*z^36-139631566925923980*z^34-368*z^66+51483*z^64-\ 89514282051683822*z^30-23353823913249982*z^42+8440923705024820*z^44-\ 2404312901911552*z^46-4550711866*z^58+87588036664*z^56-1203541448332*z^54+ 12158620311931*z^52+162139490*z^60+z^68+124961491597329414*z^32-\ 89514282051683822*z^38+51231589139277546*z^40-3743110*z^62)/(-1+z^2)/(1+ 262110915555607418*z^28-114714713509355248*z^26-531*z^2+39410343131020360*z^24-\ 10573759756172864*z^22+91635*z^4-7778202*z^6+382771790*z^8-12017400196*z^10+ 256163176440*z^12-3869332806192*z^14-351779981552573*z^18+42693360259219*z^16-\ 351779981552573*z^50+2200165484298615*z^48+2200165484298615*z^20+ 670779048389528826*z^36-754128826179875448*z^34-531*z^66+91635*z^64-\ 471834876239673206*z^30-114714713509355248*z^42+39410343131020360*z^44-\ 10573759756172864*z^46-12017400196*z^58+256163176440*z^56-3869332806192*z^54+ 42693360259219*z^52+382771790*z^60+z^68+670779048389528826*z^32-\ 471834876239673206*z^38+262110915555607418*z^40-7778202*z^62) The first , 40, terms are: [0, 164, 0, 46565, 0, 13784083, 0, 4103665132, 0, 1222984277783, 0, 364554796917367, 0, 108673739548431500, 0, 32395951910907805171, 0, 9657347166474432989861, 0, 2878890584914288352886852, 0, 858207932458307735336252225, 0, 255834965916347396444432225153, 0, 76265352147360476284836845728260, 0, 22734984350684043104050486166327525, 0, 6777383163471881450489849865306352243, 0, 2020363059840532145658451190415874608204, 0, 602277722122270824274617439365117877577463, 0, 179541222949833310357134612426811013039908119, 0, 53521904520673311238920360911491092880194341932, 0, 15955078262562753463017126538427833681636798529555] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 6403286812 z - 14506538295 z - 229 z + 23607455299 z 22 4 6 8 10 - 27751650240 z + 14020 z - 402647 z + 6544183 z - 66214592 z 12 14 18 16 + 441372453 z - 2009737709 z - 14506538295 z + 6403286812 z 20 36 34 30 42 + 23607455299 z + 6544183 z - 66214592 z - 2009737709 z - 229 z 44 32 38 40 / 2 44 + z + 441372453 z - 402647 z + 14020 z ) / ((-1 + z ) (z / 42 40 38 36 34 - 396 z + 30437 z - 1049134 z + 20032469 z - 233592310 z 32 30 28 26 + 1760897123 z - 8900025158 z + 30872602075 z - 74545499968 z 24 22 20 18 + 126250043783 z - 150451126932 z + 126250043783 z - 74545499968 z 16 14 12 10 + 30872602075 z - 8900025158 z + 1760897123 z - 233592310 z 8 6 4 2 + 20032469 z - 1049134 z + 30437 z - 396 z + 1)) And in Maple-input format, it is: -(1+6403286812*z^28-14506538295*z^26-229*z^2+23607455299*z^24-27751650240*z^22+ 14020*z^4-402647*z^6+6544183*z^8-66214592*z^10+441372453*z^12-2009737709*z^14-\ 14506538295*z^18+6403286812*z^16+23607455299*z^20+6544183*z^36-66214592*z^34-\ 2009737709*z^30-229*z^42+z^44+441372453*z^32-402647*z^38+14020*z^40)/(-1+z^2)/( z^44-396*z^42+30437*z^40-1049134*z^38+20032469*z^36-233592310*z^34+1760897123*z ^32-8900025158*z^30+30872602075*z^28-74545499968*z^26+126250043783*z^24-\ 150451126932*z^22+126250043783*z^20-74545499968*z^18+30872602075*z^16-\ 8900025158*z^14+1760897123*z^12-233592310*z^10+20032469*z^8-1049134*z^6+30437*z ^4-396*z^2+1) The first , 40, terms are: [0, 168, 0, 49883, 0, 15300531, 0, 4703098776, 0, 1445866882825, 0, 444505959716761, 0, 136655546879695480, 0, 42012349810927737891, 0, 12915959780667656142251, 0, 3970785206970437287380552, 0, 1220748239256236529962705585, 0, 375297626534030249290940580305, 0, 115378670189958900888954251359752, 0, 35471147680164066354174966112736907, 0, 10904981966567242878975767787223860675, 0, 3352545363443587171152178679164944550840, 0, 1030681247195603805735755885280308547336953, 0, 316864864799184574162570681338414109181916329, 0, 97414542873846296085836321323139713999493110872, 0, 29948391940944721869575487417738370402879611011859] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 62779269550164 z - 54964230126920 z - 292 z 24 22 4 6 + 36850959733648 z - 18864136185192 z + 28954 z - 1413149 z 8 10 12 14 + 40019650 z - 722743139 z + 8829579510 z - 75862928228 z 18 16 50 48 - 2151099211740 z + 470821373087 z - 1413149 z + 40019650 z 20 36 34 + 7335768774888 z + 7335768774888 z - 18864136185192 z 30 42 44 46 - 54964230126920 z - 75862928228 z + 8829579510 z - 722743139 z 56 54 52 32 38 + z - 292 z + 28954 z + 36850959733648 z - 2151099211740 z 40 / 2 28 + 470821373087 z ) / ((-1 + z ) (1 + 344516134659558 z / 26 2 24 22 - 298520150832972 z - 460 z + 194133629325646 z - 94628005547048 z 4 6 8 10 12 + 56351 z - 3193330 z + 102850763 z - 2092308762 z + 28598991839 z 14 18 16 50 - 273132491004 z - 9346235740880 z + 1870425146937 z - 3193330 z 48 20 36 + 102850763 z + 34478941370378 z + 34478941370378 z 34 30 42 - 94628005547048 z - 298520150832972 z - 273132491004 z 44 46 56 54 52 + 28598991839 z - 2092308762 z + z - 460 z + 56351 z 32 38 40 + 194133629325646 z - 9346235740880 z + 1870425146937 z )) And in Maple-input format, it is: -(1+62779269550164*z^28-54964230126920*z^26-292*z^2+36850959733648*z^24-\ 18864136185192*z^22+28954*z^4-1413149*z^6+40019650*z^8-722743139*z^10+ 8829579510*z^12-75862928228*z^14-2151099211740*z^18+470821373087*z^16-1413149*z ^50+40019650*z^48+7335768774888*z^20+7335768774888*z^36-18864136185192*z^34-\ 54964230126920*z^30-75862928228*z^42+8829579510*z^44-722743139*z^46+z^56-292*z^ 54+28954*z^52+36850959733648*z^32-2151099211740*z^38+470821373087*z^40)/(-1+z^2 )/(1+344516134659558*z^28-298520150832972*z^26-460*z^2+194133629325646*z^24-\ 94628005547048*z^22+56351*z^4-3193330*z^6+102850763*z^8-2092308762*z^10+ 28598991839*z^12-273132491004*z^14-9346235740880*z^18+1870425146937*z^16-\ 3193330*z^50+102850763*z^48+34478941370378*z^20+34478941370378*z^36-\ 94628005547048*z^34-298520150832972*z^30-273132491004*z^42+28598991839*z^44-\ 2092308762*z^46+z^56-460*z^54+56351*z^52+194133629325646*z^32-9346235740880*z^ 38+1870425146937*z^40) The first , 40, terms are: [0, 169, 0, 50052, 0, 15309445, 0, 4697321619, 0, 1441924384545, 0, 442659816611079, 0, 135895241017414924, 0, 41719550165733572459, 0, 12807819973517574645763, 0, 3931975968339313823443403, 0, 1207109042259638226442400755, 0, 370580150207813713104691356812, 0, 113767392182171014961856516159391, 0, 34926370228553445192154297868542825, 0, 10722328375221925122481515987691147419, 0, 3291734154859073878328365423766941666461, 0, 1010556044087787129721263055311248654175812, 0, 310238758720851919895690863678285981093980529, 0, 95242701259126418794946855520273804082439746825, 0, 29239325803573598307522021848400713216535440547449] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 2184998369576 z + 4579807413639 z + 362 z 24 22 4 6 - 6601951316087 z + 6601951316087 z - 47141 z + 2888955 z 8 10 12 14 - 94928737 z + 1800939529 z - 20761380636 z + 150570048123 z 18 16 20 + 2184998369576 z - 705185021253 z - 4579807413639 z 36 34 30 42 - 1800939529 z + 20761380636 z + 705185021253 z + 47141 z 44 46 32 38 40 / - 362 z + z - 150570048123 z + 94928737 z - 2888955 z ) / (1 / 28 26 2 24 + 34243385125290 z - 58876708043896 z - 514 z + 70460711863177 z 22 4 6 8 - 58876708043896 z + 86426 z - 6530004 z + 259090509 z 10 12 14 18 - 5855000992 z + 79577063358 z - 678005299114 z - 13734584946846 z 16 48 20 36 + 3741596958758 z + z + 34243385125290 z + 79577063358 z 34 30 42 44 - 678005299114 z - 13734584946846 z - 6530004 z + 86426 z 46 32 38 40 - 514 z + 3741596958758 z - 5855000992 z + 259090509 z ) And in Maple-input format, it is: -(-1-2184998369576*z^28+4579807413639*z^26+362*z^2-6601951316087*z^24+ 6601951316087*z^22-47141*z^4+2888955*z^6-94928737*z^8+1800939529*z^10-\ 20761380636*z^12+150570048123*z^14+2184998369576*z^18-705185021253*z^16-\ 4579807413639*z^20-1800939529*z^36+20761380636*z^34+705185021253*z^30+47141*z^ 42-362*z^44+z^46-150570048123*z^32+94928737*z^38-2888955*z^40)/(1+ 34243385125290*z^28-58876708043896*z^26-514*z^2+70460711863177*z^24-\ 58876708043896*z^22+86426*z^4-6530004*z^6+259090509*z^8-5855000992*z^10+ 79577063358*z^12-678005299114*z^14-13734584946846*z^18+3741596958758*z^16+z^48+ 34243385125290*z^20+79577063358*z^36-678005299114*z^34-13734584946846*z^30-\ 6530004*z^42+86426*z^44-514*z^46+3741596958758*z^32-5855000992*z^38+259090509*z ^40) The first , 40, terms are: [0, 152, 0, 38843, 0, 10469599, 0, 2852727604, 0, 779773674749, 0, 213387648093053, 0, 58420137962219524, 0, 15996829127789950555, 0, 4380636927059011869899, 0, 1199647965645709714549784, 0, 328530610507093239294081661, 0, 89970496139618233849319761645, 0, 24639127381734379662843397042712, 0, 6747624647776450837755400477285931, 0, 1847892354499824812775413611107965947, 0, 506060554748350896934336115940629422852, 0, 138588863434242540105469236864131661551357, 0, 37953706106347568855440838376700017760526861, 0, 10393936330716429815169299909882026385489671252, 0, 2846465444619132433972705493873234824754112009679] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 65771961918934088 z - 30059058058090014 z - 395 z 24 22 4 6 + 10894952461294490 z - 3111850739550848 z + 59388 z - 4544985 z 8 10 12 14 + 203062155 z - 5804101240 z + 112900108599 z - 1560119604221 z 18 16 50 - 119925359341255 z + 15796771715004 z - 119925359341255 z 48 20 36 + 694661893664565 z + 694661893664565 z + 159882685801169006 z 34 66 64 30 - 178569447539115536 z - 395 z + 59388 z - 114683048099017658 z 42 44 46 - 30059058058090014 z + 10894952461294490 z - 3111850739550848 z 58 56 54 - 5804101240 z + 112900108599 z - 1560119604221 z 52 60 68 32 + 15796771715004 z + 203062155 z + z + 159882685801169006 z 38 40 62 / - 114683048099017658 z + 65771961918934088 z - 4544985 z ) / (-1 / 28 26 2 - 465798596847189450 z + 192374594943137094 z + 535 z 24 22 4 6 - 63121652153259626 z + 16344700260162565 z - 101401 z + 9383899 z 8 10 12 14 - 492895973 z + 16201176547 z - 356402726575 z + 5504690183577 z 18 16 50 + 518429018750925 z - 61827243084983 z + 3310500785196787 z 48 20 36 - 16344700260162565 z - 3310500785196787 z - 1735088712244266238 z 34 66 64 + 1735088712244266238 z + 101401 z - 9383899 z 30 42 44 + 900504468358483390 z + 465798596847189450 z - 192374594943137094 z 46 58 56 + 63121652153259626 z + 356402726575 z - 5504690183577 z 54 52 60 70 + 61827243084983 z - 518429018750925 z - 16201176547 z + z 68 32 38 - 535 z - 1394874671036417330 z + 1394874671036417330 z 40 62 - 900504468358483390 z + 492895973 z ) And in Maple-input format, it is: -(1+65771961918934088*z^28-30059058058090014*z^26-395*z^2+10894952461294490*z^ 24-3111850739550848*z^22+59388*z^4-4544985*z^6+203062155*z^8-5804101240*z^10+ 112900108599*z^12-1560119604221*z^14-119925359341255*z^18+15796771715004*z^16-\ 119925359341255*z^50+694661893664565*z^48+694661893664565*z^20+ 159882685801169006*z^36-178569447539115536*z^34-395*z^66+59388*z^64-\ 114683048099017658*z^30-30059058058090014*z^42+10894952461294490*z^44-\ 3111850739550848*z^46-5804101240*z^58+112900108599*z^56-1560119604221*z^54+ 15796771715004*z^52+203062155*z^60+z^68+159882685801169006*z^32-\ 114683048099017658*z^38+65771961918934088*z^40-4544985*z^62)/(-1-\ 465798596847189450*z^28+192374594943137094*z^26+535*z^2-63121652153259626*z^24+ 16344700260162565*z^22-101401*z^4+9383899*z^6-492895973*z^8+16201176547*z^10-\ 356402726575*z^12+5504690183577*z^14+518429018750925*z^18-61827243084983*z^16+ 3310500785196787*z^50-16344700260162565*z^48-3310500785196787*z^20-\ 1735088712244266238*z^36+1735088712244266238*z^34+101401*z^66-9383899*z^64+ 900504468358483390*z^30+465798596847189450*z^42-192374594943137094*z^44+ 63121652153259626*z^46+356402726575*z^58-5504690183577*z^56+61827243084983*z^54 -518429018750925*z^52-16201176547*z^60+z^70-535*z^68-1394874671036417330*z^32+ 1394874671036417330*z^38-900504468358483390*z^40+492895973*z^62) The first , 40, terms are: [0, 140, 0, 32887, 0, 8237319, 0, 2096103020, 0, 536142657281, 0, 137402341075649, 0, 35241184836340172, 0, 9041704777073904199, 0, 2320124114007073139895, 0, 595385618995496787693804, 0, 152790649791639098625272193, 0, 39210294921439055752694093185, 0, 10062492255551219038381292559788, 0, 2582331113083357659114749031052983, 0, 662702630980192747590243283836626247, 0, 170069186522694958965943547504700629132, 0, 43644814190070733872235557208021079524161, 0, 11200558943769439215253844595137556568777857, 0, 2874397040084530987329533561400427223758714540, 0, 737655906441422373688179516877921717928520754951] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 26873773803386297083 z - 5953374780972889737 z - 425 z 24 22 4 6 + 1088081130947681465 z - 163045110953932108 z + 72207 z - 6684338 z 8 10 12 14 + 388505468 z - 15394277514 z + 438783157277 z - 9339503091831 z 18 16 50 - 1952014354481324 z + 152556834515639 z - 3255444218000602790167 z 48 20 + 5006912961059524301167 z + 19875829481592264 z 36 34 + 1779152225156624077161 z - 816031793168119943974 z 66 80 88 84 86 - 163045110953932108 z + 388505468 z + z + 72207 z - 425 z 82 64 30 - 6684338 z + 1088081130947681465 z - 100588863844618155214 z 42 44 - 6479794101082941721496 z + 7060915251668844712144 z 46 58 - 6479794101082941721496 z - 100588863844618155214 z 56 54 + 313472661549106007652 z - 816031793168119943974 z 52 60 + 1779152225156624077161 z + 26873773803386297083 z 70 68 78 - 1952014354481324 z + 19875829481592264 z - 15394277514 z 32 38 + 313472661549106007652 z - 3255444218000602790167 z 40 62 76 + 5006912961059524301167 z - 5953374780972889737 z + 438783157277 z 74 72 / - 9339503091831 z + 152556834515639 z ) / (-1 / 28 26 2 - 140839643189009796520 z + 28805121237443349687 z + 575 z 24 22 4 - 4862583092657566257 z + 673070723811771908 z - 120692 z 6 8 10 12 + 13051620 z - 859721085 z + 37942006343 z - 1191703221384 z 14 18 16 + 27758479531808 z + 6863996231089087 z - 493932151760137 z 50 48 + 42987568033512269611553 z - 60538296947583772722559 z 20 36 - 75764845127643924 z - 12922877146505593865348 z 34 66 80 + 5456928617783329135259 z + 4862583092657566257 z - 37942006343 z 90 88 84 86 82 + z - 575 z - 13051620 z + 120692 z + 859721085 z 64 30 - 28805121237443349687 z + 571361308211075573584 z 42 44 + 60538296947583772722559 z - 71829172695821438732104 z 46 58 + 71829172695821438732104 z + 1931258243657686206465 z 56 54 - 5456928617783329135259 z + 12922877146505593865348 z 52 60 - 25700438804448373329012 z - 571361308211075573584 z 70 68 78 + 75764845127643924 z - 673070723811771908 z + 1191703221384 z 32 38 - 1931258243657686206465 z + 25700438804448373329012 z 40 62 - 42987568033512269611553 z + 140839643189009796520 z 76 74 72 - 27758479531808 z + 493932151760137 z - 6863996231089087 z ) And in Maple-input format, it is: -(1+26873773803386297083*z^28-5953374780972889737*z^26-425*z^2+ 1088081130947681465*z^24-163045110953932108*z^22+72207*z^4-6684338*z^6+ 388505468*z^8-15394277514*z^10+438783157277*z^12-9339503091831*z^14-\ 1952014354481324*z^18+152556834515639*z^16-3255444218000602790167*z^50+ 5006912961059524301167*z^48+19875829481592264*z^20+1779152225156624077161*z^36-\ 816031793168119943974*z^34-163045110953932108*z^66+388505468*z^80+z^88+72207*z^ 84-425*z^86-6684338*z^82+1088081130947681465*z^64-100588863844618155214*z^30-\ 6479794101082941721496*z^42+7060915251668844712144*z^44-6479794101082941721496* z^46-100588863844618155214*z^58+313472661549106007652*z^56-\ 816031793168119943974*z^54+1779152225156624077161*z^52+26873773803386297083*z^ 60-1952014354481324*z^70+19875829481592264*z^68-15394277514*z^78+ 313472661549106007652*z^32-3255444218000602790167*z^38+5006912961059524301167*z ^40-5953374780972889737*z^62+438783157277*z^76-9339503091831*z^74+ 152556834515639*z^72)/(-1-140839643189009796520*z^28+28805121237443349687*z^26+ 575*z^2-4862583092657566257*z^24+673070723811771908*z^22-120692*z^4+13051620*z^ 6-859721085*z^8+37942006343*z^10-1191703221384*z^12+27758479531808*z^14+ 6863996231089087*z^18-493932151760137*z^16+42987568033512269611553*z^50-\ 60538296947583772722559*z^48-75764845127643924*z^20-12922877146505593865348*z^ 36+5456928617783329135259*z^34+4862583092657566257*z^66-37942006343*z^80+z^90-\ 575*z^88-13051620*z^84+120692*z^86+859721085*z^82-28805121237443349687*z^64+ 571361308211075573584*z^30+60538296947583772722559*z^42-71829172695821438732104 *z^44+71829172695821438732104*z^46+1931258243657686206465*z^58-\ 5456928617783329135259*z^56+12922877146505593865348*z^54-\ 25700438804448373329012*z^52-571361308211075573584*z^60+75764845127643924*z^70-\ 673070723811771908*z^68+1191703221384*z^78-1931258243657686206465*z^32+ 25700438804448373329012*z^38-42987568033512269611553*z^40+140839643189009796520 *z^62-27758479531808*z^76+493932151760137*z^74-6863996231089087*z^72) The first , 40, terms are: [0, 150, 0, 37765, 0, 9978357, 0, 2666149278, 0, 715211967185, 0, 192168730371657, 0, 51668163916159542, 0, 13895912519916351469, 0, 3737693029617700590825, 0, 1005408373023159261408766, 0, 270452401943284668366314041, 0, 72751711606247645927791202757, 0, 19570291699265359882927936991286, 0, 5264439144852276695197850489848109, 0, 1416143389800181434169024661231710073, 0, 380945177924543871750571137520366420654, 0, 102474967061513137890059235818834420598261, 0, 27565959765155417552724792540388652037894765, 0, 7415295434451412787295805729083517114238160326, 0, 1994728548924088397836261556278479089885930058353] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 32432265564988924513 z - 7095919279094118861 z - 417 z 24 22 4 6 + 1279019574257234009 z - 188733342382512908 z + 70345 z - 6569962 z 8 10 12 14 + 388823642 z - 15752355202 z + 459570592803 z - 10007978117727 z 18 16 50 - 2180921687845100 z + 167051862028555 z - 4092595264463132965747 z 48 20 + 6318356873838816564315 z + 22621467246761708 z 36 34 + 2224893101686789518579 z - 1013616306231289509814 z 66 80 88 84 86 - 188733342382512908 z + 388823642 z + z + 70345 z - 417 z 82 64 30 - 6569962 z + 1279019574257234009 z - 122739989612630625278 z 42 44 - 8195699697009298042280 z + 8937512167825926199368 z 46 58 - 8195699697009298042280 z - 122739989612630625278 z 56 54 + 386197967330108113046 z - 1013616306231289509814 z 52 60 + 2224893101686789518579 z + 32432265564988924513 z 70 68 78 - 2180921687845100 z + 22621467246761708 z - 15752355202 z 32 38 + 386197967330108113046 z - 4092595264463132965747 z 40 62 76 + 6318356873838816564315 z - 7095919279094118861 z + 459570592803 z 74 72 / 2 - 10007978117727 z + 167051862028555 z ) / ((-1 + z ) (1 / 28 26 2 + 138859850640927997810 z - 28971243859542628606 z - 562 z 24 22 4 + 4959872269692339763 z - 692406190510878768 z + 114632 z 6 8 10 12 - 12382724 z + 826599273 z - 37147455700 z + 1187776544034 z 14 18 16 - 28088565362146 z - 7070878276167792 z + 505431164409797 z 50 48 - 20731511245008040219646 z + 32548107122083855802791 z 20 36 + 78192720284321022 z + 11014238421707628217252 z 34 66 80 - 4875541120862984280604 z - 692406190510878768 z + 826599273 z 88 84 86 82 64 + z + 114632 z - 562 z - 12382724 z + 4959872269692339763 z 30 42 - 548815028295732206492 z - 42652796550199354169904 z 44 46 + 46673409502148467485572 z - 42652796550199354169904 z 58 56 - 548815028295732206492 z + 1795385977805269919175 z 54 52 - 4875541120862984280604 z + 11014238421707628217252 z 60 70 68 + 138859850640927997810 z - 7070878276167792 z + 78192720284321022 z 78 32 - 37147455700 z + 1795385977805269919175 z 38 40 - 20731511245008040219646 z + 32548107122083855802791 z 62 76 74 - 28971243859542628606 z + 1187776544034 z - 28088565362146 z 72 + 505431164409797 z )) And in Maple-input format, it is: -(1+32432265564988924513*z^28-7095919279094118861*z^26-417*z^2+ 1279019574257234009*z^24-188733342382512908*z^22+70345*z^4-6569962*z^6+ 388823642*z^8-15752355202*z^10+459570592803*z^12-10007978117727*z^14-\ 2180921687845100*z^18+167051862028555*z^16-4092595264463132965747*z^50+ 6318356873838816564315*z^48+22621467246761708*z^20+2224893101686789518579*z^36-\ 1013616306231289509814*z^34-188733342382512908*z^66+388823642*z^80+z^88+70345*z ^84-417*z^86-6569962*z^82+1279019574257234009*z^64-122739989612630625278*z^30-\ 8195699697009298042280*z^42+8937512167825926199368*z^44-8195699697009298042280* z^46-122739989612630625278*z^58+386197967330108113046*z^56-\ 1013616306231289509814*z^54+2224893101686789518579*z^52+32432265564988924513*z^ 60-2180921687845100*z^70+22621467246761708*z^68-15752355202*z^78+ 386197967330108113046*z^32-4092595264463132965747*z^38+6318356873838816564315*z ^40-7095919279094118861*z^62+459570592803*z^76-10007978117727*z^74+ 167051862028555*z^72)/(-1+z^2)/(1+138859850640927997810*z^28-\ 28971243859542628606*z^26-562*z^2+4959872269692339763*z^24-692406190510878768*z ^22+114632*z^4-12382724*z^6+826599273*z^8-37147455700*z^10+1187776544034*z^12-\ 28088565362146*z^14-7070878276167792*z^18+505431164409797*z^16-\ 20731511245008040219646*z^50+32548107122083855802791*z^48+78192720284321022*z^ 20+11014238421707628217252*z^36-4875541120862984280604*z^34-692406190510878768* z^66+826599273*z^80+z^88+114632*z^84-562*z^86-12382724*z^82+4959872269692339763 *z^64-548815028295732206492*z^30-42652796550199354169904*z^42+ 46673409502148467485572*z^44-42652796550199354169904*z^46-548815028295732206492 *z^58+1795385977805269919175*z^56-4875541120862984280604*z^54+ 11014238421707628217252*z^52+138859850640927997810*z^60-7070878276167792*z^70+ 78192720284321022*z^68-37147455700*z^78+1795385977805269919175*z^32-\ 20731511245008040219646*z^38+32548107122083855802791*z^40-28971243859542628606* z^62+1187776544034*z^76-28088565362146*z^74+505431164409797*z^72) The first , 40, terms are: [0, 146, 0, 37349, 0, 10136557, 0, 2778956506, 0, 763376043273, 0, 209785478067001, 0, 57656826591379274, 0, 15846537070509923461, 0, 4355317612899614486693, 0, 1197031824867690980061538, 0, 328996784292241739030306409, 0, 90422733094157109600587411641, 0, 24852129684858602983574915759650, 0, 6830454359180314744576443244481701, 0, 1877308197410627978323246105843570805, 0, 515966564336157710934286622790282489002, 0, 141810223749650728840403696202254906953593, 0, 38975664219569198250118272107053683260050377, 0, 10712220608589206963779573296578576483522594618, 0, 2944187678770200214793092230293688460882597292605] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 72725763486184256 z - 32534839640122002 z - 369 z 24 22 4 6 + 11489851662921058 z - 3185866790457272 z + 51978 z - 3836817 z 8 10 12 14 + 170438681 z - 4935288908 z + 98288162949 z - 1398852564957 z 18 16 50 - 114933815084481 z + 14634088726758 z - 114933815084481 z 48 20 36 + 688596570239217 z + 688596570239217 z + 181393212545074398 z 34 66 64 30 - 203267795337119064 z - 369 z + 51978 z - 128841215183785054 z 42 44 46 - 32534839640122002 z + 11489851662921058 z - 3185866790457272 z 58 56 54 - 4935288908 z + 98288162949 z - 1398852564957 z 52 60 68 32 + 14634088726758 z + 170438681 z + z + 181393212545074398 z 38 40 62 / - 128841215183785054 z + 72725763486184256 z - 3836817 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 367363926953435998 z - 157569336593912820 z - 530 z 24 22 4 6 + 52858387850122090 z - 13812358519637544 z + 91939 z - 7937636 z 8 10 12 14 + 401800285 z - 13040339830 z + 287549054229 z - 4488198225020 z 18 16 50 - 433947864136658 z + 51099794508443 z - 433947864136658 z 48 20 36 + 2794482390483209 z + 2794482390483209 z + 964003656968417906 z 34 66 64 30 - 1087372064239746188 z - 530 z + 91939 z - 671565680882062960 z 42 44 46 - 157569336593912820 z + 52858387850122090 z - 13812358519637544 z 58 56 54 - 13040339830 z + 287549054229 z - 4488198225020 z 52 60 68 32 + 51099794508443 z + 401800285 z + z + 964003656968417906 z 38 40 62 - 671565680882062960 z + 367363926953435998 z - 7937636 z )) And in Maple-input format, it is: -(1+72725763486184256*z^28-32534839640122002*z^26-369*z^2+11489851662921058*z^ 24-3185866790457272*z^22+51978*z^4-3836817*z^6+170438681*z^8-4935288908*z^10+ 98288162949*z^12-1398852564957*z^14-114933815084481*z^18+14634088726758*z^16-\ 114933815084481*z^50+688596570239217*z^48+688596570239217*z^20+ 181393212545074398*z^36-203267795337119064*z^34-369*z^66+51978*z^64-\ 128841215183785054*z^30-32534839640122002*z^42+11489851662921058*z^44-\ 3185866790457272*z^46-4935288908*z^58+98288162949*z^56-1398852564957*z^54+ 14634088726758*z^52+170438681*z^60+z^68+181393212545074398*z^32-\ 128841215183785054*z^38+72725763486184256*z^40-3836817*z^62)/(-1+z^2)/(1+ 367363926953435998*z^28-157569336593912820*z^26-530*z^2+52858387850122090*z^24-\ 13812358519637544*z^22+91939*z^4-7937636*z^6+401800285*z^8-13040339830*z^10+ 287549054229*z^12-4488198225020*z^14-433947864136658*z^18+51099794508443*z^16-\ 433947864136658*z^50+2794482390483209*z^48+2794482390483209*z^20+ 964003656968417906*z^36-1087372064239746188*z^34-530*z^66+91939*z^64-\ 671565680882062960*z^30-157569336593912820*z^42+52858387850122090*z^44-\ 13812358519637544*z^46-13040339830*z^58+287549054229*z^56-4488198225020*z^54+ 51099794508443*z^52+401800285*z^60+z^68+964003656968417906*z^32-\ 671565680882062960*z^38+367363926953435998*z^40-7937636*z^62) The first , 40, terms are: [0, 162, 0, 45531, 0, 13389741, 0, 3961238342, 0, 1173005486403, 0, 347407642810139, 0, 102894305224333790, 0, 30475138072838192561, 0, 9026107217181691461587, 0, 2673347334384434641590354, 0, 791790538721231802687977597, 0, 234512087285822173095186092349, 0, 69457661484487850996812688167314, 0, 20571932121599557028616166364362419, 0, 6092983584827981619622059063259743025, 0, 1804616539981292321959837294715686121662, 0, 534490338116901315201226397210392850604219, 0, 158305055512640247503253984877926795397628739, 0, 46886704611278507089144954887515099714122876134, 0, 13886878483990994142106852094578631223285991364685] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 1023816234401 z + 2048924349722 z + 358 z 24 22 4 6 - 2888482250651 z + 2888482250651 z - 45563 z + 2643181 z 8 10 12 14 - 78005670 z + 1298679659 z - 13085695481 z + 83895560954 z 18 16 20 + 1023816234401 z - 355380198839 z - 2048924349722 z 36 34 30 42 - 1298679659 z + 13085695481 z + 355380198839 z + 45563 z 44 46 32 38 40 / - 358 z + z - 83895560954 z + 78005670 z - 2643181 z ) / (1 / 28 26 2 24 + 15883630124802 z - 26515820403122 z - 526 z + 31428508146838 z 22 4 6 8 - 26515820403122 z + 87214 z - 6229636 z + 221833902 z 10 12 14 18 - 4387748622 z + 52116119652 z - 393512165906 z - 6703909486124 z 16 48 20 36 + 1967414168194 z + z + 15883630124802 z + 52116119652 z 34 30 42 44 46 - 393512165906 z - 6703909486124 z - 6229636 z + 87214 z - 526 z 32 38 40 + 1967414168194 z - 4387748622 z + 221833902 z ) And in Maple-input format, it is: -(-1-1023816234401*z^28+2048924349722*z^26+358*z^2-2888482250651*z^24+ 2888482250651*z^22-45563*z^4+2643181*z^6-78005670*z^8+1298679659*z^10-\ 13085695481*z^12+83895560954*z^14+1023816234401*z^18-355380198839*z^16-\ 2048924349722*z^20-1298679659*z^36+13085695481*z^34+355380198839*z^30+45563*z^ 42-358*z^44+z^46-83895560954*z^32+78005670*z^38-2643181*z^40)/(1+15883630124802 *z^28-26515820403122*z^26-526*z^2+31428508146838*z^24-26515820403122*z^22+87214 *z^4-6229636*z^6+221833902*z^8-4387748622*z^10+52116119652*z^12-393512165906*z^ 14-6703909486124*z^18+1967414168194*z^16+z^48+15883630124802*z^20+52116119652*z ^36-393512165906*z^34-6703909486124*z^30-6229636*z^42+87214*z^44-526*z^46+ 1967414168194*z^32-4387748622*z^38+221833902*z^40) The first , 40, terms are: [0, 168, 0, 46717, 0, 13507645, 0, 3933395448, 0, 1147761133057, 0, 335157613897921, 0, 97895770300666776, 0, 28597210593194576797, 0, 8354121510796682640349, 0, 2440532802286486616685192, 0, 712969804532286516577201345, 0, 208285313591294473975622567617, 0, 60848035128652452288260555523528, 0, 17776023865770286122710588656327453, 0, 5193052954778346238555800950196047581, 0, 1517088448122297243357503948703651560280, 0, 443199294975280121743677636393500518693057, 0, 129475388819533023471733756743854150144699137, 0, 37824690955525448990961528431455911121510500408, 0, 11050032448085498621138093116422447905106730806013] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 27857075280594612561 z - 6174367460655745627 z - 427 z 24 22 4 6 + 1128628051121383993 z - 169064830976461104 z + 72937 z - 6782450 z 8 10 12 14 + 395802466 z - 15743196030 z + 450323449475 z - 9615573837377 z 18 16 50 - 2019311130734240 z + 157486509849595 z - 3359570147665033457497 z 48 20 + 5163815120195034900171 z + 20591512405983548 z 36 34 + 1837557506680983939203 z - 843613051639623626134 z 66 80 88 84 86 - 169064830976461104 z + 395802466 z + z + 72937 z - 427 z 82 64 30 - 6782450 z + 1128628051121383993 z - 104188316077039665466 z 42 44 - 6680213073832738335728 z + 7278328061043918365288 z 46 58 - 6680213073832738335728 z - 104188316077039665466 z 56 54 + 324386940274797610254 z - 843613051639623626134 z 52 60 + 1837557506680983939203 z + 27857075280594612561 z 70 68 78 - 2019311130734240 z + 20591512405983548 z - 15743196030 z 32 38 + 324386940274797610254 z - 3359570147665033457497 z 40 62 76 + 5163815120195034900171 z - 6174367460655745627 z + 450323449475 z 74 72 / - 9615573837377 z + 157486509849595 z ) / (-1 / 28 26 2 - 147337586001471478520 z + 30077684784817175559 z + 589 z 24 22 4 - 5066955344299059935 z + 699836926511435970 z - 124518 z 6 8 10 12 + 13476678 z - 887046931 z + 39124536371 z - 1228932785184 z 14 18 16 + 28648144403956 z + 7106617383317505 z - 510466472525089 z 50 48 + 45249805381061513482907 z - 63743095665005790409455 z 20 36 - 78605722659757074 z - 13587522733885144641590 z 34 66 80 + 5732499641691324974157 z + 5066955344299059935 z - 39124536371 z 90 88 84 86 82 + z - 589 z - 13476678 z + 124518 z + 887046931 z 64 30 - 30077684784817175559 z + 598704654630694105428 z 42 44 + 63743095665005790409455 z - 75642561284722740210396 z 46 58 + 75642561284722740210396 z + 2026491891205807590461 z 56 54 - 5732499641691324974157 z + 13587522733885144641590 z 52 60 - 27040242069829129466910 z - 598704654630694105428 z 70 68 78 + 78605722659757074 z - 699836926511435970 z + 1228932785184 z 32 38 - 2026491891205807590461 z + 27040242069829129466910 z 40 62 - 45249805381061513482907 z + 147337586001471478520 z 76 74 72 - 28648144403956 z + 510466472525089 z - 7106617383317505 z ) And in Maple-input format, it is: -(1+27857075280594612561*z^28-6174367460655745627*z^26-427*z^2+ 1128628051121383993*z^24-169064830976461104*z^22+72937*z^4-6782450*z^6+ 395802466*z^8-15743196030*z^10+450323449475*z^12-9615573837377*z^14-\ 2019311130734240*z^18+157486509849595*z^16-3359570147665033457497*z^50+ 5163815120195034900171*z^48+20591512405983548*z^20+1837557506680983939203*z^36-\ 843613051639623626134*z^34-169064830976461104*z^66+395802466*z^80+z^88+72937*z^ 84-427*z^86-6782450*z^82+1128628051121383993*z^64-104188316077039665466*z^30-\ 6680213073832738335728*z^42+7278328061043918365288*z^44-6680213073832738335728* z^46-104188316077039665466*z^58+324386940274797610254*z^56-\ 843613051639623626134*z^54+1837557506680983939203*z^52+27857075280594612561*z^ 60-2019311130734240*z^70+20591512405983548*z^68-15743196030*z^78+ 324386940274797610254*z^32-3359570147665033457497*z^38+5163815120195034900171*z ^40-6174367460655745627*z^62+450323449475*z^76-9615573837377*z^74+ 157486509849595*z^72)/(-1-147337586001471478520*z^28+30077684784817175559*z^26+ 589*z^2-5066955344299059935*z^24+699836926511435970*z^22-124518*z^4+13476678*z^ 6-887046931*z^8+39124536371*z^10-1228932785184*z^12+28648144403956*z^14+ 7106617383317505*z^18-510466472525089*z^16+45249805381061513482907*z^50-\ 63743095665005790409455*z^48-78605722659757074*z^20-13587522733885144641590*z^ 36+5732499641691324974157*z^34+5066955344299059935*z^66-39124536371*z^80+z^90-\ 589*z^88-13476678*z^84+124518*z^86+887046931*z^82-30077684784817175559*z^64+ 598704654630694105428*z^30+63743095665005790409455*z^42-75642561284722740210396 *z^44+75642561284722740210396*z^46+2026491891205807590461*z^58-\ 5732499641691324974157*z^56+13587522733885144641590*z^54-\ 27040242069829129466910*z^52-598704654630694105428*z^60+78605722659757074*z^70-\ 699836926511435970*z^68+1228932785184*z^78-2026491891205807590461*z^32+ 27040242069829129466910*z^38-45249805381061513482907*z^40+147337586001471478520 *z^62-28648144403956*z^76+510466472525089*z^74-7106617383317505*z^72) The first , 40, terms are: [0, 162, 0, 43837, 0, 12342305, 0, 3503099450, 0, 996943313065, 0, 284008033585121, 0, 80940331603582042, 0, 23071115640678765705, 0, 6576575732576999573861, 0, 1874744432028425740184450, 0, 534427399855533414311468489, 0, 152348125395126105020989041017, 0, 43429638432953572561153232267970, 0, 12380426104173761660134895484905589, 0, 3529271608806836051353065247656517721, 0, 1006084868264529507502988145407740990330, 0, 286803316881666952062251961084571585669105, 0, 81758653283449848104064581372085401793107865, 0, 23306834464084325458547205851774171184936763546, 0, 6644049434948097832267603801706391884445021621713] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 123614926189875744 z - 55018806954621666 z - 389 z 24 22 4 6 + 19270086438831706 z - 5276989398138560 z + 58480 z - 4609443 z 8 10 12 14 + 217677503 z - 6667581632 z + 139674642827 z - 2077686273119 z 18 16 50 - 182664719703337 z + 22562394023536 z - 182664719703337 z 48 20 36 + 1120627269437749 z + 1120627269437749 z + 309648188087085862 z 34 66 64 30 - 347132923214420736 z - 389 z + 58480 z - 219629523313514350 z 42 44 46 - 55018806954621666 z + 19270086438831706 z - 5276989398138560 z 58 56 54 - 6667581632 z + 139674642827 z - 2077686273119 z 52 60 68 32 + 22562394023536 z + 217677503 z + z + 309648188087085862 z 38 40 62 / - 219629523313514350 z + 123614926189875744 z - 4609443 z ) / (-1 / 28 26 2 - 902042474846424210 z + 358812123993367758 z + 555 z 24 22 4 6 - 112408264197038890 z + 27564614925137829 z - 103381 z + 9593955 z 8 10 12 14 - 519846757 z + 18015002071 z - 423533216403 z + 7037671455577 z 18 16 50 + 767878040157865 z - 85169079777967 z + 5248427749259319 z 48 20 36 - 27564614925137829 z - 5248427749259319 z - 3557539654479470790 z 34 66 64 + 3557539654479470790 z + 103381 z - 9593955 z 30 42 + 1794185763837579742 z + 902042474846424210 z 44 46 58 - 358812123993367758 z + 112408264197038890 z + 423533216403 z 56 54 52 - 7037671455577 z + 85169079777967 z - 767878040157865 z 60 70 68 32 - 18015002071 z + z - 555 z - 2832725706095427090 z 38 40 62 + 2832725706095427090 z - 1794185763837579742 z + 519846757 z ) And in Maple-input format, it is: -(1+123614926189875744*z^28-55018806954621666*z^26-389*z^2+19270086438831706*z^ 24-5276989398138560*z^22+58480*z^4-4609443*z^6+217677503*z^8-6667581632*z^10+ 139674642827*z^12-2077686273119*z^14-182664719703337*z^18+22562394023536*z^16-\ 182664719703337*z^50+1120627269437749*z^48+1120627269437749*z^20+ 309648188087085862*z^36-347132923214420736*z^34-389*z^66+58480*z^64-\ 219629523313514350*z^30-55018806954621666*z^42+19270086438831706*z^44-\ 5276989398138560*z^46-6667581632*z^58+139674642827*z^56-2077686273119*z^54+ 22562394023536*z^52+217677503*z^60+z^68+309648188087085862*z^32-\ 219629523313514350*z^38+123614926189875744*z^40-4609443*z^62)/(-1-\ 902042474846424210*z^28+358812123993367758*z^26+555*z^2-112408264197038890*z^24 +27564614925137829*z^22-103381*z^4+9593955*z^6-519846757*z^8+18015002071*z^10-\ 423533216403*z^12+7037671455577*z^14+767878040157865*z^18-85169079777967*z^16+ 5248427749259319*z^50-27564614925137829*z^48-5248427749259319*z^20-\ 3557539654479470790*z^36+3557539654479470790*z^34+103381*z^66-9593955*z^64+ 1794185763837579742*z^30+902042474846424210*z^42-358812123993367758*z^44+ 112408264197038890*z^46+423533216403*z^58-7037671455577*z^56+85169079777967*z^ 54-767878040157865*z^52-18015002071*z^60+z^70-555*z^68-2832725706095427090*z^32 +2832725706095427090*z^38-1794185763837579742*z^40+519846757*z^62) The first , 40, terms are: [0, 166, 0, 47229, 0, 14035361, 0, 4197471382, 0, 1256772720941, 0, 376379482306325, 0, 112723790674001206, 0, 33760545262143594681, 0, 10111234853106948561285, 0, 3028301607370508984297158, 0, 906972449366778736814751097, 0, 271637091035547436444872006153, 0, 81354962407544468425687668409478, 0, 24365707528502665367466870725289333, 0, 7297498344383947821695997406522417929, 0, 2185591451682793760774936776950523003638, 0, 654581853710093944205613941553356393211717, 0, 196046430762499600887309657621711061162848765, 0, 58715656104570370236271028092893016872152251094, 0, 17585264155954290626716789205118391037875965883729] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 33832242098157797119 z - 7508580927867061633 z - 421 z 24 22 4 6 + 1369592802801792557 z - 203872976715477580 z + 71795 z - 6774302 z 8 10 12 14 + 405310064 z - 16610720446 z + 490034794021 z - 10772407009595 z 18 16 50 - 2370110848393148 z + 181001179506959 z - 3965696782510089716159 z 48 20 + 6067375721751371887683 z + 24559141809651296 z 36 34 + 2181846455831448406825 z - 1008241877338616928386 z 66 80 88 84 86 - 203872976715477580 z + 405310064 z + z + 71795 z - 421 z 82 64 30 - 6774302 z + 1369592802801792557 z - 126041055804291164130 z 42 44 - 7826034296981491480760 z + 8518127433260965223712 z 46 58 - 7826034296981491480760 z - 126041055804291164130 z 56 54 + 390216377691696621696 z - 1008241877338616928386 z 52 60 + 2181846455831448406825 z + 33832242098157797119 z 70 68 78 - 2370110848393148 z + 24559141809651296 z - 16610720446 z 32 38 + 390216377691696621696 z - 3965696782510089716159 z 40 62 76 + 6067375721751371887683 z - 7508580927867061633 z + 490034794021 z 74 72 / 2 - 10772407009595 z + 181001179506959 z ) / ((-1 + z ) (1 / 28 26 2 + 147778633562189295134 z - 31189690488708020950 z - 594 z 24 22 4 + 5388199990650217571 z - 756690589236453768 z + 122364 z 6 8 10 12 - 13199644 z + 880629845 z - 39701919956 z + 1277264869686 z 14 18 16 - 30422241729450 z - 7738880899744280 z + 550910369415701 z 50 48 - 20674058118219662161566 z + 32186666268183609851879 z 20 36 + 85665446750477694 z + 11105012472278921326192 z 34 66 80 - 4979368007932526153092 z - 756690589236453768 z + 880629845 z 88 84 86 82 64 + z + 122364 z - 594 z - 13199644 z + 5388199990650217571 z 30 42 - 576386937661061161084 z - 41957835809269845410192 z 44 46 + 45830742635740660953796 z - 41957835809269845410192 z 58 56 - 576386937661061161084 z + 1859299914130090946235 z 54 52 - 4979368007932526153092 z + 11105012472278921326192 z 60 70 68 + 147778633562189295134 z - 7738880899744280 z + 85665446750477694 z 78 32 - 39701919956 z + 1859299914130090946235 z 38 40 - 20674058118219662161566 z + 32186666268183609851879 z 62 76 74 - 31189690488708020950 z + 1277264869686 z - 30422241729450 z 72 + 550910369415701 z )) And in Maple-input format, it is: -(1+33832242098157797119*z^28-7508580927867061633*z^26-421*z^2+ 1369592802801792557*z^24-203872976715477580*z^22+71795*z^4-6774302*z^6+ 405310064*z^8-16610720446*z^10+490034794021*z^12-10772407009595*z^14-\ 2370110848393148*z^18+181001179506959*z^16-3965696782510089716159*z^50+ 6067375721751371887683*z^48+24559141809651296*z^20+2181846455831448406825*z^36-\ 1008241877338616928386*z^34-203872976715477580*z^66+405310064*z^80+z^88+71795*z ^84-421*z^86-6774302*z^82+1369592802801792557*z^64-126041055804291164130*z^30-\ 7826034296981491480760*z^42+8518127433260965223712*z^44-7826034296981491480760* z^46-126041055804291164130*z^58+390216377691696621696*z^56-\ 1008241877338616928386*z^54+2181846455831448406825*z^52+33832242098157797119*z^ 60-2370110848393148*z^70+24559141809651296*z^68-16610720446*z^78+ 390216377691696621696*z^32-3965696782510089716159*z^38+6067375721751371887683*z ^40-7508580927867061633*z^62+490034794021*z^76-10772407009595*z^74+ 181001179506959*z^72)/(-1+z^2)/(1+147778633562189295134*z^28-\ 31189690488708020950*z^26-594*z^2+5388199990650217571*z^24-756690589236453768*z ^22+122364*z^4-13199644*z^6+880629845*z^8-39701919956*z^10+1277264869686*z^12-\ 30422241729450*z^14-7738880899744280*z^18+550910369415701*z^16-\ 20674058118219662161566*z^50+32186666268183609851879*z^48+85665446750477694*z^ 20+11105012472278921326192*z^36-4979368007932526153092*z^34-756690589236453768* z^66+880629845*z^80+z^88+122364*z^84-594*z^86-13199644*z^82+5388199990650217571 *z^64-576386937661061161084*z^30-41957835809269845410192*z^42+ 45830742635740660953796*z^44-41957835809269845410192*z^46-576386937661061161084 *z^58+1859299914130090946235*z^56-4979368007932526153092*z^54+ 11105012472278921326192*z^52+147778633562189295134*z^60-7738880899744280*z^70+ 85665446750477694*z^68-39701919956*z^78+1859299914130090946235*z^32-\ 20674058118219662161566*z^38+32186666268183609851879*z^40-31189690488708020950* z^62+1277264869686*z^76-30422241729450*z^74+550910369415701*z^72) The first , 40, terms are: [0, 174, 0, 52367, 0, 16311379, 0, 5095838886, 0, 1592488689293, 0, 497684196732869, 0, 155537048911136918, 0, 48608733764851425235, 0, 15191296995831699461895, 0, 4747614219465133944573822, 0, 1483733814426573704504626321, 0, 463699436505006196795937212193, 0, 144916268319443055087435306601534, 0, 45289519835518606763593663621755943, 0, 14153970639792950886247203220429340899, 0, 4423427000372118252858307207970444913142, 0, 1382418186785302242595052420176696124279589, 0, 432036075874048521375665858260545562082191693, 0, 135020772036231628646035387641049473271609252486, 0, 42196959696892514579888224076574292484053653846179] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 32933792701036 z - 38525792339664 z - 312 z 24 22 4 6 + 32933792701036 z - 20547762872830 z + 34243 z - 1842058 z 8 10 12 14 + 56789088 z - 1093117318 z + 13846786410 z - 119441719136 z 18 16 50 48 - 3056295418098 z + 717967962726 z - 312 z + 34243 z 20 36 34 + 9321825227680 z + 717967962726 z - 3056295418098 z 30 42 44 46 52 - 20547762872830 z - 1093117318 z + 56789088 z - 1842058 z + z 32 38 40 / 2 + 9321825227680 z - 119441719136 z + 13846786410 z ) / ((-1 + z ) ( / 28 26 2 1 + 177806351583446 z - 210976644671258 z - 471 z 24 22 4 6 + 177806351583446 z - 106478763080804 z + 64289 z - 4090780 z 8 10 12 14 + 145778948 z - 3197507100 z + 45667636644 z - 440267529884 z 18 16 50 48 - 13709339122148 z + 2933318651492 z - 471 z + 64289 z 20 36 34 + 45332960092924 z + 2933318651492 z - 13709339122148 z 30 42 44 46 52 - 106478763080804 z - 3197507100 z + 145778948 z - 4090780 z + z 32 38 40 + 45332960092924 z - 440267529884 z + 45667636644 z )) And in Maple-input format, it is: -(1+32933792701036*z^28-38525792339664*z^26-312*z^2+32933792701036*z^24-\ 20547762872830*z^22+34243*z^4-1842058*z^6+56789088*z^8-1093117318*z^10+ 13846786410*z^12-119441719136*z^14-3056295418098*z^18+717967962726*z^16-312*z^ 50+34243*z^48+9321825227680*z^20+717967962726*z^36-3056295418098*z^34-\ 20547762872830*z^30-1093117318*z^42+56789088*z^44-1842058*z^46+z^52+ 9321825227680*z^32-119441719136*z^38+13846786410*z^40)/(-1+z^2)/(1+ 177806351583446*z^28-210976644671258*z^26-471*z^2+177806351583446*z^24-\ 106478763080804*z^22+64289*z^4-4090780*z^6+145778948*z^8-3197507100*z^10+ 45667636644*z^12-440267529884*z^14-13709339122148*z^18+2933318651492*z^16-471*z ^50+64289*z^48+45332960092924*z^20+2933318651492*z^36-13709339122148*z^34-\ 106478763080804*z^30-3197507100*z^42+145778948*z^44-4090780*z^46+z^52+ 45332960092924*z^32-440267529884*z^38+45667636644*z^40) The first , 40, terms are: [0, 160, 0, 45003, 0, 13192827, 0, 3884350464, 0, 1144307524945, 0, 337135042256625, 0, 99327929406216256, 0, 29264430929451672539, 0, 8622020163413918313963, 0, 2540259233056953475014240, 0, 748422878121269116813792481, 0, 220503797635641942315780000801, 0, 64965845190734454156690140032736, 0, 19140536752097946630991891641855147, 0, 5639273161269552062254139188255845403, 0, 1661468651573493413983851304803079355328, 0, 489509552246885222979160024226150933243057, 0, 144221560553790675177065382965881963784855953, 0, 42491220923271952487988079539368325021783617664, 0, 12518959360982449597658832798982169977230388558907] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 97726848508540376 z - 43580808061985606 z - 383 z 24 22 4 6 + 15313606644039002 z - 4214890267358336 z + 56228 z - 4309277 z 8 10 12 14 + 197484443 z - 5875687160 z + 119912587143 z - 1744602492529 z 18 16 50 - 148546715231787 z + 18607379377284 z - 148546715231787 z 48 20 36 + 901809382193141 z + 901809382193141 z + 244415938509661582 z 34 66 64 30 - 273963443127352592 z - 383 z + 56228 z - 173449213495114626 z 42 44 46 - 43580808061985606 z + 15313606644039002 z - 4214890267358336 z 58 56 54 - 5875687160 z + 119912587143 z - 1744602492529 z 52 60 68 32 + 18607379377284 z + 197484443 z + z + 244415938509661582 z 38 40 62 / - 173449213495114626 z + 97726848508540376 z - 4309277 z ) / (-1 / 28 26 2 - 738230558173343274 z + 292285784300289766 z + 567 z 24 22 4 6 - 91177841129333930 z + 22295466992875845 z - 103401 z + 9219275 z 8 10 12 14 - 477998525 z + 15907832115 z - 361772051903 z + 5864500585201 z 18 16 50 + 623425432607645 z - 69806589895911 z + 4244477695525875 z 48 20 36 - 22295466992875845 z - 4244477695525875 z - 2938112446144450334 z 34 66 64 + 2938112446144450334 z + 103401 z - 9219275 z 30 42 + 1474588599471010974 z + 738230558173343274 z 44 46 58 - 292285784300289766 z + 91177841129333930 z + 361772051903 z 56 54 52 - 5864500585201 z + 69806589895911 z - 623425432607645 z 60 70 68 32 - 15907832115 z + z - 567 z - 2335539435793623618 z 38 40 62 + 2335539435793623618 z - 1474588599471010974 z + 477998525 z ) And in Maple-input format, it is: -(1+97726848508540376*z^28-43580808061985606*z^26-383*z^2+15313606644039002*z^ 24-4214890267358336*z^22+56228*z^4-4309277*z^6+197484443*z^8-5875687160*z^10+ 119912587143*z^12-1744602492529*z^14-148546715231787*z^18+18607379377284*z^16-\ 148546715231787*z^50+901809382193141*z^48+901809382193141*z^20+ 244415938509661582*z^36-273963443127352592*z^34-383*z^66+56228*z^64-\ 173449213495114626*z^30-43580808061985606*z^42+15313606644039002*z^44-\ 4214890267358336*z^46-5875687160*z^58+119912587143*z^56-1744602492529*z^54+ 18607379377284*z^52+197484443*z^60+z^68+244415938509661582*z^32-\ 173449213495114626*z^38+97726848508540376*z^40-4309277*z^62)/(-1-\ 738230558173343274*z^28+292285784300289766*z^26+567*z^2-91177841129333930*z^24+ 22295466992875845*z^22-103401*z^4+9219275*z^6-477998525*z^8+15907832115*z^10-\ 361772051903*z^12+5864500585201*z^14+623425432607645*z^18-69806589895911*z^16+ 4244477695525875*z^50-22295466992875845*z^48-4244477695525875*z^20-\ 2938112446144450334*z^36+2938112446144450334*z^34+103401*z^66-9219275*z^64+ 1474588599471010974*z^30+738230558173343274*z^42-292285784300289766*z^44+ 91177841129333930*z^46+361772051903*z^58-5864500585201*z^56+69806589895911*z^54 -623425432607645*z^52-15907832115*z^60+z^70-567*z^68-2335539435793623618*z^32+ 2335539435793623618*z^38-1474588599471010974*z^40+477998525*z^62) The first , 40, terms are: [0, 184, 0, 57155, 0, 18291099, 0, 5877001496, 0, 1889949999513, 0, 607909665717225, 0, 195547401171504632, 0, 62903002374027250571, 0, 20234494549571210037907, 0, 6508992666659938574509720, 0, 2093800596482660733585006385, 0, 673529909891243165084099490769, 0, 216659861809385973086697706017624, 0, 69694745932931198274930080836592499, 0, 22419277739526872875878792539100779819, 0, 7211792048381949877294285233788538654904, 0, 2319876008408854882144005452125465252440137, 0, 746253449685422600780890198404161382535637817, 0, 240053437835607430928947671752820148123646010200, 0, 77219948585941106940005760406950983424311283468027] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 158661231591816 z - 138530519990698 z - 323 z 24 22 4 6 + 92105069897806 z - 46474187536288 z + 37036 z - 2085387 z 8 10 12 14 + 66853337 z - 1337352288 z + 17736792836 z - 162591229196 z 18 16 50 48 - 5039425449468 z + 1061194827408 z - 2085387 z + 66853337 z 20 36 34 + 17693380422356 z + 17693380422356 z - 46474187536288 z 30 42 44 - 138530519990698 z - 162591229196 z + 17736792836 z 46 56 54 52 32 - 1337352288 z + z - 323 z + 37036 z + 92105069897806 z 38 40 / 28 - 5039425449468 z + 1061194827408 z ) / (-1 - 1650980426493242 z / 26 2 24 + 1256596500795966 z + 491 z - 726918260960458 z 22 4 6 8 + 318650475854456 z - 70649 z + 4720965 z - 175225855 z 10 12 14 18 + 4017427821 z - 60892250960 z + 638418654464 z + 26064291187832 z 16 50 48 20 - 4776509417744 z + 175225855 z - 4017427821 z - 105333140134616 z 36 34 30 - 318650475854456 z + 726918260960458 z + 1650980426493242 z 42 44 46 58 56 + 4776509417744 z - 638418654464 z + 60892250960 z + z - 491 z 54 52 32 38 + 70649 z - 4720965 z - 1256596500795966 z + 105333140134616 z 40 - 26064291187832 z ) And in Maple-input format, it is: -(1+158661231591816*z^28-138530519990698*z^26-323*z^2+92105069897806*z^24-\ 46474187536288*z^22+37036*z^4-2085387*z^6+66853337*z^8-1337352288*z^10+ 17736792836*z^12-162591229196*z^14-5039425449468*z^18+1061194827408*z^16-\ 2085387*z^50+66853337*z^48+17693380422356*z^20+17693380422356*z^36-\ 46474187536288*z^34-138530519990698*z^30-162591229196*z^42+17736792836*z^44-\ 1337352288*z^46+z^56-323*z^54+37036*z^52+92105069897806*z^32-5039425449468*z^38 +1061194827408*z^40)/(-1-1650980426493242*z^28+1256596500795966*z^26+491*z^2-\ 726918260960458*z^24+318650475854456*z^22-70649*z^4+4720965*z^6-175225855*z^8+ 4017427821*z^10-60892250960*z^12+638418654464*z^14+26064291187832*z^18-\ 4776509417744*z^16+175225855*z^50-4017427821*z^48-105333140134616*z^20-\ 318650475854456*z^36+726918260960458*z^34+1650980426493242*z^30+4776509417744*z ^42-638418654464*z^44+60892250960*z^46+z^58-491*z^56+70649*z^54-4720965*z^52-\ 1256596500795966*z^32+105333140134616*z^38-26064291187832*z^40) The first , 40, terms are: [0, 168, 0, 48875, 0, 14764171, 0, 4480987688, 0, 1361070334097, 0, 413476978169809, 0, 125613057515410344, 0, 38161098141866704235, 0, 11593311358769078052939, 0, 3522039786542634764491816, 0, 1069993261054254277507190657, 0, 325063220164348212885510607361, 0, 98753984047264504216357617278760, 0, 30001392858723640173616146636062219, 0, 9114402647049251446671321832969999403, 0, 2768949295345514265639645631614454299304, 0, 841204903619187514306270938451873186865873, 0, 255557474838481255145329225896846781738539153, 0, 77638186207500290879766323597109385288565382952, 0, 23586427911767102690477012186923804151109364818635] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 77957633345856104 z - 35004940248959466 z - 377 z 24 22 4 6 + 12401622821033610 z - 3445438149741256 z + 54082 z - 4034721 z 8 10 12 14 + 180271865 z - 5243936916 z + 104918590709 z - 1500172394269 z 18 16 50 - 124150330966385 z + 15759364731222 z - 124150330966385 z 48 20 36 + 744921157419161 z + 744921157419161 z + 193313708199678510 z 34 66 64 30 - 216439364025283368 z - 377 z + 54082 z - 137638033836701822 z 42 44 46 - 35004940248959466 z + 12401622821033610 z - 3445438149741256 z 58 56 54 - 5243936916 z + 104918590709 z - 1500172394269 z 52 60 68 32 + 15759364731222 z + 180271865 z + z + 193313708199678510 z 38 40 62 / - 137638033836701822 z + 77957633345856104 z - 4034721 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 416001444433089198 z - 177701056439635264 z - 572 z 24 22 4 6 + 59239757234929690 z - 15349495221599592 z + 100999 z - 8661602 z 8 10 12 14 + 432839337 z - 13916259150 z + 306015212249 z - 4793239262066 z 18 16 50 - 472086750929476 z + 55012302335671 z - 472086750929476 z 48 20 36 + 3073882542051961 z + 3073882542051961 z + 1095359994053592210 z 34 66 64 30 - 1235912860358283932 z - 572 z + 100999 z - 762246525416593460 z 42 44 46 - 177701056439635264 z + 59239757234929690 z - 15349495221599592 z 58 56 54 - 13916259150 z + 306015212249 z - 4793239262066 z 52 60 68 32 + 55012302335671 z + 432839337 z + z + 1095359994053592210 z 38 40 62 - 762246525416593460 z + 416001444433089198 z - 8661602 z )) And in Maple-input format, it is: -(1+77957633345856104*z^28-35004940248959466*z^26-377*z^2+12401622821033610*z^ 24-3445438149741256*z^22+54082*z^4-4034721*z^6+180271865*z^8-5243936916*z^10+ 104918590709*z^12-1500172394269*z^14-124150330966385*z^18+15759364731222*z^16-\ 124150330966385*z^50+744921157419161*z^48+744921157419161*z^20+ 193313708199678510*z^36-216439364025283368*z^34-377*z^66+54082*z^64-\ 137638033836701822*z^30-35004940248959466*z^42+12401622821033610*z^44-\ 3445438149741256*z^46-5243936916*z^58+104918590709*z^56-1500172394269*z^54+ 15759364731222*z^52+180271865*z^60+z^68+193313708199678510*z^32-\ 137638033836701822*z^38+77957633345856104*z^40-4034721*z^62)/(-1+z^2)/(1+ 416001444433089198*z^28-177701056439635264*z^26-572*z^2+59239757234929690*z^24-\ 15349495221599592*z^22+100999*z^4-8661602*z^6+432839337*z^8-13916259150*z^10+ 306015212249*z^12-4793239262066*z^14-472086750929476*z^18+55012302335671*z^16-\ 472086750929476*z^50+3073882542051961*z^48+3073882542051961*z^20+ 1095359994053592210*z^36-1235912860358283932*z^34-572*z^66+100999*z^64-\ 762246525416593460*z^30-177701056439635264*z^42+59239757234929690*z^44-\ 15349495221599592*z^46-13916259150*z^58+306015212249*z^56-4793239262066*z^54+ 55012302335671*z^52+432839337*z^60+z^68+1095359994053592210*z^32-\ 762246525416593460*z^38+416001444433089198*z^40-8661602*z^62) The first , 40, terms are: [0, 196, 0, 64819, 0, 21961251, 0, 7456306896, 0, 2532391637833, 0, 860133301776265, 0, 292150546925060704, 0, 99231376718505894047, 0, 33704791698529668835523, 0, 11448124732583700510276276, 0, 3888454981591360251376906685, 0, 1320747514669832038790707503965, 0, 448603368100999104973237186255284, 0, 152372031553495939828838612927033011, 0, 51754484369957612658592404242684890831, 0, 17578860274896798175676735464687756927552, 0, 5970812622875581058997737691503745243542937, 0, 2028038383606774738839288460011534867215774729, 0, 688840857210358023847694676792873221678751104464, 0, 233970782011761761754437498080763409458068959645123] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 26 2 24 22 4 6 f(z) = - (-1 + z + 301 z - 301 z + 25832 z - 25832 z + 644386 z 8 10 12 14 18 - 6575978 z + 30386448 z - 62883838 z + 62883838 z + 6575978 z 16 20 / 26 18 16 - 30386448 z - 644386 z ) / (-535 z - 173047570 z + 561442110 z / 2 10 28 6 8 4 - 535 z - 173047570 z + z - 1951498 z + 1 + 26259650 z + 62005 z 12 14 24 20 22 + 561442110 z - 829175774 z + 62005 z + 26259650 z - 1951498 z ) And in Maple-input format, it is: -(-1+z^26+301*z^2-301*z^24+25832*z^22-25832*z^4+644386*z^6-6575978*z^8+30386448 *z^10-62883838*z^12+62883838*z^14+6575978*z^18-30386448*z^16-644386*z^20)/(-535 *z^26-173047570*z^18+561442110*z^16-535*z^2-173047570*z^10+z^28-1951498*z^6+1+ 26259650*z^8+62005*z^4+561442110*z^12-829175774*z^14+62005*z^24+26259650*z^20-\ 1951498*z^22) The first , 40, terms are: [0, 234, 0, 89017, 0, 34422037, 0, 13333257570, 0, 5166668796253, 0, 2002316146037989, 0, 776011027820466546, 0, 300750811259727250333, 0, 116559249319572486828337, 0, 45173834803517075521829082, 0, 17507625800700889435797963385, 0, 6785276874185302468311798420361, 0, 2629710243265846168058403090809082, 0, 1019173737290097999575858619604435009, 0, 394992227970074532267051133462928681101, 0, 153083674053848498037244513151593809953106, 0, 59329297146457195282431161163114171379555285, 0, 22993735430833230873147886935648455716526556781, 0, 8911480406744166693980725371419525306088591957378, 0, 3453744315655514583306757293078109889844929415257125] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 115742120 z + 679607968 z + 260 z - 2482464114 z 22 4 6 8 10 + 5798705203 z - 20380 z + 682641 z - 11808610 z + 115742120 z 12 14 18 16 - 679607968 z + 2482464114 z + 8820899556 z - 5798705203 z 20 36 34 30 32 38 - 8820899556 z - 260 z + 20380 z + 11808610 z - 682641 z + z ) / 28 4 26 38 40 2 / (3290347654 z + 43839 z - 14491915056 z - 428 z + z - 428 z / 36 14 34 30 6 + 43839 z - 14491915056 z - 1865900 z - 467412448 z - 1865900 z 8 16 22 32 + 39632403 z + 41137911221 z - 76536675916 z + 1 + 39632403 z 18 10 12 20 - 76536675916 z - 467412448 z + 3290347654 z + 94061094709 z 24 + 41137911221 z ) And in Maple-input format, it is: -(-1-115742120*z^28+679607968*z^26+260*z^2-2482464114*z^24+5798705203*z^22-\ 20380*z^4+682641*z^6-11808610*z^8+115742120*z^10-679607968*z^12+2482464114*z^14 +8820899556*z^18-5798705203*z^16-8820899556*z^20-260*z^36+20380*z^34+11808610*z ^30-682641*z^32+z^38)/(3290347654*z^28+43839*z^4-14491915056*z^26-428*z^38+z^40 -428*z^2+43839*z^36-14491915056*z^14-1865900*z^34-467412448*z^30-1865900*z^6+ 39632403*z^8+41137911221*z^16-76536675916*z^22+1+39632403*z^32-76536675916*z^18 -467412448*z^10+3290347654*z^12+94061094709*z^20+41137911221*z^24) The first , 40, terms are: [0, 168, 0, 48445, 0, 14552767, 0, 4390451328, 0, 1325221367995, 0, 400031680467211, 0, 120754664721928032, 0, 36451379218000892959, 0, 11003328926980618305997, 0, 3321499867111204640642472, 0, 1002638519619265665358967617, 0, 302659654432171623510545301313, 0, 91361806517656054887654325037160, 0, 27578765680175316184979647333808557, 0, 8325013979439212870885111665360447903, 0, 2513015214736373295047547791131283855712, 0, 758586770556162108227082016898650233317803, 0, 228989416812271492450299543883446814490030875, 0, 69123474132801819858104359112052927862860842496, 0, 20865831891721176716435742412504763330674290751039] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 55157108204324173 z - 21480333880259833 z - 328 z 24 22 4 6 + 6744307520074289 z - 1699886062580262 z + 39507 z - 2507539 z 8 10 12 14 + 97903287 z - 2565907478 z + 47725781337 z - 654341469314 z 18 16 50 - 54454127783014 z + 6793138844906 z - 1699886062580262 z 48 20 36 + 6744307520074289 z + 341927606913379 z + 292139177474365252 z 34 66 64 - 263319564798012528 z - 2507539 z + 97903287 z 30 42 44 - 114541230973103796 z - 114541230973103796 z + 55157108204324173 z 46 58 56 - 21480333880259833 z - 654341469314 z + 6793138844906 z 54 52 60 70 - 54454127783014 z + 341927606913379 z + 47725781337 z - 328 z 68 32 38 + 39507 z + 192776353785483563 z - 263319564798012528 z 40 62 72 / + 192776353785483563 z - 2565907478 z + z ) / (-1 / 28 26 2 - 367607799012155526 z + 129737947050842597 z + 471 z 24 22 4 6 - 36954528540796997 z + 8457564620586952 z - 71000 z + 5320796 z 8 10 12 14 - 237956765 z + 7022077013 z - 145569176914 z + 2210695242828 z 18 16 50 + 223644882749821 z - 25329114218045 z + 36954528540796997 z 48 20 36 - 129737947050842597 z - 1545623389598246 z - 2917992522888860474 z 34 66 64 + 2373384441527471361 z + 237956765 z - 7022077013 z 30 42 + 843301282749765602 z + 1569572156916041267 z 44 46 58 - 843301282749765602 z + 367607799012155526 z + 25329114218045 z 56 54 52 - 223644882749821 z + 1545623389598246 z - 8457564620586952 z 60 70 68 32 - 2210695242828 z + 71000 z - 5320796 z - 1569572156916041267 z 38 40 62 + 2917992522888860474 z - 2373384441527471361 z + 145569176914 z 74 72 + z - 471 z ) And in Maple-input format, it is: -(1+55157108204324173*z^28-21480333880259833*z^26-328*z^2+6744307520074289*z^24 -1699886062580262*z^22+39507*z^4-2507539*z^6+97903287*z^8-2565907478*z^10+ 47725781337*z^12-654341469314*z^14-54454127783014*z^18+6793138844906*z^16-\ 1699886062580262*z^50+6744307520074289*z^48+341927606913379*z^20+ 292139177474365252*z^36-263319564798012528*z^34-2507539*z^66+97903287*z^64-\ 114541230973103796*z^30-114541230973103796*z^42+55157108204324173*z^44-\ 21480333880259833*z^46-654341469314*z^58+6793138844906*z^56-54454127783014*z^54 +341927606913379*z^52+47725781337*z^60-328*z^70+39507*z^68+192776353785483563*z ^32-263319564798012528*z^38+192776353785483563*z^40-2565907478*z^62+z^72)/(-1-\ 367607799012155526*z^28+129737947050842597*z^26+471*z^2-36954528540796997*z^24+ 8457564620586952*z^22-71000*z^4+5320796*z^6-237956765*z^8+7022077013*z^10-\ 145569176914*z^12+2210695242828*z^14+223644882749821*z^18-25329114218045*z^16+ 36954528540796997*z^50-129737947050842597*z^48-1545623389598246*z^20-\ 2917992522888860474*z^36+2373384441527471361*z^34+237956765*z^66-7022077013*z^ 64+843301282749765602*z^30+1569572156916041267*z^42-843301282749765602*z^44+ 367607799012155526*z^46+25329114218045*z^58-223644882749821*z^56+ 1545623389598246*z^54-8457564620586952*z^52-2210695242828*z^60+71000*z^70-\ 5320796*z^68-1569572156916041267*z^32+2917992522888860474*z^38-\ 2373384441527471361*z^40+145569176914*z^62+z^74-471*z^72) The first , 40, terms are: [0, 143, 0, 35860, 0, 9550317, 0, 2572959657, 0, 695023588147, 0, 187864446886951, 0, 50787662281560380, 0, 13730579321137171733, 0, 3712134782905637656039, 0, 1003597726083323706969059, 0, 271328781646727024325023865, 0, 73355406270334179627136612716, 0, 19832086373682599607020071640715, 0, 5361726880963756710454122382308007, 0, 1449575938331652540489601628521575485, 0, 391901797477915975319182172644162395681, 0, 105953068637397813340371172925981190075108, 0, 28645065744383192458692277423547659004228115, 0, 7744370239202384777021722649403977298407020445, 0, 2093738270219977190884743908187177435270863914773] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 17958006826868680546 z - 4099351854643189082 z - 423 z 24 22 4 6 + 773297378162789008 z - 119750301713768438 z + 70988 z - 6432933 z 8 10 12 14 + 363358047 z - 13934165682 z + 383630110994 z - 7881856067472 z 18 16 50 - 1535218153462222 z + 124268678825694 z - 1948743000585886953686 z 48 20 + 2964058959667200759968 z + 15100667760748062 z 36 34 + 1081363880529050252086 z - 505509170798092434288 z 66 80 88 84 86 - 119750301713768438 z + 363358047 z + z + 70988 z - 423 z 82 64 30 - 6432933 z + 773297378162789008 z - 65365940970357350698 z 42 44 - 3810129110281112061668 z + 4142409443259493241416 z 46 58 - 3810129110281112061668 z - 65365940970357350698 z 56 54 + 198593776554506965594 z - 505509170798092434288 z 52 60 + 1081363880529050252086 z + 17958006826868680546 z 70 68 78 - 1535218153462222 z + 15100667760748062 z - 13934165682 z 32 38 + 198593776554506965594 z - 1948743000585886953686 z 40 62 76 + 2964058959667200759968 z - 4099351854643189082 z + 383630110994 z 74 72 / - 7881856067472 z + 124268678825694 z ) / (-1 / 28 26 2 - 96929775605884085084 z + 20333365352025236116 z + 590 z 24 22 4 - 3526511472225413836 z + 502361562862880644 z - 123414 z 6 8 10 12 + 13021106 z - 827562498 z + 35090575617 z - 1058435719308 z 14 18 16 + 23708665029524 z + 5456728562935876 z - 406542398117740 z 50 48 + 26606525365204487595590 z - 37195295713567755597776 z 20 36 - 58302805405668060 z - 8198730920933105696956 z 34 66 80 + 3520831297368459748676 z + 3526511472225413836 z - 35090575617 z 90 88 84 86 82 + z - 590 z - 13021106 z + 123414 z + 827562498 z 64 30 - 20333365352025236116 z + 384066565664459576796 z 42 44 + 37195295713567755597776 z - 43968751696865722739672 z 46 58 + 43968751696865722739672 z + 1270434431051204655084 z 56 54 - 3520831297368459748676 z + 8198730920933105696956 z 52 60 - 16079267216203723601988 z - 384066565664459576796 z 70 68 78 + 58302805405668060 z - 502361562862880644 z + 1058435719308 z 32 38 - 1270434431051204655084 z + 16079267216203723601988 z 40 62 - 26606525365204487595590 z + 96929775605884085084 z 76 74 72 - 23708665029524 z + 406542398117740 z - 5456728562935876 z ) And in Maple-input format, it is: -(1+17958006826868680546*z^28-4099351854643189082*z^26-423*z^2+ 773297378162789008*z^24-119750301713768438*z^22+70988*z^4-6432933*z^6+363358047 *z^8-13934165682*z^10+383630110994*z^12-7881856067472*z^14-1535218153462222*z^ 18+124268678825694*z^16-1948743000585886953686*z^50+2964058959667200759968*z^48 +15100667760748062*z^20+1081363880529050252086*z^36-505509170798092434288*z^34-\ 119750301713768438*z^66+363358047*z^80+z^88+70988*z^84-423*z^86-6432933*z^82+ 773297378162789008*z^64-65365940970357350698*z^30-3810129110281112061668*z^42+ 4142409443259493241416*z^44-3810129110281112061668*z^46-65365940970357350698*z^ 58+198593776554506965594*z^56-505509170798092434288*z^54+1081363880529050252086 *z^52+17958006826868680546*z^60-1535218153462222*z^70+15100667760748062*z^68-\ 13934165682*z^78+198593776554506965594*z^32-1948743000585886953686*z^38+ 2964058959667200759968*z^40-4099351854643189082*z^62+383630110994*z^76-\ 7881856067472*z^74+124268678825694*z^72)/(-1-96929775605884085084*z^28+ 20333365352025236116*z^26+590*z^2-3526511472225413836*z^24+502361562862880644*z ^22-123414*z^4+13021106*z^6-827562498*z^8+35090575617*z^10-1058435719308*z^12+ 23708665029524*z^14+5456728562935876*z^18-406542398117740*z^16+ 26606525365204487595590*z^50-37195295713567755597776*z^48-58302805405668060*z^ 20-8198730920933105696956*z^36+3520831297368459748676*z^34+3526511472225413836* z^66-35090575617*z^80+z^90-590*z^88-13021106*z^84+123414*z^86+827562498*z^82-\ 20333365352025236116*z^64+384066565664459576796*z^30+37195295713567755597776*z^ 42-43968751696865722739672*z^44+43968751696865722739672*z^46+ 1270434431051204655084*z^58-3520831297368459748676*z^56+8198730920933105696956* z^54-16079267216203723601988*z^52-384066565664459576796*z^60+58302805405668060* z^70-502361562862880644*z^68+1058435719308*z^78-1270434431051204655084*z^32+ 16079267216203723601988*z^38-26606525365204487595590*z^40+96929775605884085084* z^62-23708665029524*z^76+406542398117740*z^74-5456728562935876*z^72) The first , 40, terms are: [0, 167, 0, 46104, 0, 13179395, 0, 3796284245, 0, 1096564393813, 0, 317100046960043, 0, 91739560062417832, 0, 26545925485825821391, 0, 7681961076362329926001, 0, 2223104159770612419985969, 0, 643358469439646306900333903, 0, 186186625241945986141626207304, 0, 53882142327545503246647519242955, 0, 15593428157408137932689026379385845, 0, 4512721223955313694662273966493931061, 0, 1305976823405700268955872043124193713411, 0, 377948355032230713998065339228543234946680, 0, 109377869447619970733457103000765160755621127, 0, 31653844454046747906936710003424307110970127425, 0, 9160590508664037674953500150396660152482722269505] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1789035563619816 z - 1007669730421922 z - 301 z 24 22 4 6 + 449175771904510 z - 157745352580494 z + 31622 z - 1697933 z 8 10 12 14 + 54962031 z - 1174308750 z + 17514786242 z - 189281711146 z 18 16 50 - 9266253431684 z + 1521278303888 z - 189281711146 z 48 20 36 + 1521278303888 z + 43379698832880 z + 1789035563619816 z 34 64 30 42 - 2521563167658310 z + z - 2521563167658310 z - 157745352580494 z 44 46 58 56 + 43379698832880 z - 9266253431684 z - 1697933 z + 54962031 z 54 52 60 32 - 1174308750 z + 17514786242 z + 31622 z + 2826610161112220 z 38 40 62 / 2 - 1007669730421922 z + 449175771904510 z - 301 z ) / ((-1 + z ) (1 / 28 26 2 + 9282312735766466 z - 5068148536276733 z - 453 z 24 22 4 6 + 2166104940913534 z - 722233585928683 z + 59518 z - 3750188 z 8 10 12 14 + 138370611 z - 3313120605 z + 54717787032 z - 648549933839 z 18 16 50 - 37274338651285 z + 5669577794154 z - 648549933839 z 48 20 36 + 5669577794154 z + 186914248155212 z + 9282312735766466 z 34 64 30 - 13335654387817609 z + z - 13335654387817609 z 42 44 46 - 722233585928683 z + 186914248155212 z - 37274338651285 z 58 56 54 52 - 3750188 z + 138370611 z - 3313120605 z + 54717787032 z 60 32 38 + 59518 z + 15045813343548106 z - 5068148536276733 z 40 62 + 2166104940913534 z - 453 z )) And in Maple-input format, it is: -(1+1789035563619816*z^28-1007669730421922*z^26-301*z^2+449175771904510*z^24-\ 157745352580494*z^22+31622*z^4-1697933*z^6+54962031*z^8-1174308750*z^10+ 17514786242*z^12-189281711146*z^14-9266253431684*z^18+1521278303888*z^16-\ 189281711146*z^50+1521278303888*z^48+43379698832880*z^20+1789035563619816*z^36-\ 2521563167658310*z^34+z^64-2521563167658310*z^30-157745352580494*z^42+ 43379698832880*z^44-9266253431684*z^46-1697933*z^58+54962031*z^56-1174308750*z^ 54+17514786242*z^52+31622*z^60+2826610161112220*z^32-1007669730421922*z^38+ 449175771904510*z^40-301*z^62)/(-1+z^2)/(1+9282312735766466*z^28-\ 5068148536276733*z^26-453*z^2+2166104940913534*z^24-722233585928683*z^22+59518* z^4-3750188*z^6+138370611*z^8-3313120605*z^10+54717787032*z^12-648549933839*z^ 14-37274338651285*z^18+5669577794154*z^16-648549933839*z^50+5669577794154*z^48+ 186914248155212*z^20+9282312735766466*z^36-13335654387817609*z^34+z^64-\ 13335654387817609*z^30-722233585928683*z^42+186914248155212*z^44-37274338651285 *z^46-3750188*z^58+138370611*z^56-3313120605*z^54+54717787032*z^52+59518*z^60+ 15045813343548106*z^32-5068148536276733*z^38+2166104940913534*z^40-453*z^62) The first , 40, terms are: [0, 153, 0, 41113, 0, 11601512, 0, 3297224975, 0, 938347005495, 0, 267114561019843, 0, 76042580467532011, 0, 21648185657491615655, 0, 6162931092144282054047, 0, 1754499975084410741932680, 0, 499481576352410895030939273, 0, 142195414739104902466494675689, 0, 40481044833364288400002674130281, 0, 11524387025906103983521147537299289, 0, 3280831730101243678769373823056381257, 0, 934006886234786405793225467666222929769, 0, 265898691341102320729958357778593633130696, 0, 75697636815192254615361945452674815585333055, 0, 21550057995799153366362678458671051388268707399, 0, 6134999970423864691887747231364998632826308666971] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1471609002090992733499 z - 199941520998379788321 z - 446 z 24 22 4 + 23112865420351028907 z - 2257485338668369524 z + 82955 z 6 102 8 10 - 8819373 z - 30240153460 z + 614005779 z - 30240153460 z 12 14 18 + 1107677957701 z - 31254127713746 z - 12539755654392350 z 16 50 + 697148373979838 z - 373806828688963663802471976 z 48 20 + 237226868557302757537820626 z + 184747016786892113 z 36 34 + 951663677254956592301096 z - 234829664062197399277708 z 66 80 - 132064092921418244780334978 z + 50207829929344452023789 z 100 90 88 + 1107677957701 z - 2257485338668369524 z + 23112865420351028907 z 84 94 + 1471609002090992733499 z - 12539755654392350 z 86 96 98 - 199941520998379788321 z + 697148373979838 z - 31254127713746 z 92 82 + 184747016786892113 z - 9269232109574245532646 z 64 112 110 106 + 237226868557302757537820626 z + z - 446 z - 8819373 z 108 30 42 + 82955 z - 9269232109574245532646 z - 27526101299274304668528812 z 44 46 + 64437277481427815463090250 z - 132064092921418244780334978 z 58 56 - 627997977397393568535312576 z + 670029238854865523991954900 z 54 52 - 627997977397393568535312576 z + 517028021064098791394227546 z 60 70 + 517028021064098791394227546 z - 27526101299274304668528812 z 68 78 + 64437277481427815463090250 z - 234829664062197399277708 z 32 38 + 50207829929344452023789 z - 3351348502697021107248036 z 40 62 + 10280496972737188177244098 z - 373806828688963663802471976 z 76 74 + 951663677254956592301096 z - 3351348502697021107248036 z 72 104 / + 10280496972737188177244098 z + 614005779 z ) / (-1 / 28 26 2 - 6288080813127313437382 z + 801912457991983270011 z + 617 z 24 22 4 - 86965504013842146715 z + 7962507375278884236 z - 137786 z 6 102 8 10 + 16666770 z + 2753728393728 z - 1286855467 z + 69316480395 z 12 14 18 - 2753728393728 z + 83810589235324 z + 38724337922675401 z 16 50 - 2008907412222401 z + 3200760904646143519576859010 z 48 20 - 1905575616769722657645442530 z - 610183492109477896 z 36 34 - 5228710524509176542462220 z + 1211672329844265679337129 z 66 80 + 1905575616769722657645442530 z - 1211672329844265679337129 z 100 90 - 83810589235324 z + 86965504013842146715 z 88 84 - 801912457991983270011 z - 42182143819144340921630 z 94 86 + 610183492109477896 z + 6288080813127313437382 z 96 98 92 - 38724337922675401 z + 2008907412222401 z - 7962507375278884236 z 82 64 + 243299447335394916669745 z - 3200760904646143519576859010 z 112 114 110 106 108 - 617 z + z + 137786 z + 1286855467 z - 16666770 z 30 42 + 42182143819144340921630 z + 182730893970945167295935438 z 44 46 - 455753112139614642746541712 z + 995350186336903500880484208 z 58 56 + 6957147748199696715844299934 z - 6957147748199696715844299934 z 54 52 + 6113484235921649642760707492 z - 4719975903172442331222165532 z 60 70 - 6113484235921649642760707492 z + 455753112139614642746541712 z 68 78 - 995350186336903500880484208 z + 5228710524509176542462220 z 32 38 - 243299447335394916669745 z + 19608876748418904520389724 z 40 62 - 64066279521682435740041406 z + 4719975903172442331222165532 z 76 74 - 19608876748418904520389724 z + 64066279521682435740041406 z 72 104 - 182730893970945167295935438 z - 69316480395 z ) And in Maple-input format, it is: -(1+1471609002090992733499*z^28-199941520998379788321*z^26-446*z^2+ 23112865420351028907*z^24-2257485338668369524*z^22+82955*z^4-8819373*z^6-\ 30240153460*z^102+614005779*z^8-30240153460*z^10+1107677957701*z^12-\ 31254127713746*z^14-12539755654392350*z^18+697148373979838*z^16-\ 373806828688963663802471976*z^50+237226868557302757537820626*z^48+ 184747016786892113*z^20+951663677254956592301096*z^36-234829664062197399277708* z^34-132064092921418244780334978*z^66+50207829929344452023789*z^80+ 1107677957701*z^100-2257485338668369524*z^90+23112865420351028907*z^88+ 1471609002090992733499*z^84-12539755654392350*z^94-199941520998379788321*z^86+ 697148373979838*z^96-31254127713746*z^98+184747016786892113*z^92-\ 9269232109574245532646*z^82+237226868557302757537820626*z^64+z^112-446*z^110-\ 8819373*z^106+82955*z^108-9269232109574245532646*z^30-\ 27526101299274304668528812*z^42+64437277481427815463090250*z^44-\ 132064092921418244780334978*z^46-627997977397393568535312576*z^58+ 670029238854865523991954900*z^56-627997977397393568535312576*z^54+ 517028021064098791394227546*z^52+517028021064098791394227546*z^60-\ 27526101299274304668528812*z^70+64437277481427815463090250*z^68-\ 234829664062197399277708*z^78+50207829929344452023789*z^32-\ 3351348502697021107248036*z^38+10280496972737188177244098*z^40-\ 373806828688963663802471976*z^62+951663677254956592301096*z^76-\ 3351348502697021107248036*z^74+10280496972737188177244098*z^72+614005779*z^104) /(-1-6288080813127313437382*z^28+801912457991983270011*z^26+617*z^2-\ 86965504013842146715*z^24+7962507375278884236*z^22-137786*z^4+16666770*z^6+ 2753728393728*z^102-1286855467*z^8+69316480395*z^10-2753728393728*z^12+ 83810589235324*z^14+38724337922675401*z^18-2008907412222401*z^16+ 3200760904646143519576859010*z^50-1905575616769722657645442530*z^48-\ 610183492109477896*z^20-5228710524509176542462220*z^36+ 1211672329844265679337129*z^34+1905575616769722657645442530*z^66-\ 1211672329844265679337129*z^80-83810589235324*z^100+86965504013842146715*z^90-\ 801912457991983270011*z^88-42182143819144340921630*z^84+610183492109477896*z^94 +6288080813127313437382*z^86-38724337922675401*z^96+2008907412222401*z^98-\ 7962507375278884236*z^92+243299447335394916669745*z^82-\ 3200760904646143519576859010*z^64-617*z^112+z^114+137786*z^110+1286855467*z^106 -16666770*z^108+42182143819144340921630*z^30+182730893970945167295935438*z^42-\ 455753112139614642746541712*z^44+995350186336903500880484208*z^46+ 6957147748199696715844299934*z^58-6957147748199696715844299934*z^56+ 6113484235921649642760707492*z^54-4719975903172442331222165532*z^52-\ 6113484235921649642760707492*z^60+455753112139614642746541712*z^70-\ 995350186336903500880484208*z^68+5228710524509176542462220*z^78-\ 243299447335394916669745*z^32+19608876748418904520389724*z^38-\ 64066279521682435740041406*z^40+4719975903172442331222165532*z^62-\ 19608876748418904520389724*z^76+64066279521682435740041406*z^74-\ 182730893970945167295935438*z^72-69316480395*z^104) The first , 40, terms are: [0, 171, 0, 50676, 0, 15553083, 0, 4790976857, 0, 1476664905129, 0, 455186746463727, 0, 140316362280393404, 0, 43254330943399323455, 0, 13333724937126956756333, 0, 4110300098036803308638065, 0, 1267055406487086674411139259, 0, 390586914911574512956953228236, 0, 120403684041044522216025332029259, 0, 37116059432287627498129373425272149, 0, 11441525889649829243767223819211290741, 0, 3527004663067227693349055994226628509479, 0, 1087246754801686445406167499216142783427524, 0, 335158475464977357632052150680747086645241191, 0, 103317120221347311067275015899396139444725337957, 0, 31848895708293809998839740343253764361759392093533] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {3, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 72503389000301700 z - 28436989298659382 z - 341 z 24 22 4 6 + 8992301811093095 z - 2280463862025325 z + 43360 z - 2905861 z 8 10 12 14 + 119041279 z - 3243301262 z + 62088328844 z - 867930440350 z 18 16 50 - 73379261949909 z + 9112230148327 z - 2280463862025325 z 48 20 36 + 8992301811093095 z + 460545536590580 z + 378139430174140728 z 34 66 64 - 341199419028009964 z - 2905861 z + 119041279 z 30 42 44 - 149616004148281110 z - 149616004148281110 z + 72503389000301700 z 46 58 56 - 28436989298659382 z - 867930440350 z + 9112230148327 z 54 52 60 70 - 73379261949909 z + 460545536590580 z + 62088328844 z - 341 z 68 32 38 + 43360 z + 250572001011667890 z - 341199419028009964 z 40 62 72 / 2 + 250572001011667890 z - 3243301262 z + z ) / ((-1 + z ) (1 / 28 26 2 + 355879956289976894 z - 133511996439971664 z - 492 z 24 22 4 6 + 40077522631313287 z - 9583046042918636 z + 77171 z - 6016988 z 8 10 12 14 + 278711847 z - 8448136880 z + 178042863494 z - 2718732279804 z 18 16 50 - 269247440776464 z + 30981363490583 z - 9583046042918636 z 48 20 36 + 40077522631313287 z + 1813598343137159 z + 2019188147086349956 z 34 66 64 - 1811958429675314252 z - 6016988 z + 278711847 z 30 42 44 - 761262969894959092 z - 761262969894959092 z + 355879956289976894 z 46 58 56 - 133511996439971664 z - 2718732279804 z + 30981363490583 z 54 52 60 70 - 269247440776464 z + 1813598343137159 z + 178042863494 z - 492 z 68 32 38 + 77171 z + 1309192126912602938 z - 1811958429675314252 z 40 62 72 + 1309192126912602938 z - 8448136880 z + z )) And in Maple-input format, it is: -(1+72503389000301700*z^28-28436989298659382*z^26-341*z^2+8992301811093095*z^24 -2280463862025325*z^22+43360*z^4-2905861*z^6+119041279*z^8-3243301262*z^10+ 62088328844*z^12-867930440350*z^14-73379261949909*z^18+9112230148327*z^16-\ 2280463862025325*z^50+8992301811093095*z^48+460545536590580*z^20+ 378139430174140728*z^36-341199419028009964*z^34-2905861*z^66+119041279*z^64-\ 149616004148281110*z^30-149616004148281110*z^42+72503389000301700*z^44-\ 28436989298659382*z^46-867930440350*z^58+9112230148327*z^56-73379261949909*z^54 +460545536590580*z^52+62088328844*z^60-341*z^70+43360*z^68+250572001011667890*z ^32-341199419028009964*z^38+250572001011667890*z^40-3243301262*z^62+z^72)/(-1+z ^2)/(1+355879956289976894*z^28-133511996439971664*z^26-492*z^2+ 40077522631313287*z^24-9583046042918636*z^22+77171*z^4-6016988*z^6+278711847*z^ 8-8448136880*z^10+178042863494*z^12-2718732279804*z^14-269247440776464*z^18+ 30981363490583*z^16-9583046042918636*z^50+40077522631313287*z^48+ 1813598343137159*z^20+2019188147086349956*z^36-1811958429675314252*z^34-6016988 *z^66+278711847*z^64-761262969894959092*z^30-761262969894959092*z^42+ 355879956289976894*z^44-133511996439971664*z^46-2718732279804*z^58+ 30981363490583*z^56-269247440776464*z^54+1813598343137159*z^52+178042863494*z^ 60-492*z^70+77171*z^68+1309192126912602938*z^32-1811958429675314252*z^38+ 1309192126912602938*z^40-8448136880*z^62+z^72) The first , 40, terms are: [0, 152, 0, 40633, 0, 11415591, 0, 3232830296, 0, 917045019287, 0, 260233013445631, 0, 73853722470354672, 0, 20960006977341337567, 0, 5948570317683984705785, 0, 1688240391946211918936240, 0, 479133012915552427562571313, 0, 135980906797837100364326824513, 0, 38592221386456592768697341423152, 0, 10952710878843585942533034995087241, 0, 3108447023637571928913395076916862767, 0, 882196472429102800686233663369094979440, 0, 250372809987858404929931842598395527378671, 0, 71057350534833818304767721420827498877321223, 0, 20166515147111814059041660802708626539832920376, 0, 5723381608206858007349725360105999512450022487991] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 12041731120598749193 z - 2782905700976877567 z - 415 z 24 22 4 6 + 533256185344360881 z - 84190178121983952 z + 67689 z - 5922594 z 8 10 12 14 + 321881902 z - 11863831838 z + 314137111323 z - 6219121030717 z 18 16 50 - 1134980085156208 z + 94749958716179 z - 1272710165446970892805 z 48 20 + 1933461621582266215355 z + 10865521707254612 z 36 34 + 707685343264800454835 z - 331898561252520708598 z 66 80 88 84 86 - 84190178121983952 z + 321881902 z + z + 67689 z - 415 z 82 64 30 - 5922594 z + 533256185344360881 z - 43422115047076859786 z 42 44 - 2483810924012684400512 z + 2699912291683365745208 z 46 58 - 2483810924012684400512 z - 43422115047076859786 z 56 54 + 131020944269060057346 z - 331898561252520708598 z 52 60 + 707685343264800454835 z + 12041731120598749193 z 70 68 78 - 1134980085156208 z + 10865521707254612 z - 11863831838 z 32 38 + 131020944269060057346 z - 1272710165446970892805 z 40 62 76 + 1933461621582266215355 z - 2782905700976877567 z + 314137111323 z 74 72 / 2 - 6219121030717 z + 94749958716179 z ) / ((-1 + z ) (1 / 28 26 2 + 53484267492986121396 z - 11740901443958488444 z - 580 z 24 22 4 + 2129605410058391631 z - 317293836784388636 z + 118126 z 6 8 10 12 - 12001156 z + 729689823 z - 29501234560 z + 847300079888 z 14 18 16 - 18069659025812 z - 3778190058422604 z + 295180000948129 z 50 48 - 6802581846197375028124 z + 10527569854656733230383 z 20 36 + 38537484573153442 z + 3687626980742221917390 z 34 66 80 - 1675316974599494400084 z - 317293836784388636 z + 729689823 z 88 84 86 82 64 + z + 118126 z - 580 z - 12001156 z + 2129605410058391631 z 30 42 - 202231338179595745944 z - 13677546015965219728584 z 44 46 + 14923931779732674034876 z - 13677546015965219728584 z 58 56 - 202231338179595745944 z + 636922735075853375633 z 54 52 - 1675316974599494400084 z + 3687626980742221917390 z 60 70 68 + 53484267492986121396 z - 3778190058422604 z + 38537484573153442 z 78 32 38 - 29501234560 z + 636922735075853375633 z - 6802581846197375028124 z 40 62 + 10527569854656733230383 z - 11740901443958488444 z 76 74 72 + 847300079888 z - 18069659025812 z + 295180000948129 z )) And in Maple-input format, it is: -(1+12041731120598749193*z^28-2782905700976877567*z^26-415*z^2+ 533256185344360881*z^24-84190178121983952*z^22+67689*z^4-5922594*z^6+321881902* z^8-11863831838*z^10+314137111323*z^12-6219121030717*z^14-1134980085156208*z^18 +94749958716179*z^16-1272710165446970892805*z^50+1933461621582266215355*z^48+ 10865521707254612*z^20+707685343264800454835*z^36-331898561252520708598*z^34-\ 84190178121983952*z^66+321881902*z^80+z^88+67689*z^84-415*z^86-5922594*z^82+ 533256185344360881*z^64-43422115047076859786*z^30-2483810924012684400512*z^42+ 2699912291683365745208*z^44-2483810924012684400512*z^46-43422115047076859786*z^ 58+131020944269060057346*z^56-331898561252520708598*z^54+707685343264800454835* z^52+12041731120598749193*z^60-1134980085156208*z^70+10865521707254612*z^68-\ 11863831838*z^78+131020944269060057346*z^32-1272710165446970892805*z^38+ 1933461621582266215355*z^40-2782905700976877567*z^62+314137111323*z^76-\ 6219121030717*z^74+94749958716179*z^72)/(-1+z^2)/(1+53484267492986121396*z^28-\ 11740901443958488444*z^26-580*z^2+2129605410058391631*z^24-317293836784388636*z ^22+118126*z^4-12001156*z^6+729689823*z^8-29501234560*z^10+847300079888*z^12-\ 18069659025812*z^14-3778190058422604*z^18+295180000948129*z^16-\ 6802581846197375028124*z^50+10527569854656733230383*z^48+38537484573153442*z^20 +3687626980742221917390*z^36-1675316974599494400084*z^34-317293836784388636*z^ 66+729689823*z^80+z^88+118126*z^84-580*z^86-12001156*z^82+2129605410058391631*z ^64-202231338179595745944*z^30-13677546015965219728584*z^42+ 14923931779732674034876*z^44-13677546015965219728584*z^46-202231338179595745944 *z^58+636922735075853375633*z^56-1675316974599494400084*z^54+ 3687626980742221917390*z^52+53484267492986121396*z^60-3778190058422604*z^70+ 38537484573153442*z^68-29501234560*z^78+636922735075853375633*z^32-\ 6802581846197375028124*z^38+10527569854656733230383*z^40-11740901443958488444*z ^62+847300079888*z^76-18069659025812*z^74+295180000948129*z^72) The first , 40, terms are: [0, 166, 0, 45429, 0, 12885741, 0, 3685912382, 0, 1057713574805, 0, 303918990461437, 0, 87373869058283774, 0, 25124768736709773253, 0, 7225412090692299837645, 0, 2077972231432078911908006, 0, 597618061311108086904855513, 0, 171874135284263884584549039705, 0, 49430899347695290175552426800774, 0, 14216313231260892052041954184061581, 0, 4088609717914401638964164708479139333, 0, 1175883814273155334227076117394007831038, 0, 338184113832019102190038387554392613928925, 0, 97261733864300056892212235974734912232196885, 0, 27972470071040922694539873726396362496789240702, 0, 8044881118003643876616091883805699003695533068237] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 17929906440997612263 z - 4025293721248501141 z - 397 z 24 22 4 6 + 746739979028439093 z - 113735982009023160 z + 63339 z - 5596406 z 8 10 12 14 + 314155240 z - 12109602450 z + 337134216493 z - 7024413706159 z 18 16 50 - 1411481735966408 z + 112456929927975 z - 2085261287492556894447 z 48 20 + 3196495235007236908195 z + 14109396388380408 z 36 34 + 1145122512419434202241 z - 528597846649370305202 z 66 80 88 84 86 - 113735982009023160 z + 314155240 z + z + 63339 z - 397 z 82 64 30 - 5596406 z + 746739979028439093 z - 66337193845949922598 z 42 44 - 4128742252080139777248 z + 4496128022171654225968 z 46 58 - 4128742252080139777248 z - 66337193845949922598 z 56 54 + 204706401171047647112 z - 528597846649370305202 z 52 60 + 1145122512419434202241 z + 17929906440997612263 z 70 68 78 - 1411481735966408 z + 14109396388380408 z - 12109602450 z 32 38 + 204706401171047647112 z - 2085261287492556894447 z 40 62 76 + 3196495235007236908195 z - 4025293721248501141 z + 337134216493 z 74 72 / - 7024413706159 z + 112456929927975 z ) / (-1 / 28 26 2 - 93867724745090560432 z + 19441984537782328337 z + 555 z 24 22 4 - 3327847451765515051 z + 467556191890166614 z - 106978 z 6 8 10 12 + 10867852 z - 686118261 z + 29388812557 z - 902159253870 z 14 18 16 + 20611060505492 z + 4923404747016509 z - 360318578852503 z 50 48 + 27412527023025374380573 z - 38513043675173711097671 z 20 36 - 53464620056827446 z - 8312790876658698990832 z 34 66 80 + 3533242741942010676299 z + 3327847451765515051 z - 29388812557 z 90 88 84 86 82 + z - 555 z - 10867852 z + 106978 z + 686118261 z 64 30 - 19441984537782328337 z + 376566929840914123878 z 42 44 + 38513043675173711097671 z - 45643017348747615012100 z 46 58 + 45643017348747615012100 z + 1260599699408221251939 z 56 54 - 3533242741942010676299 z + 8312790876658698990832 z 52 60 - 16448851349392374248490 z - 376566929840914123878 z 70 68 78 + 53464620056827446 z - 467556191890166614 z + 902159253870 z 32 38 - 1260599699408221251939 z + 16448851349392374248490 z 40 62 - 27412527023025374380573 z + 93867724745090560432 z 76 74 72 - 20611060505492 z + 360318578852503 z - 4923404747016509 z ) And in Maple-input format, it is: -(1+17929906440997612263*z^28-4025293721248501141*z^26-397*z^2+ 746739979028439093*z^24-113735982009023160*z^22+63339*z^4-5596406*z^6+314155240 *z^8-12109602450*z^10+337134216493*z^12-7024413706159*z^14-1411481735966408*z^ 18+112456929927975*z^16-2085261287492556894447*z^50+3196495235007236908195*z^48 +14109396388380408*z^20+1145122512419434202241*z^36-528597846649370305202*z^34-\ 113735982009023160*z^66+314155240*z^80+z^88+63339*z^84-397*z^86-5596406*z^82+ 746739979028439093*z^64-66337193845949922598*z^30-4128742252080139777248*z^42+ 4496128022171654225968*z^44-4128742252080139777248*z^46-66337193845949922598*z^ 58+204706401171047647112*z^56-528597846649370305202*z^54+1145122512419434202241 *z^52+17929906440997612263*z^60-1411481735966408*z^70+14109396388380408*z^68-\ 12109602450*z^78+204706401171047647112*z^32-2085261287492556894447*z^38+ 3196495235007236908195*z^40-4025293721248501141*z^62+337134216493*z^76-\ 7024413706159*z^74+112456929927975*z^72)/(-1-93867724745090560432*z^28+ 19441984537782328337*z^26+555*z^2-3327847451765515051*z^24+467556191890166614*z ^22-106978*z^4+10867852*z^6-686118261*z^8+29388812557*z^10-902159253870*z^12+ 20611060505492*z^14+4923404747016509*z^18-360318578852503*z^16+ 27412527023025374380573*z^50-38513043675173711097671*z^48-53464620056827446*z^ 20-8312790876658698990832*z^36+3533242741942010676299*z^34+3327847451765515051* z^66-29388812557*z^80+z^90-555*z^88-10867852*z^84+106978*z^86+686118261*z^82-\ 19441984537782328337*z^64+376566929840914123878*z^30+38513043675173711097671*z^ 42-45643017348747615012100*z^44+45643017348747615012100*z^46+ 1260599699408221251939*z^58-3533242741942010676299*z^56+8312790876658698990832* z^54-16448851349392374248490*z^52-376566929840914123878*z^60+53464620056827446* z^70-467556191890166614*z^68+902159253870*z^78-1260599699408221251939*z^32+ 16448851349392374248490*z^38-27412527023025374380573*z^40+93867724745090560432* z^62-20611060505492*z^76+360318578852503*z^74-4923404747016509*z^72) The first , 40, terms are: [0, 158, 0, 44051, 0, 12817227, 0, 3746230702, 0, 1095609002925, 0, 320448666502541, 0, 93727949387745742, 0, 27414564414607708867, 0, 8018514629192376621259, 0, 2345344141382886737984702, 0, 685992307810500613338742065, 0, 200646652319122859856061182369, 0, 58687362392222562959661392089534, 0, 17165531875537357667393403749879147, 0, 5020765503667443968099576172958492259, 0, 1468529284508995285285726478119312164334, 0, 429531763213501116573191445686898915059421, 0, 125634222998287319759534463660109322455947261, 0, 36746893571505534486179851538971961749304737294, 0, 10748139757859174149484267158039226084285936896523] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 13721670674399674817 z - 3144804260985240469 z - 405 z 24 22 4 6 + 596726596383516913 z - 93146786331521040 z + 65173 z - 5729662 z 8 10 12 14 + 315987898 z - 11866210690 z + 320445012475 z - 6466836332471 z 18 16 50 - 1221412691031776 z + 100308228762779 z - 1485058434989805186639 z 48 20 + 2260252141787461248827 z + 11866137558990148 z 36 34 + 823537295442686443719 z - 384837372819863696066 z 66 80 88 84 86 - 93146786331521040 z + 315987898 z + z + 65173 z - 405 z 82 64 30 - 5729662 z + 596726596383516913 z - 49827280630935450134 z 42 44 - 2906779288079332894344 z + 3160810690915616668360 z 46 58 - 2906779288079332894344 z - 49827280630935450134 z 56 54 + 151216408733762956094 z - 384837372819863696066 z 52 60 + 823537295442686443719 z + 13721670674399674817 z 70 68 78 - 1221412691031776 z + 11866137558990148 z - 11866210690 z 32 38 + 151216408733762956094 z - 1485058434989805186639 z 40 62 76 + 2260252141787461248827 z - 3144804260985240469 z + 320445012475 z 74 72 / 2 - 6466836332471 z + 100308228762779 z ) / ((-1 + z ) (1 / 28 26 2 + 60086238498425400744 z - 13100326196749299430 z - 566 z 24 22 4 + 2356467540200311031 z - 347580454506703816 z + 111578 z 6 8 10 12 - 11356736 z + 702686083 z - 29057433624 z + 853926537788 z 14 18 16 - 18604716867226 z - 4031552064190640 z + 309780676228729 z 50 48 - 7782084027890904522730 z + 12059606585068369221343 z 20 36 + 41711234165671854 z + 4210317168189444694050 z 34 66 80 - 1907665144670292796552 z - 347580454506703816 z + 702686083 z 88 84 86 82 64 + z + 111578 z - 566 z - 11356736 z + 2356467540200311031 z 30 42 - 228458080709933949120 z - 15680205194319476894568 z 44 46 + 17113457914482470440004 z - 15680205194319476894568 z 58 56 - 228458080709933949120 z + 722726966152780482269 z 54 52 - 1907665144670292796552 z + 4210317168189444694050 z 60 70 68 + 60086238498425400744 z - 4031552064190640 z + 41711234165671854 z 78 32 38 - 29057433624 z + 722726966152780482269 z - 7782084027890904522730 z 40 62 + 12059606585068369221343 z - 13100326196749299430 z 76 74 72 + 853926537788 z - 18604716867226 z + 309780676228729 z )) And in Maple-input format, it is: -(1+13721670674399674817*z^28-3144804260985240469*z^26-405*z^2+ 596726596383516913*z^24-93146786331521040*z^22+65173*z^4-5729662*z^6+315987898* z^8-11866210690*z^10+320445012475*z^12-6466836332471*z^14-1221412691031776*z^18 +100308228762779*z^16-1485058434989805186639*z^50+2260252141787461248827*z^48+ 11866137558990148*z^20+823537295442686443719*z^36-384837372819863696066*z^34-\ 93146786331521040*z^66+315987898*z^80+z^88+65173*z^84-405*z^86-5729662*z^82+ 596726596383516913*z^64-49827280630935450134*z^30-2906779288079332894344*z^42+ 3160810690915616668360*z^44-2906779288079332894344*z^46-49827280630935450134*z^ 58+151216408733762956094*z^56-384837372819863696066*z^54+823537295442686443719* z^52+13721670674399674817*z^60-1221412691031776*z^70+11866137558990148*z^68-\ 11866210690*z^78+151216408733762956094*z^32-1485058434989805186639*z^38+ 2260252141787461248827*z^40-3144804260985240469*z^62+320445012475*z^76-\ 6466836332471*z^74+100308228762779*z^72)/(-1+z^2)/(1+60086238498425400744*z^28-\ 13100326196749299430*z^26-566*z^2+2356467540200311031*z^24-347580454506703816*z ^22+111578*z^4-11356736*z^6+702686083*z^8-29057433624*z^10+853926537788*z^12-\ 18604716867226*z^14-4031552064190640*z^18+309780676228729*z^16-\ 7782084027890904522730*z^50+12059606585068369221343*z^48+41711234165671854*z^20 +4210317168189444694050*z^36-1907665144670292796552*z^34-347580454506703816*z^ 66+702686083*z^80+z^88+111578*z^84-566*z^86-11356736*z^82+2356467540200311031*z ^64-228458080709933949120*z^30-15680205194319476894568*z^42+ 17113457914482470440004*z^44-15680205194319476894568*z^46-228458080709933949120 *z^58+722726966152780482269*z^56-1907665144670292796552*z^54+ 4210317168189444694050*z^52+60086238498425400744*z^60-4031552064190640*z^70+ 41711234165671854*z^68-29057433624*z^78+722726966152780482269*z^32-\ 7782084027890904522730*z^38+12059606585068369221343*z^40-13100326196749299430*z ^62+853926537788*z^76-18604716867226*z^74+309780676228729*z^72) The first , 40, terms are: [0, 162, 0, 44883, 0, 13019985, 0, 3808784290, 0, 1116418804191, 0, 327404649402247, 0, 96027945093640970, 0, 28165953761762971577, 0, 8261423124737893612063, 0, 2423182824883130629015434, 0, 710751411629687013729407505, 0, 208472770999683943126353172677, 0, 61147817719905244715068114968978, 0, 17935462877016245369040497396643267, 0, 5260708253162143140030420537304298549, 0, 1543035244234129835826403052804704449906, 0, 452592626452656417412931271458686116696387, 0, 132751397800178039206610518175027988016270963, 0, 38937739123581779908049138961385400650367492794, 0, 11420953399993422945816062857277975619092942156405] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1467682773688789917472 z - 197452719083819180744 z - 440 z 24 22 4 + 22614274295760164072 z - 2190130757691070252 z + 80754 z 6 102 8 10 - 8495704 z - 28777864060 z + 587040443 z - 28777864060 z 12 14 18 + 1052043972754 z - 29698695171600 z - 12001718920233092 z 16 50 + 664215286241901 z - 409190556617978664402410984 z 48 20 + 258515017217246991360988810 z + 177906937443989268 z 36 34 + 988902617742314260943936 z - 241559100489184781239300 z 66 80 - 143102411233057485531294108 z + 51115305343409207403500 z 100 90 88 + 1052043972754 z - 2190130757691070252 z + 22614274295760164072 z 84 94 + 1467682773688789917472 z - 12001718920233092 z 86 96 98 - 197452719083819180744 z + 664215286241901 z - 29698695171600 z 92 82 + 177906937443989268 z - 9339399820603591583196 z 64 112 110 106 + 258515017217246991360988810 z + z - 440 z - 8495704 z 108 30 42 + 80754 z - 9339399820603591583196 z - 29399947524611764893871988 z 44 46 + 69354841359501270309163356 z - 143102411233057485531294108 z 58 56 - 691066434459412692792825548 z + 737809574657577220509254866 z 54 52 - 691066434459412692792825548 z + 567823669789333611920169980 z 60 70 + 567823669789333611920169980 z - 29399947524611764893871988 z 68 78 + 69354841359501270309163356 z - 241559100489184781239300 z 32 38 + 51115305343409207403500 z - 3516607309095308869921864 z 40 62 + 10887231205772121687553600 z - 409190556617978664402410984 z 76 74 + 988902617742314260943936 z - 3516607309095308869921864 z 72 104 / 2 + 10887231205772121687553600 z + 587040443 z ) / ((-1 + z ) (1 / 28 26 2 + 5492156751764479410610 z - 705090012435286140623 z - 603 z 24 22 4 + 76890077772181884612 z - 7074071773479367629 z + 132136 z 6 102 8 10 - 15737867 z - 63906357030 z + 1199439963 z - 63906357030 z 12 14 18 + 2515898433218 z - 76003821172012 z - 34721144629853823 z 16 50 + 1810624295397421 z - 2156390573682599803864003981 z 48 20 + 1342368886280008312045091602 z + 544562861301205918 z 36 34 + 4362259511911506099560670 z - 1026295164344724907030799 z 66 80 - 729323377119540259087345509 z + 208651961976798690220568 z 100 90 88 + 2515898433218 z - 7074071773479367629 z + 76890077772181884612 z 84 94 + 5492156751764479410610 z - 34721144629853823 z 86 96 98 - 705090012435286140623 z + 1810624295397421 z - 76003821172012 z 92 82 + 544562861301205918 z - 36542550435047498780549 z 64 112 110 106 + 1342368886280008312045091602 z + z - 603 z - 15737867 z 108 30 + 132136 z - 36542550435047498780549 z 42 44 - 142780385796447328968385503 z + 345644319364436394707256050 z 46 58 - 729323377119540259087345509 z - 3704874964715939616248444555 z 56 54 + 3964047525755438560665322970 z - 3704874964715939616248444555 z 52 60 + 3024526469024358796088198438 z + 3024526469024358796088198438 z 70 68 - 142780385796447328968385503 z + 345644319364436394707256050 z 78 32 - 1026295164344724907030799 z + 208651961976798690220568 z 38 40 - 16063990030855165216393773 z + 51356922645723840500824956 z 62 76 - 2156390573682599803864003981 z + 4362259511911506099560670 z 74 72 - 16063990030855165216393773 z + 51356922645723840500824956 z 104 + 1199439963 z )) And in Maple-input format, it is: -(1+1467682773688789917472*z^28-197452719083819180744*z^26-440*z^2+ 22614274295760164072*z^24-2190130757691070252*z^22+80754*z^4-8495704*z^6-\ 28777864060*z^102+587040443*z^8-28777864060*z^10+1052043972754*z^12-\ 29698695171600*z^14-12001718920233092*z^18+664215286241901*z^16-\ 409190556617978664402410984*z^50+258515017217246991360988810*z^48+ 177906937443989268*z^20+988902617742314260943936*z^36-241559100489184781239300* z^34-143102411233057485531294108*z^66+51115305343409207403500*z^80+ 1052043972754*z^100-2190130757691070252*z^90+22614274295760164072*z^88+ 1467682773688789917472*z^84-12001718920233092*z^94-197452719083819180744*z^86+ 664215286241901*z^96-29698695171600*z^98+177906937443989268*z^92-\ 9339399820603591583196*z^82+258515017217246991360988810*z^64+z^112-440*z^110-\ 8495704*z^106+80754*z^108-9339399820603591583196*z^30-\ 29399947524611764893871988*z^42+69354841359501270309163356*z^44-\ 143102411233057485531294108*z^46-691066434459412692792825548*z^58+ 737809574657577220509254866*z^56-691066434459412692792825548*z^54+ 567823669789333611920169980*z^52+567823669789333611920169980*z^60-\ 29399947524611764893871988*z^70+69354841359501270309163356*z^68-\ 241559100489184781239300*z^78+51115305343409207403500*z^32-\ 3516607309095308869921864*z^38+10887231205772121687553600*z^40-\ 409190556617978664402410984*z^62+988902617742314260943936*z^76-\ 3516607309095308869921864*z^74+10887231205772121687553600*z^72+587040443*z^104) /(-1+z^2)/(1+5492156751764479410610*z^28-705090012435286140623*z^26-603*z^2+ 76890077772181884612*z^24-7074071773479367629*z^22+132136*z^4-15737867*z^6-\ 63906357030*z^102+1199439963*z^8-63906357030*z^10+2515898433218*z^12-\ 76003821172012*z^14-34721144629853823*z^18+1810624295397421*z^16-\ 2156390573682599803864003981*z^50+1342368886280008312045091602*z^48+ 544562861301205918*z^20+4362259511911506099560670*z^36-\ 1026295164344724907030799*z^34-729323377119540259087345509*z^66+ 208651961976798690220568*z^80+2515898433218*z^100-7074071773479367629*z^90+ 76890077772181884612*z^88+5492156751764479410610*z^84-34721144629853823*z^94-\ 705090012435286140623*z^86+1810624295397421*z^96-76003821172012*z^98+ 544562861301205918*z^92-36542550435047498780549*z^82+ 1342368886280008312045091602*z^64+z^112-603*z^110-15737867*z^106+132136*z^108-\ 36542550435047498780549*z^30-142780385796447328968385503*z^42+ 345644319364436394707256050*z^44-729323377119540259087345509*z^46-\ 3704874964715939616248444555*z^58+3964047525755438560665322970*z^56-\ 3704874964715939616248444555*z^54+3024526469024358796088198438*z^52+ 3024526469024358796088198438*z^60-142780385796447328968385503*z^70+ 345644319364436394707256050*z^68-1026295164344724907030799*z^78+ 208651961976798690220568*z^32-16063990030855165216393773*z^38+ 51356922645723840500824956*z^40-2156390573682599803864003981*z^62+ 4362259511911506099560670*z^76-16063990030855165216393773*z^74+ 51356922645723840500824956*z^72+1199439963*z^104) The first , 40, terms are: [0, 164, 0, 47071, 0, 14035987, 0, 4204121784, 0, 1260222359169, 0, 377824493503089, 0, 113278871451099468, 0, 33963414897621360359, 0, 10182973771915766043187, 0, 3053079363460921204914376, 0, 915380396023572457346685057, 0, 274451198208402500753079268961, 0, 82286512737639495548663521184688, 0, 24671308529398274899363086797458075, 0, 7397001580024864149920089436786585263, 0, 2217783962025906312407062470270109329380, 0, 664940469335697553569220623127132914700193, 0, 199363795271627574341061577026088134239438929, 0, 59773655985850464612039499772778564665597960736, 0, 17921458332229051390831409421125822438571845888779] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1847124203012742 z - 1039612125134770 z - 311 z 24 22 4 6 + 463303615155230 z - 162849592163136 z + 33757 z - 1835872 z 8 10 12 14 + 59464056 z - 1262869680 z + 18667012221 z - 199821678711 z 18 16 50 - 9634398111520 z + 1592354040769 z - 199821678711 z 48 20 36 + 1592354040769 z + 44897251741840 z + 1847124203012742 z 34 64 30 42 - 2605269424623184 z + z - 2605269424623184 z - 162849592163136 z 44 46 58 56 + 44897251741840 z - 9634398111520 z - 1835872 z + 59464056 z 54 52 60 32 - 1262869680 z + 18667012221 z + 33757 z + 2921260973375664 z 38 40 62 / 2 - 1039612125134770 z + 463303615155230 z - 311 z ) / ((-1 + z ) (1 / 28 26 2 + 9938767322099508 z - 5401329294537976 z - 482 z 24 22 4 6 + 2296344477546542 z - 761909416219012 z + 65454 z - 4135752 z 8 10 12 14 + 151245935 z - 3577285796 z + 58398688926 z - 686010944406 z 18 16 50 - 39137565141420 z + 5964395492145 z - 686010944406 z 48 20 36 + 5964395492145 z + 196489870795168 z + 9938767322099508 z 34 64 30 - 14324411507174340 z + z - 14324411507174340 z 42 44 46 - 761909416219012 z + 196489870795168 z - 39137565141420 z 58 56 54 52 - 4135752 z + 151245935 z - 3577285796 z + 58398688926 z 60 32 38 + 65454 z + 16179707861537474 z - 5401329294537976 z 40 62 + 2296344477546542 z - 482 z )) And in Maple-input format, it is: -(1+1847124203012742*z^28-1039612125134770*z^26-311*z^2+463303615155230*z^24-\ 162849592163136*z^22+33757*z^4-1835872*z^6+59464056*z^8-1262869680*z^10+ 18667012221*z^12-199821678711*z^14-9634398111520*z^18+1592354040769*z^16-\ 199821678711*z^50+1592354040769*z^48+44897251741840*z^20+1847124203012742*z^36-\ 2605269424623184*z^34+z^64-2605269424623184*z^30-162849592163136*z^42+ 44897251741840*z^44-9634398111520*z^46-1835872*z^58+59464056*z^56-1262869680*z^ 54+18667012221*z^52+33757*z^60+2921260973375664*z^32-1039612125134770*z^38+ 463303615155230*z^40-311*z^62)/(-1+z^2)/(1+9938767322099508*z^28-\ 5401329294537976*z^26-482*z^2+2296344477546542*z^24-761909416219012*z^22+65454* z^4-4135752*z^6+151245935*z^8-3577285796*z^10+58398688926*z^12-686010944406*z^ 14-39137565141420*z^18+5964395492145*z^16-686010944406*z^50+5964395492145*z^48+ 196489870795168*z^20+9938767322099508*z^36-14324411507174340*z^34+z^64-\ 14324411507174340*z^30-761909416219012*z^42+196489870795168*z^44-39137565141420 *z^46-4135752*z^58+151245935*z^56-3577285796*z^54+58398688926*z^52+65454*z^60+ 16179707861537474*z^32-5401329294537976*z^38+2296344477546542*z^40-482*z^62) The first , 40, terms are: [0, 172, 0, 50897, 0, 15607593, 0, 4809212628, 0, 1483316240945, 0, 457601750665777, 0, 141176720526115892, 0, 43555542497254684969, 0, 13437698608109216906961, 0, 4145783554514664688154572, 0, 1279052622039398830019658753, 0, 394611932813118292367900611585, 0, 121745247975919530751114660656012, 0, 37560712682100038140707684258406097, 0, 11588190596186147409116734091777694249, 0, 3575176074049328162620960802452267639668, 0, 1103009469391899772119119073766381875595569, 0, 340299292783033900868928086490250800608431921, 0, 104988771068846133161471003028218805049948108692, 0, 32391022503761167856276832545584204657544903722537] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1468996280387841601771 z - 194773893263642803069 z - 434 z 24 22 4 + 22019022288846289627 z - 2108727169822090540 z + 78471 z 6 102 8 10 - 8153489 z - 27196732644 z + 558218903 z - 27196732644 z 12 14 18 + 990947153181 z - 27956792190326 z - 11371512848547430 z 16 50 + 626505707268730 z - 482566836066873133318468608 z 48 20 + 302193050291622656495571534 z + 169733686606228185 z 36 34 + 1058544712778188816869816 z - 254102031020311798063000 z 66 80 - 165453341086081889568322546 z + 52850139265181766878189 z 100 90 88 + 990947153181 z - 2108727169822090540 z + 22019022288846289627 z 84 94 + 1468996280387841601771 z - 11371512848547430 z 86 96 98 - 194773893263642803069 z + 626505707268730 z - 27956792190326 z 92 82 + 169733686606228185 z - 9496521686662295315954 z 64 112 110 106 + 302193050291622656495571534 z + z - 434 z - 8153489 z 108 30 42 + 78471 z - 9496521686662295315954 z - 33074038726076437645667404 z 44 46 + 79160725694733985530106574 z - 165453341086081889568322546 z 58 56 - 823508380360267287684001236 z + 880374419905933788335917468 z 54 52 - 823508380360267287684001236 z + 673986032936773557508507306 z 60 70 + 673986032936773557508507306 z - 33074038726076437645667404 z 68 78 + 79160725694733985530106574 z - 254102031020311798063000 z 32 38 + 52850139265181766878189 z - 3829746840502703769039032 z 40 62 + 12056233612730664741984114 z - 482566836066873133318468608 z 76 74 + 1058544712778188816869816 z - 3829746840502703769039032 z 72 104 / + 12056233612730664741984114 z + 558218903 z ) / (-1 / 28 26 2 - 6095219318536276101922 z + 757963825666257727311 z + 593 z 24 22 4 - 80355379076387972847 z + 7214362428834643400 z - 128254 z 6 102 8 10 + 15139062 z + 2406982401268 z - 1147765087 z + 61042190767 z 12 14 18 - 2406982401268 z + 73088485750516 z + 34117040643994093 z 16 50 - 1756661778657605 z + 4109301049551045505696513450 z 48 20 - 2407790307579062508039379354 z - 544060014812310584 z 36 34 - 5683832617602630370162228 z + 1278784681737680573774297 z 66 80 + 2407790307579062508039379354 z - 1278784681737680573774297 z 100 90 - 73088485750516 z + 80355379076387972847 z 88 84 - 757963825666257727311 z - 42014527176172009102522 z 94 86 + 544060014812310584 z + 6095219318536276101922 z 96 98 92 - 34117040643994093 z + 1756661778657605 z - 7214362428834643400 z 82 64 + 249350462407103870166937 z - 4109301049551045505696513450 z 112 114 110 106 108 - 593 z + z + 128254 z + 1147765087 z - 15139062 z 30 42 + 42014527176172009102522 z + 216083537068983648222099518 z 44 46 - 552441664033151535584997632 z + 1233649743693469179688284832 z 58 56 + 9156091952767733749614389878 z - 9156091952767733749614389878 z 54 52 + 8012006534510870239049307116 z - 6134491551920030633045783580 z 60 70 - 8012006534510870239049307116 z + 552441664033151535584997632 z 68 78 - 1233649743693469179688284832 z + 5683832617602630370162228 z 32 38 - 249350462407103870166937 z + 21946924515830994233185572 z 40 62 - 73759479450309518725718030 z + 6134491551920030633045783580 z 76 74 - 21946924515830994233185572 z + 73759479450309518725718030 z 72 104 - 216083537068983648222099518 z - 61042190767 z ) And in Maple-input format, it is: -(1+1468996280387841601771*z^28-194773893263642803069*z^26-434*z^2+ 22019022288846289627*z^24-2108727169822090540*z^22+78471*z^4-8153489*z^6-\ 27196732644*z^102+558218903*z^8-27196732644*z^10+990947153181*z^12-\ 27956792190326*z^14-11371512848547430*z^18+626505707268730*z^16-\ 482566836066873133318468608*z^50+302193050291622656495571534*z^48+ 169733686606228185*z^20+1058544712778188816869816*z^36-254102031020311798063000 *z^34-165453341086081889568322546*z^66+52850139265181766878189*z^80+ 990947153181*z^100-2108727169822090540*z^90+22019022288846289627*z^88+ 1468996280387841601771*z^84-11371512848547430*z^94-194773893263642803069*z^86+ 626505707268730*z^96-27956792190326*z^98+169733686606228185*z^92-\ 9496521686662295315954*z^82+302193050291622656495571534*z^64+z^112-434*z^110-\ 8153489*z^106+78471*z^108-9496521686662295315954*z^30-\ 33074038726076437645667404*z^42+79160725694733985530106574*z^44-\ 165453341086081889568322546*z^46-823508380360267287684001236*z^58+ 880374419905933788335917468*z^56-823508380360267287684001236*z^54+ 673986032936773557508507306*z^52+673986032936773557508507306*z^60-\ 33074038726076437645667404*z^70+79160725694733985530106574*z^68-\ 254102031020311798063000*z^78+52850139265181766878189*z^32-\ 3829746840502703769039032*z^38+12056233612730664741984114*z^40-\ 482566836066873133318468608*z^62+1058544712778188816869816*z^76-\ 3829746840502703769039032*z^74+12056233612730664741984114*z^72+558218903*z^104) /(-1-6095219318536276101922*z^28+757963825666257727311*z^26+593*z^2-\ 80355379076387972847*z^24+7214362428834643400*z^22-128254*z^4+15139062*z^6+ 2406982401268*z^102-1147765087*z^8+61042190767*z^10-2406982401268*z^12+ 73088485750516*z^14+34117040643994093*z^18-1756661778657605*z^16+ 4109301049551045505696513450*z^50-2407790307579062508039379354*z^48-\ 544060014812310584*z^20-5683832617602630370162228*z^36+ 1278784681737680573774297*z^34+2407790307579062508039379354*z^66-\ 1278784681737680573774297*z^80-73088485750516*z^100+80355379076387972847*z^90-\ 757963825666257727311*z^88-42014527176172009102522*z^84+544060014812310584*z^94 +6095219318536276101922*z^86-34117040643994093*z^96+1756661778657605*z^98-\ 7214362428834643400*z^92+249350462407103870166937*z^82-\ 4109301049551045505696513450*z^64-593*z^112+z^114+128254*z^110+1147765087*z^106 -15139062*z^108+42014527176172009102522*z^30+216083537068983648222099518*z^42-\ 552441664033151535584997632*z^44+1233649743693469179688284832*z^46+ 9156091952767733749614389878*z^58-9156091952767733749614389878*z^56+ 8012006534510870239049307116*z^54-6134491551920030633045783580*z^52-\ 8012006534510870239049307116*z^60+552441664033151535584997632*z^70-\ 1233649743693469179688284832*z^68+5683832617602630370162228*z^78-\ 249350462407103870166937*z^32+21946924515830994233185572*z^38-\ 73759479450309518725718030*z^40+6134491551920030633045783580*z^62-\ 21946924515830994233185572*z^76+73759479450309518725718030*z^74-\ 216083537068983648222099518*z^72-61042190767*z^104) The first , 40, terms are: [0, 159, 0, 44504, 0, 12984059, 0, 3809295645, 0, 1118754439037, 0, 328639988559787, 0, 96544337879592296, 0, 28362061512600422159, 0, 8332010359210584382113, 0, 2447721688950840328965937, 0, 719075218883388327982441759, 0, 211245088215255490004478848776, 0, 62058163581702787049609499915675, 0, 18231030623427404605299077005796141, 0, 5355789770595192260365284576722454925, 0, 1573387959312912040400700543104483585515, 0, 462219350755759201944200483548736421815352, 0, 135787697464719872353370497516557961243669167, 0, 39890798065110932478771211438492221274394118609, 0, 11718850823618836383310806816019591786434899619185] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1379718476861905897843 z - 188078744390536753253 z - 446 z 24 22 4 + 21828565017057163575 z - 2141980750914536760 z + 82619 z 6 102 8 10 - 8744769 z - 29697662264 z + 606020759 z - 29697662264 z 12 14 18 + 1081824901865 z - 30345029488262 z - 12028991590813666 z 16 50 + 672773762373790 z - 348320590447343788650788784 z 48 20 + 220970347991703144470126306 z + 176214485565564669 z 36 34 + 886020493378207354058008 z - 218833390525180059579260 z 66 80 - 122964970685460157432149610 z + 46854576534398359460309 z 100 90 88 + 1081824901865 z - 2141980750914536760 z + 21828565017057163575 z 84 94 + 1379718476861905897843 z - 12028991590813666 z 86 96 98 - 188078744390536753253 z + 672773762373790 z - 30345029488262 z 92 82 + 176214485565564669 z - 8667525116562319186294 z 64 112 110 106 + 220970347991703144470126306 z + z - 446 z - 8744769 z 108 30 42 + 82619 z - 8667525116562319186294 z - 25613170007646601367993740 z 44 46 + 59975460910897775012643394 z - 122964970685460157432149610 z 58 56 - 585469603754242944485651080 z + 624696188970495702142475316 z 54 52 - 585469603754242944485651080 z + 481921360661149928090871130 z 60 70 + 481921360661149928090871130 z - 25613170007646601367993740 z 68 78 + 59975460910897775012643394 z - 218833390525180059579260 z 32 38 + 46854576534398359460309 z - 3118630963681074601071732 z 40 62 + 9565116163959933045729354 z - 348320590447343788650788784 z 76 74 + 886020493378207354058008 z - 3118630963681074601071732 z 72 104 / + 9565116163959933045729354 z + 606020759 z ) / (-1 / 28 26 2 - 5880279718271213530810 z + 751977810169404961763 z + 617 z 24 22 4 - 81869588656856184167 z + 7534195947217945268 z - 137930 z 6 102 8 10 + 16667702 z + 2707509023096 z - 1282315411 z + 68666115479 z 12 14 18 - 2707509023096 z + 81720390749624 z + 37139797080092053 z 16 50 - 1942295061575773 z + 3024359185328420096373253162 z 48 20 - 1797800625467065606004662546 z - 580976558538318052 z 36 34 - 4881842632690874594324076 z + 1130438265925645299239673 z 66 80 + 1797800625467065606004662546 z - 1130438265925645299239673 z 100 90 - 81720390749624 z + 81869588656856184167 z 88 84 - 751977810169404961763 z - 39379872590893516286222 z 94 86 + 580976558538318052 z + 5880279718271213530810 z 96 98 92 - 37139797080092053 z + 1942295061575773 z - 7534195947217945268 z 82 64 + 226969218107928761081441 z - 3024359185328420096373253162 z 112 114 110 106 108 - 617 z + z + 137930 z + 1282315411 z - 16667702 z 30 42 + 39379872590893516286222 z + 171426530876172957874675102 z 44 46 - 428394525941381125493595920 z + 937394599453476795799455384 z 58 56 + 6590103974352964500757625334 z - 6590103974352964500757625334 z 54 52 + 5788455159409170051950034668 z - 4465270369292241374321192316 z 60 70 - 5788455159409170051950034668 z + 428394525941381125493595920 z 68 78 - 937394599453476795799455384 z + 4881842632690874594324076 z 32 38 - 226969218107928761081441 z + 18331377672373158169089148 z 40 62 - 59990879834751916818682318 z + 4465270369292241374321192316 z 76 74 - 18331377672373158169089148 z + 59990879834751916818682318 z 72 104 - 171426530876172957874675102 z - 68666115479 z ) And in Maple-input format, it is: -(1+1379718476861905897843*z^28-188078744390536753253*z^26-446*z^2+ 21828565017057163575*z^24-2141980750914536760*z^22+82619*z^4-8744769*z^6-\ 29697662264*z^102+606020759*z^8-29697662264*z^10+1081824901865*z^12-\ 30345029488262*z^14-12028991590813666*z^18+672773762373790*z^16-\ 348320590447343788650788784*z^50+220970347991703144470126306*z^48+ 176214485565564669*z^20+886020493378207354058008*z^36-218833390525180059579260* z^34-122964970685460157432149610*z^66+46854576534398359460309*z^80+ 1081824901865*z^100-2141980750914536760*z^90+21828565017057163575*z^88+ 1379718476861905897843*z^84-12028991590813666*z^94-188078744390536753253*z^86+ 672773762373790*z^96-30345029488262*z^98+176214485565564669*z^92-\ 8667525116562319186294*z^82+220970347991703144470126306*z^64+z^112-446*z^110-\ 8744769*z^106+82619*z^108-8667525116562319186294*z^30-\ 25613170007646601367993740*z^42+59975460910897775012643394*z^44-\ 122964970685460157432149610*z^46-585469603754242944485651080*z^58+ 624696188970495702142475316*z^56-585469603754242944485651080*z^54+ 481921360661149928090871130*z^52+481921360661149928090871130*z^60-\ 25613170007646601367993740*z^70+59975460910897775012643394*z^68-\ 218833390525180059579260*z^78+46854576534398359460309*z^32-\ 3118630963681074601071732*z^38+9565116163959933045729354*z^40-\ 348320590447343788650788784*z^62+886020493378207354058008*z^76-\ 3118630963681074601071732*z^74+9565116163959933045729354*z^72+606020759*z^104)/ (-1-5880279718271213530810*z^28+751977810169404961763*z^26+617*z^2-\ 81869588656856184167*z^24+7534195947217945268*z^22-137930*z^4+16667702*z^6+ 2707509023096*z^102-1282315411*z^8+68666115479*z^10-2707509023096*z^12+ 81720390749624*z^14+37139797080092053*z^18-1942295061575773*z^16+ 3024359185328420096373253162*z^50-1797800625467065606004662546*z^48-\ 580976558538318052*z^20-4881842632690874594324076*z^36+ 1130438265925645299239673*z^34+1797800625467065606004662546*z^66-\ 1130438265925645299239673*z^80-81720390749624*z^100+81869588656856184167*z^90-\ 751977810169404961763*z^88-39379872590893516286222*z^84+580976558538318052*z^94 +5880279718271213530810*z^86-37139797080092053*z^96+1942295061575773*z^98-\ 7534195947217945268*z^92+226969218107928761081441*z^82-\ 3024359185328420096373253162*z^64-617*z^112+z^114+137930*z^110+1282315411*z^106 -16667702*z^108+39379872590893516286222*z^30+171426530876172957874675102*z^42-\ 428394525941381125493595920*z^44+937394599453476795799455384*z^46+ 6590103974352964500757625334*z^58-6590103974352964500757625334*z^56+ 5788455159409170051950034668*z^54-4465270369292241374321192316*z^52-\ 5788455159409170051950034668*z^60+428394525941381125493595920*z^70-\ 937394599453476795799455384*z^68+4881842632690874594324076*z^78-\ 226969218107928761081441*z^32+18331377672373158169089148*z^38-\ 59990879834751916818682318*z^40+4465270369292241374321192316*z^62-\ 18331377672373158169089148*z^76+59990879834751916818682318*z^74-\ 171426530876172957874675102*z^72-68666115479*z^104) The first , 40, terms are: [0, 171, 0, 50196, 0, 15307835, 0, 4695282305, 0, 1441923988161, 0, 442942361666979, 0, 136076110648443068, 0, 41804574916424068023, 0, 12843030428284993453481, 0, 3945587213920722863979805, 0, 1212148674640928064826214907, 0, 372391844177716563728976986764, 0, 114404850324413264981465867618455, 0, 35147036732615817336867182100638157, 0, 10797743178805711150776431174513552253, 0, 3317242892159224259754264364182573807119, 0, 1019111144242829205981933161884079447043780, 0, 313087572450310586932490023467619104929369743, 0, 96185610938431292475566999492279426074029712565, 0, 29549789150686624049581596911444237645705768768541] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1582980237572254690908 z - 211133767888065153987 z - 437 z 24 22 4 + 23962419420582115856 z - 2298797454012754741 z + 79716 z 6 102 8 10 - 8369897 z - 28568095527 z + 579620932 z - 28568095527 z 12 14 18 + 1052305908388 z - 29971446822747 z - 12351029921992353 z 16 50 + 676794473340596 z - 467845292226685333109985353 z 48 20 + 294891571120484408552066720 z + 184916184544185000 z 36 34 + 1098306304184436260803032 z - 266544367409206529149885 z 66 80 - 162759904946089348992139571 z + 56005464567347597388036 z 100 90 88 + 1052305908388 z - 2298797454012754741 z + 23962419420582115856 z 84 94 + 1582980237572254690908 z - 12351029921992353 z 86 96 98 - 211133767888065153987 z + 676794473340596 z - 29971446822747 z 92 82 + 184916184544185000 z - 10155345278514799870135 z 64 112 110 106 + 294891571120484408552066720 z + z - 437 z - 8369897 z 108 30 + 79716 z - 10155345278514799870135 z 42 44 - 33181002101692462969637367 z + 78601500717945598506262508 z 46 58 - 162759904946089348992139571 z - 792224668026378135978854773 z 56 54 + 846091933934120247247421482 z - 792224668026378135978854773 z 52 60 + 650292752473128396864974956 z + 650292752473128396864974956 z 70 68 - 33181002101692462969637367 z + 78601500717945598506262508 z 78 32 - 266544367409206529149885 z + 56005464567347597388036 z 38 40 - 3928895726081580898250665 z + 12228966008941636855009696 z 62 76 - 467845292226685333109985353 z + 1098306304184436260803032 z 74 72 - 3928895726081580898250665 z + 12228966008941636855009696 z 104 / 28 + 579620932 z ) / (-1 - 6646116831834688106036 z / 26 2 24 + 830514467295456665038 z + 610 z - 88291121691478763084 z 22 4 6 102 + 7930798769892955626 z - 132761 z + 15752484 z + 2569233733006 z 8 10 12 14 - 1202841498 z + 64539502168 z - 2569233733006 z + 78734633053153 z 18 16 + 37281070138001359 z - 1907589853129841 z 50 48 + 4030285236503092567343576071 z - 2378931445705614992163688361 z 20 36 - 596986887398371179 z - 5982141081514858989932977 z 34 66 + 1360232516890042355663925 z + 2378931445705614992163688361 z 80 100 - 1360232516890042355663925 z - 78734633053153 z 90 88 + 88291121691478763084 z - 830514467295456665038 z 84 94 - 45503048654120120974527 z + 596986887398371179 z 86 96 + 6646116831834688106036 z - 37281070138001359 z 98 92 + 1907589853129841 z - 7930798769892955626 z 82 64 + 267804314940618945614195 z - 4030285236503092567343576071 z 112 114 110 106 108 - 610 z + z + 132761 z + 1202841498 z - 15752484 z 30 42 + 45503048654120120974527 z + 219865320634075516085677480 z 44 46 - 556106289146177516348398658 z + 1229594414473724618414600077 z 58 56 + 8876192712724358212060863718 z - 8876192712724358212060863718 z 54 52 + 7782545512602190087421514338 z - 5982188105110630746177957527 z 60 70 - 7782545512602190087421514338 z + 556106289146177516348398658 z 68 78 - 1229594414473724618414600077 z + 5982141081514858989932977 z 32 38 - 267804314940618945614195 z + 22842112644169645199694032 z 40 62 - 75898700305951876791898962 z + 5982188105110630746177957527 z 76 74 - 22842112644169645199694032 z + 75898700305951876791898962 z 72 104 - 219865320634075516085677480 z - 64539502168 z ) And in Maple-input format, it is: -(1+1582980237572254690908*z^28-211133767888065153987*z^26-437*z^2+ 23962419420582115856*z^24-2298797454012754741*z^22+79716*z^4-8369897*z^6-\ 28568095527*z^102+579620932*z^8-28568095527*z^10+1052305908388*z^12-\ 29971446822747*z^14-12351029921992353*z^18+676794473340596*z^16-\ 467845292226685333109985353*z^50+294891571120484408552066720*z^48+ 184916184544185000*z^20+1098306304184436260803032*z^36-266544367409206529149885 *z^34-162759904946089348992139571*z^66+56005464567347597388036*z^80+ 1052305908388*z^100-2298797454012754741*z^90+23962419420582115856*z^88+ 1582980237572254690908*z^84-12351029921992353*z^94-211133767888065153987*z^86+ 676794473340596*z^96-29971446822747*z^98+184916184544185000*z^92-\ 10155345278514799870135*z^82+294891571120484408552066720*z^64+z^112-437*z^110-\ 8369897*z^106+79716*z^108-10155345278514799870135*z^30-\ 33181002101692462969637367*z^42+78601500717945598506262508*z^44-\ 162759904946089348992139571*z^46-792224668026378135978854773*z^58+ 846091933934120247247421482*z^56-792224668026378135978854773*z^54+ 650292752473128396864974956*z^52+650292752473128396864974956*z^60-\ 33181002101692462969637367*z^70+78601500717945598506262508*z^68-\ 266544367409206529149885*z^78+56005464567347597388036*z^32-\ 3928895726081580898250665*z^38+12228966008941636855009696*z^40-\ 467845292226685333109985353*z^62+1098306304184436260803032*z^76-\ 3928895726081580898250665*z^74+12228966008941636855009696*z^72+579620932*z^104) /(-1-6646116831834688106036*z^28+830514467295456665038*z^26+610*z^2-\ 88291121691478763084*z^24+7930798769892955626*z^22-132761*z^4+15752484*z^6+ 2569233733006*z^102-1202841498*z^8+64539502168*z^10-2569233733006*z^12+ 78734633053153*z^14+37281070138001359*z^18-1907589853129841*z^16+ 4030285236503092567343576071*z^50-2378931445705614992163688361*z^48-\ 596986887398371179*z^20-5982141081514858989932977*z^36+ 1360232516890042355663925*z^34+2378931445705614992163688361*z^66-\ 1360232516890042355663925*z^80-78734633053153*z^100+88291121691478763084*z^90-\ 830514467295456665038*z^88-45503048654120120974527*z^84+596986887398371179*z^94 +6646116831834688106036*z^86-37281070138001359*z^96+1907589853129841*z^98-\ 7930798769892955626*z^92+267804314940618945614195*z^82-\ 4030285236503092567343576071*z^64-610*z^112+z^114+132761*z^110+1202841498*z^106 -15752484*z^108+45503048654120120974527*z^30+219865320634075516085677480*z^42-\ 556106289146177516348398658*z^44+1229594414473724618414600077*z^46+ 8876192712724358212060863718*z^58-8876192712724358212060863718*z^56+ 7782545512602190087421514338*z^54-5982188105110630746177957527*z^52-\ 7782545512602190087421514338*z^60+556106289146177516348398658*z^70-\ 1229594414473724618414600077*z^68+5982141081514858989932977*z^78-\ 267804314940618945614195*z^32+22842112644169645199694032*z^38-\ 75898700305951876791898962*z^40+5982188105110630746177957527*z^62-\ 22842112644169645199694032*z^76+75898700305951876791898962*z^74-\ 219865320634075516085677480*z^72-64539502168*z^104) The first , 40, terms are: [0, 173, 0, 52485, 0, 16430784, 0, 5156776321, 0, 1618915191413, 0, 508262417705021, 0, 159571324242846297, 0, 50098210729739125249, 0, 15728585752309794120305, 0, 4938068964601450421732304, 0, 1550331712236720842577315797, 0, 486734478309099054306170340153, 0, 152812750050812629693845078155021, 0, 47976335394191295626053465107635557, 0, 15062413032373068786428546901810823793, 0, 4728920716726693643790128426026880152933, 0, 1484668565191478822544794055546812273208272, 0, 466119201506432333768007567901525203962642113, 0, 146340479691489762859839157207919182223120219833, 0, 45944333396099861780536408090255690887142109727505] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 77851677433128 z - 67946846493126 z - 281 z 24 22 4 6 + 45126526288898 z - 22736635897628 z + 26548 z - 1284121 z 8 10 12 14 + 37205893 z - 699351908 z + 8947341816 z - 80453259996 z 18 16 50 48 - 2462915270700 z + 520331719224 z - 1284121 z + 37205893 z 20 36 34 + 8645871072896 z + 8645871072896 z - 22736635897628 z 30 42 44 46 - 67946846493126 z - 80453259996 z + 8947341816 z - 699351908 z 56 54 52 32 38 + z - 281 z + 26548 z + 45126526288898 z - 2462915270700 z 40 / 2 28 + 520331719224 z ) / ((-1 + z ) (1 + 424498817364258 z / 26 2 24 22 - 366583033537792 z - 442 z + 236003705895562 z - 113140491041900 z 4 6 8 10 12 + 51127 z - 2876070 z + 95018213 z - 2014130436 z + 28820548588 z 14 18 16 50 - 287734486756 z - 10610356709660 z + 2051000359432 z - 2876070 z 48 20 36 + 95018213 z + 40290782455492 z + 40290782455492 z 34 30 42 - 113140491041900 z - 366583033537792 z - 287734486756 z 44 46 56 54 52 + 28820548588 z - 2014130436 z + z - 442 z + 51127 z 32 38 40 + 236003705895562 z - 10610356709660 z + 2051000359432 z )) And in Maple-input format, it is: -(1+77851677433128*z^28-67946846493126*z^26-281*z^2+45126526288898*z^24-\ 22736635897628*z^22+26548*z^4-1284121*z^6+37205893*z^8-699351908*z^10+ 8947341816*z^12-80453259996*z^14-2462915270700*z^18+520331719224*z^16-1284121*z ^50+37205893*z^48+8645871072896*z^20+8645871072896*z^36-22736635897628*z^34-\ 67946846493126*z^30-80453259996*z^42+8947341816*z^44-699351908*z^46+z^56-281*z^ 54+26548*z^52+45126526288898*z^32-2462915270700*z^38+520331719224*z^40)/(-1+z^2 )/(1+424498817364258*z^28-366583033537792*z^26-442*z^2+236003705895562*z^24-\ 113140491041900*z^22+51127*z^4-2876070*z^6+95018213*z^8-2014130436*z^10+ 28820548588*z^12-287734486756*z^14-10610356709660*z^18+2051000359432*z^16-\ 2876070*z^50+95018213*z^48+40290782455492*z^20+40290782455492*z^36-\ 113140491041900*z^34-366583033537792*z^30-287734486756*z^42+28820548588*z^44-\ 2014130436*z^46+z^56-442*z^54+51127*z^52+236003705895562*z^32-10610356709660*z^ 38+2051000359432*z^40) The first , 40, terms are: [0, 162, 0, 46745, 0, 13996933, 0, 4203565938, 0, 1262754619317, 0, 379342111272605, 0, 113957841250358354, 0, 34233986506317630973, 0, 10284205522058103992673, 0, 3089470268244474720854786, 0, 928105386479905415768574745, 0, 278811425145630187136335968489, 0, 83757525733977011298706091288514, 0, 25161533869766477642937607965305809, 0, 7558757032654302621886963332572078221, 0, 2270720385109467063433349027819331417874, 0, 682145363989966436445809836591348836156237, 0, 204922764011109857037072712702952460431274501, 0, 61560689886284552169869480463506655239901274546, 0, 18493399489135517055270239231759067567011713922037] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 126292246301600 z - 110624038175198 z - 326 z 24 22 4 6 + 74263143980568 z - 38073966352364 z + 37608 z - 2119453 z 8 10 12 14 + 67108480 z - 1310787909 z + 16855526176 z - 149358590574 z 18 16 50 48 - 4336759546384 z + 942378060809 z - 2119453 z + 67108480 z 20 36 34 + 14816868133750 z + 14816868133750 z - 38073966352364 z 30 42 44 - 110624038175198 z - 149358590574 z + 16855526176 z 46 56 54 52 32 - 1310787909 z + z - 326 z + 37608 z + 74263143980568 z 38 40 / 28 - 4336759546384 z + 942378060809 z ) / (-1 - 1288790271821186 z / 26 2 24 22 + 989200924682230 z + 487 z - 581645010780350 z + 260963487023654 z 4 6 8 10 12 - 70547 z + 4737167 z - 174814605 z + 3944990177 z - 58378761963 z 14 18 16 + 594159723551 z + 22689835257733 z - 4301181064803 z 50 48 20 + 174814605 z - 3944990177 z - 88777362666346 z 36 34 30 - 260963487023654 z + 581645010780350 z + 1288790271821186 z 42 44 46 58 56 + 4301181064803 z - 594159723551 z + 58378761963 z + z - 487 z 54 52 32 38 + 70547 z - 4737167 z - 989200924682230 z + 88777362666346 z 40 - 22689835257733 z ) And in Maple-input format, it is: -(1+126292246301600*z^28-110624038175198*z^26-326*z^2+74263143980568*z^24-\ 38073966352364*z^22+37608*z^4-2119453*z^6+67108480*z^8-1310787909*z^10+ 16855526176*z^12-149358590574*z^14-4336759546384*z^18+942378060809*z^16-2119453 *z^50+67108480*z^48+14816868133750*z^20+14816868133750*z^36-38073966352364*z^34 -110624038175198*z^30-149358590574*z^42+16855526176*z^44-1310787909*z^46+z^56-\ 326*z^54+37608*z^52+74263143980568*z^32-4336759546384*z^38+942378060809*z^40)/( -1-1288790271821186*z^28+989200924682230*z^26+487*z^2-581645010780350*z^24+ 260963487023654*z^22-70547*z^4+4737167*z^6-174814605*z^8+3944990177*z^10-\ 58378761963*z^12+594159723551*z^14+22689835257733*z^18-4301181064803*z^16+ 174814605*z^50-3944990177*z^48-88777362666346*z^20-260963487023654*z^36+ 581645010780350*z^34+1288790271821186*z^30+4301181064803*z^42-594159723551*z^44 +58378761963*z^46+z^58-487*z^56+70547*z^54-4737167*z^52-989200924682230*z^32+ 88777362666346*z^38-22689835257733*z^40) The first , 40, terms are: [0, 161, 0, 45468, 0, 13402563, 0, 3974394947, 0, 1179898287247, 0, 350364154444871, 0, 104043884209125372, 0, 30897135467896127725, 0, 9175313996065728980997, 0, 2724732621964812542822717, 0, 809146017371327975500711477, 0, 240286804851598067140570155964, 0, 71356402572936776007310727070879, 0, 21190244710984705451506199776980727, 0, 6292728541125679338593699160200365147, 0, 1868710486143686030063710059436918248747, 0, 554938745293634509871023454897416984747740, 0, 164796533926904959677420120548137369744826393, 0, 48938550109665307804783173092246013325409199817, 0, 14532961523930076014831788962487328724458330643001] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 10636521197120701938 z - 2457920843565738166 z - 395 z 24 22 4 6 + 470817539574963120 z - 74279438331511658 z + 61676 z - 5265765 z 8 10 12 14 + 282940223 z - 10386083866 z + 274920220642 z - 5450377691328 z 18 16 50 - 998612731184870 z + 83203303002878 z - 1123579213881453007214 z 48 20 + 1706807778305923959296 z + 9575401840378702 z 36 34 + 624818183463467132838 z - 293070566711697207232 z 66 80 88 84 86 - 74279438331511658 z + 282940223 z + z + 61676 z - 395 z 82 64 30 - 5265765 z + 470817539574963120 z - 38352707621748338962 z 42 44 - 2192573672771716372976 z + 2383313565326632771016 z 46 58 - 2192573672771716372976 z - 38352707621748338962 z 56 54 + 115709623293969835642 z - 293070566711697207232 z 52 60 + 624818183463467132838 z + 10636521197120701938 z 70 68 78 - 998612731184870 z + 9575401840378702 z - 10386083866 z 32 38 + 115709623293969835642 z - 1123579213881453007214 z 40 62 76 + 1706807778305923959296 z - 2457920843565738166 z + 274920220642 z 74 72 / - 5450377691328 z + 83203303002878 z ) / (-1 / 28 26 2 - 56905280797232859060 z + 12122985646265675560 z + 558 z 24 22 4 - 2140937865083367080 z + 311367395545045996 z - 106286 z 6 8 10 12 + 10492302 z - 634996298 z + 25873402329 z - 753538979492 z 14 18 16 + 16345589967856 z + 3551328687114876 z - 272035831706992 z 50 48 + 14931446154512073198382 z - 20834135380576128989416 z 20 36 - 36985891392904308 z - 4635980202049259937488 z 34 66 80 + 2003176036790306161360 z + 2140937865083367080 z - 25873402329 z 90 88 84 86 82 + z - 558 z - 10492302 z + 106286 z + 634996298 z 64 30 - 12122985646265675560 z + 222600354980891191684 z 42 44 + 20834135380576128989416 z - 24605848799690584166516 z 46 58 + 24605848799690584166516 z + 728722883484152562612 z 56 54 - 2003176036790306161360 z + 4635980202049259937488 z 52 60 - 9051376923320830511148 z - 222600354980891191684 z 70 68 78 + 36985891392904308 z - 311367395545045996 z + 753538979492 z 32 38 - 728722883484152562612 z + 9051376923320830511148 z 40 62 - 14931446154512073198382 z + 56905280797232859060 z 76 74 72 - 16345589967856 z + 272035831706992 z - 3551328687114876 z ) And in Maple-input format, it is: -(1+10636521197120701938*z^28-2457920843565738166*z^26-395*z^2+ 470817539574963120*z^24-74279438331511658*z^22+61676*z^4-5265765*z^6+282940223* z^8-10386083866*z^10+274920220642*z^12-5450377691328*z^14-998612731184870*z^18+ 83203303002878*z^16-1123579213881453007214*z^50+1706807778305923959296*z^48+ 9575401840378702*z^20+624818183463467132838*z^36-293070566711697207232*z^34-\ 74279438331511658*z^66+282940223*z^80+z^88+61676*z^84-395*z^86-5265765*z^82+ 470817539574963120*z^64-38352707621748338962*z^30-2192573672771716372976*z^42+ 2383313565326632771016*z^44-2192573672771716372976*z^46-38352707621748338962*z^ 58+115709623293969835642*z^56-293070566711697207232*z^54+624818183463467132838* z^52+10636521197120701938*z^60-998612731184870*z^70+9575401840378702*z^68-\ 10386083866*z^78+115709623293969835642*z^32-1123579213881453007214*z^38+ 1706807778305923959296*z^40-2457920843565738166*z^62+274920220642*z^76-\ 5450377691328*z^74+83203303002878*z^72)/(-1-56905280797232859060*z^28+ 12122985646265675560*z^26+558*z^2-2140937865083367080*z^24+311367395545045996*z ^22-106286*z^4+10492302*z^6-634996298*z^8+25873402329*z^10-753538979492*z^12+ 16345589967856*z^14+3551328687114876*z^18-272035831706992*z^16+ 14931446154512073198382*z^50-20834135380576128989416*z^48-36985891392904308*z^ 20-4635980202049259937488*z^36+2003176036790306161360*z^34+2140937865083367080* z^66-25873402329*z^80+z^90-558*z^88-10492302*z^84+106286*z^86+634996298*z^82-\ 12122985646265675560*z^64+222600354980891191684*z^30+20834135380576128989416*z^ 42-24605848799690584166516*z^44+24605848799690584166516*z^46+ 728722883484152562612*z^58-2003176036790306161360*z^56+4635980202049259937488*z ^54-9051376923320830511148*z^52-222600354980891191684*z^60+36985891392904308*z^ 70-311367395545045996*z^68+753538979492*z^78-728722883484152562612*z^32+ 9051376923320830511148*z^38-14931446154512073198382*z^40+56905280797232859060*z ^62-16345589967856*z^76+272035831706992*z^74-3551328687114876*z^72) The first , 40, terms are: [0, 163, 0, 46344, 0, 13761871, 0, 4111594785, 0, 1229813834701, 0, 367935340447955, 0, 110084429098374424, 0, 32937086445504725959, 0, 9854748652383993974877, 0, 2948534185917938505766469, 0, 882199561460467776353907855, 0, 263953557688339459484498021432, 0, 78974740078986245589964919173179, 0, 23629193083298043049262566005482517, 0, 7069839867716853490898855841394711513, 0, 2115291689510334117400930061449749504967, 0, 632893957361100501576648799714922852927208, 0, 189361478254800197826467366495151103821945851, 0, 56656836472869972814722982810468140950075317017, 0, 16951690220726388982052135672401175132188067828073] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1927740328806778 z - 1086314610476554 z - 309 z 24 22 4 6 + 484759802482918 z - 170577971298216 z + 33566 z - 1843713 z 8 10 12 14 + 60419349 z - 1296628326 z + 19325071498 z - 208096658488 z 18 16 50 - 10091115671468 z + 1664575807048 z - 208096658488 z 48 20 36 + 1664575807048 z + 47050647431664 z + 1927740328806778 z 34 64 30 42 - 2716657694733378 z + z - 2716657694733378 z - 170577971298216 z 44 46 58 56 + 47050647431664 z - 10091115671468 z - 1843713 z + 60419349 z 54 52 60 32 - 1296628326 z + 19325071498 z + 33566 z + 3045232107173180 z 38 40 62 / - 1086314610476554 z + 484759802482918 z - 309 z ) / (-1 / 28 26 2 - 15616700537715443 z + 7867755238793915 z + 476 z 24 22 4 6 - 3140233044853525 z + 988602826333547 z - 63419 z + 4071538 z 8 10 12 14 - 153395977 z + 3751085692 z - 63290810165 z + 767703140571 z 18 16 50 + 46788171067935 z - 6892464370325 z + 6892464370325 z 48 20 36 - 46788171067935 z - 243964059197617 z - 24629031207198523 z 34 66 64 30 + 30916871713100067 z + z - 476 z + 24629031207198523 z 42 44 46 + 3140233044853525 z - 988602826333547 z + 243964059197617 z 58 56 54 52 + 153395977 z - 3751085692 z + 63290810165 z - 767703140571 z 60 32 38 - 4071538 z - 30916871713100067 z + 15616700537715443 z 40 62 - 7867755238793915 z + 63419 z ) And in Maple-input format, it is: -(1+1927740328806778*z^28-1086314610476554*z^26-309*z^2+484759802482918*z^24-\ 170577971298216*z^22+33566*z^4-1843713*z^6+60419349*z^8-1296628326*z^10+ 19325071498*z^12-208096658488*z^14-10091115671468*z^18+1664575807048*z^16-\ 208096658488*z^50+1664575807048*z^48+47050647431664*z^20+1927740328806778*z^36-\ 2716657694733378*z^34+z^64-2716657694733378*z^30-170577971298216*z^42+ 47050647431664*z^44-10091115671468*z^46-1843713*z^58+60419349*z^56-1296628326*z ^54+19325071498*z^52+33566*z^60+3045232107173180*z^32-1086314610476554*z^38+ 484759802482918*z^40-309*z^62)/(-1-15616700537715443*z^28+7867755238793915*z^26 +476*z^2-3140233044853525*z^24+988602826333547*z^22-63419*z^4+4071538*z^6-\ 153395977*z^8+3751085692*z^10-63290810165*z^12+767703140571*z^14+46788171067935 *z^18-6892464370325*z^16+6892464370325*z^50-46788171067935*z^48-243964059197617 *z^20-24629031207198523*z^36+30916871713100067*z^34+z^66-476*z^64+ 24629031207198523*z^30+3140233044853525*z^42-988602826333547*z^44+ 243964059197617*z^46+153395977*z^58-3751085692*z^56+63290810165*z^54-\ 767703140571*z^52-4071538*z^60-30916871713100067*z^32+15616700537715443*z^38-\ 7867755238793915*z^40+63419*z^62) The first , 40, terms are: [0, 167, 0, 49639, 0, 15265016, 0, 4705062093, 0, 1450461910553, 0, 447149671931463, 0, 137847837178633799, 0, 42495900311691859697, 0, 13100688370958371813581, 0, 4038696313418583076042008, 0, 1245054263678441125637119543, 0, 383826858773631411466354737215, 0, 118326615806282882595123686294929, 0, 36477874562781478755952921348925329, 0, 11245444007259596856892516591124185519, 0, 3466759300977456077413754721736507418679, 0, 1068736818497794839487328710156828867402392, 0, 329471500052150951580468636914834564414152685, 0, 101569879008372992231627898138977744630557971105, 0, 31312087146058380002098017351089349764313209247399] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1388608866048586329164 z - 186984140915950107036 z - 440 z 24 22 4 + 21455011304919019000 z - 2083649413618492004 z + 80362 z 6 102 8 10 - 8408916 z - 28187194112 z + 578010043 z - 28187194112 z 12 14 18 + 1024937469286 z - 28776784192916 z - 11510354420072420 z 16 50 + 640188234644101 z - 396785038575903277203540884 z 48 20 + 250121063351157531393397858 z + 169875755361853616 z 36 34 + 940015261662193041477316 z - 229108132536356972207140 z 66 80 - 138086760046847818542788124 z + 48403095796408789120724 z 100 90 88 + 1024937469286 z - 2083649413618492004 z + 21455011304919019000 z 84 94 + 1388608866048586329164 z - 11510354420072420 z 86 96 98 - 186984140915950107036 z + 640188234644101 z - 28776784192916 z 92 82 + 169875755361853616 z - 8836293998285841686620 z 64 112 110 106 + 250121063351157531393397858 z + z - 440 z - 8408916 z 108 30 42 + 80362 z - 8836293998285841686620 z - 28195734720448261324674460 z 44 46 + 66724397764654608520037456 z - 138086760046847818542788124 z 58 56 - 671928323044625213050156204 z + 717628240292948555554939874 z 54 52 - 671928323044625213050156204 z + 551528094033773039250040352 z 60 70 + 551528094033773039250040352 z - 28195734720448261324674460 z 68 78 + 66724397764654608520037456 z - 229108132536356972207140 z 32 38 + 48403095796408789120724 z - 3351784266543253269976580 z 40 62 + 10408270272320838828875040 z - 396785038575903277203540884 z 76 74 + 940015261662193041477316 z - 3351784266543253269976580 z 72 104 / + 10408270272320838828875040 z + 578010043 z ) / (-1 / 28 26 2 - 5847982694693554844933 z + 738811998921712891267 z + 600 z 24 22 4 - 79511820409899871645 z + 7239051999602698735 z - 131595 z 6 102 8 10 + 15701267 z + 2519856832440 z - 1198546866 z + 63936020261 z 12 14 18 - 2519856832440 z + 76237203667610 z + 35047259740506656 z 16 50 - 1820588338606725 z + 3386293499037905488076054839 z 48 20 - 2000134074731524614841570579 z - 552840263037433153 z 36 34 - 5107019862917018829930149 z + 1167812258128201662763775 z 66 80 + 2000134074731524614841570579 z - 1167812258128201662763775 z 100 90 - 76237203667610 z + 79511820409899871645 z 88 84 - 738811998921712891267 z - 39658042747492409463359 z 94 86 + 552840263037433153 z + 5847982694693554844933 z 96 98 92 - 35047259740506656 z + 1820588338606725 z - 7239051999602698735 z 82 64 + 231503397994220744362501 z - 3386293499037905488076054839 z 112 114 110 106 108 - 600 z + z + 131595 z + 1198546866 z - 15701267 z 30 42 + 39658042747492409463359 z + 185683960702842551803954287 z 44 46 - 468704879747764285705648397 z + 1034842857852555504789841859 z 58 56 + 7452219106565346410079189797 z - 7452219106565346410079189797 z 54 52 + 6534718100245896885944064785 z - 5024216309889940336662414767 z 60 70 - 6534718100245896885944064785 z + 468704879747764285705648397 z 68 78 - 1034842857852555504789841859 z + 5107019862917018829930149 z 32 38 - 231503397994220744362501 z + 19412379315719270813739943 z 40 62 - 64273555977331945471740009 z + 5024216309889940336662414767 z 76 74 - 19412379315719270813739943 z + 64273555977331945471740009 z 72 104 - 185683960702842551803954287 z - 63936020261 z ) And in Maple-input format, it is: -(1+1388608866048586329164*z^28-186984140915950107036*z^26-440*z^2+ 21455011304919019000*z^24-2083649413618492004*z^22+80362*z^4-8408916*z^6-\ 28187194112*z^102+578010043*z^8-28187194112*z^10+1024937469286*z^12-\ 28776784192916*z^14-11510354420072420*z^18+640188234644101*z^16-\ 396785038575903277203540884*z^50+250121063351157531393397858*z^48+ 169875755361853616*z^20+940015261662193041477316*z^36-229108132536356972207140* z^34-138086760046847818542788124*z^66+48403095796408789120724*z^80+ 1024937469286*z^100-2083649413618492004*z^90+21455011304919019000*z^88+ 1388608866048586329164*z^84-11510354420072420*z^94-186984140915950107036*z^86+ 640188234644101*z^96-28776784192916*z^98+169875755361853616*z^92-\ 8836293998285841686620*z^82+250121063351157531393397858*z^64+z^112-440*z^110-\ 8408916*z^106+80362*z^108-8836293998285841686620*z^30-\ 28195734720448261324674460*z^42+66724397764654608520037456*z^44-\ 138086760046847818542788124*z^46-671928323044625213050156204*z^58+ 717628240292948555554939874*z^56-671928323044625213050156204*z^54+ 551528094033773039250040352*z^52+551528094033773039250040352*z^60-\ 28195734720448261324674460*z^70+66724397764654608520037456*z^68-\ 229108132536356972207140*z^78+48403095796408789120724*z^32-\ 3351784266543253269976580*z^38+10408270272320838828875040*z^40-\ 396785038575903277203540884*z^62+940015261662193041477316*z^76-\ 3351784266543253269976580*z^74+10408270272320838828875040*z^72+578010043*z^104) /(-1-5847982694693554844933*z^28+738811998921712891267*z^26+600*z^2-\ 79511820409899871645*z^24+7239051999602698735*z^22-131595*z^4+15701267*z^6+ 2519856832440*z^102-1198546866*z^8+63936020261*z^10-2519856832440*z^12+ 76237203667610*z^14+35047259740506656*z^18-1820588338606725*z^16+ 3386293499037905488076054839*z^50-2000134074731524614841570579*z^48-\ 552840263037433153*z^20-5107019862917018829930149*z^36+ 1167812258128201662763775*z^34+2000134074731524614841570579*z^66-\ 1167812258128201662763775*z^80-76237203667610*z^100+79511820409899871645*z^90-\ 738811998921712891267*z^88-39658042747492409463359*z^84+552840263037433153*z^94 +5847982694693554844933*z^86-35047259740506656*z^96+1820588338606725*z^98-\ 7239051999602698735*z^92+231503397994220744362501*z^82-\ 3386293499037905488076054839*z^64-600*z^112+z^114+131595*z^110+1198546866*z^106 -15701267*z^108+39658042747492409463359*z^30+185683960702842551803954287*z^42-\ 468704879747764285705648397*z^44+1034842857852555504789841859*z^46+ 7452219106565346410079189797*z^58-7452219106565346410079189797*z^56+ 6534718100245896885944064785*z^54-5024216309889940336662414767*z^52-\ 6534718100245896885944064785*z^60+468704879747764285705648397*z^70-\ 1034842857852555504789841859*z^68+5107019862917018829930149*z^78-\ 231503397994220744362501*z^32+19412379315719270813739943*z^38-\ 64273555977331945471740009*z^40+5024216309889940336662414767*z^62-\ 19412379315719270813739943*z^76+64273555977331945471740009*z^74-\ 185683960702842551803954287*z^72-63936020261*z^104) The first , 40, terms are: [0, 160, 0, 44767, 0, 13097351, 0, 3858963132, 0, 1138711921733, 0, 336131401056361, 0, 99229373923407224, 0, 29294088003943742111, 0, 8648121060173073701047, 0, 2553077683758479931728028, 0, 753713711869506398268683637, 0, 222509640033128536059388162877, 0, 65688788595348367163801523154820, 0, 19392494467455917593295816094366231, 0, 5725008027252874896143054060893075695, 0, 1690123824668450717040529574233259720240, 0, 498954504388881734160331301892111597860337, 0, 147300211865778217721664756734133782572197437, 0, 43485632908240971103341223272784165202364717796, 0, 12837729460666054816573807980329322908651638116055] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 809488458808 z + 1655017555290 z + 334 z 24 22 4 6 - 2359966332303 z + 2359966332303 z - 38208 z + 2038309 z 8 10 12 14 - 57397604 z + 940040426 z - 9543787766 z + 62615000756 z 18 16 20 36 + 809488458808 z - 273181125171 z - 1655017555290 z - 940040426 z 34 30 42 44 46 + 9543787766 z + 273181125171 z + 38208 z - 334 z + z 32 38 40 / - 62615000756 z + 57397604 z - 2038309 z ) / (1 / 28 26 2 24 + 12857388814169 z - 21913914044382 z - 502 z + 26164327620806 z 22 4 6 8 - 21913914044382 z + 74723 z - 4864534 z + 163066471 z 10 12 14 18 - 3138840052 z + 37346644292 z - 288303556128 z - 5261695402242 z 16 48 20 36 + 1490102166453 z + z + 12857388814169 z + 37346644292 z 34 30 42 44 46 - 288303556128 z - 5261695402242 z - 4864534 z + 74723 z - 502 z 32 38 40 + 1490102166453 z - 3138840052 z + 163066471 z ) And in Maple-input format, it is: -(-1-809488458808*z^28+1655017555290*z^26+334*z^2-2359966332303*z^24+ 2359966332303*z^22-38208*z^4+2038309*z^6-57397604*z^8+940040426*z^10-9543787766 *z^12+62615000756*z^14+809488458808*z^18-273181125171*z^16-1655017555290*z^20-\ 940040426*z^36+9543787766*z^34+273181125171*z^30+38208*z^42-334*z^44+z^46-\ 62615000756*z^32+57397604*z^38-2038309*z^40)/(1+12857388814169*z^28-\ 21913914044382*z^26-502*z^2+26164327620806*z^24-21913914044382*z^22+74723*z^4-\ 4864534*z^6+163066471*z^8-3138840052*z^10+37346644292*z^12-288303556128*z^14-\ 5261695402242*z^18+1490102166453*z^16+z^48+12857388814169*z^20+37346644292*z^36 -288303556128*z^34-5261695402242*z^30-4864534*z^42+74723*z^44-502*z^46+ 1490102166453*z^32-3138840052*z^38+163066471*z^40) The first , 40, terms are: [0, 168, 0, 47821, 0, 14278903, 0, 4306253568, 0, 1302207335179, 0, 394093626586915, 0, 119293722334687920, 0, 36113097293856732799, 0, 10932522431700078618517, 0, 3309622845669643627050072, 0, 1001929962445838328400152025, 0, 303316779920911122468079377961, 0, 91823865660949167856158182811384, 0, 27798075214558879239750263438777605, 0, 8415382969006058020037460533717015407, 0, 2547610589089695641121228860989419316560, 0, 771244724628155304877931603553893700766195, 0, 233480904813440299071403690381769382621523003, 0, 70682276554681253269198154114103547329949298784, 0, 21397827899797047629122165688846524691254118963719] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 11685385544872349866 z - 2665410025182437065 z - 389 z 24 22 4 6 + 503445913970799442 z - 78256998918475183 z + 59775 z - 5063084 z 8 10 12 14 + 272177279 z - 10056770345 z + 269026588264 z - 5403616466831 z 18 16 50 - 1020054564180943 z + 83697982758884 z - 1290422409126082157123 z 48 20 + 1968455559898885427133 z + 9934308074949750 z 36 34 + 713439847445244345691 z - 332166418743046216366 z 66 80 88 84 86 - 78256998918475183 z + 272177279 z + z + 59775 z - 389 z 82 64 30 - 5063084 z + 503445913970799442 z - 42634257102643760249 z 42 44 - 2535049204759126983334 z + 2757894628570898748540 z 46 58 - 2535049204759126983334 z - 42634257102643760249 z 56 54 + 129975410860818442285 z - 332166418743046216366 z 52 60 + 713439847445244345691 z + 11685385544872349866 z 70 68 78 - 1020054564180943 z + 9934308074949750 z - 10056770345 z 32 38 + 129975410860818442285 z - 1290422409126082157123 z 40 62 76 + 1968455559898885427133 z - 2665410025182437065 z + 269026588264 z 74 72 / 2 - 5403616466831 z + 83697982758884 z ) / ((-1 + z ) (1 / 28 26 2 + 51158779474724597630 z - 11092259991176337188 z - 560 z 24 22 4 + 1983652428160534371 z - 290863662114054944 z + 103924 z 6 8 10 12 - 10041588 z + 600110386 z - 24311473432 z + 706986539966 z 14 18 16 - 15345274458508 z - 3338300697907860 z + 255662224601441 z 50 48 - 6760274658355600851120 z + 10497685096995743105379 z 20 36 + 34707834823781564 z + 3647298615357706104888 z 34 66 80 - 1646851465678139943960 z - 290863662114054944 z + 600110386 z 88 84 86 82 64 + z + 103924 z - 560 z - 10041588 z + 1983652428160534371 z 30 42 - 195514020802439480196 z - 13666558893547198026836 z 44 46 + 14922090700824469980760 z - 13666558893547198026836 z 58 56 - 195514020802439480196 z + 621379057158890757238 z 54 52 - 1646851465678139943960 z + 3647298615357706104888 z 60 70 68 + 51158779474724597630 z - 3338300697907860 z + 34707834823781564 z 78 32 38 - 24311473432 z + 621379057158890757238 z - 6760274658355600851120 z 40 62 + 10497685096995743105379 z - 11092259991176337188 z 76 74 72 + 706986539966 z - 15345274458508 z + 255662224601441 z )) And in Maple-input format, it is: -(1+11685385544872349866*z^28-2665410025182437065*z^26-389*z^2+ 503445913970799442*z^24-78256998918475183*z^22+59775*z^4-5063084*z^6+272177279* z^8-10056770345*z^10+269026588264*z^12-5403616466831*z^14-1020054564180943*z^18 +83697982758884*z^16-1290422409126082157123*z^50+1968455559898885427133*z^48+ 9934308074949750*z^20+713439847445244345691*z^36-332166418743046216366*z^34-\ 78256998918475183*z^66+272177279*z^80+z^88+59775*z^84-389*z^86-5063084*z^82+ 503445913970799442*z^64-42634257102643760249*z^30-2535049204759126983334*z^42+ 2757894628570898748540*z^44-2535049204759126983334*z^46-42634257102643760249*z^ 58+129975410860818442285*z^56-332166418743046216366*z^54+713439847445244345691* z^52+11685385544872349866*z^60-1020054564180943*z^70+9934308074949750*z^68-\ 10056770345*z^78+129975410860818442285*z^32-1290422409126082157123*z^38+ 1968455559898885427133*z^40-2665410025182437065*z^62+269026588264*z^76-\ 5403616466831*z^74+83697982758884*z^72)/(-1+z^2)/(1+51158779474724597630*z^28-\ 11092259991176337188*z^26-560*z^2+1983652428160534371*z^24-290863662114054944*z ^22+103924*z^4-10041588*z^6+600110386*z^8-24311473432*z^10+706986539966*z^12-\ 15345274458508*z^14-3338300697907860*z^18+255662224601441*z^16-\ 6760274658355600851120*z^50+10497685096995743105379*z^48+34707834823781564*z^20 +3647298615357706104888*z^36-1646851465678139943960*z^34-290863662114054944*z^ 66+600110386*z^80+z^88+103924*z^84-560*z^86-10041588*z^82+1983652428160534371*z ^64-195514020802439480196*z^30-13666558893547198026836*z^42+ 14922090700824469980760*z^44-13666558893547198026836*z^46-195514020802439480196 *z^58+621379057158890757238*z^56-1646851465678139943960*z^54+ 3647298615357706104888*z^52+51158779474724597630*z^60-3338300697907860*z^70+ 34707834823781564*z^68-24311473432*z^78+621379057158890757238*z^32-\ 6760274658355600851120*z^38+10497685096995743105379*z^40-11092259991176337188*z ^62+706986539966*z^76-15345274458508*z^74+255662224601441*z^72) The first , 40, terms are: [0, 172, 0, 51783, 0, 16161443, 0, 5063127920, 0, 1587076274549, 0, 497527067644445, 0, 155970558130387168, 0, 48895600884996647079, 0, 15328412313345231302639, 0, 4805345304054666028018764, 0, 1506440681132723757620428405, 0, 472258160293190156545331682965, 0, 148049487035697151203460405603980, 0, 46412433826949549789031698185373295, 0, 14549959320449689875790771670511183991, 0, 4561306071921081639457370321631129333200, 0, 1429936168447988506628071460350113060745501, 0, 448274554172768583278306272314538829797954165, 0, 140530801551026172430514476089838316596964925216, 0, 44055380794517722306445988768269373711103337489523] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 58147973383029516 z - 26565792875861534 z - 387 z 24 22 4 6 + 9624279777529770 z - 2747218037732152 z + 56472 z - 4191513 z 8 10 12 14 + 183447287 z - 5178387352 z + 100017921423 z - 1377245922449 z 18 16 50 - 105736333013131 z + 13927529752180 z - 105736333013131 z 48 20 36 + 612830679860905 z + 612830679860905 z + 141401522291195542 z 34 66 64 30 - 157935453926601776 z - 387 z + 56472 z - 101412777227455498 z 42 44 46 - 26565792875861534 z + 9624279777529770 z - 2747218037732152 z 58 56 54 - 5178387352 z + 100017921423 z - 1377245922449 z 52 60 68 32 + 13927529752180 z + 183447287 z + z + 141401522291195542 z 38 40 62 / - 101412777227455498 z + 58147973383029516 z - 4191513 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 310159268399378946 z - 135089615544050524 z - 564 z 24 22 4 6 + 46148695523588486 z - 12308130069019604 z + 104539 z - 9059066 z 8 10 12 14 + 446001277 z - 13929070994 z + 295389561529 z - 4448926821266 z 18 16 50 - 405414593316088 z + 49082566435255 z - 405414593316088 z 48 20 36 + 2546804972996661 z + 2546804972996661 z + 798460460077152198 z 34 66 64 30 - 898366110311543124 z - 564 z + 104539 z - 560373929769228556 z 42 44 46 - 135089615544050524 z + 46148695523588486 z - 12308130069019604 z 58 56 54 - 13929070994 z + 295389561529 z - 4448926821266 z 52 60 68 32 + 49082566435255 z + 446001277 z + z + 798460460077152198 z 38 40 62 - 560373929769228556 z + 310159268399378946 z - 9059066 z )) And in Maple-input format, it is: -(1+58147973383029516*z^28-26565792875861534*z^26-387*z^2+9624279777529770*z^24 -2747218037732152*z^22+56472*z^4-4191513*z^6+183447287*z^8-5178387352*z^10+ 100017921423*z^12-1377245922449*z^14-105736333013131*z^18+13927529752180*z^16-\ 105736333013131*z^50+612830679860905*z^48+612830679860905*z^20+ 141401522291195542*z^36-157935453926601776*z^34-387*z^66+56472*z^64-\ 101412777227455498*z^30-26565792875861534*z^42+9624279777529770*z^44-\ 2747218037732152*z^46-5178387352*z^58+100017921423*z^56-1377245922449*z^54+ 13927529752180*z^52+183447287*z^60+z^68+141401522291195542*z^32-\ 101412777227455498*z^38+58147973383029516*z^40-4191513*z^62)/(-1+z^2)/(1+ 310159268399378946*z^28-135089615544050524*z^26-564*z^2+46148695523588486*z^24-\ 12308130069019604*z^22+104539*z^4-9059066*z^6+446001277*z^8-13929070994*z^10+ 295389561529*z^12-4448926821266*z^14-405414593316088*z^18+49082566435255*z^16-\ 405414593316088*z^50+2546804972996661*z^48+2546804972996661*z^20+ 798460460077152198*z^36-898366110311543124*z^34-564*z^66+104539*z^64-\ 560373929769228556*z^30-135089615544050524*z^42+46148695523588486*z^44-\ 12308130069019604*z^46-13929070994*z^58+295389561529*z^56-4448926821266*z^54+ 49082566435255*z^52+446001277*z^60+z^68+798460460077152198*z^32-\ 560373929769228556*z^38+310159268399378946*z^40-9059066*z^62) The first , 40, terms are: [0, 178, 0, 51939, 0, 15609293, 0, 4719814462, 0, 1430256072811, 0, 433779500117955, 0, 131603493955032774, 0, 39932072494685491609, 0, 12117086775483862720907, 0, 3676911461324143863296514, 0, 1115761798966275369473295109, 0, 338579858978687291786726469861, 0, 102742769092064276647619263782338, 0, 31177523475968882470700045186210091, 0, 9460891071463254473190969586989416873, 0, 2870929318783976530381128165252211693798, 0, 871190178677978699937863371460555452601347, 0, 264364688720642910687231225951172904727119675, 0, 80222080839657760673768450922741642527984762942, 0, 24343577412474417741769204647651828532299114981869] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 18517304508608313997 z - 4128690227167048791 z - 399 z 24 22 4 6 + 760199039150437969 z - 114884198750367848 z + 63637 z - 5592314 z 8 10 12 14 + 311843010 z - 11959361198 z + 332204501431 z - 6927539647317 z 18 16 50 - 1404908764358376 z + 111302550705763 z - 2199864863629898770181 z 48 20 + 3377980841660245842219 z + 14142521570943188 z 36 34 + 1205047516587684529031 z - 554409052568855977294 z 66 80 88 84 86 - 114884198750367848 z + 311843010 z + z + 63637 z - 399 z 82 64 30 - 5592314 z + 760199039150437969 z - 68926809531497017066 z 42 44 - 4367550744507351389264 z + 4757759962392819925208 z 46 58 - 4367550744507351389264 z - 68926809531497017066 z 56 54 + 213796359369036550398 z - 554409052568855977294 z 52 60 + 1205047516587684529031 z + 18517304508608313997 z 70 68 78 - 1404908764358376 z + 14142521570943188 z - 11959361198 z 32 38 + 213796359369036550398 z - 2199864863629898770181 z 40 62 76 + 3377980841660245842219 z - 4128690227167048791 z + 332204501431 z 74 72 / 2 - 6927539647317 z + 111302550705763 z ) / ((-1 + z ) (1 / 28 26 2 + 80569241012911741284 z - 17032221884299008172 z - 576 z 24 22 4 + 2963223107863116211 z - 421839125599506960 z + 110894 z 6 8 10 12 - 11079592 z + 684512155 z - 28701212760 z + 864759160736 z 14 18 16 - 19454154483828 z - 4538797823391752 z + 335814927985957 z 50 48 - 11624342069938300915424 z + 18206244132316732184839 z 20 36 + 48778310874942806 z + 6197952477266565247406 z 34 66 80 - 2757227276581242471840 z - 421839125599506960 z + 684512155 z 88 84 86 82 64 + z + 110894 z - 576 z - 11079592 z + 2963223107863116211 z 30 42 - 315067098878786893056 z - 23825691969763859990152 z 44 46 + 26059956359085322475732 z - 23825691969763859990152 z 58 56 - 315067098878786893056 z + 1022030669223876883365 z 54 52 - 2757227276581242471840 z + 6197952477266565247406 z 60 70 68 + 80569241012911741284 z - 4538797823391752 z + 48778310874942806 z 78 32 - 28701212760 z + 1022030669223876883365 z 38 40 - 11624342069938300915424 z + 18206244132316732184839 z 62 76 74 - 17032221884299008172 z + 864759160736 z - 19454154483828 z 72 + 335814927985957 z )) And in Maple-input format, it is: -(1+18517304508608313997*z^28-4128690227167048791*z^26-399*z^2+ 760199039150437969*z^24-114884198750367848*z^22+63637*z^4-5592314*z^6+311843010 *z^8-11959361198*z^10+332204501431*z^12-6927539647317*z^14-1404908764358376*z^ 18+111302550705763*z^16-2199864863629898770181*z^50+3377980841660245842219*z^48 +14142521570943188*z^20+1205047516587684529031*z^36-554409052568855977294*z^34-\ 114884198750367848*z^66+311843010*z^80+z^88+63637*z^84-399*z^86-5592314*z^82+ 760199039150437969*z^64-68926809531497017066*z^30-4367550744507351389264*z^42+ 4757759962392819925208*z^44-4367550744507351389264*z^46-68926809531497017066*z^ 58+213796359369036550398*z^56-554409052568855977294*z^54+1205047516587684529031 *z^52+18517304508608313997*z^60-1404908764358376*z^70+14142521570943188*z^68-\ 11959361198*z^78+213796359369036550398*z^32-2199864863629898770181*z^38+ 3377980841660245842219*z^40-4128690227167048791*z^62+332204501431*z^76-\ 6927539647317*z^74+111302550705763*z^72)/(-1+z^2)/(1+80569241012911741284*z^28-\ 17032221884299008172*z^26-576*z^2+2963223107863116211*z^24-421839125599506960*z ^22+110894*z^4-11079592*z^6+684512155*z^8-28701212760*z^10+864759160736*z^12-\ 19454154483828*z^14-4538797823391752*z^18+335814927985957*z^16-\ 11624342069938300915424*z^50+18206244132316732184839*z^48+48778310874942806*z^ 20+6197952477266565247406*z^36-2757227276581242471840*z^34-421839125599506960*z ^66+684512155*z^80+z^88+110894*z^84-576*z^86-11079592*z^82+2963223107863116211* z^64-315067098878786893056*z^30-23825691969763859990152*z^42+ 26059956359085322475732*z^44-23825691969763859990152*z^46-315067098878786893056 *z^58+1022030669223876883365*z^56-2757227276581242471840*z^54+ 6197952477266565247406*z^52+80569241012911741284*z^60-4538797823391752*z^70+ 48778310874942806*z^68-28701212760*z^78+1022030669223876883365*z^32-\ 11624342069938300915424*z^38+18206244132316732184839*z^40-17032221884299008172* z^62+864759160736*z^76-19454154483828*z^74+335814927985957*z^72) The first , 40, terms are: [0, 178, 0, 54873, 0, 17418233, 0, 5541784902, 0, 1763666026973, 0, 561311224298393, 0, 178646849942987526, 0, 56857523739064007433, 0, 18095922755884618701525, 0, 5759351369963107036577322, 0, 1833016724579898863112250921, 0, 583390405730598176784495431049, 0, 185674446608500327219758974667242, 0, 59094218546389435951633973318422893, 0, 18807793585195228447013296273204060137, 0, 5985917205613420575516882211363996839910, 0, 1905125374238399579363073326743617061422561, 0, 606340276167951923978196503636411287624895941, 0, 192978654042899244498163534570056871348297078246, 0, 61418913405474093478313202054561864053588115685449] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 74707707987541828 z - 33532243510754062 z - 375 z 24 22 4 6 + 11878807253126298 z - 3301415993902784 z + 53494 z - 3965521 z 8 10 12 14 + 176134895 z - 5097678344 z + 101571011899 z - 1447519351149 z 18 16 50 - 119241774380283 z + 15166695677446 z - 119241774380283 z 48 20 36 + 714429707445957 z + 714429707445957 z + 185422740035304886 z 34 66 64 30 - 207636032381152368 z - 375 z + 53494 z - 131966583346140994 z 42 44 46 - 33532243510754062 z + 11878807253126298 z - 3301415993902784 z 58 56 54 - 5097678344 z + 101571011899 z - 1447519351149 z 52 60 68 32 + 15166695677446 z + 176134895 z + z + 185422740035304886 z 38 40 62 / - 131966583346140994 z + 74707707987541828 z - 3965521 z ) / (-1 / 28 26 2 - 562513736390663338 z + 224752092510443806 z + 565 z 24 22 4 6 - 70888729678355110 z + 17557893629154425 z - 99847 z + 8592887 z 8 10 12 14 - 432130857 z + 14010477581 z - 311335709349 z + 4941816559649 z 18 16 50 + 506227701636559 z - 57697472192751 z + 3391407720299981 z 48 20 36 - 17557893629154425 z - 3391407720299981 z - 2207153475103646318 z 34 66 64 + 2207153475103646318 z + 99847 z - 8592887 z 30 42 + 1115740777713530266 z + 562513736390663338 z 44 46 58 - 224752092510443806 z + 70888729678355110 z + 311335709349 z 56 54 52 - 4941816559649 z + 57697472192751 z - 506227701636559 z 60 70 68 32 - 14010477581 z + z - 565 z - 1758747350906005462 z 38 40 62 + 1758747350906005462 z - 1115740777713530266 z + 432130857 z ) And in Maple-input format, it is: -(1+74707707987541828*z^28-33532243510754062*z^26-375*z^2+11878807253126298*z^ 24-3301415993902784*z^22+53494*z^4-3965521*z^6+176134895*z^8-5097678344*z^10+ 101571011899*z^12-1447519351149*z^14-119241774380283*z^18+15166695677446*z^16-\ 119241774380283*z^50+714429707445957*z^48+714429707445957*z^20+ 185422740035304886*z^36-207636032381152368*z^34-375*z^66+53494*z^64-\ 131966583346140994*z^30-33532243510754062*z^42+11878807253126298*z^44-\ 3301415993902784*z^46-5097678344*z^58+101571011899*z^56-1447519351149*z^54+ 15166695677446*z^52+176134895*z^60+z^68+185422740035304886*z^32-\ 131966583346140994*z^38+74707707987541828*z^40-3965521*z^62)/(-1-\ 562513736390663338*z^28+224752092510443806*z^26+565*z^2-70888729678355110*z^24+ 17557893629154425*z^22-99847*z^4+8592887*z^6-432130857*z^8+14010477581*z^10-\ 311335709349*z^12+4941816559649*z^14+506227701636559*z^18-57697472192751*z^16+ 3391407720299981*z^50-17557893629154425*z^48-3391407720299981*z^20-\ 2207153475103646318*z^36+2207153475103646318*z^34+99847*z^66-8592887*z^64+ 1115740777713530266*z^30+562513736390663338*z^42-224752092510443806*z^44+ 70888729678355110*z^46+311335709349*z^58-4941816559649*z^56+57697472192751*z^54 -506227701636559*z^52-14010477581*z^60+z^70-565*z^68-1758747350906005462*z^32+ 1758747350906005462*z^38-1115740777713530266*z^40+432130857*z^62) The first , 40, terms are: [0, 190, 0, 60997, 0, 20119741, 0, 6653938774, 0, 2201527892429, 0, 728467635495585, 0, 241049379262040110, 0, 79763492474204073433, 0, 26393859543487605959441, 0, 8733770793429504720477110, 0, 2890019124327863252685596997, 0, 956312103573716166531411977213, 0, 316445256955833529579252311490502, 0, 104712259117133762388620653287017849, 0, 34649459810653764526237045298612684641, 0, 11465563587260179293899121073562620392702, 0, 3793973963655235814265334508090781846725881, 0, 1255432262655247878481072848171446449389651701, 0, 415424613140818707709117258125399645434979746694, 0, 137464691912771828121555530552761457377762895613749] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1529456215957530476560 z - 205569208611228380783 z - 443 z 24 22 4 + 23508451580620852704 z - 2272119735587278513 z + 81652 z 6 102 8 10 - 8615097 z - 29334872303 z + 596807640 z - 29334872303 z 12 14 18 + 1075538801044 z - 30456421925277 z - 12384962685993291 z 16 50 + 683321812361216 z - 420552110097704623245682503 z 48 20 + 266074502467379034581830912 z + 184109737144460412 z 36 34 + 1028825086948164789819212 z - 251598466376993238416559 z 66 80 - 147538859992323563388164873 z + 53276760107595427551864 z 100 90 88 + 1075538801044 z - 2272119735587278513 z + 23508451580620852704 z 84 94 + 1529456215957530476560 z - 12384962685993291 z 86 96 98 - 205569208611228380783 z + 683321812361216 z - 30456421925277 z 92 82 + 184109737144460412 z - 9736107752230284200009 z 64 112 110 106 + 266074502467379034581830912 z + z - 443 z - 8615097 z 108 30 42 + 81652 z - 9736107752230284200009 z - 30428296105806624252536275 z 44 46 + 71640088368156458402590868 z - 147538859992323563388164873 z 58 56 - 709006867095388476846172141 z + 756790281875777170119975422 z 54 52 - 709006867095388476846172141 z + 582957635995065223231165012 z 60 70 + 582957635995065223231165012 z - 30428296105806624252536275 z 68 78 + 71640088368156458402590868 z - 251598466376993238416559 z 32 38 + 53276760107595427551864 z - 3653068801382189071330333 z 40 62 + 11289721755472788945388152 z - 420552110097704623245682503 z 76 74 + 1028825086948164789819212 z - 3653068801382189071330333 z 72 104 / 2 + 11289721755472788945388152 z + 596807640 z ) / ((-1 + z ) (1 / 28 26 2 + 5730957771127819507008 z - 733817390790172425136 z - 623 z 24 22 4 + 79824685465120146442 z - 7328192097128067082 z + 137776 z 6 102 8 10 - 16423736 z - 66350951598 z + 1248965730 z - 66350951598 z 12 14 18 + 2605444704640 z - 78571835177575 z - 35872551578393593 z 16 50 + 1870351054196832 z - 2284109869559163737767745961 z 48 20 + 1421674272070953746204522228 z + 563183836981182348 z 36 34 + 4594917250153171958003852 z - 1078881668888082468722599 z 66 80 - 772198597173347146543761187 z + 218843583558053242001804 z 100 90 88 + 2605444704640 z - 7328192097128067082 z + 79824685465120146442 z 84 94 + 5730957771127819507008 z - 35872551578393593 z 86 96 98 - 733817390790172425136 z + 1870351054196832 z - 78571835177575 z 92 82 + 563183836981182348 z - 38231715146099669309697 z 64 112 110 106 + 1421674272070953746204522228 z + z - 623 z - 16423736 z 108 30 + 137776 z - 38231715146099669309697 z 42 44 - 150996768497193578708020194 z + 365793960233496181803697736 z 46 58 - 772198597173347146543761187 z - 3924609280257092208246140274 z 56 54 + 4199165270766468013028490476 z - 3924609280257092208246140274 z 52 60 + 3203852900783025085503265960 z + 3203852900783025085503265960 z 70 68 - 150996768497193578708020194 z + 365793960233496181803697736 z 78 32 - 1078881668888082468722599 z + 218843583558053242001804 z 38 40 - 16948924701089366253011912 z + 54257954101739822614372678 z 62 76 - 2284109869559163737767745961 z + 4594917250153171958003852 z 74 72 - 16948924701089366253011912 z + 54257954101739822614372678 z 104 + 1248965730 z )) And in Maple-input format, it is: -(1+1529456215957530476560*z^28-205569208611228380783*z^26-443*z^2+ 23508451580620852704*z^24-2272119735587278513*z^22+81652*z^4-8615097*z^6-\ 29334872303*z^102+596807640*z^8-29334872303*z^10+1075538801044*z^12-\ 30456421925277*z^14-12384962685993291*z^18+683321812361216*z^16-\ 420552110097704623245682503*z^50+266074502467379034581830912*z^48+ 184109737144460412*z^20+1028825086948164789819212*z^36-251598466376993238416559 *z^34-147538859992323563388164873*z^66+53276760107595427551864*z^80+ 1075538801044*z^100-2272119735587278513*z^90+23508451580620852704*z^88+ 1529456215957530476560*z^84-12384962685993291*z^94-205569208611228380783*z^86+ 683321812361216*z^96-30456421925277*z^98+184109737144460412*z^92-\ 9736107752230284200009*z^82+266074502467379034581830912*z^64+z^112-443*z^110-\ 8615097*z^106+81652*z^108-9736107752230284200009*z^30-\ 30428296105806624252536275*z^42+71640088368156458402590868*z^44-\ 147538859992323563388164873*z^46-709006867095388476846172141*z^58+ 756790281875777170119975422*z^56-709006867095388476846172141*z^54+ 582957635995065223231165012*z^52+582957635995065223231165012*z^60-\ 30428296105806624252536275*z^70+71640088368156458402590868*z^68-\ 251598466376993238416559*z^78+53276760107595427551864*z^32-\ 3653068801382189071330333*z^38+11289721755472788945388152*z^40-\ 420552110097704623245682503*z^62+1028825086948164789819212*z^76-\ 3653068801382189071330333*z^74+11289721755472788945388152*z^72+596807640*z^104) /(-1+z^2)/(1+5730957771127819507008*z^28-733817390790172425136*z^26-623*z^2+ 79824685465120146442*z^24-7328192097128067082*z^22+137776*z^4-16423736*z^6-\ 66350951598*z^102+1248965730*z^8-66350951598*z^10+2605444704640*z^12-\ 78571835177575*z^14-35872551578393593*z^18+1870351054196832*z^16-\ 2284109869559163737767745961*z^50+1421674272070953746204522228*z^48+ 563183836981182348*z^20+4594917250153171958003852*z^36-\ 1078881668888082468722599*z^34-772198597173347146543761187*z^66+ 218843583558053242001804*z^80+2605444704640*z^100-7328192097128067082*z^90+ 79824685465120146442*z^88+5730957771127819507008*z^84-35872551578393593*z^94-\ 733817390790172425136*z^86+1870351054196832*z^96-78571835177575*z^98+ 563183836981182348*z^92-38231715146099669309697*z^82+ 1421674272070953746204522228*z^64+z^112-623*z^110-16423736*z^106+137776*z^108-\ 38231715146099669309697*z^30-150996768497193578708020194*z^42+ 365793960233496181803697736*z^44-772198597173347146543761187*z^46-\ 3924609280257092208246140274*z^58+4199165270766468013028490476*z^56-\ 3924609280257092208246140274*z^54+3203852900783025085503265960*z^52+ 3203852900783025085503265960*z^60-150996768497193578708020194*z^70+ 365793960233496181803697736*z^68-1078881668888082468722599*z^78+ 218843583558053242001804*z^32-16948924701089366253011912*z^38+ 54257954101739822614372678*z^40-2284109869559163737767745961*z^62+ 4594917250153171958003852*z^76-16948924701089366253011912*z^74+ 54257954101739822614372678*z^72+1248965730*z^104) The first , 40, terms are: [0, 181, 0, 56197, 0, 17963124, 0, 5760432619, 0, 1848368397323, 0, 593168495936431, 0, 190361968100194687, 0, 61092112345856285151, 0, 19606078839565564512607, 0, 6292112734493560779430916, 0, 2019306667934337691076704993, 0, 648049337860930752008758010169, 0, 207976308239145025012634691711657, 0, 66745141614525570909042344949919465, 0, 21420295262479799924614168565781377569, 0, 6874343780762616497201189561545018110105, 0, 2206160178352247105283015624313758618565156, 0, 708015613967377967059658157411290536375421479, 0, 227221085096502703925928027104319985652700171719, 0, 72921303561559238014731138242580028074680834865455] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 58623943289368857 z - 23147030524943405 z - 342 z 24 22 4 6 + 7373919038140253 z - 1885195079720534 z + 43067 z - 2803351 z 8 10 12 14 + 111078647 z - 2936567030 z + 54841739331 z - 751790915182 z 18 16 50 - 61831076579446 z + 7772860620954 z - 1885195079720534 z 48 20 36 + 7373919038140253 z + 384126887846897 z + 301557113023035572 z 34 66 64 - 272358070122001604 z - 2803351 z + 111078647 z 30 42 44 - 120288673000834038 z - 120288673000834038 z + 58623943289368857 z 46 58 56 - 23147030524943405 z - 751790915182 z + 7772860620954 z 54 52 60 70 - 61831076579446 z + 384126887846897 z + 54841739331 z - 342 z 68 32 38 + 43067 z + 200573181010150899 z - 272358070122001604 z 40 62 72 / 2 + 200573181010150899 z - 2936567030 z + z ) / ((-1 + z ) (1 / 28 26 2 + 301873099200363060 z - 113347757088189922 z - 536 z 24 22 4 6 + 34058664680396459 z - 8156525415422800 z + 82262 z - 6099342 z 8 10 12 14 + 268747263 z - 7818014456 z + 159762511672 z - 2387359747840 z 18 16 50 - 231007462167016 z + 26818591856073 z - 8156525415422800 z 48 20 36 + 34058664680396459 z + 1548004935040954 z + 1708570346954743122 z 34 66 64 - 1533549673726707176 z - 6099342 z + 268747263 z 30 42 44 - 645185131849565168 z - 645185131849565168 z + 301873099200363060 z 46 58 56 - 113347757088189922 z - 2387359747840 z + 26818591856073 z 54 52 60 70 - 231007462167016 z + 1548004935040954 z + 159762511672 z - 536 z 68 32 38 + 82262 z + 1108680573856781681 z - 1533549673726707176 z 40 62 72 + 1108680573856781681 z - 7818014456 z + z )) And in Maple-input format, it is: -(1+58623943289368857*z^28-23147030524943405*z^26-342*z^2+7373919038140253*z^24 -1885195079720534*z^22+43067*z^4-2803351*z^6+111078647*z^8-2936567030*z^10+ 54841739331*z^12-751790915182*z^14-61831076579446*z^18+7772860620954*z^16-\ 1885195079720534*z^50+7373919038140253*z^48+384126887846897*z^20+ 301557113023035572*z^36-272358070122001604*z^34-2803351*z^66+111078647*z^64-\ 120288673000834038*z^30-120288673000834038*z^42+58623943289368857*z^44-\ 23147030524943405*z^46-751790915182*z^58+7772860620954*z^56-61831076579446*z^54 +384126887846897*z^52+54841739331*z^60-342*z^70+43067*z^68+200573181010150899*z ^32-272358070122001604*z^38+200573181010150899*z^40-2936567030*z^62+z^72)/(-1+z ^2)/(1+301873099200363060*z^28-113347757088189922*z^26-536*z^2+ 34058664680396459*z^24-8156525415422800*z^22+82262*z^4-6099342*z^6+268747263*z^ 8-7818014456*z^10+159762511672*z^12-2387359747840*z^14-231007462167016*z^18+ 26818591856073*z^16-8156525415422800*z^50+34058664680396459*z^48+ 1548004935040954*z^20+1708570346954743122*z^36-1533549673726707176*z^34-6099342 *z^66+268747263*z^64-645185131849565168*z^30-645185131849565168*z^42+ 301873099200363060*z^44-113347757088189922*z^46-2387359747840*z^58+ 26818591856073*z^56-231007462167016*z^54+1548004935040954*z^52+159762511672*z^ 60-536*z^70+82262*z^68+1108680573856781681*z^32-1533549673726707176*z^38+ 1108680573856781681*z^40-7818014456*z^62+z^72) The first , 40, terms are: [0, 195, 0, 64984, 0, 22129051, 0, 7544399977, 0, 2572362084001, 0, 877093787959783, 0, 299061954091786792, 0, 101970966484608102587, 0, 34768980199696596594881, 0, 11855158907089620650740989, 0, 4042246639372594426418650599, 0, 1378282487579926824760274418248, 0, 469952178928982229199777680526075, 0, 160239321386692013276867401511227917, 0, 54636708307632520256416744149561194517, 0, 18629446685526752491013935562122259647695, 0, 6352071611907881069128988168182336552355128, 0, 2165862166703851431553796236721649040531746479, 0, 738492764528234591075057279828444378593033648477, 0, 251803448827280524446812919088654434108397464976517] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 81511784486116860 z - 31838871497977510 z - 345 z 24 22 4 6 + 10016827083589231 z - 2525117417437617 z + 44596 z - 3027281 z 8 10 12 14 + 125238807 z - 3439913006 z + 66338824484 z - 934062787070 z 18 16 50 - 80134172875289 z + 9878363587663 z - 2525117417437617 z 48 20 36 + 10016827083589231 z + 506550507107728 z + 428092860657376824 z 34 66 64 - 386112618644078356 z - 3027281 z + 125238807 z 30 42 44 - 168731184849145254 z - 168731184849145254 z + 81511784486116860 z 46 58 56 - 31838871497977510 z - 934062787070 z + 9878363587663 z 54 52 60 70 - 80134172875289 z + 506550507107728 z + 66338824484 z - 345 z 68 32 38 + 44596 z + 283197327025498410 z - 386112618644078356 z 40 62 72 / 2 + 283197327025498410 z - 3439913006 z + z ) / ((-1 + z ) (1 / 28 26 2 + 409465172552967946 z - 152205379353704084 z - 512 z 24 22 4 6 + 45216840326446823 z - 10693290356760740 z + 80991 z - 6335648 z 8 10 12 14 + 294310947 z - 8954447380 z + 189678844186 z - 2915815840068 z 18 16 50 - 293979154519532 z + 33502760352391 z - 10693290356760740 z 48 20 36 + 45216840326446823 z + 2001345787127567 z + 2365312378830640348 z 34 66 64 - 2120059154591318160 z - 6335648 z + 294310947 z 30 42 44 - 882581726784302420 z - 882581726784302420 z + 409465172552967946 z 46 58 56 - 152205379353704084 z - 2915815840068 z + 33502760352391 z 54 52 60 70 - 293979154519532 z + 2001345787127567 z + 189678844186 z - 512 z 68 32 38 + 80991 z + 1526463241764916590 z - 2120059154591318160 z 40 62 72 + 1526463241764916590 z - 8954447380 z + z )) And in Maple-input format, it is: -(1+81511784486116860*z^28-31838871497977510*z^26-345*z^2+10016827083589231*z^ 24-2525117417437617*z^22+44596*z^4-3027281*z^6+125238807*z^8-3439913006*z^10+ 66338824484*z^12-934062787070*z^14-80134172875289*z^18+9878363587663*z^16-\ 2525117417437617*z^50+10016827083589231*z^48+506550507107728*z^20+ 428092860657376824*z^36-386112618644078356*z^34-3027281*z^66+125238807*z^64-\ 168731184849145254*z^30-168731184849145254*z^42+81511784486116860*z^44-\ 31838871497977510*z^46-934062787070*z^58+9878363587663*z^56-80134172875289*z^54 +506550507107728*z^52+66338824484*z^60-345*z^70+44596*z^68+283197327025498410*z ^32-386112618644078356*z^38+283197327025498410*z^40-3439913006*z^62+z^72)/(-1+z ^2)/(1+409465172552967946*z^28-152205379353704084*z^26-512*z^2+ 45216840326446823*z^24-10693290356760740*z^22+80991*z^4-6335648*z^6+294310947*z ^8-8954447380*z^10+189678844186*z^12-2915815840068*z^14-293979154519532*z^18+ 33502760352391*z^16-10693290356760740*z^50+45216840326446823*z^48+ 2001345787127567*z^20+2365312378830640348*z^36-2120059154591318160*z^34-6335648 *z^66+294310947*z^64-882581726784302420*z^30-882581726784302420*z^42+ 409465172552967946*z^44-152205379353704084*z^46-2915815840068*z^58+ 33502760352391*z^56-293979154519532*z^54+2001345787127567*z^52+189678844186*z^ 60-512*z^70+80991*z^68+1526463241764916590*z^32-2120059154591318160*z^38+ 1526463241764916590*z^40-8954447380*z^62+z^72) The first , 40, terms are: [0, 168, 0, 49277, 0, 14975955, 0, 4569029148, 0, 1394819629923, 0, 425854719086339, 0, 130021379442241660, 0, 39698137582400501179, 0, 12120650599169746290029, 0, 3700682487647226074115048, 0, 1129894092430642848483221129, 0, 344979791466191624839761693801, 0, 105329391112507266201364468322776, 0, 32159218921699059705681485221477981, 0, 9818867752248710218020429876094253419, 0, 2997901291477417551109354279934437546412, 0, 915320623544037251874590295962667183944515, 0, 279466120604998577356488152500097159405009731, 0, 85326726566723650707202107549023653254527527500, 0, 26051996037413607223550035577520364512970436769731] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 17835941447773499579 z - 3973741472456365601 z - 401 z 24 22 4 6 + 732298791544738129 z - 110961552121063328 z + 64131 z - 5647414 z 8 10 12 14 + 314621784 z - 12017062698 z + 331692916805 z - 6864516996295 z 18 16 50 - 1371259968715960 z + 109428241050647 z - 2152225918802596150535 z 48 20 + 3313510585578636477719 z + 13720496503314972 z 36 34 + 1175069120222107962205 z - 538659609172263017650 z 66 80 88 84 86 - 110961552121063328 z + 314621784 z + z + 64131 z - 401 z 82 64 30 - 5647414 z + 732298791544738129 z - 66529431994956829726 z 42 44 - 4291414925073639873000 z + 4677534068394966422008 z 46 58 - 4291414925073639873000 z - 66529431994956829726 z 56 54 + 206987155712791885024 z - 538659609172263017650 z 52 60 + 1175069120222107962205 z + 17835941447773499579 z 70 68 78 - 1371259968715960 z + 13720496503314972 z - 12017062698 z 32 38 + 206987155712791885024 z - 2152225918802596150535 z 40 62 76 + 3313510585578636477719 z - 3973741472456365601 z + 331692916805 z 74 72 / - 6864516996295 z + 109428241050647 z ) / (-1 / 28 26 2 - 94770130629250918948 z + 19355725503071473823 z + 579 z 24 22 4 - 3272813081220582657 z + 455453115919902712 z - 112952 z 6 8 10 12 + 11407368 z - 708449625 z + 29745788099 z - 896285746360 z 14 18 16 + 20182579588760 z + 4760049612917131 z - 349619299978437 z 50 48 + 29912932210075744267617 z - 42284882502602187800059 z 20 36 - 51770483944262236 z - 8890107310546420622540 z 34 66 80 + 3729299480960935898091 z + 3272813081220582657 z - 29745788099 z 90 88 84 86 82 + z - 579 z - 11407368 z + 112952 z + 708449625 z 64 30 - 19355725503071473823 z + 385910108503558559436 z 42 44 + 42284882502602187800059 z - 50270565624283513005636 z 46 58 + 50270565624283513005636 z + 1311480448164574111793 z 56 54 - 3729299480960935898091 z + 8890107310546420622540 z 52 60 - 17790372461687115025124 z - 385910108503558559436 z 70 68 78 + 51770483944262236 z - 455453115919902712 z + 896285746360 z 32 38 - 1311480448164574111793 z + 17790372461687115025124 z 40 62 - 29912932210075744267617 z + 94770130629250918948 z 76 74 72 - 20182579588760 z + 349619299978437 z - 4760049612917131 z ) And in Maple-input format, it is: -(1+17835941447773499579*z^28-3973741472456365601*z^26-401*z^2+ 732298791544738129*z^24-110961552121063328*z^22+64131*z^4-5647414*z^6+314621784 *z^8-12017062698*z^10+331692916805*z^12-6864516996295*z^14-1371259968715960*z^ 18+109428241050647*z^16-2152225918802596150535*z^50+3313510585578636477719*z^48 +13720496503314972*z^20+1175069120222107962205*z^36-538659609172263017650*z^34-\ 110961552121063328*z^66+314621784*z^80+z^88+64131*z^84-401*z^86-5647414*z^82+ 732298791544738129*z^64-66529431994956829726*z^30-4291414925073639873000*z^42+ 4677534068394966422008*z^44-4291414925073639873000*z^46-66529431994956829726*z^ 58+206987155712791885024*z^56-538659609172263017650*z^54+1175069120222107962205 *z^52+17835941447773499579*z^60-1371259968715960*z^70+13720496503314972*z^68-\ 12017062698*z^78+206987155712791885024*z^32-2152225918802596150535*z^38+ 3313510585578636477719*z^40-3973741472456365601*z^62+331692916805*z^76-\ 6864516996295*z^74+109428241050647*z^72)/(-1-94770130629250918948*z^28+ 19355725503071473823*z^26+579*z^2-3272813081220582657*z^24+455453115919902712*z ^22-112952*z^4+11407368*z^6-708449625*z^8+29745788099*z^10-896285746360*z^12+ 20182579588760*z^14+4760049612917131*z^18-349619299978437*z^16+ 29912932210075744267617*z^50-42284882502602187800059*z^48-51770483944262236*z^ 20-8890107310546420622540*z^36+3729299480960935898091*z^34+3272813081220582657* z^66-29745788099*z^80+z^90-579*z^88-11407368*z^84+112952*z^86+708449625*z^82-\ 19355725503071473823*z^64+385910108503558559436*z^30+42284882502602187800059*z^ 42-50270565624283513005636*z^44+50270565624283513005636*z^46+ 1311480448164574111793*z^58-3729299480960935898091*z^56+8890107310546420622540* z^54-17790372461687115025124*z^52-385910108503558559436*z^60+51770483944262236* z^70-455453115919902712*z^68+896285746360*z^78-1311480448164574111793*z^32+ 17790372461687115025124*z^38-29912932210075744267617*z^40+94770130629250918948* z^62-20182579588760*z^76+349619299978437*z^74-4760049612917131*z^72) The first , 40, terms are: [0, 178, 0, 54241, 0, 17060037, 0, 5387815654, 0, 1702951704281, 0, 538357744523149, 0, 170199172735176550, 0, 53808138970106267653, 0, 17011374950661033969665, 0, 5378127217200361536362890, 0, 1700289106729545353276813193, 0, 537544576637031092958109207813, 0, 169944142060100986075147997618786, 0, 53727658528570343328951566150775133, 0, 16985941714774125050810670773413352785, 0, 5370087285737378100535568256603236491870, 0, 1697747345487890218783916809999098909055649, 0, 536741005453730310542251176471642012426665037, 0, 169690094171764754981679019725730808814966221534, 0, 53647341580854277706824190873475442676101531004353] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 21378397216446126337 z - 4651710944673976879 z - 391 z 24 22 4 6 + 835288033674922377 z - 123099835797771072 z + 61177 z - 5325026 z 8 10 12 14 + 296592374 z - 11437408670 z + 321175928435 z - 6799931719133 z 18 16 50 - 1435161745631808 z + 111304685061579 z - 2785630090861046072333 z 48 20 + 4318146512924388298475 z + 14786834064623748 z 36 34 + 1506287085105620260883 z - 681914394728843390678 z 66 80 88 84 86 - 123099835797771072 z + 296592374 z + z + 61177 z - 391 z 82 64 30 - 5325026 z + 835288033674922377 z - 81437898899057830186 z 42 44 - 5615448718242389300608 z + 6129018828377489411352 z 46 58 - 5615448718242389300608 z - 81437898899057830186 z 56 54 + 258035759593816135770 z - 681914394728843390678 z 52 60 + 1506287085105620260883 z + 21378397216446126337 z 70 68 78 - 1435161745631808 z + 14786834064623748 z - 11437408670 z 32 38 + 258035759593816135770 z - 2785630090861046072333 z 40 62 76 + 4318146512924388298475 z - 4651710944673976879 z + 321175928435 z 74 72 / 2 - 6799931719133 z + 111304685061579 z ) / ((-1 + z ) (1 / 28 26 2 + 92505172940039160164 z - 19008327827143644508 z - 572 z 24 22 4 + 3212903587229270175 z - 444501397976018212 z + 107134 z 6 8 10 12 - 10511524 z + 643960703 z - 26998504456 z + 819227030624 z 14 18 16 - 18674071190692 z - 4536956120811540 z + 328281191833745 z 50 48 - 14900480174001767652468 z + 23609573189599920347055 z 20 36 + 50009280585636466 z + 7820647318711457887358 z 34 66 80 - 3412102808280916776036 z - 444501397976018212 z + 643960703 z 88 84 86 82 64 + z + 107134 z - 572 z - 10511524 z + 3212903587229270175 z 30 42 - 371693542105570119328 z - 31117039152259031633336 z 44 46 + 34116590181254377272156 z - 31117039152259031633336 z 58 56 - 371693542105570119328 z + 1236489069718365678097 z 54 52 - 3412102808280916776036 z + 7820647318711457887358 z 60 70 68 + 92505172940039160164 z - 4536956120811540 z + 50009280585636466 z 78 32 - 26998504456 z + 1236489069718365678097 z 38 40 - 14900480174001767652468 z + 23609573189599920347055 z 62 76 74 - 19008327827143644508 z + 819227030624 z - 18674071190692 z 72 + 328281191833745 z )) And in Maple-input format, it is: -(1+21378397216446126337*z^28-4651710944673976879*z^26-391*z^2+ 835288033674922377*z^24-123099835797771072*z^22+61177*z^4-5325026*z^6+296592374 *z^8-11437408670*z^10+321175928435*z^12-6799931719133*z^14-1435161745631808*z^ 18+111304685061579*z^16-2785630090861046072333*z^50+4318146512924388298475*z^48 +14786834064623748*z^20+1506287085105620260883*z^36-681914394728843390678*z^34-\ 123099835797771072*z^66+296592374*z^80+z^88+61177*z^84-391*z^86-5325026*z^82+ 835288033674922377*z^64-81437898899057830186*z^30-5615448718242389300608*z^42+ 6129018828377489411352*z^44-5615448718242389300608*z^46-81437898899057830186*z^ 58+258035759593816135770*z^56-681914394728843390678*z^54+1506287085105620260883 *z^52+21378397216446126337*z^60-1435161745631808*z^70+14786834064623748*z^68-\ 11437408670*z^78+258035759593816135770*z^32-2785630090861046072333*z^38+ 4318146512924388298475*z^40-4651710944673976879*z^62+321175928435*z^76-\ 6799931719133*z^74+111304685061579*z^72)/(-1+z^2)/(1+92505172940039160164*z^28-\ 19008327827143644508*z^26-572*z^2+3212903587229270175*z^24-444501397976018212*z ^22+107134*z^4-10511524*z^6+643960703*z^8-26998504456*z^10+819227030624*z^12-\ 18674071190692*z^14-4536956120811540*z^18+328281191833745*z^16-\ 14900480174001767652468*z^50+23609573189599920347055*z^48+50009280585636466*z^ 20+7820647318711457887358*z^36-3412102808280916776036*z^34-444501397976018212*z ^66+643960703*z^80+z^88+107134*z^84-572*z^86-10511524*z^82+3212903587229270175* z^64-371693542105570119328*z^30-31117039152259031633336*z^42+ 34116590181254377272156*z^44-31117039152259031633336*z^46-371693542105570119328 *z^58+1236489069718365678097*z^56-3412102808280916776036*z^54+ 7820647318711457887358*z^52+92505172940039160164*z^60-4536956120811540*z^70+ 50009280585636466*z^68-26998504456*z^78+1236489069718365678097*z^32-\ 14900480174001767652468*z^38+23609573189599920347055*z^40-19008327827143644508* z^62+819227030624*z^76-18674071190692*z^74+328281191833745*z^72) The first , 40, terms are: [0, 182, 0, 57757, 0, 18785901, 0, 6118261734, 0, 1992802661757, 0, 649089011442869, 0, 211419304693031926, 0, 68862863362250669925, 0, 22429806189693336305013, 0, 7305769500318546666677606, 0, 2379613428550653221781176585, 0, 775080581086548037518667180745, 0, 252456932698990833199963692969862, 0, 82229518353424218476316474167686485, 0, 26783553203900586955446455255691474821, 0, 8723858981430037454812448706763616233974, 0, 2841509300446147112848293000395459136571093, 0, 925527925395054422361715751114086443895076989, 0, 301460192494030504396682539597181138938305442406, 0, 98190713823947830176871173353901532841054676369837] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 14436780852152556589 z - 3311567840886283539 z - 411 z 24 22 4 6 + 628671981885386713 z - 98136317723983936 z + 66845 z - 5914522 z 8 10 12 14 + 327782866 z - 12359853782 z + 334926771687 z - 6777972342233 z 18 16 50 - 1284954645406112 z + 105361716794491 z - 1552083448446896810169 z 48 20 + 2359953175275342871723 z + 12495932668478116 z 36 34 + 861782088724991475087 z - 403278381319546276654 z 66 80 88 84 86 - 98136317723983936 z + 327782866 z + z + 66845 z - 411 z 82 64 30 - 5914522 z + 628671981885386713 z - 52362707790249840930 z 42 44 - 3033108085197256815968 z + 3297474263333725826040 z 46 58 - 3033108085197256815968 z - 52362707790249840930 z 56 54 + 158693168526669978254 z - 403278381319546276654 z 52 60 + 861782088724991475087 z + 14436780852152556589 z 70 68 78 - 1284954645406112 z + 12495932668478116 z - 12359853782 z 32 38 + 158693168526669978254 z - 1552083448446896810169 z 40 62 76 + 2359953175275342871723 z - 3311567840886283539 z + 334926771687 z 74 72 / - 6777972342233 z + 105361716794491 z ) / (-1 / 28 26 2 - 78805206717684343628 z + 16565115416922081131 z + 605 z 24 22 4 - 2883746039506378835 z + 413164617904822182 z - 121194 z 6 8 10 12 + 12319798 z - 760266675 z + 31457692451 z - 929098321560 z 14 18 16 + 20435944593624 z + 4571240229302509 z - 345013373862285 z 50 48 + 21735198893426682479499 z - 30414284801258749962943 z 20 36 - 48335532579846798 z - 6679392166798940121342 z 34 66 80 + 2864443518339194426213 z + 2883746039506378835 z - 31457692451 z 90 88 84 86 82 + z - 605 z - 12319798 z + 121194 z + 760266675 z 64 30 - 16565115416922081131 z + 312040402432915283380 z 42 44 + 30414284801258749962943 z - 35971067777820737835500 z 46 58 + 35971067777820737835500 z + 1032536890461259186405 z 56 54 - 2864443518339194426213 z + 6679392166798940121342 z 52 60 - 13118476249758444268498 z - 312040402432915283380 z 70 68 78 + 48335532579846798 z - 413164617904822182 z + 929098321560 z 32 38 - 1032536890461259186405 z + 13118476249758444268498 z 40 62 - 21735198893426682479499 z + 78805206717684343628 z 76 74 72 - 20435944593624 z + 345013373862285 z - 4571240229302509 z ) And in Maple-input format, it is: -(1+14436780852152556589*z^28-3311567840886283539*z^26-411*z^2+ 628671981885386713*z^24-98136317723983936*z^22+66845*z^4-5914522*z^6+327782866* z^8-12359853782*z^10+334926771687*z^12-6777972342233*z^14-1284954645406112*z^18 +105361716794491*z^16-1552083448446896810169*z^50+2359953175275342871723*z^48+ 12495932668478116*z^20+861782088724991475087*z^36-403278381319546276654*z^34-\ 98136317723983936*z^66+327782866*z^80+z^88+66845*z^84-411*z^86-5914522*z^82+ 628671981885386713*z^64-52362707790249840930*z^30-3033108085197256815968*z^42+ 3297474263333725826040*z^44-3033108085197256815968*z^46-52362707790249840930*z^ 58+158693168526669978254*z^56-403278381319546276654*z^54+861782088724991475087* z^52+14436780852152556589*z^60-1284954645406112*z^70+12495932668478116*z^68-\ 12359853782*z^78+158693168526669978254*z^32-1552083448446896810169*z^38+ 2359953175275342871723*z^40-3311567840886283539*z^62+334926771687*z^76-\ 6777972342233*z^74+105361716794491*z^72)/(-1-78805206717684343628*z^28+ 16565115416922081131*z^26+605*z^2-2883746039506378835*z^24+413164617904822182*z ^22-121194*z^4+12319798*z^6-760266675*z^8+31457692451*z^10-929098321560*z^12+ 20435944593624*z^14+4571240229302509*z^18-345013373862285*z^16+ 21735198893426682479499*z^50-30414284801258749962943*z^48-48335532579846798*z^ 20-6679392166798940121342*z^36+2864443518339194426213*z^34+2883746039506378835* z^66-31457692451*z^80+z^90-605*z^88-12319798*z^84+121194*z^86+760266675*z^82-\ 16565115416922081131*z^64+312040402432915283380*z^30+30414284801258749962943*z^ 42-35971067777820737835500*z^44+35971067777820737835500*z^46+ 1032536890461259186405*z^58-2864443518339194426213*z^56+6679392166798940121342* z^54-13118476249758444268498*z^52-312040402432915283380*z^60+48335532579846798* z^70-413164617904822182*z^68+929098321560*z^78-1032536890461259186405*z^32+ 13118476249758444268498*z^38-21735198893426682479499*z^40+78805206717684343628* z^62-20435944593624*z^76+345013373862285*z^74-4571240229302509*z^72) The first , 40, terms are: [0, 194, 0, 63021, 0, 21021345, 0, 7037703654, 0, 2358161918217, 0, 790335082627165, 0, 264894816940830966, 0, 88785496394537558233, 0, 29758582792081342840721, 0, 9974310662378074414643482, 0, 3343132949226502838845485409, 0, 1120532439472638943354958552337, 0, 375573747326972477144156147311738, 0, 125882692319543733303325768753991785, 0, 42192651527948485649266237050338469449, 0, 14141895209678724614990576054843762521846, 0, 4740000755930082571482747925708727220371781, 0, 1588726746560586734972468254521715377277588241, 0, 532500479476191186175278560407303749642949648166, 0, 178480510419195268917285971609084849974164933748097] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1702066327846671783183 z - 224588004156516533429 z - 434 z 24 22 4 + 25217026329264342819 z - 2393597016498768028 z + 78707 z 6 102 8 10 - 8243457 z - 28259874796 z + 571330807 z - 28259874796 z 12 14 18 + 1046904812721 z - 30036716309838 z - 12601329331130982 z 16 50 + 684068090848154 z - 550943131089568621527812632 z 48 20 + 345781575466866796855008262 z + 190554616767241965 z 36 34 + 1230679754568979709583080 z - 295739573366762553905152 z 66 80 - 189821465061949405665762586 z + 61507486052461730882813 z 100 90 88 + 1046904812721 z - 2393597016498768028 z + 25217026329264342819 z 84 94 + 1702066327846671783183 z - 12601329331130982 z 86 96 98 - 224588004156516533429 z + 684068090848154 z - 30036716309838 z 92 82 + 190554616767241965 z - 11036520522343546474794 z 64 112 110 106 + 345781575466866796855008262 z + z - 434 z - 8243457 z 108 30 + 78707 z - 11036520522343546474794 z 42 44 - 38170971590931558395543908 z + 91085456531490829729945726 z 46 58 - 189821465061949405665762586 z - 937633573257745266718749228 z 56 54 + 1002025192266829693671069468 z - 937633573257745266718749228 z 52 60 + 768192157899388337476062338 z + 768192157899388337476062338 z 70 68 - 38170971590931558395543908 z + 91085456531490829729945726 z 78 32 - 295739573366762553905152 z + 61507486052461730882813 z 38 40 - 4443938092995194518547520 z + 13954381434939810333790850 z 62 76 - 550943131089568621527812632 z + 1230679754568979709583080 z 74 72 - 4443938092995194518547520 z + 13954381434939810333790850 z 104 / 2 28 + 571330807 z ) / ((-1 + z ) (1 + 6298344946164750483146 z / 26 2 24 - 791367805194795054248 z - 612 z + 84494907176489523355 z 22 4 6 102 - 7616286360445933076 z + 132078 z - 15547800 z - 63020011300 z 8 10 12 14 + 1179946083 z - 63020011300 z + 2499916048504 z - 76392909906624 z 18 16 - 35996709005420176 z + 1846268005284725 z 50 48 - 2996752691717516006939271400 z + 1848057763979461598540963466 z 20 36 + 574968703483822956 z + 5447943413114759655133028 z 34 66 - 1255457000088416063383344 z - 992341797182914106564375616 z 80 100 + 249867467028114256759201 z + 2499916048504 z 90 88 - 7616286360445933076 z + 84494907176489523355 z 84 94 + 6298344946164750483146 z - 35996709005420176 z 86 96 98 - 791367805194795054248 z + 1846268005284725 z - 76392909906624 z 92 82 + 574968703483822956 z - 42825449173424908163508 z 64 112 110 106 + 1848057763979461598540963466 z + z - 612 z - 15547800 z 108 30 + 132078 z - 42825449173424908163508 z 42 44 - 188594707914634188763162760 z + 463810915491799292817207056 z 46 58 - 992341797182914106564375616 z - 5205771149092729839399052496 z 56 54 + 5577744496571880590986640542 z - 5205771149092729839399052496 z 52 60 + 4232113447868596281516386212 z + 4232113447868596281516386212 z 70 68 - 188594707914634188763162760 z + 463810915491799292817207056 z 78 32 - 1255457000088416063383344 z + 249867467028114256759201 z 38 40 - 20464608139090605185654576 z + 66665465595576881394767782 z 62 76 - 2996752691717516006939271400 z + 5447943413114759655133028 z 74 72 - 20464608139090605185654576 z + 66665465595576881394767782 z 104 + 1179946083 z )) And in Maple-input format, it is: -(1+1702066327846671783183*z^28-224588004156516533429*z^26-434*z^2+ 25217026329264342819*z^24-2393597016498768028*z^22+78707*z^4-8243457*z^6-\ 28259874796*z^102+571330807*z^8-28259874796*z^10+1046904812721*z^12-\ 30036716309838*z^14-12601329331130982*z^18+684068090848154*z^16-\ 550943131089568621527812632*z^50+345781575466866796855008262*z^48+ 190554616767241965*z^20+1230679754568979709583080*z^36-295739573366762553905152 *z^34-189821465061949405665762586*z^66+61507486052461730882813*z^80+ 1046904812721*z^100-2393597016498768028*z^90+25217026329264342819*z^88+ 1702066327846671783183*z^84-12601329331130982*z^94-224588004156516533429*z^86+ 684068090848154*z^96-30036716309838*z^98+190554616767241965*z^92-\ 11036520522343546474794*z^82+345781575466866796855008262*z^64+z^112-434*z^110-\ 8243457*z^106+78707*z^108-11036520522343546474794*z^30-\ 38170971590931558395543908*z^42+91085456531490829729945726*z^44-\ 189821465061949405665762586*z^46-937633573257745266718749228*z^58+ 1002025192266829693671069468*z^56-937633573257745266718749228*z^54+ 768192157899388337476062338*z^52+768192157899388337476062338*z^60-\ 38170971590931558395543908*z^70+91085456531490829729945726*z^68-\ 295739573366762553905152*z^78+61507486052461730882813*z^32-\ 4443938092995194518547520*z^38+13954381434939810333790850*z^40-\ 550943131089568621527812632*z^62+1230679754568979709583080*z^76-\ 4443938092995194518547520*z^74+13954381434939810333790850*z^72+571330807*z^104) /(-1+z^2)/(1+6298344946164750483146*z^28-791367805194795054248*z^26-612*z^2+ 84494907176489523355*z^24-7616286360445933076*z^22+132078*z^4-15547800*z^6-\ 63020011300*z^102+1179946083*z^8-63020011300*z^10+2499916048504*z^12-\ 76392909906624*z^14-35996709005420176*z^18+1846268005284725*z^16-\ 2996752691717516006939271400*z^50+1848057763979461598540963466*z^48+ 574968703483822956*z^20+5447943413114759655133028*z^36-\ 1255457000088416063383344*z^34-992341797182914106564375616*z^66+ 249867467028114256759201*z^80+2499916048504*z^100-7616286360445933076*z^90+ 84494907176489523355*z^88+6298344946164750483146*z^84-35996709005420176*z^94-\ 791367805194795054248*z^86+1846268005284725*z^96-76392909906624*z^98+ 574968703483822956*z^92-42825449173424908163508*z^82+ 1848057763979461598540963466*z^64+z^112-612*z^110-15547800*z^106+132078*z^108-\ 42825449173424908163508*z^30-188594707914634188763162760*z^42+ 463810915491799292817207056*z^44-992341797182914106564375616*z^46-\ 5205771149092729839399052496*z^58+5577744496571880590986640542*z^56-\ 5205771149092729839399052496*z^54+4232113447868596281516386212*z^52+ 4232113447868596281516386212*z^60-188594707914634188763162760*z^70+ 463810915491799292817207056*z^68-1255457000088416063383344*z^78+ 249867467028114256759201*z^32-20464608139090605185654576*z^38+ 66665465595576881394767782*z^40-2996752691717516006939271400*z^62+ 5447943413114759655133028*z^76-20464608139090605185654576*z^74+ 66665465595576881394767782*z^72+1179946083*z^104) The first , 40, terms are: [0, 179, 0, 55744, 0, 17855983, 0, 5731581305, 0, 1840154752457, 0, 590805125016287, 0, 189686037668791728, 0, 60901308945094993195, 0, 19553202996603817187833, 0, 6277824834852213445797933, 0, 2015582035197394610351620731, 0, 647130343386653458811108034240, 0, 207770100162662419436170590749863, 0, 66707449222360299792241159892065693, 0, 21417344354513451590683273893197032397, 0, 6876333071450724487374598130687713752391, 0, 2207741339302080414620471428297026338251568, 0, 708825731769714824893097171076446027725596003, 0, 227578253427869017132822702010400256397566356485, 0, 73067129354871747906685410475383764993071205641469] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 89708164910705934 z - 40675772767810900 z - 390 z 24 22 4 6 + 14572026139810846 z - 4094530731687456 z + 58173 z - 4498712 z 8 10 12 14 + 207289778 z - 6177298820 z + 125677884394 z - 1813995116952 z 18 16 50 - 150107771071878 z + 19108934996725 z - 150107771071878 z 48 20 36 + 894203004700465 z + 894203004700465 z + 219755604470336468 z 34 66 64 30 - 245651957433100344 z - 390 z + 58173 z - 157201023199900976 z 42 44 46 - 40675772767810900 z + 14572026139810846 z - 4094530731687456 z 58 56 54 - 6177298820 z + 125677884394 z - 1813995116952 z 52 60 68 32 + 19108934996725 z + 207289778 z + z + 219755604470336468 z 38 40 62 / - 157201023199900976 z + 89708164910705934 z - 4498712 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 475718239537440826 z - 205967983550516512 z - 581 z 24 22 4 6 + 69690833310422444 z - 18321448744956992 z + 106927 z - 9557934 z 8 10 12 14 + 495525654 z - 16404283916 z + 368187773356 z - 5835564292544 z 18 16 50 - 575306737520539 z + 67236660568151 z - 575306737520539 z 48 20 36 + 3714600971485939 z + 3714600971485939 z + 1230889080853333342 z 34 66 64 30 - 1385587326175076072 z - 581 z + 106927 z - 862416108495284946 z 42 44 46 - 205967983550516512 z + 69690833310422444 z - 18321448744956992 z 58 56 54 - 16404283916 z + 368187773356 z - 5835564292544 z 52 60 68 32 + 67236660568151 z + 495525654 z + z + 1230889080853333342 z 38 40 62 - 862416108495284946 z + 475718239537440826 z - 9557934 z )) And in Maple-input format, it is: -(1+89708164910705934*z^28-40675772767810900*z^26-390*z^2+14572026139810846*z^ 24-4094530731687456*z^22+58173*z^4-4498712*z^6+207289778*z^8-6177298820*z^10+ 125677884394*z^12-1813995116952*z^14-150107771071878*z^18+19108934996725*z^16-\ 150107771071878*z^50+894203004700465*z^48+894203004700465*z^20+ 219755604470336468*z^36-245651957433100344*z^34-390*z^66+58173*z^64-\ 157201023199900976*z^30-40675772767810900*z^42+14572026139810846*z^44-\ 4094530731687456*z^46-6177298820*z^58+125677884394*z^56-1813995116952*z^54+ 19108934996725*z^52+207289778*z^60+z^68+219755604470336468*z^32-\ 157201023199900976*z^38+89708164910705934*z^40-4498712*z^62)/(-1+z^2)/(1+ 475718239537440826*z^28-205967983550516512*z^26-581*z^2+69690833310422444*z^24-\ 18321448744956992*z^22+106927*z^4-9557934*z^6+495525654*z^8-16404283916*z^10+ 368187773356*z^12-5835564292544*z^14-575306737520539*z^18+67236660568151*z^16-\ 575306737520539*z^50+3714600971485939*z^48+3714600971485939*z^20+ 1230889080853333342*z^36-1385587326175076072*z^34-581*z^66+106927*z^64-\ 862416108495284946*z^30-205967983550516512*z^42+69690833310422444*z^44-\ 18321448744956992*z^46-16404283916*z^58+368187773356*z^56-5835564292544*z^54+ 67236660568151*z^52+495525654*z^60+z^68+1230889080853333342*z^32-\ 862416108495284946*z^38+475718239537440826*z^40-9557934*z^62) The first , 40, terms are: [0, 192, 0, 62409, 0, 20846651, 0, 6981143612, 0, 2338764598479, 0, 783576280727463, 0, 262533220573664132, 0, 87960828861029289835, 0, 29471001314685754383145, 0, 9874170212362710988304296, 0, 3308311262533121437501915817, 0, 1108439840135238274592802574905, 0, 371379470930971658992522255141336, 0, 124429587106835499477507542636247625, 0, 41689763074512773892397936615114841259, 0, 13968031123518033004913928380727806924148, 0, 4679947284045597969984999606686713285415191, 0, 1568002418374655701072699164337798874049070751, 0, 525354546709385689289253816202078913030120950348, 0, 176018478360803623266479351158251400279741025045563] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 23092369302257972968 z - 5279571813985642550 z - 423 z 24 22 4 6 + 994521311226054760 z - 153201950703131594 z + 71662 z - 6652261 z 8 10 12 14 + 388695889 z - 15474061040 z + 441761560730 z - 9374037289324 z 18 16 50 - 1915400124440872 z + 151821947032502 z - 2446166816893131019392 z 48 20 + 3707050465249406348302 z + 19128893223936120 z 36 34 + 1363802147140979253142 z - 640966706359405434116 z 66 80 88 84 86 - 153201950703131594 z + 388695889 z + z + 71662 z - 423 z 82 64 30 - 6652261 z + 994521311226054760 z - 83748184898704791784 z 42 44 - 4754194069913268718516 z + 5164722923945492499508 z 46 58 - 4754194069913268718516 z - 83748184898704791784 z 56 54 + 253190230485211312922 z - 640966706359405434116 z 52 60 + 1363802147140979253142 z + 23092369302257972968 z 70 68 78 - 1915400124440872 z + 19128893223936120 z - 15474061040 z 32 38 + 253190230485211312922 z - 2446166816893131019392 z 40 62 76 + 3707050465249406348302 z - 5279571813985642550 z + 441761560730 z 74 72 / 2 - 9374037289324 z + 151821947032502 z ) / ((-1 + z ) (1 / 28 26 2 + 104028469963563625428 z - 22501982921433763966 z - 623 z 24 22 4 + 3994104107641733852 z - 577775539427532402 z + 128043 z 6 8 10 12 - 13507677 z + 872314897 z - 37869633272 z + 1170684839462 z 14 18 16 - 26790097091916 z - 6317303546550712 z + 466643506215422 z 50 48 - 13393946311970165682792 z + 20687872907430768623446 z 20 36 + 67548032768993556 z + 7273891200065221379058 z 34 66 80 - 3307200368812482555716 z - 577775539427532402 z + 872314897 z 88 84 86 82 64 + z + 128043 z - 623 z - 13507677 z + 3994104107641733852 z 30 42 - 396909651439651073064 z - 26839260150537507583460 z 44 46 + 29269640097617141520354 z - 26839260150537507583460 z 58 56 - 396909651439651073064 z + 1255727839208630905434 z 54 52 - 3307200368812482555716 z + 7273891200065221379058 z 60 70 68 + 104028469963563625428 z - 6317303546550712 z + 67548032768993556 z 78 32 - 37869633272 z + 1255727839208630905434 z 38 40 - 13393946311970165682792 z + 20687872907430768623446 z 62 76 74 - 22501982921433763966 z + 1170684839462 z - 26790097091916 z 72 + 466643506215422 z )) And in Maple-input format, it is: -(1+23092369302257972968*z^28-5279571813985642550*z^26-423*z^2+ 994521311226054760*z^24-153201950703131594*z^22+71662*z^4-6652261*z^6+388695889 *z^8-15474061040*z^10+441761560730*z^12-9374037289324*z^14-1915400124440872*z^ 18+151821947032502*z^16-2446166816893131019392*z^50+3707050465249406348302*z^48 +19128893223936120*z^20+1363802147140979253142*z^36-640966706359405434116*z^34-\ 153201950703131594*z^66+388695889*z^80+z^88+71662*z^84-423*z^86-6652261*z^82+ 994521311226054760*z^64-83748184898704791784*z^30-4754194069913268718516*z^42+ 5164722923945492499508*z^44-4754194069913268718516*z^46-83748184898704791784*z^ 58+253190230485211312922*z^56-640966706359405434116*z^54+1363802147140979253142 *z^52+23092369302257972968*z^60-1915400124440872*z^70+19128893223936120*z^68-\ 15474061040*z^78+253190230485211312922*z^32-2446166816893131019392*z^38+ 3707050465249406348302*z^40-5279571813985642550*z^62+441761560730*z^76-\ 9374037289324*z^74+151821947032502*z^72)/(-1+z^2)/(1+104028469963563625428*z^28 -22501982921433763966*z^26-623*z^2+3994104107641733852*z^24-577775539427532402* z^22+128043*z^4-13507677*z^6+872314897*z^8-37869633272*z^10+1170684839462*z^12-\ 26790097091916*z^14-6317303546550712*z^18+466643506215422*z^16-\ 13393946311970165682792*z^50+20687872907430768623446*z^48+67548032768993556*z^ 20+7273891200065221379058*z^36-3307200368812482555716*z^34-577775539427532402*z ^66+872314897*z^80+z^88+128043*z^84-623*z^86-13507677*z^82+3994104107641733852* z^64-396909651439651073064*z^30-26839260150537507583460*z^42+ 29269640097617141520354*z^44-26839260150537507583460*z^46-396909651439651073064 *z^58+1255727839208630905434*z^56-3307200368812482555716*z^54+ 7273891200065221379058*z^52+104028469963563625428*z^60-6317303546550712*z^70+ 67548032768993556*z^68-37869633272*z^78+1255727839208630905434*z^32-\ 13393946311970165682792*z^38+20687872907430768623446*z^40-22501982921433763966* z^62+1170684839462*z^76-26790097091916*z^74+466643506215422*z^72) The first , 40, terms are: [0, 201, 0, 68420, 0, 23815673, 0, 8301305267, 0, 2893920616545, 0, 1008866928036703, 0, 351707977932804140, 0, 122611368475023969903, 0, 42744406943538101465787, 0, 14901427002271794238031315, 0, 5194890827520181111443076223, 0, 1811027274712494745820768179788, 0, 631354902210249990973609489573615, 0, 220101054318106502696380085260974729, 0, 76730970081735061569497582924854820315, 0, 26749720885885367295278314811579590362281, 0, 9325407546795800612371765018529448823676324, 0, 3250995637854824590280385103782177538231833849, 0, 1133352358523237352257854276229935899202777738009, 0, 395105903438789359175787159318367389041474279690985] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 24773695594920107838 z - 5478737294937180661 z - 407 z 24 22 4 6 + 998700360202802918 z - 149108683134513135 z + 66473 z - 6009958 z 8 10 12 14 + 345525395 z - 13658223069 z + 390319868500 z - 8351229161115 z 18 16 50 - 1767072017540415 z + 137257423354976 z - 3001704094540162169913 z 48 20 + 4614512510135796239973 z + 18092689436673982 z 36 34 + 1641299715015817624489 z - 753102251536794640028 z 66 80 88 84 86 - 149108683134513135 z + 345525395 z + z + 66473 z - 407 z 82 64 30 - 6009958 z + 998700360202802918 z - 92814744501163321621 z 42 44 - 5970031271928718805118 z + 6504685881826845333500 z 46 58 - 5970031271928718805118 z - 92814744501163321621 z 56 54 + 289336191566234751009 z - 753102251536794640028 z 52 60 + 1641299715015817624489 z + 24773695594920107838 z 70 68 78 - 1767072017540415 z + 18092689436673982 z - 13658223069 z 32 38 + 289336191566234751009 z - 3001704094540162169913 z 40 62 76 + 4614512510135796239973 z - 5478737294937180661 z + 390319868500 z 74 72 / 2 - 8351229161115 z + 137257423354976 z ) / ((-1 + z ) (1 / 28 26 2 + 107993946440851515558 z - 22608292353428460930 z - 578 z 24 22 4 + 3887474459289006775 z - 545835216903004474 z + 114144 z 6 8 10 12 - 11775668 z + 751868158 z - 32528751554 z + 1008666772734 z 14 18 16 - 23287282975158 z - 5675957386896978 z + 411397722524189 z 50 48 - 15959572334953193459382 z + 25029365785907277891619 z 20 36 + 62118612281156532 z + 8491038365848940496644 z 34 66 80 - 3765081239943741044712 z - 545835216903004474 z + 751868158 z 88 84 86 82 64 + z + 114144 z - 578 z - 11775668 z + 3887474459289006775 z 30 42 - 425649244785587438554 z - 32777770333033388410060 z 44 46 + 35859292835535546105320 z - 32777770333033388410060 z 58 56 - 425649244785587438554 z + 1389238458740470855578 z 54 52 - 3765081239943741044712 z + 8491038365848940496644 z 60 70 68 + 107993946440851515558 z - 5675957386896978 z + 62118612281156532 z 78 32 - 32528751554 z + 1389238458740470855578 z 38 40 - 15959572334953193459382 z + 25029365785907277891619 z 62 76 74 - 22608292353428460930 z + 1008666772734 z - 23287282975158 z 72 + 411397722524189 z )) And in Maple-input format, it is: -(1+24773695594920107838*z^28-5478737294937180661*z^26-407*z^2+ 998700360202802918*z^24-149108683134513135*z^22+66473*z^4-6009958*z^6+345525395 *z^8-13658223069*z^10+390319868500*z^12-8351229161115*z^14-1767072017540415*z^ 18+137257423354976*z^16-3001704094540162169913*z^50+4614512510135796239973*z^48 +18092689436673982*z^20+1641299715015817624489*z^36-753102251536794640028*z^34-\ 149108683134513135*z^66+345525395*z^80+z^88+66473*z^84-407*z^86-6009958*z^82+ 998700360202802918*z^64-92814744501163321621*z^30-5970031271928718805118*z^42+ 6504685881826845333500*z^44-5970031271928718805118*z^46-92814744501163321621*z^ 58+289336191566234751009*z^56-753102251536794640028*z^54+1641299715015817624489 *z^52+24773695594920107838*z^60-1767072017540415*z^70+18092689436673982*z^68-\ 13658223069*z^78+289336191566234751009*z^32-3001704094540162169913*z^38+ 4614512510135796239973*z^40-5478737294937180661*z^62+390319868500*z^76-\ 8351229161115*z^74+137257423354976*z^72)/(-1+z^2)/(1+107993946440851515558*z^28 -22608292353428460930*z^26-578*z^2+3887474459289006775*z^24-545835216903004474* z^22+114144*z^4-11775668*z^6+751868158*z^8-32528751554*z^10+1008666772734*z^12-\ 23287282975158*z^14-5675957386896978*z^18+411397722524189*z^16-\ 15959572334953193459382*z^50+25029365785907277891619*z^48+62118612281156532*z^ 20+8491038365848940496644*z^36-3765081239943741044712*z^34-545835216903004474*z ^66+751868158*z^80+z^88+114144*z^84-578*z^86-11775668*z^82+3887474459289006775* z^64-425649244785587438554*z^30-32777770333033388410060*z^42+ 35859292835535546105320*z^44-32777770333033388410060*z^46-425649244785587438554 *z^58+1389238458740470855578*z^56-3765081239943741044712*z^54+ 8491038365848940496644*z^52+107993946440851515558*z^60-5675957386896978*z^70+ 62118612281156532*z^68-32528751554*z^78+1389238458740470855578*z^32-\ 15959572334953193459382*z^38+25029365785907277891619*z^40-22608292353428460930* z^62+1008666772734*z^76-23287282975158*z^74+411397722524189*z^72) The first , 40, terms are: [0, 172, 0, 51339, 0, 15872951, 0, 4927655104, 0, 1530822337433, 0, 475630768359493, 0, 147784279266811352, 0, 45918690273089243087, 0, 14267616426873790966503, 0, 4433161116043873567252228, 0, 1377449361659662653979467585, 0, 427994095344925519523880285089, 0, 132984160140751880710501404934500, 0, 41320165534641206399544909313166103, 0, 12838792818911100923351856723757707503, 0, 3989204760531741851374315622913923482920, 0, 1239505524088785132208013518127329077885781, 0, 385132886497123910698180234558658953780064665, 0, 119666542326031778860727125107146547630027484432, 0, 37182182707156283981907830355474207968532678390503] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1626371881523829485568 z - 216872371343899479136 z - 440 z 24 22 4 + 24599643848034033544 z - 2357853197876043344 z + 80642 z 6 102 8 10 - 8491732 z - 29075842484 z + 589126011 z - 29075842484 z 12 14 18 + 1072354873234 z - 30582509942920 z - 12637618941877872 z 16 50 + 691543317128349 z - 474562305400135365919650208 z 48 20 + 299471512745580600316046938 z + 189452809397950844 z 36 34 + 1125489348158291125322832 z - 273448832543339878706944 z 66 80 - 165516473185701603369462704 z + 57504066065561598484492 z 100 90 88 + 1072354873234 z - 2357853197876043344 z + 24599643848034033544 z 84 94 + 1626371881523829485568 z - 12637618941877872 z 86 96 98 - 216872371343899479136 z + 691543317128349 z - 30582509942920 z 92 82 + 189452809397950844 z - 10432273654989169291456 z 64 112 110 106 + 299471512745580600316046938 z + z - 440 z - 8491732 z 108 30 + 80642 z - 10432273654989169291456 z 42 44 - 33849642436546466059246864 z + 80055861622536378215585604 z 46 58 - 165516473185701603369462704 z - 802465380242706891879090616 z 56 54 + 856871809898297376227833458 z - 802465380242706891879090616 z 52 60 + 659055114662830340195786396 z + 659055114662830340195786396 z 70 68 - 33849642436546466059246864 z + 80055861622536378215585604 z 78 32 - 273448832543339878706944 z + 57504066065561598484492 z 38 40 - 4020663644816657840262208 z + 12495526703010221202565248 z 62 76 - 474562305400135365919650208 z + 1125489348158291125322832 z 74 72 - 4020663644816657840262208 z + 12495526703010221202565248 z 104 / 2 28 + 589126011 z ) / ((-1 + z ) (1 + 6080236166433974070382 z / 26 2 24 - 771940971888950647847 z - 623 z + 83213964403803689712 z 22 4 6 102 - 7566745571010422041 z + 135876 z - 16062095 z - 64938196150 z 8 10 12 14 + 1218968835 z - 64938196150 z + 2565534249922 z - 78003145427720 z 18 16 - 36307463396877747 z + 1874298661741609 z 50 48 - 2561353703411029676327040077 z + 1590779033449727269176325622 z 20 36 + 575783316579086838 z + 5014124493392380285844002 z 34 66 - 1170087980888110435093051 z - 861665659379991889012409777 z 80 100 + 235750598334103792304704 z + 2565534249922 z 90 88 - 7566745571010422041 z + 83213964403803689712 z 84 94 + 6080236166433974070382 z - 36307463396877747 z 86 96 98 - 771940971888950647847 z + 1874298661741609 z - 78003145427720 z 92 82 + 575783316579086838 z - 40884574744895750833881 z 64 112 110 106 + 1590779033449727269176325622 z + z - 623 z - 16062095 z 108 30 + 135876 z - 40884574744895750833881 z 42 44 - 167274616826358777217520707 z + 406813606946467576278220106 z 46 58 - 861665659379991889012409777 z - 4412027482438958796612984167 z 56 54 + 4722172692754706584211526266 z - 4412027482438958796612984167 z 52 60 + 3598357640177152803754396778 z + 3598357640177152803754396778 z 70 68 - 167274616826358777217520707 z + 406813606946467576278220106 z 78 32 - 1170087980888110435093051 z + 235750598334103792304704 z 38 40 - 18598778106442581661932757 z + 59839326192807178816375280 z 62 76 - 2561353703411029676327040077 z + 5014124493392380285844002 z 74 72 - 18598778106442581661932757 z + 59839326192807178816375280 z 104 + 1218968835 z )) And in Maple-input format, it is: -(1+1626371881523829485568*z^28-216872371343899479136*z^26-440*z^2+ 24599643848034033544*z^24-2357853197876043344*z^22+80642*z^4-8491732*z^6-\ 29075842484*z^102+589126011*z^8-29075842484*z^10+1072354873234*z^12-\ 30582509942920*z^14-12637618941877872*z^18+691543317128349*z^16-\ 474562305400135365919650208*z^50+299471512745580600316046938*z^48+ 189452809397950844*z^20+1125489348158291125322832*z^36-273448832543339878706944 *z^34-165516473185701603369462704*z^66+57504066065561598484492*z^80+ 1072354873234*z^100-2357853197876043344*z^90+24599643848034033544*z^88+ 1626371881523829485568*z^84-12637618941877872*z^94-216872371343899479136*z^86+ 691543317128349*z^96-30582509942920*z^98+189452809397950844*z^92-\ 10432273654989169291456*z^82+299471512745580600316046938*z^64+z^112-440*z^110-\ 8491732*z^106+80642*z^108-10432273654989169291456*z^30-\ 33849642436546466059246864*z^42+80055861622536378215585604*z^44-\ 165516473185701603369462704*z^46-802465380242706891879090616*z^58+ 856871809898297376227833458*z^56-802465380242706891879090616*z^54+ 659055114662830340195786396*z^52+659055114662830340195786396*z^60-\ 33849642436546466059246864*z^70+80055861622536378215585604*z^68-\ 273448832543339878706944*z^78+57504066065561598484492*z^32-\ 4020663644816657840262208*z^38+12495526703010221202565248*z^40-\ 474562305400135365919650208*z^62+1125489348158291125322832*z^76-\ 4020663644816657840262208*z^74+12495526703010221202565248*z^72+589126011*z^104) /(-1+z^2)/(1+6080236166433974070382*z^28-771940971888950647847*z^26-623*z^2+ 83213964403803689712*z^24-7566745571010422041*z^22+135876*z^4-16062095*z^6-\ 64938196150*z^102+1218968835*z^8-64938196150*z^10+2565534249922*z^12-\ 78003145427720*z^14-36307463396877747*z^18+1874298661741609*z^16-\ 2561353703411029676327040077*z^50+1590779033449727269176325622*z^48+ 575783316579086838*z^20+5014124493392380285844002*z^36-\ 1170087980888110435093051*z^34-861665659379991889012409777*z^66+ 235750598334103792304704*z^80+2565534249922*z^100-7566745571010422041*z^90+ 83213964403803689712*z^88+6080236166433974070382*z^84-36307463396877747*z^94-\ 771940971888950647847*z^86+1874298661741609*z^96-78003145427720*z^98+ 575783316579086838*z^92-40884574744895750833881*z^82+ 1590779033449727269176325622*z^64+z^112-623*z^110-16062095*z^106+135876*z^108-\ 40884574744895750833881*z^30-167274616826358777217520707*z^42+ 406813606946467576278220106*z^44-861665659379991889012409777*z^46-\ 4412027482438958796612984167*z^58+4722172692754706584211526266*z^56-\ 4412027482438958796612984167*z^54+3598357640177152803754396778*z^52+ 3598357640177152803754396778*z^60-167274616826358777217520707*z^70+ 406813606946467576278220106*z^68-1170087980888110435093051*z^78+ 235750598334103792304704*z^32-18598778106442581661932757*z^38+ 59839326192807178816375280*z^40-2561353703411029676327040077*z^62+ 5014124493392380285844002*z^76-18598778106442581661932757*z^74+ 59839326192807178816375280*z^72+1218968835*z^104) The first , 40, terms are: [0, 184, 0, 58959, 0, 19380839, 0, 6380320740, 0, 2100706802669, 0, 691662524436397, 0, 227731891623672256, 0, 74981407078935838555, 0, 24687853917244103739011, 0, 8128550259266100130849860, 0, 2676349658904100234860937201, 0, 881196187517064713468921838625, 0, 290136499297089572972732954491004, 0, 95528316415260102899906490777289419, 0, 31452985954046064382942530041248706211, 0, 10355990375934755447325799801923949549160, 0, 3409741028185671802919010701134552812963277, 0, 1122667505206460449565983360350486716753183437, 0, 369641658069601623535905328680898294788037664316, 0, 121705629446642730799821541472065467845394193160831] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 11278891008765956014 z - 2606760782534327452 z - 401 z 24 22 4 6 + 499217647271770264 z - 78705999185762948 z + 63278 z - 5437509 z 8 10 12 14 + 293643167 z - 10827516910 z + 287778947856 z - 5725969141636 z 18 16 50 - 1054897431712626 z + 87678050469672 z - 1187863642780566507490 z 48 20 + 1803610086074899677456 z + 10133577355040770 z 36 34 + 660965804401841393248 z - 310236625840059994252 z 66 80 88 84 86 - 78705999185762948 z + 293643167 z + z + 63278 z - 401 z 82 64 30 - 5437509 z + 499217647271770264 z - 40651513226787318302 z 42 44 - 2316233878954224272202 z + 2517473902948314892180 z 46 58 - 2316233878954224272202 z - 40651513226787318302 z 56 54 + 122571016068123705512 z - 310236625840059994252 z 52 60 + 660965804401841393248 z + 11278891008765956014 z 70 68 78 - 1054897431712626 z + 10133577355040770 z - 10827516910 z 32 38 + 122571016068123705512 z - 1187863642780566507490 z 40 62 76 + 1803610086074899677456 z - 2606760782534327452 z + 287778947856 z 74 72 / 2 - 5725969141636 z + 87678050469672 z ) / ((-1 + z ) (1 / 28 26 2 + 50546215708091148504 z - 11060129632714276894 z - 595 z 24 22 4 + 1998250049447489084 z - 296326149249781154 z + 114251 z 6 8 10 12 - 11170561 z + 666664917 z - 26781241492 z + 769577970858 z 14 18 16 - 16480753923844 z - 3488068102271052 z + 270800846005818 z 50 48 - 6487173149327619264380 z + 10047377055383766872674 z 20 36 + 35794801424212320 z + 3512804916573949701022 z 34 66 80 - 1593655356784719729644 z - 296326149249781154 z + 666664917 z 88 84 86 82 64 + z + 114251 z - 595 z - 11170561 z + 1998250049447489084 z 30 42 - 191626626940075671684 z - 13059938924236987509676 z 44 46 + 14252342604898777290850 z - 13059938924236987509676 z 58 56 - 191626626940075671684 z + 604820211512535710670 z 54 52 - 1593655356784719729644 z + 3512804916573949701022 z 60 70 68 + 50546215708091148504 z - 3488068102271052 z + 35794801424212320 z 78 32 38 - 26781241492 z + 604820211512535710670 z - 6487173149327619264380 z 40 62 + 10047377055383766872674 z - 11060129632714276894 z 76 74 72 + 769577970858 z - 16480753923844 z + 270800846005818 z )) And in Maple-input format, it is: -(1+11278891008765956014*z^28-2606760782534327452*z^26-401*z^2+ 499217647271770264*z^24-78705999185762948*z^22+63278*z^4-5437509*z^6+293643167* z^8-10827516910*z^10+287778947856*z^12-5725969141636*z^14-1054897431712626*z^18 +87678050469672*z^16-1187863642780566507490*z^50+1803610086074899677456*z^48+ 10133577355040770*z^20+660965804401841393248*z^36-310236625840059994252*z^34-\ 78705999185762948*z^66+293643167*z^80+z^88+63278*z^84-401*z^86-5437509*z^82+ 499217647271770264*z^64-40651513226787318302*z^30-2316233878954224272202*z^42+ 2517473902948314892180*z^44-2316233878954224272202*z^46-40651513226787318302*z^ 58+122571016068123705512*z^56-310236625840059994252*z^54+660965804401841393248* z^52+11278891008765956014*z^60-1054897431712626*z^70+10133577355040770*z^68-\ 10827516910*z^78+122571016068123705512*z^32-1187863642780566507490*z^38+ 1803610086074899677456*z^40-2606760782534327452*z^62+287778947856*z^76-\ 5725969141636*z^74+87678050469672*z^72)/(-1+z^2)/(1+50546215708091148504*z^28-\ 11060129632714276894*z^26-595*z^2+1998250049447489084*z^24-296326149249781154*z ^22+114251*z^4-11170561*z^6+666664917*z^8-26781241492*z^10+769577970858*z^12-\ 16480753923844*z^14-3488068102271052*z^18+270800846005818*z^16-\ 6487173149327619264380*z^50+10047377055383766872674*z^48+35794801424212320*z^20 +3512804916573949701022*z^36-1593655356784719729644*z^34-296326149249781154*z^ 66+666664917*z^80+z^88+114251*z^84-595*z^86-11170561*z^82+1998250049447489084*z ^64-191626626940075671684*z^30-13059938924236987509676*z^42+ 14252342604898777290850*z^44-13059938924236987509676*z^46-191626626940075671684 *z^58+604820211512535710670*z^56-1593655356784719729644*z^54+ 3512804916573949701022*z^52+50546215708091148504*z^60-3488068102271052*z^70+ 35794801424212320*z^68-26781241492*z^78+604820211512535710670*z^32-\ 6487173149327619264380*z^38+10047377055383766872674*z^40-11060129632714276894*z ^62+769577970858*z^76-16480753923844*z^74+270800846005818*z^72) The first , 40, terms are: [0, 195, 0, 64652, 0, 21984925, 0, 7494337737, 0, 2555772731415, 0, 871661923800543, 0, 297291058114917316, 0, 101395190281367533677, 0, 34582249989781882696243, 0, 11794763097780557213925683, 0, 4022770138803678442445461253, 0, 1372022448640235670890771711972, 0, 467947592998315698557608637364719, 0, 159600121787617059016239136638618463, 0, 54433870925899179033578750844349908513, 0, 18565438865963524162194865168678853674301, 0, 6332004585102429918605702107642435445787564, 0, 2159619406536718782631555098194417517072222635, 0, 736568636110013106285784733716947905726123634353, 0, 251217114487331124965263462613103189522367629836465] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {3, 7}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 10579715766528829 z + 6243181645631460 z + 354 z 24 22 4 6 - 2821519876289445 z + 973010633734777 z - 46999 z + 3234555 z 8 10 12 14 - 132646714 z + 3513645337 z - 63341007999 z + 805532407806 z 18 16 50 + 50287172696097 z - 7413506184819 z + 63341007999 z 48 20 36 - 805532407806 z - 254747766914820 z - 6243181645631460 z 34 30 42 + 10579715766528829 z + 13763917948705215 z + 254747766914820 z 44 46 58 56 - 50287172696097 z + 7413506184819 z + 46999 z - 3234555 z 54 52 60 32 + 132646714 z - 3513645337 z - 354 z - 13763917948705215 z 38 40 62 / + 2821519876289445 z - 973010633734777 z + z ) / (1 / 28 26 2 + 88812628097708352 z - 45962338047311198 z - 522 z 24 22 4 6 + 18238691787723608 z - 5534513051343862 z + 85560 z - 6885028 z 8 10 12 14 + 323273188 z - 9706879666 z + 197323272296 z - 2823295824222 z 18 16 50 - 223422589232880 z + 29231520625838 z - 2823295824222 z 48 20 36 + 29231520625838 z + 1278686562477376 z + 88812628097708352 z 34 64 30 - 131792142759095920 z + z - 131792142759095920 z 42 44 46 - 5534513051343862 z + 1278686562477376 z - 223422589232880 z 58 56 54 52 - 6885028 z + 323273188 z - 9706879666 z + 197323272296 z 60 32 38 + 85560 z + 150311642473508847 z - 45962338047311198 z 40 62 + 18238691787723608 z - 522 z ) And in Maple-input format, it is: -(-1-10579715766528829*z^28+6243181645631460*z^26+354*z^2-2821519876289445*z^24 +973010633734777*z^22-46999*z^4+3234555*z^6-132646714*z^8+3513645337*z^10-\ 63341007999*z^12+805532407806*z^14+50287172696097*z^18-7413506184819*z^16+ 63341007999*z^50-805532407806*z^48-254747766914820*z^20-6243181645631460*z^36+ 10579715766528829*z^34+13763917948705215*z^30+254747766914820*z^42-\ 50287172696097*z^44+7413506184819*z^46+46999*z^58-3234555*z^56+132646714*z^54-\ 3513645337*z^52-354*z^60-13763917948705215*z^32+2821519876289445*z^38-\ 973010633734777*z^40+z^62)/(1+88812628097708352*z^28-45962338047311198*z^26-522 *z^2+18238691787723608*z^24-5534513051343862*z^22+85560*z^4-6885028*z^6+ 323273188*z^8-9706879666*z^10+197323272296*z^12-2823295824222*z^14-\ 223422589232880*z^18+29231520625838*z^16-2823295824222*z^50+29231520625838*z^48 +1278686562477376*z^20+88812628097708352*z^36-131792142759095920*z^34+z^64-\ 131792142759095920*z^30-5534513051343862*z^42+1278686562477376*z^44-\ 223422589232880*z^46-6885028*z^58+323273188*z^56-9706879666*z^54+197323272296*z ^52+85560*z^60+150311642473508847*z^32-45962338047311198*z^38+18238691787723608 *z^40-522*z^62) The first , 40, terms are: [0, 168, 0, 49135, 0, 14924863, 0, 4552846116, 0, 1389793583797, 0, 424301582135449, 0, 129542166776970660, 0, 39550353216211921663, 0, 12075084751836250786999, 0, 3686635053697475539376472, 0, 1125563861036772255917053825, 0, 343645087747267658425317756373, 0, 104918033417827234781662514473656, 0, 32032448995849691501235299005320571, 0, 9779803866432841451830452491059878835, 0, 2985864854813144704357159305455217304020, 0, 911612242235515134440854641520335500492269, 0, 278323675251429984067224662800176793616284369, 0, 84974800267648751687284721696232684057802576020, 0, 25943594895420649597933957595605476517346615876779] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 745258 z + 11895744 z + 279 z - 102404246 z 22 4 6 8 10 + 498844698 z - 22832 z + 745258 z - 11895744 z + 102404246 z 12 14 18 16 - 498844698 z + 1408829850 z + 2354782167 z - 2354782167 z 20 34 30 32 / 30 - 1408829850 z + z + 22832 z - 279 z ) / (-2081582 z / 32 24 26 34 28 + 50803 z + 2621890652 z - 429220328 z - 447 z + 40613018 z 18 16 6 36 - 26741949143 z + 20639753613 z - 2081582 z + z + 1 12 10 4 8 2 + 2621890652 z - 429220328 z + 50803 z + 40613018 z - 447 z 22 20 14 - 9514886572 z + 20639753613 z - 9514886572 z ) And in Maple-input format, it is: -(-1-745258*z^28+11895744*z^26+279*z^2-102404246*z^24+498844698*z^22-22832*z^4+ 745258*z^6-11895744*z^8+102404246*z^10-498844698*z^12+1408829850*z^14+ 2354782167*z^18-2354782167*z^16-1408829850*z^20+z^34+22832*z^30-279*z^32)/(-\ 2081582*z^30+50803*z^32+2621890652*z^24-429220328*z^26-447*z^34+40613018*z^28-\ 26741949143*z^18+20639753613*z^16-2081582*z^6+z^36+1+2621890652*z^12-429220328* z^10+50803*z^4+40613018*z^8-447*z^2-9514886572*z^22+20639753613*z^20-9514886572 *z^14) The first , 40, terms are: [0, 168, 0, 47125, 0, 13866295, 0, 4125130992, 0, 1231082549347, 0, 367744797346123, 0, 109882678627786560, 0, 32835913382356920559, 0, 9812511211713453786061, 0, 2932341904922684686368504, 0, 876294458200958784966119017, 0, 261870049513452863735332007449, 0, 78256728420091438365444707632728, 0, 23386087669743362161805619108919261, 0, 6988652836846143168705140843853185791, 0, 2088475398212935508329893893893514806944, 0, 624115920254246820748591769705071324535803, 0, 186509586109770677904055303849582234235082931, 0, 55736161484586360632535163121795755564166402448, 0, 16656085952370515525347252347209942537534791505959] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {3, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 29683066020664260 z - 13393157671457670 z - 331 z 24 22 4 6 + 4787630374465314 z - 1349664544785712 z + 40370 z - 2605497 z 8 10 12 14 + 103276019 z - 2726424732 z + 50440409575 z - 677120208261 z 18 16 50 - 51213219739455 z + 6763279434762 z - 51213219739455 z 48 20 36 + 298197957995853 z + 298197957995853 z + 73386016189971126 z 34 66 64 30 - 82151114164879208 z - 331 z + 40370 z - 52291452091990962 z 42 44 46 - 13393157671457670 z + 4787630374465314 z - 1349664544785712 z 58 56 54 52 - 2726424732 z + 50440409575 z - 677120208261 z + 6763279434762 z 60 68 32 38 + 103276019 z + z + 73386016189971126 z - 52291452091990962 z 40 62 / 2 + 29683066020664260 z - 2605497 z ) / ((-1 + z ) (1 / 28 26 2 + 153716270527109854 z - 66106238294161012 z - 502 z 24 22 4 6 + 22295400438471106 z - 5880922734288824 z + 74391 z - 5508720 z 8 10 12 14 + 244732605 z - 7156139646 z + 145627943753 z - 2140194515288 z 18 16 50 - 191762040969254 z + 23311781904331 z - 191762040969254 z 48 20 36 + 1207697488735133 z + 1207697488735133 z + 402866322853408538 z 34 66 64 30 - 454416814385067540 z - 502 z + 74391 z - 280719694675827240 z 42 44 46 - 66106238294161012 z + 22295400438471106 z - 5880922734288824 z 58 56 54 - 7156139646 z + 145627943753 z - 2140194515288 z 52 60 68 32 + 23311781904331 z + 244732605 z + z + 402866322853408538 z 38 40 62 - 280719694675827240 z + 153716270527109854 z - 5508720 z )) And in Maple-input format, it is: -(1+29683066020664260*z^28-13393157671457670*z^26-331*z^2+4787630374465314*z^24 -1349664544785712*z^22+40370*z^4-2605497*z^6+103276019*z^8-2726424732*z^10+ 50440409575*z^12-677120208261*z^14-51213219739455*z^18+6763279434762*z^16-\ 51213219739455*z^50+298197957995853*z^48+298197957995853*z^20+73386016189971126 *z^36-82151114164879208*z^34-331*z^66+40370*z^64-52291452091990962*z^30-\ 13393157671457670*z^42+4787630374465314*z^44-1349664544785712*z^46-2726424732*z ^58+50440409575*z^56-677120208261*z^54+6763279434762*z^52+103276019*z^60+z^68+ 73386016189971126*z^32-52291452091990962*z^38+29683066020664260*z^40-2605497*z^ 62)/(-1+z^2)/(1+153716270527109854*z^28-66106238294161012*z^26-502*z^2+ 22295400438471106*z^24-5880922734288824*z^22+74391*z^4-5508720*z^6+244732605*z^ 8-7156139646*z^10+145627943753*z^12-2140194515288*z^14-191762040969254*z^18+ 23311781904331*z^16-191762040969254*z^50+1207697488735133*z^48+1207697488735133 *z^20+402866322853408538*z^36-454416814385067540*z^34-502*z^66+74391*z^64-\ 280719694675827240*z^30-66106238294161012*z^42+22295400438471106*z^44-\ 5880922734288824*z^46-7156139646*z^58+145627943753*z^56-2140194515288*z^54+ 23311781904331*z^52+244732605*z^60+z^68+402866322853408538*z^32-\ 280719694675827240*z^38+153716270527109854*z^40-5508720*z^62) The first , 40, terms are: [0, 172, 0, 51993, 0, 16248497, 0, 5092412028, 0, 1596500194105, 0, 500530555098601, 0, 156925759759863516, 0, 49199211978574490145, 0, 15424890435634722489833, 0, 4835997120828152079914444, 0, 1516177264538531485563573905, 0, 475350468689119949788914718193, 0, 149031431465318899550475941528460, 0, 46724194100284294366652037909792969, 0, 14648925349887367929684368729817578433, 0, 4592717285738809064479893411106536112156, 0, 1439904400010236687019225133855925207827721, 0, 451437472018336405151181183720079756363980057, 0, 141534251260609717726594466148035161271429271868, 0, 44373685220104525196737756353954062373870191560209] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 5}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 405652146352 z + 806357446462 z + 326 z - 1134017380563 z 22 4 6 8 10 + 1134017380563 z - 35352 z + 1667937 z - 41735236 z + 615759062 z 12 14 18 16 - 5711796830 z + 34761150584 z + 405652146352 z - 142947454095 z 20 36 34 30 - 806357446462 z - 615759062 z + 5711796830 z + 142947454095 z 42 44 46 32 38 40 + 35352 z - 326 z + z - 34761150584 z + 41735236 z - 1667937 z / 28 26 2 ) / (1 + 6474467914929 z - 10814968753712 z - 492 z / 24 22 4 6 + 12823772520758 z - 10814968753712 z + 71227 z - 4049688 z 8 10 12 14 + 119295219 z - 2072013240 z + 22753355604 z - 164751808988 z 18 16 48 20 - 2735599257656 z + 808169359665 z + z + 6474467914929 z 36 34 30 42 + 22753355604 z - 164751808988 z - 2735599257656 z - 4049688 z 44 46 32 38 40 + 71227 z - 492 z + 808169359665 z - 2072013240 z + 119295219 z ) And in Maple-input format, it is: -(-1-405652146352*z^28+806357446462*z^26+326*z^2-1134017380563*z^24+ 1134017380563*z^22-35352*z^4+1667937*z^6-41735236*z^8+615759062*z^10-5711796830 *z^12+34761150584*z^14+405652146352*z^18-142947454095*z^16-806357446462*z^20-\ 615759062*z^36+5711796830*z^34+142947454095*z^30+35352*z^42-326*z^44+z^46-\ 34761150584*z^32+41735236*z^38-1667937*z^40)/(1+6474467914929*z^28-\ 10814968753712*z^26-492*z^2+12823772520758*z^24-10814968753712*z^22+71227*z^4-\ 4049688*z^6+119295219*z^8-2072013240*z^10+22753355604*z^12-164751808988*z^14-\ 2735599257656*z^18+808169359665*z^16+z^48+6474467914929*z^20+22753355604*z^36-\ 164751808988*z^34-2735599257656*z^30-4049688*z^42+71227*z^44-492*z^46+ 808169359665*z^32-2072013240*z^38+119295219*z^40) The first , 40, terms are: [0, 166, 0, 45797, 0, 13090193, 0, 3773080262, 0, 1091097121253, 0, 315949342839309, 0, 91540947364804006, 0, 26528622445043005865, 0, 7688756217036975654061, 0, 2228511935408195135218694, 0, 645923516026465757584863033, 0, 187219138118054940317793149961, 0, 54265103838885527602332671518918, 0, 15728653849045802031939512924351325, 0, 4558927511050178773347790578369593529, 0, 1321398800980968052080149610291292705638, 0, 383005638170228991970778057758175041499133, 0, 111013665198067840692974621208473331741526645, 0, 32177160893750609987804465946724795774053157894, 0, 9326506675374268671167797780310515342746871489281] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 23439451871540207870 z - 5232771410487814593 z - 411 z 24 22 4 6 + 962685758205357418 z - 145004387617995619 z + 67501 z - 6115958 z 8 10 12 14 + 351273023 z - 13838899041 z + 393529622900 z - 8369659878095 z 18 16 50 - 1746395979571131 z + 136641851350972 z - 2726930611044443573893 z 48 20 + 4172745302272374737541 z + 17741887432291454 z 36 34 + 1500313998831088690885 z - 693581174959209346892 z 66 80 88 84 86 - 145004387617995619 z + 351273023 z + z + 67501 z - 411 z 82 64 30 - 6115958 z + 962685758205357418 z - 86993341972660749473 z 42 44 - 5383062642236253901934 z + 5859471637659762795260 z 46 58 - 5383062642236253901934 z - 86993341972660749473 z 56 54 + 268732164304540285237 z - 693581174959209346892 z 52 60 + 1500313998831088690885 z + 23439451871540207870 z 70 68 78 - 1746395979571131 z + 17741887432291454 z - 13838899041 z 32 38 + 268732164304540285237 z - 2726930611044443573893 z 40 62 76 + 4172745302272374737541 z - 5232771410487814593 z + 393529622900 z 74 72 / - 8369659878095 z + 136641851350972 z ) / (-1 / 28 26 2 - 124601096366011171886 z + 25579319438101295835 z + 585 z 24 22 4 - 4327316902218902067 z + 599328092487625036 z - 116768 z 6 8 10 12 + 12148872 z - 780796522 z + 33973213274 z - 1059560454150 z 14 18 16 + 24620742176382 z + 6098463427890593 z - 438212258043905 z 50 48 + 36677042852310251991171 z - 51459866271572468919523 z 20 36 - 67415334769895276 z - 11152479652164043222100 z 34 66 80 + 4741328158097132678790 z + 4327316902218902067 z - 33973213274 z 90 88 84 86 82 + z - 585 z - 12148872 z + 116768 z + 780796522 z 64 30 - 25579319438101295835 z + 502894558845489328326 z 42 44 + 51459866271572468919523 z - 60939445246940169326456 z 46 58 + 60939445246940169326456 z + 1689359989080190991478 z 56 54 - 4741328158097132678790 z + 11152479652164043222100 z 52 60 - 22041694668652311273532 z - 502894558845489328326 z 70 68 78 + 67415334769895276 z - 599328092487625036 z + 1059560454150 z 32 38 - 1689359989080190991478 z + 22041694668652311273532 z 40 62 - 36677042852310251991171 z + 124601096366011171886 z 76 74 72 - 24620742176382 z + 438212258043905 z - 6098463427890593 z ) And in Maple-input format, it is: -(1+23439451871540207870*z^28-5232771410487814593*z^26-411*z^2+ 962685758205357418*z^24-145004387617995619*z^22+67501*z^4-6115958*z^6+351273023 *z^8-13838899041*z^10+393529622900*z^12-8369659878095*z^14-1746395979571131*z^ 18+136641851350972*z^16-2726930611044443573893*z^50+4172745302272374737541*z^48 +17741887432291454*z^20+1500313998831088690885*z^36-693581174959209346892*z^34-\ 145004387617995619*z^66+351273023*z^80+z^88+67501*z^84-411*z^86-6115958*z^82+ 962685758205357418*z^64-86993341972660749473*z^30-5383062642236253901934*z^42+ 5859471637659762795260*z^44-5383062642236253901934*z^46-86993341972660749473*z^ 58+268732164304540285237*z^56-693581174959209346892*z^54+1500313998831088690885 *z^52+23439451871540207870*z^60-1746395979571131*z^70+17741887432291454*z^68-\ 13838899041*z^78+268732164304540285237*z^32-2726930611044443573893*z^38+ 4172745302272374737541*z^40-5232771410487814593*z^62+393529622900*z^76-\ 8369659878095*z^74+136641851350972*z^72)/(-1-124601096366011171886*z^28+ 25579319438101295835*z^26+585*z^2-4327316902218902067*z^24+599328092487625036*z ^22-116768*z^4+12148872*z^6-780796522*z^8+33973213274*z^10-1059560454150*z^12+ 24620742176382*z^14+6098463427890593*z^18-438212258043905*z^16+ 36677042852310251991171*z^50-51459866271572468919523*z^48-67415334769895276*z^ 20-11152479652164043222100*z^36+4741328158097132678790*z^34+4327316902218902067 *z^66-33973213274*z^80+z^90-585*z^88-12148872*z^84+116768*z^86+780796522*z^82-\ 25579319438101295835*z^64+502894558845489328326*z^30+51459866271572468919523*z^ 42-60939445246940169326456*z^44+60939445246940169326456*z^46+ 1689359989080190991478*z^58-4741328158097132678790*z^56+11152479652164043222100 *z^54-22041694668652311273532*z^52-502894558845489328326*z^60+67415334769895276 *z^70-599328092487625036*z^68+1059560454150*z^78-1689359989080190991478*z^32+ 22041694668652311273532*z^38-36677042852310251991171*z^40+124601096366011171886 *z^62-24620742176382*z^76+438212258043905*z^74-6098463427890593*z^72) The first , 40, terms are: [0, 174, 0, 52523, 0, 16441237, 0, 5169498210, 0, 1626717014295, 0, 511975502765379, 0, 161139750827156050, 0, 50717742087105470821, 0, 15963127798605080252151, 0, 5024308168045414998227478, 0, 1581374000355257518161312337, 0, 497728982355699702017214724449, 0, 156657527740348768378635369078630, 0, 49307116727074345548968991295708135, 0, 15519150568696085156932979379564281845, 0, 4884569416728347069512882624397366819538, 0, 1537392029392983909223158842388525359945571, 0, 483885896668058586572227292352527392139791767, 0, 152300491038014733959467135775313735386465384898, 0, 47935762811325816516505875537825512784389605244901] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 77457513239472863 z - 29744561185489769 z - 324 z 24 22 4 6 + 9169731426451023 z - 2258706362944348 z + 38733 z - 2488267 z 8 10 12 14 + 99938573 z - 2721302096 z + 52825271415 z - 756281897150 z 18 16 50 - 68087577984414 z + 8181524785674 z - 2258706362944348 z 48 20 36 + 9169731426451023 z + 441823227529165 z + 419577880943909492 z 34 66 64 - 377694386306189004 z - 2488267 z + 99938573 z 30 42 44 - 162508395062650088 z - 162508395062650088 z + 77457513239472863 z 46 58 56 - 29744561185489769 z - 756281897150 z + 8181524785674 z 54 52 60 70 - 68087577984414 z + 441823227529165 z + 52825271415 z - 324 z 68 32 38 + 38733 z + 275408653062836259 z - 377694386306189004 z 40 62 72 / 2 + 275408653062836259 z - 2721302096 z + z ) / ((-1 + z ) (1 / 28 26 2 + 386451421784705796 z - 140868216035294286 z - 496 z 24 22 4 6 + 40885728115265323 z - 9416979049433360 z + 71174 z - 5182994 z 8 10 12 14 + 231134047 z - 6924693816 z + 147182909968 z - 2299554857840 z 18 16 50 - 244294563703928 z + 27066816071697 z - 9416979049433360 z 48 20 36 + 40885728115265323 z + 1712753165791202 z + 2313418286259876962 z 34 66 64 - 2068869437255247912 z - 5182994 z + 231134047 z 30 42 44 - 845968248427028760 z - 845968248427028760 z + 386451421784705796 z 46 58 56 - 140868216035294286 z - 2299554857840 z + 27066816071697 z 54 52 60 70 - 244294563703928 z + 1712753165791202 z + 147182909968 z - 496 z 68 32 38 + 71174 z + 1479583278880885169 z - 2068869437255247912 z 40 62 72 + 1479583278880885169 z - 6924693816 z + z )) And in Maple-input format, it is: -(1+77457513239472863*z^28-29744561185489769*z^26-324*z^2+9169731426451023*z^24 -2258706362944348*z^22+38733*z^4-2488267*z^6+99938573*z^8-2721302096*z^10+ 52825271415*z^12-756281897150*z^14-68087577984414*z^18+8181524785674*z^16-\ 2258706362944348*z^50+9169731426451023*z^48+441823227529165*z^20+ 419577880943909492*z^36-377694386306189004*z^34-2488267*z^66+99938573*z^64-\ 162508395062650088*z^30-162508395062650088*z^42+77457513239472863*z^44-\ 29744561185489769*z^46-756281897150*z^58+8181524785674*z^56-68087577984414*z^54 +441823227529165*z^52+52825271415*z^60-324*z^70+38733*z^68+275408653062836259*z ^32-377694386306189004*z^38+275408653062836259*z^40-2721302096*z^62+z^72)/(-1+z ^2)/(1+386451421784705796*z^28-140868216035294286*z^26-496*z^2+ 40885728115265323*z^24-9416979049433360*z^22+71174*z^4-5182994*z^6+231134047*z^ 8-6924693816*z^10+147182909968*z^12-2299554857840*z^14-244294563703928*z^18+ 27066816071697*z^16-9416979049433360*z^50+40885728115265323*z^48+ 1712753165791202*z^20+2313418286259876962*z^36-2068869437255247912*z^34-5182994 *z^66+231134047*z^64-845968248427028760*z^30-845968248427028760*z^42+ 386451421784705796*z^44-140868216035294286*z^46-2299554857840*z^58+ 27066816071697*z^56-244294563703928*z^54+1712753165791202*z^52+147182909968*z^ 60-496*z^70+71174*z^68+1479583278880885169*z^32-2068869437255247912*z^38+ 1479583278880885169*z^40-6924693816*z^62+z^72) The first , 40, terms are: [0, 173, 0, 53044, 0, 16729859, 0, 5285669039, 0, 1670202282919, 0, 527769599153051, 0, 166770939083919948, 0, 52698282883076041521, 0, 16652236313567699742729, 0, 5261973634617505699791541, 0, 1662741629488030147724805077, 0, 525413070941350295442653410380, 0, 166026332788022148199001479010479, 0, 52462994743691784536700896507853627, 0, 16577887201731339388968940381000469315, 0, 5238479911716766476259711390017260097671, 0, 1655317800847036528641771964645973185998868, 0, 523067200405295763869364261792339786879608865, 0, 165285056440419700105696817501750602758310719693, 0, 52228757340059999544226395008967608676285337604469] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1537659234676114053014 z - 204300474207090680633 z - 437 z 24 22 4 + 23118042498613159408 z - 2213388844453753411 z + 79338 z 6 102 8 10 - 8274587 z - 27866529925 z + 569017508 z - 27866529925 z 12 14 18 + 1021004796526 z - 28961720240555 z - 11884321674522577 z 16 50 + 652186393520052 z - 481628798624805388885661333 z 48 20 + 302521781612977926146393224 z + 177884056612001230 z 36 34 + 1090116346493146237710630 z - 262969923527462178661197 z 66 80 - 166253840249060334513998179 z + 54940092167611973642012 z 100 90 88 + 1021004796526 z - 2213388844453753411 z + 23118042498613159408 z 84 94 + 1537659234676114053014 z - 11884321674522577 z 86 96 98 - 204300474207090680633 z + 652186393520052 z - 28961720240555 z 92 82 + 177884056612001230 z - 9910176125773347020623 z 64 112 110 106 + 302521781612977926146393224 z + z - 437 z - 8274587 z 108 30 42 + 79338 z - 9910176125773347020623 z - 33538314297796346099422261 z 44 46 + 79888879496814162534506214 z - 166253840249060334513998179 z 58 56 - 818952542980752077125477891 z + 875100905404186172922602738 z 54 52 - 818952542980752077125477891 z + 671174774607084927800192938 z 60 70 + 671174774607084927800192938 z - 33538314297796346099422261 z 68 78 + 79888879496814162534506214 z - 262969923527462178661197 z 32 38 + 54940092167611973642012 z - 3923683975085521688384903 z 40 62 + 12287760939489491371031704 z - 481628798624805388885661333 z 76 74 + 1090116346493146237710630 z - 3923683975085521688384903 z 72 104 / 2 + 12287760939489491371031704 z + 569017508 z ) / ((-1 + z ) (1 / 28 26 2 + 5700703477295794227372 z - 722314266388418021888 z - 599 z 24 22 4 + 77784569751372480518 z - 7072515428703488734 z + 129632 z 6 102 8 10 - 15279764 z - 61477250282 z + 1156924942 z - 61477250282 z 12 14 18 + 2422094553468 z - 73435379717095 z - 34017205396932789 z 16 50 + 1759930978225408 z - 2534270300493946959080698565 z 48 20 + 1567832516891650246611645168 z + 538610957109694540 z 36 34 + 4779590532019942442029204 z - 1109560262760573217941611 z 66 80 - 845234843419380507855949535 z + 222532763569988756471608 z 100 90 88 + 2422094553468 z - 7072515428703488734 z + 77784569751372480518 z 84 94 + 5700703477295794227372 z - 34017205396932789 z 86 96 98 - 722314266388418021888 z + 1759930978225408 z - 73435379717095 z 92 82 + 538610957109694540 z - 38445333468848282923257 z 64 112 110 106 + 1567832516891650246611645168 z + z - 599 z - 15279764 z 108 30 + 129632 z - 38445333468848282923257 z 42 44 - 162270841723764064053009130 z + 396926241247107686594306820 z 46 58 - 845234843419380507855949535 z - 4385989332637427260709936978 z 56 54 + 4697162006766991898603496836 z - 4385989332637427260709936978 z 52 60 + 3570691171658271155288727616 z + 3570691171658271155288727616 z 70 68 - 162270841723764064053009130 z + 396926241247107686594306820 z 78 32 - 1109560262760573217941611 z + 222532763569988756471608 z 38 40 - 17829516487265717879555620 z + 57704734275655565561290846 z 62 76 - 2534270300493946959080698565 z + 4779590532019942442029204 z 74 72 - 17829516487265717879555620 z + 57704734275655565561290846 z 104 + 1156924942 z )) And in Maple-input format, it is: -(1+1537659234676114053014*z^28-204300474207090680633*z^26-437*z^2+ 23118042498613159408*z^24-2213388844453753411*z^22+79338*z^4-8274587*z^6-\ 27866529925*z^102+569017508*z^8-27866529925*z^10+1021004796526*z^12-\ 28961720240555*z^14-11884321674522577*z^18+652186393520052*z^16-\ 481628798624805388885661333*z^50+302521781612977926146393224*z^48+ 177884056612001230*z^20+1090116346493146237710630*z^36-262969923527462178661197 *z^34-166253840249060334513998179*z^66+54940092167611973642012*z^80+ 1021004796526*z^100-2213388844453753411*z^90+23118042498613159408*z^88+ 1537659234676114053014*z^84-11884321674522577*z^94-204300474207090680633*z^86+ 652186393520052*z^96-28961720240555*z^98+177884056612001230*z^92-\ 9910176125773347020623*z^82+302521781612977926146393224*z^64+z^112-437*z^110-\ 8274587*z^106+79338*z^108-9910176125773347020623*z^30-\ 33538314297796346099422261*z^42+79888879496814162534506214*z^44-\ 166253840249060334513998179*z^46-818952542980752077125477891*z^58+ 875100905404186172922602738*z^56-818952542980752077125477891*z^54+ 671174774607084927800192938*z^52+671174774607084927800192938*z^60-\ 33538314297796346099422261*z^70+79888879496814162534506214*z^68-\ 262969923527462178661197*z^78+54940092167611973642012*z^32-\ 3923683975085521688384903*z^38+12287760939489491371031704*z^40-\ 481628798624805388885661333*z^62+1090116346493146237710630*z^76-\ 3923683975085521688384903*z^74+12287760939489491371031704*z^72+569017508*z^104) /(-1+z^2)/(1+5700703477295794227372*z^28-722314266388418021888*z^26-599*z^2+ 77784569751372480518*z^24-7072515428703488734*z^22+129632*z^4-15279764*z^6-\ 61477250282*z^102+1156924942*z^8-61477250282*z^10+2422094553468*z^12-\ 73435379717095*z^14-34017205396932789*z^18+1759930978225408*z^16-\ 2534270300493946959080698565*z^50+1567832516891650246611645168*z^48+ 538610957109694540*z^20+4779590532019942442029204*z^36-\ 1109560262760573217941611*z^34-845234843419380507855949535*z^66+ 222532763569988756471608*z^80+2422094553468*z^100-7072515428703488734*z^90+ 77784569751372480518*z^88+5700703477295794227372*z^84-34017205396932789*z^94-\ 722314266388418021888*z^86+1759930978225408*z^96-73435379717095*z^98+ 538610957109694540*z^92-38445333468848282923257*z^82+ 1567832516891650246611645168*z^64+z^112-599*z^110-15279764*z^106+129632*z^108-\ 38445333468848282923257*z^30-162270841723764064053009130*z^42+ 396926241247107686594306820*z^44-845234843419380507855949535*z^46-\ 4385989332637427260709936978*z^58+4697162006766991898603496836*z^56-\ 4385989332637427260709936978*z^54+3570691171658271155288727616*z^52+ 3570691171658271155288727616*z^60-162270841723764064053009130*z^70+ 396926241247107686594306820*z^68-1109560262760573217941611*z^78+ 222532763569988756471608*z^32-17829516487265717879555620*z^38+ 57704734275655565561290846*z^40-2534270300493946959080698565*z^62+ 4779590532019942442029204*z^76-17829516487265717879555620*z^74+ 57704734275655565561290846*z^72+1156924942*z^104) The first , 40, terms are: [0, 163, 0, 46907, 0, 14051356, 0, 4230612433, 0, 1274952132957, 0, 384299498363099, 0, 115841579271726391, 0, 34919119865847424817, 0, 10525992141584933327705, 0, 3172948358606799698745052, 0, 956451633683173351209183883, 0, 288312208884627116485164918039, 0, 86908660430320162130408328458805, 0, 26197694840554894313850187665873437, 0, 7897017532573829751492564062341521935, 0, 2380472262665192424824948822445454467003, 0, 717568141393236418611806055260491600989724, 0, 216303313262596676615876171146306942789151609, 0, 65202341951202704442112555058776264961002547305, 0, 19654555132778008794231788749863595039645870440047] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 66956361925221331 z - 25934303239724973 z - 334 z 24 22 4 6 + 8088857447486599 z - 2022777484090428 z + 41045 z - 2658087 z 8 10 12 14 + 105836793 z - 2825563640 z + 53453692125 z - 744148594998 z 18 16 50 - 63533630066746 z + 7831214016350 z - 2022777484090428 z 48 20 36 + 8088857447486599 z + 403163912345703 z + 358076421416517932 z 34 66 64 - 322556879182671720 z - 2658087 z + 105836793 z 30 42 44 - 139632352422830822 z - 139632352422830822 z + 66956361925221331 z 46 58 56 - 25934303239724973 z - 744148594998 z + 7831214016350 z 54 52 60 70 - 63533630066746 z + 403163912345703 z + 53453692125 z - 334 z 68 32 38 + 41045 z + 235716114473355379 z - 322556879182671720 z 40 62 72 / 2 + 235716114473355379 z - 2825563640 z + z ) / ((-1 + z ) (1 / 28 26 2 + 338750181485158420 z - 124501269852965862 z - 514 z 24 22 4 6 + 36547781986291055 z - 8544308351727414 z + 76606 z - 5642342 z 8 10 12 14 + 249453395 z - 7318527422 z + 151369939252 z - 2296634895496 z 18 16 50 - 231015197553340 z + 26271302981305 z - 8544308351727414 z 48 20 36 + 36547781986291055 z + 1583452279692934 z + 2004067145491458082 z 34 66 64 - 1793327240525176004 z - 5642342 z + 249453395 z 30 42 44 - 737370517853950098 z - 737370517853950098 z + 338750181485158420 z 46 58 56 - 124501269852965862 z - 2296634895496 z + 26271302981305 z 54 52 60 70 - 231015197553340 z + 1583452279692934 z + 151369939252 z - 514 z 68 32 38 + 76606 z + 1285035292990554489 z - 1793327240525176004 z 40 62 72 + 1285035292990554489 z - 7318527422 z + z )) And in Maple-input format, it is: -(1+66956361925221331*z^28-25934303239724973*z^26-334*z^2+8088857447486599*z^24 -2022777484090428*z^22+41045*z^4-2658087*z^6+105836793*z^8-2825563640*z^10+ 53453692125*z^12-744148594998*z^14-63533630066746*z^18+7831214016350*z^16-\ 2022777484090428*z^50+8088857447486599*z^48+403163912345703*z^20+ 358076421416517932*z^36-322556879182671720*z^34-2658087*z^66+105836793*z^64-\ 139632352422830822*z^30-139632352422830822*z^42+66956361925221331*z^44-\ 25934303239724973*z^46-744148594998*z^58+7831214016350*z^56-63533630066746*z^54 +403163912345703*z^52+53453692125*z^60-334*z^70+41045*z^68+235716114473355379*z ^32-322556879182671720*z^38+235716114473355379*z^40-2825563640*z^62+z^72)/(-1+z ^2)/(1+338750181485158420*z^28-124501269852965862*z^26-514*z^2+ 36547781986291055*z^24-8544308351727414*z^22+76606*z^4-5642342*z^6+249453395*z^ 8-7318527422*z^10+151369939252*z^12-2296634895496*z^14-231015197553340*z^18+ 26271302981305*z^16-8544308351727414*z^50+36547781986291055*z^48+ 1583452279692934*z^20+2004067145491458082*z^36-1793327240525176004*z^34-5642342 *z^66+249453395*z^64-737370517853950098*z^30-737370517853950098*z^42+ 338750181485158420*z^44-124501269852965862*z^46-2296634895496*z^58+ 26271302981305*z^56-231015197553340*z^54+1583452279692934*z^52+151369939252*z^ 60-514*z^70+76606*z^68+1285035292990554489*z^32-1793327240525176004*z^38+ 1285035292990554489*z^40-7318527422*z^62+z^72) The first , 40, terms are: [0, 181, 0, 57140, 0, 18529241, 0, 6021792959, 0, 1957599085465, 0, 636419021116011, 0, 206902760481681516, 0, 67265144234890633931, 0, 21868247837844653594975, 0, 7109481317806033893177307, 0, 2311329440298211794712506895, 0, 751425251828059243783079733260, 0, 244292267208021457218752485074863, 0, 79420689782872713300913907809812077, 0, 25820080339542752720942061076242161123, 0, 8394242741583494485299975819552088154397, 0, 2729012081993142751400429086992217723700852, 0, 887216056639791050751931948822087382476267273, 0, 288438565865405853740456878810688120900204840813, 0, 93772881651385649863098507786277120747564811630949] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1624945089950622580465 z - 216765772610420908985 z - 440 z 24 22 4 + 24599497160513205841 z - 2359023106657324474 z + 80561 z 6 102 8 10 - 8479413 z - 29061955030 z + 588436745 z - 29061955030 z 12 14 18 + 1072654960005 z - 30609007769354 z - 12652431533878814 z 16 50 + 692352041179522 z - 477336270498807185626712064 z 48 20 + 300979116528840082099886854 z + 189627220334672085 z 36 34 + 1125005414938123714211736 z - 273206835359586426527284 z 66 80 - 166193037721165326485631906 z + 57440383046188417405513 z 100 90 88 + 1072654960005 z - 2359023106657324474 z + 24599497160513205841 z 84 94 + 1624945089950622580465 z - 12652431533878814 z 86 96 98 - 216765772610420908985 z + 692352041179522 z - 30609007769354 z 92 82 + 189627220334672085 z - 10420824398911476851900 z 64 112 110 106 + 300979116528840082099886854 z + z - 440 z - 8479413 z 108 30 + 80561 z - 10420824398911476851900 z 42 44 - 33918799208335363126692636 z + 80301488239072766347997406 z 46 58 - 166193037721165326485631906 z - 807968107585135465504316984 z 56 54 + 862861683561632079090383084 z - 807968107585135465504316984 z 52 60 + 663317129290913276430391158 z + 663317129290913276430391158 z 70 68 - 33918799208335363126692636 z + 80301488239072766347997406 z 78 32 - 273206835359586426527284 z + 57440383046188417405513 z 38 40 - 4021617647076460589367116 z + 12509008085225169694863494 z 62 76 - 477336270498807185626712064 z + 1125005414938123714211736 z 74 72 - 4021617647076460589367116 z + 12509008085225169694863494 z 104 / 2 28 + 588436745 z ) / ((-1 + z ) (1 + 6067472026166672997630 z / 26 2 24 - 771660863061108757836 z - 616 z + 83318462074906254747 z 22 4 6 102 - 7586289829061248608 z + 134082 z - 15876308 z - 64639284408 z 8 10 12 14 + 1208857795 z - 64639284408 z + 2562302982080 z - 78105277200060 z 18 16 - 36438144870543428 z + 1879788816500637 z 50 48 - 2539652597661878275234630888 z + 1576852724383428244247294570 z 20 36 + 577757067896890372 z + 4975226002697920846655748 z 34 66 - 1162246064870964074102392 z - 853901458371329559130309824 z 80 100 + 234485070207135590840865 z + 2562302982080 z 90 88 - 7586289829061248608 z + 83318462074906254747 z 84 94 + 6067472026166672997630 z - 36438144870543428 z 86 96 98 - 771660863061108757836 z + 1879788816500637 z - 78105277200060 z 92 82 + 577757067896890372 z - 40728956505836392782192 z 64 112 110 106 + 1576852724383428244247294570 z + z - 616 z - 15876308 z 108 30 + 134082 z - 40728956505836392782192 z 42 44 - 165737936770947056833752232 z + 403080498284479331077048648 z 46 58 - 853901458371329559130309824 z - 4376492013741695034381067488 z 56 54 + 4684422126780923440713886862 z - 4376492013741695034381067488 z 52 60 + 3568766159157542490987012196 z + 3568766159157542490987012196 z 70 68 - 165737936770947056833752232 z + 403080498284479331077048648 z 78 32 - 1162246064870964074102392 z + 234485070207135590840865 z 38 40 - 18440400659855436788168168 z + 59301853852858467167358790 z 62 76 - 2539652597661878275234630888 z + 4975226002697920846655748 z 74 72 - 18440400659855436788168168 z + 59301853852858467167358790 z 104 + 1208857795 z )) And in Maple-input format, it is: -(1+1624945089950622580465*z^28-216765772610420908985*z^26-440*z^2+ 24599497160513205841*z^24-2359023106657324474*z^22+80561*z^4-8479413*z^6-\ 29061955030*z^102+588436745*z^8-29061955030*z^10+1072654960005*z^12-\ 30609007769354*z^14-12652431533878814*z^18+692352041179522*z^16-\ 477336270498807185626712064*z^50+300979116528840082099886854*z^48+ 189627220334672085*z^20+1125005414938123714211736*z^36-273206835359586426527284 *z^34-166193037721165326485631906*z^66+57440383046188417405513*z^80+ 1072654960005*z^100-2359023106657324474*z^90+24599497160513205841*z^88+ 1624945089950622580465*z^84-12652431533878814*z^94-216765772610420908985*z^86+ 692352041179522*z^96-30609007769354*z^98+189627220334672085*z^92-\ 10420824398911476851900*z^82+300979116528840082099886854*z^64+z^112-440*z^110-\ 8479413*z^106+80561*z^108-10420824398911476851900*z^30-\ 33918799208335363126692636*z^42+80301488239072766347997406*z^44-\ 166193037721165326485631906*z^46-807968107585135465504316984*z^58+ 862861683561632079090383084*z^56-807968107585135465504316984*z^54+ 663317129290913276430391158*z^52+663317129290913276430391158*z^60-\ 33918799208335363126692636*z^70+80301488239072766347997406*z^68-\ 273206835359586426527284*z^78+57440383046188417405513*z^32-\ 4021617647076460589367116*z^38+12509008085225169694863494*z^40-\ 477336270498807185626712064*z^62+1125005414938123714211736*z^76-\ 4021617647076460589367116*z^74+12509008085225169694863494*z^72+588436745*z^104) /(-1+z^2)/(1+6067472026166672997630*z^28-771660863061108757836*z^26-616*z^2+ 83318462074906254747*z^24-7586289829061248608*z^22+134082*z^4-15876308*z^6-\ 64639284408*z^102+1208857795*z^8-64639284408*z^10+2562302982080*z^12-\ 78105277200060*z^14-36438144870543428*z^18+1879788816500637*z^16-\ 2539652597661878275234630888*z^50+1576852724383428244247294570*z^48+ 577757067896890372*z^20+4975226002697920846655748*z^36-\ 1162246064870964074102392*z^34-853901458371329559130309824*z^66+ 234485070207135590840865*z^80+2562302982080*z^100-7586289829061248608*z^90+ 83318462074906254747*z^88+6067472026166672997630*z^84-36438144870543428*z^94-\ 771660863061108757836*z^86+1879788816500637*z^96-78105277200060*z^98+ 577757067896890372*z^92-40728956505836392782192*z^82+ 1576852724383428244247294570*z^64+z^112-616*z^110-15876308*z^106+134082*z^108-\ 40728956505836392782192*z^30-165737936770947056833752232*z^42+ 403080498284479331077048648*z^44-853901458371329559130309824*z^46-\ 4376492013741695034381067488*z^58+4684422126780923440713886862*z^56-\ 4376492013741695034381067488*z^54+3568766159157542490987012196*z^52+ 3568766159157542490987012196*z^60-165737936770947056833752232*z^70+ 403080498284479331077048648*z^68-1162246064870964074102392*z^78+ 234485070207135590840865*z^32-18440400659855436788168168*z^38+ 59301853852858467167358790*z^40-2539652597661878275234630888*z^62+ 4975226002697920846655748*z^76-18440400659855436788168168*z^74+ 59301853852858467167358790*z^72+1208857795*z^104) The first , 40, terms are: [0, 177, 0, 55072, 0, 17668855, 0, 5681136951, 0, 1827034516999, 0, 587580048709067, 0, 188968042310869888, 0, 60772880629449171821, 0, 19544803135952514496741, 0, 6285687437141321465981573, 0, 2021502407984712842589208929, 0, 650123320187604323066376860168, 0, 209082279495213772412618530816895, 0, 67241703599517353215104719083280303, 0, 21625202833480717092673470162000232815, 0, 6954752371750391661751086741620925764971, 0, 2236676387491914205936089119561892649239768, 0, 719324139085748073660972484524962032951478981, 0, 231337541704755831734730431947292634552413337537, 0, 74399085605578542158302737894204231875717593828369] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 24077825271110 z - 27951070224950 z - 324 z 24 22 4 6 + 24077825271110 z - 15364293178674 z + 36083 z - 1946383 z 8 10 12 14 + 58879851 z - 1093858710 z + 13224733837 z - 108147700418 z 18 16 50 48 - 2481281123810 z + 614843450858 z - 324 z + 36083 z 20 36 34 + 7224400419262 z + 614843450858 z - 2481281123810 z 30 42 44 46 52 - 15364293178674 z - 1093858710 z + 58879851 z - 1946383 z + z 32 38 40 / 2 + 7224400419262 z - 108147700418 z + 13224733837 z ) / ((-1 + z ) ( / 28 26 2 1 + 128007218817889 z - 150317225617328 z - 482 z 24 22 4 6 + 128007218817889 z - 78970971688036 z + 68450 z - 4422978 z 8 10 12 14 + 155545343 z - 3296400302 z + 44851182900 z - 408361967632 z 18 16 50 48 - 11249972632572 z + 2558734047605 z - 482 z + 68450 z 20 36 34 + 35180049181365 z + 2558734047605 z - 11249972632572 z 30 42 44 46 52 - 78970971688036 z - 3296400302 z + 155545343 z - 4422978 z + z 32 38 40 + 35180049181365 z - 408361967632 z + 44851182900 z )) And in Maple-input format, it is: -(1+24077825271110*z^28-27951070224950*z^26-324*z^2+24077825271110*z^24-\ 15364293178674*z^22+36083*z^4-1946383*z^6+58879851*z^8-1093858710*z^10+ 13224733837*z^12-108147700418*z^14-2481281123810*z^18+614843450858*z^16-324*z^ 50+36083*z^48+7224400419262*z^20+614843450858*z^36-2481281123810*z^34-\ 15364293178674*z^30-1093858710*z^42+58879851*z^44-1946383*z^46+z^52+ 7224400419262*z^32-108147700418*z^38+13224733837*z^40)/(-1+z^2)/(1+ 128007218817889*z^28-150317225617328*z^26-482*z^2+128007218817889*z^24-\ 78970971688036*z^22+68450*z^4-4422978*z^6+155545343*z^8-3296400302*z^10+ 44851182900*z^12-408361967632*z^14-11249972632572*z^18+2558734047605*z^16-482*z ^50+68450*z^48+35180049181365*z^20+2558734047605*z^36-11249972632572*z^34-\ 78970971688036*z^30-3296400302*z^42+155545343*z^44-4422978*z^46+z^52+ 35180049181365*z^32-408361967632*z^38+44851182900*z^40) The first , 40, terms are: [0, 159, 0, 43948, 0, 12811741, 0, 3771695949, 0, 1112902614395, 0, 328568078589347, 0, 97019002746074228, 0, 28648672780517485809, 0, 8459728053529025429415, 0, 2498097478844047024087735, 0, 737670879755552026088997105, 0, 217829136809799029498388926644, 0, 64323448557703515688401475652291, 0, 18994272947176311252301707986269003, 0, 5608878471533779685310648570383754653, 0, 1656263328390169311081933155724477393533, 0, 489083196831612877446207949699926615906284, 0, 144422912307925296697702947545593742016491807, 0, 42647095085035354387461843145152573613565890449, 0, 12593394566924863538001955447254297764084050201457] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 70427066779494392 z - 27535699373329460 z - 338 z 24 22 4 6 + 8673906613259663 z - 2189913490016186 z + 42204 z - 2786478 z 8 10 12 14 + 112948824 z - 3059081782 z + 58458321222 z - 818369998542 z 18 16 50 - 69763862691494 z + 8622612342968 z - 2189913490016186 z 48 20 36 + 8673906613259663 z + 440083606585740 z + 369367337178001556 z 34 66 64 - 333168109163275836 z - 2786478 z + 112948824 z 30 42 44 - 145687322928778764 z - 145687322928778764 z + 70427066779494392 z 46 58 56 - 27535699373329460 z - 818369998542 z + 8622612342968 z 54 52 60 70 - 69763862691494 z + 440083606585740 z + 58458321222 z - 338 z 68 32 38 + 42204 z + 244418223625667920 z - 333168109163275836 z 40 62 72 / 2 + 244418223625667920 z - 3059081782 z + z ) / ((-1 + z ) (1 / 28 26 2 + 352131825798045444 z - 131220577764146226 z - 509 z 24 22 4 6 + 39077995153656343 z - 9260807298301153 z + 77054 z - 5822383 z 8 10 12 14 + 263936660 z - 7904975269 z + 165944992640 z - 2539501638217 z 18 16 50 - 255260332210907 z + 29120540758284 z - 9260807298301153 z 48 20 36 + 39077995153656343 z + 1735915771368846 z + 2023053672541828656 z 34 66 64 - 1813962984689699618 z - 5822383 z + 263936660 z 30 42 44 - 757302664947777062 z - 757302664947777062 z + 352131825798045444 z 46 58 56 - 131220577764146226 z - 2539501638217 z + 29120540758284 z 54 52 60 70 - 255260332210907 z + 1735915771368846 z + 165944992640 z - 509 z 68 32 38 + 77054 z + 1307503044348183792 z - 1813962984689699618 z 40 62 72 + 1307503044348183792 z - 7904975269 z + z )) And in Maple-input format, it is: -(1+70427066779494392*z^28-27535699373329460*z^26-338*z^2+8673906613259663*z^24 -2189913490016186*z^22+42204*z^4-2786478*z^6+112948824*z^8-3059081782*z^10+ 58458321222*z^12-818369998542*z^14-69763862691494*z^18+8622612342968*z^16-\ 2189913490016186*z^50+8673906613259663*z^48+440083606585740*z^20+ 369367337178001556*z^36-333168109163275836*z^34-2786478*z^66+112948824*z^64-\ 145687322928778764*z^30-145687322928778764*z^42+70427066779494392*z^44-\ 27535699373329460*z^46-818369998542*z^58+8622612342968*z^56-69763862691494*z^54 +440083606585740*z^52+58458321222*z^60-338*z^70+42204*z^68+244418223625667920*z ^32-333168109163275836*z^38+244418223625667920*z^40-3059081782*z^62+z^72)/(-1+z ^2)/(1+352131825798045444*z^28-131220577764146226*z^26-509*z^2+ 39077995153656343*z^24-9260807298301153*z^22+77054*z^4-5822383*z^6+263936660*z^ 8-7904975269*z^10+165944992640*z^12-2539501638217*z^14-255260332210907*z^18+ 29120540758284*z^16-9260807298301153*z^50+39077995153656343*z^48+ 1735915771368846*z^20+2023053672541828656*z^36-1813962984689699618*z^34-5822383 *z^66+263936660*z^64-757302664947777062*z^30-757302664947777062*z^42+ 352131825798045444*z^44-131220577764146226*z^46-2539501638217*z^58+ 29120540758284*z^56-255260332210907*z^54+1735915771368846*z^52+165944992640*z^ 60-509*z^70+77054*z^68+1307503044348183792*z^32-1813962984689699618*z^38+ 1307503044348183792*z^40-7904975269*z^62+z^72) The first , 40, terms are: [0, 172, 0, 52361, 0, 16476233, 0, 5199495532, 0, 1641408356649, 0, 518195096477873, 0, 163596143593105644, 0, 51647973454419279857, 0, 16305479611638658569425, 0, 5147707719501515865948140, 0, 1625152739457359679848828457, 0, 513067480104870080110657014169, 0, 161977537723537899556978299776172, 0, 51136982452927416168516612674270273, 0, 16144158079862179374480712995861848993, 0, 5096777862236334816030713594353155089708, 0, 1609073972670439660502619081258988823750497, 0, 507991346593513780094336209629679920371507193, 0, 160374981260569371872459790864390400813597404332, 0, 50631048711364770995632474431390196236616603212697] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 7}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 32374321727958 z - 37889880747316 z - 310 z 24 22 4 6 + 32374321727958 z - 20167295541476 z + 32857 z - 1741868 z 8 10 12 14 + 53596096 z - 1036544140 z + 13226904800 z - 114969374096 z 18 16 50 48 - 2978633591044 z + 695813546384 z - 310 z + 32857 z 20 36 34 + 9123461185440 z + 695813546384 z - 2978633591044 z 30 42 44 46 52 - 20167295541476 z - 1036544140 z + 53596096 z - 1741868 z + z 32 38 40 / + 9123461185440 z - 114969374096 z + 13226904800 z ) / (-1 / 28 26 2 24 - 367357824783684 z + 367357824783684 z + 464 z - 269242046887958 z 22 4 6 8 + 144361916639180 z - 61644 z + 3916745 z - 141761300 z 10 12 14 18 + 3186133204 z - 46858927456 z + 467472797692 z + 15968949528000 z 16 50 48 20 - 3245426807844 z + 61644 z - 3916745 z - 56412232354636 z 36 34 30 - 15968949528000 z + 56412232354636 z + 269242046887958 z 42 44 46 54 52 + 46858927456 z - 3186133204 z + 141761300 z + z - 464 z 32 38 40 - 144361916639180 z + 3245426807844 z - 467472797692 z ) And in Maple-input format, it is: -(1+32374321727958*z^28-37889880747316*z^26-310*z^2+32374321727958*z^24-\ 20167295541476*z^22+32857*z^4-1741868*z^6+53596096*z^8-1036544140*z^10+ 13226904800*z^12-114969374096*z^14-2978633591044*z^18+695813546384*z^16-310*z^ 50+32857*z^48+9123461185440*z^20+695813546384*z^36-2978633591044*z^34-\ 20167295541476*z^30-1036544140*z^42+53596096*z^44-1741868*z^46+z^52+ 9123461185440*z^32-114969374096*z^38+13226904800*z^40)/(-1-367357824783684*z^28 +367357824783684*z^26+464*z^2-269242046887958*z^24+144361916639180*z^22-61644*z ^4+3916745*z^6-141761300*z^8+3186133204*z^10-46858927456*z^12+467472797692*z^14 +15968949528000*z^18-3245426807844*z^16+61644*z^50-3916745*z^48-56412232354636* z^20-15968949528000*z^36+56412232354636*z^34+269242046887958*z^30+46858927456*z ^42-3186133204*z^44+141761300*z^46+z^54-464*z^52-144361916639180*z^32+ 3245426807844*z^38-467472797692*z^40) The first , 40, terms are: [0, 154, 0, 42669, 0, 12480117, 0, 3675499978, 0, 1083549598713, 0, 319484148599225, 0, 94202197384512586, 0, 27776315880928579701, 0, 8190088142472659371693, 0, 2414918980719165254144794, 0, 712059957427800333904544129, 0, 209957099594884619448171235649, 0, 61907685217235909195268047236250, 0, 18254021877828015295130162827197037, 0, 5382357837381063968728478779497722677, 0, 1587035234405495637407473241797212100554, 0, 467951204907555139628935621161355601929209, 0, 137979501290962911105477961028162091553991481, 0, 40684461492655564490407475708534690168615446346, 0, 11996168934231315616178649028034944863037195673397] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 6}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 956502315 z - 3849172654 z - 274 z + 10178635663 z 22 4 6 8 10 - 18082482340 z + 22917 z - 809762 z + 14637796 z - 151409658 z 12 14 18 16 + 956502315 z - 3849172654 z - 18082482340 z + 10178635663 z 20 36 34 30 32 + 21868164792 z + 22917 z - 809762 z - 151409658 z + 14637796 z 38 40 / 20 38 28 - 274 z + z ) / (-229864624576 z + 48962 z - 21984514022 z / 12 42 36 30 - 4426295681 z + z - 2090483 z + 4426295681 z - 1 16 22 2 14 - 71699320466 z + 229864624576 z + 454 z + 21984514022 z 8 10 26 6 - 45476488 z + 571181624 z + 71699320466 z + 2090483 z 32 4 24 40 18 - 571181624 z - 48962 z - 156188641931 z - 454 z + 156188641931 z 34 + 45476488 z ) And in Maple-input format, it is: -(1+956502315*z^28-3849172654*z^26-274*z^2+10178635663*z^24-18082482340*z^22+ 22917*z^4-809762*z^6+14637796*z^8-151409658*z^10+956502315*z^12-3849172654*z^14 -18082482340*z^18+10178635663*z^16+21868164792*z^20+22917*z^36-809762*z^34-\ 151409658*z^30+14637796*z^32-274*z^38+z^40)/(-229864624576*z^20+48962*z^38-\ 21984514022*z^28-4426295681*z^12+z^42-2090483*z^36+4426295681*z^30-1-\ 71699320466*z^16+229864624576*z^22+454*z^2+21984514022*z^14-45476488*z^8+ 571181624*z^10+71699320466*z^26+2090483*z^6-571181624*z^32-48962*z^4-\ 156188641931*z^24-454*z^40+156188641931*z^18+45476488*z^34) The first , 40, terms are: [0, 180, 0, 55675, 0, 17744011, 0, 5675269892, 0, 1816411909537, 0, 581439435254561, 0, 186126485556179556, 0, 59581966203015093163, 0, 19073137171743677450715, 0, 6105617243955698401658708, 0, 1954506186169240836583534849, 0, 625668845399633968850180674049, 0, 200286654610166615684236485433876, 0, 64114977659371836876727196668706331, 0, 20524234971802213408639576612182904427, 0, 6570137533710575785555919006007463744420, 0, 2103206637008250215988048387019243396757281, 0, 673270252756022737715395991290738785881270177, 0, 215524630471461855471026122050403372367132433860, 0, 68992898690708535155492948052779599919747819009995] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 88128893166460 z - 77034102426878 z - 295 z 24 22 4 6 + 51389005860798 z - 26070338076884 z + 29466 z - 1478283 z 8 10 12 14 + 43710897 z - 829177564 z + 10626940848 z - 95259806356 z 18 16 50 48 - 2873913707100 z + 612212233304 z - 1478283 z + 43710897 z 20 36 34 + 9998109363456 z + 9998109363456 z - 26070338076884 z 30 42 44 46 - 77034102426878 z - 95259806356 z + 10626940848 z - 829177564 z 56 54 52 32 38 + z - 295 z + 29466 z + 51389005860798 z - 2873913707100 z 40 / 2 28 + 612212233304 z ) / ((-1 + z ) (1 + 509907306028722 z / 26 2 24 22 - 440199478073298 z - 486 z + 283139894045502 z - 135540776729106 z 4 6 8 10 12 + 59047 z - 3393456 z + 112848145 z - 2393062694 z + 34219522680 z 14 18 16 50 - 341776436730 z - 12655574441134 z + 2440442567840 z - 3393456 z 48 20 36 + 112848145 z + 48172406133400 z + 48172406133400 z 34 30 42 - 135540776729106 z - 440199478073298 z - 341776436730 z 44 46 56 54 52 + 34219522680 z - 2393062694 z + z - 486 z + 59047 z 32 38 40 + 283139894045502 z - 12655574441134 z + 2440442567840 z )) And in Maple-input format, it is: -(1+88128893166460*z^28-77034102426878*z^26-295*z^2+51389005860798*z^24-\ 26070338076884*z^22+29466*z^4-1478283*z^6+43710897*z^8-829177564*z^10+ 10626940848*z^12-95259806356*z^14-2873913707100*z^18+612212233304*z^16-1478283* z^50+43710897*z^48+9998109363456*z^20+9998109363456*z^36-26070338076884*z^34-\ 77034102426878*z^30-95259806356*z^42+10626940848*z^44-829177564*z^46+z^56-295*z ^54+29466*z^52+51389005860798*z^32-2873913707100*z^38+612212233304*z^40)/(-1+z^ 2)/(1+509907306028722*z^28-440199478073298*z^26-486*z^2+283139894045502*z^24-\ 135540776729106*z^22+59047*z^4-3393456*z^6+112848145*z^8-2393062694*z^10+ 34219522680*z^12-341776436730*z^14-12655574441134*z^18+2440442567840*z^16-\ 3393456*z^50+112848145*z^48+48172406133400*z^20+48172406133400*z^36-\ 135540776729106*z^34-440199478073298*z^30-341776436730*z^42+34219522680*z^44-\ 2393062694*z^46+z^56-486*z^54+59047*z^52+283139894045502*z^32-12655574441134*z^ 38+2440442567840*z^40) The first , 40, terms are: [0, 192, 0, 63437, 0, 21437703, 0, 7253916312, 0, 2454781249939, 0, 830725933632531, 0, 281127471022525432, 0, 95136872566105696999, 0, 32195447390627752824461, 0, 10895321734480774728713408, 0, 3687106263792456916852498113, 0, 1247760546447605910375671789761, 0, 422256986886976337226377085585408, 0, 142896778939393640609533803399065357, 0, 48357966985450700408879789600804374631, 0, 16364910310244249862021408388180809016824, 0, 5538079992959883938102986475145713708559571, 0, 1874152037926127533642938921067254847931205587, 0, 634235306410840450172851306560373070608852423192, 0, 214632759646956735772651430510293766092135505942407] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 1435927 z + 24622120 z + 336 z - 204369027 z 22 4 6 8 10 + 888310948 z - 35925 z + 1435927 z - 24622120 z + 204369027 z 12 14 18 16 - 888310948 z + 2245487560 z + 3519292636 z - 3519292636 z 20 34 30 32 / 30 - 2245487560 z + z + 35925 z - 336 z ) / (1 - 3899750 z / 28 16 4 6 36 + 82858760 z + 35465268352 z + 81176 z - 3899750 z + z 26 18 24 22 - 894948728 z - 44635733836 z + 5253106443 z - 17584848448 z 8 12 10 32 34 + 82858760 z + 5253106443 z - 894948728 z + 81176 z - 584 z 20 2 14 + 35465268352 z - 584 z - 17584848448 z ) And in Maple-input format, it is: -(-1-1435927*z^28+24622120*z^26+336*z^2-204369027*z^24+888310948*z^22-35925*z^4 +1435927*z^6-24622120*z^8+204369027*z^10-888310948*z^12+2245487560*z^14+ 3519292636*z^18-3519292636*z^16-2245487560*z^20+z^34+35925*z^30-336*z^32)/(1-\ 3899750*z^30+82858760*z^28+35465268352*z^16+81176*z^4-3899750*z^6+z^36-\ 894948728*z^26-44635733836*z^18+5253106443*z^24-17584848448*z^22+82858760*z^8+ 5253106443*z^12-894948728*z^10+81176*z^32-584*z^34+35465268352*z^20-584*z^2-\ 17584848448*z^14) The first , 40, terms are: [0, 248, 0, 99581, 0, 40487479, 0, 16470001840, 0, 6700352091227, 0, 2725894222452467, 0, 1108976424383117728, 0, 451165772013376382527, 0, 183548192622776967914885, 0, 74673089039922361448274920, 0, 30379325849078990256893344537, 0, 12359250922584843563601409255657, 0, 5028126179367306475530159298845704, 0, 2045597507879984860988428661924534645, 0, 832212441663782155557818853113073294895, 0, 338569804379053359918935285316515369082304, 0, 137740685790562355471785363862231855266099907, 0, 56037178380074201618151625887350980218759918731, 0, 22797660275750142149641078861429483366084658403472, 0, 9274794503811144635584780629019763142921869217741895] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 92811975673141180 z - 41051592282972470 z - 375 z 24 22 4 6 + 14291915229612754 z - 3896322416622344 z + 53436 z - 3990033 z 8 10 12 14 + 179791955 z - 5301972176 z + 107859985019 z - 1570461643649 z 18 16 50 - 135024707571119 z + 16812150245920 z - 135024707571119 z 48 20 36 + 826151949406033 z + 826151949406033 z + 234570033073339870 z 34 66 64 30 - 263295217223767328 z - 375 z + 53436 z - 165784872365291282 z 42 44 46 - 41051592282972470 z + 14291915229612754 z - 3896322416622344 z 58 56 54 - 5301972176 z + 107859985019 z - 1570461643649 z 52 60 68 32 + 16812150245920 z + 179791955 z + z + 234570033073339870 z 38 40 62 / - 165784872365291282 z + 92811975673141180 z - 3990033 z ) / (-1 / 28 26 2 - 704456645814002598 z + 276054706183454874 z + 573 z 24 22 4 6 - 85144183216440746 z + 20580471286255385 z - 99879 z + 8565517 z 8 10 12 14 - 433431343 z + 14244805311 z - 322534936619 z + 5234423696715 z 18 16 50 + 563610110267391 z - 62619093426521 z + 3874949377519229 z 48 20 36 - 20580471286255385 z - 3874949377519229 z - 2854006481961155850 z 34 66 64 + 2854006481961155850 z + 99879 z - 8565517 z 30 42 + 1419169543981907510 z + 704456645814002598 z 44 46 58 - 276054706183454874 z + 85144183216440746 z + 322534936619 z 56 54 52 - 5234423696715 z + 62619093426521 z - 563610110267391 z 60 70 68 32 - 14244805311 z + z - 573 z - 2261493338338265898 z 38 40 62 + 2261493338338265898 z - 1419169543981907510 z + 433431343 z ) And in Maple-input format, it is: -(1+92811975673141180*z^28-41051592282972470*z^26-375*z^2+14291915229612754*z^ 24-3896322416622344*z^22+53436*z^4-3990033*z^6+179791955*z^8-5301972176*z^10+ 107859985019*z^12-1570461643649*z^14-135024707571119*z^18+16812150245920*z^16-\ 135024707571119*z^50+826151949406033*z^48+826151949406033*z^20+ 234570033073339870*z^36-263295217223767328*z^34-375*z^66+53436*z^64-\ 165784872365291282*z^30-41051592282972470*z^42+14291915229612754*z^44-\ 3896322416622344*z^46-5301972176*z^58+107859985019*z^56-1570461643649*z^54+ 16812150245920*z^52+179791955*z^60+z^68+234570033073339870*z^32-\ 165784872365291282*z^38+92811975673141180*z^40-3990033*z^62)/(-1-\ 704456645814002598*z^28+276054706183454874*z^26+573*z^2-85144183216440746*z^24+ 20580471286255385*z^22-99879*z^4+8565517*z^6-433431343*z^8+14244805311*z^10-\ 322534936619*z^12+5234423696715*z^14+563610110267391*z^18-62619093426521*z^16+ 3874949377519229*z^50-20580471286255385*z^48-3874949377519229*z^20-\ 2854006481961155850*z^36+2854006481961155850*z^34+99879*z^66-8565517*z^64+ 1419169543981907510*z^30+704456645814002598*z^42-276054706183454874*z^44+ 85144183216440746*z^46+322534936619*z^58-5234423696715*z^56+62619093426521*z^54 -563610110267391*z^52-14244805311*z^60+z^70-573*z^68-2261493338338265898*z^32+ 2261493338338265898*z^38-1419169543981907510*z^40+433431343*z^62) The first , 40, terms are: [0, 198, 0, 67011, 0, 23196745, 0, 8041076194, 0, 2787776252215, 0, 966514385755039, 0, 335088577935377050, 0, 116174565585441536837, 0, 40277500399966116021227, 0, 13964132711569440646007766, 0, 4841338234288998635805333573, 0, 1678482752151941706147603622357, 0, 581926775837574514956722027609462, 0, 201752905715626529453194704216205243, 0, 69947348454961016934561251844549502661, 0, 24250612592306437364271802001636283694010, 0, 8407641234332895448977868896402462789073455, 0, 2914913215333870275053762259891601625527452151, 0, 1010594864375447453635834870713257020063342517922, 0, 350371316212599902637437735944277514620593351762681] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 128182525698608 z - 112288771817854 z - 330 z 24 22 4 6 + 75401549198032 z - 38678871851968 z + 38262 z - 2147175 z 8 10 12 14 + 68020760 z - 1331949627 z + 17162911610 z - 152204748726 z 18 16 50 48 - 4414011024048 z + 960044050861 z - 2147175 z + 68020760 z 20 36 34 + 15065108076530 z + 15065108076530 z - 38678871851968 z 30 42 44 - 112288771817854 z - 152204748726 z + 17162911610 z 46 56 54 52 32 - 1331949627 z + z - 330 z + 38262 z + 75401549198032 z 38 40 / 28 - 4414011024048 z + 960044050861 z ) / (-1 - 1359243058675178 z / 26 2 24 + 1041377692373858 z + 517 z - 610254628660914 z 22 4 6 8 + 272558056961626 z - 74989 z + 4925913 z - 179307189 z 10 12 14 18 + 4023259953 z - 59516852309 z + 607554266865 z + 23441335236061 z 16 50 48 20 - 4419196725553 z + 179307189 z - 4023259953 z - 92233843702386 z 36 34 30 - 272558056961626 z + 610254628660914 z + 1359243058675178 z 42 44 46 58 56 + 4419196725553 z - 607554266865 z + 59516852309 z + z - 517 z 54 52 32 38 + 74989 z - 4925913 z - 1041377692373858 z + 92233843702386 z 40 - 23441335236061 z ) And in Maple-input format, it is: -(1+128182525698608*z^28-112288771817854*z^26-330*z^2+75401549198032*z^24-\ 38678871851968*z^22+38262*z^4-2147175*z^6+68020760*z^8-1331949627*z^10+ 17162911610*z^12-152204748726*z^14-4414011024048*z^18+960044050861*z^16-2147175 *z^50+68020760*z^48+15065108076530*z^20+15065108076530*z^36-38678871851968*z^34 -112288771817854*z^30-152204748726*z^42+17162911610*z^44-1331949627*z^46+z^56-\ 330*z^54+38262*z^52+75401549198032*z^32-4414011024048*z^38+960044050861*z^40)/( -1-1359243058675178*z^28+1041377692373858*z^26+517*z^2-610254628660914*z^24+ 272558056961626*z^22-74989*z^4+4925913*z^6-179307189*z^8+4023259953*z^10-\ 59516852309*z^12+607554266865*z^14+23441335236061*z^18-4419196725553*z^16+ 179307189*z^50-4023259953*z^48-92233843702386*z^20-272558056961626*z^36+ 610254628660914*z^34+1359243058675178*z^30+4419196725553*z^42-607554266865*z^44 +59516852309*z^46+z^58-517*z^56+74989*z^54-4925913*z^52-1041377692373858*z^32+ 92233843702386*z^38-23441335236061*z^40) The first , 40, terms are: [0, 187, 0, 59952, 0, 19750979, 0, 6525374917, 0, 2156991870017, 0, 713085232442287, 0, 235746866619832368, 0, 77938694528720190503, 0, 25766828338293329939389, 0, 8518614241721348555256165, 0, 2816287421839004579017270895, 0, 931075732617678888504973626672, 0, 307817311937556083404481124457975, 0, 101765618318079093663254364977573993, 0, 33644115105531651707153118252570584157, 0, 11122877254717734011546477323634692181691, 0, 3677267124940248817204961818378161581620016, 0, 1215719026520437116166561038922797182177128243, 0, 401921508889291610692029646658272172801228733721, 0, 132876837315138924937736801446946706172200026348393] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 6}, {4, 5}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 14076984706 z - 33236860870 z - 246 z + 55433556067 z 22 4 6 8 10 - 65696948744 z + 17118 z - 560722 z + 10296701 z - 116095708 z 12 14 18 16 + 848727999 z - 4167640850 z - 33236860870 z + 14076984706 z 20 36 34 30 + 55433556067 z + 10296701 z - 116095708 z - 4167640850 z 42 44 32 38 40 / 2 - 246 z + z + 848727999 z - 560722 z + 17118 z ) / ((-1 + z ) ( / 44 42 40 38 36 34 z - 431 z + 36398 z - 1390403 z + 29617853 z - 386200172 z 32 30 28 26 + 3245192715 z - 18116619813 z + 68406999154 z - 176337394473 z 24 22 20 + 311326456503 z - 376368925288 z + 311326456503 z 18 16 14 12 - 176337394473 z + 68406999154 z - 18116619813 z + 3245192715 z 10 8 6 4 2 - 386200172 z + 29617853 z - 1390403 z + 36398 z - 431 z + 1)) And in Maple-input format, it is: -(1+14076984706*z^28-33236860870*z^26-246*z^2+55433556067*z^24-65696948744*z^22 +17118*z^4-560722*z^6+10296701*z^8-116095708*z^10+848727999*z^12-4167640850*z^ 14-33236860870*z^18+14076984706*z^16+55433556067*z^20+10296701*z^36-116095708*z ^34-4167640850*z^30-246*z^42+z^44+848727999*z^32-560722*z^38+17118*z^40)/(-1+z^ 2)/(z^44-431*z^42+36398*z^40-1390403*z^38+29617853*z^36-386200172*z^34+ 3245192715*z^32-18116619813*z^30+68406999154*z^28-176337394473*z^26+ 311326456503*z^24-376368925288*z^22+311326456503*z^20-176337394473*z^18+ 68406999154*z^16-18116619813*z^14+3245192715*z^12-386200172*z^10+29617853*z^8-\ 1390403*z^6+36398*z^4-431*z^2+1) The first , 40, terms are: [0, 186, 0, 60641, 0, 20212797, 0, 6743254346, 0, 2249723602901, 0, 750567108952461, 0, 250408996332446058, 0, 83543050224198081317, 0, 27872166512755161142521, 0, 9298890380919469553744026, 0, 3102355257415495080379992137, 0, 1035027594578680638502572631993, 0, 345312523115677689241788250830554, 0, 115205371571811777528122603466984137, 0, 38435552580735919480312620027957131093, 0, 12823114773477944442057475775120190314794, 0, 4278129529903065479884077030724585786492957, 0, 1427296924182841469612148261212978335471035717, 0, 476183924666712597309626145663034441849915048330, 0, 158867525228370663953295534578335490842807500433421] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 5}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 100237378 z + 578313178 z + 258 z - 2080422451 z 22 4 6 8 10 + 4802150316 z - 18930 z + 617029 z - 10432299 z + 100237378 z 12 14 18 16 - 578313178 z + 2080422451 z + 7257466272 z - 4802150316 z 20 36 34 30 32 38 - 7257466272 z - 258 z + 18930 z + 10432299 z - 617029 z + z ) / 38 28 4 18 / (-476 z + 2888249239 z + 43517 z + 1 - 70732773628 z / 30 26 22 2 - 404036204 z - 13003622292 z - 70732773628 z - 476 z 10 40 16 20 6 - 404036204 z + z + 37604554459 z + 87229921736 z - 1693256 z 12 36 24 34 + 2888249239 z + 43517 z + 37604554459 z - 1693256 z 32 8 14 + 34447244 z + 34447244 z - 13003622292 z ) And in Maple-input format, it is: -(-1-100237378*z^28+578313178*z^26+258*z^2-2080422451*z^24+4802150316*z^22-\ 18930*z^4+617029*z^6-10432299*z^8+100237378*z^10-578313178*z^12+2080422451*z^14 +7257466272*z^18-4802150316*z^16-7257466272*z^20-258*z^36+18930*z^34+10432299*z ^30-617029*z^32+z^38)/(-476*z^38+2888249239*z^28+43517*z^4+1-70732773628*z^18-\ 404036204*z^30-13003622292*z^26-70732773628*z^22-476*z^2-404036204*z^10+z^40+ 37604554459*z^16+87229921736*z^20-1693256*z^6+2888249239*z^12+43517*z^36+ 37604554459*z^24-1693256*z^34+34447244*z^32+34447244*z^8-13003622292*z^14) The first , 40, terms are: [0, 218, 0, 79181, 0, 29279677, 0, 10836521538, 0, 4010888551049, 0, 1484546234017737, 0, 549473944527465682, 0, 203376376089261776237, 0, 75275544937467490812029, 0, 27861680782409810701999498, 0, 10312422935213731373201239841, 0, 3816929338399116760547912889633, 0, 1412757182853180852781653152592298, 0, 522902752645801989819920500628985181, 0, 193541602225196222773953094431088991309, 0, 71635407544448771524475505772226637872946, 0, 26514359471346810844783520539311279673934985, 0, 9813739912620580244375702235597792695846513225, 0, 3632352166630337337910717002691070584573395020514, 0, 1344439773205778146391130050545565848570647508175197] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 1930941301218728 z - 1094175254174730 z - 317 z 24 22 4 6 + 491828935340702 z - 174538409394956 z + 35082 z - 1946563 z 8 10 12 14 + 64125578 z - 1378313883 z + 20510378670 z - 219927038953 z 18 16 50 - 10512268670428 z + 1747967019441 z - 219927038953 z 48 20 36 + 1747967019441 z + 48580457028640 z + 1930941301218728 z 34 64 30 42 - 2711735409429514 z + z - 2711735409429514 z - 174538409394956 z 44 46 58 56 + 48580457028640 z - 10512268670428 z - 1946563 z + 64125578 z 54 52 60 32 - 1378313883 z + 20510378670 z + 35082 z + 3036126836923980 z 38 40 62 / 2 - 1094175254174730 z + 491828935340702 z - 317 z ) / ((-1 + z ) (1 / 28 26 2 + 10653798421483690 z - 5814227107030548 z - 510 z 24 22 4 6 + 2484285738731386 z - 828442053219784 z + 68845 z - 4369934 z 8 10 12 14 + 161331428 z - 3853959774 z + 63435891921 z - 749333629326 z 18 16 50 - 42848543892224 z + 6531684340637 z - 749333629326 z 48 20 36 + 6531684340637 z + 214558511795816 z + 10653798421483690 z 34 64 30 - 15312477941952844 z + z - 15312477941952844 z 42 44 46 - 828442053219784 z + 214558511795816 z - 42848543892224 z 58 56 54 52 - 4369934 z + 161331428 z - 3853959774 z + 63435891921 z 60 32 38 + 68845 z + 17279024621292328 z - 5814227107030548 z 40 62 + 2484285738731386 z - 510 z )) And in Maple-input format, it is: -(1+1930941301218728*z^28-1094175254174730*z^26-317*z^2+491828935340702*z^24-\ 174538409394956*z^22+35082*z^4-1946563*z^6+64125578*z^8-1378313883*z^10+ 20510378670*z^12-219927038953*z^14-10512268670428*z^18+1747967019441*z^16-\ 219927038953*z^50+1747967019441*z^48+48580457028640*z^20+1930941301218728*z^36-\ 2711735409429514*z^34+z^64-2711735409429514*z^30-174538409394956*z^42+ 48580457028640*z^44-10512268670428*z^46-1946563*z^58+64125578*z^56-1378313883*z ^54+20510378670*z^52+35082*z^60+3036126836923980*z^32-1094175254174730*z^38+ 491828935340702*z^40-317*z^62)/(-1+z^2)/(1+10653798421483690*z^28-\ 5814227107030548*z^26-510*z^2+2484285738731386*z^24-828442053219784*z^22+68845* z^4-4369934*z^6+161331428*z^8-3853959774*z^10+63435891921*z^12-749333629326*z^ 14-42848543892224*z^18+6531684340637*z^16-749333629326*z^50+6531684340637*z^48+ 214558511795816*z^20+10653798421483690*z^36-15312477941952844*z^34+z^64-\ 15312477941952844*z^30-828442053219784*z^42+214558511795816*z^44-42848543892224 *z^46-4369934*z^58+161331428*z^56-3853959774*z^54+63435891921*z^52+68845*z^60+ 17279024621292328*z^32-5814227107030548*z^38+2484285738731386*z^40-510*z^62) The first , 40, terms are: [0, 194, 0, 64861, 0, 22181317, 0, 7595765674, 0, 2601445576689, 0, 890976956439233, 0, 305154391907907146, 0, 104513652833097176669, 0, 35795338452386223642229, 0, 12259702377301975761540450, 0, 4198879221790855670183914577, 0, 1438092556143062733915530366417, 0, 492538625429721400925651604547074, 0, 168691713553732255349487690616696133, 0, 57775964670709473717663755581804366637, 0, 19787943481703761338120405096159054528874, 0, 6777259531136008525261311543102547774642113, 0, 2321173334401498734900276546218988720277520113, 0, 794988833404388691926824516671821649759528874570, 0, 272279211496555772453966824941664531192047185610805] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2374098194904348 z - 1314233419677974 z - 305 z 24 22 4 6 + 573006192877040 z - 196219983745328 z + 32362 z - 1765315 z 8 10 12 14 + 58176031 z - 1266429470 z + 19264599636 z - 212681236764 z 18 16 50 - 10930150002866 z + 1749716053436 z - 212681236764 z 48 20 36 + 1749716053436 z + 52543017436152 z + 2374098194904348 z 34 64 30 42 - 3383403683201680 z + z - 3383403683201680 z - 196219983745328 z 44 46 58 56 + 52543017436152 z - 10930150002866 z - 1765315 z + 58176031 z 54 52 60 32 - 1266429470 z + 19264599636 z + 32362 z + 3807169642304944 z 38 40 62 / 2 - 1314233419677974 z + 573006192877040 z - 305 z ) / ((-1 + z ) (1 / 28 26 2 + 12950502718791774 z - 6878067736928047 z - 487 z 24 22 4 6 + 2838802772418586 z - 910026275505849 z + 62516 z - 3884396 z 8 10 12 14 + 142972983 z - 3449597735 z + 57931314860 z - 703855280493 z 18 16 50 - 43313906051819 z + 6350400104374 z - 703855280493 z 48 20 36 + 6350400104374 z + 226064700960064 z + 12950502718791774 z 34 64 30 - 18940694946927667 z + z - 18940694946927667 z 42 44 46 - 910026275505849 z + 226064700960064 z - 43313906051819 z 58 56 54 52 - 3884396 z + 142972983 z - 3449597735 z + 57931314860 z 60 32 38 + 62516 z + 21502092911412254 z - 6878067736928047 z 40 62 + 2838802772418586 z - 487 z )) And in Maple-input format, it is: -(1+2374098194904348*z^28-1314233419677974*z^26-305*z^2+573006192877040*z^24-\ 196219983745328*z^22+32362*z^4-1765315*z^6+58176031*z^8-1266429470*z^10+ 19264599636*z^12-212681236764*z^14-10930150002866*z^18+1749716053436*z^16-\ 212681236764*z^50+1749716053436*z^48+52543017436152*z^20+2374098194904348*z^36-\ 3383403683201680*z^34+z^64-3383403683201680*z^30-196219983745328*z^42+ 52543017436152*z^44-10930150002866*z^46-1765315*z^58+58176031*z^56-1266429470*z ^54+19264599636*z^52+32362*z^60+3807169642304944*z^32-1314233419677974*z^38+ 573006192877040*z^40-305*z^62)/(-1+z^2)/(1+12950502718791774*z^28-\ 6878067736928047*z^26-487*z^2+2838802772418586*z^24-910026275505849*z^22+62516* z^4-3884396*z^6+142972983*z^8-3449597735*z^10+57931314860*z^12-703855280493*z^ 14-43313906051819*z^18+6350400104374*z^16-703855280493*z^50+6350400104374*z^48+ 226064700960064*z^20+12950502718791774*z^36-18940694946927667*z^34+z^64-\ 18940694946927667*z^30-910026275505849*z^42+226064700960064*z^44-43313906051819 *z^46-3884396*z^58+142972983*z^56-3449597735*z^54+57931314860*z^52+62516*z^60+ 21502092911412254*z^32-6878067736928047*z^38+2838802772418586*z^40-487*z^62) The first , 40, terms are: [0, 183, 0, 58663, 0, 19279592, 0, 6346099455, 0, 2089213338811, 0, 687807888100465, 0, 226439754525124817, 0, 74548406110552132227, 0, 24542797988348355172191, 0, 8079970708112590349356712, 0, 2660084915599170961591737895, 0, 875752154918198854361269323023, 0, 288314794901307015988373475021825, 0, 94918888286393535169439501788919457, 0, 31249160684289742746588004989456350399, 0, 10287836921627798899624447923507927713351, 0, 3386957800092823516067720239296184326802152, 0, 1115052972456580509563100494224906986569266879, 0, 367097320005044251627786258125057435761078724115, 0, 120855641555749815810503888370494412024066161642097] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 953933994931 z + 1938635794536 z + 376 z 24 22 4 6 - 2756374379187 z + 2756374379187 z - 46145 z + 2550839 z 8 10 12 14 - 72013832 z + 1166403959 z - 11666168277 z + 75420982296 z 18 16 20 36 + 953933994931 z - 324958685653 z - 1938635794536 z - 1166403959 z 34 30 42 44 46 + 11666168277 z + 324958685653 z + 46145 z - 376 z + z 32 38 40 / - 75420982296 z + 72013832 z - 2550839 z ) / (1 / 28 26 2 24 + 15776392607896 z - 26562217673808 z - 560 z + 31581893518806 z 22 4 6 8 - 26562217673808 z + 91464 z - 6399700 z + 221330776 z 10 12 14 18 - 4273631344 z + 50181074708 z - 378899533584 z - 6581547313964 z 16 48 20 36 + 1909223996104 z + z + 15776392607896 z + 50181074708 z 34 30 42 44 46 - 378899533584 z - 6581547313964 z - 6399700 z + 91464 z - 560 z 32 38 40 + 1909223996104 z - 4273631344 z + 221330776 z ) And in Maple-input format, it is: -(-1-953933994931*z^28+1938635794536*z^26+376*z^2-2756374379187*z^24+ 2756374379187*z^22-46145*z^4+2550839*z^6-72013832*z^8+1166403959*z^10-\ 11666168277*z^12+75420982296*z^14+953933994931*z^18-324958685653*z^16-\ 1938635794536*z^20-1166403959*z^36+11666168277*z^34+324958685653*z^30+46145*z^ 42-376*z^44+z^46-75420982296*z^32+72013832*z^38-2550839*z^40)/(1+15776392607896 *z^28-26562217673808*z^26-560*z^2+31581893518806*z^24-26562217673808*z^22+91464 *z^4-6399700*z^6+221330776*z^8-4273631344*z^10+50181074708*z^12-378899533584*z^ 14-6581547313964*z^18+1909223996104*z^16+z^48+15776392607896*z^20+50181074708*z ^36-378899533584*z^34-6581547313964*z^30-6399700*z^42+91464*z^44-560*z^46+ 1909223996104*z^32-4273631344*z^38+221330776*z^40) The first , 40, terms are: [0, 184, 0, 57721, 0, 19343245, 0, 6581051512, 0, 2247957734341, 0, 768690400303261, 0, 262932666326811256, 0, 89944297773654323461, 0, 30768954933380938145089, 0, 10525788771549872148078520, 0, 3600786200886703491227782633, 0, 1231800110845265572261847392153, 0, 421389008258617356428537604345784, 0, 144153829580323251879528995414599729, 0, 49313879583406163831054641686023850133, 0, 16869886385430914639985840165573508228216, 0, 5771054098895466046580482160310718547985517, 0, 1974231755891489836602090759770197255060685557, 0, 675368998356656314536319970262740103672190466680, 0, 231038368514090492910219020487714146522564956161277] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 870251 z + 13718346 z + 314 z - 116461625 z 22 4 6 8 10 + 557904454 z - 26803 z + 870251 z - 13718346 z + 116461625 z 12 14 18 16 - 557904454 z + 1549501436 z + 2560136290 z - 2560136290 z 20 34 30 32 / 10 - 1549501436 z + z + 26803 z - 314 z ) / (-536517002 z / 12 16 20 36 34 + 3182197695 z + 23400368532 z + 23400368532 z + z - 498 z 30 32 28 26 24 - 2598510 z + 61242 z + 51312242 z - 536517002 z + 3182197695 z 22 8 14 18 - 11125438788 z + 51312242 z - 11125438788 z - 29939556260 z 4 6 2 + 61242 z - 2598510 z + 1 - 498 z ) And in Maple-input format, it is: -(-1-870251*z^28+13718346*z^26+314*z^2-116461625*z^24+557904454*z^22-26803*z^4+ 870251*z^6-13718346*z^8+116461625*z^10-557904454*z^12+1549501436*z^14+ 2560136290*z^18-2560136290*z^16-1549501436*z^20+z^34+26803*z^30-314*z^32)/(-\ 536517002*z^10+3182197695*z^12+23400368532*z^16+23400368532*z^20+z^36-498*z^34-\ 2598510*z^30+61242*z^32+51312242*z^28-536517002*z^26+3182197695*z^24-\ 11125438788*z^22+51312242*z^8-11125438788*z^14-29939556260*z^18+61242*z^4-\ 2598510*z^6+1-498*z^2) The first , 40, terms are: [0, 184, 0, 57193, 0, 18941845, 0, 6370957048, 0, 2152295323693, 0, 728054887094869, 0, 246373619692326904, 0, 83382360691765230013, 0, 28220779695560382682177, 0, 9551426619647938275136696, 0, 3232725216988506644224146649, 0, 1094132022014099315468272467241, 0, 370314555426834631331787759788728, 0, 125334857312822893187310789674271889, 0, 42420225180802396678813570966956783469, 0, 14357342846355015042943729171781023226872, 0, 4859316355821291735196147940537448382134277, 0, 1644660555461123926159528571100650295090884221, 0, 556643804421046361372332509616416477315431495416, 0, 188398951983436498239700236948286542392596369145317] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 30480201124604 z - 35324754653888 z - 370 z 24 22 4 6 + 30480201124604 z - 19542796616467 z + 45158 z - 2566604 z 8 10 12 14 + 79136654 z - 1471462681 z + 17655704091 z - 142859877938 z 18 16 50 48 - 3208652193838 z + 803331851388 z - 370 z + 45158 z 20 36 34 + 9256892182178 z + 803331851388 z - 3208652193838 z 30 42 44 46 52 - 19542796616467 z - 1471462681 z + 79136654 z - 2566604 z + z 32 38 40 / + 9256892182178 z - 142859877938 z + 17655704091 z ) / (-1 / 28 26 2 24 - 376645293185106 z + 376645293185106 z + 558 z - 281297427412265 z 22 4 6 8 + 156401247136256 z - 88962 z + 6236527 z - 230407408 z 10 12 14 18 + 5035609840 z - 70115375581 z + 653671077806 z + 19383717609359 z 16 50 48 20 - 4223555452130 z + 88962 z - 6236527 z - 64325832462396 z 36 34 30 - 19383717609359 z + 64325832462396 z + 281297427412265 z 42 44 46 54 52 + 70115375581 z - 5035609840 z + 230407408 z + z - 558 z 32 38 40 - 156401247136256 z + 4223555452130 z - 653671077806 z ) And in Maple-input format, it is: -(1+30480201124604*z^28-35324754653888*z^26-370*z^2+30480201124604*z^24-\ 19542796616467*z^22+45158*z^4-2566604*z^6+79136654*z^8-1471462681*z^10+ 17655704091*z^12-142859877938*z^14-3208652193838*z^18+803331851388*z^16-370*z^ 50+45158*z^48+9256892182178*z^20+803331851388*z^36-3208652193838*z^34-\ 19542796616467*z^30-1471462681*z^42+79136654*z^44-2566604*z^46+z^52+ 9256892182178*z^32-142859877938*z^38+17655704091*z^40)/(-1-376645293185106*z^28 +376645293185106*z^26+558*z^2-281297427412265*z^24+156401247136256*z^22-88962*z ^4+6236527*z^6-230407408*z^8+5035609840*z^10-70115375581*z^12+653671077806*z^14 +19383717609359*z^18-4223555452130*z^16+88962*z^50-6236527*z^48-64325832462396* z^20-19383717609359*z^36+64325832462396*z^34+281297427412265*z^30+70115375581*z ^42-5035609840*z^44+230407408*z^46+z^54-558*z^52-156401247136256*z^32+ 4223555452130*z^38-653671077806*z^40) The first , 40, terms are: [0, 188, 0, 61100, 0, 21038867, 0, 7325305908, 0, 2557160364765, 0, 893247423795913, 0, 312073924255484944, 0, 109033946276124670103, 0, 38095245953058255654051, 0, 13310092687534679706560484, 0, 4650414904955489550630197973, 0, 1624809345461126876644812148356, 0, 567692473996503843301848265281004, 0, 198346193768121049291115447623373175, 0, 69300218902037427178835628939012848647, 0, 24212818269480745992639271638524434249980, 0, 8459721743673935865257101627313619519447652, 0, 2955743986024115524065149917652184046358750181, 0, 1032708022292587589463055365435011648226378070804, 0, 360818076380967169790853502666763998497640772866147] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 43984883513558 z - 51496850418632 z - 348 z 24 22 4 6 + 43984883513558 z - 27365309913112 z + 39337 z - 2166792 z 8 10 12 14 + 68250368 z - 1341303328 z + 17334448080 z - 152325001024 z 18 16 50 48 - 4011777594176 z + 930484385072 z - 348 z + 39337 z 20 36 34 + 12346410350976 z + 930484385072 z - 4011777594176 z 30 42 44 46 52 - 27365309913112 z - 1341303328 z + 68250368 z - 2166792 z + z 32 38 40 / 2 + 12346410350976 z - 152325001024 z + 17334448080 z ) / ((-1 + z ) / 28 26 2 (1 + 245331196859934 z - 290578277879546 z - 535 z 24 22 4 6 + 245331196859934 z - 147576500346414 z + 76537 z - 5081978 z 8 10 12 14 + 188318686 z - 4264955194 z + 62345152996 z - 609941929428 z 18 16 50 48 - 19154334777070 z + 4093325463604 z - 535 z + 76537 z 20 36 34 + 63146837363130 z + 4093325463604 z - 19154334777070 z 30 42 44 46 52 - 147576500346414 z - 4264955194 z + 188318686 z - 5081978 z + z 32 38 40 + 63146837363130 z - 609941929428 z + 62345152996 z )) And in Maple-input format, it is: -(1+43984883513558*z^28-51496850418632*z^26-348*z^2+43984883513558*z^24-\ 27365309913112*z^22+39337*z^4-2166792*z^6+68250368*z^8-1341303328*z^10+ 17334448080*z^12-152325001024*z^14-4011777594176*z^18+930484385072*z^16-348*z^ 50+39337*z^48+12346410350976*z^20+930484385072*z^36-4011777594176*z^34-\ 27365309913112*z^30-1341303328*z^42+68250368*z^44-2166792*z^46+z^52+ 12346410350976*z^32-152325001024*z^38+17334448080*z^40)/(-1+z^2)/(1+ 245331196859934*z^28-290578277879546*z^26-535*z^2+245331196859934*z^24-\ 147576500346414*z^22+76537*z^4-5081978*z^6+188318686*z^8-4264955194*z^10+ 62345152996*z^12-609941929428*z^14-19154334777070*z^18+4093325463604*z^16-535*z ^50+76537*z^48+63146837363130*z^20+4093325463604*z^36-19154334777070*z^34-\ 147576500346414*z^30-4264955194*z^42+188318686*z^44-5081978*z^46+z^52+ 63146837363130*z^32-609941929428*z^38+62345152996*z^40) The first , 40, terms are: [0, 188, 0, 63033, 0, 22287875, 0, 7932872148, 0, 2826157691043, 0, 1006987853489435, 0, 358807914524882004, 0, 127850206435776547179, 0, 45555532400199425098593, 0, 16232330082223508196990332, 0, 5783897821485688506154085241, 0, 2060916328157103475261216604809, 0, 734344942619909067270134915331452, 0, 261661517952269738368549595626426513, 0, 93235135158451562589603804707844392731, 0, 33221508825849096326692462223685658232788, 0, 11837475719757036435758467061198968371028523, 0, 4217924963926318298793515749204382151300435859, 0, 1502929460849462841958714112559241782465289981780, 0, 535523268812877816242092136461905282296171662631251] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 84858496837706196 z - 38789170498590178 z - 451 z 24 22 4 6 + 14063325979129382 z - 4018980987343216 z + 73542 z - 5836075 z 8 10 12 14 + 264568025 z - 7590907960 z + 147473703169 z - 2031696148723 z 18 16 50 - 155299879994747 z + 20507344384902 z - 155299879994747 z 48 20 36 + 898056418270649 z + 898056418270649 z + 206232303854145478 z 34 66 64 30 - 230328608372853488 z - 451 z + 73542 z - 147943043326774722 z 42 44 46 - 38789170498590178 z + 14063325979129382 z - 4018980987343216 z 58 56 54 - 7590907960 z + 147473703169 z - 2031696148723 z 52 60 68 32 + 20507344384902 z + 264568025 z + z + 206232303854145478 z 38 40 62 / - 147943043326774722 z + 84858496837706196 z - 5836075 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 480500355429612342 z - 210522588139496700 z - 654 z 24 22 4 6 + 72456014487399426 z - 19493964712005648 z + 138779 z - 13181078 z 8 10 12 14 + 687212441 z - 22192611680 z + 478559694597 z - 7244780967182 z 18 16 50 - 654045735439030 z + 79732636285407 z - 654045735439030 z 48 20 36 + 4071857320344333 z + 4071857320344333 z + 1228532873826582130 z 34 66 64 30 - 1381055431426334304 z - 654 z + 138779 z - 864435732016781468 z 42 44 46 - 210522588139496700 z + 72456014487399426 z - 19493964712005648 z 58 56 54 - 22192611680 z + 478559694597 z - 7244780967182 z 52 60 68 32 + 79732636285407 z + 687212441 z + z + 1228532873826582130 z 38 40 62 - 864435732016781468 z + 480500355429612342 z - 13181078 z )) And in Maple-input format, it is: -(1+84858496837706196*z^28-38789170498590178*z^26-451*z^2+14063325979129382*z^ 24-4018980987343216*z^22+73542*z^4-5836075*z^6+264568025*z^8-7590907960*z^10+ 147473703169*z^12-2031696148723*z^14-155299879994747*z^18+20507344384902*z^16-\ 155299879994747*z^50+898056418270649*z^48+898056418270649*z^20+ 206232303854145478*z^36-230328608372853488*z^34-451*z^66+73542*z^64-\ 147943043326774722*z^30-38789170498590178*z^42+14063325979129382*z^44-\ 4018980987343216*z^46-7590907960*z^58+147473703169*z^56-2031696148723*z^54+ 20507344384902*z^52+264568025*z^60+z^68+206232303854145478*z^32-\ 147943043326774722*z^38+84858496837706196*z^40-5836075*z^62)/(-1+z^2)/(1+ 480500355429612342*z^28-210522588139496700*z^26-654*z^2+72456014487399426*z^24-\ 19493964712005648*z^22+138779*z^4-13181078*z^6+687212441*z^8-22192611680*z^10+ 478559694597*z^12-7244780967182*z^14-654045735439030*z^18+79732636285407*z^16-\ 654045735439030*z^50+4071857320344333*z^48+4071857320344333*z^20+ 1228532873826582130*z^36-1381055431426334304*z^34-654*z^66+138779*z^64-\ 864435732016781468*z^30-210522588139496700*z^42+72456014487399426*z^44-\ 19493964712005648*z^46-22192611680*z^58+478559694597*z^56-7244780967182*z^54+ 79732636285407*z^52+687212441*z^60+z^68+1228532873826582130*z^32-\ 864435732016781468*z^38+480500355429612342*z^40-13181078*z^62) The first , 40, terms are: [0, 204, 0, 67729, 0, 23401945, 0, 8166041652, 0, 2860303117913, 0, 1003470670360789, 0, 352285746678966532, 0, 123712649976777246517, 0, 43449900232896302504157, 0, 15261159207452909064279484, 0, 5360393506179758034793300157, 0, 1882826597641894099192025346277, 0, 661341726501461389571374483890716, 0, 232296340119109596380468574839330149, 0, 81594178549940498327197464360975297373, 0, 28659996909096201705269071197131281616420, 0, 10066840139729319045903902995486435768269437, 0, 3535983538407737856431130705691883059516599025, 0, 1242016340544270659315707075264569982149237866644, 0, 436258990751804203344020623035020657021980538382321] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 89942523684985244 z - 40804521363420882 z - 423 z 24 22 4 6 + 14647145626487966 z - 4133122917044608 z + 65154 z - 5082175 z 8 10 12 14 + 232622665 z - 6823370888 z + 136120702153 z - 1926121075775 z 18 16 50 - 154300983917319 z + 19931559654626 z - 154300983917319 z 48 20 36 + 909231485425633 z + 909231485425633 z + 220432953523361646 z 34 66 64 30 - 246444095719141872 z - 423 z + 65154 z - 157635996192561410 z 42 44 46 - 40804521363420882 z + 14647145626487966 z - 4133122917044608 z 58 56 54 - 6823370888 z + 136120702153 z - 1926121075775 z 52 60 68 32 + 19931559654626 z + 232622665 z + z + 220432953523361646 z 38 40 62 / - 157635996192561410 z + 89942523684985244 z - 5082175 z ) / (-1 / 28 26 2 - 712082397143360378 z + 290061319163376574 z + 623 z 24 22 4 6 - 93539947702108098 z + 23723920019398461 z - 120549 z + 11226051 z 8 10 12 14 - 597102449 z + 20021913743 z - 451664164819 z + 7174344881997 z 18 16 50 + 715507740253601 z - 82948157732039 z + 4691702691192275 z 48 20 36 - 23723920019398461 z - 4691702691192275 z - 2707229716103981214 z 34 66 64 + 2707229716103981214 z + 120549 z - 11226051 z 30 42 + 1390833710992787318 z + 712082397143360378 z 44 46 58 - 290061319163376574 z + 93539947702108098 z + 451664164819 z 56 54 52 - 7174344881997 z + 82948157732039 z - 715507740253601 z 60 70 68 32 - 20021913743 z + z - 623 z - 2169065905014123842 z 38 40 62 + 2169065905014123842 z - 1390833710992787318 z + 597102449 z ) And in Maple-input format, it is: -(1+89942523684985244*z^28-40804521363420882*z^26-423*z^2+14647145626487966*z^ 24-4133122917044608*z^22+65154*z^4-5082175*z^6+232622665*z^8-6823370888*z^10+ 136120702153*z^12-1926121075775*z^14-154300983917319*z^18+19931559654626*z^16-\ 154300983917319*z^50+909231485425633*z^48+909231485425633*z^20+ 220432953523361646*z^36-246444095719141872*z^34-423*z^66+65154*z^64-\ 157635996192561410*z^30-40804521363420882*z^42+14647145626487966*z^44-\ 4133122917044608*z^46-6823370888*z^58+136120702153*z^56-1926121075775*z^54+ 19931559654626*z^52+232622665*z^60+z^68+220432953523361646*z^32-\ 157635996192561410*z^38+89942523684985244*z^40-5082175*z^62)/(-1-\ 712082397143360378*z^28+290061319163376574*z^26+623*z^2-93539947702108098*z^24+ 23723920019398461*z^22-120549*z^4+11226051*z^6-597102449*z^8+20021913743*z^10-\ 451664164819*z^12+7174344881997*z^14+715507740253601*z^18-82948157732039*z^16+ 4691702691192275*z^50-23723920019398461*z^48-4691702691192275*z^20-\ 2707229716103981214*z^36+2707229716103981214*z^34+120549*z^66-11226051*z^64+ 1390833710992787318*z^30+712082397143360378*z^42-290061319163376574*z^44+ 93539947702108098*z^46+451664164819*z^58-7174344881997*z^56+82948157732039*z^54 -715507740253601*z^52-20021913743*z^60+z^70-623*z^68-2169065905014123842*z^32+ 2169065905014123842*z^38-1390833710992787318*z^40+597102449*z^62) The first , 40, terms are: [0, 200, 0, 69205, 0, 25148791, 0, 9205833664, 0, 3374249678923, 0, 1237091482264723, 0, 453574854351952144, 0, 166303307680193030527, 0, 60975284964216327519101, 0, 22356665004579711773779480, 0, 8197100347514711468009034281, 0, 3005477602136557219357380912345, 0, 1101962314766647166794564079958392, 0, 404035932278386916040999850592409773, 0, 148140306106137665937938117228947036783, 0, 54315838126549837716149048067201731853424, 0, 19914973507142602445612156905889925922725763, 0, 7301851236610756770083533060253259957409642843, 0, 2677233362250038244077650150343122322702026915936, 0, 981611134448866234282087315803279537246972756793223] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 77811028570156 z - 68171618679785 z - 343 z 24 22 4 6 + 45799873203871 z - 23525433313785 z + 36829 z - 1847420 z 8 10 12 14 + 52590016 z - 945218173 z + 11444477326 z - 97350796470 z 18 16 50 48 - 2712890099166 z + 598540274491 z - 1847420 z + 52590016 z 20 36 34 + 9192815267754 z + 9192815267754 z - 23525433313785 z 30 42 44 46 - 68171618679785 z - 97350796470 z + 11444477326 z - 945218173 z 56 54 52 32 38 + z - 343 z + 36829 z + 45799873203871 z - 2712890099166 z 40 / 2 28 + 598540274491 z ) / ((-1 + z ) (1 + 453456349706217 z / 26 2 24 22 - 393735456717928 z - 534 z + 257648001692563 z - 126877444105266 z 4 6 8 10 12 + 76083 z - 4627996 z + 153163737 z - 3119918454 z + 42093101205 z 14 18 16 50 - 394164388120 z - 12934915498484 z + 2641855906643 z - 4627996 z 48 20 36 + 153163737 z + 46884914573869 z + 46884914573869 z 34 30 42 - 126877444105266 z - 393735456717928 z - 394164388120 z 44 46 56 54 52 + 42093101205 z - 3119918454 z + z - 534 z + 76083 z 32 38 40 + 257648001692563 z - 12934915498484 z + 2641855906643 z )) And in Maple-input format, it is: -(1+77811028570156*z^28-68171618679785*z^26-343*z^2+45799873203871*z^24-\ 23525433313785*z^22+36829*z^4-1847420*z^6+52590016*z^8-945218173*z^10+ 11444477326*z^12-97350796470*z^14-2712890099166*z^18+598540274491*z^16-1847420* z^50+52590016*z^48+9192815267754*z^20+9192815267754*z^36-23525433313785*z^34-\ 68171618679785*z^30-97350796470*z^42+11444477326*z^44-945218173*z^46+z^56-343*z ^54+36829*z^52+45799873203871*z^32-2712890099166*z^38+598540274491*z^40)/(-1+z^ 2)/(1+453456349706217*z^28-393735456717928*z^26-534*z^2+257648001692563*z^24-\ 126877444105266*z^22+76083*z^4-4627996*z^6+153163737*z^8-3119918454*z^10+ 42093101205*z^12-394164388120*z^14-12934915498484*z^18+2641855906643*z^16-\ 4627996*z^50+153163737*z^48+46884914573869*z^20+46884914573869*z^36-\ 126877444105266*z^34-393735456717928*z^30-394164388120*z^42+42093101205*z^44-\ 3119918454*z^46+z^56-534*z^54+76083*z^52+257648001692563*z^32-12934915498484*z^ 38+2641855906643*z^40) The first , 40, terms are: [0, 192, 0, 62932, 0, 21814815, 0, 7647246432, 0, 2687960111175, 0, 945432706625649, 0, 332591751706861312, 0, 117006711584947298261, 0, 41163731351347529790919, 0, 14481710949051583936292840, 0, 5094778557807625654719079427, 0, 1792383037094931308026599523084, 0, 630574431450800065320275828289384, 0, 221841040545008710389916091875983777, 0, 78045421687454455530735893730430205465, 0, 27456992789611604143464194884859522151928, 0, 9659585878115186726256424203683269109595340, 0, 3398318237392910529177868806957163587880456675, 0, 1195555067106323813793497839791502809251641617176, 0, 420605669816342498316100867191190478988566030694511] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 2889404477592807267771 z - 385620530948712668974 z - 509 z 24 22 4 + 43746248658975110269 z - 4188161141511269964 z + 104656 z 6 102 8 10 - 12065996 z - 46660647188 z + 897553325 z - 46660647188 z 12 14 18 + 1786563782739 z - 52295280955604 z - 22231736407010014 z 16 50 + 1203249319282065 z - 838294899081299035972816154 z 48 20 + 528962374309697366409691534 z + 335424635519783265 z 36 34 + 1990830740211927103332190 z - 484126341760414365329246 z 66 80 - 292339128753594067646234734 z + 101919498380050684935265 z 100 90 88 + 1786563782739 z - 4188161141511269964 z + 43746248658975110269 z 84 94 + 2889404477592807267771 z - 22231736407010014 z 86 96 98 - 385620530948712668974 z + 1203249319282065 z - 52295280955604 z 92 82 + 335424635519783265 z - 18512419107378017589274 z 64 112 110 106 + 528962374309697366409691534 z + z - 509 z - 12065996 z 108 30 + 104656 z - 18512419107378017589274 z 42 44 - 59792985406923754933801298 z + 141397491976287246572296462 z 46 58 - 292339128753594067646234734 z - 1417705210184211342762528978 z 56 54 + 1513853072953558272007300090 z - 1417705210184211342762528978 z 52 60 + 1164282247423836945164928430 z + 1164282247423836945164928430 z 70 68 - 59792985406923754933801298 z + 141397491976287246572296462 z 78 32 - 484126341760414365329246 z + 101919498380050684935265 z 38 40 - 7107228639919873452586145 z + 22078146745243770412582861 z 62 76 - 838294899081299035972816154 z + 1990830740211927103332190 z 74 72 - 7107228639919873452586145 z + 22078146745243770412582861 z 104 / 2 28 + 897553325 z ) / ((-1 + z ) (1 + 11794685212182079047126 z / 26 2 24 - 1502811242378346634174 z - 709 z + 162375645750467837372 z 22 4 6 102 - 14768443746261842856 z + 177216 z - 23496136 z - 111268739502 z 8 10 12 + 1951583778 z - 111268739502 z + 4617255279916 z 14 18 16 - 145262431658980 z - 70127801941114138 z + 3570229744263023 z 50 48 - 4798950680608760452791179322 z + 2984168653741161698821370467 z 20 36 + 1120501808926987523 z + 9559583217236173397434722 z 34 66 - 2240126390219171508301866 z - 1619036844455842902066769530 z 80 100 + 453343717877428769407129 z + 4617255279916 z 90 88 - 14768443746261842856 z + 162375645750467837372 z 84 94 + 11794685212182079047126 z - 70127801941114138 z 86 96 98 - 1502811242378346634174 z + 3570229744263023 z - 145262431658980 z 92 82 + 1120501808926987523 z - 78973479371378324759796 z 64 112 110 106 + 2984168653741161698821370467 z + z - 709 z - 23496136 z 108 30 + 177216 z - 78973479371378324759796 z 42 44 - 315718022773826734868330858 z + 765941099484029989381968746 z 46 58 - 1619036844455842902066769530 z - 8254932200114866305805875034 z 56 54 + 8833704815694690653709572319 z - 8254932200114866305805875034 z 52 60 + 6736024631915194346151188602 z + 6736024631915194346151188602 z 70 68 - 315718022773826734868330858 z + 765941099484029989381968746 z 78 32 - 2240126390219171508301866 z + 453343717877428769407129 z 38 40 - 35324781073083888853979401 z + 113271468438271466070378434 z 62 76 - 4798950680608760452791179322 z + 9559583217236173397434722 z 74 72 - 35324781073083888853979401 z + 113271468438271466070378434 z 104 + 1951583778 z )) And in Maple-input format, it is: -(1+2889404477592807267771*z^28-385620530948712668974*z^26-509*z^2+ 43746248658975110269*z^24-4188161141511269964*z^22+104656*z^4-12065996*z^6-\ 46660647188*z^102+897553325*z^8-46660647188*z^10+1786563782739*z^12-\ 52295280955604*z^14-22231736407010014*z^18+1203249319282065*z^16-\ 838294899081299035972816154*z^50+528962374309697366409691534*z^48+ 335424635519783265*z^20+1990830740211927103332190*z^36-484126341760414365329246 *z^34-292339128753594067646234734*z^66+101919498380050684935265*z^80+ 1786563782739*z^100-4188161141511269964*z^90+43746248658975110269*z^88+ 2889404477592807267771*z^84-22231736407010014*z^94-385620530948712668974*z^86+ 1203249319282065*z^96-52295280955604*z^98+335424635519783265*z^92-\ 18512419107378017589274*z^82+528962374309697366409691534*z^64+z^112-509*z^110-\ 12065996*z^106+104656*z^108-18512419107378017589274*z^30-\ 59792985406923754933801298*z^42+141397491976287246572296462*z^44-\ 292339128753594067646234734*z^46-1417705210184211342762528978*z^58+ 1513853072953558272007300090*z^56-1417705210184211342762528978*z^54+ 1164282247423836945164928430*z^52+1164282247423836945164928430*z^60-\ 59792985406923754933801298*z^70+141397491976287246572296462*z^68-\ 484126341760414365329246*z^78+101919498380050684935265*z^32-\ 7107228639919873452586145*z^38+22078146745243770412582861*z^40-\ 838294899081299035972816154*z^62+1990830740211927103332190*z^76-\ 7107228639919873452586145*z^74+22078146745243770412582861*z^72+897553325*z^104) /(-1+z^2)/(1+11794685212182079047126*z^28-1502811242378346634174*z^26-709*z^2+ 162375645750467837372*z^24-14768443746261842856*z^22+177216*z^4-23496136*z^6-\ 111268739502*z^102+1951583778*z^8-111268739502*z^10+4617255279916*z^12-\ 145262431658980*z^14-70127801941114138*z^18+3570229744263023*z^16-\ 4798950680608760452791179322*z^50+2984168653741161698821370467*z^48+ 1120501808926987523*z^20+9559583217236173397434722*z^36-\ 2240126390219171508301866*z^34-1619036844455842902066769530*z^66+ 453343717877428769407129*z^80+4617255279916*z^100-14768443746261842856*z^90+ 162375645750467837372*z^88+11794685212182079047126*z^84-70127801941114138*z^94-\ 1502811242378346634174*z^86+3570229744263023*z^96-145262431658980*z^98+ 1120501808926987523*z^92-78973479371378324759796*z^82+ 2984168653741161698821370467*z^64+z^112-709*z^110-23496136*z^106+177216*z^108-\ 78973479371378324759796*z^30-315718022773826734868330858*z^42+ 765941099484029989381968746*z^44-1619036844455842902066769530*z^46-\ 8254932200114866305805875034*z^58+8833704815694690653709572319*z^56-\ 8254932200114866305805875034*z^54+6736024631915194346151188602*z^52+ 6736024631915194346151188602*z^60-315718022773826734868330858*z^70+ 765941099484029989381968746*z^68-2240126390219171508301866*z^78+ 453343717877428769407129*z^32-35324781073083888853979401*z^38+ 113271468438271466070378434*z^40-4798950680608760452791179322*z^62+ 9559583217236173397434722*z^76-35324781073083888853979401*z^74+ 113271468438271466070378434*z^72+1951583778*z^104) The first , 40, terms are: [0, 201, 0, 69441, 0, 25147541, 0, 9180281348, 0, 3357093374265, 0, 1228125127347209, 0, 449326914042032041, 0, 164396282396632757293, 0, 60148376185565482796445, 0, 22006774547055176687639445, 0, 8051726613058077803500839477, 0, 2945924964207103411148767225217, 0, 1077840130117013839897046801372436, 0, 394354698648247099665876101521300589, 0, 144284503960601906605211155467088369453, 0, 52790085068838184262215848769834199721621, 0, 19314569515553070814312400473751254893374465, 0, 7066717075580576977502370242348180174287569889, 0, 2585534727361699115477943164248353927117230205117, 0, 945982378366368147793538601437060458261667515633829] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 88234383490824 z - 77269548434424 z - 338 z 24 22 4 6 + 51840480333883 z - 26561444243822 z + 35910 z - 1838818 z 8 10 12 14 + 53817627 z - 993197700 z + 12296549148 z - 106440518212 z 18 16 50 48 - 3032047032734 z + 662916644521 z - 1838818 z + 53817627 z 20 36 34 + 10337536925978 z + 10337536925978 z - 26561444243822 z 30 42 44 46 - 77269548434424 z - 106440518212 z + 12296549148 z - 993197700 z 56 54 52 32 38 + z - 338 z + 35910 z + 51840480333883 z - 3032047032734 z 40 / 2 28 + 662916644521 z ) / ((-1 + z ) (1 + 523310507383128 z / 26 2 24 22 - 453850984170480 z - 548 z + 295922530401691 z - 144836202319836 z 4 6 8 10 12 + 74002 z - 4507396 z + 152439755 z - 3192311176 z + 44264334004 z 14 18 16 50 - 424702432776 z - 14454648962940 z + 2904930711193 z - 4507396 z 48 20 36 + 152439755 z + 53042842783630 z + 53042842783630 z 34 30 42 - 144836202319836 z - 453850984170480 z - 424702432776 z 44 46 56 54 52 + 44264334004 z - 3192311176 z + z - 548 z + 74002 z 32 38 40 + 295922530401691 z - 14454648962940 z + 2904930711193 z )) And in Maple-input format, it is: -(1+88234383490824*z^28-77269548434424*z^26-338*z^2+51840480333883*z^24-\ 26561444243822*z^22+35910*z^4-1838818*z^6+53817627*z^8-993197700*z^10+ 12296549148*z^12-106440518212*z^14-3032047032734*z^18+662916644521*z^16-1838818 *z^50+53817627*z^48+10337536925978*z^20+10337536925978*z^36-26561444243822*z^34 -77269548434424*z^30-106440518212*z^42+12296549148*z^44-993197700*z^46+z^56-338 *z^54+35910*z^52+51840480333883*z^32-3032047032734*z^38+662916644521*z^40)/(-1+ z^2)/(1+523310507383128*z^28-453850984170480*z^26-548*z^2+295922530401691*z^24-\ 144836202319836*z^22+74002*z^4-4507396*z^6+152439755*z^8-3192311176*z^10+ 44264334004*z^12-424702432776*z^14-14454648962940*z^18+2904930711193*z^16-\ 4507396*z^50+152439755*z^48+53042842783630*z^20+53042842783630*z^36-\ 144836202319836*z^34-453850984170480*z^30-424702432776*z^42+44264334004*z^44-\ 3192311176*z^46+z^56-548*z^54+74002*z^52+295922530401691*z^32-14454648962940*z^ 38+2904930711193*z^40) The first , 40, terms are: [0, 211, 0, 77199, 0, 29394781, 0, 11246094773, 0, 4305640155399, 0, 1648623690111099, 0, 631266901855849481, 0, 241716192584081592121, 0, 92554741440124166775275, 0, 35439830147605213033647063, 0, 13570148398061638651160571205, 0, 5196100747249613291565664503373, 0, 1989621792800137778492554589700159, 0, 761839516042700474239075290107469539, 0, 291713455446378740762764756840119183697, 0, 111699036734943140754799306534532977308593, 0, 42770309612305148998416249113011845540140483, 0, 16377038135730354335691192426662318128592540127, 0, 6270877637556401553013127830513259450463790193005, 0, 2401161065834649213064028884579210115679069513940773] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 10425125647643337 z + 6264394465670549 z + 409 z 24 22 4 6 - 2906431609351201 z + 1036246524333066 z - 59969 z + 4346101 z 8 10 12 14 - 180630309 z + 4728007311 z - 82885809639 z + 1015176639075 z 18 16 50 + 58107294313707 z - 8952646925415 z + 82885809639 z 48 20 36 - 1015176639075 z - 282069261903778 z - 6264394465670549 z 34 30 42 + 10425125647643337 z + 13438085554902003 z + 282069261903778 z 44 46 58 56 - 58107294313707 z + 8952646925415 z + 59969 z - 4346101 z 54 52 60 32 + 180630309 z - 4728007311 z - 409 z - 13438085554902003 z 38 40 62 / + 2906431609351201 z - 1036246524333066 z + z ) / (1 / 28 26 2 + 93859250915404820 z - 49878799672883340 z - 617 z 24 22 4 6 + 20497512208874422 z - 6483913798274578 z + 113193 z - 9763956 z 8 10 12 14 + 474265948 z - 14310778191 z + 286067794215 z - 3964716072411 z 18 16 50 - 287411494002888 z + 39386581866336 z - 3964716072411 z 48 20 36 + 39386581866336 z + 1568390250804010 z + 93859250915404820 z 34 64 30 - 136995624595382187 z + z - 136995624595382187 z 42 44 46 - 6483913798274578 z + 1568390250804010 z - 287411494002888 z 58 56 54 52 - 9763956 z + 474265948 z - 14310778191 z + 286067794215 z 60 32 38 + 113193 z + 155369804207828487 z - 49878799672883340 z 40 62 + 20497512208874422 z - 617 z ) And in Maple-input format, it is: -(-1-10425125647643337*z^28+6264394465670549*z^26+409*z^2-2906431609351201*z^24 +1036246524333066*z^22-59969*z^4+4346101*z^6-180630309*z^8+4728007311*z^10-\ 82885809639*z^12+1015176639075*z^14+58107294313707*z^18-8952646925415*z^16+ 82885809639*z^50-1015176639075*z^48-282069261903778*z^20-6264394465670549*z^36+ 10425125647643337*z^34+13438085554902003*z^30+282069261903778*z^42-\ 58107294313707*z^44+8952646925415*z^46+59969*z^58-4346101*z^56+180630309*z^54-\ 4728007311*z^52-409*z^60-13438085554902003*z^32+2906431609351201*z^38-\ 1036246524333066*z^40+z^62)/(1+93859250915404820*z^28-49878799672883340*z^26-\ 617*z^2+20497512208874422*z^24-6483913798274578*z^22+113193*z^4-9763956*z^6+ 474265948*z^8-14310778191*z^10+286067794215*z^12-3964716072411*z^14-\ 287411494002888*z^18+39386581866336*z^16-3964716072411*z^50+39386581866336*z^48 +1568390250804010*z^20+93859250915404820*z^36-136995624595382187*z^34+z^64-\ 136995624595382187*z^30-6483913798274578*z^42+1568390250804010*z^44-\ 287411494002888*z^46-9763956*z^58+474265948*z^56-14310778191*z^54+286067794215* z^52+113193*z^60+155369804207828487*z^32-49878799672883340*z^38+ 20497512208874422*z^40-617*z^62) The first , 40, terms are: [0, 208, 0, 75112, 0, 28217815, 0, 10645506448, 0, 4018544061889, 0, 1517112774886165, 0, 572765390491605904, 0, 216240911796824524255, 0, 81639336612721620948499, 0, 30822026435597406706951756, 0, 11636515113491680869086098489, 0, 4393237618669376350581749237752, 0, 1658618293706065824982698542244172, 0, 626193000535300067411497945960197367, 0, 236412244748445562580909087932789362103, 0, 89254829456387436712844649232677213838084, 0, 33697174145321252195470537823371693204656424, 0, 12721995574904137972999681014709371376448616913, 0, 4803048787117632458190674155135913203906014500788, 0, 1813337971673459032324033963069014570951254762002027] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 376572054762 z + 732642529046 z + 379 z - 1018929076402 z 22 4 6 8 10 + 1018929076402 z - 44789 z + 2054547 z - 48533627 z + 674784373 z 12 14 18 16 - 5937709339 z + 34545816041 z + 376572054762 z - 136790997334 z 20 36 34 30 - 732642529046 z - 674784373 z + 5937709339 z + 136790997334 z 42 44 46 32 38 40 + 44789 z - 379 z + z - 34545816041 z + 48533627 z - 2054547 z / 28 26 2 ) / (1 + 6661917178136 z - 10991270981732 z - 602 z / 24 22 4 6 + 12978557011460 z - 10991270981732 z + 99164 z - 5534282 z 8 10 12 14 + 155031934 z - 2548845842 z + 26600143244 z - 184358917570 z 18 16 48 20 - 2872341132324 z + 872574800255 z + z + 6661917178136 z 36 34 30 42 + 26600143244 z - 184358917570 z - 2872341132324 z - 5534282 z 44 46 32 38 40 + 99164 z - 602 z + 872574800255 z - 2548845842 z + 155031934 z ) And in Maple-input format, it is: -(-1-376572054762*z^28+732642529046*z^26+379*z^2-1018929076402*z^24+ 1018929076402*z^22-44789*z^4+2054547*z^6-48533627*z^8+674784373*z^10-5937709339 *z^12+34545816041*z^14+376572054762*z^18-136790997334*z^16-732642529046*z^20-\ 674784373*z^36+5937709339*z^34+136790997334*z^30+44789*z^42-379*z^44+z^46-\ 34545816041*z^32+48533627*z^38-2054547*z^40)/(1+6661917178136*z^28-\ 10991270981732*z^26-602*z^2+12978557011460*z^24-10991270981732*z^22+99164*z^4-\ 5534282*z^6+155031934*z^8-2548845842*z^10+26600143244*z^12-184358917570*z^14-\ 2872341132324*z^18+872574800255*z^16+z^48+6661917178136*z^20+26600143244*z^36-\ 184358917570*z^34-2872341132324*z^30-5534282*z^42+99164*z^44-602*z^46+ 872574800255*z^32-2548845842*z^38+155031934*z^40) The first , 40, terms are: [0, 223, 0, 79871, 0, 29448505, 0, 10935318745, 0, 4072160912479, 0, 1518192427019935, 0, 566291552199495233, 0, 211271619311919738049, 0, 78827673023142362939807, 0, 29412462927829544091524639, 0, 10974642069156782313435567001, 0, 4094981523043221385053708584057, 0, 1527969240004084660850460271995199, 0, 570135030598841108896046119598393567, 0, 212736018719560484095023583033332615553, 0, 79378777496097625776731617811451635118721, 0, 29618824407233406981424675078277060000527071, 0, 11051754813030496310172004430299114313933238463, 0, 4123772244546158312866401938684958218923308656121, 0, 1538714702117947943337537893711577994245932583214617] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3190563942996532581028 z - 420700769809890218692 z - 506 z 24 22 4 + 47123984360993981068 z - 4452527503944654116 z + 103604 z 6 102 8 10 - 11915694 z - 46297329046 z + 886879430 z - 46297329046 z 12 14 18 + 1785905461144 z - 52803579130602 z - 23009826331939972 z 16 50 + 1229403869683763 z - 1003818396116843765941198872 z 48 20 + 631407120982664309458570315 z + 351840758255742244 z 36 34 + 2290132637567976648414128 z - 551891893800147987567286 z 66 80 - 347548918252519746278172582 z + 115046962995149378346881 z 100 90 88 + 1785905461144 z - 4452527503944654116 z + 47123984360993981068 z 84 94 + 3190563942996532581028 z - 23009826331939972 z 86 96 98 - 420700769809890218692 z + 1229403869683763 z - 52803579130602 z 92 82 + 351840758255742244 z - 20675912107606526642212 z 64 112 110 106 + 631407120982664309458570315 z + z - 506 z - 11915694 z 108 30 + 103604 z - 20675912107606526642212 z 42 44 - 70330932639751413522371074 z + 167278117814352910725666308 z 46 58 - 347548918252519746278172582 z - 1703819772995726807177692632 z 56 54 + 1820202180043103222253767208 z - 1703819772995726807177692632 z 52 60 + 1397341231587466321200201528 z + 1397341231587466321200201528 z 70 68 - 70330932639751413522371074 z + 167278117814352910725666308 z 78 32 - 551891893800147987567286 z + 115046962995149378346881 z 38 40 - 8243399995342168699891450 z + 25798434934856282651715738 z 62 76 - 1003818396116843765941198872 z + 2290132637567976648414128 z 74 72 - 8243399995342168699891450 z + 25798434934856282651715738 z 104 / 28 + 886879430 z ) / (-1 - 14598049983634904464240 z / 26 2 24 + 1803565797653005437604 z + 717 z - 189179140263761912948 z 22 4 6 102 + 16724071034069164048 z - 178116 z + 23462396 z + 4676866569796 z 8 10 12 - 1945683814 z + 111473315934 z - 4676866569796 z 14 18 16 + 149524066014924 z + 75365098667218427 z - 3749423834330095 z 50 48 + 9365087485442914506038957771 z - 5517812105002696526310842767 z 20 36 - 1235040587320531328 z - 13561858616896149602167044 z 34 66 + 3064401160986581060516469 z + 5517812105002696526310842767 z 80 100 - 3064401160986581060516469 z - 149524066014924 z 90 88 + 189179140263761912948 z - 1803565797653005437604 z 84 94 - 100923284499762408612544 z + 1235040587320531328 z 86 96 + 14598049983634904464240 z - 75365098667218427 z 98 92 + 3749423834330095 z - 16724071034069164048 z 82 64 + 598968683118162218033273 z - 9365087485442914506038957771 z 112 114 110 106 108 - 717 z + z + 178116 z + 1945683814 z - 23462396 z 30 42 + 100923284499762408612544 z + 505586814057447759700513530 z 44 46 - 1283182972533140782924961300 z + 2845353096203264027459413228 z 58 56 + 20680374575282073541934051176 z - 20680374575282073541934051176 z 54 52 + 18124391747581069734173213440 z - 13919342972557576412450845760 z 60 70 - 18124391747581069734173213440 z + 1283182972533140782924961300 z 68 78 - 2845353096203264027459413228 z + 13561858616896149602167044 z 32 38 - 598968683118162218033273 z + 52067646432152032982969900 z 40 62 - 173825996476241876330620610 z + 13919342972557576412450845760 z 76 74 - 52067646432152032982969900 z + 173825996476241876330620610 z 72 104 - 505586814057447759700513530 z - 111473315934 z ) And in Maple-input format, it is: -(1+3190563942996532581028*z^28-420700769809890218692*z^26-506*z^2+ 47123984360993981068*z^24-4452527503944654116*z^22+103604*z^4-11915694*z^6-\ 46297329046*z^102+886879430*z^8-46297329046*z^10+1785905461144*z^12-\ 52803579130602*z^14-23009826331939972*z^18+1229403869683763*z^16-\ 1003818396116843765941198872*z^50+631407120982664309458570315*z^48+ 351840758255742244*z^20+2290132637567976648414128*z^36-551891893800147987567286 *z^34-347548918252519746278172582*z^66+115046962995149378346881*z^80+ 1785905461144*z^100-4452527503944654116*z^90+47123984360993981068*z^88+ 3190563942996532581028*z^84-23009826331939972*z^94-420700769809890218692*z^86+ 1229403869683763*z^96-52803579130602*z^98+351840758255742244*z^92-\ 20675912107606526642212*z^82+631407120982664309458570315*z^64+z^112-506*z^110-\ 11915694*z^106+103604*z^108-20675912107606526642212*z^30-\ 70330932639751413522371074*z^42+167278117814352910725666308*z^44-\ 347548918252519746278172582*z^46-1703819772995726807177692632*z^58+ 1820202180043103222253767208*z^56-1703819772995726807177692632*z^54+ 1397341231587466321200201528*z^52+1397341231587466321200201528*z^60-\ 70330932639751413522371074*z^70+167278117814352910725666308*z^68-\ 551891893800147987567286*z^78+115046962995149378346881*z^32-\ 8243399995342168699891450*z^38+25798434934856282651715738*z^40-\ 1003818396116843765941198872*z^62+2290132637567976648414128*z^76-\ 8243399995342168699891450*z^74+25798434934856282651715738*z^72+886879430*z^104) /(-1-14598049983634904464240*z^28+1803565797653005437604*z^26+717*z^2-\ 189179140263761912948*z^24+16724071034069164048*z^22-178116*z^4+23462396*z^6+ 4676866569796*z^102-1945683814*z^8+111473315934*z^10-4676866569796*z^12+ 149524066014924*z^14+75365098667218427*z^18-3749423834330095*z^16+ 9365087485442914506038957771*z^50-5517812105002696526310842767*z^48-\ 1235040587320531328*z^20-13561858616896149602167044*z^36+ 3064401160986581060516469*z^34+5517812105002696526310842767*z^66-\ 3064401160986581060516469*z^80-149524066014924*z^100+189179140263761912948*z^90 -1803565797653005437604*z^88-100923284499762408612544*z^84+1235040587320531328* z^94+14598049983634904464240*z^86-75365098667218427*z^96+3749423834330095*z^98-\ 16724071034069164048*z^92+598968683118162218033273*z^82-\ 9365087485442914506038957771*z^64-717*z^112+z^114+178116*z^110+1945683814*z^106 -23462396*z^108+100923284499762408612544*z^30+505586814057447759700513530*z^42-\ 1283182972533140782924961300*z^44+2845353096203264027459413228*z^46+ 20680374575282073541934051176*z^58-20680374575282073541934051176*z^56+ 18124391747581069734173213440*z^54-13919342972557576412450845760*z^52-\ 18124391747581069734173213440*z^60+1283182972533140782924961300*z^70-\ 2845353096203264027459413228*z^68+13561858616896149602167044*z^78-\ 598968683118162218033273*z^32+52067646432152032982969900*z^38-\ 173825996476241876330620610*z^40+13919342972557576412450845760*z^62-\ 52067646432152032982969900*z^76+173825996476241876330620610*z^74-\ 505586814057447759700513530*z^72-111473315934*z^104) The first , 40, terms are: [0, 211, 0, 76775, 0, 29011901, 0, 11018438289, 0, 4188698649731, 0, 1592675521281971, 0, 605613341784186657, 0, 230286298253878492929, 0, 87567268477704893613603, 0, 33297815760308227548804987, 0, 12661633672813561892346189945, 0, 4814639287998028510447009345141, 0, 1830786787039471555701495866657919, 0, 696164357151298768823780042957330035, 0, 264719417779364513180023224784696969313, 0, 100660669331105052926580537499195445379169, 0, 38276641870024292745871758419369821578737139, 0, 14554853674114411069907895601474073311692441615, 0, 5534544179566671119813520931534083818748613364997, 0, 2104533646398639943030668913886956555457266503326313] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 36208378280001170272 z - 8130187596285978880 z - 470 z 24 22 4 6 + 1503849588284732693 z - 227551600617889710 z + 85520 z - 8366710 z 8 10 12 14 + 507514947 z - 20751949920 z + 604540911368 z - 13046640540656 z 18 16 50 - 2750887263280414 z + 214642760513423 z - 4114036689077553498700 z 48 20 + 6279822273233732052702 z + 27927002555467024 z 36 34 + 2271115923185925901664 z - 1054327010835283594140 z 66 80 88 84 86 - 227551600617889710 z + 507514947 z + z + 85520 z - 470 z 82 64 30 - 8366710 z + 1503849588284732693 z - 133613534597363739680 z 42 44 - 8089185065865184628624 z + 8800667820196010157104 z 46 58 - 8089185065865184628624 z - 133613534597363739680 z 56 54 + 410510326891184384314 z - 1054327010835283594140 z 52 60 + 2271115923185925901664 z + 36208378280001170272 z 70 68 78 - 2750887263280414 z + 27927002555467024 z - 20751949920 z 32 38 + 410510326891184384314 z - 4114036689077553498700 z 40 62 76 + 6279822273233732052702 z - 8130187596285978880 z + 604540911368 z 74 72 / - 13046640540656 z + 214642760513423 z ) / (-1 / 28 26 2 - 210838728156720392392 z + 43441779846208299373 z + 689 z 24 22 4 - 7373261380084986973 z + 1023644677031780878 z - 154894 z 6 8 10 12 + 17581526 z - 1201412743 z + 54521163363 z - 1748744788968 z 14 18 16 + 41366311847152 z + 10408552296152795 z - 744013111117543 z 50 48 + 61009170368962881486126 z - 85500674376881937476646 z 20 36 - 115243712595059702 z - 18624726449671624345572 z 34 66 80 + 7939802056191365284962 z + 7373261380084986973 z - 54521163363 z 90 88 84 86 82 + z - 689 z - 17581526 z + 154894 z + 1201412743 z 64 30 - 43441779846208299373 z + 847804731648968270328 z 42 44 + 85500674376881937476646 z - 101192433378053001245288 z 46 58 + 101192433378053001245288 z + 2838002338358043579170 z 56 54 - 7939802056191365284962 z + 18624726449671624345572 z 52 60 - 36727245225673111490868 z - 847804731648968270328 z 70 68 78 + 115243712595059702 z - 1023644677031780878 z + 1748744788968 z 32 38 - 2838002338358043579170 z + 36727245225673111490868 z 40 62 - 61009170368962881486126 z + 210838728156720392392 z 76 74 72 - 41366311847152 z + 744013111117543 z - 10408552296152795 z ) And in Maple-input format, it is: -(1+36208378280001170272*z^28-8130187596285978880*z^26-470*z^2+ 1503849588284732693*z^24-227551600617889710*z^22+85520*z^4-8366710*z^6+ 507514947*z^8-20751949920*z^10+604540911368*z^12-13046640540656*z^14-\ 2750887263280414*z^18+214642760513423*z^16-4114036689077553498700*z^50+ 6279822273233732052702*z^48+27927002555467024*z^20+2271115923185925901664*z^36-\ 1054327010835283594140*z^34-227551600617889710*z^66+507514947*z^80+z^88+85520*z ^84-470*z^86-8366710*z^82+1503849588284732693*z^64-133613534597363739680*z^30-\ 8089185065865184628624*z^42+8800667820196010157104*z^44-8089185065865184628624* z^46-133613534597363739680*z^58+410510326891184384314*z^56-\ 1054327010835283594140*z^54+2271115923185925901664*z^52+36208378280001170272*z^ 60-2750887263280414*z^70+27927002555467024*z^68-20751949920*z^78+ 410510326891184384314*z^32-4114036689077553498700*z^38+6279822273233732052702*z ^40-8130187596285978880*z^62+604540911368*z^76-13046640540656*z^74+ 214642760513423*z^72)/(-1-210838728156720392392*z^28+43441779846208299373*z^26+ 689*z^2-7373261380084986973*z^24+1023644677031780878*z^22-154894*z^4+17581526*z ^6-1201412743*z^8+54521163363*z^10-1748744788968*z^12+41366311847152*z^14+ 10408552296152795*z^18-744013111117543*z^16+61009170368962881486126*z^50-\ 85500674376881937476646*z^48-115243712595059702*z^20-18624726449671624345572*z^ 36+7939802056191365284962*z^34+7373261380084986973*z^66-54521163363*z^80+z^90-\ 689*z^88-17581526*z^84+154894*z^86+1201412743*z^82-43441779846208299373*z^64+ 847804731648968270328*z^30+85500674376881937476646*z^42-\ 101192433378053001245288*z^44+101192433378053001245288*z^46+ 2838002338358043579170*z^58-7939802056191365284962*z^56+18624726449671624345572 *z^54-36727245225673111490868*z^52-847804731648968270328*z^60+ 115243712595059702*z^70-1023644677031780878*z^68+1748744788968*z^78-\ 2838002338358043579170*z^32+36727245225673111490868*z^38-\ 61009170368962881486126*z^40+210838728156720392392*z^62-41366311847152*z^76+ 744013111117543*z^74-10408552296152795*z^72) The first , 40, terms are: [0, 219, 0, 81517, 0, 31458243, 0, 12204691627, 0, 4740192517429, 0, 1841503425182627, 0, 715440995542058905, 0, 277959077762058541721, 0, 107991417992702532984179, 0, 41956373163052491454313349, 0, 16300717038362939386178412395, 0, 6333087597839854405268554794179, 0, 2460505191395676494387452826084029, 0, 955945376821956053243571676045194347, 0, 371399973929714736636765460213561330433, 0, 144294793401563385972357714252982232066497, 0, 56060820853296775711990188140308630806275531, 0, 21780520008232160627187276065423636337650100765, 0, 8462078232339166401791410329996431130972759024259, 0, 3287651901019368494457967890604811882461733340311339] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3462564375008446640489 z - 455644480884386473726 z - 503 z 24 22 4 + 50858937595422328799 z - 4780970246359548988 z + 102722 z 6 102 8 10 - 11851440 z - 46880154196 z + 888839327 z - 46880154196 z 12 14 18 + 1829268398977 z - 54714246318612 z - 24338929599783982 z 16 50 + 1287825973851651 z - 1061334790326925224318553418 z 48 20 + 669233931504860392619488338 z + 375264243321945915 z 36 34 + 2475840207174030259162336 z - 598067804741280237013682 z 66 80 - 369467535804437363432516798 z + 124868160669871313959299 z 100 90 88 + 1829268398977 z - 4780970246359548988 z + 50858937595422328799 z 84 94 + 3462564375008446640489 z - 24338929599783982 z 86 96 98 - 455644480884386473726 z + 1287825973851651 z - 54714246318612 z 92 82 + 375264243321945915 z - 22453263476944647162026 z 64 112 110 106 + 669233931504860392619488338 z + z - 503 z - 11851440 z 108 30 + 102722 z - 22453263476944647162026 z 42 44 - 75283036554338942463742746 z + 178422977801410477458472826 z 46 58 - 369467535804437363432516798 z - 1796073296520590096287035442 z 56 54 + 1918016820865559343459410534 z - 1796073296520590096287035442 z 52 60 + 1474681919787513591470910106 z + 1474681919787513591470910106 z 70 68 - 75283036554338942463742746 z + 178422977801410477458472826 z 78 32 - 598067804741280237013682 z + 124868160669871313959299 z 38 40 - 8885280606359426339931223 z + 27713459636440396361825081 z 62 76 - 1061334790326925224318553418 z + 2475840207174030259162336 z 74 72 - 8885280606359426339931223 z + 27713459636440396361825081 z 104 / 2 28 + 888839327 z ) / ((-1 + z ) (1 + 14069591709727159553414 z / 26 2 24 - 1763739863900315635038 z - 723 z + 187050992740970188260 z 22 4 6 102 - 16661338722595160160 z + 177004 z - 23117708 z - 109973053554 z 8 10 12 + 1914033390 z - 109973053554 z + 4636342168252 z 14 18 16 - 148965123017916 z - 75484952747920186 z + 3749375860865575 z 50 48 - 6096066704422008901422710962 z + 3788099507580903044996530339 z 20 36 + 1235616601124217987 z + 11902604485632116581108654 z 34 66 - 2767803780849548972640422 z - 2052900588464867406077759818 z 80 100 + 554793780696922857467621 z + 4636342168252 z 90 88 - 16661338722595160160 z + 187050992740970188260 z 84 94 + 14069591709727159553414 z - 75484952747920186 z 86 96 98 - 1763739863900315635038 z + 3749375860865575 z - 148965123017916 z 92 82 + 1235616601124217987 z - 95525583945867271210448 z 64 112 110 106 + 3788099507580903044996530339 z + z - 723 z - 23117708 z 108 30 + 177004 z - 95525583945867271210448 z 42 44 - 398665567351561749090445850 z + 969550275968409239990595794 z 46 58 - 2052900588464867406077759818 z - 10492676451710984184177276530 z 56 54 + 11229056987057583321047427423 z - 10492676451710984184177276530 z 52 60 + 8560230680856963568269394202 z + 8560230680856963568269394202 z 70 68 - 398665567351561749090445850 z + 969550275968409239990595794 z 78 32 - 2767803780849548972640422 z + 554793780696922857467621 z 38 40 - 44248887313838053170855283 z + 142542695021725945708717514 z 62 76 - 6096066704422008901422710962 z + 11902604485632116581108654 z 74 72 - 44248887313838053170855283 z + 142542695021725945708717514 z 104 + 1914033390 z )) And in Maple-input format, it is: -(1+3462564375008446640489*z^28-455644480884386473726*z^26-503*z^2+ 50858937595422328799*z^24-4780970246359548988*z^22+102722*z^4-11851440*z^6-\ 46880154196*z^102+888839327*z^8-46880154196*z^10+1829268398977*z^12-\ 54714246318612*z^14-24338929599783982*z^18+1287825973851651*z^16-\ 1061334790326925224318553418*z^50+669233931504860392619488338*z^48+ 375264243321945915*z^20+2475840207174030259162336*z^36-598067804741280237013682 *z^34-369467535804437363432516798*z^66+124868160669871313959299*z^80+ 1829268398977*z^100-4780970246359548988*z^90+50858937595422328799*z^88+ 3462564375008446640489*z^84-24338929599783982*z^94-455644480884386473726*z^86+ 1287825973851651*z^96-54714246318612*z^98+375264243321945915*z^92-\ 22453263476944647162026*z^82+669233931504860392619488338*z^64+z^112-503*z^110-\ 11851440*z^106+102722*z^108-22453263476944647162026*z^30-\ 75283036554338942463742746*z^42+178422977801410477458472826*z^44-\ 369467535804437363432516798*z^46-1796073296520590096287035442*z^58+ 1918016820865559343459410534*z^56-1796073296520590096287035442*z^54+ 1474681919787513591470910106*z^52+1474681919787513591470910106*z^60-\ 75283036554338942463742746*z^70+178422977801410477458472826*z^68-\ 598067804741280237013682*z^78+124868160669871313959299*z^32-\ 8885280606359426339931223*z^38+27713459636440396361825081*z^40-\ 1061334790326925224318553418*z^62+2475840207174030259162336*z^76-\ 8885280606359426339931223*z^74+27713459636440396361825081*z^72+888839327*z^104) /(-1+z^2)/(1+14069591709727159553414*z^28-1763739863900315635038*z^26-723*z^2+ 187050992740970188260*z^24-16661338722595160160*z^22+177004*z^4-23117708*z^6-\ 109973053554*z^102+1914033390*z^8-109973053554*z^10+4636342168252*z^12-\ 148965123017916*z^14-75484952747920186*z^18+3749375860865575*z^16-\ 6096066704422008901422710962*z^50+3788099507580903044996530339*z^48+ 1235616601124217987*z^20+11902604485632116581108654*z^36-\ 2767803780849548972640422*z^34-2052900588464867406077759818*z^66+ 554793780696922857467621*z^80+4636342168252*z^100-16661338722595160160*z^90+ 187050992740970188260*z^88+14069591709727159553414*z^84-75484952747920186*z^94-\ 1763739863900315635038*z^86+3749375860865575*z^96-148965123017916*z^98+ 1235616601124217987*z^92-95525583945867271210448*z^82+ 3788099507580903044996530339*z^64+z^112-723*z^110-23117708*z^106+177004*z^108-\ 95525583945867271210448*z^30-398665567351561749090445850*z^42+ 969550275968409239990595794*z^44-2052900588464867406077759818*z^46-\ 10492676451710984184177276530*z^58+11229056987057583321047427423*z^56-\ 10492676451710984184177276530*z^54+8560230680856963568269394202*z^52+ 8560230680856963568269394202*z^60-398665567351561749090445850*z^70+ 969550275968409239990595794*z^68-2767803780849548972640422*z^78+ 554793780696922857467621*z^32-44248887313838053170855283*z^38+ 142542695021725945708717514*z^40-6096066704422008901422710962*z^62+ 11902604485632116581108654*z^76-44248887313838053170855283*z^74+ 142542695021725945708717514*z^72+1914033390*z^104) The first , 40, terms are: [0, 221, 0, 84999, 0, 33704881, 0, 13395536152, 0, 5325024553939, 0, 2116868912467497, 0, 841526405295871547, 0, 334535136352034133139, 0, 132989012642338797068311, 0, 52867623675342009323397195, 0, 21016665836272774065172095909, 0, 8354834439560620894209707572547, 0, 3321328847316497579610966927801192, 0, 1320340383992603671013112837880754961, 0, 524879892881785273659758403265848767491, 0, 208657483548740701615285639093499976043061, 0, 82948396445243378284191265581612755731754557, 0, 32974788901976337221412216387566297224922715333, 0, 13108592205850365221944160464997945233361218278549, 0, 5211108102316982024675248105108523262629017912626571] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 40375719531643140 z - 18176961953470938 z - 371 z 24 22 4 6 + 6476815192747710 z - 1817883952029424 z + 47966 z - 3203655 z 8 10 12 14 + 129797109 z - 3480485360 z + 65197958725 z - 884665317111 z 18 16 50 - 68110915475795 z + 8920528041662 z - 68110915475795 z 48 20 36 + 399376751041249 z + 399376751041249 z + 100046605636940070 z 34 66 64 30 - 112024763876203392 z - 371 z + 47966 z - 71231276112545698 z 42 44 46 - 18176961953470938 z + 6476815192747710 z - 1817883952029424 z 58 56 54 52 - 3480485360 z + 65197958725 z - 884665317111 z + 8920528041662 z 60 68 32 38 + 129797109 z + z + 100046605636940070 z - 71231276112545698 z 40 62 / 28 + 40375719531643140 z - 3203655 z ) / (-1 - 321340689636325954 z / 26 2 24 + 129329973117203822 z + 583 z - 41227197871885314 z 22 4 6 8 + 10361318030796205 z - 92641 z + 7218905 z - 334577223 z 10 12 14 + 10157051905 z - 213994181341 z + 3251680350091 z 18 16 50 + 311927828035661 z - 36622092815829 z + 2039949624788827 z 48 20 36 - 10361318030796205 z - 2039949624788827 z - 1250213610137024466 z 34 66 64 + 1250213610137024466 z + 92641 z - 7218905 z 30 42 44 + 634426341205411266 z + 321340689636325954 z - 129329973117203822 z 46 58 56 + 41227197871885314 z + 213994181341 z - 3251680350091 z 54 52 60 70 + 36622092815829 z - 311927828035661 z - 10157051905 z + z 68 32 38 - 583 z - 997399826649827646 z + 997399826649827646 z 40 62 - 634426341205411266 z + 334577223 z ) And in Maple-input format, it is: -(1+40375719531643140*z^28-18176961953470938*z^26-371*z^2+6476815192747710*z^24 -1817883952029424*z^22+47966*z^4-3203655*z^6+129797109*z^8-3480485360*z^10+ 65197958725*z^12-884665317111*z^14-68110915475795*z^18+8920528041662*z^16-\ 68110915475795*z^50+399376751041249*z^48+399376751041249*z^20+ 100046605636940070*z^36-112024763876203392*z^34-371*z^66+47966*z^64-\ 71231276112545698*z^30-18176961953470938*z^42+6476815192747710*z^44-\ 1817883952029424*z^46-3480485360*z^58+65197958725*z^56-884665317111*z^54+ 8920528041662*z^52+129797109*z^60+z^68+100046605636940070*z^32-\ 71231276112545698*z^38+40375719531643140*z^40-3203655*z^62)/(-1-\ 321340689636325954*z^28+129329973117203822*z^26+583*z^2-41227197871885314*z^24+ 10361318030796205*z^22-92641*z^4+7218905*z^6-334577223*z^8+10157051905*z^10-\ 213994181341*z^12+3251680350091*z^14+311927828035661*z^18-36622092815829*z^16+ 2039949624788827*z^50-10361318030796205*z^48-2039949624788827*z^20-\ 1250213610137024466*z^36+1250213610137024466*z^34+92641*z^66-7218905*z^64+ 634426341205411266*z^30+321340689636325954*z^42-129329973117203822*z^44+ 41227197871885314*z^46+213994181341*z^58-3251680350091*z^56+36622092815829*z^54 -311927828035661*z^52-10157051905*z^60+z^70-583*z^68-997399826649827646*z^32+ 997399826649827646*z^38-634426341205411266*z^40+334577223*z^62) The first , 40, terms are: [0, 212, 0, 78921, 0, 30386301, 0, 11729520868, 0, 4528762751877, 0, 1748589291597169, 0, 675144784482282980, 0, 260679034600862074601, 0, 100650352274456554104405, 0, 38861941652517053258539220, 0, 15004920252873854152637656605, 0, 5793525033123551142969225810853, 0, 2236928403739448686911558294246292, 0, 863696739868851789427721183955564413, 0, 333480524997164179653669028701734126161, 0, 128759615984276032284401583399540912437156, 0, 49715163152508589194518524520322152232637849, 0, 19195439722205884082234348218939570135872389469, 0, 7411519600137268208995397584048868486799377487780, 0, 2861649619814305807238857216887266590628847205281045] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 23997549575772807177 z - 5434695156328822413 z - 461 z 24 22 4 6 + 1017753751131741694 z - 156588016661341930 z + 81330 z - 7658542 z 8 10 12 14 + 445124563 z - 17413770058 z + 485897319379 z - 10073505054166 z 18 16 50 - 1985580013847394 z + 159859528258711 z - 2703777537341054264506 z 48 20 + 4130845908833265510704 z + 19633345424460611 z 36 34 + 1491469004464203069296 z - 692292146666249021172 z 66 80 88 84 86 - 156588016661341930 z + 445124563 z + z + 81330 z - 461 z 82 64 30 - 7658542 z + 1017753751131741694 z - 88078477397611024274 z 42 44 - 5324671027400724505722 z + 5794461120155486657892 z 46 58 - 5324671027400724505722 z - 88078477397611024274 z 56 54 + 269831897256470248226 z - 692292146666249021172 z 52 60 + 1491469004464203069296 z + 23997549575772807177 z 70 68 78 - 1985580013847394 z + 19633345424460611 z - 17413770058 z 32 38 + 269831897256470248226 z - 2703777537341054264506 z 40 62 76 + 4130845908833265510704 z - 5434695156328822413 z + 485897319379 z 74 72 / - 10073505054166 z + 159859528258711 z ) / (-1 / 28 26 2 - 140491358651373879986 z + 29173469967526829621 z + 685 z 24 22 4 - 5011437517001029362 z + 707577198849049809 z - 150514 z 6 8 10 12 + 16467669 z - 1076176966 z + 46577747697 z - 1426057041454 z 14 18 16 + 32303137836041 z + 7562825117259977 z - 558940178745450 z 50 48 + 40717226835883880552691 z - 57169836830032013678871 z 20 36 - 81444976147761130 z - 12370322225210804368839 z 34 66 80 + 5263756724406815317991 z + 5011437517001029362 z - 46577747697 z 90 88 84 86 82 + z - 685 z - 16467669 z + 150514 z + 1076176966 z 64 30 - 29173469967526829621 z + 562541397837677137022 z 42 44 + 57169836830032013678871 z - 67731377840699179118139 z 46 58 + 67731377840699179118139 z + 1880379429754042097455 z 56 54 - 5263756724406815317991 z + 12370322225210804368839 z 52 60 - 24453037066891695618583 z - 562541397837677137022 z 70 68 78 + 81444976147761130 z - 707577198849049809 z + 1426057041454 z 32 38 - 1880379429754042097455 z + 24453037066891695618583 z 40 62 - 40717226835883880552691 z + 140491358651373879986 z 76 74 72 - 32303137836041 z + 558940178745450 z - 7562825117259977 z ) And in Maple-input format, it is: -(1+23997549575772807177*z^28-5434695156328822413*z^26-461*z^2+ 1017753751131741694*z^24-156588016661341930*z^22+81330*z^4-7658542*z^6+ 445124563*z^8-17413770058*z^10+485897319379*z^12-10073505054166*z^14-\ 1985580013847394*z^18+159859528258711*z^16-2703777537341054264506*z^50+ 4130845908833265510704*z^48+19633345424460611*z^20+1491469004464203069296*z^36-\ 692292146666249021172*z^34-156588016661341930*z^66+445124563*z^80+z^88+81330*z^ 84-461*z^86-7658542*z^82+1017753751131741694*z^64-88078477397611024274*z^30-\ 5324671027400724505722*z^42+5794461120155486657892*z^44-5324671027400724505722* z^46-88078477397611024274*z^58+269831897256470248226*z^56-692292146666249021172 *z^54+1491469004464203069296*z^52+23997549575772807177*z^60-1985580013847394*z^ 70+19633345424460611*z^68-17413770058*z^78+269831897256470248226*z^32-\ 2703777537341054264506*z^38+4130845908833265510704*z^40-5434695156328822413*z^ 62+485897319379*z^76-10073505054166*z^74+159859528258711*z^72)/(-1-\ 140491358651373879986*z^28+29173469967526829621*z^26+685*z^2-\ 5011437517001029362*z^24+707577198849049809*z^22-150514*z^4+16467669*z^6-\ 1076176966*z^8+46577747697*z^10-1426057041454*z^12+32303137836041*z^14+ 7562825117259977*z^18-558940178745450*z^16+40717226835883880552691*z^50-\ 57169836830032013678871*z^48-81444976147761130*z^20-12370322225210804368839*z^ 36+5263756724406815317991*z^34+5011437517001029362*z^66-46577747697*z^80+z^90-\ 685*z^88-16467669*z^84+150514*z^86+1076176966*z^82-29173469967526829621*z^64+ 562541397837677137022*z^30+57169836830032013678871*z^42-67731377840699179118139 *z^44+67731377840699179118139*z^46+1880379429754042097455*z^58-\ 5263756724406815317991*z^56+12370322225210804368839*z^54-\ 24453037066891695618583*z^52-562541397837677137022*z^60+81444976147761130*z^70-\ 707577198849049809*z^68+1426057041454*z^78-1880379429754042097455*z^32+ 24453037066891695618583*z^38-40717226835883880552691*z^40+140491358651373879986 *z^62-32303137836041*z^76+558940178745450*z^74-7562825117259977*z^72) The first , 40, terms are: [0, 224, 0, 84256, 0, 32809351, 0, 12850403304, 0, 5039859863345, 0, 1977250825180645, 0, 775782115653856416, 0, 304387123151147993439, 0, 119430397969969660519939, 0, 46860184754311938043692772, 0, 18386253336893835475659994329, 0, 7214105912580812165844304468336, 0, 2830556289919150198533774497000364, 0, 1110608721582608090610374706503639095, 0, 435763011728307825776408227180751531575, 0, 170977769893029527682772444515803544927652, 0, 67085541940603778898901297509423458478144896, 0, 26321959517608064376306714087087395500274393361, 0, 10327792439430615427468612517945343285653580534092, 0, 4052255175026351016486584408862960075060283430509531] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 101483733631630518 z - 39806816594100230 z - 387 z 24 22 4 6 + 12578984254000711 z - 3184565234920175 z + 52955 z - 3719230 z 8 10 12 14 + 156772295 z - 4348119667 z + 84249557174 z - 1188314583611 z 18 16 50 - 101740254467274 z + 12563962324103 z - 3184565234920175 z 48 20 36 + 12578984254000711 z + 641249531447639 z + 528265180399218308 z 34 66 64 - 476753541416487578 z - 3719230 z + 156772295 z 30 42 44 - 209306634907999544 z - 209306634907999544 z + 101483733631630518 z 46 58 56 - 39806816594100230 z - 1188314583611 z + 12563962324103 z 54 52 60 70 - 101740254467274 z + 641249531447639 z + 84249557174 z - 387 z 68 32 38 + 52955 z + 350305285136003722 z - 476753541416487578 z 40 62 72 / + 350305285136003722 z - 4348119667 z + z ) / (-1 / 28 26 2 - 764916139130873024 z + 269991209816704519 z + 607 z 24 22 4 6 - 76755428176103177 z + 17480439920504978 z - 102622 z + 8385680 z 8 10 12 14 - 401702396 z + 12524865990 z - 271306339400 z + 4265270007404 z 18 16 50 + 451909666209764 z - 50185502687218 z + 76755428176103177 z 48 20 36 - 269991209816704519 z - 3166423141789880 z - 6038211329498335772 z 34 66 64 + 4915666793215872408 z + 401702396 z - 12524865990 z 30 42 + 1752414004093745592 z + 3255905209631979360 z 44 46 58 - 1752414004093745592 z + 764916139130873024 z + 50185502687218 z 56 54 52 - 451909666209764 z + 3166423141789880 z - 17480439920504978 z 60 70 68 32 - 4265270007404 z + 102622 z - 8385680 z - 3255905209631979360 z 38 40 62 + 6038211329498335772 z - 4915666793215872408 z + 271306339400 z 74 72 + z - 607 z ) And in Maple-input format, it is: -(1+101483733631630518*z^28-39806816594100230*z^26-387*z^2+12578984254000711*z^ 24-3184565234920175*z^22+52955*z^4-3719230*z^6+156772295*z^8-4348119667*z^10+ 84249557174*z^12-1188314583611*z^14-101740254467274*z^18+12563962324103*z^16-\ 3184565234920175*z^50+12578984254000711*z^48+641249531447639*z^20+ 528265180399218308*z^36-476753541416487578*z^34-3719230*z^66+156772295*z^64-\ 209306634907999544*z^30-209306634907999544*z^42+101483733631630518*z^44-\ 39806816594100230*z^46-1188314583611*z^58+12563962324103*z^56-101740254467274*z ^54+641249531447639*z^52+84249557174*z^60-387*z^70+52955*z^68+ 350305285136003722*z^32-476753541416487578*z^38+350305285136003722*z^40-\ 4348119667*z^62+z^72)/(-1-764916139130873024*z^28+269991209816704519*z^26+607*z ^2-76755428176103177*z^24+17480439920504978*z^22-102622*z^4+8385680*z^6-\ 401702396*z^8+12524865990*z^10-271306339400*z^12+4265270007404*z^14+ 451909666209764*z^18-50185502687218*z^16+76755428176103177*z^50-\ 269991209816704519*z^48-3166423141789880*z^20-6038211329498335772*z^36+ 4915666793215872408*z^34+401702396*z^66-12524865990*z^64+1752414004093745592*z^ 30+3255905209631979360*z^42-1752414004093745592*z^44+764916139130873024*z^46+ 50185502687218*z^58-451909666209764*z^56+3166423141789880*z^54-\ 17480439920504978*z^52-4265270007404*z^60+102622*z^70-8385680*z^68-\ 3255905209631979360*z^32+6038211329498335772*z^38-4915666793215872408*z^40+ 271306339400*z^62+z^74-607*z^72) The first , 40, terms are: [0, 220, 0, 83873, 0, 33000521, 0, 13024020740, 0, 5142135480961, 0, 2030333418178193, 0, 801668927036973356, 0, 316536157558715600841, 0, 124983215116860782720161, 0, 49349195006487297089412676, 0, 19485360956204438082952186321, 0, 7693728172072251486710183506481, 0, 3037842271744642440598532836932916, 0, 1199481637741428443061377670093850849, 0, 473611224870101142399867667554483028681, 0, 187003773351148324868243411889852252368860, 0, 73837800734477225344185291620440803043204721, 0, 29154603244645967542467117450994886942453848737, 0, 11511595441599262576242289841662239484775609661908, 0, 4545314113831568533751387429288185661348251484962633] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 18971671377647578807 z - 4319346273765972334 z - 446 z 24 22 4 6 + 813669944310374453 z - 126002677311012776 z + 75595 z - 6874744 z 8 10 12 14 + 388921708 z - 14907907040 z + 409530566077 z - 8385883810826 z 18 16 50 - 1621774761909064 z + 131724141427719 z - 2101879055676535306442 z 48 20 + 3206660732648840625411 z + 15911023302552952 z 36 34 + 1161827470580029164289 z - 540738851193012433688 z 66 80 88 84 86 - 126002677311012776 z + 388921708 z + z + 75595 z - 446 z 82 64 30 - 6874744 z + 813669944310374453 z - 69306818201445170192 z 42 44 - 4129898664121555088272 z + 4493019819746313693072 z 46 58 - 4129898664121555088272 z - 69306818201445170192 z 56 54 + 211470870742126552404 z - 540738851193012433688 z 52 60 + 1161827470580029164289 z + 18971671377647578807 z 70 68 78 - 1621774761909064 z + 15911023302552952 z - 14907907040 z 32 38 + 211470870742126552404 z - 2101879055676535306442 z 40 62 76 + 3206660732648840625411 z - 4319346273765972334 z + 409530566077 z 74 72 / 2 - 8385883810826 z + 131724141427719 z ) / ((-1 + z ) (1 / 28 26 2 + 91420432427023239845 z - 19712295705457891245 z - 677 z 24 22 4 + 3503565405363789537 z - 510176880334392956 z + 140757 z 6 8 10 12 - 14688718 z + 925639082 z - 38923437118 z + 1162719755839 z 14 18 16 - 25743354311719 z - 5753243114479932 z + 435427293471123 z 50 48 - 12316399825335882035591 z + 19162332111311157815819 z 20 36 + 60391308242523076 z + 6627423031880870258375 z 34 66 80 - 2982584065439604181106 z - 510176880334392956 z + 925639082 z 88 84 86 82 64 + z + 140757 z - 677 z - 14688718 z + 3503565405363789537 z 30 42 - 351134310139709123138 z - 24975954805088721008520 z 44 46 + 27281178230034767126840 z - 24975954805088721008520 z 58 56 - 351134310139709123138 z + 1120972174431881198070 z 54 52 - 2982584065439604181106 z + 6627423031880870258375 z 60 70 68 + 91420432427023239845 z - 5753243114479932 z + 60391308242523076 z 78 32 - 38923437118 z + 1120972174431881198070 z 38 40 - 12316399825335882035591 z + 19162332111311157815819 z 62 76 74 - 19712295705457891245 z + 1162719755839 z - 25743354311719 z 72 + 435427293471123 z )) And in Maple-input format, it is: -(1+18971671377647578807*z^28-4319346273765972334*z^26-446*z^2+ 813669944310374453*z^24-126002677311012776*z^22+75595*z^4-6874744*z^6+388921708 *z^8-14907907040*z^10+409530566077*z^12-8385883810826*z^14-1621774761909064*z^ 18+131724141427719*z^16-2101879055676535306442*z^50+3206660732648840625411*z^48 +15911023302552952*z^20+1161827470580029164289*z^36-540738851193012433688*z^34-\ 126002677311012776*z^66+388921708*z^80+z^88+75595*z^84-446*z^86-6874744*z^82+ 813669944310374453*z^64-69306818201445170192*z^30-4129898664121555088272*z^42+ 4493019819746313693072*z^44-4129898664121555088272*z^46-69306818201445170192*z^ 58+211470870742126552404*z^56-540738851193012433688*z^54+1161827470580029164289 *z^52+18971671377647578807*z^60-1621774761909064*z^70+15911023302552952*z^68-\ 14907907040*z^78+211470870742126552404*z^32-2101879055676535306442*z^38+ 3206660732648840625411*z^40-4319346273765972334*z^62+409530566077*z^76-\ 8385883810826*z^74+131724141427719*z^72)/(-1+z^2)/(1+91420432427023239845*z^28-\ 19712295705457891245*z^26-677*z^2+3503565405363789537*z^24-510176880334392956*z ^22+140757*z^4-14688718*z^6+925639082*z^8-38923437118*z^10+1162719755839*z^12-\ 25743354311719*z^14-5753243114479932*z^18+435427293471123*z^16-\ 12316399825335882035591*z^50+19162332111311157815819*z^48+60391308242523076*z^ 20+6627423031880870258375*z^36-2982584065439604181106*z^34-510176880334392956*z ^66+925639082*z^80+z^88+140757*z^84-677*z^86-14688718*z^82+3503565405363789537* z^64-351134310139709123138*z^30-24975954805088721008520*z^42+ 27281178230034767126840*z^44-24975954805088721008520*z^46-351134310139709123138 *z^58+1120972174431881198070*z^56-2982584065439604181106*z^54+ 6627423031880870258375*z^52+91420432427023239845*z^60-5753243114479932*z^70+ 60391308242523076*z^68-38923437118*z^78+1120972174431881198070*z^32-\ 12316399825335882035591*z^38+19162332111311157815819*z^40-19712295705457891245* z^62+1162719755839*z^76-25743354311719*z^74+435427293471123*z^72) The first , 40, terms are: [0, 232, 0, 91457, 0, 37149889, 0, 15141527512, 0, 6174742666945, 0, 2518315389681697, 0, 1027091326086407544, 0, 418899095858000543393, 0, 170848058600340346247713, 0, 69680414023783826034077640, 0, 28419171052619486683411283681, 0, 11590764748433858177726023772833, 0, 4727295785820068865959162444534792, 0, 1928028558493103058399802969302562273, 0, 786346844132948782418087837641320073313, 0, 320711722114831652883175459882654173573688, 0, 130802341828393491926877977168901015144987873, 0, 53347762018093217032732145351233545794662993153, 0, 21757895711630294437449291011772065522135852508952, 0, 8873962241145642684139339493584965312747442402346497] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 26 2 24 22 4 6 f(z) = - (-1 + z + 323 z - 323 z + 27124 z - 27124 z + 674004 z 8 10 12 14 18 - 6906849 z + 32245979 z - 67773240 z + 67773240 z + 6906849 z 16 20 / 14 2 10 - 32245979 z - 674004 z ) / (-925386456 z - 570 z - 194104646 z / 6 8 22 20 - 2198548 z + 1 + 29747265 z - 2198548 z + 29747265 z 18 16 28 26 24 4 - 194104646 z + 622588091 z + z - 570 z + 67587 z + 67587 z 12 + 622588091 z ) And in Maple-input format, it is: -(-1+z^26+323*z^2-323*z^24+27124*z^22-27124*z^4+674004*z^6-6906849*z^8+32245979 *z^10-67773240*z^12+67773240*z^14+6906849*z^18-32245979*z^16-674004*z^20)/(-\ 925386456*z^14-570*z^2-194104646*z^10-2198548*z^6+1+29747265*z^8-2198548*z^22+ 29747265*z^20-194104646*z^18+622588091*z^16+z^28-570*z^26+67587*z^24+67587*z^4+ 622588091*z^12) The first , 40, terms are: [0, 247, 0, 100327, 0, 42016945, 0, 17689058641, 0, 7456352173063, 0, 3144009537849559, 0, 1325792638568261473, 0, 559082671841479774369, 0, 235764633146028686786839, 0, 99421849758961543903427527, 0, 41926167128190016715485936657, 0, 17680254777664019110822538523121, 0, 7455759419525734575408825245980711, 0, 3144092068404825869922375633430878391, 0, 1325862919055434011956132233829141694145, 0, 559116095353255684178302659442918872783169, 0, 235779131927453946743493293532723085717244215, 0, 99428007019791279458802208663730007498674165671, 0, 41928768246645983463130018695968606108717687598961, 0, 17681352159994907213788300752327478509823905622040209] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 102667451020448124 z - 45978758414178162 z - 421 z 24 22 4 6 + 16248931760659242 z - 4505452231378192 z + 64526 z - 5017171 z 8 10 12 14 + 229649263 z - 6769248556 z + 136337337611 z - 1955059835615 z 18 16 50 - 162038483244409 z + 20559519061110 z - 162038483244409 z 48 20 36 + 972880846347365 z + 972880846347365 z + 255586786049315830 z 34 66 64 30 - 286323674198222728 z - 421 z + 64526 z - 181686697722889278 z 42 44 46 - 45978758414178162 z + 16248931760659242 z - 4505452231378192 z 58 56 54 - 6769248556 z + 136337337611 z - 1955059835615 z 52 60 68 32 + 20559519061110 z + 229649263 z + z + 255586786049315830 z 38 40 62 / - 181686697722889278 z + 102667451020448124 z - 5017171 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 594781310275073666 z - 253008131055445776 z - 652 z 24 22 4 6 + 83981150105188082 z - 21670953057249592 z + 124697 z - 11240174 z 8 10 12 14 + 579185089 z - 18974006042 z + 422054155661 z - 6658704292382 z 18 16 50 - 661598466023188 z + 76798586569573 z - 661598466023188 z 48 20 36 + 4323417331198861 z + 4323417331198861 z + 1575447340683649074 z 34 66 64 30 - 1779111094456957276 z - 652 z + 124697 z - 1093707541843426372 z 42 44 46 - 253008131055445776 z + 83981150105188082 z - 21670953057249592 z 58 56 54 - 18974006042 z + 422054155661 z - 6658704292382 z 52 60 68 32 + 76798586569573 z + 579185089 z + z + 1575447340683649074 z 38 40 62 - 1093707541843426372 z + 594781310275073666 z - 11240174 z )) And in Maple-input format, it is: -(1+102667451020448124*z^28-45978758414178162*z^26-421*z^2+16248931760659242*z^ 24-4505452231378192*z^22+64526*z^4-5017171*z^6+229649263*z^8-6769248556*z^10+ 136337337611*z^12-1955059835615*z^14-162038483244409*z^18+20559519061110*z^16-\ 162038483244409*z^50+972880846347365*z^48+972880846347365*z^20+ 255586786049315830*z^36-286323674198222728*z^34-421*z^66+64526*z^64-\ 181686697722889278*z^30-45978758414178162*z^42+16248931760659242*z^44-\ 4505452231378192*z^46-6769248556*z^58+136337337611*z^56-1955059835615*z^54+ 20559519061110*z^52+229649263*z^60+z^68+255586786049315830*z^32-\ 181686697722889278*z^38+102667451020448124*z^40-5017171*z^62)/(-1+z^2)/(1+ 594781310275073666*z^28-253008131055445776*z^26-652*z^2+83981150105188082*z^24-\ 21670953057249592*z^22+124697*z^4-11240174*z^6+579185089*z^8-18974006042*z^10+ 422054155661*z^12-6658704292382*z^14-661598466023188*z^18+76798586569573*z^16-\ 661598466023188*z^50+4323417331198861*z^48+4323417331198861*z^20+ 1575447340683649074*z^36-1779111094456957276*z^34-652*z^66+124697*z^64-\ 1093707541843426372*z^30-253008131055445776*z^42+83981150105188082*z^44-\ 21670953057249592*z^46-18974006042*z^58+422054155661*z^56-6658704292382*z^54+ 76798586569573*z^52+579185089*z^60+z^68+1575447340683649074*z^32-\ 1093707541843426372*z^38+594781310275073666*z^40-11240174*z^62) The first , 40, terms are: [0, 232, 0, 90673, 0, 36476201, 0, 14729063448, 0, 5952115342137, 0, 2405701282853481, 0, 972364420814980120, 0, 393025145189827633337, 0, 158859253910881531293217, 0, 64210331070613610062164904, 0, 25953584527636037655165427601, 0, 10490345648400994637294128698609, 0, 4240160057994523715193103730698152, 0, 1713857478547678558218553582141085249, 0, 692735042434618838345233526502521994137, 0, 280001018205246439978420676959062230150552, 0, 113175406749689110517763851504474371035102857, 0, 45745093268263164523565061684027568601817112601, 0, 18490002538733837931707813722739265301094123812248, 0, 7473592673154375780526463616931072631761934795608649] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 126054166368588 z - 109883171058245 z - 331 z 24 22 4 6 + 72706883751759 z - 36397450431373 z + 34541 z - 1771676 z 8 10 12 14 + 53178224 z - 1023752529 z + 13351438438 z - 122174109542 z 18 16 50 48 - 3858390844798 z + 803234925283 z - 1771676 z + 53178224 z 20 36 34 + 13711606429778 z + 13711606429778 z - 36397450431373 z 30 42 44 - 109883171058245 z - 122174109542 z + 13351438438 z 46 56 54 52 32 - 1023752529 z + z - 331 z + 34541 z + 72706883751759 z 38 40 / 2 - 3858390844798 z + 803234925283 z ) / ((-1 + z ) (1 / 28 26 2 24 + 771845881638577 z - 664900804271456 z - 562 z + 424919781065543 z 22 4 6 8 - 201251708192966 z + 71455 z - 4235768 z + 144684809 z 10 12 14 18 - 3146035290 z + 46079734493 z - 470987596024 z - 18192355762300 z 16 50 48 20 + 3437733578351 z - 4235768 z + 144684809 z + 70492587954993 z 36 34 30 + 70492587954993 z - 201251708192966 z - 664900804271456 z 42 44 46 56 54 - 470987596024 z + 46079734493 z - 3146035290 z + z - 562 z 52 32 38 40 + 71455 z + 424919781065543 z - 18192355762300 z + 3437733578351 z )) And in Maple-input format, it is: -(1+126054166368588*z^28-109883171058245*z^26-331*z^2+72706883751759*z^24-\ 36397450431373*z^22+34541*z^4-1771676*z^6+53178224*z^8-1023752529*z^10+ 13351438438*z^12-122174109542*z^14-3858390844798*z^18+803234925283*z^16-1771676 *z^50+53178224*z^48+13711606429778*z^20+13711606429778*z^36-36397450431373*z^34 -109883171058245*z^30-122174109542*z^42+13351438438*z^44-1023752529*z^46+z^56-\ 331*z^54+34541*z^52+72706883751759*z^32-3858390844798*z^38+803234925283*z^40)/( -1+z^2)/(1+771845881638577*z^28-664900804271456*z^26-562*z^2+424919781065543*z^ 24-201251708192966*z^22+71455*z^4-4235768*z^6+144684809*z^8-3146035290*z^10+ 46079734493*z^12-470987596024*z^14-18192355762300*z^18+3437733578351*z^16-\ 4235768*z^50+144684809*z^48+70492587954993*z^20+70492587954993*z^36-\ 201251708192966*z^34-664900804271456*z^30-470987596024*z^42+46079734493*z^44-\ 3146035290*z^46+z^56-562*z^54+71455*z^52+424919781065543*z^32-18192355762300*z^ 38+3437733578351*z^40) The first , 40, terms are: [0, 232, 0, 93140, 0, 38265423, 0, 15739303152, 0, 6474358850311, 0, 2663241457867681, 0, 1095530691309267120, 0, 450649173141699622741, 0, 185375617555868076965631, 0, 76254704668193535473061960, 0, 31367555567088455240428426371, 0, 12903119178600978578862786517244, 0, 5307729006210057286191098488350848, 0, 2183347050695167937498553784804822665, 0, 898125043347529639502324893488280416049, 0, 369445889617583339794120119323412198777840, 0, 151972452350950271999592576665051958188554332, 0, 62514232591593684609564442766050484988261331363, 0, 25715379439235833665230309038751107318246785600024, 0, 10578082978703888960459822887798246223142370921899447] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 97773110686670034 z - 37923665093210499 z - 381 z 24 22 4 6 + 11834617941030444 z - 2956882423592210 z + 50670 z - 3444019 z 8 10 12 14 + 141785832 z - 3881504370 z + 74877078236 z - 1058279037126 z 18 16 50 - 92129390614480 z + 11264459439217 z - 2956882423592210 z 48 20 36 + 11834617941030444 z + 587687963450706 z + 520333835967252746 z 34 66 64 - 468907369275314430 z - 3444019 z + 141785832 z 30 42 44 - 203543529577104995 z - 203543529577104995 z + 97773110686670034 z 46 58 56 - 37923665093210499 z - 1058279037126 z + 11264459439217 z 54 52 60 70 - 92129390614480 z + 587687963450706 z + 74877078236 z - 381 z 68 32 38 + 50670 z + 343050964346375477 z - 468907369275314430 z 40 62 72 / 2 + 343050964346375477 z - 3881504370 z + z ) / ((-1 + z ) (1 / 28 26 2 + 544300021228492263 z - 200397919230965254 z - 620 z 24 22 4 6 + 58862203986041009 z - 13743620885392556 z + 100686 z - 7812116 z 8 10 12 14 + 358792531 z - 10852035958 z + 230094331416 z - 3560856252344 z 18 16 50 - 367587042407174 z + 41352867015913 z - 13743620885392556 z 48 20 36 + 58862203986041009 z + 2537301160677811 z + 3199693873370307247 z 34 66 64 - 2864751614247053230 z - 7812116 z + 358792531 z 30 42 - 1182151754126683894 z - 1182151754126683894 z 44 46 58 + 544300021228492263 z - 200397919230965254 z - 3560856252344 z 56 54 52 + 41352867015913 z - 367587042407174 z + 2537301160677811 z 60 70 68 32 + 230094331416 z - 620 z + 100686 z + 2055828077169955013 z 38 40 62 - 2864751614247053230 z + 2055828077169955013 z - 10852035958 z 72 + z )) And in Maple-input format, it is: -(1+97773110686670034*z^28-37923665093210499*z^26-381*z^2+11834617941030444*z^ 24-2956882423592210*z^22+50670*z^4-3444019*z^6+141785832*z^8-3881504370*z^10+ 74877078236*z^12-1058279037126*z^14-92129390614480*z^18+11264459439217*z^16-\ 2956882423592210*z^50+11834617941030444*z^48+587687963450706*z^20+ 520333835967252746*z^36-468907369275314430*z^34-3444019*z^66+141785832*z^64-\ 203543529577104995*z^30-203543529577104995*z^42+97773110686670034*z^44-\ 37923665093210499*z^46-1058279037126*z^58+11264459439217*z^56-92129390614480*z^ 54+587687963450706*z^52+74877078236*z^60-381*z^70+50670*z^68+343050964346375477 *z^32-468907369275314430*z^38+343050964346375477*z^40-3881504370*z^62+z^72)/(-1 +z^2)/(1+544300021228492263*z^28-200397919230965254*z^26-620*z^2+ 58862203986041009*z^24-13743620885392556*z^22+100686*z^4-7812116*z^6+358792531* z^8-10852035958*z^10+230094331416*z^12-3560856252344*z^14-367587042407174*z^18+ 41352867015913*z^16-13743620885392556*z^50+58862203986041009*z^48+ 2537301160677811*z^20+3199693873370307247*z^36-2864751614247053230*z^34-7812116 *z^66+358792531*z^64-1182151754126683894*z^30-1182151754126683894*z^42+ 544300021228492263*z^44-200397919230965254*z^46-3560856252344*z^58+ 41352867015913*z^56-367587042407174*z^54+2537301160677811*z^52+230094331416*z^ 60-620*z^70+100686*z^68+2055828077169955013*z^32-2864751614247053230*z^38+ 2055828077169955013*z^40-10852035958*z^62+z^72) The first , 40, terms are: [0, 240, 0, 98404, 0, 41264227, 0, 17330423008, 0, 7279874484353, 0, 3058091045333653, 0, 1284632219015827472, 0, 539644200604756043491, 0, 226692038636698306757775, 0, 95228080439229189545980000, 0, 40003113415194217883670719189, 0, 16804382450312841050529450261820, 0, 7059132288744249320335286337306624, 0, 2965378157632376855263896503912834919, 0, 1245686758385810746487889084545090822239, 0, 523284187557800129674427502210922213768320, 0, 219819580729005556920714982241193539619121324, 0, 92341120218808302457132176818987205611388294037, 0, 38790368241928343287973875638548256834235934655968, 0, 16294936262186727134939644286544384469930155114120599] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 15077482126 z - 35042909816 z - 288 z + 57891053834 z 22 4 6 8 10 - 68389918160 z + 21659 z - 705320 z + 12509883 z - 135552872 z 12 14 18 16 + 956415761 z - 4563383536 z - 35042909816 z + 15077482126 z 20 36 34 30 + 57891053834 z + 12509883 z - 135552872 z - 4563383536 z 42 44 32 38 40 / 2 - 288 z + z + 956415761 z - 705320 z + 21659 z ) / ((-1 + z ) ( / 44 42 40 38 36 34 z - 518 z + 49243 z - 1919668 z + 39900411 z - 500406686 z 32 30 28 26 + 4046643265 z - 21888760592 z + 80735510014 z - 204810361820 z 24 22 20 + 358188568938 z - 431638187896 z + 358188568938 z 18 16 14 12 - 204810361820 z + 80735510014 z - 21888760592 z + 4046643265 z 10 8 6 4 2 - 500406686 z + 39900411 z - 1919668 z + 49243 z - 518 z + 1)) And in Maple-input format, it is: -(1+15077482126*z^28-35042909816*z^26-288*z^2+57891053834*z^24-68389918160*z^22 +21659*z^4-705320*z^6+12509883*z^8-135552872*z^10+956415761*z^12-4563383536*z^ 14-35042909816*z^18+15077482126*z^16+57891053834*z^20+12509883*z^36-135552872*z ^34-4563383536*z^30-288*z^42+z^44+956415761*z^32-705320*z^38+21659*z^40)/(-1+z^ 2)/(z^44-518*z^42+49243*z^40-1919668*z^38+39900411*z^36-500406686*z^34+ 4046643265*z^32-21888760592*z^30+80735510014*z^28-204810361820*z^26+ 358188568938*z^24-431638187896*z^22+358188568938*z^20-204810361820*z^18+ 80735510014*z^16-21888760592*z^14+4046643265*z^12-500406686*z^10+39900411*z^8-\ 1919668*z^6+49243*z^4-518*z^2+1) The first , 40, terms are: [0, 231, 0, 91787, 0, 37406253, 0, 15271940645, 0, 6236229389155, 0, 2546588278250127, 0, 1039911629420801545, 0, 424653062641390509049, 0, 173409186793966116262591, 0, 70812502816700506011061171, 0, 28916637293080517010062909877, 0, 11808252485551402917500707789725, 0, 4821958561447012837967748321454139, 0, 1969070732251431202482260739304945079, 0, 804079815121042569537909890776815274001, 0, 328349986872153577256796560222330109777905, 0, 134083348259045431080308690257106149651986967, 0, 54753601337454955776806640116198176920815652059, 0, 22358905101541594783570390665471363449784876824253, 0, 9130369968883949029383316003301570859049784984486037] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 693759648 z - 2624468948 z - 296 z + 6637565222 z 22 4 6 8 10 - 11482309236 z + 22966 z - 750596 z + 12462681 z - 118494860 z 12 14 18 16 + 693759648 z - 2624468948 z - 11482309236 z + 6637565222 z 20 36 34 30 32 + 13764445268 z + 22966 z - 750596 z - 118494860 z + 12462681 z 38 40 / 2 40 38 36 34 - 296 z + z ) / ((-1 + z ) (z - 546 z + 53450 z - 2113030 z / 32 30 28 26 + 42185297 z - 475386036 z + 3235037280 z - 13886172076 z 24 22 20 18 + 38661291694 z - 71030936952 z + 86914662316 z - 71030936952 z 16 14 12 10 + 38661291694 z - 13886172076 z + 3235037280 z - 475386036 z 8 6 4 2 + 42185297 z - 2113030 z + 53450 z - 546 z + 1)) And in Maple-input format, it is: -(1+693759648*z^28-2624468948*z^26-296*z^2+6637565222*z^24-11482309236*z^22+ 22966*z^4-750596*z^6+12462681*z^8-118494860*z^10+693759648*z^12-2624468948*z^14 -11482309236*z^18+6637565222*z^16+13764445268*z^20+22966*z^36-750596*z^34-\ 118494860*z^30+12462681*z^32-296*z^38+z^40)/(-1+z^2)/(z^40-546*z^38+53450*z^36-\ 2113030*z^34+42185297*z^32-475386036*z^30+3235037280*z^28-13886172076*z^26+ 38661291694*z^24-71030936952*z^22+86914662316*z^20-71030936952*z^18+38661291694 *z^16-13886172076*z^14+3235037280*z^12-475386036*z^10+42185297*z^8-2113030*z^6+ 53450*z^4-546*z^2+1) The first , 40, terms are: [0, 251, 0, 106267, 0, 45990937, 0, 19931000441, 0, 8638436133531, 0, 3744088153846587, 0, 1622772098890430497, 0, 703346056824092986337, 0, 304846061540974613929403, 0, 132127166270407313977891995, 0, 57266897206331025389999535545, 0, 24820766298738666858499346269401, 0, 10757880550756035255242524670414619, 0, 4662708336698246434797568929552725115, 0, 2020923074070397250526478544597188408385, 0, 875913691441015664693615676143410626919361, 0, 379640771436459658772805466642414163635710331, 0, 164544882384197531645574860032571019997395235611, 0, 71317467342574230126001700776745979594655985865561, 0, 30910600648662930480346455923555703648527315124516665] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 35218241610536313 z + 21054563724637181 z + 465 z 24 22 4 6 - 9666528382259409 z + 3381258343859894 z - 79405 z + 6682085 z 8 10 12 14 - 321762645 z + 9678731803 z - 192314672467 z + 2622618533375 z 18 16 50 + 175151964583803 z - 25240290366107 z + 192314672467 z 48 20 36 - 2622618533375 z - 891816376271374 z - 21054563724637181 z 34 30 42 + 35218241610536313 z + 45486762288146223 z + 891816376271374 z 44 46 58 56 - 175151964583803 z + 25240290366107 z + 79405 z - 6682085 z 54 52 60 32 + 321762645 z - 9678731803 z - 465 z - 45486762288146223 z 38 40 62 / + 9666528382259409 z - 3381258343859894 z + z ) / (1 / 28 26 2 + 334984172998319468 z - 175620197732354652 z - 681 z 24 22 4 6 + 70609848933850066 z - 21621076922844770 z + 150641 z - 15172300 z 8 10 12 14 + 845027084 z - 28967819815 z + 651896478919 z - 10050897283911 z 18 16 50 - 862154950549604 z + 109478103910244 z - 10050897283911 z 48 20 36 + 109478103910244 z + 4997305590903010 z + 334984172998319468 z 34 64 30 - 492497288354787783 z + z - 492497288354787783 z 42 44 46 - 21621076922844770 z + 4997305590903010 z - 862154950549604 z 58 56 54 52 - 15172300 z + 845027084 z - 28967819815 z + 651896478919 z 60 32 38 + 150641 z + 559818678944354727 z - 175620197732354652 z 40 62 + 70609848933850066 z - 681 z ) And in Maple-input format, it is: -(-1-35218241610536313*z^28+21054563724637181*z^26+465*z^2-9666528382259409*z^ 24+3381258343859894*z^22-79405*z^4+6682085*z^6-321762645*z^8+9678731803*z^10-\ 192314672467*z^12+2622618533375*z^14+175151964583803*z^18-25240290366107*z^16+ 192314672467*z^50-2622618533375*z^48-891816376271374*z^20-21054563724637181*z^ 36+35218241610536313*z^34+45486762288146223*z^30+891816376271374*z^42-\ 175151964583803*z^44+25240290366107*z^46+79405*z^58-6682085*z^56+321762645*z^54 -9678731803*z^52-465*z^60-45486762288146223*z^32+9666528382259409*z^38-\ 3381258343859894*z^40+z^62)/(1+334984172998319468*z^28-175620197732354652*z^26-\ 681*z^2+70609848933850066*z^24-21621076922844770*z^22+150641*z^4-15172300*z^6+ 845027084*z^8-28967819815*z^10+651896478919*z^12-10050897283911*z^14-\ 862154950549604*z^18+109478103910244*z^16-10050897283911*z^50+109478103910244*z ^48+4997305590903010*z^20+334984172998319468*z^36-492497288354787783*z^34+z^64-\ 492497288354787783*z^30-21621076922844770*z^42+4997305590903010*z^44-\ 862154950549604*z^46-15172300*z^58+845027084*z^56-28967819815*z^54+651896478919 *z^52+150641*z^60+559818678944354727*z^32-175620197732354652*z^38+ 70609848933850066*z^40-681*z^62) The first , 40, terms are: [0, 216, 0, 75860, 0, 27612419, 0, 10130383440, 0, 3726962627929, 0, 1372648075309657, 0, 505772054259449664, 0, 186392342406942526571, 0, 68696215369021735442387, 0, 25319221779449686573816020, 0, 9331964794052523888400905541, 0, 3439520810514270397415091371212, 0, 1267720895158616339134725381457068, 0, 467250423910016836590575369302065551, 0, 172216952948388321638633500684122988487, 0, 63474918991517975928158887727468120229924, 0, 23395290001919060221704506664740162132061484, 0, 8622927211496500685121378890420887115884468141, 0, 3178198447114547900506183618763335394335992364860, 0, 1171405621068346740207866386787661466141297388612723] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 2338215836391 z + 4813218965242 z + 422 z 24 22 4 6 - 6878184231527 z + 6878184231527 z - 60681 z + 3832975 z 8 10 12 14 - 124011178 z + 2264167859 z - 24892879285 z + 172319647094 z 18 16 20 + 2338215836391 z - 776326075121 z - 4813218965242 z 36 34 30 42 - 2264167859 z + 24892879285 z + 776326075121 z + 60681 z 44 46 32 38 40 / - 422 z + z - 172319647094 z + 124011178 z - 3832975 z ) / (1 / 28 26 2 24 + 42535942744576 z - 73193382932514 z - 654 z + 87629902833070 z 22 4 6 8 - 73193382932514 z + 125696 z - 9655612 z + 369812900 z 10 12 14 - 7959556830 z + 103688970600 z - 858710233074 z 18 16 48 20 - 17066334807812 z + 4671485266780 z + z + 42535942744576 z 36 34 30 42 + 103688970600 z - 858710233074 z - 17066334807812 z - 9655612 z 44 46 32 38 + 125696 z - 654 z + 4671485266780 z - 7959556830 z 40 + 369812900 z ) And in Maple-input format, it is: -(-1-2338215836391*z^28+4813218965242*z^26+422*z^2-6878184231527*z^24+ 6878184231527*z^22-60681*z^4+3832975*z^6-124011178*z^8+2264167859*z^10-\ 24892879285*z^12+172319647094*z^14+2338215836391*z^18-776326075121*z^16-\ 4813218965242*z^20-2264167859*z^36+24892879285*z^34+776326075121*z^30+60681*z^ 42-422*z^44+z^46-172319647094*z^32+124011178*z^38-3832975*z^40)/(1+ 42535942744576*z^28-73193382932514*z^26-654*z^2+87629902833070*z^24-\ 73193382932514*z^22+125696*z^4-9655612*z^6+369812900*z^8-7959556830*z^10+ 103688970600*z^12-858710233074*z^14-17066334807812*z^18+4671485266780*z^16+z^48 +42535942744576*z^20+103688970600*z^36-858710233074*z^34-17066334807812*z^30-\ 9655612*z^42+125696*z^44-654*z^46+4671485266780*z^32-7959556830*z^38+369812900* z^40) The first , 40, terms are: [0, 232, 0, 86713, 0, 33371467, 0, 12919762432, 0, 5012030594023, 0, 1945827722156719, 0, 755652974532029872, 0, 293487438147457432459, 0, 113992320919314786362185, 0, 44276062752488277058035448, 0, 17197496055524623246293326329, 0, 6679785763752249839540423353801, 0, 2594539402118920034782246038220888, 0, 1007762444569989016940720868939232297, 0, 391431824363831536320940109573113764427, 0, 152038689778159489906079669624441859408592, 0, 59054379968664983056828014973807518223905055, 0, 22937712943996192491126190397181293546234240503, 0, 8909392953940442176717509269103258146506896138208, 0, 3460557863812688334845312322021374310398704901975179] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 3497101014779 z + 7392531143433 z + 411 z 24 22 4 6 - 10704240181915 z + 10704240181915 z - 58665 z + 3876539 z 8 10 12 14 - 135053301 z + 2671197399 z - 31692421261 z + 234535380711 z 18 16 20 + 3497101014779 z - 1115294077217 z - 7392531143433 z 36 34 30 42 - 2671197399 z + 31692421261 z + 1115294077217 z + 58665 z 44 46 32 38 40 / - 411 z + z - 234535380711 z + 135053301 z - 3876539 z ) / (1 / 28 26 2 24 + 63254026508324 z - 110588297548729 z - 651 z + 133110495525970 z 22 4 6 8 - 110588297548729 z + 118048 z - 9387967 z + 388762450 z 10 12 14 - 9140924491 z + 128988523794 z - 1139945639161 z 18 16 48 20 - 24720598883813 z + 6519341503010 z + z + 63254026508324 z 36 34 30 42 + 128988523794 z - 1139945639161 z - 24720598883813 z - 9387967 z 44 46 32 38 + 118048 z - 651 z + 6519341503010 z - 9140924491 z 40 + 388762450 z ) And in Maple-input format, it is: -(-1-3497101014779*z^28+7392531143433*z^26+411*z^2-10704240181915*z^24+ 10704240181915*z^22-58665*z^4+3876539*z^6-135053301*z^8+2671197399*z^10-\ 31692421261*z^12+234535380711*z^14+3497101014779*z^18-1115294077217*z^16-\ 7392531143433*z^20-2671197399*z^36+31692421261*z^34+1115294077217*z^30+58665*z^ 42-411*z^44+z^46-234535380711*z^32+135053301*z^38-3876539*z^40)/(1+ 63254026508324*z^28-110588297548729*z^26-651*z^2+133110495525970*z^24-\ 110588297548729*z^22+118048*z^4-9387967*z^6+388762450*z^8-9140924491*z^10+ 128988523794*z^12-1139945639161*z^14-24720598883813*z^18+6519341503010*z^16+z^ 48+63254026508324*z^20+128988523794*z^36-1139945639161*z^34-24720598883813*z^30 -9387967*z^42+118048*z^44-651*z^46+6519341503010*z^32-9140924491*z^38+388762450 *z^40) The first , 40, terms are: [0, 240, 0, 96857, 0, 40233815, 0, 16757841360, 0, 6982290391051, 0, 2909397276368683, 0, 1212308186707865232, 0, 505154341184185647119, 0, 210491893918214187587105, 0, 87709516368920559936161712, 0, 36547533235631051341525390609, 0, 15228931252546880168815664650321, 0, 6345718218173056154250529445599536, 0, 2644186847570759998840468689652718705, 0, 1101801852075833059227870466743641426879, 0, 459107994724648718562586833129703122271248, 0, 191304952359741405778945555001415154254328075, 0, 79714544764920507010894302998348620930287233131, 0, 33216122053815392398167912212722908306827102020176, 0, 13840771060635491582868739112752031479694563744035047] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 242198246 z + 1529543861 z + 319 z - 5876177259 z 22 4 6 8 10 + 14128635628 z - 29326 z + 1145512 z - 22432416 z + 242198246 z 12 14 18 16 - 1529543861 z + 5876177259 z + 21765547032 z - 14128635628 z 20 36 34 30 32 38 - 21765547032 z - 319 z + 29326 z + 22432416 z - 1145512 z + z / 40 38 36 34 32 ) / (z - 523 z + 64901 z - 3192690 z + 77195722 z / 30 28 26 24 - 1017752834 z + 7858307651 z - 37201423429 z + 110976274647 z 22 20 18 - 212207649836 z + 263042853132 z - 212207649836 z 16 14 12 10 + 110976274647 z - 37201423429 z + 7858307651 z - 1017752834 z 8 6 4 2 + 77195722 z - 3192690 z + 64901 z - 523 z + 1) And in Maple-input format, it is: -(-1-242198246*z^28+1529543861*z^26+319*z^2-5876177259*z^24+14128635628*z^22-\ 29326*z^4+1145512*z^6-22432416*z^8+242198246*z^10-1529543861*z^12+5876177259*z^ 14+21765547032*z^18-14128635628*z^16-21765547032*z^20-319*z^36+29326*z^34+ 22432416*z^30-1145512*z^32+z^38)/(z^40-523*z^38+64901*z^36-3192690*z^34+ 77195722*z^32-1017752834*z^30+7858307651*z^28-37201423429*z^26+110976274647*z^ 24-212207649836*z^22+263042853132*z^20-212207649836*z^18+110976274647*z^16-\ 37201423429*z^14+7858307651*z^12-1017752834*z^10+77195722*z^8-3192690*z^6+64901 *z^4-523*z^2+1) The first , 40, terms are: [0, 204, 0, 71117, 0, 26001565, 0, 9579799532, 0, 3534789747201, 0, 1304682769572513, 0, 481586627446629292, 0, 177766469862509321629, 0, 65618535178142354862605, 0, 24221636339543032552339212, 0, 8940884477870984706391575713, 0, 3300331005941207084044826383841, 0, 1218244664552601170612359004174988, 0, 449688246981018662201373545944101901, 0, 165992534512103592907326885941988830749, 0, 61272496446665471688611606402976124714028, 0, 22617395606885430633864550793222354405594081, 0, 8348714573509817797656459058653022963893630401, 0, 3081744522731517135050657957703142241246499834732, 0, 1137558269571325899878391286860727618847548536218909] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 122084914393282404 z - 55216207740070786 z - 435 z 24 22 4 6 + 19721581023133958 z - 5523015543553456 z + 69562 z - 5613495 z 8 10 12 14 + 265418277 z - 8038968988 z + 165338355101 z - 2404659721391 z 18 16 50 - 200985786966923 z + 25473005465538 z - 200985786966923 z 48 20 36 + 1201885508840761 z + 1201885508840761 z + 299991632386313214 z 34 66 64 30 - 335476789544432936 z - 435 z + 69562 z - 214344635377867994 z 42 44 46 - 55216207740070786 z + 19721581023133958 z - 5523015543553456 z 58 56 54 - 8038968988 z + 165338355101 z - 2404659721391 z 52 60 68 32 + 25473005465538 z + 265418277 z + z + 299991632386313214 z 38 40 62 / - 214344635377867994 z + 122084914393282404 z - 5613495 z ) / (-1 / 28 26 2 - 991920075684691194 z + 400483435934864406 z + 671 z 24 22 4 6 - 127687118866815210 z + 31934084317603173 z - 133825 z + 12664529 z 8 10 12 14 - 686645567 z + 23577365033 z - 545828988525 z + 8896135976667 z 18 16 50 + 929050649407125 z - 105370797534949 z + 6210916175013691 z 48 20 36 - 31934084317603173 z - 6210916175013691 z - 3819687436414300090 z 34 66 64 + 3819687436414300090 z + 133825 z - 12664529 z 30 42 + 1950009391866547354 z + 991920075684691194 z 44 46 58 - 400483435934864406 z + 127687118866815210 z + 545828988525 z 56 54 52 - 8896135976667 z + 105370797534949 z - 929050649407125 z 60 70 68 32 - 23577365033 z + z - 671 z - 3054010826801711062 z 38 40 62 + 3054010826801711062 z - 1950009391866547354 z + 686645567 z ) And in Maple-input format, it is: -(1+122084914393282404*z^28-55216207740070786*z^26-435*z^2+19721581023133958*z^ 24-5523015543553456*z^22+69562*z^4-5613495*z^6+265418277*z^8-8038968988*z^10+ 165338355101*z^12-2404659721391*z^14-200985786966923*z^18+25473005465538*z^16-\ 200985786966923*z^50+1201885508840761*z^48+1201885508840761*z^20+ 299991632386313214*z^36-335476789544432936*z^34-435*z^66+69562*z^64-\ 214344635377867994*z^30-55216207740070786*z^42+19721581023133958*z^44-\ 5523015543553456*z^46-8038968988*z^58+165338355101*z^56-2404659721391*z^54+ 25473005465538*z^52+265418277*z^60+z^68+299991632386313214*z^32-\ 214344635377867994*z^38+122084914393282404*z^40-5613495*z^62)/(-1-\ 991920075684691194*z^28+400483435934864406*z^26+671*z^2-127687118866815210*z^24 +31934084317603173*z^22-133825*z^4+12664529*z^6-686645567*z^8+23577365033*z^10-\ 545828988525*z^12+8896135976667*z^14+929050649407125*z^18-105370797534949*z^16+ 6210916175013691*z^50-31934084317603173*z^48-6210916175013691*z^20-\ 3819687436414300090*z^36+3819687436414300090*z^34+133825*z^66-12664529*z^64+ 1950009391866547354*z^30+991920075684691194*z^42-400483435934864406*z^44+ 127687118866815210*z^46+545828988525*z^58-8896135976667*z^56+105370797534949*z^ 54-929050649407125*z^52-23577365033*z^60+z^70-671*z^68-3054010826801711062*z^32 +3054010826801711062*z^38-1950009391866547354*z^40+686645567*z^62) The first , 40, terms are: [0, 236, 0, 94093, 0, 38604737, 0, 15879384356, 0, 6533921543281, 0, 2688688781552357, 0, 1106401414887522612, 0, 455287962071687029421, 0, 187352677367494910851209, 0, 77096330518537722276876508, 0, 31725430777790589627077620709, 0, 13055134529611104109194980332285, 0, 5372237153280995880171869948203964, 0, 2210695874424110870265526624437487153, 0, 909709700129198738338286790677517796789, 0, 374348976765563615362664342562724148579668, 0, 154046017522259585923608837741183995582150813, 0, 63390517905336709692922860174361722146513149689, 0, 26085437487713677751655016623066879273050096609156, 0, 10734256027717126438287135237312868665821032193196409] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 205156861375364 z - 179338540127753 z - 377 z 24 22 4 6 + 119650329884237 z - 60706595910327 z + 47727 z - 2837608 z 8 10 12 14 + 92685298 z - 1852667631 z + 24345446218 z - 220575005074 z 18 16 50 48 - 6689662331264 z + 1423233847413 z - 2837608 z + 92685298 z 20 36 34 + 23280797174194 z + 23280797174194 z - 60706595910327 z 30 42 44 - 179338540127753 z - 220575005074 z + 24345446218 z 46 56 54 52 32 - 1852667631 z + z - 377 z + 47727 z + 119650329884237 z 38 40 / 28 - 6689662331264 z + 1423233847413 z ) / (-1 - 2337902235284885 z / 26 2 24 + 1783523821930895 z + 587 z - 1036246659854541 z 22 4 6 8 + 456999602450277 z - 95313 z + 6845993 z - 261656195 z 10 12 14 18 + 6037311205 z - 91122066695 z + 947423529085 z + 37971279004537 z 16 50 48 20 - 7022238352615 z + 261656195 z - 6037311205 z - 152177808382495 z 36 34 30 - 456999602450277 z + 1036246659854541 z + 2337902235284885 z 42 44 46 58 56 + 7022238352615 z - 947423529085 z + 91122066695 z + z - 587 z 54 52 32 38 + 95313 z - 6845993 z - 1783523821930895 z + 152177808382495 z 40 - 37971279004537 z ) And in Maple-input format, it is: -(1+205156861375364*z^28-179338540127753*z^26-377*z^2+119650329884237*z^24-\ 60706595910327*z^22+47727*z^4-2837608*z^6+92685298*z^8-1852667631*z^10+ 24345446218*z^12-220575005074*z^14-6689662331264*z^18+1423233847413*z^16-\ 2837608*z^50+92685298*z^48+23280797174194*z^20+23280797174194*z^36-\ 60706595910327*z^34-179338540127753*z^30-220575005074*z^42+24345446218*z^44-\ 1852667631*z^46+z^56-377*z^54+47727*z^52+119650329884237*z^32-6689662331264*z^ 38+1423233847413*z^40)/(-1-2337902235284885*z^28+1783523821930895*z^26+587*z^2-\ 1036246659854541*z^24+456999602450277*z^22-95313*z^4+6845993*z^6-261656195*z^8+ 6037311205*z^10-91122066695*z^12+947423529085*z^14+37971279004537*z^18-\ 7022238352615*z^16+261656195*z^50-6037311205*z^48-152177808382495*z^20-\ 456999602450277*z^36+1036246659854541*z^34+2337902235284885*z^30+7022238352615* z^42-947423529085*z^44+91122066695*z^46+z^58-587*z^56+95313*z^54-6845993*z^52-\ 1783523821930895*z^32+152177808382495*z^38-37971279004537*z^40) The first , 40, terms are: [0, 210, 0, 75684, 0, 28419163, 0, 10737067222, 0, 4061311753131, 0, 1536563173191463, 0, 581375131599647720, 0, 219971920409237592197, 0, 83229846374554604683981, 0, 31491342400485104602995090, 0, 11915254074871544512024483801, 0, 4508327454938436593764193891972, 0, 1705797999628524932424614058569230, 0, 645416032590299092867085049325558255, 0, 244203507886912265385934888100255145855, 0, 92398314044333185992992999061418722953182, 0, 34960384116704644464992478063284069695122292, 0, 13227822068328848085692343217836328241823839689, 0, 5004958643686836708700031519204616174500857726402, 0, 1893706378542553508983077854214549434444744819566781] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 117217315401556508 z - 53044525905919914 z - 455 z 24 22 4 6 + 18976458543272094 z - 5331284203466176 z + 75018 z - 6053371 z 8 10 12 14 + 282006177 z - 8373928308 z + 168832295537 z - 2412558766787 z 18 16 50 - 196587761938255 z + 25191493646106 z - 196587761938255 z 48 20 36 + 1166314270022945 z + 1166314270022945 z + 288082988300269422 z 34 66 64 30 - 322188268510960152 z - 455 z + 75018 z - 205799484060560322 z 42 44 46 - 53044525905919914 z + 18976458543272094 z - 5331284203466176 z 58 56 54 - 8373928308 z + 168832295537 z - 2412558766787 z 52 60 68 32 + 25191493646106 z + 282006177 z + z + 288082988300269422 z 38 40 62 / - 205799484060560322 z + 117217315401556508 z - 6053371 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 700016628178237298 z - 300609002042437000 z - 688 z 24 22 4 6 + 100856580332849154 z - 26318046797387560 z + 147961 z - 14011346 z 8 10 12 14 + 734924453 z - 24167712318 z + 536081160033 z - 8405850776002 z 18 16 50 - 820770174947192 z + 96170733305029 z - 820770174947192 z 48 20 36 + 5308490611430797 z + 5308490611430797 z + 1831696077365283082 z 34 66 64 30 - 2065105457900656852 z - 688 z + 147961 z - 1277665521829248412 z 42 44 46 - 300609002042437000 z + 100856580332849154 z - 26318046797387560 z 58 56 54 - 24167712318 z + 536081160033 z - 8405850776002 z 52 60 68 32 + 96170733305029 z + 734924453 z + z + 1831696077365283082 z 38 40 62 - 1277665521829248412 z + 700016628178237298 z - 14011346 z )) And in Maple-input format, it is: -(1+117217315401556508*z^28-53044525905919914*z^26-455*z^2+18976458543272094*z^ 24-5331284203466176*z^22+75018*z^4-6053371*z^6+282006177*z^8-8373928308*z^10+ 168832295537*z^12-2412558766787*z^14-196587761938255*z^18+25191493646106*z^16-\ 196587761938255*z^50+1166314270022945*z^48+1166314270022945*z^20+ 288082988300269422*z^36-322188268510960152*z^34-455*z^66+75018*z^64-\ 205799484060560322*z^30-53044525905919914*z^42+18976458543272094*z^44-\ 5331284203466176*z^46-8373928308*z^58+168832295537*z^56-2412558766787*z^54+ 25191493646106*z^52+282006177*z^60+z^68+288082988300269422*z^32-\ 205799484060560322*z^38+117217315401556508*z^40-6053371*z^62)/(-1+z^2)/(1+ 700016628178237298*z^28-300609002042437000*z^26-688*z^2+100856580332849154*z^24 -26318046797387560*z^22+147961*z^4-14011346*z^6+734924453*z^8-24167712318*z^10+ 536081160033*z^12-8405850776002*z^14-820770174947192*z^18+96170733305029*z^16-\ 820770174947192*z^50+5308490611430797*z^48+5308490611430797*z^20+ 1831696077365283082*z^36-2065105457900656852*z^34-688*z^66+147961*z^64-\ 1277665521829248412*z^30-300609002042437000*z^42+100856580332849154*z^44-\ 26318046797387560*z^46-24167712318*z^58+536081160033*z^56-8405850776002*z^54+ 96170733305029*z^52+734924453*z^60+z^68+1831696077365283082*z^32-\ 1277665521829248412*z^38+700016628178237298*z^40-14011346*z^62) The first , 40, terms are: [0, 234, 0, 87595, 0, 33675025, 0, 13027531286, 0, 5051772492991, 0, 1960789535860055, 0, 761340221677812422, 0, 295658357032526808481, 0, 114822426118267286744875, 0, 44593672898159117598508090, 0, 17319035932227150961893569673, 0, 6726292418156921956716092714457, 0, 2612332132559192314961339591509594, 0, 1014568292595595109846772695655206731, 0, 394034524541849384505303647278223354337, 0, 153033779355372855921065705487599990726054, 0, 59434737270506349519852232993289469931743687, 0, 23083060864877509156737278313794568412225202319, 0, 8964920621785774534559871431258404885376997298614, 0, 3481765368961446434826009981911355226168914946959249] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 59906616674758514813 z - 13042473519444681497 z - 473 z 24 22 4 6 + 2330818568783016473 z - 339552609459507948 z + 87389 z - 8763386 z 8 10 12 14 + 550428722 z - 23483526410 z + 717004454647 z - 16246099979787 z 18 16 50 - 3768018031759436 z + 280551889383963 z - 7526049891822138160387 z 48 20 + 11594512890011513912523 z + 39987057356531028 z 36 34 + 4101528619333035021871 z - 1872900148325244940358 z 66 80 88 84 86 - 339552609459507948 z + 550428722 z + z + 87389 z - 473 z 82 64 30 - 8763386 z + 2330818568783016473 z - 227177707230582597430 z 42 44 - 15018235181886555549768 z + 16369434934349614292056 z 46 58 - 15018235181886555549768 z - 227177707230582597430 z 56 54 + 714712059600453366862 z - 1872900148325244940358 z 52 60 + 4101528619333035021871 z + 59906616674758514813 z 70 68 78 - 3768018031759436 z + 39987057356531028 z - 23483526410 z 32 38 + 714712059600453366862 z - 7526049891822138160387 z 40 62 + 11594512890011513912523 z - 13042473519444681497 z 76 74 72 / + 717004454647 z - 16246099979787 z + 280551889383963 z ) / ( / 2 28 26 (-1 + z ) (1 + 284838129933238722712 z - 58627383257216829786 z 2 24 22 4 - 694 z + 9867650780582097603 z - 1349278725939937076 z + 156530 z 6 8 10 12 - 18018152 z + 1263683111 z - 59324689468 z + 1975589037852 z 14 18 16 - 48533768075318 z - 13068159780927972 z + 904690406504965 z 50 48 - 43816180719387454426618 z + 68888860788971219219191 z 20 36 + 148674419181763030 z + 23222786276093924241050 z 34 66 80 - 10241273476361964191648 z - 1349278725939937076 z + 1263683111 z 88 84 86 82 64 + z + 156530 z - 694 z - 18018152 z + 9867650780582097603 z 30 42 - 1137590679134744376060 z - 90342153499795211841320 z 44 46 + 98880380114532306972180 z - 90342153499795211841320 z 58 56 - 1137590679134744376060 z + 3750495672515469575353 z 54 52 - 10241273476361964191648 z + 23222786276093924241050 z 60 70 + 284838129933238722712 z - 13068159780927972 z 68 78 32 + 148674419181763030 z - 59324689468 z + 3750495672515469575353 z 38 40 - 43816180719387454426618 z + 68888860788971219219191 z 62 76 74 - 58627383257216829786 z + 1975589037852 z - 48533768075318 z 72 + 904690406504965 z )) And in Maple-input format, it is: -(1+59906616674758514813*z^28-13042473519444681497*z^26-473*z^2+ 2330818568783016473*z^24-339552609459507948*z^22+87389*z^4-8763386*z^6+ 550428722*z^8-23483526410*z^10+717004454647*z^12-16246099979787*z^14-\ 3768018031759436*z^18+280551889383963*z^16-7526049891822138160387*z^50+ 11594512890011513912523*z^48+39987057356531028*z^20+4101528619333035021871*z^36 -1872900148325244940358*z^34-339552609459507948*z^66+550428722*z^80+z^88+87389* z^84-473*z^86-8763386*z^82+2330818568783016473*z^64-227177707230582597430*z^30-\ 15018235181886555549768*z^42+16369434934349614292056*z^44-\ 15018235181886555549768*z^46-227177707230582597430*z^58+714712059600453366862*z ^56-1872900148325244940358*z^54+4101528619333035021871*z^52+ 59906616674758514813*z^60-3768018031759436*z^70+39987057356531028*z^68-\ 23483526410*z^78+714712059600453366862*z^32-7526049891822138160387*z^38+ 11594512890011513912523*z^40-13042473519444681497*z^62+717004454647*z^76-\ 16246099979787*z^74+280551889383963*z^72)/(-1+z^2)/(1+284838129933238722712*z^ 28-58627383257216829786*z^26-694*z^2+9867650780582097603*z^24-\ 1349278725939937076*z^22+156530*z^4-18018152*z^6+1263683111*z^8-59324689468*z^ 10+1975589037852*z^12-48533768075318*z^14-13068159780927972*z^18+ 904690406504965*z^16-43816180719387454426618*z^50+68888860788971219219191*z^48+ 148674419181763030*z^20+23222786276093924241050*z^36-10241273476361964191648*z^ 34-1349278725939937076*z^66+1263683111*z^80+z^88+156530*z^84-694*z^86-18018152* z^82+9867650780582097603*z^64-1137590679134744376060*z^30-\ 90342153499795211841320*z^42+98880380114532306972180*z^44-\ 90342153499795211841320*z^46-1137590679134744376060*z^58+3750495672515469575353 *z^56-10241273476361964191648*z^54+23222786276093924241050*z^52+ 284838129933238722712*z^60-13068159780927972*z^70+148674419181763030*z^68-\ 59324689468*z^78+3750495672515469575353*z^32-43816180719387454426618*z^38+ 68888860788971219219191*z^40-58627383257216829786*z^62+1975589037852*z^76-\ 48533768075318*z^74+904690406504965*z^72) The first , 40, terms are: [0, 222, 0, 84455, 0, 33203793, 0, 13101790078, 0, 5172820887671, 0, 2042549681949899, 0, 806543466001736926, 0, 318482117009634029269, 0, 125760074938021725686603, 0, 49659303588518506443532558, 0, 19609137166835015620623033349, 0, 7743126403392173799457905064989, 0, 3057554554249662962014411707224398, 0, 1207346925566142636434356902277850755, 0, 476749170943585583815628238854186570701, 0, 188255560343609448351809660948380596666046, 0, 74337110918005018658133454840080026885823875, 0, 29353746840502050314253296589950285064098099775, 0, 11591013464687163394740188442727099768595155200222, 0, 4576982756872139742402210027747539319595696698013081] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 55069351142512328681 z - 11994073697651378917 z - 477 z 24 22 4 6 + 2148863519174171405 z - 314577707096463084 z + 88429 z - 8849768 z 8 10 12 14 + 551941324 z - 23294684524 z + 702074083139 z - 15692523635839 z 18 16 50 - 3550672655359692 z + 267470308541903 z - 7019460419331277816575 z 48 20 + 10844078269392207626403 z + 37318326463453728 z 36 34 + 3812359173784170095719 z - 1734485149722749562960 z 66 80 88 84 86 - 314577707096463084 z + 551941324 z + z + 88429 z - 477 z 82 64 30 - 8849768 z + 2148863519174171405 z - 209117300271975696604 z 42 44 - 14071453695832074305224 z + 15347001617757696065280 z 46 58 - 14071453695832074305224 z - 209117300271975696604 z 56 54 + 659634729746231549652 z - 1734485149722749562960 z 52 60 + 3812359173784170095719 z + 55069351142512328681 z 70 68 78 - 3550672655359692 z + 37318326463453728 z - 23294684524 z 32 38 + 659634729746231549652 z - 7019460419331277816575 z 40 62 + 10844078269392207626403 z - 11994073697651378917 z 76 74 72 / + 702074083139 z - 15692523635839 z + 267470308541903 z ) / (-1 / 28 26 2 - 318085765767240877950 z + 63425248687657138129 z + 711 z 24 22 4 - 10411520620946077187 z + 1397713800923055226 z - 162764 z 6 8 10 12 + 18832762 z - 1317648227 z + 61444066651 z - 2029897200840 z 14 18 16 + 49530640533286 z + 13295742060364261 z - 919832409933887 z 50 48 + 106949473950948618031369 z - 151543211968424702333943 z 20 36 - 152175427259833706 z - 31476569088860834094426 z 34 66 80 + 13095333523393205646797 z + 10411520620946077187 z - 61444066651 z 90 88 84 86 82 + z - 711 z - 18832762 z + 162764 z + 1317648227 z 64 30 - 63425248687657138129 z + 1320215108758611041348 z 42 44 + 151543211968424702333943 z - 180363908183962052694140 z 46 58 + 180363908183962052694140 z + 4553711680806206126805 z 56 54 - 13095333523393205646797 z + 31476569088860834094426 z 52 60 - 63358494814653263436144 z - 1320215108758611041348 z 70 68 78 + 152175427259833706 z - 1397713800923055226 z + 2029897200840 z 32 38 - 4553711680806206126805 z + 63358494814653263436144 z 40 62 - 106949473950948618031369 z + 318085765767240877950 z 76 74 72 - 49530640533286 z + 919832409933887 z - 13295742060364261 z ) And in Maple-input format, it is: -(1+55069351142512328681*z^28-11994073697651378917*z^26-477*z^2+ 2148863519174171405*z^24-314577707096463084*z^22+88429*z^4-8849768*z^6+ 551941324*z^8-23294684524*z^10+702074083139*z^12-15692523635839*z^14-\ 3550672655359692*z^18+267470308541903*z^16-7019460419331277816575*z^50+ 10844078269392207626403*z^48+37318326463453728*z^20+3812359173784170095719*z^36 -1734485149722749562960*z^34-314577707096463084*z^66+551941324*z^80+z^88+88429* z^84-477*z^86-8849768*z^82+2148863519174171405*z^64-209117300271975696604*z^30-\ 14071453695832074305224*z^42+15347001617757696065280*z^44-\ 14071453695832074305224*z^46-209117300271975696604*z^58+659634729746231549652*z ^56-1734485149722749562960*z^54+3812359173784170095719*z^52+ 55069351142512328681*z^60-3550672655359692*z^70+37318326463453728*z^68-\ 23294684524*z^78+659634729746231549652*z^32-7019460419331277816575*z^38+ 10844078269392207626403*z^40-11994073697651378917*z^62+702074083139*z^76-\ 15692523635839*z^74+267470308541903*z^72)/(-1-318085765767240877950*z^28+ 63425248687657138129*z^26+711*z^2-10411520620946077187*z^24+1397713800923055226 *z^22-162764*z^4+18832762*z^6-1317648227*z^8+61444066651*z^10-2029897200840*z^ 12+49530640533286*z^14+13295742060364261*z^18-919832409933887*z^16+ 106949473950948618031369*z^50-151543211968424702333943*z^48-152175427259833706* z^20-31476569088860834094426*z^36+13095333523393205646797*z^34+ 10411520620946077187*z^66-61444066651*z^80+z^90-711*z^88-18832762*z^84+162764*z ^86+1317648227*z^82-63425248687657138129*z^64+1320215108758611041348*z^30+ 151543211968424702333943*z^42-180363908183962052694140*z^44+ 180363908183962052694140*z^46+4553711680806206126805*z^58-\ 13095333523393205646797*z^56+31476569088860834094426*z^54-\ 63358494814653263436144*z^52-1320215108758611041348*z^60+152175427259833706*z^ 70-1397713800923055226*z^68+2029897200840*z^78-4553711680806206126805*z^32+ 63358494814653263436144*z^38-106949473950948618031369*z^40+ 318085765767240877950*z^62-49530640533286*z^76+919832409933887*z^74-\ 13295742060364261*z^72) The first , 40, terms are: [0, 234, 0, 92039, 0, 37335947, 0, 15206381926, 0, 6197957750605, 0, 2526610480086085, 0, 1030012275969670134, 0, 419903681568285989819, 0, 171181832371370559699879, 0, 69785600166556989773442426, 0, 28449458236882327027168276089, 0, 11597975614769735705046269650937, 0, 4728140613617827666003074719309946, 0, 1927518596857503131556030944384713847, 0, 785790492636656721678814110645883549323, 0, 320342796876197182792257324270320860404662, 0, 130593979531699831790255986576220635088907045, 0, 53239178955410645240664054707902563971172688333, 0, 21703988085911956711493860442789141399162001683046, 0, 8848053408712360467894099951327306148219348311139035] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 173227326633005852 z - 76906224938252742 z - 439 z 24 22 4 6 + 26852848166821978 z - 7327455774260176 z + 71206 z - 5888469 z 8 10 12 14 + 286343751 z - 8928298676 z + 189102376067 z - 2833478779289 z 18 16 50 - 251610593883571 z + 30936927454782 z - 251610593883571 z 48 20 36 + 1550041231522517 z + 1550041231522517 z + 435223292720389254 z 34 66 64 30 - 488095230700405752 z - 439 z + 71206 z - 308348886051503330 z 42 44 46 - 76906224938252742 z + 26852848166821978 z - 7327455774260176 z 58 56 54 - 8928298676 z + 189102376067 z - 2833478779289 z 52 60 68 32 + 30936927454782 z + 286343751 z + z + 435223292720389254 z 38 40 62 / - 308348886051503330 z + 173227326633005852 z - 5888469 z ) / (-1 / 28 26 2 - 1430879029460439994 z + 563134856014631578 z + 673 z 24 22 4 6 - 174225008577051434 z + 42139776139945893 z - 136609 z + 13212903 z 8 10 12 14 - 731119803 z + 25649181807 z - 608723445451 z + 10212950570871 z 18 16 50 + 1141317081809653 z - 124981360638835 z + 7910118500325621 z 48 20 36 - 42139776139945893 z - 7910118500325621 z - 5740005617715789046 z 34 66 64 + 5740005617715789046 z + 136609 z - 13212903 z 30 42 + 2869942485662443606 z + 1430879029460439994 z 44 46 58 - 563134856014631578 z + 174225008577051434 z + 608723445451 z 56 54 52 - 10212950570871 z + 124981360638835 z - 1141317081809653 z 60 70 68 32 - 25649181807 z + z - 673 z - 4557232505448017310 z 38 40 62 + 4557232505448017310 z - 2869942485662443606 z + 731119803 z ) And in Maple-input format, it is: -(1+173227326633005852*z^28-76906224938252742*z^26-439*z^2+26852848166821978*z^ 24-7327455774260176*z^22+71206*z^4-5888469*z^6+286343751*z^8-8928298676*z^10+ 189102376067*z^12-2833478779289*z^14-251610593883571*z^18+30936927454782*z^16-\ 251610593883571*z^50+1550041231522517*z^48+1550041231522517*z^20+ 435223292720389254*z^36-488095230700405752*z^34-439*z^66+71206*z^64-\ 308348886051503330*z^30-76906224938252742*z^42+26852848166821978*z^44-\ 7327455774260176*z^46-8928298676*z^58+189102376067*z^56-2833478779289*z^54+ 30936927454782*z^52+286343751*z^60+z^68+435223292720389254*z^32-\ 308348886051503330*z^38+173227326633005852*z^40-5888469*z^62)/(-1-\ 1430879029460439994*z^28+563134856014631578*z^26+673*z^2-174225008577051434*z^ 24+42139776139945893*z^22-136609*z^4+13212903*z^6-731119803*z^8+25649181807*z^ 10-608723445451*z^12+10212950570871*z^14+1141317081809653*z^18-124981360638835* z^16+7910118500325621*z^50-42139776139945893*z^48-7910118500325621*z^20-\ 5740005617715789046*z^36+5740005617715789046*z^34+136609*z^66-13212903*z^64+ 2869942485662443606*z^30+1430879029460439994*z^42-563134856014631578*z^44+ 174225008577051434*z^46+608723445451*z^58-10212950570871*z^56+124981360638835*z ^54-1141317081809653*z^52-25649181807*z^60+z^70-673*z^68-4557232505448017310*z^ 32+4557232505448017310*z^38-2869942485662443606*z^40+731119803*z^62) The first , 40, terms are: [0, 234, 0, 92079, 0, 37327095, 0, 15189358074, 0, 6185490607513, 0, 2519292954364985, 0, 1026122097604120826, 0, 417948781992296958647, 0, 170234640180637516006287, 0, 69338271082790718488455146, 0, 28242174123964000926644369089, 0, 11503321423028130520474582222465, 0, 4685418479313772550220554001956458, 0, 1908418060238215551956102520907440655, 0, 777317866033845436526241475227970506359, 0, 316609383179700305118738574284073000119418, 0, 128958185446930518810476332745601433387154489, 0, 52525965676750149956015463798129100125943082009, 0, 21394353997119547264714755160348932610110579257210, 0, 8714135514823519899114071049459056872894349988310455] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 60469670645905921797 z - 13264418489290479941 z - 477 z 24 22 4 6 + 2388709567987600376 z - 350570551461266312 z + 88900 z - 9001944 z 8 10 12 14 + 570533137 z - 24513171848 z + 751717775389 z - 17058070985790 z 18 16 50 - 3937430745841576 z + 294211943617041 z - 7393219279756621109610 z 48 20 + 11360565389005598895756 z + 41560438846628477 z 36 34 + 4043658498710546409688 z - 1854973446329958043184 z 66 80 88 84 86 - 350570551461266312 z + 570533137 z + z + 88900 z - 477 z 82 64 30 - 9001944 z + 2388709567987600376 z - 227698962233274146510 z 42 44 - 14692456873657066588934 z + 16006089435068269153636 z 46 58 - 14692456873657066588934 z - 227698962233274146510 z 56 54 + 711798664671795907978 z - 1854973446329958043184 z 52 60 + 4043658498710546409688 z + 60469670645905921797 z 70 68 78 - 3937430745841576 z + 41560438846628477 z - 24513171848 z 32 38 + 711798664671795907978 z - 7393219279756621109610 z 40 62 + 11360565389005598895756 z - 13264418489290479941 z 76 74 72 / + 751717775389 z - 17058070985790 z + 294211943617041 z ) / (-1 / 28 26 2 - 351596584322290132220 z + 70499368157732250297 z + 723 z 24 22 4 - 11608788297405176104 z + 1558523103093364089 z - 164470 z 6 8 10 12 + 19064907 z - 1347855162 z + 63844053397 z - 2146494533328 z 14 18 16 + 53265215872699 z + 14667248818976159 z - 1003566312426028 z 50 48 + 112741104635667870844163 z - 159041332389123472445079 z 20 36 - 169096132248066638 z - 33649758684701308892699 z 34 66 80 + 14120686588501429839463 z + 11608788297405176104 z - 63844053397 z 90 88 84 86 82 + z - 723 z - 19064907 z + 164470 z + 1347855162 z 64 30 - 70499368157732250297 z + 1448562182040847246166 z 42 44 + 159041332389123472445079 z - 188853694995981957460223 z 46 58 + 188853694995981957460223 z + 4954146726151915051559 z 56 54 - 14120686588501429839463 z + 33649758684701308892699 z 52 60 - 67211488967163827204171 z - 1448562182040847246166 z 70 68 78 + 169096132248066638 z - 1558523103093364089 z + 2146494533328 z 32 38 - 4954146726151915051559 z + 67211488967163827204171 z 40 62 - 112741104635667870844163 z + 351596584322290132220 z 76 74 72 - 53265215872699 z + 1003566312426028 z - 14667248818976159 z ) And in Maple-input format, it is: -(1+60469670645905921797*z^28-13264418489290479941*z^26-477*z^2+ 2388709567987600376*z^24-350570551461266312*z^22+88900*z^4-9001944*z^6+ 570533137*z^8-24513171848*z^10+751717775389*z^12-17058070985790*z^14-\ 3937430745841576*z^18+294211943617041*z^16-7393219279756621109610*z^50+ 11360565389005598895756*z^48+41560438846628477*z^20+4043658498710546409688*z^36 -1854973446329958043184*z^34-350570551461266312*z^66+570533137*z^80+z^88+88900* z^84-477*z^86-9001944*z^82+2388709567987600376*z^64-227698962233274146510*z^30-\ 14692456873657066588934*z^42+16006089435068269153636*z^44-\ 14692456873657066588934*z^46-227698962233274146510*z^58+711798664671795907978*z ^56-1854973446329958043184*z^54+4043658498710546409688*z^52+ 60469670645905921797*z^60-3937430745841576*z^70+41560438846628477*z^68-\ 24513171848*z^78+711798664671795907978*z^32-7393219279756621109610*z^38+ 11360565389005598895756*z^40-13264418489290479941*z^62+751717775389*z^76-\ 17058070985790*z^74+294211943617041*z^72)/(-1-351596584322290132220*z^28+ 70499368157732250297*z^26+723*z^2-11608788297405176104*z^24+1558523103093364089 *z^22-164470*z^4+19064907*z^6-1347855162*z^8+63844053397*z^10-2146494533328*z^ 12+53265215872699*z^14+14667248818976159*z^18-1003566312426028*z^16+ 112741104635667870844163*z^50-159041332389123472445079*z^48-169096132248066638* z^20-33649758684701308892699*z^36+14120686588501429839463*z^34+ 11608788297405176104*z^66-63844053397*z^80+z^90-723*z^88-19064907*z^84+164470*z ^86+1347855162*z^82-70499368157732250297*z^64+1448562182040847246166*z^30+ 159041332389123472445079*z^42-188853694995981957460223*z^44+ 188853694995981957460223*z^46+4954146726151915051559*z^58-\ 14120686588501429839463*z^56+33649758684701308892699*z^54-\ 67211488967163827204171*z^52-1448562182040847246166*z^60+169096132248066638*z^ 70-1558523103093364089*z^68+2146494533328*z^78-4954146726151915051559*z^32+ 67211488967163827204171*z^38-112741104635667870844163*z^40+ 351596584322290132220*z^62-53265215872699*z^76+1003566312426028*z^74-\ 14667248818976159*z^72) The first , 40, terms are: [0, 246, 0, 102288, 0, 43557567, 0, 18581458678, 0, 7928351298617, 0, 3382967895697767, 0, 1443492990624810200, 0, 615930592480458599495, 0, 262814257092270550795797, 0, 112141426227979133976986514, 0, 47850141996880702562664278387, 0, 20417397634300664986508134193608, 0, 8711993503228137259013239248120010, 0, 3717360662770940645198621963056528273, 0, 1586177755074150412484688463947511380241, 0, 676813497245781916436260844383381901820946, 0, 288792670675622770588145921400258798939544488, 0, 123226275739702049212340854949161832408234394283, 0, 52579987563925759055035483749329547800793929898586, 0, 22435597242770931913510750036215503853357337458457133] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 199472391885184 z - 174729271505555 z - 381 z 24 22 4 6 + 117277854893239 z - 60065103138053 z + 48695 z - 2886360 z 8 10 12 14 + 94858070 z - 1911672217 z + 25219604082 z - 227967942862 z 18 16 50 48 - 6782432480510 z + 1459471132391 z - 2886360 z + 94858070 z 20 36 34 + 23307853240692 z + 23307853240692 z - 60065103138053 z 30 42 44 - 174729271505555 z - 227967942862 z + 25219604082 z 46 56 54 52 32 - 1911672217 z + z - 381 z + 48695 z + 117277854893239 z 38 40 / 2 - 6782432480510 z + 1459471132391 z ) / ((-1 + z ) (1 / 28 26 2 + 1256838275295423 z - 1088261643167026 z - 626 z 24 22 4 6 + 705943975563545 z - 342264317558352 z + 100663 z - 6973234 z 8 10 12 14 + 261963761 z - 5982205636 z + 88901764189 z - 900337134566 z 18 16 50 48 - 32921050877570 z + 6416993170579 z - 6973234 z + 261963761 z 20 36 34 + 123484279588753 z + 123484279588753 z - 342264317558352 z 30 42 44 - 1088261643167026 z - 900337134566 z + 88901764189 z 46 56 54 52 32 - 5982205636 z + z - 626 z + 100663 z + 705943975563545 z 38 40 - 32921050877570 z + 6416993170579 z )) And in Maple-input format, it is: -(1+199472391885184*z^28-174729271505555*z^26-381*z^2+117277854893239*z^24-\ 60065103138053*z^22+48695*z^4-2886360*z^6+94858070*z^8-1911672217*z^10+ 25219604082*z^12-227967942862*z^14-6782432480510*z^18+1459471132391*z^16-\ 2886360*z^50+94858070*z^48+23307853240692*z^20+23307853240692*z^36-\ 60065103138053*z^34-174729271505555*z^30-227967942862*z^42+25219604082*z^44-\ 1911672217*z^46+z^56-381*z^54+48695*z^52+117277854893239*z^32-6782432480510*z^ 38+1459471132391*z^40)/(-1+z^2)/(1+1256838275295423*z^28-1088261643167026*z^26-\ 626*z^2+705943975563545*z^24-342264317558352*z^22+100663*z^4-6973234*z^6+ 261963761*z^8-5982205636*z^10+88901764189*z^12-900337134566*z^14-32921050877570 *z^18+6416993170579*z^16-6973234*z^50+261963761*z^48+123484279588753*z^20+ 123484279588753*z^36-342264317558352*z^34-1088261643167026*z^30-900337134566*z^ 42+88901764189*z^44-5982205636*z^46+z^56-626*z^54+100663*z^52+705943975563545*z ^32-32921050877570*z^38+6416993170579*z^40) The first , 40, terms are: [0, 246, 0, 101648, 0, 43003739, 0, 18233619818, 0, 7733895384981, 0, 3280620802576727, 0, 1391620292900778960, 0, 590319322977065405623, 0, 250411102960035339653485, 0, 106223409447002015927398418, 0, 45059556079663246046834421427, 0, 19114088141099836219397234379456, 0, 8108121737418749094537815417604430, 0, 3439433660121512653926296862190326385, 0, 1458994362260464380517208122786084038193, 0, 618899725797806736264732871896984458008862, 0, 262534853116758184600070537873467044819779936, 0, 111366262139240004805056185955348644374034478723, 0, 47241134636522669200231248833577033494799776960578, 0, 20039505312262902249130729587969505081953507293943261] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 158135849970245772 z - 72064489887498450 z - 443 z 24 22 4 6 + 25926339797285582 z - 7298062701465472 z + 72594 z - 6075751 z 8 10 12 14 + 300364185 z - 9515872484 z + 203629653273 z - 3055904745567 z 18 16 50 - 264512428948211 z + 33092759370146 z - 264512428948211 z 48 20 36 + 1589817477931313 z + 1589817477931313 z + 384424422315101966 z 34 66 64 30 - 429252204721383096 z - 443 z + 72594 z - 275853493445286778 z 42 44 46 - 72064489887498450 z + 25926339797285582 z - 7298062701465472 z 58 56 54 - 9515872484 z + 203629653273 z - 3055904745567 z 52 60 68 32 + 33092759370146 z + 300364185 z + z + 384424422315101966 z 38 40 62 / - 275853493445286778 z + 158135849970245772 z - 6075751 z ) / ((1 / 28 26 2 + 939542420837347506 z - 406904178561000328 z - 712 z 24 22 4 6 + 137417868553863554 z - 35927950787536776 z + 143505 z - 13755210 z 8 10 12 14 + 760410669 z - 26718845862 z + 632287973993 z - 10475103251498 z 18 16 50 - 1096945352458336 z + 124965010401837 z - 1096945352458336 z 48 20 36 + 7206953454849293 z + 7206953454849293 z + 2425185697152144506 z 34 66 64 30 - 2728715570127338692 z - 712 z + 143505 z - 1701181176631553468 z 42 44 46 - 406904178561000328 z + 137417868553863554 z - 35927950787536776 z 58 56 54 - 26718845862 z + 632287973993 z - 10475103251498 z 52 60 68 32 + 124965010401837 z + 760410669 z + z + 2425185697152144506 z 38 40 62 - 1701181176631553468 z + 939542420837347506 z - 13755210 z ) 2 (-1 + z )) And in Maple-input format, it is: -(1+158135849970245772*z^28-72064489887498450*z^26-443*z^2+25926339797285582*z^ 24-7298062701465472*z^22+72594*z^4-6075751*z^6+300364185*z^8-9515872484*z^10+ 203629653273*z^12-3055904745567*z^14-264512428948211*z^18+33092759370146*z^16-\ 264512428948211*z^50+1589817477931313*z^48+1589817477931313*z^20+ 384424422315101966*z^36-429252204721383096*z^34-443*z^66+72594*z^64-\ 275853493445286778*z^30-72064489887498450*z^42+25926339797285582*z^44-\ 7298062701465472*z^46-9515872484*z^58+203629653273*z^56-3055904745567*z^54+ 33092759370146*z^52+300364185*z^60+z^68+384424422315101966*z^32-\ 275853493445286778*z^38+158135849970245772*z^40-6075751*z^62)/(1+ 939542420837347506*z^28-406904178561000328*z^26-712*z^2+137417868553863554*z^24 -35927950787536776*z^22+143505*z^4-13755210*z^6+760410669*z^8-26718845862*z^10+ 632287973993*z^12-10475103251498*z^14-1096945352458336*z^18+124965010401837*z^ 16-1096945352458336*z^50+7206953454849293*z^48+7206953454849293*z^20+ 2425185697152144506*z^36-2728715570127338692*z^34-712*z^66+143505*z^64-\ 1701181176631553468*z^30-406904178561000328*z^42+137417868553863554*z^44-\ 35927950787536776*z^46-26718845862*z^58+632287973993*z^56-10475103251498*z^54+ 124965010401837*z^52+760410669*z^60+z^68+2425185697152144506*z^32-\ 1701181176631553468*z^38+939542420837347506*z^40-13755210*z^62)/(-1+z^2) The first , 40, terms are: [0, 270, 0, 120887, 0, 55076805, 0, 25114652842, 0, 11452848446583, 0, 5222795300786655, 0, 2381732428588968010, 0, 1086133021838775881173, 0, 495305420947294110374903, 0, 225872390184089740658345870, 0, 103003792322823504824253364569, 0, 46972457443847981926714031279177, 0, 21420684701710967310458948315314510, 0, 9768399569966703767148327074922675543, 0, 4454648928710215927964823770576298259797, 0, 2031437896856476684980261556304607226099530, 0, 926389485417838975734647575780952574313566831, 0, 422458141605386309200211313550544168116998157735, 0, 192652101754133802519350451064086013214463315933738, 0, 87854460963267976488652711927494934217228671808679301] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 178894311400641068 z - 77560141091060054 z - 419 z 24 22 4 6 + 26337039798806706 z - 6972126136302816 z + 64262 z - 5096229 z 8 10 12 14 + 241883627 z - 7477871172 z + 159065201119 z - 2417610703025 z 18 16 50 - 225301573651303 z + 26972560550158 z - 225301573651303 z 48 20 36 + 1429907186066861 z + 1429907186066861 z + 462900687198347670 z 34 66 64 30 - 521146214177376056 z - 419 z + 64262 z - 324255976488109322 z 42 44 46 - 77560141091060054 z + 26337039798806706 z - 6972126136302816 z 58 56 54 - 7477871172 z + 159065201119 z - 2417610703025 z 52 60 68 32 + 26972560550158 z + 241883627 z + z + 462900687198347670 z 38 40 62 / - 324255976488109322 z + 178894311400641068 z - 5096229 z ) / (-1 / 28 26 2 - 1482488720345237774 z + 565322195109335538 z + 671 z 24 22 4 6 - 168874825567903918 z + 39387292106237969 z - 126373 z + 11527429 z 8 10 12 14 - 614995323 z + 21198834229 z - 502012339093 z + 8508927639643 z 18 16 50 + 996787745722893 z - 106221351168429 z + 7135935592016495 z 48 20 36 - 39387292106237969 z - 7135935592016495 z - 6276867171889458550 z 34 66 64 + 6276867171889458550 z + 126373 z - 11527429 z 30 42 + 3051561668853055886 z + 1482488720345237774 z 44 46 58 - 565322195109335538 z + 168874825567903918 z + 502012339093 z 56 54 52 - 8508927639643 z + 106221351168429 z - 996787745722893 z 60 70 68 32 - 21198834229 z + z - 671 z - 4935873254156239914 z 38 40 62 + 4935873254156239914 z - 3051561668853055886 z + 614995323 z ) And in Maple-input format, it is: -(1+178894311400641068*z^28-77560141091060054*z^26-419*z^2+26337039798806706*z^ 24-6972126136302816*z^22+64262*z^4-5096229*z^6+241883627*z^8-7477871172*z^10+ 159065201119*z^12-2417610703025*z^14-225301573651303*z^18+26972560550158*z^16-\ 225301573651303*z^50+1429907186066861*z^48+1429907186066861*z^20+ 462900687198347670*z^36-521146214177376056*z^34-419*z^66+64262*z^64-\ 324255976488109322*z^30-77560141091060054*z^42+26337039798806706*z^44-\ 6972126136302816*z^46-7477871172*z^58+159065201119*z^56-2417610703025*z^54+ 26972560550158*z^52+241883627*z^60+z^68+462900687198347670*z^32-\ 324255976488109322*z^38+178894311400641068*z^40-5096229*z^62)/(-1-\ 1482488720345237774*z^28+565322195109335538*z^26+671*z^2-168874825567903918*z^ 24+39387292106237969*z^22-126373*z^4+11527429*z^6-614995323*z^8+21198834229*z^ 10-502012339093*z^12+8508927639643*z^14+996787745722893*z^18-106221351168429*z^ 16+7135935592016495*z^50-39387292106237969*z^48-7135935592016495*z^20-\ 6276867171889458550*z^36+6276867171889458550*z^34+126373*z^66-11527429*z^64+ 3051561668853055886*z^30+1482488720345237774*z^42-565322195109335538*z^44+ 168874825567903918*z^46+502012339093*z^58-8508927639643*z^56+106221351168429*z^ 54-996787745722893*z^52-21198834229*z^60+z^70-671*z^68-4935873254156239914*z^32 +4935873254156239914*z^38-3051561668853055886*z^40+614995323*z^62) The first , 40, terms are: [0, 252, 0, 106981, 0, 46369455, 0, 20126194804, 0, 8736787600279, 0, 3792703808540383, 0, 1646443936500896708, 0, 714735043680237659935, 0, 310272453206811564123557, 0, 134691864453542956625414252, 0, 58470863824447168497774890745, 0, 25382690561861839980562186478537, 0, 11018838067790941722848203999632972, 0, 4783369677396813945774320284248029029, 0, 2076500746256129271448316986674261095615, 0, 901426324956123173865531272957731689470052, 0, 391316699880288538884997433673678635271573967, 0, 169873849216299968432358468297225158653218251559, 0, 73743657391545467751241023315440938270907478887508, 0, 32012737867364648717588070241900394069047937910314127] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 26712670264030 z - 30830301559480 z - 388 z 24 22 4 6 + 26712670264030 z - 17338913346120 z + 49377 z - 2798328 z 8 10 12 14 + 84945504 z - 1545113456 z + 18039906360 z - 141478373664 z 18 16 50 48 - 2981707855696 z + 769989792520 z - 388 z + 49377 z 20 36 34 + 8378013293216 z + 769989792520 z - 2981707855696 z 30 42 44 46 52 - 17338913346120 z - 1545113456 z + 84945504 z - 2798328 z + z 32 38 40 / + 8378013293216 z - 141478373664 z + 18039906360 z ) / (-1 / 28 26 2 24 - 380262374045072 z + 380262374045072 z + 672 z - 283924816421724 z 22 4 6 8 + 157787627693680 z - 109136 z + 7224035 z - 252119312 z 10 12 14 18 + 5292784560 z - 71964187566 z + 662980038032 z + 19542241354514 z 16 50 48 20 - 4263346819440 z + 109136 z - 7224035 z - 64861998015056 z 36 34 30 - 19542241354514 z + 64861998015056 z + 283924816421724 z 42 44 46 54 52 + 71964187566 z - 5292784560 z + 252119312 z + z - 672 z 32 38 40 - 157787627693680 z + 4263346819440 z - 662980038032 z ) And in Maple-input format, it is: -(1+26712670264030*z^28-30830301559480*z^26-388*z^2+26712670264030*z^24-\ 17338913346120*z^22+49377*z^4-2798328*z^6+84945504*z^8-1545113456*z^10+ 18039906360*z^12-141478373664*z^14-2981707855696*z^18+769989792520*z^16-388*z^ 50+49377*z^48+8378013293216*z^20+769989792520*z^36-2981707855696*z^34-\ 17338913346120*z^30-1545113456*z^42+84945504*z^44-2798328*z^46+z^52+ 8378013293216*z^32-141478373664*z^38+18039906360*z^40)/(-1-380262374045072*z^28 +380262374045072*z^26+672*z^2-283924816421724*z^24+157787627693680*z^22-109136* z^4+7224035*z^6-252119312*z^8+5292784560*z^10-71964187566*z^12+662980038032*z^ 14+19542241354514*z^18-4263346819440*z^16+109136*z^50-7224035*z^48-\ 64861998015056*z^20-19542241354514*z^36+64861998015056*z^34+283924816421724*z^ 30+71964187566*z^42-5292784560*z^44+252119312*z^46+z^54-672*z^52-\ 157787627693680*z^32+4263346819440*z^38-662980038032*z^40) The first , 40, terms are: [0, 284, 0, 131089, 0, 61522891, 0, 28921305780, 0, 13599892562595, 0, 6395614850386011, 0, 3007706630504864244, 0, 1414457892027733120691, 0, 665188715096390870149113, 0, 312823800231202298276213276, 0, 147114241357073125449884730873, 0, 69184634121128821048429712038153, 0, 32536031601628711209074246040043420, 0, 15300989391415747043810360689322351017, 0, 7195723167465874488769710411800730570851, 0, 3383992405920655425861923580543276243556340, 0, 1591418171163587223720550475559440398242373867, 0, 748409420505930442880915716520857994981590589459, 0, 351960704515882526419158853074820093691644928230836, 0, 165519479217110060868775931916135829056459267614094107] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3193731980205587862249 z - 421051287583535420922 z - 509 z 24 22 4 + 47159284866270312739 z - 4456018814815129948 z + 104454 z 6 102 8 10 - 12017384 z - 46580015044 z + 893619031 z - 46580015044 z 12 14 18 + 1794029990017 z - 52972335388912 z - 23042701480033054 z 16 50 + 1232049260130263 z - 1007195347377751606477678002 z 48 20 + 633428425330751154266255026 z + 352189507550593511 z 36 34 + 2294451690086515052628420 z - 552803756000534264168810 z 66 80 - 348595026413072888400647998 z + 115210335872622579073787 z 100 90 88 + 1794029990017 z - 4456018814815129948 z + 47159284866270312739 z 84 94 + 3193731980205587862249 z - 23042701480033054 z 86 96 98 - 421051287583535420922 z + 1232049260130263 z - 52972335388912 z 92 82 + 352189507550593511 z - 20700638492135790754682 z 64 112 110 106 + 633428425330751154266255026 z + z - 509 z - 12017384 z 108 30 + 104454 z - 20700638492135790754682 z 42 44 - 70512211679964556187768138 z + 167746405103712901430670314 z 46 58 - 348595026413072888400647998 z - 1709898377021721970940436162 z 56 54 + 1826744920811852785061444878 z - 1709898377021721970940436162 z 52 60 + 1402216512691987193468886338 z + 1402216512691987193468886338 z 70 68 - 70512211679964556187768138 z + 167746405103712901430670314 z 78 32 - 552803756000534264168810 z + 115210335872622579073787 z 38 40 - 8260864954057066159581601 z + 25859050568442840187575825 z 62 76 - 1007195347377751606477678002 z + 2294451690086515052628420 z 74 72 - 8260864954057066159581601 z + 25859050568442840187575825 z 104 / 28 + 893619031 z ) / (-1 - 14648892097696334358824 z / 26 2 24 + 1805931938332256521014 z + 736 z - 189177618135235354940 z 22 4 6 102 + 16719002452144641495 z - 184339 z + 24285020 z + 4758513713698 z 8 10 12 - 2005253266 z + 114170654476 z - 4758513713698 z 14 18 16 + 151222419141696 z + 75576140344215625 z - 3773426290795647 z 50 48 + 9750204944648518374909956293 z - 5731717921203168904644472221 z 20 36 - 1235768210270372789 z - 13796806569898871015725372 z 34 66 + 3105467067692125310515763 z + 5731717921203168904644472221 z 80 100 - 3105467067692125310515763 z - 151222419141696 z 90 88 + 189177618135235354940 z - 1805931938332256521014 z 84 94 - 101562194282513789665134 z + 1235768210270372789 z 86 96 + 14648892097696334358824 z - 75576140344215625 z 98 92 + 3773426290795647 z - 16719002452144641495 z 82 64 + 604771886873031407067897 z - 9750204944648518374909956293 z 112 114 110 106 108 - 736 z + z + 184339 z + 2005253266 z - 24285020 z 30 42 + 101562194282513789665134 z + 520305346828205211272535060 z 44 46 - 1325126987143155743327239784 z + 2947591895256010454549512488 z 58 56 + 21607074044189940568447348257 z - 21607074044189940568447348257 z 54 52 + 18925178154303736740639583016 z - 14517083323655278356050893432 z 60 70 - 18925178154303736740639583016 z + 1325126987143155743327239784 z 68 78 - 2947591895256010454549512488 z + 13796806569898871015725372 z 32 38 - 604771886873031407067897 z + 53178125577816795778902089 z 40 62 - 178222522210516609995081073 z + 14517083323655278356050893432 z 76 74 - 53178125577816795778902089 z + 178222522210516609995081073 z 72 104 - 520305346828205211272535060 z - 114170654476 z ) And in Maple-input format, it is: -(1+3193731980205587862249*z^28-421051287583535420922*z^26-509*z^2+ 47159284866270312739*z^24-4456018814815129948*z^22+104454*z^4-12017384*z^6-\ 46580015044*z^102+893619031*z^8-46580015044*z^10+1794029990017*z^12-\ 52972335388912*z^14-23042701480033054*z^18+1232049260130263*z^16-\ 1007195347377751606477678002*z^50+633428425330751154266255026*z^48+ 352189507550593511*z^20+2294451690086515052628420*z^36-552803756000534264168810 *z^34-348595026413072888400647998*z^66+115210335872622579073787*z^80+ 1794029990017*z^100-4456018814815129948*z^90+47159284866270312739*z^88+ 3193731980205587862249*z^84-23042701480033054*z^94-421051287583535420922*z^86+ 1232049260130263*z^96-52972335388912*z^98+352189507550593511*z^92-\ 20700638492135790754682*z^82+633428425330751154266255026*z^64+z^112-509*z^110-\ 12017384*z^106+104454*z^108-20700638492135790754682*z^30-\ 70512211679964556187768138*z^42+167746405103712901430670314*z^44-\ 348595026413072888400647998*z^46-1709898377021721970940436162*z^58+ 1826744920811852785061444878*z^56-1709898377021721970940436162*z^54+ 1402216512691987193468886338*z^52+1402216512691987193468886338*z^60-\ 70512211679964556187768138*z^70+167746405103712901430670314*z^68-\ 552803756000534264168810*z^78+115210335872622579073787*z^32-\ 8260864954057066159581601*z^38+25859050568442840187575825*z^40-\ 1007195347377751606477678002*z^62+2294451690086515052628420*z^76-\ 8260864954057066159581601*z^74+25859050568442840187575825*z^72+893619031*z^104) /(-1-14648892097696334358824*z^28+1805931938332256521014*z^26+736*z^2-\ 189177618135235354940*z^24+16719002452144641495*z^22-184339*z^4+24285020*z^6+ 4758513713698*z^102-2005253266*z^8+114170654476*z^10-4758513713698*z^12+ 151222419141696*z^14+75576140344215625*z^18-3773426290795647*z^16+ 9750204944648518374909956293*z^50-5731717921203168904644472221*z^48-\ 1235768210270372789*z^20-13796806569898871015725372*z^36+ 3105467067692125310515763*z^34+5731717921203168904644472221*z^66-\ 3105467067692125310515763*z^80-151222419141696*z^100+189177618135235354940*z^90 -1805931938332256521014*z^88-101562194282513789665134*z^84+1235768210270372789* z^94+14648892097696334358824*z^86-75576140344215625*z^96+3773426290795647*z^98-\ 16719002452144641495*z^92+604771886873031407067897*z^82-\ 9750204944648518374909956293*z^64-736*z^112+z^114+184339*z^110+2005253266*z^106 -24285020*z^108+101562194282513789665134*z^30+520305346828205211272535060*z^42-\ 1325126987143155743327239784*z^44+2947591895256010454549512488*z^46+ 21607074044189940568447348257*z^58-21607074044189940568447348257*z^56+ 18925178154303736740639583016*z^54-14517083323655278356050893432*z^52-\ 18925178154303736740639583016*z^60+1325126987143155743327239784*z^70-\ 2947591895256010454549512488*z^68+13796806569898871015725372*z^78-\ 604771886873031407067897*z^32+53178125577816795778902089*z^38-\ 178222522210516609995081073*z^40+14517083323655278356050893432*z^62-\ 53178125577816795778902089*z^76+178222522210516609995081073*z^74-\ 520305346828205211272535060*z^72-114170654476*z^104) The first , 40, terms are: [0, 227, 0, 87187, 0, 34592315, 0, 13789044752, 0, 5501760369477, 0, 2195632211357073, 0, 876269442083291625, 0, 349719829568287594783, 0, 139573787947222591582927, 0, 55704169165637538945012921, 0, 22231644470881406448419507721, 0, 8872693669636077064706600421189, 0, 3541109769662217704994584735622912, 0, 1413263985528601502226102215877577627, 0, 564036481026305254963927148881251353675, 0, 225108086817693587111461654574754978016035, 0, 89841087334129124463652492755690816751311201, 0, 35855757505255754888691634079180641789632276001, 0, 14310104479204629485048967132038046895274340396899, 0, 5711191296844666265384275597737296986147451284664667] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {4, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 133750766256842984 z - 51856534791123820 z - 382 z 24 22 4 6 + 16143872780699252 z - 4011804807575280 z + 51542 z - 3636638 z 8 10 12 14 + 156980873 z - 4509417172 z + 90870096856 z - 1332239003060 z 18 16 50 - 121970225693712 z + 14596208837956 z - 4011804807575280 z 48 20 36 + 16143872780699252 z + 789759493422792 z + 709805528905776452 z 34 66 64 - 639853554979317668 z - 3636638 z + 156980873 z 30 42 44 - 278259105117539916 z - 278259105117539916 z + 133750766256842984 z 46 58 56 - 51856534791123820 z - 1332239003060 z + 14596208837956 z 54 52 60 70 - 121970225693712 z + 789759493422792 z + 90870096856 z - 382 z 68 32 38 + 51542 z + 468502456491383358 z - 639853554979317668 z 40 62 72 / 2 + 468502456491383358 z - 4509417172 z + z ) / ((-1 + z ) (1 / 28 26 2 + 737848058637970456 z - 271766996656465644 z - 610 z 24 22 4 6 + 79710636827823960 z - 18530907013337288 z + 100198 z - 8109594 z 8 10 12 14 + 392746957 z - 12519540260 z + 278021363048 z - 4467887201044 z 18 16 50 - 484666171938072 z + 53404479689456 z - 18530907013337288 z 48 20 36 + 79710636827823960 z + 3392309100419496 z + 4318224555500823268 z 34 66 64 - 3867845549232029412 z - 8109594 z + 392746957 z 30 42 - 1600373715518902940 z - 1600373715518902940 z 44 46 58 + 737848058637970456 z - 271766996656465644 z - 4467887201044 z 56 54 52 + 53404479689456 z - 484666171938072 z + 3392309100419496 z 60 70 68 32 + 278021363048 z - 610 z + 100198 z + 2778889365800178154 z 38 40 62 - 3867845549232029412 z + 2778889365800178154 z - 12519540260 z 72 + z )) And in Maple-input format, it is: -(1+133750766256842984*z^28-51856534791123820*z^26-382*z^2+16143872780699252*z^ 24-4011804807575280*z^22+51542*z^4-3636638*z^6+156980873*z^8-4509417172*z^10+ 90870096856*z^12-1332239003060*z^14-121970225693712*z^18+14596208837956*z^16-\ 4011804807575280*z^50+16143872780699252*z^48+789759493422792*z^20+ 709805528905776452*z^36-639853554979317668*z^34-3636638*z^66+156980873*z^64-\ 278259105117539916*z^30-278259105117539916*z^42+133750766256842984*z^44-\ 51856534791123820*z^46-1332239003060*z^58+14596208837956*z^56-121970225693712*z ^54+789759493422792*z^52+90870096856*z^60-382*z^70+51542*z^68+ 468502456491383358*z^32-639853554979317668*z^38+468502456491383358*z^40-\ 4509417172*z^62+z^72)/(-1+z^2)/(1+737848058637970456*z^28-271766996656465644*z^ 26-610*z^2+79710636827823960*z^24-18530907013337288*z^22+100198*z^4-8109594*z^6 +392746957*z^8-12519540260*z^10+278021363048*z^12-4467887201044*z^14-\ 484666171938072*z^18+53404479689456*z^16-18530907013337288*z^50+ 79710636827823960*z^48+3392309100419496*z^20+4318224555500823268*z^36-\ 3867845549232029412*z^34-8109594*z^66+392746957*z^64-1600373715518902940*z^30-\ 1600373715518902940*z^42+737848058637970456*z^44-271766996656465644*z^46-\ 4467887201044*z^58+53404479689456*z^56-484666171938072*z^54+3392309100419496*z^ 52+278021363048*z^60-610*z^70+100198*z^68+2778889365800178154*z^32-\ 3867845549232029412*z^38+2778889365800178154*z^40-12519540260*z^62+z^72) The first , 40, terms are: [0, 229, 0, 90653, 0, 36877105, 0, 15029530221, 0, 6126384758233, 0, 2497295960422393, 0, 1017973542336875237, 0, 414956955022980077213, 0, 169149070294098455886529, 0, 68950303661814545103278065, 0, 28106240065526620267443755957, 0, 11456957963897680328506917047209, 0, 4670204391662518212518123638993317, 0, 1903717298138426051706217279966568509, 0, 776013049386389364974413010831637206729, 0, 316326512033500769858455150702327540886713, 0, 128944303571186362100957269527199216680408269, 0, 52561618425765753700528426191321950744447677013, 0, 21425713699795814758555762612126313502154084512825, 0, 8733772309426945572028707627631491194187859722238373] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 37460234917201721344 z - 8416867955283402288 z - 474 z 24 22 4 6 + 1557045718637288261 z - 235454441442192914 z + 86568 z - 8468134 z 8 10 12 14 + 513990975 z - 21068034900 z + 616034931440 z - 13350260082980 z 18 16 50 - 2835346030267806 z + 220506471431179 z - 4232028998770154071612 z 48 20 + 6455048243204885636550 z + 28853982934129928 z 36 34 + 2338593142685532491344 z - 1086935270599652296084 z 66 80 88 84 86 - 235454441442192914 z + 513990975 z + z + 86568 z - 474 z 82 64 30 - 8468134 z + 1557045718637288261 z - 138090080943478607248 z 42 44 - 8311000792165240811976 z + 9040558767570501231392 z 46 58 - 8311000792165240811976 z - 138090080943478607248 z 56 54 + 423743188175599040426 z - 1086935270599652296084 z 52 60 + 2338593142685532491344 z + 37460234917201721344 z 70 68 78 - 2835346030267806 z + 28853982934129928 z - 21068034900 z 32 38 + 423743188175599040426 z - 4232028998770154071612 z 40 62 76 + 6455048243204885636550 z - 8416867955283402288 z + 616034931440 z 74 72 / - 13350260082980 z + 220506471431179 z ) / (-1 / 28 26 2 - 221994345526147034776 z + 45534439199631625981 z + 727 z 24 22 4 - 7693190215140316347 z + 1063322996548667792 z - 164456 z 6 8 10 12 + 18489948 z - 1250486789 z + 56331771159 z - 1799517731896 z 14 18 16 + 42513002764176 z + 10731305688022897 z - 765275046030907 z 50 48 + 65505967401187253879706 z - 91926704052764535078910 z 20 36 - 119223278917131156 z - 19906504144593569786792 z 34 66 80 + 8459855250558679942494 z + 7693190215140316347 z - 56331771159 z 90 88 84 86 82 + z - 727 z - 18489948 z + 164456 z + 1250486789 z 64 30 - 45534439199631625981 z + 896515163257107092264 z 42 44 + 91926704052764535078910 z - 108872213988005489455224 z 46 58 + 108872213988005489455224 z + 3013060137910537090994 z 56 54 - 8459855250558679942494 z + 19906504144593569786792 z 52 60 - 39356097798310585895536 z - 896515163257107092264 z 70 68 78 + 119223278917131156 z - 1063322996548667792 z + 1799517731896 z 32 38 - 3013060137910537090994 z + 39356097798310585895536 z 40 62 - 65505967401187253879706 z + 221994345526147034776 z 76 74 72 - 42513002764176 z + 765275046030907 z - 10731305688022897 z ) And in Maple-input format, it is: -(1+37460234917201721344*z^28-8416867955283402288*z^26-474*z^2+ 1557045718637288261*z^24-235454441442192914*z^22+86568*z^4-8468134*z^6+ 513990975*z^8-21068034900*z^10+616034931440*z^12-13350260082980*z^14-\ 2835346030267806*z^18+220506471431179*z^16-4232028998770154071612*z^50+ 6455048243204885636550*z^48+28853982934129928*z^20+2338593142685532491344*z^36-\ 1086935270599652296084*z^34-235454441442192914*z^66+513990975*z^80+z^88+86568*z ^84-474*z^86-8468134*z^82+1557045718637288261*z^64-138090080943478607248*z^30-\ 8311000792165240811976*z^42+9040558767570501231392*z^44-8311000792165240811976* z^46-138090080943478607248*z^58+423743188175599040426*z^56-\ 1086935270599652296084*z^54+2338593142685532491344*z^52+37460234917201721344*z^ 60-2835346030267806*z^70+28853982934129928*z^68-21068034900*z^78+ 423743188175599040426*z^32-4232028998770154071612*z^38+6455048243204885636550*z ^40-8416867955283402288*z^62+616034931440*z^76-13350260082980*z^74+ 220506471431179*z^72)/(-1-221994345526147034776*z^28+45534439199631625981*z^26+ 727*z^2-7693190215140316347*z^24+1063322996548667792*z^22-164456*z^4+18489948*z ^6-1250486789*z^8+56331771159*z^10-1799517731896*z^12+42513002764176*z^14+ 10731305688022897*z^18-765275046030907*z^16+65505967401187253879706*z^50-\ 91926704052764535078910*z^48-119223278917131156*z^20-19906504144593569786792*z^ 36+8459855250558679942494*z^34+7693190215140316347*z^66-56331771159*z^80+z^90-\ 727*z^88-18489948*z^84+164456*z^86+1250486789*z^82-45534439199631625981*z^64+ 896515163257107092264*z^30+91926704052764535078910*z^42-\ 108872213988005489455224*z^44+108872213988005489455224*z^46+ 3013060137910537090994*z^58-8459855250558679942494*z^56+19906504144593569786792 *z^54-39356097798310585895536*z^52-896515163257107092264*z^60+ 119223278917131156*z^70-1063322996548667792*z^68+1799517731896*z^78-\ 3013060137910537090994*z^32+39356097798310585895536*z^38-\ 65505967401187253879706*z^40+221994345526147034776*z^62-42513002764176*z^76+ 765275046030907*z^74-10731305688022897*z^72) The first , 40, terms are: [0, 253, 0, 106043, 0, 45507707, 0, 19586156411, 0, 8434740382811, 0, 3632893540342261, 0, 1564757641356857081, 0, 673976260927316664105, 0, 290297193703159815079013, 0, 125037776269515518635010267, 0, 53856693131297959819724617723, 0, 23197337164352239367176877636091, 0, 9991635631095246147197961442694139, 0, 4303631144666989819852728031434535949, 0, 1853674585209605689955985161384855362833, 0, 798420996722996421332122394011509452732657, 0, 343898596389662300565651567784101371343150317, 0, 148125168406839009062267958407604258145128934395, 0, 63800974316014561081653593875807690524153764115579, 0, 27480571785702633731496010721398520718148287433569915] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3529194895879215 z - 1972519386918412 z - 348 z 24 22 4 6 + 869356507643342 z - 300624303742062 z + 39582 z - 2284446 z 8 10 12 14 + 79166088 z - 1799664183 z + 28344130952 z - 320916202582 z 18 16 50 - 16865962944905 z + 2681768534032 z - 320916202582 z 48 20 36 + 2681768534032 z + 81015511783744 z + 3529194895879215 z 34 64 30 42 - 4996594934426208 z + z - 4996594934426208 z - 300624303742062 z 44 46 58 56 + 81015511783744 z - 16865962944905 z - 2284446 z + 79166088 z 54 52 60 32 - 1799664183 z + 28344130952 z + 39582 z + 5609300398641884 z 38 40 62 / 2 - 1972519386918412 z + 869356507643342 z - 348 z ) / ((-1 + z ) (1 / 28 26 2 + 21295746200445012 z - 11432062044642038 z - 598 z 24 22 4 6 + 4772922991981461 z - 1544955652286248 z + 82232 z - 5417124 z 8 10 12 14 + 210665323 z - 5336650882 z + 93249444002 z - 1166710946686 z 18 16 50 - 73857129408978 z + 10727899374011 z - 1166710946686 z 48 20 36 + 10727899374011 z + 385879604042178 z + 21295746200445012 z 34 64 30 - 30910854508297788 z + z - 30910854508297788 z 42 44 46 - 1544955652286248 z + 385879604042178 z - 73857129408978 z 58 56 54 52 - 5417124 z + 210665323 z - 5336650882 z + 93249444002 z 60 32 38 + 82232 z + 34995065690518269 z - 11432062044642038 z 40 62 + 4772922991981461 z - 598 z )) And in Maple-input format, it is: -(1+3529194895879215*z^28-1972519386918412*z^26-348*z^2+869356507643342*z^24-\ 300624303742062*z^22+39582*z^4-2284446*z^6+79166088*z^8-1799664183*z^10+ 28344130952*z^12-320916202582*z^14-16865962944905*z^18+2681768534032*z^16-\ 320916202582*z^50+2681768534032*z^48+81015511783744*z^20+3529194895879215*z^36-\ 4996594934426208*z^34+z^64-4996594934426208*z^30-300624303742062*z^42+ 81015511783744*z^44-16865962944905*z^46-2284446*z^58+79166088*z^56-1799664183*z ^54+28344130952*z^52+39582*z^60+5609300398641884*z^32-1972519386918412*z^38+ 869356507643342*z^40-348*z^62)/(-1+z^2)/(1+21295746200445012*z^28-\ 11432062044642038*z^26-598*z^2+4772922991981461*z^24-1544955652286248*z^22+ 82232*z^4-5417124*z^6+210665323*z^8-5336650882*z^10+93249444002*z^12-\ 1166710946686*z^14-73857129408978*z^18+10727899374011*z^16-1166710946686*z^50+ 10727899374011*z^48+385879604042178*z^20+21295746200445012*z^36-\ 30910854508297788*z^34+z^64-30910854508297788*z^30-1544955652286248*z^42+ 385879604042178*z^44-73857129408978*z^46-5417124*z^58+210665323*z^56-5336650882 *z^54+93249444002*z^52+82232*z^60+34995065690518269*z^32-11432062044642038*z^38 +4772922991981461*z^40-598*z^62) The first , 40, terms are: [0, 251, 0, 107101, 0, 46578079, 0, 20272515488, 0, 8823671978523, 0, 3840535983728737, 0, 1671607695716843715, 0, 727573524586070566561, 0, 316679107891476621218009, 0, 137835770527341915296031467, 0, 59993536559405163238932159905, 0, 26112412004045154403105188427611, 0, 11365525351115699576369355741920000, 0, 4946887575412137143048528894869544031, 0, 2153151387882365157005163412943751953565, 0, 937167224535826224593382920352433399423427, 0, 407905552617913568288573569851938397162373465, 0, 177542423060021109378678681674734814381014829609, 0, 77276006133580703728994500198056177405953650736851, 0, 33634671764834550166216894583288300346006468828674397] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {3, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 121195012466247100 z - 47199010371935379 z - 395 z 24 22 4 6 + 14801186652197072 z - 3717906039525012 z + 55212 z - 3958915 z 8 10 12 14 + 170024836 z - 4791195680 z + 94063637988 z - 1341324923004 z 18 16 50 - 116905769562238 z + 14315316985133 z - 3717906039525012 z 48 20 36 + 14801186652197072 z + 742764643928990 z + 640629858136526290 z 34 66 64 - 577540629991588098 z - 3958915 z + 170024836 z 30 42 44 - 251529845402233693 z - 251529845402233693 z + 121195012466247100 z 46 58 56 - 47199010371935379 z - 1341324923004 z + 14315316985133 z 54 52 60 70 - 116905769562238 z + 742764643928990 z + 94063637988 z - 395 z 68 32 38 + 55212 z + 423037341550818201 z - 577540629991588098 z 40 62 72 / 2 + 423037341550818201 z - 4791195680 z + z ) / ((-1 + z ) (1 / 28 26 2 + 675840560692773803 z - 249657461257291004 z - 626 z 24 22 4 6 + 73630193755255411 z - 17269072059030784 z + 107798 z - 8905856 z 8 10 12 14 + 428709861 z - 13368005332 z + 288475767144 z - 4502749039964 z 18 16 50 - 465447377741912 z + 52425259028051 z - 17269072059030784 z 48 20 36 + 73630193755255411 z + 3202021195867327 z + 3952128561017751427 z 34 66 64 - 3539420943635926964 z - 8905856 z + 428709861 z 30 42 - 1464234854860680946 z - 1464234854860680946 z 44 46 58 + 675840560692773803 z - 249657461257291004 z - 4502749039964 z 56 54 52 + 52425259028051 z - 465447377741912 z + 3202021195867327 z 60 70 68 32 + 288475767144 z - 626 z + 107798 z + 2542262620034976385 z 38 40 62 - 3539420943635926964 z + 2542262620034976385 z - 13368005332 z 72 + z )) And in Maple-input format, it is: -(1+121195012466247100*z^28-47199010371935379*z^26-395*z^2+14801186652197072*z^ 24-3717906039525012*z^22+55212*z^4-3958915*z^6+170024836*z^8-4791195680*z^10+ 94063637988*z^12-1341324923004*z^14-116905769562238*z^18+14315316985133*z^16-\ 3717906039525012*z^50+14801186652197072*z^48+742764643928990*z^20+ 640629858136526290*z^36-577540629991588098*z^34-3958915*z^66+170024836*z^64-\ 251529845402233693*z^30-251529845402233693*z^42+121195012466247100*z^44-\ 47199010371935379*z^46-1341324923004*z^58+14315316985133*z^56-116905769562238*z ^54+742764643928990*z^52+94063637988*z^60-395*z^70+55212*z^68+ 423037341550818201*z^32-577540629991588098*z^38+423037341550818201*z^40-\ 4791195680*z^62+z^72)/(-1+z^2)/(1+675840560692773803*z^28-249657461257291004*z^ 26-626*z^2+73630193755255411*z^24-17269072059030784*z^22+107798*z^4-8905856*z^6 +428709861*z^8-13368005332*z^10+288475767144*z^12-4502749039964*z^14-\ 465447377741912*z^18+52425259028051*z^16-17269072059030784*z^50+ 73630193755255411*z^48+3202021195867327*z^20+3952128561017751427*z^36-\ 3539420943635926964*z^34-8905856*z^66+428709861*z^64-1464234854860680946*z^30-\ 1464234854860680946*z^42+675840560692773803*z^44-249657461257291004*z^46-\ 4502749039964*z^58+52425259028051*z^56-465447377741912*z^54+3202021195867327*z^ 52+288475767144*z^60-626*z^70+107798*z^68+2542262620034976385*z^32-\ 3539420943635926964*z^38+2542262620034976385*z^40-13368005332*z^62+z^72) The first , 40, terms are: [0, 232, 0, 92252, 0, 37742375, 0, 15485715124, 0, 6356370397725, 0, 2609239904822853, 0, 1071082734609042960, 0, 439675922351393569087, 0, 180485555907034478677439, 0, 74088744595117433570302392, 0, 30413193347710259151570075849, 0, 12484518877031619698401559625060, 0, 5124855184792231260045856053622884, 0, 2103736709792522079340778133410530807, 0, 863577210391860989225894804598853873583, 0, 354495690852007025733050719841261738082668, 0, 145519350580859034783113173795773503496987492, 0, 59735229341097030760613923775003133058089425105, 0, 24521121144302480890891989981449043227252710006032, 0, 10065842030004264404429201772869326188159101650104895] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {4, 6}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 31248607061801885881 z - 6905365979238586771 z - 455 z 24 22 4 6 + 1258980170329716092 z - 188181709253395518 z + 79284 z - 7433246 z 8 10 12 14 + 435519255 z - 17364205522 z + 497545528021 z - 10639075457386 z 18 16 50 - 2240219707161174 z + 174479750206241 z - 3830984043375575298502 z 48 20 + 5901330629998121351928 z + 22878738924255679 z 36 34 + 2089468845481021135644 z - 956127478122451917208 z 66 80 88 84 86 - 188181709253395518 z + 435519255 z + z + 79284 z - 455 z 82 64 30 - 7433246 z + 1258980170329716092 z - 117261287344040923506 z 42 44 - 7644904040350759254950 z + 8333347601201179323252 z 46 58 - 7644904040350759254950 z - 117261287344040923506 z 56 54 + 366368730745471272314 z - 956127478122451917208 z 52 60 + 2089468845481021135644 z + 31248607061801885881 z 70 68 78 - 2240219707161174 z + 22878738924255679 z - 17364205522 z 32 38 + 366368730745471272314 z - 3830984043375575298502 z 40 62 76 + 5901330629998121351928 z - 6905365979238586771 z + 497545528021 z 74 72 / 2 - 10639075457386 z + 174479750206241 z ) / ((-1 + z ) (1 / 28 26 2 + 149776925002213968688 z - 31213290963895428078 z - 688 z 24 22 4 + 5348915498550176355 z - 749420125878446331 z + 147074 z 6 8 10 12 - 15763027 z + 1024369259 z - 44636934258 z + 1386865096704 z 14 18 16 - 32008006979125 z - 7786580688505252 z + 564862194196865 z 50 48 - 22799565144573109248438 z + 35910166430895141446137 z 20 36 + 85207563065979710 z + 12064189610069182640353 z 34 66 80 - 5316648246297314768026 z - 749420125878446331 z + 1024369259 z 88 84 86 82 64 + z + 147074 z - 688 z - 15763027 z + 5348915498550176355 z 30 42 - 593585070748598571982 z - 47156101361583794740526 z 44 46 + 51638069524509011840005 z - 47156101361583794740526 z 58 56 - 593585070748598571982 z + 1949267944854671559041 z 54 52 - 5316648246297314768026 z + 12064189610069182640353 z 60 70 68 + 149776925002213968688 z - 7786580688505252 z + 85207563065979710 z 78 32 - 44636934258 z + 1949267944854671559041 z 38 40 - 22799565144573109248438 z + 35910166430895141446137 z 62 76 74 - 31213290963895428078 z + 1386865096704 z - 32008006979125 z 72 + 564862194196865 z )) And in Maple-input format, it is: -(1+31248607061801885881*z^28-6905365979238586771*z^26-455*z^2+ 1258980170329716092*z^24-188181709253395518*z^22+79284*z^4-7433246*z^6+ 435519255*z^8-17364205522*z^10+497545528021*z^12-10639075457386*z^14-\ 2240219707161174*z^18+174479750206241*z^16-3830984043375575298502*z^50+ 5901330629998121351928*z^48+22878738924255679*z^20+2089468845481021135644*z^36-\ 956127478122451917208*z^34-188181709253395518*z^66+435519255*z^80+z^88+79284*z^ 84-455*z^86-7433246*z^82+1258980170329716092*z^64-117261287344040923506*z^30-\ 7644904040350759254950*z^42+8333347601201179323252*z^44-7644904040350759254950* z^46-117261287344040923506*z^58+366368730745471272314*z^56-\ 956127478122451917208*z^54+2089468845481021135644*z^52+31248607061801885881*z^ 60-2240219707161174*z^70+22878738924255679*z^68-17364205522*z^78+ 366368730745471272314*z^32-3830984043375575298502*z^38+5901330629998121351928*z ^40-6905365979238586771*z^62+497545528021*z^76-10639075457386*z^74+ 174479750206241*z^72)/(-1+z^2)/(1+149776925002213968688*z^28-\ 31213290963895428078*z^26-688*z^2+5348915498550176355*z^24-749420125878446331*z ^22+147074*z^4-15763027*z^6+1024369259*z^8-44636934258*z^10+1386865096704*z^12-\ 32008006979125*z^14-7786580688505252*z^18+564862194196865*z^16-\ 22799565144573109248438*z^50+35910166430895141446137*z^48+85207563065979710*z^ 20+12064189610069182640353*z^36-5316648246297314768026*z^34-749420125878446331* z^66+1024369259*z^80+z^88+147074*z^84-688*z^86-15763027*z^82+ 5348915498550176355*z^64-593585070748598571982*z^30-47156101361583794740526*z^ 42+51638069524509011840005*z^44-47156101361583794740526*z^46-\ 593585070748598571982*z^58+1949267944854671559041*z^56-5316648246297314768026*z ^54+12064189610069182640353*z^52+149776925002213968688*z^60-7786580688505252*z^ 70+85207563065979710*z^68-44636934258*z^78+1949267944854671559041*z^32-\ 22799565144573109248438*z^38+35910166430895141446137*z^40-31213290963895428078* z^62+1386865096704*z^76-32008006979125*z^74+564862194196865*z^72) The first , 40, terms are: [0, 234, 0, 92748, 0, 37803919, 0, 15460620818, 0, 6326921254943, 0, 2589501260090339, 0, 1059869570677177840, 0, 433801913592729403521, 0, 177554261946504631480633, 0, 72672628720188693460350062, 0, 29744773531569481999796700093, 0, 12174481378658739342456067325796, 0, 4982992966621697572064770418654974, 0, 2039529910967909145121568822544359407, 0, 834775863866087383036069908179779488855, 0, 341672234944195451935660690944727982709790, 0, 139845821118893173682070653315145443310191892, 0, 57238638918499751944586924970886699003857585029, 0, 23427670265943136735773654911177927893104491442302, 0, 9588902609499404388059499781828008396582683391647393] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 157851217563793844 z - 70650138876945126 z - 441 z 24 22 4 6 + 24910840797047294 z - 6872652391558496 z + 71238 z - 5877145 z 8 10 12 14 + 285447829 z - 8874806712 z + 186941399317 z - 2778799587441 z 18 16 50 - 241589755279041 z + 30039567379462 z - 241589755279041 z 48 20 36 + 1470902059546465 z + 1470902059546465 z + 392722524196312822 z 34 66 64 30 - 439874107973350352 z - 441 z + 71238 z - 279281488764091798 z 42 44 46 - 70650138876945126 z + 24910840797047294 z - 6872652391558496 z 58 56 54 - 8874806712 z + 186941399317 z - 2778799587441 z 52 60 68 32 + 30039567379462 z + 285447829 z + z + 392722524196312822 z 38 40 62 / - 279281488764091798 z + 157851217563793844 z - 5877145 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 921938801408693294 z - 391752704848330036 z - 678 z 24 22 4 6 + 129631445176549338 z - 33248928603717736 z + 138251 z - 13321880 z 8 10 12 14 + 729441897 z - 25158568010 z + 583641513917 z - 9521968141216 z 18 16 50 - 990674874325462 z + 112734947281783 z - 990674874325462 z 48 20 36 + 6568187023650741 z + 6568187023650741 z + 2440975366262756202 z 34 66 64 30 - 2756006380200192060 z - 678 z + 138251 z - 1695222095556402296 z 42 44 46 - 391752704848330036 z + 129631445176549338 z - 33248928603717736 z 58 56 54 - 25158568010 z + 583641513917 z - 9521968141216 z 52 60 68 32 + 112734947281783 z + 729441897 z + z + 2440975366262756202 z 38 40 62 - 1695222095556402296 z + 921938801408693294 z - 13321880 z )) And in Maple-input format, it is: -(1+157851217563793844*z^28-70650138876945126*z^26-441*z^2+24910840797047294*z^ 24-6872652391558496*z^22+71238*z^4-5877145*z^6+285447829*z^8-8874806712*z^10+ 186941399317*z^12-2778799587441*z^14-241589755279041*z^18+30039567379462*z^16-\ 241589755279041*z^50+1470902059546465*z^48+1470902059546465*z^20+ 392722524196312822*z^36-439874107973350352*z^34-441*z^66+71238*z^64-\ 279281488764091798*z^30-70650138876945126*z^42+24910840797047294*z^44-\ 6872652391558496*z^46-8874806712*z^58+186941399317*z^56-2778799587441*z^54+ 30039567379462*z^52+285447829*z^60+z^68+392722524196312822*z^32-\ 279281488764091798*z^38+157851217563793844*z^40-5877145*z^62)/(-1+z^2)/(1+ 921938801408693294*z^28-391752704848330036*z^26-678*z^2+129631445176549338*z^24 -33248928603717736*z^22+138251*z^4-13321880*z^6+729441897*z^8-25158568010*z^10+ 583641513917*z^12-9521968141216*z^14-990674874325462*z^18+112734947281783*z^16-\ 990674874325462*z^50+6568187023650741*z^48+6568187023650741*z^20+ 2440975366262756202*z^36-2756006380200192060*z^34-678*z^66+138251*z^64-\ 1695222095556402296*z^30-391752704848330036*z^42+129631445176549338*z^44-\ 33248928603717736*z^46-25158568010*z^58+583641513917*z^56-9521968141216*z^54+ 112734947281783*z^52+729441897*z^60+z^68+2440975366262756202*z^32-\ 1695222095556402296*z^38+921938801408693294*z^40-13321880*z^62) The first , 40, terms are: [0, 238, 0, 93911, 0, 38283453, 0, 15693698498, 0, 6441629224915, 0, 2644842496214395, 0, 1086016225141004770, 0, 445944378467277413285, 0, 183116294902401390226335, 0, 75192367428262427916901646, 0, 30875972049083610908863360073, 0, 12678490431561753596195198380089, 0, 5206123450139826920878883594512334, 0, 2137771971157740873247908849700429455, 0, 877825708259644293789385466473705859669, 0, 360458451440214421689014712439299956983970, 0, 148013773120679298793768917204314379718494379, 0, 60778369729383916006009025102208519771269817763, 0, 24957206002457528207745589820183432057498205940482, 0, 10248088822107236233695165713629360540166593163791629] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 33826741898666876653 z - 7418111446441044653 z - 449 z 24 22 4 6 + 1340421185143892753 z - 198324393114451308 z + 77285 z - 7218732 z 8 10 12 14 + 424266352 z - 17044938184 z + 493501493383 z - 10679520763695 z 18 16 50 - 2306281110493788 z + 177369079075331 z - 4235264419920868325667 z 48 20 + 6534139562926330322923 z + 23841218349440512 z 36 34 + 2304718119815184482855 z - 1051337352740578663692 z 66 80 88 84 86 - 198324393114451308 z + 424266352 z + z + 77285 z - 449 z 82 64 30 - 7218732 z + 1340421185143892753 z - 127748166073215953536 z 42 44 - 8472191702882935852600 z + 9237809054172288158784 z 46 58 - 8472191702882935852600 z - 127748166073215953536 z 56 54 + 401207002454972219696 z - 1051337352740578663692 z 52 60 + 2304718119815184482855 z + 33826741898666876653 z 70 68 78 - 2306281110493788 z + 23841218349440512 z - 17044938184 z 32 38 + 401207002454972219696 z - 4235264419920868325667 z 40 62 76 + 6534139562926330322923 z - 7418111446441044653 z + 493501493383 z 74 72 / 2 - 10679520763695 z + 177369079075331 z ) / ((-1 + z ) (1 / 28 26 2 + 161712258754753619440 z - 33443981141390211002 z - 694 z 24 22 4 + 5680240044855938619 z - 787769920852368340 z + 144502 z 6 8 10 12 - 15303032 z + 994166759 z - 43614348476 z + 1369590738588 z 14 18 16 - 32007997745142 z - 7993195659155076 z + 572352091080109 z 50 48 - 25146480784297389873498 z + 39669929901808814011191 z 20 36 + 88555559166793394 z + 13274875615046299609118 z 34 66 80 - 5831549550122982548720 z - 787769920852368340 z + 994166759 z 88 84 86 82 64 + z + 144502 z - 694 z - 15303032 z + 5680240044855938619 z 30 42 - 644998004175724947356 z - 52141529376331454372456 z 44 46 + 57114635448352987350588 z - 52141529376331454372456 z 58 56 - 644998004175724947356 z + 2129192633560358765033 z 54 52 - 5831549550122982548720 z + 13274875615046299609118 z 60 70 68 + 161712258754753619440 z - 7993195659155076 z + 88555559166793394 z 78 32 - 43614348476 z + 2129192633560358765033 z 38 40 - 25146480784297389873498 z + 39669929901808814011191 z 62 76 74 - 33443981141390211002 z + 1369590738588 z - 32007997745142 z 72 + 572352091080109 z )) And in Maple-input format, it is: -(1+33826741898666876653*z^28-7418111446441044653*z^26-449*z^2+ 1340421185143892753*z^24-198324393114451308*z^22+77285*z^4-7218732*z^6+ 424266352*z^8-17044938184*z^10+493501493383*z^12-10679520763695*z^14-\ 2306281110493788*z^18+177369079075331*z^16-4235264419920868325667*z^50+ 6534139562926330322923*z^48+23841218349440512*z^20+2304718119815184482855*z^36-\ 1051337352740578663692*z^34-198324393114451308*z^66+424266352*z^80+z^88+77285*z ^84-449*z^86-7218732*z^82+1340421185143892753*z^64-127748166073215953536*z^30-\ 8472191702882935852600*z^42+9237809054172288158784*z^44-8472191702882935852600* z^46-127748166073215953536*z^58+401207002454972219696*z^56-\ 1051337352740578663692*z^54+2304718119815184482855*z^52+33826741898666876653*z^ 60-2306281110493788*z^70+23841218349440512*z^68-17044938184*z^78+ 401207002454972219696*z^32-4235264419920868325667*z^38+6534139562926330322923*z ^40-7418111446441044653*z^62+493501493383*z^76-10679520763695*z^74+ 177369079075331*z^72)/(-1+z^2)/(1+161712258754753619440*z^28-\ 33443981141390211002*z^26-694*z^2+5680240044855938619*z^24-787769920852368340*z ^22+144502*z^4-15303032*z^6+994166759*z^8-43614348476*z^10+1369590738588*z^12-\ 32007997745142*z^14-7993195659155076*z^18+572352091080109*z^16-\ 25146480784297389873498*z^50+39669929901808814011191*z^48+88555559166793394*z^ 20+13274875615046299609118*z^36-5831549550122982548720*z^34-787769920852368340* z^66+994166759*z^80+z^88+144502*z^84-694*z^86-15303032*z^82+5680240044855938619 *z^64-644998004175724947356*z^30-52141529376331454372456*z^42+ 57114635448352987350588*z^44-52141529376331454372456*z^46-644998004175724947356 *z^58+2129192633560358765033*z^56-5831549550122982548720*z^54+ 13274875615046299609118*z^52+161712258754753619440*z^60-7993195659155076*z^70+ 88555559166793394*z^68-43614348476*z^78+2129192633560358765033*z^32-\ 25146480784297389873498*z^38+39669929901808814011191*z^40-33443981141390211002* z^62+1369590738588*z^76-32007997745142*z^74+572352091080109*z^72) The first , 40, terms are: [0, 246, 0, 103059, 0, 44136591, 0, 18926066106, 0, 8116400891805, 0, 3480730059557753, 0, 1492717300460485834, 0, 640154537014536187835, 0, 274531443596237345286015, 0, 117733311651103377469449062, 0, 50490146020183537767838642541, 0, 21652791460924672676615907225173, 0, 9285839218290011039291315972169126, 0, 3982249131414246369275952496832488711, 0, 1707794823047837401042487377993707826883, 0, 732390933209453143583851260517017241218954, 0, 314087191159256187373013443939540671687229633, 0, 134696866355250838700765194857996973936833591125, 0, 57764997480348928096687248155942476091352543275322, 0, 24772624814479516410241024806247855091186652563343431] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3569524535704730 z - 1973835335662160 z - 352 z 24 22 4 6 + 858898711774674 z - 293179960825568 z + 40565 z - 2325054 z 8 10 12 14 + 79253936 z - 1769052166 z + 27437853841 z - 307569771088 z 18 16 50 - 16143050114800 z + 2560703976333 z - 307569771088 z 48 20 36 + 2560703976333 z + 78130161134400 z + 3569524535704730 z 34 64 30 42 - 5089654848516844 z + z - 5089654848516844 z - 293179960825568 z 44 46 58 56 + 78130161134400 z - 16143050114800 z - 2325054 z + 79253936 z 54 52 60 32 - 1769052166 z + 27437853841 z + 40565 z + 5727961933862432 z 38 40 62 / 2 - 1973835335662160 z + 858898711774674 z - 352 z ) / ((-1 + z ) (1 / 28 26 2 + 21550899679594786 z - 11416429174722950 z - 595 z 24 22 4 6 + 4693689055001306 z - 1496535768144256 z + 83509 z - 5478330 z 8 10 12 14 + 209622554 z - 5210547890 z + 89579669577 z - 1108824886439 z 18 16 50 - 70070114153072 z + 10153321920645 z - 1108824886439 z 48 20 36 + 10153321920645 z + 369107622770168 z + 21550899679594786 z 34 64 30 - 31564686592395428 z + z - 31564686592395428 z 42 44 46 - 1496535768144256 z + 369107622770168 z - 70070114153072 z 58 56 54 52 - 5478330 z + 209622554 z - 5210547890 z + 89579669577 z 60 32 38 + 83509 z + 35849834685137116 z - 11416429174722950 z 40 62 + 4693689055001306 z - 595 z )) And in Maple-input format, it is: -(1+3569524535704730*z^28-1973835335662160*z^26-352*z^2+858898711774674*z^24-\ 293179960825568*z^22+40565*z^4-2325054*z^6+79253936*z^8-1769052166*z^10+ 27437853841*z^12-307569771088*z^14-16143050114800*z^18+2560703976333*z^16-\ 307569771088*z^50+2560703976333*z^48+78130161134400*z^20+3569524535704730*z^36-\ 5089654848516844*z^34+z^64-5089654848516844*z^30-293179960825568*z^42+ 78130161134400*z^44-16143050114800*z^46-2325054*z^58+79253936*z^56-1769052166*z ^54+27437853841*z^52+40565*z^60+5727961933862432*z^32-1973835335662160*z^38+ 858898711774674*z^40-352*z^62)/(-1+z^2)/(1+21550899679594786*z^28-\ 11416429174722950*z^26-595*z^2+4693689055001306*z^24-1496535768144256*z^22+ 83509*z^4-5478330*z^6+209622554*z^8-5210547890*z^10+89579669577*z^12-\ 1108824886439*z^14-70070114153072*z^18+10153321920645*z^16-1108824886439*z^50+ 10153321920645*z^48+369107622770168*z^20+21550899679594786*z^36-\ 31564686592395428*z^34+z^64-31564686592395428*z^30-1496535768144256*z^42+ 369107622770168*z^44-70070114153072*z^46-5478330*z^58+209622554*z^56-5210547890 *z^54+89579669577*z^52+83509*z^60+35849834685137116*z^32-11416429174722950*z^38 +4693689055001306*z^40-595*z^62) The first , 40, terms are: [0, 244, 0, 101885, 0, 43438869, 0, 18541871652, 0, 7915407335313, 0, 3379076809805201, 0, 1442525521967372996, 0, 615813255397604660645, 0, 262890305727583013946029, 0, 112227712677312934392922324, 0, 47909942756971862493441632129, 0, 20452725626897851487771912930177, 0, 8731256217414367029246197795538580, 0, 3727368005854809027299484562167902285, 0, 1591211150505731471433544180232253888389, 0, 679287079117686006765437892016190479924996, 0, 289987243810843739391897719875111658665917073, 0, 123795379241183504285153169069638319180033228433, 0, 52848172630189937631173839537190843806214420648676, 0, 22560852977468983467368461637939731708240520973614901] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3532765592370337614180 z - 464787585822208377484 z - 509 z 24 22 4 + 51871229113654234948 z - 4875623906174241912 z + 104563 z 6 102 8 10 - 12093862 z - 47866913166 z + 907647792 z - 47866913166 z 12 14 18 + 1867077334195 z - 55824008828637 z - 24821396168548536 z 16 50 + 1313573134390857 z - 1086899379231686511462351310 z 48 20 + 685155663984564017469495870 z + 382681118837030696 z 36 34 + 2528993239253068840356004 z - 610699453883301885818496 z 66 80 - 378127999661784441986091208 z + 127465954777358446133104 z 100 90 88 + 1867077334195 z - 4875623906174241912 z + 51871229113654234948 z 84 94 + 3532765592370337614180 z - 24821396168548536 z 86 96 98 - 464787585822208377484 z + 1313573134390857 z - 55824008828637 z 92 82 + 382681118837030696 z - 22914037809825102135312 z 64 112 110 106 + 685155663984564017469495870 z + z - 509 z - 12093862 z 108 30 + 104563 z - 22914037809825102135312 z 42 44 - 76988065372156170708908936 z + 182536256319322608411046104 z 46 58 - 378127999661784441986091208 z - 1839993477480789862415927996 z 56 54 + 1965010988184133572743408320 z - 1839993477480789862415927996 z 52 60 + 1510535092407431792374370386 z + 1510535092407431792374370386 z 70 68 - 76988065372156170708908936 z + 182536256319322608411046104 z 78 32 - 610699453883301885818496 z + 127465954777358446133104 z 38 40 - 9079372417913283402741452 z + 28329826325880253464451764 z 62 76 - 1086899379231686511462351310 z + 2528993239253068840356004 z 74 72 - 9079372417913283402741452 z + 28329826325880253464451764 z 104 / 2 28 + 907647792 z ) / ((-1 + z ) (1 + 14449085267441139099728 z / 26 2 24 - 1804668152074775320464 z - 764 z + 190916454120834758332 z 22 4 6 102 - 16987922682553223712 z + 189534 z - 24731676 z - 115747013148 z 8 10 12 + 2033297263 z - 115747013148 z + 4834340643638 z 14 18 16 - 154053232983356 z - 77207880351878768 z + 3852321210918673 z 50 48 - 6625440880386185598730063016 z + 4103772097842178945087770134 z 20 36 + 1260640615334632200 z + 12492048325765535323234720 z 34 66 - 2887413752634716242969152 z - 2215068826935793323892218096 z 80 100 + 575437382959428257734820 z + 4834340643638 z 90 88 - 16987922682553223712 z + 190916454120834758332 z 84 94 + 14449085267441139099728 z - 77207880351878768 z 86 96 98 - 1804668152074775320464 z + 3852321210918673 z - 154053232983356 z 92 82 + 1260640615334632200 z - 98555657017945616592640 z 64 112 110 106 + 4103772097842178945087770134 z + z - 764 z - 24731676 z 108 30 + 189534 z - 98555657017945616592640 z 42 44 - 425899054167254644680713600 z + 1041241167460327090033896184 z 46 58 - 2215068826935793323892218096 z - 11447425258642058714229559688 z 56 54 + 12256766684921130261569946586 z - 11447425258642058714229559688 z 52 60 + 9325650431052182922295051900 z + 9325650431052182922295051900 z 70 68 - 425899054167254644680713600 z + 1041241167460327090033896184 z 78 32 - 2887413752634716242969152 z + 575437382959428257734820 z 38 40 - 46723781343845215744685712 z + 151416426829193945633891868 z 62 76 - 6625440880386185598730063016 z + 12492048325765535323234720 z 74 72 - 46723781343845215744685712 z + 151416426829193945633891868 z 104 + 2033297263 z )) And in Maple-input format, it is: -(1+3532765592370337614180*z^28-464787585822208377484*z^26-509*z^2+ 51871229113654234948*z^24-4875623906174241912*z^22+104563*z^4-12093862*z^6-\ 47866913166*z^102+907647792*z^8-47866913166*z^10+1867077334195*z^12-\ 55824008828637*z^14-24821396168548536*z^18+1313573134390857*z^16-\ 1086899379231686511462351310*z^50+685155663984564017469495870*z^48+ 382681118837030696*z^20+2528993239253068840356004*z^36-610699453883301885818496 *z^34-378127999661784441986091208*z^66+127465954777358446133104*z^80+ 1867077334195*z^100-4875623906174241912*z^90+51871229113654234948*z^88+ 3532765592370337614180*z^84-24821396168548536*z^94-464787585822208377484*z^86+ 1313573134390857*z^96-55824008828637*z^98+382681118837030696*z^92-\ 22914037809825102135312*z^82+685155663984564017469495870*z^64+z^112-509*z^110-\ 12093862*z^106+104563*z^108-22914037809825102135312*z^30-\ 76988065372156170708908936*z^42+182536256319322608411046104*z^44-\ 378127999661784441986091208*z^46-1839993477480789862415927996*z^58+ 1965010988184133572743408320*z^56-1839993477480789862415927996*z^54+ 1510535092407431792374370386*z^52+1510535092407431792374370386*z^60-\ 76988065372156170708908936*z^70+182536256319322608411046104*z^68-\ 610699453883301885818496*z^78+127465954777358446133104*z^32-\ 9079372417913283402741452*z^38+28329826325880253464451764*z^40-\ 1086899379231686511462351310*z^62+2528993239253068840356004*z^76-\ 9079372417913283402741452*z^74+28329826325880253464451764*z^72+907647792*z^104) /(-1+z^2)/(1+14449085267441139099728*z^28-1804668152074775320464*z^26-764*z^2+ 190916454120834758332*z^24-16987922682553223712*z^22+189534*z^4-24731676*z^6-\ 115747013148*z^102+2033297263*z^8-115747013148*z^10+4834340643638*z^12-\ 154053232983356*z^14-77207880351878768*z^18+3852321210918673*z^16-\ 6625440880386185598730063016*z^50+4103772097842178945087770134*z^48+ 1260640615334632200*z^20+12492048325765535323234720*z^36-\ 2887413752634716242969152*z^34-2215068826935793323892218096*z^66+ 575437382959428257734820*z^80+4834340643638*z^100-16987922682553223712*z^90+ 190916454120834758332*z^88+14449085267441139099728*z^84-77207880351878768*z^94-\ 1804668152074775320464*z^86+3852321210918673*z^96-154053232983356*z^98+ 1260640615334632200*z^92-98555657017945616592640*z^82+ 4103772097842178945087770134*z^64+z^112-764*z^110-24731676*z^106+189534*z^108-\ 98555657017945616592640*z^30-425899054167254644680713600*z^42+ 1041241167460327090033896184*z^44-2215068826935793323892218096*z^46-\ 11447425258642058714229559688*z^58+12256766684921130261569946586*z^56-\ 11447425258642058714229559688*z^54+9325650431052182922295051900*z^52+ 9325650431052182922295051900*z^60-425899054167254644680713600*z^70+ 1041241167460327090033896184*z^68-2887413752634716242969152*z^78+ 575437382959428257734820*z^32-46723781343845215744685712*z^38+ 151416426829193945633891868*z^40-6625440880386185598730063016*z^62+ 12492048325765535323234720*z^76-46723781343845215744685712*z^74+ 151416426829193945633891868*z^72+2033297263*z^104) The first , 40, terms are: [0, 256, 0, 110105, 0, 48341385, 0, 21257846848, 0, 9349991771901, 0, 4112613246113441, 0, 1808951737435535728, 0, 795676464428178458437, 0, 349982328896707998227701, 0, 153941507079004930314427872, 0, 67711955044129013469030649669, 0, 29783447926960630333216898415901, 0, 13100401103082745269394241444949504, 0, 5762278077643406378531453912924900957, 0, 2534567329883359829794887417993898201837, 0, 1114842335471667962689347185533714984334320, 0, 490369073374464900758298151751401586050078009, 0, 215691331833395467864070268674586060936721700709, 0, 94872929705616915260563730642807226006338820667648, 0, 41730340827416277295266178272806040459954773996140257] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3797171267144991228449 z - 488848622982985814890 z - 497 z 24 22 4 + 53423465387125300219 z - 4922264629806886176 z + 99870 z 6 102 8 10 - 11388148 z - 44772850208 z + 849124567 z - 44772850208 z 12 14 18 + 1754965740169 z - 52942994414708 z - 24184013960186654 z 16 50 + 1261044824672119 z - 1426823480692991198589613958 z 48 20 + 890654806491725870377521530 z + 379214913457042371 z 36 34 + 2968370703279675660214524 z - 701495348070282750277634 z 66 80 - 485535156228257778404374906 z + 143221418637250948707667 z 100 90 88 + 1754965740169 z - 4922264629806886176 z + 53423465387125300219 z 84 94 + 3797171267144991228449 z - 24184013960186654 z 86 96 98 - 488848622982985814890 z + 1261044824672119 z - 52942994414708 z 92 82 + 379214913457042371 z - 25179933058144027107498 z 64 112 110 106 + 890654806491725870377521530 z + z - 497 z - 11388148 z 108 30 + 99870 z - 25179933058144027107498 z 42 44 - 95829787462353117679193390 z + 231000363686032120100932186 z 46 58 - 485535156228257778404374906 z - 2443356307276180448569220038 z 56 54 + 2613174501914504430619143646 z - 2443356307276180448569220038 z 52 60 + 1997169710837615130427315346 z + 1997169710837615130427315346 z 70 68 - 95829787462353117679193390 z + 231000363686032120100932186 z 78 32 - 701495348070282750277634 z + 143221418637250948707667 z 38 40 - 10880634683239040024680881 z + 34624960349556803226655113 z 62 76 - 1426823480692991198589613958 z + 2968370703279675660214524 z 74 72 - 10880634683239040024680881 z + 34624960349556803226655113 z 104 / 28 + 849124567 z ) / (-1 - 17061089956170132718868 z / 26 2 24 + 2051165267560081602458 z + 736 z - 209409921640993200592 z 22 4 6 102 + 18029786159990165795 z - 177775 z + 22905824 z + 4552566142326 z 8 10 12 - 1879828990 z + 107677100208 z - 4552566142326 z 14 18 16 + 147552633542028 z + 77361777057930117 z - 3767511339199627 z 50 48 + 13818603696227413507285263445 z - 8047688244106056295572089101 z 20 36 - 1298151699290533969 z - 17604939000102444406696496 z 34 66 + 3879456743421010074399303 z + 8047688244106056295572089101 z 80 100 - 3879456743421010074399303 z - 147552633542028 z 90 88 + 209409921640993200592 z - 2051165267560081602458 z 84 94 - 121198420545276455463986 z + 1298151699290533969 z 86 96 + 17061089956170132718868 z - 77361777057930117 z 98 92 + 3767511339199627 z - 18029786159990165795 z 82 64 + 738790577678291551352037 z - 13818603696227413507285263445 z 112 114 110 106 108 - 736 z + z + 177775 z + 1879828990 z - 22905824 z 30 42 + 121198420545276455463986 z + 701756390830357134106381988 z 44 46 - 1814776156491402116296870704 z + 4091408934180654962877019872 z 58 56 + 31065869474889755109672946561 z - 31065869474889755109672946561 z 54 52 + 27144122004326294657877496544 z - 20721756618652544433481438544 z 60 70 - 27144122004326294657877496544 z + 1814776156491402116296870704 z 68 78 - 4091408934180654962877019872 z + 17604939000102444406696496 z 32 38 - 738790577678291551352037 z + 69222174166651957001941269 z 40 62 - 236329964916482952344500241 z + 20721756618652544433481438544 z 76 74 - 69222174166651957001941269 z + 236329964916482952344500241 z 72 104 - 701756390830357134106381988 z - 107677100208 z ) And in Maple-input format, it is: -(1+3797171267144991228449*z^28-488848622982985814890*z^26-497*z^2+ 53423465387125300219*z^24-4922264629806886176*z^22+99870*z^4-11388148*z^6-\ 44772850208*z^102+849124567*z^8-44772850208*z^10+1754965740169*z^12-\ 52942994414708*z^14-24184013960186654*z^18+1261044824672119*z^16-\ 1426823480692991198589613958*z^50+890654806491725870377521530*z^48+ 379214913457042371*z^20+2968370703279675660214524*z^36-701495348070282750277634 *z^34-485535156228257778404374906*z^66+143221418637250948707667*z^80+ 1754965740169*z^100-4922264629806886176*z^90+53423465387125300219*z^88+ 3797171267144991228449*z^84-24184013960186654*z^94-488848622982985814890*z^86+ 1261044824672119*z^96-52942994414708*z^98+379214913457042371*z^92-\ 25179933058144027107498*z^82+890654806491725870377521530*z^64+z^112-497*z^110-\ 11388148*z^106+99870*z^108-25179933058144027107498*z^30-\ 95829787462353117679193390*z^42+231000363686032120100932186*z^44-\ 485535156228257778404374906*z^46-2443356307276180448569220038*z^58+ 2613174501914504430619143646*z^56-2443356307276180448569220038*z^54+ 1997169710837615130427315346*z^52+1997169710837615130427315346*z^60-\ 95829787462353117679193390*z^70+231000363686032120100932186*z^68-\ 701495348070282750277634*z^78+143221418637250948707667*z^32-\ 10880634683239040024680881*z^38+34624960349556803226655113*z^40-\ 1426823480692991198589613958*z^62+2968370703279675660214524*z^76-\ 10880634683239040024680881*z^74+34624960349556803226655113*z^72+849124567*z^104 )/(-1-17061089956170132718868*z^28+2051165267560081602458*z^26+736*z^2-\ 209409921640993200592*z^24+18029786159990165795*z^22-177775*z^4+22905824*z^6+ 4552566142326*z^102-1879828990*z^8+107677100208*z^10-4552566142326*z^12+ 147552633542028*z^14+77361777057930117*z^18-3767511339199627*z^16+ 13818603696227413507285263445*z^50-8047688244106056295572089101*z^48-\ 1298151699290533969*z^20-17604939000102444406696496*z^36+ 3879456743421010074399303*z^34+8047688244106056295572089101*z^66-\ 3879456743421010074399303*z^80-147552633542028*z^100+209409921640993200592*z^90 -2051165267560081602458*z^88-121198420545276455463986*z^84+1298151699290533969* z^94+17061089956170132718868*z^86-77361777057930117*z^96+3767511339199627*z^98-\ 18029786159990165795*z^92+738790577678291551352037*z^82-\ 13818603696227413507285263445*z^64-736*z^112+z^114+177775*z^110+1879828990*z^ 106-22905824*z^108+121198420545276455463986*z^30+701756390830357134106381988*z^ 42-1814776156491402116296870704*z^44+4091408934180654962877019872*z^46+ 31065869474889755109672946561*z^58-31065869474889755109672946561*z^56+ 27144122004326294657877496544*z^54-20721756618652544433481438544*z^52-\ 27144122004326294657877496544*z^60+1814776156491402116296870704*z^70-\ 4091408934180654962877019872*z^68+17604939000102444406696496*z^78-\ 738790577678291551352037*z^32+69222174166651957001941269*z^38-\ 236329964916482952344500241*z^40+20721756618652544433481438544*z^62-\ 69222174166651957001941269*z^76+236329964916482952344500241*z^74-\ 701756390830357134106381988*z^72-107677100208*z^104) The first , 40, terms are: [0, 239, 0, 97999, 0, 41156715, 0, 17313357528, 0, 7284369099049, 0, 3064857857924569, 0, 1289524595718717229, 0, 542561569303728759455, 0, 228280302695362643514067, 0, 96047895258473499281477149, 0, 40411713487750144000815618717, 0, 17003043978243119177706112385705, 0, 7153953138251275412139950361150072, 0, 3009993126514297112478160649087938331, 0, 1266440867947697123762749646309719507363, 0, 532849214132753567254125782663486855908607, 0, 224193874493332762203225283332459600113294677, 0, 94328549291638438368842921541687849658531052829, 0, 39688306522976273696636647998410617576194173700255, 0, 16698673800142371419964158691238489445575787349210187] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 43602915584811990855 z - 9629162190983474546 z - 466 z 24 22 4 6 + 1749119825689098601 z - 259533337800566712 z + 83903 z - 8185108 z 8 10 12 14 + 500031236 z - 20753362100 z + 616704462885 z - 13610160973366 z 18 16 50 - 3005292198753384 z + 229168542011083 z - 5248624509245889995910 z 48 20 + 8057297257012709134219 z + 31192153230443048 z 36 34 + 2874947813544287979285 z - 1321633971979615335804 z 66 80 88 84 86 - 259533337800566712 z + 500031236 z + z + 83903 z - 466 z 82 64 30 - 8185108 z + 1749119825689098601 z - 163341093117410673468 z 42 44 - 10414661180330480631648 z + 11343807221991887250096 z 46 58 - 10414661180330480631648 z - 163341093117410673468 z 56 54 + 508620330232682690428 z - 1321633971979615335804 z 52 60 + 2874947813544287979285 z + 43602915584811990855 z 70 68 78 - 3005292198753384 z + 31192153230443048 z - 20753362100 z 32 38 + 508620330232682690428 z - 5248624509245889995910 z 40 62 76 + 8057297257012709134219 z - 9629162190983474546 z + 616704462885 z 74 72 / 2 - 13610160973366 z + 229168542011083 z ) / ((-1 + z ) (1 / 28 26 2 + 210187394268700158757 z - 43788798334706644001 z - 713 z 24 22 4 + 7474029308277047165 z - 1038505305627644728 z + 155553 z 6 8 10 12 - 17233142 z + 1165093650 z - 52871104922 z + 1706849198871 z 14 18 16 - 40764087301027 z - 10455914208313912 z + 740686096489903 z 50 48 - 31179703517555360801139 z + 48870954037463177383251 z 20 36 + 116535780068214564 z + 16597818334136427771811 z 34 66 80 - 7361743394080380386722 z - 1038505305627644728 z + 1165093650 z 88 84 86 82 64 + z + 155553 z - 713 z - 17233142 z + 7474029308277047165 z 30 42 - 830816937962991315214 z - 63973419955538899582288 z 44 46 + 69977193195318422768376 z - 63973419955538899582288 z 58 56 - 830816937962991315214 z + 2715373938548547564622 z 54 52 - 7361743394080380386722 z + 16597818334136427771811 z 60 70 + 210187394268700158757 z - 10455914208313912 z 68 78 32 + 116535780068214564 z - 52871104922 z + 2715373938548547564622 z 38 40 - 31179703517555360801139 z + 48870954037463177383251 z 62 76 74 - 43788798334706644001 z + 1706849198871 z - 40764087301027 z 72 + 740686096489903 z )) And in Maple-input format, it is: -(1+43602915584811990855*z^28-9629162190983474546*z^26-466*z^2+ 1749119825689098601*z^24-259533337800566712*z^22+83903*z^4-8185108*z^6+ 500031236*z^8-20753362100*z^10+616704462885*z^12-13610160973366*z^14-\ 3005292198753384*z^18+229168542011083*z^16-5248624509245889995910*z^50+ 8057297257012709134219*z^48+31192153230443048*z^20+2874947813544287979285*z^36-\ 1321633971979615335804*z^34-259533337800566712*z^66+500031236*z^80+z^88+83903*z ^84-466*z^86-8185108*z^82+1749119825689098601*z^64-163341093117410673468*z^30-\ 10414661180330480631648*z^42+11343807221991887250096*z^44-\ 10414661180330480631648*z^46-163341093117410673468*z^58+508620330232682690428*z ^56-1321633971979615335804*z^54+2874947813544287979285*z^52+ 43602915584811990855*z^60-3005292198753384*z^70+31192153230443048*z^68-\ 20753362100*z^78+508620330232682690428*z^32-5248624509245889995910*z^38+ 8057297257012709134219*z^40-9629162190983474546*z^62+616704462885*z^76-\ 13610160973366*z^74+229168542011083*z^72)/(-1+z^2)/(1+210187394268700158757*z^ 28-43788798334706644001*z^26-713*z^2+7474029308277047165*z^24-\ 1038505305627644728*z^22+155553*z^4-17233142*z^6+1165093650*z^8-52871104922*z^ 10+1706849198871*z^12-40764087301027*z^14-10455914208313912*z^18+ 740686096489903*z^16-31179703517555360801139*z^50+48870954037463177383251*z^48+ 116535780068214564*z^20+16597818334136427771811*z^36-7361743394080380386722*z^ 34-1038505305627644728*z^66+1165093650*z^80+z^88+155553*z^84-713*z^86-17233142* z^82+7474029308277047165*z^64-830816937962991315214*z^30-\ 63973419955538899582288*z^42+69977193195318422768376*z^44-\ 63973419955538899582288*z^46-830816937962991315214*z^58+2715373938548547564622* z^56-7361743394080380386722*z^54+16597818334136427771811*z^52+ 210187394268700158757*z^60-10455914208313912*z^70+116535780068214564*z^68-\ 52871104922*z^78+2715373938548547564622*z^32-31179703517555360801139*z^38+ 48870954037463177383251*z^40-43788798334706644001*z^62+1706849198871*z^76-\ 40764087301027*z^74+740686096489903*z^72) The first , 40, terms are: [0, 248, 0, 104709, 0, 45211845, 0, 19548901540, 0, 8453660185601, 0, 3655717412903169, 0, 1580887807511037276, 0, 683643284065847948085, 0, 295636506470572167606005, 0, 127845831571907161308111616, 0, 55285989033378771888181279489, 0, 23908019103281194633139160882113, 0, 10338846920148312657647554589730416, 0, 4470958266201283700587627598340288725, 0, 1933433000072712673116735163686828100245, 0, 836098872590529673893713112199147028134380, 0, 361564804532075987533891209903438687530451265, 0, 156356038935052478649134290134517830054382295361, 0, 67615018400638684925186684595803459147770530688020, 0, 29239617122928962287230692372423170754950663258117093] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 180084323607528 z - 157786531511616 z - 380 z 24 22 4 6 + 106008135667483 z - 54413745672404 z + 48126 z - 2857316 z 8 10 12 14 + 93320843 z - 1854674160 z + 24057767084 z - 213949164496 z 18 16 50 48 - 6213835839052 z + 1351043218009 z - 2857316 z + 93320843 z 20 36 34 + 21204428762466 z + 21204428762466 z - 54413745672404 z 30 42 44 - 157786531511616 z - 213949164496 z + 24057767084 z 46 56 54 52 32 - 1854674160 z + z - 380 z + 48126 z + 106008135667483 z 38 40 / 28 - 6213835839052 z + 1351043218009 z ) / (-1 - 2105754172426432 z / 26 2 24 + 1613156751548323 z + 611 z - 944974820387049 z 22 4 6 8 + 421675185075772 z - 98612 z + 6919148 z - 261689365 z 10 12 14 18 + 6004415799 z - 90030588304 z + 926901823408 z + 36107889707615 z 16 50 48 20 - 6780411936101 z + 261689365 z - 6004415799 z - 142454716023460 z 36 34 30 - 421675185075772 z + 944974820387049 z + 2105754172426432 z 42 44 46 58 56 + 6780411936101 z - 926901823408 z + 90030588304 z + z - 611 z 54 52 32 38 + 98612 z - 6919148 z - 1613156751548323 z + 142454716023460 z 40 - 36107889707615 z ) And in Maple-input format, it is: -(1+180084323607528*z^28-157786531511616*z^26-380*z^2+106008135667483*z^24-\ 54413745672404*z^22+48126*z^4-2857316*z^6+93320843*z^8-1854674160*z^10+ 24057767084*z^12-213949164496*z^14-6213835839052*z^18+1351043218009*z^16-\ 2857316*z^50+93320843*z^48+21204428762466*z^20+21204428762466*z^36-\ 54413745672404*z^34-157786531511616*z^30-213949164496*z^42+24057767084*z^44-\ 1854674160*z^46+z^56-380*z^54+48126*z^52+106008135667483*z^32-6213835839052*z^ 38+1351043218009*z^40)/(-1-2105754172426432*z^28+1613156751548323*z^26+611*z^2-\ 944974820387049*z^24+421675185075772*z^22-98612*z^4+6919148*z^6-261689365*z^8+ 6004415799*z^10-90030588304*z^12+926901823408*z^14+36107889707615*z^18-\ 6780411936101*z^16+261689365*z^50-6004415799*z^48-142454716023460*z^20-\ 421675185075772*z^36+944974820387049*z^34+2105754172426432*z^30+6780411936101*z ^42-926901823408*z^44+90030588304*z^46+z^58-611*z^56+98612*z^54-6919148*z^52-\ 1613156751548323*z^32+142454716023460*z^38-36107889707615*z^40) The first , 40, terms are: [0, 231, 0, 90655, 0, 36672665, 0, 14897282121, 0, 6056829395215, 0, 2463013170494007, 0, 1001628886428160913, 0, 407334478427711257585, 0, 165651913691556459390231, 0, 67366185729066435619978927, 0, 27396022100180283911912794729, 0, 11141228166581725472746821064057, 0, 4530839014364541545711684148576127, 0, 1842570845303249168514253608164355335, 0, 749324200261553145207946248761245926049, 0, 304730077850673228666012019955237932202849, 0, 123925558944776395617571671558616781915458247, 0, 50397204857974586636773916379791625618478887615, 0, 20495193074984686549508125974574288561920067258809, 0, 8334845957523483990653672929861607157426175868490793] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 149618674235431996 z - 65863363876553368 z - 422 z 24 22 4 6 + 22774859907914796 z - 6152090391327092 z + 64505 z - 5097524 z 8 10 12 14 + 240790820 z - 7379667048 z + 154962133620 z - 2316847616348 z 18 16 50 - 207493128014584 z + 25360756165020 z - 207493128014584 z 48 20 36 + 1288896515629484 z + 1288896515629484 z + 379968542590705854 z 34 66 64 30 - 426734619936718020 z - 422 z + 64505 z - 268083907709128444 z 42 44 46 - 65863363876553368 z + 22774859907914796 z - 6152090391327092 z 58 56 54 - 7379667048 z + 154962133620 z - 2316847616348 z 52 60 68 32 + 25360756165020 z + 240790820 z + z + 379968542590705854 z 38 40 62 / - 268083907709128444 z + 149618674235431996 z - 5097524 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 857099037139208824 z - 358465215679096456 z - 654 z 24 22 4 6 + 116478880622920688 z - 29303443087941416 z + 123853 z - 11345000 z 8 10 12 14 + 604005240 z - 20589830008 z + 477501317664 z - 7849096456952 z 18 16 50 - 840745128248504 z + 94127788123368 z - 840745128248504 z 48 20 36 + 5677634102204048 z + 5677634102204048 z + 2315836386659218618 z 34 66 64 30 - 2621830137136479604 z - 654 z + 123853 z - 1595654298700215672 z 42 44 46 - 358465215679096456 z + 116478880622920688 z - 29303443087941416 z 58 56 54 - 20589830008 z + 477501317664 z - 7849096456952 z 52 60 68 32 + 94127788123368 z + 604005240 z + z + 2315836386659218618 z 38 40 62 - 1595654298700215672 z + 857099037139208824 z - 11345000 z )) And in Maple-input format, it is: -(1+149618674235431996*z^28-65863363876553368*z^26-422*z^2+22774859907914796*z^ 24-6152090391327092*z^22+64505*z^4-5097524*z^6+240790820*z^8-7379667048*z^10+ 154962133620*z^12-2316847616348*z^14-207493128014584*z^18+25360756165020*z^16-\ 207493128014584*z^50+1288896515629484*z^48+1288896515629484*z^20+ 379968542590705854*z^36-426734619936718020*z^34-422*z^66+64505*z^64-\ 268083907709128444*z^30-65863363876553368*z^42+22774859907914796*z^44-\ 6152090391327092*z^46-7379667048*z^58+154962133620*z^56-2316847616348*z^54+ 25360756165020*z^52+240790820*z^60+z^68+379968542590705854*z^32-\ 268083907709128444*z^38+149618674235431996*z^40-5097524*z^62)/(-1+z^2)/(1+ 857099037139208824*z^28-358465215679096456*z^26-654*z^2+116478880622920688*z^24 -29303443087941416*z^22+123853*z^4-11345000*z^6+604005240*z^8-20589830008*z^10+ 477501317664*z^12-7849096456952*z^14-840745128248504*z^18+94127788123368*z^16-\ 840745128248504*z^50+5677634102204048*z^48+5677634102204048*z^20+ 2315836386659218618*z^36-2621830137136479604*z^34-654*z^66+123853*z^64-\ 1595654298700215672*z^30-358465215679096456*z^42+116478880622920688*z^44-\ 29303443087941416*z^46-20589830008*z^58+477501317664*z^56-7849096456952*z^54+ 94127788123368*z^52+604005240*z^60+z^68+2315836386659218618*z^32-\ 1595654298700215672*z^38+857099037139208824*z^40-11345000*z^62) The first , 40, terms are: [0, 233, 0, 92613, 0, 38022713, 0, 15671593553, 0, 6463402294893, 0, 2665987913531345, 0, 1099673950174465529, 0, 453598153040894925705, 0, 187102203567148254626881, 0, 77176766445280049671380061, 0, 31834223715114659879670363937, 0, 13131125468456654769438618864745, 0, 5416386393810148422695239058705237, 0, 2234175710315208869602835343476848121, 0, 921562965001845477361636612351937562225, 0, 380130485952977287481826958263958339192721, 0, 156797952867673809866162312610416908125639897, 0, 64676733206128070047575877194059342404084651317, 0, 26678153264838783309265026406389979794910230572041, 0, 11004326074323061735825396714628349595434691411336577] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 3230035534235743 z - 1815649155367840 z - 360 z 24 22 4 6 + 807032114690020 z - 282333538863884 z + 42524 z - 2490684 z 8 10 12 14 + 85922332 z - 1920685447 z + 29560470252 z - 326410408258 z 18 16 50 - 16381751090585 z + 2662299002976 z - 326410408258 z 48 20 36 + 2662299002976 z + 77240500902340 z + 3230035534235743 z 34 64 30 42 - 4558111470216828 z + z - 4558111470216828 z - 282333538863884 z 44 46 58 56 + 77240500902340 z - 16381751090585 z - 2490684 z + 85922332 z 54 52 60 32 - 1920685447 z + 29560470252 z + 42524 z + 5111608310647172 z 38 40 62 / - 1815649155367840 z + 807032114690020 z - 360 z ) / (-1 / 28 26 2 - 29777883141712992 z + 14876260912867703 z + 601 z 24 22 4 6 - 5869959240923281 z + 1821291967416830 z - 87698 z + 5964376 z 8 10 12 14 - 234676011 z + 5950093527 z - 103663241504 z + 1294535630472 z 18 16 50 + 82949560442385 z - 11933624217593 z + 11933624217593 z 48 20 36 - 82949560442385 z - 441581141990786 z - 47223076455598976 z 34 66 64 30 + 59441589491085681 z + z - 601 z + 47223076455598976 z 42 44 46 + 5869959240923281 z - 1821291967416830 z + 441581141990786 z 58 56 54 52 + 234676011 z - 5950093527 z + 103663241504 z - 1294535630472 z 60 32 38 - 5964376 z - 59441589491085681 z + 29777883141712992 z 40 62 - 14876260912867703 z + 87698 z ) And in Maple-input format, it is: -(1+3230035534235743*z^28-1815649155367840*z^26-360*z^2+807032114690020*z^24-\ 282333538863884*z^22+42524*z^4-2490684*z^6+85922332*z^8-1920685447*z^10+ 29560470252*z^12-326410408258*z^14-16381751090585*z^18+2662299002976*z^16-\ 326410408258*z^50+2662299002976*z^48+77240500902340*z^20+3230035534235743*z^36-\ 4558111470216828*z^34+z^64-4558111470216828*z^30-282333538863884*z^42+ 77240500902340*z^44-16381751090585*z^46-2490684*z^58+85922332*z^56-1920685447*z ^54+29560470252*z^52+42524*z^60+5111608310647172*z^32-1815649155367840*z^38+ 807032114690020*z^40-360*z^62)/(-1-29777883141712992*z^28+14876260912867703*z^ 26+601*z^2-5869959240923281*z^24+1821291967416830*z^22-87698*z^4+5964376*z^6-\ 234676011*z^8+5950093527*z^10-103663241504*z^12+1294535630472*z^14+ 82949560442385*z^18-11933624217593*z^16+11933624217593*z^50-82949560442385*z^48 -441581141990786*z^20-47223076455598976*z^36+59441589491085681*z^34+z^66-601*z^ 64+47223076455598976*z^30+5869959240923281*z^42-1821291967416830*z^44+ 441581141990786*z^46+234676011*z^58-5950093527*z^56+103663241504*z^54-\ 1294535630472*z^52-5964376*z^60-59441589491085681*z^32+29777883141712992*z^38-\ 14876260912867703*z^40+87698*z^62) The first , 40, terms are: [0, 241, 0, 99667, 0, 42238341, 0, 17933307312, 0, 7615623617715, 0, 3234170372719573, 0, 1373479109588848123, 0, 583285890209497906051, 0, 247708505547720446471651, 0, 105196277627656684059769915, 0, 44674512976152598376353329317, 0, 18972269315497085102957430127347, 0, 8057099652869108082131851446158352, 0, 3421670530660297532553369852286506933, 0, 1453107163224469989297269794166736361987, 0, 617102204579297101717299263904145101505937, 0, 262069543481981657195249046406724765276507505, 0, 111295090361371479021195639724346762271167179217, 0, 47264542739196533825537849890421035935187688368817, 0, 20072197192992218211075120549590431074269392884306307] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 137653999210871688 z - 53090147647461544 z - 390 z 24 22 4 6 + 16438782318970988 z - 4064538602101396 z + 53474 z - 3775194 z 8 10 12 14 + 161827873 z - 4607351520 z + 92121264696 z - 1343476166664 z 18 16 50 - 122754397960540 z + 14684785031980 z - 4064538602101396 z 48 20 36 + 16438782318970988 z + 796862409253304 z + 738887977039871052 z 34 66 64 - 665540140863218032 z - 3775194 z + 161827873 z 30 42 44 - 287712114946819760 z - 287712114946819760 z + 137653999210871688 z 46 58 56 - 53090147647461544 z - 1343476166664 z + 14684785031980 z 54 52 60 70 - 122754397960540 z + 796862409253304 z + 92121264696 z - 390 z 68 32 38 + 53474 z + 486182635466633670 z - 665540140863218032 z 40 62 72 / 2 + 486182635466633670 z - 4607351520 z + z ) / ((-1 + z ) (1 / 28 26 2 + 784443521982407896 z - 285502983210594936 z - 650 z 24 22 4 6 + 82685202311755832 z - 18985124662786388 z + 107798 z - 8610478 z 8 10 12 14 + 408850141 z - 12794781904 z + 280345760168 z - 4473445380536 z 18 16 50 - 486985754900796 z + 53424283910704 z - 18985124662786388 z 48 20 36 + 82685202311755832 z + 3437289095898376 z + 4707658787433026756 z 34 66 64 - 4209361308182925816 z - 8610478 z + 408850141 z 30 42 - 1719101949475940672 z - 1719101949475940672 z 44 46 58 + 784443521982407896 z - 285502983210594936 z - 4473445380536 z 56 54 52 + 53424283910704 z - 486985754900796 z + 3437289095898376 z 60 70 68 32 + 280345760168 z - 650 z + 107798 z + 3008993875183156154 z 38 40 62 - 4209361308182925816 z + 3008993875183156154 z - 12794781904 z 72 + z )) And in Maple-input format, it is: -(1+137653999210871688*z^28-53090147647461544*z^26-390*z^2+16438782318970988*z^ 24-4064538602101396*z^22+53474*z^4-3775194*z^6+161827873*z^8-4607351520*z^10+ 92121264696*z^12-1343476166664*z^14-122754397960540*z^18+14684785031980*z^16-\ 4064538602101396*z^50+16438782318970988*z^48+796862409253304*z^20+ 738887977039871052*z^36-665540140863218032*z^34-3775194*z^66+161827873*z^64-\ 287712114946819760*z^30-287712114946819760*z^42+137653999210871688*z^44-\ 53090147647461544*z^46-1343476166664*z^58+14684785031980*z^56-122754397960540*z ^54+796862409253304*z^52+92121264696*z^60-390*z^70+53474*z^68+ 486182635466633670*z^32-665540140863218032*z^38+486182635466633670*z^40-\ 4607351520*z^62+z^72)/(-1+z^2)/(1+784443521982407896*z^28-285502983210594936*z^ 26-650*z^2+82685202311755832*z^24-18985124662786388*z^22+107798*z^4-8610478*z^6 +408850141*z^8-12794781904*z^10+280345760168*z^12-4473445380536*z^14-\ 486985754900796*z^18+53424283910704*z^16-18985124662786388*z^50+ 82685202311755832*z^48+3437289095898376*z^20+4707658787433026756*z^36-\ 4209361308182925816*z^34-8610478*z^66+408850141*z^64-1719101949475940672*z^30-\ 1719101949475940672*z^42+784443521982407896*z^44-285502983210594936*z^46-\ 4473445380536*z^58+53424283910704*z^56-486985754900796*z^54+3437289095898376*z^ 52+280345760168*z^60-650*z^70+107798*z^68+3008993875183156154*z^32-\ 4209361308182925816*z^38+3008993875183156154*z^40-12794781904*z^62+z^72) The first , 40, terms are: [0, 261, 0, 114937, 0, 51462141, 0, 23057003305, 0, 10330834431965, 0, 4628811927097857, 0, 2073976619861790901, 0, 929262050682372470525, 0, 416363404159860630873865, 0, 186555002850557024139665589, 0, 83587483304869567689204527041, 0, 37452050379974593440391192089093, 0, 16780695173613498134429377414260705, 0, 7518726682596402328602167524477493693, 0, 3368826520160034185838238276612595602969, 0, 1509430067354779582594641452261220636715817, 0, 676312393826211090823293106429804336728029389, 0, 303027257728152997826607964803770745452639781393, 0, 135773822518237700050571238814547062428350135640117, 0, 60834563264772717138122099080004676053516106742697937] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 7}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 35951917408991118437 z - 7815527282958511721 z - 449 z 24 22 4 6 + 1400417455384762676 z - 205573362638442264 z + 77100 z - 7200704 z 8 10 12 14 + 423922509 z - 17081764980 z + 496505856525 z - 10794710175942 z 18 16 50 - 2357606023475120 z + 180238430257541 z - 4679913560743065244670 z 48 20 + 7253059659688889830972 z + 24533582990797801 z 36 34 + 2531339360099570367768 z - 1146376146112498895056 z 66 80 88 84 86 - 205573362638442264 z + 423922509 z + z + 77100 z - 449 z 82 64 30 - 7200704 z + 1400417455384762676 z - 136980046294158573166 z 42 44 - 9430963605108757267442 z + 10293086885508562294556 z 46 58 - 9430963605108757267442 z - 136980046294158573166 z 56 54 + 433937315685011927658 z - 1146376146112498895056 z 52 60 + 2531339360099570367768 z + 35951917408991118437 z 70 68 78 - 2357606023475120 z + 24533582990797801 z - 17081764980 z 32 38 + 433937315685011927658 z - 4679913560743065244670 z 40 62 76 + 7253059659688889830972 z - 7815527282958511721 z + 496505856525 z 74 72 / - 10794710175942 z + 180238430257541 z ) / (-1 / 28 26 2 - 207724295098193154080 z + 41129074213388346529 z + 707 z 24 22 4 - 6719224109704893628 z + 900479014661747089 z - 147218 z 6 8 10 12 + 15578891 z - 1013016846 z + 44613146393 z - 1411627956196 z 14 18 16 + 33381024627263 z + 8648869467629271 z - 606636207887032 z 50 48 + 73503990102038676027575 z - 104634351842044544665627 z 20 36 - 98262851134376094 z - 21319337728918923821919 z 34 66 80 + 8789462293606897331051 z + 6719224109704893628 z - 44613146393 z 90 88 84 86 82 + z - 707 z - 15578891 z + 147218 z + 1013016846 z 64 30 - 41129074213388346529 z + 869552524058573813806 z 42 44 + 104634351842044544665627 z - 124831544358313674195295 z 46 58 + 124831544358313674195295 z + 3027456165351745550275 z 56 54 - 8789462293606897331051 z + 21319337728918923821919 z 52 60 - 43260646969463249209759 z - 869552524058573813806 z 70 68 78 + 98262851134376094 z - 900479014661747089 z + 1411627956196 z 32 38 - 3027456165351745550275 z + 43260646969463249209759 z 40 62 - 73503990102038676027575 z + 207724295098193154080 z 76 74 72 - 33381024627263 z + 606636207887032 z - 8648869467629271 z ) And in Maple-input format, it is: -(1+35951917408991118437*z^28-7815527282958511721*z^26-449*z^2+ 1400417455384762676*z^24-205573362638442264*z^22+77100*z^4-7200704*z^6+ 423922509*z^8-17081764980*z^10+496505856525*z^12-10794710175942*z^14-\ 2357606023475120*z^18+180238430257541*z^16-4679913560743065244670*z^50+ 7253059659688889830972*z^48+24533582990797801*z^20+2531339360099570367768*z^36-\ 1146376146112498895056*z^34-205573362638442264*z^66+423922509*z^80+z^88+77100*z ^84-449*z^86-7200704*z^82+1400417455384762676*z^64-136980046294158573166*z^30-\ 9430963605108757267442*z^42+10293086885508562294556*z^44-9430963605108757267442 *z^46-136980046294158573166*z^58+433937315685011927658*z^56-\ 1146376146112498895056*z^54+2531339360099570367768*z^52+35951917408991118437*z^ 60-2357606023475120*z^70+24533582990797801*z^68-17081764980*z^78+ 433937315685011927658*z^32-4679913560743065244670*z^38+7253059659688889830972*z ^40-7815527282958511721*z^62+496505856525*z^76-10794710175942*z^74+ 180238430257541*z^72)/(-1-207724295098193154080*z^28+41129074213388346529*z^26+ 707*z^2-6719224109704893628*z^24+900479014661747089*z^22-147218*z^4+15578891*z^ 6-1013016846*z^8+44613146393*z^10-1411627956196*z^12+33381024627263*z^14+ 8648869467629271*z^18-606636207887032*z^16+73503990102038676027575*z^50-\ 104634351842044544665627*z^48-98262851134376094*z^20-21319337728918923821919*z^ 36+8789462293606897331051*z^34+6719224109704893628*z^66-44613146393*z^80+z^90-\ 707*z^88-15578891*z^84+147218*z^86+1013016846*z^82-41129074213388346529*z^64+ 869552524058573813806*z^30+104634351842044544665627*z^42-\ 124831544358313674195295*z^44+124831544358313674195295*z^46+ 3027456165351745550275*z^58-8789462293606897331051*z^56+21319337728918923821919 *z^54-43260646969463249209759*z^52-869552524058573813806*z^60+98262851134376094 *z^70-900479014661747089*z^68+1411627956196*z^78-3027456165351745550275*z^32+ 43260646969463249209759*z^38-73503990102038676027575*z^40+207724295098193154080 *z^62-33381024627263*z^76+606636207887032*z^74-8648869467629271*z^72) The first , 40, terms are: [0, 258, 0, 112288, 0, 49783559, 0, 22096420970, 0, 9808629184681, 0, 4354128004527151, 0, 1932835785035167592, 0, 858003079930294300379, 0, 380875250959205449129597, 0, 169073935754795378313013122, 0, 75053434694115354664953546675, 0, 33316892014575398417385311102472, 0, 14789666829408158137416777729790770, 0, 6565265596493921123800818138166970005, 0, 2914380212190723650725236251992391800381, 0, 1293719484211746366438782436727684288815970, 0, 574293668625686732929935354950383931457952360, 0, 254934104223144729121698806409899194989540841051, 0, 113167532652039973514834279189163456987542474057170, 0, 50236081537920156013083742673648930338920207802068229] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 1936226146502 z + 3894416000306 z + 411 z 24 22 4 6 - 5497049202502 z + 5497049202502 z - 57065 z + 3535455 z 8 10 12 14 - 113798363 z + 2064872037 z - 22411145503 z + 151682432741 z 18 16 20 + 1936226146502 z - 662677306010 z - 3894416000306 z 36 34 30 42 - 2064872037 z + 22411145503 z + 662677306010 z + 57065 z 44 46 32 38 40 / - 411 z + z - 151682432741 z + 113798363 z - 3535455 z ) / (1 / 28 26 2 24 + 36295944693752 z - 61538424940008 z - 708 z + 73286153307532 z 22 4 6 8 - 61538424940008 z + 123972 z - 8988404 z + 336311546 z 10 12 14 18 - 7202564772 z + 93837953604 z - 773772088020 z - 14867398176232 z 16 48 20 36 + 4153621638335 z + z + 36295944693752 z + 93837953604 z 34 30 42 44 - 773772088020 z - 14867398176232 z - 8988404 z + 123972 z 46 32 38 40 - 708 z + 4153621638335 z - 7202564772 z + 336311546 z ) And in Maple-input format, it is: -(-1-1936226146502*z^28+3894416000306*z^26+411*z^2-5497049202502*z^24+ 5497049202502*z^22-57065*z^4+3535455*z^6-113798363*z^8+2064872037*z^10-\ 22411145503*z^12+151682432741*z^14+1936226146502*z^18-662677306010*z^16-\ 3894416000306*z^20-2064872037*z^36+22411145503*z^34+662677306010*z^30+57065*z^ 42-411*z^44+z^46-151682432741*z^32+113798363*z^38-3535455*z^40)/(1+ 36295944693752*z^28-61538424940008*z^26-708*z^2+73286153307532*z^24-\ 61538424940008*z^22+123972*z^4-8988404*z^6+336311546*z^8-7202564772*z^10+ 93837953604*z^12-773772088020*z^14-14867398176232*z^18+4153621638335*z^16+z^48+ 36295944693752*z^20+93837953604*z^36-773772088020*z^34-14867398176232*z^30-\ 8988404*z^42+123972*z^44-708*z^46+4153621638335*z^32-7202564772*z^38+336311546* z^40) The first , 40, terms are: [0, 297, 0, 143369, 0, 70138517, 0, 34331371173, 0, 16805310217609, 0, 8226315298655593, 0, 4026844530767429265, 0, 1971171998721614967665, 0, 964904232385924143332201, 0, 472328235605321986119575049, 0, 231208399147875154432207729285, 0, 113178336261789567553059241643701, 0, 55401688901424243199402066123514505, 0, 27119563996185751409223829832899746857, 0, 13275240627720934783012759304679391549217, 0, 6498335067221354341399239374686447719026913, 0, 3180986306019421295386222302749551996216217641, 0, 1557117903957281633248601860103790966473181084553, 0, 762221504140468512763571990268834556477102799574645, 0, 373113442403844459998570617318845417853903028106767045] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 43469575366425922496 z - 9607102285285913248 z - 468 z 24 22 4 6 + 1746265776804145573 z - 259241665141838836 z + 84328 z - 8213304 z 8 10 12 14 + 500859191 z - 20762224376 z + 616570457664 z - 13603755483816 z 18 16 50 - 3003441228205912 z + 229043201986915 z - 5213462249580349179952 z 48 20 + 8000021344831416794998 z + 31167536858138472 z 36 34 + 2857280516861717663792 z - 1314407085890881448312 z 66 80 88 84 86 - 259241665141838836 z + 500859191 z + z + 84328 z - 468 z 82 64 30 - 8213304 z + 1746265776804145573 z - 162707471376914089472 z 42 44 - 10338018389434807315392 z + 11259370368552857988608 z 46 58 - 10338018389434807315392 z - 162707471376914089472 z 56 54 + 506230272995074480714 z - 1314407085890881448312 z 52 60 + 2857280516861717663792 z + 43469575366425922496 z 70 68 78 - 3003441228205912 z + 31167536858138472 z - 20762224376 z 32 38 + 506230272995074480714 z - 5213462249580349179952 z 40 62 76 + 8000021344831416794998 z - 9607102285285913248 z + 616570457664 z 74 72 / 2 - 13603755483816 z + 229043201986915 z ) / ((-1 + z ) (1 / 28 26 2 + 211622120843801578672 z - 44004676513752478872 z - 734 z 24 22 4 + 7497252599518978965 z - 1040067940332812494 z + 160354 z 6 8 10 12 - 17648910 z + 1184111075 z - 53384374144 z + 1715133619628 z 14 18 16 - 40835073489496 z - 10453487099856350 z + 740792835081615 z 50 48 - 31629382257025010919828 z + 49614717850900747443070 z 20 36 + 116572265565522458 z + 16819192119072998305540 z 34 66 80 - 7450040179419368159956 z - 1040067940332812494 z + 1184111075 z 88 84 86 82 64 + z + 160354 z - 734 z - 17648910 z + 7497252599518978965 z 30 42 - 838032944420771839016 z - 64978194165396365034600 z 44 46 + 71087736403144230615624 z - 64978194165396365034600 z 58 56 - 838032944420771839016 z + 2743689929677781898362 z 54 52 - 7450040179419368159956 z + 16819192119072998305540 z 60 70 + 211622120843801578672 z - 10453487099856350 z 68 78 32 + 116572265565522458 z - 53384374144 z + 2743689929677781898362 z 38 40 - 31629382257025010919828 z + 49614717850900747443070 z 62 76 74 - 44004676513752478872 z + 1715133619628 z - 40835073489496 z 72 + 740792835081615 z )) And in Maple-input format, it is: -(1+43469575366425922496*z^28-9607102285285913248*z^26-468*z^2+ 1746265776804145573*z^24-259241665141838836*z^22+84328*z^4-8213304*z^6+ 500859191*z^8-20762224376*z^10+616570457664*z^12-13603755483816*z^14-\ 3003441228205912*z^18+229043201986915*z^16-5213462249580349179952*z^50+ 8000021344831416794998*z^48+31167536858138472*z^20+2857280516861717663792*z^36-\ 1314407085890881448312*z^34-259241665141838836*z^66+500859191*z^80+z^88+84328*z ^84-468*z^86-8213304*z^82+1746265776804145573*z^64-162707471376914089472*z^30-\ 10338018389434807315392*z^42+11259370368552857988608*z^44-\ 10338018389434807315392*z^46-162707471376914089472*z^58+506230272995074480714*z ^56-1314407085890881448312*z^54+2857280516861717663792*z^52+ 43469575366425922496*z^60-3003441228205912*z^70+31167536858138472*z^68-\ 20762224376*z^78+506230272995074480714*z^32-5213462249580349179952*z^38+ 8000021344831416794998*z^40-9607102285285913248*z^62+616570457664*z^76-\ 13603755483816*z^74+229043201986915*z^72)/(-1+z^2)/(1+211622120843801578672*z^ 28-44004676513752478872*z^26-734*z^2+7497252599518978965*z^24-\ 1040067940332812494*z^22+160354*z^4-17648910*z^6+1184111075*z^8-53384374144*z^ 10+1715133619628*z^12-40835073489496*z^14-10453487099856350*z^18+ 740792835081615*z^16-31629382257025010919828*z^50+49614717850900747443070*z^48+ 116572265565522458*z^20+16819192119072998305540*z^36-7450040179419368159956*z^ 34-1040067940332812494*z^66+1184111075*z^80+z^88+160354*z^84-734*z^86-17648910* z^82+7497252599518978965*z^64-838032944420771839016*z^30-\ 64978194165396365034600*z^42+71087736403144230615624*z^44-\ 64978194165396365034600*z^46-838032944420771839016*z^58+2743689929677781898362* z^56-7450040179419368159956*z^54+16819192119072998305540*z^52+ 211622120843801578672*z^60-10453487099856350*z^70+116572265565522458*z^68-\ 53384374144*z^78+2743689929677781898362*z^32-31629382257025010919828*z^38+ 49614717850900747443070*z^40-44004676513752478872*z^62+1715133619628*z^76-\ 40835073489496*z^74+740792835081615*z^72) The first , 40, terms are: [0, 267, 0, 119485, 0, 54406939, 0, 24795673179, 0, 11301391050821, 0, 5151010107587219, 0, 2347759422591627457, 0, 1070076669552223149185, 0, 487726347869858750440739, 0, 222299017398363259822443349, 0, 101320860387024227405579316699, 0, 46180666347414109092378723982427, 0, 21048517908612861828213489333041069, 0, 9593627402843113814653434419616332667, 0, 4372644532232159923583463680242821655809, 0, 1992991743623194018317442673167425532614913, 0, 908378456302897233224747794115372457733931803, 0, 414026511908745303422337310348853767371192956749, 0, 188707637630458695584536902425290894610228195748315, 0, 86010367635387965098203689273926369306826416609529819] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 4}, {3, 7}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 861115 z + 15015440 z + 304 z - 140759747 z 22 4 6 8 10 + 739897004 z - 24769 z + 861115 z - 15015440 z + 140759747 z 12 14 18 16 - 739897004 z + 2222784320 z + 3836169492 z - 3836169492 z 20 34 30 32 / 32 - 2222784320 z + z + 24769 z - 304 z ) / (56252 z / 10 22 30 14 - 609054952 z - 16024029280 z - 2418398 z - 16024029280 z 24 36 34 4 2 8 + 4078019107 z + z - 536 z + 56252 z - 536 z + 1 + 52066788 z 20 16 12 6 + 36897050048 z + 36897050048 z + 4078019107 z - 2418398 z 18 28 26 - 48864622908 z + 52066788 z - 609054952 z ) And in Maple-input format, it is: -(-1-861115*z^28+15015440*z^26+304*z^2-140759747*z^24+739897004*z^22-24769*z^4+ 861115*z^6-15015440*z^8+140759747*z^10-739897004*z^12+2222784320*z^14+ 3836169492*z^18-3836169492*z^16-2222784320*z^20+z^34+24769*z^30-304*z^32)/( 56252*z^32-609054952*z^10-16024029280*z^22-2418398*z^30-16024029280*z^14+ 4078019107*z^24+z^36-536*z^34+56252*z^4-536*z^2+1+52066788*z^8+36897050048*z^20 +36897050048*z^16+4078019107*z^12-2418398*z^6-48864622908*z^18+52066788*z^28-\ 609054952*z^26) The first , 40, terms are: [0, 232, 0, 92869, 0, 38284603, 0, 15820497208, 0, 6539184019783, 0, 2702958005077255, 0, 1117265843764492408, 0, 461821268027053790635, 0, 190893596251017983627125, 0, 78905775578270610804633640, 0, 32615664155908662176504879233, 0, 13481668998850847090837949451777, 0, 5572641358096804710177571180124584, 0, 2303448609268676384316544048369549717, 0, 952129368209517055101185286705680228683, 0, 393562213699614507210151143060423842978296, 0, 162678750623368645391609636034399494797966087, 0, 67243182864549726581870935174733112452390336071, 0, 27794937104131805344582205314319097838123571015608, 0, 11489023804522078770655664535788059002654197156752027] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 5}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 1122558 z + 18531342 z + 335 z - 158338418 z 22 4 6 8 10 + 742615354 z - 31714 z + 1122558 z - 18531342 z + 158338418 z 12 14 18 16 - 742615354 z + 2018666982 z + 3299278128 z - 3299278128 z 20 34 30 32 / 20 - 2018666982 z + z + 31714 z - 335 z ) / (32640355642 z / 22 26 10 18 - 15529006040 z - 726481768 z - 726481768 z + 1 - 41739830460 z 16 14 2 4 8 + 32640355642 z - 15529006040 z - 570 z + 72285 z + 66395792 z 24 6 28 30 34 36 + 4414552536 z - 3197992 z + 66395792 z - 3197992 z - 570 z + z 32 12 + 72285 z + 4414552536 z ) And in Maple-input format, it is: -(-1-1122558*z^28+18531342*z^26+335*z^2-158338418*z^24+742615354*z^22-31714*z^4 +1122558*z^6-18531342*z^8+158338418*z^10-742615354*z^12+2018666982*z^14+ 3299278128*z^18-3299278128*z^16-2018666982*z^20+z^34+31714*z^30-335*z^32)/( 32640355642*z^20-15529006040*z^22-726481768*z^26-726481768*z^10+1-41739830460*z ^18+32640355642*z^16-15529006040*z^14-570*z^2+72285*z^4+66395792*z^8+4414552536 *z^24-3197992*z^6+66395792*z^28-3197992*z^30-570*z^34+z^36+72285*z^32+ 4414552536*z^12) The first , 40, terms are: [0, 235, 0, 93379, 0, 38314489, 0, 15793021385, 0, 6516049779283, 0, 2689046331299803, 0, 1109772599979751857, 0, 458009779004453905745, 0, 189023877655545831010043, 0, 78011540588176483729108019, 0, 32195939539084056734320478505, 0, 13287502711690336573414475806873, 0, 5483850826876368735046983541089507, 0, 2263225874499744020215646252606817227, 0, 934050090493595892906908718870292980321, 0, 385489394347918850985885883613098496072097, 0, 159094329812053751818290285777311448730571275, 0, 65659408921714671437622954832185043723722959011, 0, 27098124647477741063855924550485062395110809806681, 0, 11183596859453150030210779452541646823929422860080745] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 143370206107707058 z - 55563120217210407 z - 393 z 24 22 4 6 + 17300540421165126 z - 4303241926378682 z + 54574 z - 3921595 z 8 10 12 14 + 170812118 z - 4918662750 z + 98966627846 z - 1446055931398 z 18 16 50 - 131426727588344 z + 15782652390785 z - 4303241926378682 z 48 20 36 + 17300540421165126 z + 848676071584212 z + 762610956868149634 z 34 66 64 - 687307777031987454 z - 3921595 z + 170812118 z 30 42 44 - 298489909165241595 z - 298489909165241595 z + 143370206107707058 z 46 58 56 - 55563120217210407 z - 1446055931398 z + 15782652390785 z 54 52 60 70 - 131426727588344 z + 848676071584212 z + 98966627846 z - 393 z 68 32 38 + 54574 z + 502956143678456193 z - 687307777031987454 z 40 62 72 / + 502956143678456193 z - 4918662750 z + z ) / (-1 / 28 26 2 - 1100688750246511603 z + 381390276529857767 z + 643 z 24 22 4 6 - 106084863245588319 z + 23561772945007863 z - 108232 z + 8858522 z 8 10 12 14 - 431612741 z + 13840196797 z - 310139419356 z + 5054283938428 z 18 16 50 + 573972572482779 z - 61623230895595 z + 106084863245588319 z 48 20 36 - 381390276529857767 z - 4149183026247979 z - 9027999746680211827 z 34 66 64 + 7320758757575546001 z + 431612741 z - 13840196797 z 30 42 + 2559943675234253717 z + 4811296871713136059 z 44 46 58 - 2559943675234253717 z + 1100688750246511603 z + 61623230895595 z 56 54 52 - 573972572482779 z + 4149183026247979 z - 23561772945007863 z 60 70 68 32 - 5054283938428 z + 108232 z - 8858522 z - 4811296871713136059 z 38 40 62 + 9027999746680211827 z - 7320758757575546001 z + 310139419356 z 74 72 + z - 643 z ) And in Maple-input format, it is: -(1+143370206107707058*z^28-55563120217210407*z^26-393*z^2+17300540421165126*z^ 24-4303241926378682*z^22+54574*z^4-3921595*z^6+170812118*z^8-4918662750*z^10+ 98966627846*z^12-1446055931398*z^14-131426727588344*z^18+15782652390785*z^16-\ 4303241926378682*z^50+17300540421165126*z^48+848676071584212*z^20+ 762610956868149634*z^36-687307777031987454*z^34-3921595*z^66+170812118*z^64-\ 298489909165241595*z^30-298489909165241595*z^42+143370206107707058*z^44-\ 55563120217210407*z^46-1446055931398*z^58+15782652390785*z^56-131426727588344*z ^54+848676071584212*z^52+98966627846*z^60-393*z^70+54574*z^68+ 502956143678456193*z^32-687307777031987454*z^38+502956143678456193*z^40-\ 4918662750*z^62+z^72)/(-1-1100688750246511603*z^28+381390276529857767*z^26+643* z^2-106084863245588319*z^24+23561772945007863*z^22-108232*z^4+8858522*z^6-\ 431612741*z^8+13840196797*z^10-310139419356*z^12+5054283938428*z^14+ 573972572482779*z^18-61623230895595*z^16+106084863245588319*z^50-\ 381390276529857767*z^48-4149183026247979*z^20-9027999746680211827*z^36+ 7320758757575546001*z^34+431612741*z^66-13840196797*z^64+2559943675234253717*z^ 30+4811296871713136059*z^42-2559943675234253717*z^44+1100688750246511603*z^46+ 61623230895595*z^58-573972572482779*z^56+4149183026247979*z^54-\ 23561772945007863*z^52-5054283938428*z^60+108232*z^70-8858522*z^68-\ 4811296871713136059*z^32+9027999746680211827*z^38-7320758757575546001*z^40+ 310139419356*z^62+z^74-643*z^72) The first , 40, terms are: [0, 250, 0, 107092, 0, 46739083, 0, 20416278902, 0, 8918698089551, 0, 3896093973223923, 0, 1701993012969297352, 0, 743508903955559123029, 0, 324798923906937409671905, 0, 141887125546783161504084790, 0, 61982829745070790381913674561, 0, 27076954082740697385323324247604, 0, 11828460323904812063765293978637402, 0, 5167216120644793291937556013979676899, 0, 2257277930204863410026841412865714255275, 0, 986082938128433662688965726030006721282434, 0, 430766432372710942622046146706988008773473284, 0, 188178612654329215272134007381370052001540627537, 0, 82205082846077799412865305417041689758558577921678, 0, 35910965387676048466582787666540184955309991482687409] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 295251123595860 z - 256374654774408 z - 364 z 24 22 4 6 + 167712518964186 z - 82441442646152 z + 43382 z - 2530268 z 8 10 12 14 + 84823365 z - 1786051576 z + 25000733140 z - 241855579928 z 18 16 50 48 - 8278437687448 z + 1662157904608 z - 2530268 z + 84823365 z 20 36 34 + 30312858424204 z + 30312858424204 z - 82441442646152 z 30 42 44 - 256374654774408 z - 241855579928 z + 25000733140 z 46 56 54 52 32 - 1786051576 z + z - 364 z + 43382 z + 167712518964186 z 38 40 / 28 - 8278437687448 z + 1662157904608 z ) / (-1 - 3355215581687124 z / 26 2 24 + 2527074049959088 z + 596 z - 1432394253849964 z 22 4 6 8 + 609803268838378 z - 88289 z + 6048875 z - 234367292 z 10 12 14 + 5676897459 z - 91291151926 z + 1014711969512 z 18 16 50 48 + 46001525833714 z - 8020174040878 z + 234367292 z - 5676897459 z 20 36 34 - 194244832017880 z - 609803268838378 z + 1432394253849964 z 30 42 44 + 3355215581687124 z + 8020174040878 z - 1014711969512 z 46 58 56 54 52 + 91291151926 z + z - 596 z + 88289 z - 6048875 z 32 38 40 - 2527074049959088 z + 194244832017880 z - 46001525833714 z ) And in Maple-input format, it is: -(1+295251123595860*z^28-256374654774408*z^26-364*z^2+167712518964186*z^24-\ 82441442646152*z^22+43382*z^4-2530268*z^6+84823365*z^8-1786051576*z^10+ 25000733140*z^12-241855579928*z^14-8278437687448*z^18+1662157904608*z^16-\ 2530268*z^50+84823365*z^48+30312858424204*z^20+30312858424204*z^36-\ 82441442646152*z^34-256374654774408*z^30-241855579928*z^42+25000733140*z^44-\ 1786051576*z^46+z^56-364*z^54+43382*z^52+167712518964186*z^32-8278437687448*z^ 38+1662157904608*z^40)/(-1-3355215581687124*z^28+2527074049959088*z^26+596*z^2-\ 1432394253849964*z^24+609803268838378*z^22-88289*z^4+6048875*z^6-234367292*z^8+ 5676897459*z^10-91291151926*z^12+1014711969512*z^14+46001525833714*z^18-\ 8020174040878*z^16+234367292*z^50-5676897459*z^48-194244832017880*z^20-\ 609803268838378*z^36+1432394253849964*z^34+3355215581687124*z^30+8020174040878* z^42-1014711969512*z^44+91291151926*z^46+z^58-596*z^56+88289*z^54-6048875*z^52-\ 2527074049959088*z^32+194244832017880*z^38-46001525833714*z^40) The first , 40, terms are: [0, 232, 0, 93365, 0, 38681099, 0, 16064627592, 0, 6673673343735, 0, 2772525587683719, 0, 1151830261056224488, 0, 478521810849247102283, 0, 198799386229791944382645, 0, 82590167704015280807359048, 0, 34311654311352104637162331601, 0, 14254597789559160578356676300209, 0, 5921998289814130137912560458379912, 0, 2460263296275718533587475382322595125, 0, 1022103552008646472863302463388298641483, 0, 424627588685444385493371452771063295649064, 0, 176409316569221662799614825290099817340113639, 0, 73288330296110904368500254200798119669076744855, 0, 30447254499079931624984558973624539619200234341704, 0, 12649153047779790583106553582935034876287386518486539] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 22 4 6 8 10 f(z) = - (-1 + 316 z + z - 24885 z + 427703 z - 2182548 z + 4650059 z 12 14 18 16 20 / 24 - 4650059 z + 2182548 z + 24885 z - 427703 z - 316 z ) / (z / 22 20 18 16 14 - 608 z + 62464 z - 1526900 z + 13007936 z - 44654048 z 12 10 8 6 4 + 67143910 z - 44654048 z + 13007936 z - 1526900 z + 62464 z 2 - 608 z + 1) And in Maple-input format, it is: -(-1+316*z^2+z^22-24885*z^4+427703*z^6-2182548*z^8+4650059*z^10-4650059*z^12+ 2182548*z^14+24885*z^18-427703*z^16-316*z^20)/(z^24-608*z^22+62464*z^20-1526900 *z^18+13007936*z^16-44654048*z^14+67143910*z^12-44654048*z^10+13007936*z^8-\ 1526900*z^6+62464*z^4-608*z^2+1) The first , 40, terms are: [0, 292, 0, 139957, 0, 67953565, 0, 33008522884, 0, 16034472459289, 0, 7789065605009449, 0, 3783697608198850372, 0, 1838008640096069041549, 0, 892850384068364789861701, 0, 433720383820488139725807460, 0, 210688571006807640798323926897, 0, 102346294104994112806900149942673, 0, 49716811250690547120383833320042724, 0, 24150960643662839045139646139210069797, 0, 11731824413908596492590401292747964578989, 0, 5698974302081521399051204163858433469600708, 0, 2768393640232345061972957485902414489882842377, 0, 1344803984197599488153324748424927043583366494521, 0, 653266114193918365833616298414885698615295710127620, 0, 317337411971346462835135118670106114206086499819381373] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 6}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 16 14 12 10 8 6 4 f(z) = - (z - 215 z + 2959 z - 12519 z + 19676 z - 12519 z + 2959 z 2 / 18 16 14 12 10 - 215 z + 1) / (z - 482 z + 11154 z - 81418 z + 222658 z / 8 6 4 2 - 222658 z + 81418 z - 11154 z + 482 z - 1) And in Maple-input format, it is: -(z^16-215*z^14+2959*z^12-12519*z^10+19676*z^8-12519*z^6+2959*z^4-215*z^2+1)/(z ^18-482*z^16+11154*z^14-81418*z^12+222658*z^10-222658*z^8+81418*z^6-11154*z^4+ 482*z^2-1) The first , 40, terms are: [0, 267, 0, 120499, 0, 55171299, 0, 25270055896, 0, 11574537820861, 0, 5301530192317885, 0, 2428280337731523437, 0, 1112234616436197059471, 0, 509441114679035103327855, 0, 233341280239100061891158717, 0, 106878207303559851632089977053, 0, 48953837849513170169253414202237, 0, 22422515315866524906319034474901624, 0, 10270271242794271739923679798863467939, 0, 4704131981389651130594616537711015569875, 0, 2154651729754339244992496453196833500024283, 0, 986903448904066391641301941984387880599527137, 0, 452035196226254388998486230734885034883621888289, 0, 207047425818826089506302691727386522496739882224443, 0, 94834731667101245168307129606596388668823367005625235] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {3, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 18 16 14 8 12 10 f(z) = - (z - 343 z + 23908 z - 1002515 z - 336582 z + 1002515 z 6 4 2 / 18 16 + 336582 z - 23908 z + 343 z - 1) / (-611 z + 68153 z / 14 8 12 10 6 20 - 1541296 z + 8308673 z + 8308673 z - 14634307 z - 1541296 z + z 4 2 + 68153 z - 611 z + 1) And in Maple-input format, it is: -(z^18-343*z^16+23908*z^14-1002515*z^8-336582*z^12+1002515*z^10+336582*z^6-\ 23908*z^4+343*z^2-1)/(-611*z^18+68153*z^16-1541296*z^14+8308673*z^8+8308673*z^ 12-14634307*z^10-1541296*z^6+z^20+68153*z^4-611*z^2+1) The first , 40, terms are: [0, 268, 0, 119503, 0, 55956043, 0, 26450415484, 0, 12529608065461, 0, 5938171189440013, 0, 2814596964001033180, 0, 1334107448756031568915, 0, 632365186006502108504263, 0, 299740675556948230437514540, 0, 142076923948887982365432918889, 0, 67344392681087403887158617334297, 0, 31921209819343512898233719076099052, 0, 15130638195687403334665989790730939479, 0, 7171915278352144118549079620477046565891, 0, 3399484417302870584185967811438752886673948, 0, 1611353990599335614169402750149473453055844061, 0, 763781022155026023693787714745863770635910826981, 0, 362031839812309449062509027291897412082085792974652, 0, 171602919208636045065350448906209791829140177739424091] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 249699249218720664 z - 111234609194589718 z - 507 z 24 22 4 6 + 38993108464622946 z - 10686040054727808 z + 90668 z - 8024793 z 8 10 12 14 + 407333603 z - 13003299872 z + 278412338519 z - 4185154904661 z 18 16 50 - 370077081566591 z + 45648219419452 z - 370077081566591 z 48 20 36 + 2270466343022029 z + 2270466343022029 z + 624664191203649430 z 34 66 64 30 - 700149665478672576 z - 507 z + 90668 z - 443303185049346114 z 42 44 46 - 111234609194589718 z + 38993108464622946 z - 10686040054727808 z 58 56 54 - 13003299872 z + 278412338519 z - 4185154904661 z 52 60 68 32 + 45648219419452 z + 407333603 z + z + 624664191203649430 z 38 40 62 / - 443303185049346114 z + 249699249218720664 z - 8024793 z ) / (-1 / 28 26 2 - 2222323030715829818 z + 880429412736898614 z + 783 z 24 22 4 6 - 274445843681633514 z + 66913036300812261 z - 180313 z + 19105283 z 8 10 12 14 - 1120532941 z + 40651643219 z - 981133773039 z + 16558767370257 z 18 16 50 + 1839666234178301 z - 202452848401791 z + 12660403989631275 z 48 20 36 - 66913036300812261 z - 12660403989631275 z - 8816013613549917422 z 34 66 64 + 8816013613549917422 z + 180313 z - 19105283 z 30 42 + 4433452888774930158 z + 2222323030715829818 z 44 46 58 - 880429412736898614 z + 274445843681633514 z + 981133773039 z 56 54 52 - 16558767370257 z + 202452848401791 z - 1839666234178301 z 60 70 68 32 - 40651643219 z + z - 783 z - 7013231515035775922 z 38 40 62 + 7013231515035775922 z - 4433452888774930158 z + 1120532941 z ) And in Maple-input format, it is: -(1+249699249218720664*z^28-111234609194589718*z^26-507*z^2+38993108464622946*z ^24-10686040054727808*z^22+90668*z^4-8024793*z^6+407333603*z^8-13003299872*z^10 +278412338519*z^12-4185154904661*z^14-370077081566591*z^18+45648219419452*z^16-\ 370077081566591*z^50+2270466343022029*z^48+2270466343022029*z^20+ 624664191203649430*z^36-700149665478672576*z^34-507*z^66+90668*z^64-\ 443303185049346114*z^30-111234609194589718*z^42+38993108464622946*z^44-\ 10686040054727808*z^46-13003299872*z^58+278412338519*z^56-4185154904661*z^54+ 45648219419452*z^52+407333603*z^60+z^68+624664191203649430*z^32-\ 443303185049346114*z^38+249699249218720664*z^40-8024793*z^62)/(-1-\ 2222323030715829818*z^28+880429412736898614*z^26+783*z^2-274445843681633514*z^ 24+66913036300812261*z^22-180313*z^4+19105283*z^6-1120532941*z^8+40651643219*z^ 10-981133773039*z^12+16558767370257*z^14+1839666234178301*z^18-202452848401791* z^16+12660403989631275*z^50-66913036300812261*z^48-12660403989631275*z^20-\ 8816013613549917422*z^36+8816013613549917422*z^34+180313*z^66-19105283*z^64+ 4433452888774930158*z^30+2222323030715829818*z^42-880429412736898614*z^44+ 274445843681633514*z^46+981133773039*z^58-16558767370257*z^56+202452848401791*z ^54-1839666234178301*z^52-40651643219*z^60+z^70-783*z^68-7013231515035775922*z^ 32+7013231515035775922*z^38-4433452888774930158*z^40+1120532941*z^62) The first , 40, terms are: [0, 276, 0, 126463, 0, 60334631, 0, 28998951924, 0, 13961553692649, 0, 6724529897803769, 0, 3239166030138077940, 0, 1560325033855848258231, 0, 751622121403373004078703, 0, 362063441865801099898517396, 0, 174409427749540227886842794225, 0, 84014699457116966385642059488657, 0, 40470690001059325715289005877102228, 0, 19495121316544032531643420297387719439, 0, 9390987790457237587140872989871551299095, 0, 4523729310462100752400621860238244521793780, 0, 2179123999805591796313268205317621934764630361, 0, 1049705028911918243825519568900991348014665575305, 0, 505653027467458633695114981375295852489106333227508, 0, 243577935843908633435362304269297006771434668149837255] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 233256933452073816 z - 102776422458431798 z - 483 z 24 22 4 6 + 35549921668207706 z - 9595321827462520 z + 81206 z - 6883351 z 8 10 12 14 + 341616397 z - 10825972140 z + 232484072545 z - 3528766216299 z 18 16 50 - 321471031214163 z + 39023727708130 z - 321471031214163 z 48 20 36 + 2005558137574857 z + 2005558137574857 z + 591267761355333310 z 34 66 64 30 - 663842008518536536 z - 483 z + 81206 z - 417501672220735490 z 42 44 46 - 102776422458431798 z + 35549921668207706 z - 9595321827462520 z 58 56 54 - 10825972140 z + 232484072545 z - 3528766216299 z 52 60 68 32 + 39023727708130 z + 341616397 z + z + 591267761355333310 z 38 40 62 / - 417501672220735490 z + 233256933452073816 z - 6883351 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 1440836884350406574 z - 604498333747773620 z - 762 z 24 22 4 6 + 196982388527480906 z - 49649186665067048 z + 161283 z - 16126076 z 8 10 12 14 + 915102589 z - 32602159126 z + 778655381125 z - 13042582379012 z 18 16 50 - 1421441623731594 z + 158186438428843 z - 1421441623731594 z 48 20 36 + 9621584477392169 z + 9621584477392169 z + 3874594394331607586 z 34 66 64 30 - 4383515279383745292 z - 762 z + 161283 z - 2674854659522378160 z 42 44 46 - 604498333747773620 z + 196982388527480906 z - 49649186665067048 z 58 56 54 - 32602159126 z + 778655381125 z - 13042582379012 z 52 60 68 32 + 158186438428843 z + 915102589 z + z + 3874594394331607586 z 38 40 62 - 2674854659522378160 z + 1440836884350406574 z - 16126076 z )) And in Maple-input format, it is: -(1+233256933452073816*z^28-102776422458431798*z^26-483*z^2+35549921668207706*z ^24-9595321827462520*z^22+81206*z^4-6883351*z^6+341616397*z^8-10825972140*z^10+ 232484072545*z^12-3528766216299*z^14-321471031214163*z^18+39023727708130*z^16-\ 321471031214163*z^50+2005558137574857*z^48+2005558137574857*z^20+ 591267761355333310*z^36-663842008518536536*z^34-483*z^66+81206*z^64-\ 417501672220735490*z^30-102776422458431798*z^42+35549921668207706*z^44-\ 9595321827462520*z^46-10825972140*z^58+232484072545*z^56-3528766216299*z^54+ 39023727708130*z^52+341616397*z^60+z^68+591267761355333310*z^32-\ 417501672220735490*z^38+233256933452073816*z^40-6883351*z^62)/(-1+z^2)/(1+ 1440836884350406574*z^28-604498333747773620*z^26-762*z^2+196982388527480906*z^ 24-49649186665067048*z^22+161283*z^4-16126076*z^6+915102589*z^8-32602159126*z^ 10+778655381125*z^12-13042582379012*z^14-1421441623731594*z^18+158186438428843* z^16-1421441623731594*z^50+9621584477392169*z^48+9621584477392169*z^20+ 3874594394331607586*z^36-4383515279383745292*z^34-762*z^66+161283*z^64-\ 2674854659522378160*z^30-604498333747773620*z^42+196982388527480906*z^44-\ 49649186665067048*z^46-32602159126*z^58+778655381125*z^56-13042582379012*z^54+ 158186438428843*z^52+915102589*z^60+z^68+3874594394331607586*z^32-\ 2674854659522378160*z^38+1440836884350406574*z^40-16126076*z^62) The first , 40, terms are: [0, 280, 0, 132801, 0, 65358571, 0, 32319699880, 0, 15993826196679, 0, 7915700092487295, 0, 3917732650659424008, 0, 1939017191300701495239, 0, 959685103305738663944857, 0, 474980615064055396792606264, 0, 235083974077745466374280051293, 0, 116351011488681829340157446939581, 0, 57586051677475403589277343374089048, 0, 28501285081996707557887208242118600681, 0, 14106250171198526000445259645183440475047, 0, 6981660417076690283396977983421218911988648, 0, 3455459926473745630450356696104193996168713263, 0, 1710224014084323593161191138513649163667540515159, 0, 846447720589111401033216608616182351293209688273608, 0, 418935611820485883356551181214899882879852958644282219] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 971841 z + 16638488 z + 344 z - 154042721 z 22 4 6 8 10 + 801417054 z - 28417 z + 971841 z - 16638488 z + 154042721 z 12 14 18 16 - 801417054 z + 2377942544 z + 4067601278 z - 4067601278 z 20 34 30 32 / 4 2 - 2377942544 z + z + 28417 z - 344 z ) / (67752 z - 624 z / 20 6 10 22 + 41567209888 z + 1 - 2964706 z - 738173536 z - 18520371248 z 30 34 36 14 32 8 - 2964706 z - 624 z + z - 18520371248 z + 67752 z + 63872696 z 16 28 18 26 + 41567209888 z + 63872696 z - 54429362236 z - 738173536 z 12 24 + 4841930495 z + 4841930495 z ) And in Maple-input format, it is: -(-1-971841*z^28+16638488*z^26+344*z^2-154042721*z^24+801417054*z^22-28417*z^4+ 971841*z^6-16638488*z^8+154042721*z^10-801417054*z^12+2377942544*z^14+ 4067601278*z^18-4067601278*z^16-2377942544*z^20+z^34+28417*z^30-344*z^32)/( 67752*z^4-624*z^2+41567209888*z^20+1-2964706*z^6-738173536*z^10-18520371248*z^ 22-2964706*z^30-624*z^34+z^36-18520371248*z^14+67752*z^32+63872696*z^8+ 41567209888*z^16+63872696*z^28-54429362236*z^18-738173536*z^26+4841930495*z^12+ 4841930495*z^24) The first , 40, terms are: [0, 280, 0, 135385, 0, 67502545, 0, 33731867032, 0, 16859329096873, 0, 8426500344602137, 0, 4211675047131416728, 0, 2105050565119322582689, 0, 1052131965794408816997769, 0, 525869398782951093418023448, 0, 262836444104619097798454343697, 0, 131369112769287400511121985602289, 0, 65660010918988543761752332535400472, 0, 32817737312832993913269715628127663913, 0, 16402736875309676220011158626002978500609, 0, 8198303692784923832141986070629261751630488, 0, 4097620046585182791101212959648610258045519417, 0, 2048044409595798740174361444913276591271296088713, 0, 1023639540999450587741475739282288336774160121790360, 0, 511628510098746720699007924830754779584273659889735281] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 1355539 z + 20842564 z + 388 z - 166057337 z 22 4 6 8 10 + 734630726 z - 39819 z + 1355539 z - 20842564 z + 166057337 z 12 14 18 16 - 734630726 z + 1907926200 z + 3039917090 z - 3039917090 z 20 34 30 32 / 36 34 - 1907926200 z + z + 39819 z - 388 z ) / (z - 680 z / 32 30 28 26 24 + 97272 z - 4247958 z + 83451832 z - 854012520 z + 4903549823 z 22 20 18 16 - 16512562672 z + 33655107408 z - 42542797780 z + 33655107408 z 14 12 10 8 - 16512562672 z + 4903549823 z - 854012520 z + 83451832 z 6 4 2 - 4247958 z + 97272 z - 680 z + 1) And in Maple-input format, it is: -(-1-1355539*z^28+20842564*z^26+388*z^2-166057337*z^24+734630726*z^22-39819*z^4 +1355539*z^6-20842564*z^8+166057337*z^10-734630726*z^12+1907926200*z^14+ 3039917090*z^18-3039917090*z^16-1907926200*z^20+z^34+39819*z^30-388*z^32)/(z^36 -680*z^34+97272*z^32-4247958*z^30+83451832*z^28-854012520*z^26+4903549823*z^24-\ 16512562672*z^22+33655107408*z^20-42542797780*z^18+33655107408*z^16-16512562672 *z^14+4903549823*z^12-854012520*z^10+83451832*z^8-4247958*z^6+97272*z^4-680*z^2 +1) The first , 40, terms are: [0, 292, 0, 141107, 0, 70441755, 0, 35352427764, 0, 17763377116905, 0, 8927998267800601, 0, 4487575827937034068, 0, 2255674641566452180555, 0, 1133816503954391391939075, 0, 569914212734386897875991236, 0, 286468118501779581926638563537, 0, 143993578946570520120071354002225, 0, 72378563951815853353713793509437316, 0, 36381181533021462080310433059399442531, 0, 18287049347313313833931841184945799916523, 0, 9192009709304825682482022832532368185246356, 0, 4620375922579972848297429612312669592258530745, 0, 2322438111064238004513369144509701016418517343561, 0, 1167376609634007359797106352927404137297863841302964, 0, 586783407587493965249822844661320777699429721912216955] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 400020295087420 z - 348060681633782 z - 421 z 24 22 4 6 + 229006626498478 z - 113525547887680 z + 55434 z - 3442221 z 8 10 12 14 + 119246521 z - 2545801280 z + 35739367300 z - 344676546340 z 18 16 50 48 - 11622709665300 z + 2353555353432 z - 3442221 z + 119246521 z 20 36 34 + 42149978012948 z + 42149978012948 z - 113525547887680 z 30 42 44 - 348060681633782 z - 344676546340 z + 35739367300 z 46 56 54 52 32 - 2545801280 z + z - 421 z + 55434 z + 229006626498478 z 38 40 / 2 - 11622709665300 z + 2353555353432 z ) / ((-1 + z ) (1 / 28 26 2 + 2651241767531774 z - 2280661461086456 z - 708 z 24 22 4 6 + 1451132368835286 z - 681954291960656 z + 117205 z - 8617188 z 8 10 12 14 + 346248689 z - 8464812272 z + 134534400332 z - 1453947701872 z 18 16 50 48 - 59843831021424 z + 11021145136620 z - 8617188 z + 346248689 z 20 36 34 + 236015160804460 z + 236015160804460 z - 681954291960656 z 30 42 44 - 2280661461086456 z - 1453947701872 z + 134534400332 z 46 56 54 52 32 - 8464812272 z + z - 708 z + 117205 z + 1451132368835286 z 38 40 - 59843831021424 z + 11021145136620 z )) And in Maple-input format, it is: -(1+400020295087420*z^28-348060681633782*z^26-421*z^2+229006626498478*z^24-\ 113525547887680*z^22+55434*z^4-3442221*z^6+119246521*z^8-2545801280*z^10+ 35739367300*z^12-344676546340*z^14-11622709665300*z^18+2353555353432*z^16-\ 3442221*z^50+119246521*z^48+42149978012948*z^20+42149978012948*z^36-\ 113525547887680*z^34-348060681633782*z^30-344676546340*z^42+35739367300*z^44-\ 2545801280*z^46+z^56-421*z^54+55434*z^52+229006626498478*z^32-11622709665300*z^ 38+2353555353432*z^40)/(-1+z^2)/(1+2651241767531774*z^28-2280661461086456*z^26-\ 708*z^2+1451132368835286*z^24-681954291960656*z^22+117205*z^4-8617188*z^6+ 346248689*z^8-8464812272*z^10+134534400332*z^12-1453947701872*z^14-\ 59843831021424*z^18+11021145136620*z^16-8617188*z^50+346248689*z^48+ 236015160804460*z^20+236015160804460*z^36-681954291960656*z^34-2280661461086456 *z^30-1453947701872*z^42+134534400332*z^44-8464812272*z^46+z^56-708*z^54+117205 *z^52+1451132368835286*z^32-59843831021424*z^38+11021145136620*z^40) The first , 40, terms are: [0, 288, 0, 141713, 0, 71807745, 0, 36481772064, 0, 18540350679505, 0, 9422771725344561, 0, 4788969501649692576, 0, 2433917552306246212513, 0, 1237000033573392753011441, 0, 628685680611772605302254240, 0, 319519543589655764342552166817, 0, 162390749348819203759377730671201, 0, 82532527367433691284313153181500832, 0, 41945850370384788274515902401506780721, 0, 21318314359429413569896127033799557403745, 0, 10834695759283500167664837465737087105800864, 0, 5506562583561669191075057285417330810299268209, 0, 2798623252591147609078224931791965336770571908241, 0, 1422355960018509092390780267027627500449516462052128, 0, 722889897783512092222301944213194330772458856971974209] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 2305563229048 z + 4636704712193 z + 494 z 24 22 4 6 - 6550162197699 z + 6550162197699 z - 78395 z + 5026579 z 8 10 12 14 - 157047645 z + 2717672403 z - 28176954624 z + 184494572747 z 18 16 20 + 2305563229048 z - 793060637121 z - 4636704712193 z 36 34 30 42 - 2717672403 z + 28176954624 z + 793060637121 z + 78395 z 44 46 32 38 40 / - 494 z + z - 184494572747 z + 157047645 z - 5026579 z ) / (1 / 28 26 2 24 + 47182901871486 z - 80160762105764 z - 798 z + 95564629353513 z 22 4 6 8 - 80160762105764 z + 176782 z - 13813716 z + 511792909 z 10 12 14 - 10525157820 z + 130933212978 z - 1039281979094 z 18 16 48 20 - 19336640187310 z + 5450562899702 z + z + 47182901871486 z 36 34 30 42 + 130933212978 z - 1039281979094 z - 19336640187310 z - 13813716 z 44 46 32 38 + 176782 z - 798 z + 5450562899702 z - 10525157820 z 40 + 511792909 z ) And in Maple-input format, it is: -(-1-2305563229048*z^28+4636704712193*z^26+494*z^2-6550162197699*z^24+ 6550162197699*z^22-78395*z^4+5026579*z^6-157047645*z^8+2717672403*z^10-\ 28176954624*z^12+184494572747*z^14+2305563229048*z^18-793060637121*z^16-\ 4636704712193*z^20-2717672403*z^36+28176954624*z^34+793060637121*z^30+78395*z^ 42-494*z^44+z^46-184494572747*z^32+157047645*z^38-5026579*z^40)/(1+ 47182901871486*z^28-80160762105764*z^26-798*z^2+95564629353513*z^24-\ 80160762105764*z^22+176782*z^4-13813716*z^6+511792909*z^8-10525157820*z^10+ 130933212978*z^12-1039281979094*z^14-19336640187310*z^18+5450562899702*z^16+z^ 48+47182901871486*z^20+130933212978*z^36-1039281979094*z^34-19336640187310*z^30 -13813716*z^42+176782*z^44-798*z^46+5450562899702*z^32-10525157820*z^38+ 511792909*z^40) The first , 40, terms are: [0, 304, 0, 144205, 0, 70120999, 0, 34308333292, 0, 16826148878659, 0, 8260096390242403, 0, 4056513642289273372, 0, 1992453871413280275811, 0, 978702784346633997136333, 0, 480755588124427316154774304, 0, 236157791699333269028091553693, 0, 116006413171517070377279367175885, 0, 56985247245994209113539446538237984, 0, 27992595227813915422199984264408483245, 0, 13750674708195137323370136396670041285763, 0, 6754681867083029866792978265034762106075324, 0, 3318072028042770168249310807980756961149291235, 0, 1629921644497999417254884725017259596416222221747, 0, 800659106727043152877013587059262496292549604647148, 0, 393304186848324635016841009309460530291712329143067447] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 83132856217776413527 z - 18270607819731573567 z - 551 z 24 22 4 + 3302516916745815797 z - 487460568586192060 z + 113683 z 6 8 10 12 - 12297234 z + 809198152 z - 35356952858 z + 1087741159169 z 14 18 16 - 24566810366953 z - 5567736545521684 z + 420096796776123 z 50 48 - 10214498086260512535217 z + 15717288013336179080239 z 20 36 + 58232656159846212 z + 5577510565186247690677 z 34 66 80 - 2554328536552005288622 z - 487460568586192060 z + 809198152 z 88 84 86 82 64 + z + 113683 z - 551 z - 12297234 z + 3302516916745815797 z 30 42 - 312905650596510430406 z - 20345165666883433750176 z 44 46 + 22171096686811426537160 z - 20345165666883433750176 z 58 56 - 312905650596510430406 z + 978820652120999075744 z 54 52 - 2554328536552005288622 z + 5577510565186247690677 z 60 70 68 + 83132856217776413527 z - 5567736545521684 z + 58232656159846212 z 78 32 - 35356952858 z + 978820652120999075744 z 38 40 - 10214498086260512535217 z + 15717288013336179080239 z 62 76 74 - 18270607819731573567 z + 1087741159169 z - 24566810366953 z 72 / 28 + 420096796776123 z ) / (-1 - 528258882750086722940 z / 26 2 24 + 106205072345683193199 z + 839 z - 17568588646158122437 z 22 4 6 8 + 2374146674610264180 z - 217396 z + 27540568 z - 2052446905 z 10 12 14 + 99822950767 z - 3386817800036 z + 83929283191248 z 18 16 + 22741019488756967 z - 1570380302225397 z 50 48 + 170425618950972839510541 z - 240730736652905114186251 z 20 36 - 259732165184202832 z - 50679867416136572856492 z 34 66 80 + 21229060249049143828587 z + 17568588646158122437 z - 99822950767 z 90 88 84 86 82 + z - 839 z - 27540568 z + 217396 z + 2052446905 z 64 30 - 106205072345683193199 z + 2174255338125794687008 z 42 44 + 240730736652905114186251 z - 286055586425444780589444 z 46 58 + 286055586425444780589444 z + 7438601904100082404277 z 56 54 - 21229060249049143828587 z + 50679867416136572856492 z 52 60 - 101423053124115254962280 z - 2174255338125794687008 z 70 68 78 + 259732165184202832 z - 2374146674610264180 z + 3386817800036 z 32 38 - 7438601904100082404277 z + 101423053124115254962280 z 40 62 - 170425618950972839510541 z + 528258882750086722940 z 76 74 72 - 83929283191248 z + 1570380302225397 z - 22741019488756967 z ) And in Maple-input format, it is: -(1+83132856217776413527*z^28-18270607819731573567*z^26-551*z^2+ 3302516916745815797*z^24-487460568586192060*z^22+113683*z^4-12297234*z^6+ 809198152*z^8-35356952858*z^10+1087741159169*z^12-24566810366953*z^14-\ 5567736545521684*z^18+420096796776123*z^16-10214498086260512535217*z^50+ 15717288013336179080239*z^48+58232656159846212*z^20+5577510565186247690677*z^36 -2554328536552005288622*z^34-487460568586192060*z^66+809198152*z^80+z^88+113683 *z^84-551*z^86-12297234*z^82+3302516916745815797*z^64-312905650596510430406*z^ 30-20345165666883433750176*z^42+22171096686811426537160*z^44-\ 20345165666883433750176*z^46-312905650596510430406*z^58+978820652120999075744*z ^56-2554328536552005288622*z^54+5577510565186247690677*z^52+ 83132856217776413527*z^60-5567736545521684*z^70+58232656159846212*z^68-\ 35356952858*z^78+978820652120999075744*z^32-10214498086260512535217*z^38+ 15717288013336179080239*z^40-18270607819731573567*z^62+1087741159169*z^76-\ 24566810366953*z^74+420096796776123*z^72)/(-1-528258882750086722940*z^28+ 106205072345683193199*z^26+839*z^2-17568588646158122437*z^24+ 2374146674610264180*z^22-217396*z^4+27540568*z^6-2052446905*z^8+99822950767*z^ 10-3386817800036*z^12+83929283191248*z^14+22741019488756967*z^18-\ 1570380302225397*z^16+170425618950972839510541*z^50-240730736652905114186251*z^ 48-259732165184202832*z^20-50679867416136572856492*z^36+21229060249049143828587 *z^34+17568588646158122437*z^66-99822950767*z^80+z^90-839*z^88-27540568*z^84+ 217396*z^86+2052446905*z^82-106205072345683193199*z^64+2174255338125794687008*z ^30+240730736652905114186251*z^42-286055586425444780589444*z^44+ 286055586425444780589444*z^46+7438601904100082404277*z^58-\ 21229060249049143828587*z^56+50679867416136572856492*z^54-\ 101423053124115254962280*z^52-2174255338125794687008*z^60+259732165184202832*z^ 70-2374146674610264180*z^68+3386817800036*z^78-7438601904100082404277*z^32+ 101423053124115254962280*z^38-170425618950972839510541*z^40+ 528258882750086722940*z^62-83929283191248*z^76+1570380302225397*z^74-\ 22741019488756967*z^72) The first , 40, terms are: [0, 288, 0, 137919, 0, 68347327, 0, 34048803260, 0, 16980239321909, 0, 8470049872921761, 0, 4225231297983355060, 0, 2107754718349771049871, 0, 1051455282699644270027119, 0, 524519706271287913351723272, 0, 261657309989948322532136185313, 0, 130528079810027486373211189975869, 0, 65114098078785401388187663073156248, 0, 32482250425066800544409233369059533091, 0, 16203811835311016484682326021884006690259, 0, 8083292092826716966063339520537703651585540, 0, 4032360516366036769658318529503705494605830197, 0, 2011548159747475558701150808852435736957705868201, 0, 1003463351693203524915036865510528349308991681869948, 0, 500578966162076527856364937615826656219665533581339011] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 163887606233504852 z - 74048825558937150 z - 491 z 24 22 4 6 + 26405266552714618 z - 7377453114959760 z + 83914 z - 7015801 z 8 10 12 14 + 338823107 z - 10401039324 z + 215845947007 z - 3159696004397 z 18 16 50 - 266636716831591 z + 33645142104546 z - 266636716831591 z 48 20 36 + 1600490583758069 z + 1600490583758069 z + 403023691692319566 z 34 66 64 30 - 450725891337512232 z - 491 z + 83914 z - 287891885226653130 z 42 44 46 - 74048825558937150 z + 26405266552714618 z - 7377453114959760 z 58 56 54 - 10401039324 z + 215845947007 z - 3159696004397 z 52 60 68 32 + 33645142104546 z + 338823107 z + z + 403023691692319566 z 38 40 62 / - 287891885226653130 z + 163887606233504852 z - 7015801 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 1041528111845682918 z - 447531706565275796 z - 790 z 24 22 4 6 + 150022700785365810 z - 39021315738270520 z + 170547 z - 16670964 z 8 10 12 14 + 914697165 z - 31474326274 z + 726313339561 z - 11755228426684 z 18 16 50 - 1195490623733094 z + 137732434360295 z - 1195490623733094 z 48 20 36 + 7818770364702189 z + 7818770364702189 z + 2719247646830426730 z 34 66 64 30 - 3064507972593968524 z - 790 z + 170547 z - 1898741406052322880 z 42 44 46 - 447531706565275796 z + 150022700785365810 z - 39021315738270520 z 58 56 54 - 31474326274 z + 726313339561 z - 11755228426684 z 52 60 68 32 + 137732434360295 z + 914697165 z + z + 2719247646830426730 z 38 40 62 - 1898741406052322880 z + 1041528111845682918 z - 16670964 z )) And in Maple-input format, it is: -(1+163887606233504852*z^28-74048825558937150*z^26-491*z^2+26405266552714618*z^ 24-7377453114959760*z^22+83914*z^4-7015801*z^6+338823107*z^8-10401039324*z^10+ 215845947007*z^12-3159696004397*z^14-266636716831591*z^18+33645142104546*z^16-\ 266636716831591*z^50+1600490583758069*z^48+1600490583758069*z^20+ 403023691692319566*z^36-450725891337512232*z^34-491*z^66+83914*z^64-\ 287891885226653130*z^30-74048825558937150*z^42+26405266552714618*z^44-\ 7377453114959760*z^46-10401039324*z^58+215845947007*z^56-3159696004397*z^54+ 33645142104546*z^52+338823107*z^60+z^68+403023691692319566*z^32-\ 287891885226653130*z^38+163887606233504852*z^40-7015801*z^62)/(-1+z^2)/(1+ 1041528111845682918*z^28-447531706565275796*z^26-790*z^2+150022700785365810*z^ 24-39021315738270520*z^22+170547*z^4-16670964*z^6+914697165*z^8-31474326274*z^ 10+726313339561*z^12-11755228426684*z^14-1195490623733094*z^18+137732434360295* z^16-1195490623733094*z^50+7818770364702189*z^48+7818770364702189*z^20+ 2719247646830426730*z^36-3064507972593968524*z^34-790*z^66+170547*z^64-\ 1898741406052322880*z^30-447531706565275796*z^42+150022700785365810*z^44-\ 39021315738270520*z^46-31474326274*z^58+726313339561*z^56-11755228426684*z^54+ 137732434360295*z^52+914697165*z^60+z^68+2719247646830426730*z^32-\ 1898741406052322880*z^38+1041528111845682918*z^40-16670964*z^62) The first , 40, terms are: [0, 300, 0, 149877, 0, 76977317, 0, 39669490476, 0, 20456237093249, 0, 10550030486322273, 0, 5441200115553111724, 0, 2806329034671658876453, 0, 1447381879956362343504949, 0, 746496573951776875253653164, 0, 385010469292619761054391629409, 0, 198571660121063131746179086035745, 0, 102414628944198097151349176974146988, 0, 52821012949510698426337375135436507189, 0, 27242781996135790977689981086529835951781, 0, 14050642528002200018815106871758303916636588, 0, 7246710540780633247722259372508819188299556385, 0, 3737538234092429880314107692518274120155853501121, 0, 1927659725429664086450133287682788959115563352038444, 0, 994203078151594000384448456985512984259815689876535077] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 216427709281946420 z - 97648729655005562 z - 497 z 24 22 4 6 + 34745607757546210 z - 9675407382820296 z + 87180 z - 7633731 z 8 10 12 14 + 386705799 z - 12367292624 z + 265022329807 z - 3973350242275 z 18 16 50 - 345142218432921 z + 43038994992104 z - 345142218432921 z 48 20 36 + 2088198588599217 z + 2088198588599217 z + 533001016090334070 z 34 66 64 30 - 596191117475983488 z - 497 z + 87180 z - 380537012431397062 z 42 44 46 - 97648729655005562 z + 34745607757546210 z - 9675407382820296 z 58 56 54 - 12367292624 z + 265022329807 z - 3973350242275 z 52 60 68 32 + 43038994992104 z + 386705799 z + z + 533001016090334070 z 38 40 62 / - 380537012431397062 z + 216427709281946420 z - 7633731 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 1379382072232902802 z - 592154330108971332 z - 812 z 24 22 4 6 + 198123166732213094 z - 51347394405553188 z + 176271 z - 17903786 z 8 10 12 14 + 1028027473 z - 36905523574 z + 882578825085 z - 14690641871402 z 18 16 50 - 1548129372219720 z + 175753158234659 z - 1548129372219720 z 48 20 36 + 10225227509620053 z + 10225227509620053 z + 3603339760051863678 z 34 66 64 30 - 4061012180800971292 z - 812 z + 176271 z - 2515676209739628884 z 42 44 46 - 592154330108971332 z + 198123166732213094 z - 51347394405553188 z 58 56 54 - 36905523574 z + 882578825085 z - 14690641871402 z 52 60 68 32 + 175753158234659 z + 1028027473 z + z + 3603339760051863678 z 38 40 62 - 2515676209739628884 z + 1379382072232902802 z - 17903786 z )) And in Maple-input format, it is: -(1+216427709281946420*z^28-97648729655005562*z^26-497*z^2+34745607757546210*z^ 24-9675407382820296*z^22+87180*z^4-7633731*z^6+386705799*z^8-12367292624*z^10+ 265022329807*z^12-3973350242275*z^14-345142218432921*z^18+43038994992104*z^16-\ 345142218432921*z^50+2088198588599217*z^48+2088198588599217*z^20+ 533001016090334070*z^36-596191117475983488*z^34-497*z^66+87180*z^64-\ 380537012431397062*z^30-97648729655005562*z^42+34745607757546210*z^44-\ 9675407382820296*z^46-12367292624*z^58+265022329807*z^56-3973350242275*z^54+ 43038994992104*z^52+386705799*z^60+z^68+533001016090334070*z^32-\ 380537012431397062*z^38+216427709281946420*z^40-7633731*z^62)/(-1+z^2)/(1+ 1379382072232902802*z^28-592154330108971332*z^26-812*z^2+198123166732213094*z^ 24-51347394405553188*z^22+176271*z^4-17903786*z^6+1028027473*z^8-36905523574*z^ 10+882578825085*z^12-14690641871402*z^14-1548129372219720*z^18+175753158234659* z^16-1548129372219720*z^50+10225227509620053*z^48+10225227509620053*z^20+ 3603339760051863678*z^36-4061012180800971292*z^34-812*z^66+176271*z^64-\ 2515676209739628884*z^30-592154330108971332*z^42+198123166732213094*z^44-\ 51347394405553188*z^46-36905523574*z^58+882578825085*z^56-14690641871402*z^54+ 175753158234659*z^52+1028027473*z^60+z^68+3603339760051863678*z^32-\ 2515676209739628884*z^38+1379382072232902802*z^40-17903786*z^62) The first , 40, terms are: [0, 316, 0, 167005, 0, 90263163, 0, 48864277656, 0, 26457097940663, 0, 14325207667844567, 0, 7756407556406483320, 0, 4199720974837066851287, 0, 2273946640653730540631405, 0, 1231232593862932877520900972, 0, 666653154509248790113336285933, 0, 360960577776030472157593606551853, 0, 195442769345339614058619649807887436, 0, 105822847261589580011321240464139336221, 0, 57297975463935547988113587162928612767431, 0, 31024094297428240948908547761726596562541848, 0, 16798052971025798954196019589868128521195325255, 0, 9095336705470882232018498950363424194946213775735, 0, 4924686803201228191527734702304109754560109036373848, 0, 2666480735675935008425961333583136302412809199488226235] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 161201461350192 z - 140932834907250 z - 387 z 24 22 4 6 + 94048250595546 z - 47711524467764 z + 43472 z - 2350035 z 8 10 12 14 + 73095629 z - 1434838220 z + 18809447488 z - 171091049284 z 18 16 50 48 - 5238606318708 z + 1109772923288 z - 2350035 z + 73095629 z 20 36 34 + 18277857198488 z + 18277857198488 z - 47711524467764 z 30 42 44 - 140932834907250 z - 171091049284 z + 18809447488 z 46 56 54 52 32 - 1434838220 z + z - 387 z + 43472 z + 94048250595546 z 38 40 / 2 - 5238606318708 z + 1109772923288 z ) / ((-1 + z ) (1 / 28 26 2 + 1086822234303026 z - 939568140632360 z - 694 z 24 22 4 6 + 606625084308410 z - 291855229503468 z + 96591 z - 6127586 z 8 10 12 14 + 219362229 z - 4896361476 z + 72309250588 z - 735164525588 z 18 16 50 48 - 27460478870540 z + 5289767316264 z - 6127586 z + 219362229 z 20 36 34 + 104225460884116 z + 104225460884116 z - 291855229503468 z 30 42 44 - 939568140632360 z - 735164525588 z + 72309250588 z 46 56 54 52 32 - 4896361476 z + z - 694 z + 96591 z + 606625084308410 z 38 40 - 27460478870540 z + 5289767316264 z )) And in Maple-input format, it is: -(1+161201461350192*z^28-140932834907250*z^26-387*z^2+94048250595546*z^24-\ 47711524467764*z^22+43472*z^4-2350035*z^6+73095629*z^8-1434838220*z^10+ 18809447488*z^12-171091049284*z^14-5238606318708*z^18+1109772923288*z^16-\ 2350035*z^50+73095629*z^48+18277857198488*z^20+18277857198488*z^36-\ 47711524467764*z^34-140932834907250*z^30-171091049284*z^42+18809447488*z^44-\ 1434838220*z^46+z^56-387*z^54+43472*z^52+94048250595546*z^32-5238606318708*z^38 +1109772923288*z^40)/(-1+z^2)/(1+1086822234303026*z^28-939568140632360*z^26-694 *z^2+606625084308410*z^24-291855229503468*z^22+96591*z^4-6127586*z^6+219362229* z^8-4896361476*z^10+72309250588*z^12-735164525588*z^14-27460478870540*z^18+ 5289767316264*z^16-6127586*z^50+219362229*z^48+104225460884116*z^20+ 104225460884116*z^36-291855229503468*z^34-939568140632360*z^30-735164525588*z^ 42+72309250588*z^44-4896361476*z^46+z^56-694*z^54+96591*z^52+606625084308410*z^ 32-27460478870540*z^38+5289767316264*z^40) The first , 40, terms are: [0, 308, 0, 160247, 0, 85282027, 0, 45446031700, 0, 24219965748989, 0, 12907852612931893, 0, 6879148751285392980, 0, 3666193828275578956019, 0, 1953872160797138764631855, 0, 1041302397181533636120704692, 0, 554954773480407365177428191753, 0, 295759235206785213236313623801593, 0, 157622799893275324192170521601478836, 0, 84003960278113309726725932154475041343, 0, 44769318570567461094948563232876338513635, 0, 23859492798164685376296090086298897663551316, 0, 12715748525150196456404315326462550973946926661, 0, 6776768555922399950147697165420463432731388434669, 0, 3611631039234956543848791057200474664848000174970452, 0, 1924793307595841741618280634517852943479949158048258299] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 8950474 z - 66771685 z - 338 z + 302605402 z 22 4 6 8 10 - 868261842 z + 25286 z - 678107 z + 8950474 z - 66771685 z 12 14 18 16 + 302605402 z - 868261842 z - 1988247260 z + 1618252999 z 20 36 34 30 32 / 2 + 1618252999 z + z - 338 z - 678107 z + 25286 z ) / ((-1 + z ) / 36 34 32 30 28 26 (z - 630 z + 64931 z - 2157672 z + 34503397 z - 308297728 z 24 22 20 18 + 1646900027 z - 5416142450 z + 11062861097 z - 14039905376 z 16 14 12 10 + 11062861097 z - 5416142450 z + 1646900027 z - 308297728 z 8 6 4 2 + 34503397 z - 2157672 z + 64931 z - 630 z + 1)) And in Maple-input format, it is: -(1+8950474*z^28-66771685*z^26-338*z^2+302605402*z^24-868261842*z^22+25286*z^4-\ 678107*z^6+8950474*z^8-66771685*z^10+302605402*z^12-868261842*z^14-1988247260*z ^18+1618252999*z^16+1618252999*z^20+z^36-338*z^34-678107*z^30+25286*z^32)/(-1+z ^2)/(z^36-630*z^34+64931*z^32-2157672*z^30+34503397*z^28-308297728*z^26+ 1646900027*z^24-5416142450*z^22+11062861097*z^20-14039905376*z^18+11062861097*z ^16-5416142450*z^14+1646900027*z^12-308297728*z^10+34503397*z^8-2157672*z^6+ 64931*z^4-630*z^2+1) The first , 40, terms are: [0, 293, 0, 144608, 0, 73582771, 0, 37573595497, 0, 19195719219923, 0, 9807478925236829, 0, 5010891222181566208, 0, 2560196153901434602003, 0, 1308071867607706713918383, 0, 668328504983708028653676659, 0, 341466706703449617288748871215, 0, 174464370442498891047248853003520, 0, 89138460527992684505462439183235625, 0, 45543196729915072091585348840062286975, 0, 23269223588768277996522129309445715222197, 0, 11888861680818604151243937763255822023515791, 0, 6074333830969293850706505009159385457242961504, 0, 3103537788616764648678666322916379541196828345537, 0, 1585679528554205004656054615060035866836641305456997, 0, 810165604072291942920802093634222939942655101903472013] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 59156178383184296870 z - 13188397931566784600 z - 539 z 24 22 4 + 2420040922934994504 z - 362897144818678328 z + 107522 z 6 8 10 12 - 11192339 z + 709417167 z - 29976207038 z + 895873232720 z 14 18 16 - 19733955106988 z - 4290913753271866 z + 330152832626104 z 50 48 - 6885534557950192662466 z + 10535016002969574163680 z 20 36 + 44082418045001034 z + 3788844638007717691072 z 34 66 80 - 1751769971832227076196 z - 362897144818678328 z + 709417167 z 88 84 86 82 64 + z + 107522 z - 539 z - 11192339 z + 2420040922934994504 z 30 42 - 219686899339415427814 z - 13589729602700449552290 z 44 46 + 14792070847030827859340 z - 13589729602700449552290 z 58 56 - 219686899339415427814 z + 678759940310506911768 z 54 52 - 1751769971832227076196 z + 3788844638007717691072 z 60 70 68 + 59156178383184296870 z - 4290913753271866 z + 44082418045001034 z 78 32 38 - 29976207038 z + 678759940310506911768 z - 6885534557950192662466 z 40 62 + 10535016002969574163680 z - 13188397931566784600 z 76 74 72 / + 895873232720 z - 19733955106988 z + 330152832626104 z ) / ( / 2 28 26 (-1 + z ) (1 + 316013458446679595352 z - 66252651925239256678 z 2 24 22 4 - 847 z + 11389280328935374412 z - 1595134419215739434 z + 212155 z 6 8 10 12 - 25512677 z + 1802468453 z - 83480307812 z + 2711807149906 z 14 18 16 - 64639940958476 z - 16341437091747132 z + 1167178445107234 z 50 48 - 45927955778971608186124 z + 71844787120664604138258 z 20 36 + 180537884012437856 z + 24515014641395674365638 z 34 66 80 - 10910334907570727534948 z - 1595134419215739434 z + 1802468453 z 88 84 86 82 64 + z + 212155 z - 847 z - 25512677 z + 11389280328935374412 z 30 42 - 1242247702231199899444 z - 93932823788503914425564 z 44 46 + 102706247593136260966690 z - 93932823788503914425564 z 58 56 - 1242247702231199899444 z + 4040716365007955967030 z 54 52 - 10910334907570727534948 z + 24515014641395674365638 z 60 70 + 316013458446679595352 z - 16341437091747132 z 68 78 32 + 180537884012437856 z - 83480307812 z + 4040716365007955967030 z 38 40 - 45927955778971608186124 z + 71844787120664604138258 z 62 76 74 - 66252651925239256678 z + 2711807149906 z - 64639940958476 z 72 + 1167178445107234 z )) And in Maple-input format, it is: -(1+59156178383184296870*z^28-13188397931566784600*z^26-539*z^2+ 2420040922934994504*z^24-362897144818678328*z^22+107522*z^4-11192339*z^6+ 709417167*z^8-29976207038*z^10+895873232720*z^12-19733955106988*z^14-\ 4290913753271866*z^18+330152832626104*z^16-6885534557950192662466*z^50+ 10535016002969574163680*z^48+44082418045001034*z^20+3788844638007717691072*z^36 -1751769971832227076196*z^34-362897144818678328*z^66+709417167*z^80+z^88+107522 *z^84-539*z^86-11192339*z^82+2420040922934994504*z^64-219686899339415427814*z^ 30-13589729602700449552290*z^42+14792070847030827859340*z^44-\ 13589729602700449552290*z^46-219686899339415427814*z^58+678759940310506911768*z ^56-1751769971832227076196*z^54+3788844638007717691072*z^52+ 59156178383184296870*z^60-4290913753271866*z^70+44082418045001034*z^68-\ 29976207038*z^78+678759940310506911768*z^32-6885534557950192662466*z^38+ 10535016002969574163680*z^40-13188397931566784600*z^62+895873232720*z^76-\ 19733955106988*z^74+330152832626104*z^72)/(-1+z^2)/(1+316013458446679595352*z^ 28-66252651925239256678*z^26-847*z^2+11389280328935374412*z^24-\ 1595134419215739434*z^22+212155*z^4-25512677*z^6+1802468453*z^8-83480307812*z^ 10+2711807149906*z^12-64639940958476*z^14-16341437091747132*z^18+ 1167178445107234*z^16-45927955778971608186124*z^50+71844787120664604138258*z^48 +180537884012437856*z^20+24515014641395674365638*z^36-10910334907570727534948*z ^34-1595134419215739434*z^66+1802468453*z^80+z^88+212155*z^84-847*z^86-25512677 *z^82+11389280328935374412*z^64-1242247702231199899444*z^30-\ 93932823788503914425564*z^42+102706247593136260966690*z^44-\ 93932823788503914425564*z^46-1242247702231199899444*z^58+4040716365007955967030 *z^56-10910334907570727534948*z^54+24515014641395674365638*z^52+ 316013458446679595352*z^60-16341437091747132*z^70+180537884012437856*z^68-\ 83480307812*z^78+4040716365007955967030*z^32-45927955778971608186124*z^38+ 71844787120664604138258*z^40-66252651925239256678*z^62+2711807149906*z^76-\ 64639940958476*z^74+1167178445107234*z^72) The first , 40, terms are: [0, 309, 0, 156552, 0, 81470971, 0, 42571903429, 0, 22265228642171, 0, 11647119218006149, 0, 6092981594788383520, 0, 3187467707606262956575, 0, 1667488209719747707512779, 0, 872328358373567811986767147, 0, 456349170721241965058447789775, 0, 238734152913603512638694331216112, 0, 124891201311959045991387364644321805, 0, 65335487188339967368150819019244893523, 0, 34179556628487148373457906594660825668333, 0, 17880667026278333057126741494228068677026547, 0, 9354078427222586426440694958019518522555626392, 0, 4893485410494496905375898771708176845161422896213, 0, 2559974202595600642493992829240538989524411304153361, 0, 1339222939931930010996382271234491360969750932179454353] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 4736386474697778 z - 2648986622177820 z - 411 z 24 22 4 6 + 1169582376368474 z - 405904526937674 z + 52200 z - 3202145 z 8 10 12 14 + 113645529 z - 2588367800 z + 40393416138 z - 451236391922 z 18 16 50 - 23118490935686 z + 3719548140932 z - 451236391922 z 48 20 36 + 3719548140932 z + 110055964084184 z + 4736386474697778 z 34 64 30 42 - 6704823827093552 z + z - 6704823827093552 z - 405904526937674 z 44 46 58 56 + 110055964084184 z - 23118490935686 z - 3202145 z + 113645529 z 54 52 60 32 - 2588367800 z + 40393416138 z + 52200 z + 7527004862203612 z 38 40 62 / 2 - 2648986622177820 z + 1169582376368474 z - 411 z ) / ((-1 + z ) (1 / 28 26 2 + 31117572650936662 z - 16718234327466431 z - 713 z 24 22 4 6 + 6995982078100714 z - 2274962539620769 z + 112396 z - 8043334 z 8 10 12 14 + 325391167 z - 8341011959 z + 145194595548 z - 1796211409573 z 18 16 50 - 110727199603995 z + 16292150595062 z - 1796211409573 z 48 20 36 + 16292150595062 z + 572476158442288 z + 31117572650936662 z 34 64 30 - 45162306806043847 z + z - 45162306806043847 z 42 44 46 - 2274962539620769 z + 572476158442288 z - 110727199603995 z 58 56 54 52 - 8043334 z + 325391167 z - 8341011959 z + 145194595548 z 60 32 38 + 112396 z + 51131141826178542 z - 16718234327466431 z 40 62 + 6995982078100714 z - 713 z )) And in Maple-input format, it is: -(1+4736386474697778*z^28-2648986622177820*z^26-411*z^2+1169582376368474*z^24-\ 405904526937674*z^22+52200*z^4-3202145*z^6+113645529*z^8-2588367800*z^10+ 40393416138*z^12-451236391922*z^14-23118490935686*z^18+3719548140932*z^16-\ 451236391922*z^50+3719548140932*z^48+110055964084184*z^20+4736386474697778*z^36 -6704823827093552*z^34+z^64-6704823827093552*z^30-405904526937674*z^42+ 110055964084184*z^44-23118490935686*z^46-3202145*z^58+113645529*z^56-2588367800 *z^54+40393416138*z^52+52200*z^60+7527004862203612*z^32-2648986622177820*z^38+ 1169582376368474*z^40-411*z^62)/(-1+z^2)/(1+31117572650936662*z^28-\ 16718234327466431*z^26-713*z^2+6995982078100714*z^24-2274962539620769*z^22+ 112396*z^4-8043334*z^6+325391167*z^8-8341011959*z^10+145194595548*z^12-\ 1796211409573*z^14-110727199603995*z^18+16292150595062*z^16-1796211409573*z^50+ 16292150595062*z^48+572476158442288*z^20+31117572650936662*z^36-\ 45162306806043847*z^34+z^64-45162306806043847*z^30-2274962539620769*z^42+ 572476158442288*z^44-110727199603995*z^46-8043334*z^58+325391167*z^56-\ 8341011959*z^54+145194595548*z^52+112396*z^60+51131141826178542*z^32-\ 16718234327466431*z^38+6995982078100714*z^40-713*z^62) The first , 40, terms are: [0, 303, 0, 155433, 0, 81660720, 0, 42976280101, 0, 22621218576247, 0, 11907234834583825, 0, 6267678709353379273, 0, 3299154364138116182855, 0, 1736595050809484333340261, 0, 914101629652655944379335376, 0, 481160987580830802210217427449, 0, 253271505564632678511444375834415, 0, 133315994411504980458137706446148041, 0, 70174314818078893982571944088403850009, 0, 36938061947669126309544252960017732775791, 0, 19443302353389607046755508918293118855237369, 0, 10234484065268660222813622160497264322315937936, 0, 5387184860805175855982141193544615853677297540933, 0, 2835683805789055872581123385295940491886346180945575, 0, 1492635365999389522384772789786659511487901406508083385] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 7332872323482863198540 z - 948205209508596227781 z - 581 z 24 22 4 + 103794694231888018904 z - 9546293938728033747 z + 132208 z 6 102 8 10 - 16655587 z - 75332322313 z + 1343708496 z - 75332322313 z 12 14 18 + 3094505563976 z - 96682710348879 z - 46103109060075305 z 16 50 + 2362127217599632 z - 2493265526536943447418486273 z 48 20 + 1566233393779431774299533992 z + 731051522522464752 z 36 34 + 5536555933539129664142688 z - 1322024434637132955383985 z 66 80 - 860496860711258404294296803 z + 272495041791924840810128 z 100 90 88 + 3094505563976 z - 9546293938728033747 z + 103794694231888018904 z 84 94 + 7332872323482863198540 z - 46103109060075305 z 86 96 98 - 948205209508596227781 z + 2362127217599632 z - 96682710348879 z 92 82 + 731051522522464752 z - 48306276125301197335367 z 64 112 110 106 + 1566233393779431774299533992 z + z - 581 z - 16655587 z 108 30 + 132208 z - 48306276125301197335367 z 42 44 - 173081214293096842555179685 z + 413094459103944661234320616 z 46 58 - 860496860711258404294296803 z - 4237523573097468352569243287 z 56 54 + 4527667473846338244194786862 z - 4237523573097468352569243287 z 52 60 + 3473636557179788641709429360 z + 3473636557179788641709429360 z 70 68 - 173081214293096842555179685 z + 413094459103944661234320616 z 78 32 - 1322024434637132955383985 z + 272495041791924840810128 z 38 40 - 20076811630048939598510675 z + 63200639415183958999399176 z 62 76 - 2493265526536943447418486273 z + 5536555933539129664142688 z 74 72 - 20076811630048939598510675 z + 63200639415183958999399176 z 104 / 2 28 + 1343708496 z ) / ((-1 + z ) (1 + 32498229774731952046848 z / 26 2 24 - 3984869162184224184964 z - 895 z + 412915834730970022502 z 22 4 6 102 - 35890630767470515922 z + 248172 z - 35545960 z - 192543588698 z 8 10 12 + 3162959186 z - 192543588698 z + 8512215705852 z 14 18 16 - 284674823998363 z - 154033192456743921 z + 7418251465269236 z 50 48 - 16552587432733776117757513845 z + 10215365765817494081881840380 z 20 36 + 2593235778726546212 z + 29720421596322984735522260 z 34 66 - 6789033157505150436263323 z - 5487974939637193996755562499 z 80 100 + 1335257730015186280821912 z + 8512215705852 z 90 88 - 35890630767470515922 z + 412915834730970022502 z 84 94 + 32498229774731952046848 z - 154033192456743921 z 86 96 98 - 3984869162184224184964 z + 7418251465269236 z - 284674823998363 z 92 82 + 2593235778726546212 z - 225345389487596046142585 z 64 112 110 106 + 10215365765817494081881840380 z + z - 895 z - 35545960 z 108 30 + 248172 z - 225345389487596046142585 z 42 44 - 1041850995644855735413786634 z + 2564814057663819398273931324 z 46 58 - 5487974939637193996755562499 z - 28717906617662776213268842702 z 56 54 + 30764094520774565289207742532 z - 28717906617662776213268842702 z 52 60 + 23358933088436857976395038940 z + 23358933088436857976395038940 z 70 68 - 1041850995644855735413786634 z + 2564814057663819398273931324 z 78 32 - 6789033157505150436263323 z + 1335257730015186280821912 z 38 40 - 112334860393397804659047676 z + 367424480155932090512362778 z 62 76 - 16552587432733776117757513845 z + 29720421596322984735522260 z 74 72 - 112334860393397804659047676 z + 367424480155932090512362778 z 104 + 3162959186 z )) And in Maple-input format, it is: -(1+7332872323482863198540*z^28-948205209508596227781*z^26-581*z^2+ 103794694231888018904*z^24-9546293938728033747*z^22+132208*z^4-16655587*z^6-\ 75332322313*z^102+1343708496*z^8-75332322313*z^10+3094505563976*z^12-\ 96682710348879*z^14-46103109060075305*z^18+2362127217599632*z^16-\ 2493265526536943447418486273*z^50+1566233393779431774299533992*z^48+ 731051522522464752*z^20+5536555933539129664142688*z^36-\ 1322024434637132955383985*z^34-860496860711258404294296803*z^66+ 272495041791924840810128*z^80+3094505563976*z^100-9546293938728033747*z^90+ 103794694231888018904*z^88+7332872323482863198540*z^84-46103109060075305*z^94-\ 948205209508596227781*z^86+2362127217599632*z^96-96682710348879*z^98+ 731051522522464752*z^92-48306276125301197335367*z^82+ 1566233393779431774299533992*z^64+z^112-581*z^110-16655587*z^106+132208*z^108-\ 48306276125301197335367*z^30-173081214293096842555179685*z^42+ 413094459103944661234320616*z^44-860496860711258404294296803*z^46-\ 4237523573097468352569243287*z^58+4527667473846338244194786862*z^56-\ 4237523573097468352569243287*z^54+3473636557179788641709429360*z^52+ 3473636557179788641709429360*z^60-173081214293096842555179685*z^70+ 413094459103944661234320616*z^68-1322024434637132955383985*z^78+ 272495041791924840810128*z^32-20076811630048939598510675*z^38+ 63200639415183958999399176*z^40-2493265526536943447418486273*z^62+ 5536555933539129664142688*z^76-20076811630048939598510675*z^74+ 63200639415183958999399176*z^72+1343708496*z^104)/(-1+z^2)/(1+ 32498229774731952046848*z^28-3984869162184224184964*z^26-895*z^2+ 412915834730970022502*z^24-35890630767470515922*z^22+248172*z^4-35545960*z^6-\ 192543588698*z^102+3162959186*z^8-192543588698*z^10+8512215705852*z^12-\ 284674823998363*z^14-154033192456743921*z^18+7418251465269236*z^16-\ 16552587432733776117757513845*z^50+10215365765817494081881840380*z^48+ 2593235778726546212*z^20+29720421596322984735522260*z^36-\ 6789033157505150436263323*z^34-5487974939637193996755562499*z^66+ 1335257730015186280821912*z^80+8512215705852*z^100-35890630767470515922*z^90+ 412915834730970022502*z^88+32498229774731952046848*z^84-154033192456743921*z^94 -3984869162184224184964*z^86+7418251465269236*z^96-284674823998363*z^98+ 2593235778726546212*z^92-225345389487596046142585*z^82+ 10215365765817494081881840380*z^64+z^112-895*z^110-35545960*z^106+248172*z^108-\ 225345389487596046142585*z^30-1041850995644855735413786634*z^42+ 2564814057663819398273931324*z^44-5487974939637193996755562499*z^46-\ 28717906617662776213268842702*z^58+30764094520774565289207742532*z^56-\ 28717906617662776213268842702*z^54+23358933088436857976395038940*z^52+ 23358933088436857976395038940*z^60-1041850995644855735413786634*z^70+ 2564814057663819398273931324*z^68-6789033157505150436263323*z^78+ 1335257730015186280821912*z^32-112334860393397804659047676*z^38+ 367424480155932090512362778*z^40-16552587432733776117757513845*z^62+ 29720421596322984735522260*z^76-112334860393397804659047676*z^74+ 367424480155932090512362778*z^72+3162959186*z^104) The first , 40, terms are: [0, 315, 0, 165381, 0, 88863816, 0, 47851384539, 0, 25774310936145, 0, 13883450295040779, 0, 7478429213973415171, 0, 4028318091234161805021, 0, 2169887334665046495097775, 0, 1168828040330723619883286856, 0, 629599044240225057655631504153, 0, 339138815130704774428970295463127, 0, 182679972259113830424832345626201801, 0, 98402101959561389025947809962522647497, 0, 53005119008561530231468078423200849412703, 0, 28551652710294974335754963867850916935406321, 0, 15379587627331816134369695766169664746361935816, 0, 8284344096882771462022390242696631666429519283751, 0, 4462431553989790942357793472830312724508933642951301, 0, 2403726250523165374469384364465604356589552809565953827] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 200224311081169835 z - 77882563306796451 z - 452 z 24 22 4 6 + 24337026263454455 z - 6072214827960708 z + 68537 z - 5181985 z 8 10 12 14 + 232597325 z - 6820005696 z + 138718054543 z - 2039454041882 z 18 16 50 - 186049045812946 z + 22326079650562 z - 6072214827960708 z 48 20 36 + 24337026263454455 z + 1200221042958365 z + 1056035946535346852 z 34 66 64 - 952341102085166428 z - 5181985 z + 232597325 z 30 42 44 - 415451004570043056 z - 415451004570043056 z + 200224311081169835 z 46 58 56 - 77882563306796451 z - 2039454041882 z + 22326079650562 z 54 52 60 70 - 186049045812946 z + 1200221042958365 z + 138718054543 z - 452 z 68 32 38 + 68537 z + 698132983537314651 z - 952341102085166428 z 40 62 72 / 2 + 698132983537314651 z - 6820005696 z + z ) / ((-1 + z ) (1 / 28 26 2 + 1237745112786786432 z - 455593775500382098 z - 776 z 24 22 4 6 + 133494269067749259 z - 30988886392747076 z + 143370 z - 12370938 z 8 10 12 14 + 621307071 z - 20227528336 z + 454883742988 z - 7368110023068 z 18 16 50 - 806577802900344 z + 88539282079633 z - 30988886392747076 z 48 20 36 + 133494269067749259 z + 5661249023162550 z + 7249916876981358754 z 34 66 64 - 6493487007645591316 z - 12370938 z + 621307071 z 30 42 - 2685737381287230412 z - 2685737381287230412 z 44 46 58 + 1237745112786786432 z - 455593775500382098 z - 7368110023068 z 56 54 52 + 88539282079633 z - 806577802900344 z + 5661249023162550 z 60 70 68 32 + 454883742988 z - 776 z + 143370 z + 4664673046647382145 z 38 40 62 - 6493487007645591316 z + 4664673046647382145 z - 20227528336 z 72 + z )) And in Maple-input format, it is: -(1+200224311081169835*z^28-77882563306796451*z^26-452*z^2+24337026263454455*z^ 24-6072214827960708*z^22+68537*z^4-5181985*z^6+232597325*z^8-6820005696*z^10+ 138718054543*z^12-2039454041882*z^14-186049045812946*z^18+22326079650562*z^16-\ 6072214827960708*z^50+24337026263454455*z^48+1200221042958365*z^20+ 1056035946535346852*z^36-952341102085166428*z^34-5181985*z^66+232597325*z^64-\ 415451004570043056*z^30-415451004570043056*z^42+200224311081169835*z^44-\ 77882563306796451*z^46-2039454041882*z^58+22326079650562*z^56-186049045812946*z ^54+1200221042958365*z^52+138718054543*z^60-452*z^70+68537*z^68+ 698132983537314651*z^32-952341102085166428*z^38+698132983537314651*z^40-\ 6820005696*z^62+z^72)/(-1+z^2)/(1+1237745112786786432*z^28-455593775500382098*z ^26-776*z^2+133494269067749259*z^24-30988886392747076*z^22+143370*z^4-12370938* z^6+621307071*z^8-20227528336*z^10+454883742988*z^12-7368110023068*z^14-\ 806577802900344*z^18+88539282079633*z^16-30988886392747076*z^50+ 133494269067749259*z^48+5661249023162550*z^20+7249916876981358754*z^36-\ 6493487007645591316*z^34-12370938*z^66+621307071*z^64-2685737381287230412*z^30-\ 2685737381287230412*z^42+1237745112786786432*z^44-455593775500382098*z^46-\ 7368110023068*z^58+88539282079633*z^56-806577802900344*z^54+5661249023162550*z^ 52+454883742988*z^60-776*z^70+143370*z^68+4664673046647382145*z^32-\ 6493487007645591316*z^38+4664673046647382145*z^40-20227528336*z^62+z^72) The first , 40, terms are: [0, 325, 0, 176916, 0, 97948605, 0, 54270401765, 0, 30071267345989, 0, 16662604105806353, 0, 9232818998268240748, 0, 5115944300048808342897, 0, 2834766544595317524055421, 0, 1570756228927001984480497785, 0, 870362723872122485565656170293, 0, 482271696384243755931581009451580, 0, 267228803296671896077132315360991061, 0, 148072619328088279999581083641483557593, 0, 82047669727958079209485795314841747206313, 0, 45462963634568401278517853619479208017198833, 0, 25191221996811861506880898168089035768369187492, 0, 13958563519825898625446399817434236854419235942793, 0, 7734499563446070301307881301123104410115949773116957, 0, 4285719186790189276254629500058523861424941589540351749] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 67201652448097174354 z - 15006702351799764694 z - 533 z 24 22 4 + 2752132609920380736 z - 411300572925954394 z + 105592 z 6 8 10 12 - 11072071 z + 714457807 z - 30885399246 z + 945011565706 z 14 18 16 - 21271963554352 z - 4779992041862402 z + 362487138690470 z 50 48 - 7664783493457930528298 z + 11691924222662445670416 z 20 36 + 49626252945457870 z + 4234102537836008709598 z 34 66 80 - 1966350247545332398288 z - 411300572925954394 z + 714457807 z 88 84 86 82 64 + z + 105592 z - 533 z - 11072071 z + 2752132609920380736 z 30 42 - 248762749262369809670 z - 15053373198621413320052 z 44 46 + 16374545891741889173136 z - 15053373198621413320052 z 58 56 - 248762749262369809670 z + 765389011994482374642 z 54 52 - 1966350247545332398288 z + 4234102537836008709598 z 60 70 68 + 67201652448097174354 z - 4779992041862402 z + 49626252945457870 z 78 32 38 - 30885399246 z + 765389011994482374642 z - 7664783493457930528298 z 40 62 + 11691924222662445670416 z - 15006702351799764694 z 76 74 72 / + 945011565706 z - 21271963554352 z + 362487138690470 z ) / ( / 2 28 26 (-1 + z ) (1 + 359065860611317314148 z - 75369284927320138308 z 2 24 22 4 - 869 z + 12936361382027729580 z - 1802838613856275572 z + 210517 z 6 8 10 12 - 25085725 z + 1790239385 z - 84612296008 z + 2815638020600 z 14 18 16 - 68738361746144 z - 18065437466269576 z + 1267930551581576 z 50 48 - 51002940783302863736824 z + 79492637304812112238126 z 20 36 + 202235631023635924 z + 27350909851509311043624 z 34 66 80 - 12236027594911215639328 z - 1802838613856275572 z + 1790239385 z 88 84 86 82 64 + z + 210517 z - 869 z - 25085725 z + 12936361382027729580 z 30 42 - 1406903236394500090424 z - 103690637290176883513542 z 44 46 + 113285123244704727022078 z - 103690637290176883513542 z 58 56 - 1406903236394500090424 z + 4555384655983850986584 z 54 52 - 12236027594911215639328 z + 27350909851509311043624 z 60 70 + 359065860611317314148 z - 18065437466269576 z 68 78 32 + 202235631023635924 z - 84612296008 z + 4555384655983850986584 z 38 40 - 51002940783302863736824 z + 79492637304812112238126 z 62 76 74 - 75369284927320138308 z + 2815638020600 z - 68738361746144 z 72 + 1267930551581576 z )) And in Maple-input format, it is: -(1+67201652448097174354*z^28-15006702351799764694*z^26-533*z^2+ 2752132609920380736*z^24-411300572925954394*z^22+105592*z^4-11072071*z^6+ 714457807*z^8-30885399246*z^10+945011565706*z^12-21271963554352*z^14-\ 4779992041862402*z^18+362487138690470*z^16-7664783493457930528298*z^50+ 11691924222662445670416*z^48+49626252945457870*z^20+4234102537836008709598*z^36 -1966350247545332398288*z^34-411300572925954394*z^66+714457807*z^80+z^88+105592 *z^84-533*z^86-11072071*z^82+2752132609920380736*z^64-248762749262369809670*z^ 30-15053373198621413320052*z^42+16374545891741889173136*z^44-\ 15053373198621413320052*z^46-248762749262369809670*z^58+765389011994482374642*z ^56-1966350247545332398288*z^54+4234102537836008709598*z^52+ 67201652448097174354*z^60-4779992041862402*z^70+49626252945457870*z^68-\ 30885399246*z^78+765389011994482374642*z^32-7664783493457930528298*z^38+ 11691924222662445670416*z^40-15006702351799764694*z^62+945011565706*z^76-\ 21271963554352*z^74+362487138690470*z^72)/(-1+z^2)/(1+359065860611317314148*z^ 28-75369284927320138308*z^26-869*z^2+12936361382027729580*z^24-\ 1802838613856275572*z^22+210517*z^4-25085725*z^6+1790239385*z^8-84612296008*z^ 10+2815638020600*z^12-68738361746144*z^14-18065437466269576*z^18+ 1267930551581576*z^16-51002940783302863736824*z^50+79492637304812112238126*z^48 +202235631023635924*z^20+27350909851509311043624*z^36-12236027594911215639328*z ^34-1802838613856275572*z^66+1790239385*z^80+z^88+210517*z^84-869*z^86-25085725 *z^82+12936361382027729580*z^64-1406903236394500090424*z^30-\ 103690637290176883513542*z^42+113285123244704727022078*z^44-\ 103690637290176883513542*z^46-1406903236394500090424*z^58+ 4555384655983850986584*z^56-12236027594911215639328*z^54+ 27350909851509311043624*z^52+359065860611317314148*z^60-18065437466269576*z^70+ 202235631023635924*z^68-84612296008*z^78+4555384655983850986584*z^32-\ 51002940783302863736824*z^38+79492637304812112238126*z^40-75369284927320138308* z^62+2815638020600*z^76-68738361746144*z^74+1267930551581576*z^72) The first , 40, terms are: [0, 337, 0, 187396, 0, 106021609, 0, 60049875225, 0, 34016074745585, 0, 19269208134203457, 0, 10915525974529901364, 0, 6183375759522240407145, 0, 3502729799907949467916777, 0, 1984210024411657106803109513, 0, 1124006032903763975455056830073, 0, 636721691125142571144643793555316, 0, 360687131648415946823797634705771985, 0, 204320362806529896501723320692366627633, 0, 115742445444715509333573649294682477874809, 0, 65565240260510124963545114463632154546565977, 0, 37141091272966348641562606894624394964279830916, 0, 21039512025973146905830894339543286446106698540609, 0, 11918364569251787379622061695155519867608831332994561, 0, 6751459531487218154632592039829842554257465807752165057] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 56769647266579589253 z - 12281767270114740061 z - 509 z 24 22 4 6 + 2188110559452302209 z - 319017413479656884 z + 94997 z - 9429854 z 8 10 12 14 + 580841550 z - 24209671542 z + 721902245343 z - 16005739025927 z 18 16 50 - 3593133218652916 z + 271367135942707 z - 7490760735358699007647 z 48 20 + 11621653841884404414795 z + 37762477281400668 z 36 34 + 4045488224184867118471 z - 1828309747875233036626 z 66 80 88 84 86 - 319017413479656884 z + 580841550 z + z + 94997 z - 509 z 82 64 30 - 9429854 z + 2188110559452302209 z - 217167517207289943834 z 42 44 - 15120635098194653602328 z + 16506208032240541482120 z 46 58 - 15120635098194653602328 z - 217167517207289943834 z 56 54 + 690235617949219379218 z - 1828309747875233036626 z 52 60 + 4045488224184867118471 z + 56769647266579589253 z 70 68 78 - 3593133218652916 z + 37762477281400668 z - 24209671542 z 32 38 + 690235617949219379218 z - 7490760735358699007647 z 40 62 + 11621653841884404414795 z - 12281767270114740061 z 76 74 72 / + 721902245343 z - 16005739025927 z + 271367135942707 z ) / (-1 / 28 26 2 - 357378656746822734016 z + 70373619097402713867 z + 833 z 24 22 4 - 11422136662580218371 z + 1518785847044564646 z - 189890 z 6 8 10 12 + 21487414 z - 1470907499 z + 67406685235 z - 2199764655112 z 14 18 16 + 53285360344660 z + 14278867309075173 z - 986769272831373 z 50 48 + 128744028092150727639111 z - 183449975094129338360759 z 20 36 - 164157695709689590 z - 37209177635277765471670 z 34 66 80 + 15300368834389208397861 z + 11422136662580218371 z - 67406685235 z 90 88 84 86 82 + z - 833 z - 21487414 z + 189890 z + 1470907499 z 64 30 - 70373619097402713867 z + 1502953294998819555948 z 42 44 + 183449975094129338360759 z - 218967094481352120468164 z 46 58 + 218967094481352120468164 z + 5253102438443831708381 z 56 54 - 15300368834389208397861 z + 37209177635277765471670 z 52 60 - 75658414292078068682506 z - 1502953294998819555948 z 70 68 78 + 164157695709689590 z - 1518785847044564646 z + 2199764655112 z 32 38 - 5253102438443831708381 z + 75658414292078068682506 z 40 62 - 128744028092150727639111 z + 357378656746822734016 z 76 74 72 - 53285360344660 z + 986769272831373 z - 14278867309075173 z ) And in Maple-input format, it is: -(1+56769647266579589253*z^28-12281767270114740061*z^26-509*z^2+ 2188110559452302209*z^24-319017413479656884*z^22+94997*z^4-9429854*z^6+ 580841550*z^8-24209671542*z^10+721902245343*z^12-16005739025927*z^14-\ 3593133218652916*z^18+271367135942707*z^16-7490760735358699007647*z^50+ 11621653841884404414795*z^48+37762477281400668*z^20+4045488224184867118471*z^36 -1828309747875233036626*z^34-319017413479656884*z^66+580841550*z^80+z^88+94997* z^84-509*z^86-9429854*z^82+2188110559452302209*z^64-217167517207289943834*z^30-\ 15120635098194653602328*z^42+16506208032240541482120*z^44-\ 15120635098194653602328*z^46-217167517207289943834*z^58+690235617949219379218*z ^56-1828309747875233036626*z^54+4045488224184867118471*z^52+ 56769647266579589253*z^60-3593133218652916*z^70+37762477281400668*z^68-\ 24209671542*z^78+690235617949219379218*z^32-7490760735358699007647*z^38+ 11621653841884404414795*z^40-12281767270114740061*z^62+721902245343*z^76-\ 16005739025927*z^74+271367135942707*z^72)/(-1-357378656746822734016*z^28+ 70373619097402713867*z^26+833*z^2-11422136662580218371*z^24+1518785847044564646 *z^22-189890*z^4+21487414*z^6-1470907499*z^8+67406685235*z^10-2199764655112*z^ 12+53285360344660*z^14+14278867309075173*z^18-986769272831373*z^16+ 128744028092150727639111*z^50-183449975094129338360759*z^48-164157695709689590* z^20-37209177635277765471670*z^36+15300368834389208397861*z^34+ 11422136662580218371*z^66-67406685235*z^80+z^90-833*z^88-21487414*z^84+189890*z ^86+1470907499*z^82-70373619097402713867*z^64+1502953294998819555948*z^30+ 183449975094129338360759*z^42-218967094481352120468164*z^44+ 218967094481352120468164*z^46+5253102438443831708381*z^58-\ 15300368834389208397861*z^56+37209177635277765471670*z^54-\ 75658414292078068682506*z^52-1502953294998819555948*z^60+164157695709689590*z^ 70-1518785847044564646*z^68+2199764655112*z^78-5253102438443831708381*z^32+ 75658414292078068682506*z^38-128744028092150727639111*z^40+ 357378656746822734016*z^62-53285360344660*z^76+986769272831373*z^74-\ 14278867309075173*z^72) The first , 40, terms are: [0, 324, 0, 174999, 0, 96307367, 0, 53065332788, 0, 29242515239377, 0, 16114789979455529, 0, 8880457038797944436, 0, 4893798432347795699775, 0, 2696850356404205575172239, 0, 1486167027408272451293236932, 0, 818989614692341428670635743641, 0, 451324768104106888810120409998025, 0, 248713833058905897785169189821804484, 0, 137059995654029480620584475650292039935, 0, 75530348182263629681465007693318049488751, 0, 41622892728921021809263575490158025037395956, 0, 22937338974561943213656692279340886835269466457, 0, 12640195929207752211966446740071925957569443307361, 0, 6965696993271714314467272806350494393687555318852788, 0, 3838622033536448607586147438563063834196255580103743031] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 40354260966966 z - 46766997020100 z - 430 z 24 22 4 6 + 40354260966966 z - 25877365616428 z + 56569 z - 3311524 z 8 10 12 14 + 103686064 z - 1944755524 z + 23429875120 z - 189760124432 z 18 16 50 48 - 4254107600204 z + 1066294271936 z - 430 z + 56569 z 20 36 34 + 12262590831536 z + 1066294271936 z - 4254107600204 z 30 42 44 46 52 - 25877365616428 z - 1944755524 z + 103686064 z - 3311524 z + z 32 38 40 / 2 + 12262590831536 z - 189760124432 z + 23429875120 z ) / ((-1 + z ) / 28 26 2 (1 + 284546505947962 z - 334913960033790 z - 761 z 24 22 4 6 + 284546505947962 z - 174345513296148 z + 129237 z - 8867516 z 8 10 12 14 + 317456088 z - 6774901436 z + 92857690508 z - 854552550680 z 18 16 50 48 - 24228046698004 z + 5428705505988 z - 761 z + 129237 z 20 36 34 + 76823284716056 z + 5428705505988 z - 24228046698004 z 30 42 44 46 52 - 174345513296148 z - 6774901436 z + 317456088 z - 8867516 z + z 32 38 40 + 76823284716056 z - 854552550680 z + 92857690508 z )) And in Maple-input format, it is: -(1+40354260966966*z^28-46766997020100*z^26-430*z^2+40354260966966*z^24-\ 25877365616428*z^22+56569*z^4-3311524*z^6+103686064*z^8-1944755524*z^10+ 23429875120*z^12-189760124432*z^14-4254107600204*z^18+1066294271936*z^16-430*z^ 50+56569*z^48+12262590831536*z^20+1066294271936*z^36-4254107600204*z^34-\ 25877365616428*z^30-1944755524*z^42+103686064*z^44-3311524*z^46+z^52+ 12262590831536*z^32-189760124432*z^38+23429875120*z^40)/(-1+z^2)/(1+ 284546505947962*z^28-334913960033790*z^26-761*z^2+284546505947962*z^24-\ 174345513296148*z^22+129237*z^4-8867516*z^6+317456088*z^8-6774901436*z^10+ 92857690508*z^12-854552550680*z^14-24228046698004*z^18+5428705505988*z^16-761*z ^50+129237*z^48+76823284716056*z^20+5428705505988*z^36-24228046698004*z^34-\ 174345513296148*z^30-6774901436*z^42+317456088*z^44-8867516*z^46+z^52+ 76823284716056*z^32-854552550680*z^38+92857690508*z^40) The first , 40, terms are: [0, 332, 0, 179555, 0, 99346803, 0, 55124757452, 0, 30602399632417, 0, 16990404373852241, 0, 9433203153524990988, 0, 5237402959211715732867, 0, 2907856931910351243704403, 0, 1614470544947041267029107084, 0, 896369818383069501987158891537, 0, 497673281573285868526773823173681, 0, 276313068859436746464788148369728780, 0, 153411715795013228073327222948723465011, 0, 85175683657766092932713554448509277696291, 0, 47290371853284693033065261342488177674492812, 0, 26256076546556283066754800442318260768229845297, 0, 14577630257539031610371333294667976920743264069569, 0, 8093642762989085200222668464532544683252855796804684, 0, 4493669548314135074739312157637848670996063330905846611] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 24 22 4 6 8 f(z) = - (1 - 281 z + z - 281 z + 9347 z - 109620 z + 583945 z 10 12 14 18 16 - 1533315 z + 2102150 z - 1533315 z - 109620 z + 583945 z 20 / 2 24 22 20 18 + 9347 z ) / ((-1 + z ) (z - 624 z + 27836 z - 441904 z / 16 14 12 10 8 + 3147356 z - 10647640 z + 16361006 z - 10647640 z + 3147356 z 6 4 2 - 441904 z + 27836 z - 624 z + 1)) And in Maple-input format, it is: -(1-281*z^2+z^24-281*z^22+9347*z^4-109620*z^6+583945*z^8-1533315*z^10+2102150*z ^12-1533315*z^14-109620*z^18+583945*z^16+9347*z^20)/(-1+z^2)/(z^24-624*z^22+ 27836*z^20-441904*z^18+3147356*z^16-10647640*z^14+16361006*z^12-10647640*z^10+ 3147356*z^8-441904*z^6+27836*z^4-624*z^2+1) The first , 40, terms are: [0, 344, 0, 195887, 0, 112999255, 0, 65208175600, 0, 37629944468321, 0, 21715272516366833, 0, 12531325100856080224, 0, 7231505326780350975511, 0, 4173115681824561700199183, 0, 2408197699780824459641360936, 0, 1389708937734081844116512564881, 0, 801965275439704925951193360180913, 0, 462793528592925422882232878482843400, 0, 267066239233438804977751861816817520527, 0, 154117055947490363031171710645674056653527, 0, 88936988074934213102719595277343361706779392, 0, 51323247769123980262418474274847796625303057425, 0, 29617325913393183556961267957586707618736984574913, 0, 17091396830655476962150094877918566408387978704362832, 0, 9863005406941314739289932971701613943802230410199670551] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 16 14 8 12 6 f(z) = - (-295 z + 11755 z - 291878 z - 112983 z + 112983 z 10 4 18 2 / 20 16 + 291878 z - 11755 z + z + 295 z - 1) / (z + 36080 z / 14 8 12 6 10 - 575113 z + 2834567 z + 2834567 z - 575113 z - 4622128 z 4 18 2 + 36080 z - 671 z - 671 z + 1) And in Maple-input format, it is: -(-295*z^16+11755*z^14-291878*z^8-112983*z^12+112983*z^6+291878*z^10-11755*z^4+ z^18+295*z^2-1)/(z^20+36080*z^16-575113*z^14+2834567*z^8+2834567*z^12-575113*z^ 6-4622128*z^10+36080*z^4-671*z^18-671*z^2+1) The first , 40, terms are: [0, 376, 0, 227971, 0, 139864591, 0, 85837646680, 0, 52680794097781, 0, 32331584028045421, 0, 19842740775357405208, 0, 12178010251334319844375, 0, 7473964174871337072220987, 0, 4586967766859060744902845880, 0, 2815142379855998111637220668745, 0, 1727726686051690494231253815560761, 0, 1060351164848667433443807171481509688, 0, 650765310203866083400967339334827405899, 0, 399391732667332926665175516146068070937223, 0, 245117177608995864086338536267079642737361048, 0, 150434838392222657261603440561412330671585418749, 0, 92325804428908362135508324436766943941347008170469, 0, 56662766780262787151611657115356005125308519293265624, 0, 34775425560106481639939041821376513713587108062312539295] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 4930437 z + 144739577 z + 495 z - 2106216115 z 22 4 6 8 10 + 15427813780 z - 78435 z + 4930437 z - 144739577 z + 2106216115 z 12 14 18 16 - 15427813780 z + 55880737824 z + 104676780192 z - 104676780192 z 20 34 30 32 / 36 34 - 55880737824 z + z + 78435 z - 495 z ) / (z - 815 z / 32 30 28 26 + 180154 z - 13779103 z + 484043710 z - 8696613439 z 24 22 20 + 83713359571 z - 432925448442 z + 1173930076912 z 18 16 14 - 1642624609898 z + 1173930076912 z - 432925448442 z 12 10 8 6 + 83713359571 z - 8696613439 z + 484043710 z - 13779103 z 4 2 + 180154 z - 815 z + 1) And in Maple-input format, it is: -(-1-4930437*z^28+144739577*z^26+495*z^2-2106216115*z^24+15427813780*z^22-78435 *z^4+4930437*z^6-144739577*z^8+2106216115*z^10-15427813780*z^12+55880737824*z^ 14+104676780192*z^18-104676780192*z^16-55880737824*z^20+z^34+78435*z^30-495*z^ 32)/(z^36-815*z^34+180154*z^32-13779103*z^30+484043710*z^28-8696613439*z^26+ 83713359571*z^24-432925448442*z^22+1173930076912*z^20-1642624609898*z^18+ 1173930076912*z^16-432925448442*z^14+83713359571*z^12-8696613439*z^10+484043710 *z^8-13779103*z^6+180154*z^4-815*z^2+1) The first , 40, terms are: [0, 320, 0, 159081, 0, 80850401, 0, 41304007168, 0, 21140932594633, 0, 10828536433576505, 0, 5547987718469058144, 0, 2842805925713602176017, 0, 1456721538656273305358713, 0, 746470436167478433322099360, 0, 382517460106433561661082760465, 0, 196015720141278399397626739762545, 0, 100445600542252008319888513993549664, 0, 51472004510843746178080389179731946329, 0, 26376143625165812275229079773861205143473, 0, 13516104397496485213747206687420407093680928, 0, 6926148268528222708688116399465559369932872153, 0, 3549212759106038075554138777632976011803076007721, 0, 1818746979638881508575884328443467082637170768497728, 0, 931992755556383292785280232157521260794246055427530817] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 4016138087719 z + 8383333975114 z + 470 z 24 22 4 6 - 12056765814859 z + 12056765814859 z - 72749 z + 4994963 z 8 10 12 14 - 173256518 z + 3349476867 z - 38689263761 z + 279485933274 z 18 16 20 + 4016138087719 z - 1302609390905 z - 8383333975114 z 36 34 30 42 - 3349476867 z + 38689263761 z + 1302609390905 z + 72749 z 44 46 32 38 40 / - 470 z + z - 279485933274 z + 173256518 z - 4994963 z ) / (1 / 28 26 2 24 + 81001864902470 z - 141178947472154 z - 774 z + 169723553902934 z 22 4 6 8 - 141178947472154 z + 154954 z - 12904780 z + 533182858 z 10 12 14 - 12283006310 z + 169872067844 z - 1481600073050 z 18 16 48 20 - 31775925053316 z + 8414765732102 z + z + 81001864902470 z 36 34 30 42 + 169872067844 z - 1481600073050 z - 31775925053316 z - 12904780 z 44 46 32 38 + 154954 z - 774 z + 8414765732102 z - 12283006310 z 40 + 533182858 z ) And in Maple-input format, it is: -(-1-4016138087719*z^28+8383333975114*z^26+470*z^2-12056765814859*z^24+ 12056765814859*z^22-72749*z^4+4994963*z^6-173256518*z^8+3349476867*z^10-\ 38689263761*z^12+279485933274*z^14+4016138087719*z^18-1302609390905*z^16-\ 8383333975114*z^20-3349476867*z^36+38689263761*z^34+1302609390905*z^30+72749*z^ 42-470*z^44+z^46-279485933274*z^32+173256518*z^38-4994963*z^40)/(1+ 81001864902470*z^28-141178947472154*z^26-774*z^2+169723553902934*z^24-\ 141178947472154*z^22+154954*z^4-12904780*z^6+533182858*z^8-12283006310*z^10+ 169872067844*z^12-1481600073050*z^14-31775925053316*z^18+8414765732102*z^16+z^ 48+81001864902470*z^20+169872067844*z^36-1481600073050*z^34-31775925053316*z^30 -12904780*z^42+154954*z^44-774*z^46+8414765732102*z^32-12283006310*z^38+ 533182858*z^40) The first , 40, terms are: [0, 304, 0, 153091, 0, 79296235, 0, 41216349856, 0, 21436637605945, 0, 11150597053118185, 0, 5800307888335165120, 0, 3017216364398831316091, 0, 1569503926409351230711603, 0, 816429105592893201103319440, 0, 424692486745582792220556083089, 0, 220917786407732760415302313445617, 0, 114917664011549851053050533517812432, 0, 59778208561579495185710465460604880467, 0, 31095604406072018784011212060328920255195, 0, 16175402988457276814683284414453181618692736, 0, 8414168717359723936347689507652907957302997513, 0, 4376907039338396805153600252032924037202046012953, 0, 2276792381342759000392452442073338319250697751953120, 0, 1184348559645085296140773418889029608667790816672203659] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 48806462716586419 z + 29174673863030756 z + 510 z 24 22 4 6 - 13391494297211749 z + 4681310722883081 z - 91597 z + 8026359 z 8 10 12 14 - 401017150 z + 12457753995 z - 254136558625 z + 3534974394450 z 18 16 50 + 241139631649775 z - 34478041560679 z + 254136558625 z 48 20 36 - 3534974394450 z - 1232643869901220 z - 29174673863030756 z 34 30 42 + 48806462716586419 z + 63040601920402723 z + 1232643869901220 z 44 46 58 56 - 241139631649775 z + 34478041560679 z + 91597 z - 8026359 z 54 52 60 32 + 401017150 z - 12457753995 z - 510 z - 63040601920402723 z 38 40 62 / + 13391494297211749 z - 4681310722883081 z + z ) / (1 / 28 26 2 + 529303160724806704 z - 275807740731888942 z - 842 z 24 22 4 6 + 109951203970207064 z - 33297416501244166 z + 191768 z - 19649640 z 8 10 12 14 + 1118535964 z - 39321025934 z + 908430546744 z - 14367746340618 z 18 16 50 - 1287383905647420 z + 160190417665198 z - 14367746340618 z 48 20 36 + 160190417665198 z + 7590055298890288 z + 529303160724806704 z 34 64 30 - 781061850045048940 z + z - 781061850045048940 z 42 44 46 - 33297416501244166 z + 7590055298890288 z - 1287383905647420 z 58 56 54 52 - 19649640 z + 1118535964 z - 39321025934 z + 908430546744 z 60 32 38 + 191768 z + 888926556939807231 z - 275807740731888942 z 40 62 + 109951203970207064 z - 842 z ) And in Maple-input format, it is: -(-1-48806462716586419*z^28+29174673863030756*z^26+510*z^2-13391494297211749*z^ 24+4681310722883081*z^22-91597*z^4+8026359*z^6-401017150*z^8+12457753995*z^10-\ 254136558625*z^12+3534974394450*z^14+241139631649775*z^18-34478041560679*z^16+ 254136558625*z^50-3534974394450*z^48-1232643869901220*z^20-29174673863030756*z^ 36+48806462716586419*z^34+63040601920402723*z^30+1232643869901220*z^42-\ 241139631649775*z^44+34478041560679*z^46+91597*z^58-8026359*z^56+401017150*z^54 -12457753995*z^52-510*z^60-63040601920402723*z^32+13391494297211749*z^38-\ 4681310722883081*z^40+z^62)/(1+529303160724806704*z^28-275807740731888942*z^26-\ 842*z^2+109951203970207064*z^24-33297416501244166*z^22+191768*z^4-19649640*z^6+ 1118535964*z^8-39321025934*z^10+908430546744*z^12-14367746340618*z^14-\ 1287383905647420*z^18+160190417665198*z^16-14367746340618*z^50+160190417665198* z^48+7590055298890288*z^20+529303160724806704*z^36-781061850045048940*z^34+z^64 -781061850045048940*z^30-33297416501244166*z^42+7590055298890288*z^44-\ 1287383905647420*z^46-19649640*z^58+1118535964*z^56-39321025934*z^54+ 908430546744*z^52+191768*z^60+888926556939807231*z^32-275807740731888942*z^38+ 109951203970207064*z^40-842*z^62) The first , 40, terms are: [0, 332, 0, 179373, 0, 98988371, 0, 54756368584, 0, 30302184625411, 0, 16770771153467387, 0, 9281983211590644864, 0, 5137246693955009634995, 0, 2843285569764476426663569, 0, 1573659008206643446126512004, 0, 870965201449765779183721958537, 0, 482048770365385152112334036970669, 0, 266797131671561327441969274674192516, 0, 147662879524354363971748565003854279597, 0, 81726238416365954757431063675261526948343, 0, 45232614096128802308626062664297826461785584, 0, 25034669619277943580502232111104373495068982487, 0, 13855813896034833332781898072169720060772448254647, 0, 7668708300978091393288121724714620430427458061183912, 0, 4244361785367453294713829105449231948534010139600573279] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 420 z + 50411 z + 420 z - 2095317 z + 35075414 z 4 6 8 10 12 - 50411 z + 2095317 z - 35075414 z + 263716815 z - 946566751 z 14 18 16 20 30 + 1750889958 z + 946566751 z - 1750889958 z - 263716815 z + z ) / 32 30 28 26 24 / (z - 796 z + 122204 z - 6302576 z + 138925946 z / 22 20 18 16 - 1424454524 z + 7313051920 z - 19330108300 z + 26625569475 z 14 12 10 8 - 19330108300 z + 7313051920 z - 1424454524 z + 138925946 z 6 4 2 - 6302576 z + 122204 z - 796 z + 1) And in Maple-input format, it is: -(-1-420*z^28+50411*z^26+420*z^2-2095317*z^24+35075414*z^22-50411*z^4+2095317*z ^6-35075414*z^8+263716815*z^10-946566751*z^12+1750889958*z^14+946566751*z^18-\ 1750889958*z^16-263716815*z^20+z^30)/(z^32-796*z^30+122204*z^28-6302576*z^26+ 138925946*z^24-1424454524*z^22+7313051920*z^20-19330108300*z^18+26625569475*z^ 16-19330108300*z^14+7313051920*z^12-1424454524*z^10+138925946*z^8-6302576*z^6+ 122204*z^4-796*z^2+1) The first , 40, terms are: [0, 376, 0, 227503, 0, 139350943, 0, 85387492060, 0, 52321980571129, 0, 32060796522770629, 0, 19645561750535721844, 0, 12038007138650008682707, 0, 7376404796167140881550595, 0, 4519963071412848603590901568, 0, 2769650897922940826002909548637, 0, 1697130258628802611939845954524389, 0, 1039932908842200997920463521058195888, 0, 637228904142474028461564595000596534427, 0, 390468147341068642620045564189624239769835, 0, 239263117377170949648419257035058626427873156, 0, 146610779206626457930672752240698259957405859693, 0, 89837166777737016415434740095570756343058407907025, 0, 55048589048670429744424356324600227439822265008140236, 0, 33731553041367317327100688771189799072938044815378639623] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 302388426 z + 1846055375 z + 365 z - 6797699715 z 22 4 6 8 10 + 15808754852 z - 34578 z + 1404486 z - 28164514 z + 302388426 z 12 14 18 16 - 1846055375 z + 6797699715 z + 23930949592 z - 15808754852 z 20 36 34 30 32 38 - 23930949592 z - 365 z + 34578 z + 28164514 z - 1404486 z + z / 40 38 36 34 32 ) / (z - 705 z + 83721 z - 4127354 z + 102006606 z / 30 28 26 24 - 1367859202 z + 10588054143 z - 49636210879 z + 146015479031 z 22 20 18 - 276246483460 z + 341100036196 z - 276246483460 z 16 14 12 10 + 146015479031 z - 49636210879 z + 10588054143 z - 1367859202 z 8 6 4 2 + 102006606 z - 4127354 z + 83721 z - 705 z + 1) And in Maple-input format, it is: -(-1-302388426*z^28+1846055375*z^26+365*z^2-6797699715*z^24+15808754852*z^22-\ 34578*z^4+1404486*z^6-28164514*z^8+302388426*z^10-1846055375*z^12+6797699715*z^ 14+23930949592*z^18-15808754852*z^16-23930949592*z^20-365*z^36+34578*z^34+ 28164514*z^30-1404486*z^32+z^38)/(z^40-705*z^38+83721*z^36-4127354*z^34+ 102006606*z^32-1367859202*z^30+10588054143*z^28-49636210879*z^26+146015479031*z ^24-276246483460*z^22+341100036196*z^20-276246483460*z^18+146015479031*z^16-\ 49636210879*z^14+10588054143*z^12-1367859202*z^10+102006606*z^8-4127354*z^6+ 83721*z^4-705*z^2+1) The first , 40, terms are: [0, 340, 0, 190557, 0, 108600413, 0, 61939126836, 0, 35327828663521, 0, 20149764176253121, 0, 11492726442128610388, 0, 6555052539131671652573, 0, 3738774611045667016239069, 0, 2132467361598535789014076788, 0, 1216285420061163428296133368673, 0, 693727017676409761317797409636513, 0, 395677829493361675069094813737158836, 0, 225680910161129643175722811317644601181, 0, 128720563586720668040739279102592144125533, 0, 73417744895894047989843702051578507965995924, 0, 41874935250474988664403753248396794114100386561, 0, 23884010666875432773230962528983520785186584501409, 0, 13622611285803692532455021434004305789809734373130740, 0, 7769864987603591564089410198503871416136888355849322973] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 284054035698396 z - 248283349572314 z - 441 z 24 22 4 6 + 165598726259930 z - 83971399275312 z + 61082 z - 3733341 z 8 10 12 14 + 123917685 z - 2505481792 z + 33193340976 z - 302439646624 z 18 16 50 48 - 9230053544752 z + 1958864111720 z - 3733341 z + 123917685 z 20 36 34 + 32171826378432 z + 32171826378432 z - 83971399275312 z 30 42 44 - 248283349572314 z - 302439646624 z + 33193340976 z 46 56 54 52 32 - 2505481792 z + z - 441 z + 61082 z + 165598726259930 z 38 40 / 2 - 9230053544752 z + 1958864111720 z ) / ((-1 + z ) (1 / 28 26 2 + 2037606335028554 z - 1757702680608714 z - 758 z 24 22 4 6 + 1127833867240786 z - 537573497666754 z + 135867 z - 9730664 z 8 10 12 14 + 368300029 z - 8431379286 z + 126175144572 z - 1295559572146 z 18 16 50 48 - 49430638650742 z + 9416336453440 z - 9730664 z + 368300029 z 20 36 34 + 189831755913364 z + 189831755913364 z - 537573497666754 z 30 42 44 - 1757702680608714 z - 1295559572146 z + 126175144572 z 46 56 54 52 32 - 8431379286 z + z - 758 z + 135867 z + 1127833867240786 z 38 40 - 49430638650742 z + 9416336453440 z )) And in Maple-input format, it is: -(1+284054035698396*z^28-248283349572314*z^26-441*z^2+165598726259930*z^24-\ 83971399275312*z^22+61082*z^4-3733341*z^6+123917685*z^8-2505481792*z^10+ 33193340976*z^12-302439646624*z^14-9230053544752*z^18+1958864111720*z^16-\ 3733341*z^50+123917685*z^48+32171826378432*z^20+32171826378432*z^36-\ 83971399275312*z^34-248283349572314*z^30-302439646624*z^42+33193340976*z^44-\ 2505481792*z^46+z^56-441*z^54+61082*z^52+165598726259930*z^32-9230053544752*z^ 38+1958864111720*z^40)/(-1+z^2)/(1+2037606335028554*z^28-1757702680608714*z^26-\ 758*z^2+1127833867240786*z^24-537573497666754*z^22+135867*z^4-9730664*z^6+ 368300029*z^8-8431379286*z^10+126175144572*z^12-1295559572146*z^14-\ 49430638650742*z^18+9416336453440*z^16-9730664*z^50+368300029*z^48+ 189831755913364*z^20+189831755913364*z^36-537573497666754*z^34-1757702680608714 *z^30-1295559572146*z^42+126175144572*z^44-8431379286*z^46+z^56-758*z^54+135867 *z^52+1127833867240786*z^32-49430638650742*z^38+9416336453440*z^40) The first , 40, terms are: [0, 318, 0, 165819, 0, 88543061, 0, 47432606274, 0, 25426291193879, 0, 13631700253517023, 0, 7308534212003563538, 0, 3918442002614616129805, 0, 2100860373567594341929907, 0, 1126370045801712576673671886, 0, 603899994006226973958485220425, 0, 323779214087876576867678249158073, 0, 173593278450776153091470329129655598, 0, 93071528478132864543025883332856626755, 0, 49900027765456340502822747207409032911773, 0, 26753753933050208199398674599511543695706866, 0, 14343946918821562132343712232603492439055620559, 0, 7690465185751984483110060453469820112886155474727, 0, 4123220415412883410683565807611744024354473452506210, 0, 2210652565670056626053221196870101370128796533959072709] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 106130402224859305325 z - 22670259725015545359 z - 535 z 24 22 4 + 3972615333750196033 z - 567179057374899288 z + 107061 z 6 8 10 12 - 11427222 z + 754903474 z - 33568961538 z + 1060939736423 z 14 18 16 - 24757303778237 z - 6031586566685608 z + 438715001946387 z 50 48 - 14370148861486335215037 z + 22317772896561653240043 z 20 36 + 65418334669181460 z + 7746514520035297272311 z 34 66 80 - 3490360335493162601682 z - 567179057374899288 z + 754903474 z 88 84 86 82 64 + z + 107061 z - 535 z - 11427222 z + 3972615333750196033 z 30 42 - 409861096066723216406 z - 29051639193614789299200 z 44 46 + 31718444047791401559160 z - 29051639193614789299200 z 58 56 - 409861096066723216406 z + 1311574665750847131870 z 54 52 - 3490360335493162601682 z + 7746514520035297272311 z 60 70 68 + 106130402224859305325 z - 6031586566685608 z + 65418334669181460 z 78 32 - 33568961538 z + 1311574665750847131870 z 38 40 - 14370148861486335215037 z + 22317772896561653240043 z 62 76 74 - 22670259725015545359 z + 1060939736423 z - 24757303778237 z 72 / 2 28 + 438715001946387 z ) / ((-1 + z ) (1 + 556495502293363684816 z / 26 2 24 - 111413735851749753512 z - 860 z + 18238317407416794319 z 22 4 6 8 - 2426380831500202928 z + 210266 z - 25555452 z + 1868929743 z 10 12 14 - 90796384892 z + 3114487032324 z - 78608373972992 z 18 16 - 22287715669286992 z + 1503734829519041 z 50 48 - 95887394258472344249964 z + 152661048788640601251023 z 20 36 + 260284454238794226 z + 49971134018868203267090 z 34 66 80 - 21589676974720549912388 z - 2426380831500202928 z + 1868929743 z 88 84 86 82 64 + z + 210266 z - 860 z - 25555452 z + 18238317407416794319 z 30 42 - 2283359090142736513204 z - 201759477941486123630640 z 44 46 + 221406808522471633350364 z - 201759477941486123630640 z 58 56 - 2283359090142736513204 z + 7723188112967172887105 z 54 52 - 21589676974720549912388 z + 49971134018868203267090 z 60 70 + 556495502293363684816 z - 22287715669286992 z 68 78 32 + 260284454238794226 z - 90796384892 z + 7723188112967172887105 z 38 40 - 95887394258472344249964 z + 152661048788640601251023 z 62 76 74 - 111413735851749753512 z + 3114487032324 z - 78608373972992 z 72 + 1503734829519041 z )) And in Maple-input format, it is: -(1+106130402224859305325*z^28-22670259725015545359*z^26-535*z^2+ 3972615333750196033*z^24-567179057374899288*z^22+107061*z^4-11427222*z^6+ 754903474*z^8-33568961538*z^10+1060939736423*z^12-24757303778237*z^14-\ 6031586566685608*z^18+438715001946387*z^16-14370148861486335215037*z^50+ 22317772896561653240043*z^48+65418334669181460*z^20+7746514520035297272311*z^36 -3490360335493162601682*z^34-567179057374899288*z^66+754903474*z^80+z^88+107061 *z^84-535*z^86-11427222*z^82+3972615333750196033*z^64-409861096066723216406*z^ 30-29051639193614789299200*z^42+31718444047791401559160*z^44-\ 29051639193614789299200*z^46-409861096066723216406*z^58+1311574665750847131870* z^56-3490360335493162601682*z^54+7746514520035297272311*z^52+ 106130402224859305325*z^60-6031586566685608*z^70+65418334669181460*z^68-\ 33568961538*z^78+1311574665750847131870*z^32-14370148861486335215037*z^38+ 22317772896561653240043*z^40-22670259725015545359*z^62+1060939736423*z^76-\ 24757303778237*z^74+438715001946387*z^72)/(-1+z^2)/(1+556495502293363684816*z^ 28-111413735851749753512*z^26-860*z^2+18238317407416794319*z^24-\ 2426380831500202928*z^22+210266*z^4-25555452*z^6+1868929743*z^8-90796384892*z^ 10+3114487032324*z^12-78608373972992*z^14-22287715669286992*z^18+ 1503734829519041*z^16-95887394258472344249964*z^50+152661048788640601251023*z^ 48+260284454238794226*z^20+49971134018868203267090*z^36-21589676974720549912388 *z^34-2426380831500202928*z^66+1868929743*z^80+z^88+210266*z^84-860*z^86-\ 25555452*z^82+18238317407416794319*z^64-2283359090142736513204*z^30-\ 201759477941486123630640*z^42+221406808522471633350364*z^44-\ 201759477941486123630640*z^46-2283359090142736513204*z^58+ 7723188112967172887105*z^56-21589676974720549912388*z^54+ 49971134018868203267090*z^52+556495502293363684816*z^60-22287715669286992*z^70+ 260284454238794226*z^68-90796384892*z^78+7723188112967172887105*z^32-\ 95887394258472344249964*z^38+152661048788640601251023*z^40-\ 111413735851749753512*z^62+3114487032324*z^76-78608373972992*z^74+ 1503734829519041*z^72) The first , 40, terms are: [0, 326, 0, 176621, 0, 97582101, 0, 53988946062, 0, 29874624962061, 0, 16531313441526349, 0, 9147725600822666622, 0, 5061963417526699303645, 0, 2801075973171285274920117, 0, 1549996708415519157282242774, 0, 857702475813863077861686102481, 0, 474616193093480334270882970508609, 0, 262632482824800695281026828252403766, 0, 145329683307258628244732688565468707797, 0, 80419286384614727313523665872969220098237, 0, 44500624204468622343027768810324159122606654, 0, 24624759104629408396858328010115032769591602909, 0, 13626297873370914762096501305693221680806130844189, 0, 7540215640076088312039588785740526158675456855500878, 0, 4172435714175616297291423194958919402698029001589367509] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 96938059042564 z - 112523988461364 z - 470 z 24 22 4 6 + 96938059042564 z - 61849238703238 z + 72027 z - 4959994 z 8 10 12 14 + 179008296 z - 3746507542 z + 48834280166 z - 417419782704 z 18 16 50 48 - 9926198646058 z + 2430917473530 z - 470 z + 72027 z 20 36 34 + 29040612364280 z + 2430917473530 z - 9926198646058 z 30 42 44 46 52 - 61849238703238 z - 3746507542 z + 179008296 z - 4959994 z + z 32 38 40 / + 29040612364280 z - 417419782704 z + 48834280166 z ) / (-1 / 28 26 2 - 1509640288714000 z + 1509640288714000 z + 804 z 24 22 4 6 - 1114121454755484 z + 604949021926344 z - 160764 z + 13125651 z 8 10 12 14 - 547753896 z + 13288593872 z - 202842145558 z + 2050037782016 z 18 16 50 48 + 69071988835994 z - 14199749177776 z + 160764 z - 13125651 z 20 36 34 - 240184688706080 z - 69071988835994 z + 240184688706080 z 30 42 44 + 1114121454755484 z + 202842145558 z - 13288593872 z 46 54 52 32 38 + 547753896 z + z - 804 z - 604949021926344 z + 14199749177776 z 40 - 2050037782016 z ) And in Maple-input format, it is: -(1+96938059042564*z^28-112523988461364*z^26-470*z^2+96938059042564*z^24-\ 61849238703238*z^22+72027*z^4-4959994*z^6+179008296*z^8-3746507542*z^10+ 48834280166*z^12-417419782704*z^14-9926198646058*z^18+2430917473530*z^16-470*z^ 50+72027*z^48+29040612364280*z^20+2430917473530*z^36-9926198646058*z^34-\ 61849238703238*z^30-3746507542*z^42+179008296*z^44-4959994*z^46+z^52+ 29040612364280*z^32-417419782704*z^38+48834280166*z^40)/(-1-1509640288714000*z^ 28+1509640288714000*z^26+804*z^2-1114121454755484*z^24+604949021926344*z^22-\ 160764*z^4+13125651*z^6-547753896*z^8+13288593872*z^10-202842145558*z^12+ 2050037782016*z^14+69071988835994*z^18-14199749177776*z^16+160764*z^50-13125651 *z^48-240184688706080*z^20-69071988835994*z^36+240184688706080*z^34+ 1114121454755484*z^30+202842145558*z^42-13288593872*z^44+547753896*z^46+z^54-\ 804*z^52-604949021926344*z^32+14199749177776*z^38-2050037782016*z^40) The first , 40, terms are: [0, 334, 0, 179799, 0, 99028877, 0, 54729232506, 0, 30268595762011, 0, 16743077916227139, 0, 9261770626972014634, 0, 5123376436520055640325, 0, 2834126969251698172430959, 0, 1567770602733604382780973374, 0, 867252919634478739507300609737, 0, 479743427280533157381005807542521, 0, 265382510540991042054616074889429982, 0, 146803213904717797246649867519824734335, 0, 81208002644440869656465993568976784258261, 0, 44922311429276931128034680230724517134889034, 0, 24849940873542021475917385360290331396368586675, 0, 13746388860518926468422021953626324252858801235563, 0, 7604171280178078241425153187833266736867665000754010, 0, 4206444430243626181851780419354791700653188542665554749] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 106373578165966424159 z - 23005357504896680133 z - 537 z 24 22 4 + 4074647715806904041 z - 586602052247789556 z + 107791 z 6 8 10 12 - 11538720 z + 766255454 z - 34316835832 z + 1092759020533 z 14 18 16 - 25663674861975 z - 6285446355584220 z + 456659344270467 z 50 48 - 13550083971882522952459 z + 20890959310259925049099 z 20 36 + 68024886400776220 z + 7375706669074882311189 z 34 66 80 - 3362100276378942373816 z - 586602052247789556 z + 766255454 z 88 84 86 82 64 + z + 107791 z - 537 z - 11538720 z + 4074647715806904041 z 30 42 - 405361006631941037864 z - 27071271590151239544120 z 44 46 + 29510892560698316886104 z - 27071271590151239544120 z 58 56 - 405361006631941037864 z + 1279786147546540186602 z 54 52 - 3362100276378942373816 z + 7375706669074882311189 z 60 70 68 + 106373578165966424159 z - 6285446355584220 z + 68024886400776220 z 78 32 - 34316835832 z + 1279786147546540186602 z 38 40 - 13550083971882522952459 z + 20890959310259925049099 z 62 76 74 - 23005357504896680133 z + 1092759020533 z - 25663674861975 z 72 / 2 28 + 456659344270467 z ) / ((-1 + z ) (1 + 564496084374358779136 z / 26 2 24 - 114313675063066303950 z - 886 z + 18887751120530506959 z 22 4 6 8 - 2529386179663406264 z + 216178 z - 26139604 z + 1909968171 z 10 12 14 - 93112262068 z + 3212955670788 z - 81611111382018 z 18 16 - 23321562392639472 z + 1569184511550417 z 50 48 - 91404250109752148025690 z + 144370468814064243273999 z 20 36 + 272283786458545566 z + 48133310299570176691850 z 34 66 80 - 21051115277353623714388 z - 2529386179663406264 z + 1909968171 z 88 84 86 82 64 + z + 216178 z - 886 z - 26139604 z + 18887751120530506959 z 30 42 - 2286644337295576119556 z - 189853017944361618786488 z 44 46 + 207987066660361884425124 z - 189853017944361618786488 z 58 56 - 2286644337295576119556 z + 7630924003892465903509 z 54 52 - 21051115277353623714388 z + 48133310299570176691850 z 60 70 + 564496084374358779136 z - 23321562392639472 z 68 78 32 + 272283786458545566 z - 93112262068 z + 7630924003892465903509 z 38 40 - 91404250109752148025690 z + 144370468814064243273999 z 62 76 74 - 114313675063066303950 z + 3212955670788 z - 81611111382018 z 72 + 1569184511550417 z )) And in Maple-input format, it is: -(1+106373578165966424159*z^28-23005357504896680133*z^26-537*z^2+ 4074647715806904041*z^24-586602052247789556*z^22+107791*z^4-11538720*z^6+ 766255454*z^8-34316835832*z^10+1092759020533*z^12-25663674861975*z^14-\ 6285446355584220*z^18+456659344270467*z^16-13550083971882522952459*z^50+ 20890959310259925049099*z^48+68024886400776220*z^20+7375706669074882311189*z^36 -3362100276378942373816*z^34-586602052247789556*z^66+766255454*z^80+z^88+107791 *z^84-537*z^86-11538720*z^82+4074647715806904041*z^64-405361006631941037864*z^ 30-27071271590151239544120*z^42+29510892560698316886104*z^44-\ 27071271590151239544120*z^46-405361006631941037864*z^58+1279786147546540186602* z^56-3362100276378942373816*z^54+7375706669074882311189*z^52+ 106373578165966424159*z^60-6285446355584220*z^70+68024886400776220*z^68-\ 34316835832*z^78+1279786147546540186602*z^32-13550083971882522952459*z^38+ 20890959310259925049099*z^40-23005357504896680133*z^62+1092759020533*z^76-\ 25663674861975*z^74+456659344270467*z^72)/(-1+z^2)/(1+564496084374358779136*z^ 28-114313675063066303950*z^26-886*z^2+18887751120530506959*z^24-\ 2529386179663406264*z^22+216178*z^4-26139604*z^6+1909968171*z^8-93112262068*z^ 10+3212955670788*z^12-81611111382018*z^14-23321562392639472*z^18+ 1569184511550417*z^16-91404250109752148025690*z^50+144370468814064243273999*z^ 48+272283786458545566*z^20+48133310299570176691850*z^36-21051115277353623714388 *z^34-2529386179663406264*z^66+1909968171*z^80+z^88+216178*z^84-886*z^86-\ 26139604*z^82+18887751120530506959*z^64-2286644337295576119556*z^30-\ 189853017944361618786488*z^42+207987066660361884425124*z^44-\ 189853017944361618786488*z^46-2286644337295576119556*z^58+ 7630924003892465903509*z^56-21051115277353623714388*z^54+ 48133310299570176691850*z^52+564496084374358779136*z^60-23321562392639472*z^70+ 272283786458545566*z^68-93112262068*z^78+7630924003892465903509*z^32-\ 91404250109752148025690*z^38+144370468814064243273999*z^40-\ 114313675063066303950*z^62+3212955670788*z^76-81611111382018*z^74+ 1569184511550417*z^72) The first , 40, terms are: [0, 350, 0, 201177, 0, 117288661, 0, 68421429358, 0, 39915906757813, 0, 23286358650223273, 0, 13584928849905013334, 0, 7925253830906919131701, 0, 4623480144888665135122649, 0, 2697272432055671772423523374, 0, 1573550301060441440473490829193, 0, 917986822762027311520453332457405, 0, 535540431214497715472261470209029174, 0, 312426656193845017353205739788424802533, 0, 182265259187085598126041403839159881043489, 0, 106330954955149884428104038352019648304337838, 0, 62031963919513831075543605590462046285117068861, 0, 36188563803785340216675468202074691924930283109873, 0, 21111892441126930531113465845728047709379240167875238, 0, 12316377208622209814378784694855582185670821404032365145] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 211436307398186188 z - 95747351667685014 z - 501 z 24 22 4 6 + 34208303722073614 z - 9564160595509320 z + 87820 z - 7630391 z 8 10 12 14 + 384488691 z - 12278457332 z + 263324007647 z - 3952657536963 z 18 16 50 - 343156870772185 z + 42832410534272 z - 343156870772185 z 48 20 36 + 2071295596551381 z + 2071295596551381 z + 518255225395913626 z 34 66 64 30 - 579332220930165048 z - 501 z + 87820 z - 370689483703947442 z 42 44 46 - 95747351667685014 z + 34208303722073614 z - 9564160595509320 z 58 56 54 - 12278457332 z + 263324007647 z - 3952657536963 z 52 60 68 32 + 42832410534272 z + 384488691 z + z + 518255225395913626 z 38 40 62 / - 370689483703947442 z + 211436307398186188 z - 7630391 z ) / (-1 / 28 26 2 - 2020679970991238002 z + 807141526002595182 z + 867 z 24 22 4 6 - 253618222568949522 z + 62242228895881873 z - 188725 z + 18809167 z 8 10 12 14 - 1063059865 z + 37938420491 z - 911955479935 z + 15416961700093 z 18 16 50 + 1720404028554477 z - 189043578100651 z + 11825503746013563 z 48 20 36 - 62242228895881873 z - 11825503746013563 z - 7892625147918435866 z 34 66 64 + 7892625147918435866 z + 188725 z - 18809167 z 30 42 + 4001972241390568954 z + 2020679970991238002 z 44 46 58 - 807141526002595182 z + 253618222568949522 z + 911955479935 z 56 54 52 - 15416961700093 z + 189043578100651 z - 1720404028554477 z 60 70 68 32 - 37938420491 z + z - 867 z - 6296641659070002070 z 38 40 62 + 6296641659070002070 z - 4001972241390568954 z + 1063059865 z ) And in Maple-input format, it is: -(1+211436307398186188*z^28-95747351667685014*z^26-501*z^2+34208303722073614*z^ 24-9564160595509320*z^22+87820*z^4-7630391*z^6+384488691*z^8-12278457332*z^10+ 263324007647*z^12-3952657536963*z^14-343156870772185*z^18+42832410534272*z^16-\ 343156870772185*z^50+2071295596551381*z^48+2071295596551381*z^20+ 518255225395913626*z^36-579332220930165048*z^34-501*z^66+87820*z^64-\ 370689483703947442*z^30-95747351667685014*z^42+34208303722073614*z^44-\ 9564160595509320*z^46-12278457332*z^58+263324007647*z^56-3952657536963*z^54+ 42832410534272*z^52+384488691*z^60+z^68+518255225395913626*z^32-\ 370689483703947442*z^38+211436307398186188*z^40-7630391*z^62)/(-1-\ 2020679970991238002*z^28+807141526002595182*z^26+867*z^2-253618222568949522*z^ 24+62242228895881873*z^22-188725*z^4+18809167*z^6-1063059865*z^8+37938420491*z^ 10-911955479935*z^12+15416961700093*z^14+1720404028554477*z^18-189043578100651* z^16+11825503746013563*z^50-62242228895881873*z^48-11825503746013563*z^20-\ 7892625147918435866*z^36+7892625147918435866*z^34+188725*z^66-18809167*z^64+ 4001972241390568954*z^30+2020679970991238002*z^42-807141526002595182*z^44+ 253618222568949522*z^46+911955479935*z^58-15416961700093*z^56+189043578100651*z ^54-1720404028554477*z^52-37938420491*z^60+z^70-867*z^68-6296641659070002070*z^ 32+6296641659070002070*z^38-4001972241390568954*z^40+1063059865*z^62) The first , 40, terms are: [0, 366, 0, 216417, 0, 129738965, 0, 77845968278, 0, 46714671874609, 0, 28033594614736321, 0, 16823085820186179150, 0, 10095614627682574627137, 0, 6058427386326355311818881, 0, 3635691763371803731543774654, 0, 2181796329107147457140039682661, 0, 1309306600653407572931506624171877, 0, 785721266355830716910397165275971166, 0, 471515157796570452197944066858150099777, 0, 282958542109365471479386071728840226331361, 0, 169804798910044358588126641883610372737835470, 0, 101900686644539653144393958849558478367441799713, 0, 61151098233269877428780946170530239356440581701745, 0, 36697071808548083116353973024346915830307402982854678, 0, 22022091478793140542685968234952364775330341198006984789] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 1235153 z + 19134520 z + 376 z - 151358137 z 22 4 6 8 10 + 669382726 z - 36265 z + 1235153 z - 19134520 z + 151358137 z 12 14 18 16 - 669382726 z + 1756730832 z + 2826594694 z - 2826594694 z 20 34 30 32 / 8 4 - 1756730832 z + z + 36265 z - 376 z ) / (73163744 z + 90416 z / 26 20 10 24 - 761665656 z + 33672710768 z + 1 - 761665656 z + 4520626367 z 22 18 28 2 - 15881552928 z - 43289074076 z + 73163744 z - 744 z 16 30 14 32 + 33672710768 z - 3754994 z - 15881552928 z + 90416 z 12 34 6 36 + 4520626367 z - 744 z - 3754994 z + z ) And in Maple-input format, it is: -(-1-1235153*z^28+19134520*z^26+376*z^2-151358137*z^24+669382726*z^22-36265*z^4 +1235153*z^6-19134520*z^8+151358137*z^10-669382726*z^12+1756730832*z^14+ 2826594694*z^18-2826594694*z^16-1756730832*z^20+z^34+36265*z^30-376*z^32)/( 73163744*z^8+90416*z^4-761665656*z^26+33672710768*z^20+1-761665656*z^10+ 4520626367*z^24-15881552928*z^22-43289074076*z^18+73163744*z^28-744*z^2+ 33672710768*z^16-3754994*z^30-15881552928*z^14+90416*z^32+4520626367*z^12-744*z ^34-3754994*z^6+z^36) The first , 40, terms are: [0, 368, 0, 219641, 0, 132659657, 0, 80167532720, 0, 48448525483249, 0, 29279618221024657, 0, 17695002447064099632, 0, 10693894480404277257897, 0, 6462806732021183882775065, 0, 3905768008080241499398439152, 0, 2360433225041372524817924302945, 0, 1426517140436146602802025727410593, 0, 862109179950660382922719359985763824, 0, 521011782535249205020218832169913482201, 0, 314871113605493984865895042753201685737193, 0, 190290932962648634771982569458790984714324528, 0, 115001464418752839972438956083197513856351465041, 0, 69500614730044029132197347341012265432815537899569, 0, 42002382076330753929299317552000760147920192155214256, 0, 25383949579994648054847635701831128273179112928839275913] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 49993613418562 z - 58113070465686 z - 424 z 24 22 4 6 + 49993613418562 z - 31777869098782 z + 54993 z - 3276327 z 8 10 12 14 + 106308665 z - 2070798182 z + 25813206913 z - 215294565286 z 18 16 50 48 - 5059179635222 z + 1240770961266 z - 424 z + 54993 z 20 36 34 + 14850648611866 z + 1240770961266 z - 5059179635222 z 30 42 44 46 52 - 31777869098782 z - 2070798182 z + 106308665 z - 3276327 z + z 32 38 40 / 2 + 14850648611866 z - 215294565286 z + 25813206913 z ) / ((-1 + z ) / 28 26 2 (1 + 352670269732131 z - 416750311133316 z - 754 z 24 22 4 6 + 352670269732131 z - 213646680298324 z + 123700 z - 8582728 z 8 10 12 14 + 317485167 z - 7037152278 z + 100059990446 z - 952184747504 z 18 16 50 48 - 28541734684100 z + 6231381590067 z - 754 z + 123700 z 20 36 34 + 92519048927759 z + 6231381590067 z - 28541734684100 z 30 42 44 46 52 - 213646680298324 z - 7037152278 z + 317485167 z - 8582728 z + z 32 38 40 + 92519048927759 z - 952184747504 z + 100059990446 z )) And in Maple-input format, it is: -(1+49993613418562*z^28-58113070465686*z^26-424*z^2+49993613418562*z^24-\ 31777869098782*z^22+54993*z^4-3276327*z^6+106308665*z^8-2070798182*z^10+ 25813206913*z^12-215294565286*z^14-5059179635222*z^18+1240770961266*z^16-424*z^ 50+54993*z^48+14850648611866*z^20+1240770961266*z^36-5059179635222*z^34-\ 31777869098782*z^30-2070798182*z^42+106308665*z^44-3276327*z^46+z^52+ 14850648611866*z^32-215294565286*z^38+25813206913*z^40)/(-1+z^2)/(1+ 352670269732131*z^28-416750311133316*z^26-754*z^2+352670269732131*z^24-\ 213646680298324*z^22+123700*z^4-8582728*z^6+317485167*z^8-7037152278*z^10+ 100059990446*z^12-952184747504*z^14-28541734684100*z^18+6231381590067*z^16-754* z^50+123700*z^48+92519048927759*z^20+6231381590067*z^36-28541734684100*z^34-\ 213646680298324*z^30-7037152278*z^42+317485167*z^44-8582728*z^46+z^52+ 92519048927759*z^32-952184747504*z^38+100059990446*z^40) The first , 40, terms are: [0, 331, 0, 180444, 0, 100471047, 0, 56060731347, 0, 31290206543697, 0, 17465383724034917, 0, 9748794172748414948, 0, 5441569005587157455777, 0, 3037368277062037529541035, 0, 1695394522421949097085793043, 0, 946333251582088536516794678041, 0, 528223143193235446593996399016996, 0, 294842951533302486267772421515024605, 0, 164575080040164368216513219515948342953, 0, 91862317988041763926258370705690902198091, 0, 51275596913454000194710751878426776580864767, 0, 28620950313635706990174980705746012134100986268, 0, 15975607231608344944503678329032180542572680262915, 0, 8917245011847989267163816370044764712053872765821833, 0, 4977416973797399586575277598026417223019436666762747961] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {4, 6}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 8303917256824372169006 z - 1054339089665487349671 z - 575 z 24 22 4 + 113241999547978841108 z - 10214174560705559693 z + 129134 z 6 102 8 10 - 16125253 z - 73081599075 z + 1297992896 z - 73081599075 z 12 14 18 + 3031509417506 z - 96039935380929 z - 47438629899203595 z 16 50 + 2386024648654896 z - 3210485219769185009871495375 z 48 20 + 2006200880250947229061829572 z + 766961660460281770 z 36 34 + 6680054134968534174269594 z - 1572504425982331277880007 z 66 80 - 1094873226930323286747317785 z + 319185301197313832210096 z 100 90 + 3031509417506 z - 10214174560705559693 z 88 84 + 113241999547978841108 z + 8303917256824372169006 z 94 86 - 47438629899203595 z - 1054339089665487349671 z 96 98 92 + 2386024648654896 z - 96039935380929 z + 766961660460281770 z 82 64 112 - 55662520362725866193109 z + 2006200880250947229061829572 z + z 110 106 108 30 - 575 z - 16125253 z + 129134 z - 55662520362725866193109 z 42 44 - 216413303232100863993113139 z + 521389292345010662197390306 z 46 58 - 1094873226930323286747317785 z - 5489780744504534873163727533 z 56 54 + 5870159196546392486480518634 z - 5489780744504534873163727533 z 52 60 + 4489862520146358608966169494 z + 4489862520146358608966169494 z 70 68 - 216413303232100863993113139 z + 521389292345010662197390306 z 78 32 - 1572504425982331277880007 z + 319185301197313832210096 z 38 40 - 24542759339520352922220769 z + 78184536942549757805522836 z 62 76 - 3210485219769185009871495375 z + 6680054134968534174269594 z 74 72 - 24542759339520352922220769 z + 78184536942549757805522836 z 104 / 2 28 + 1297992896 z ) / ((-1 + z ) (1 + 36416323354672055333356 z / 26 2 24 - 4372410059115572598104 z - 915 z + 443546489165233891318 z 22 4 6 102 - 37746999547210015766 z + 248280 z - 34846196 z - 185865921226 z 8 10 12 + 3063483222 z - 185865921226 z + 8251124970324 z 14 18 16 - 278717519475435 z - 155664434871793689 z + 7367822163167288 z 50 48 - 21833674609911819508632559569 z + 13377774469031592207346168928 z 20 36 + 2671904041864732332 z + 35975053011032307812548020 z 34 66 - 8072000967465999594363871 z - 7121852809538400854697033515 z 80 100 + 1557822871956543001576896 z + 8251124970324 z 90 88 - 37746999547210015766 z + 443546489165233891318 z 84 94 + 36416323354672055333356 z - 155664434871793689 z 86 96 98 - 4372410059115572598104 z + 7367822163167288 z - 278717519475435 z 92 82 + 2671904041864732332 z - 257749474522995803759405 z 64 112 110 106 + 13377774469031592207346168928 z + z - 915 z - 34846196 z 108 30 + 248280 z - 257749474522995803759405 z 42 44 - 1320785229289628445495005810 z + 3292443268220263122767036364 z 46 58 - 7121852809538400854697033515 z - 38199753882026178160696834722 z 56 54 + 40965021457633153704748526908 z - 38199753882026178160696834722 z 52 60 + 30973075036460716714954910344 z + 30973075036460716714954910344 z 70 68 - 1320785229289628445495005810 z + 3292443268220263122767036364 z 78 32 - 8072000967465999594363871 z + 1557822871956543001576896 z 38 40 - 138270680856055679975557548 z + 459298131435845957303975358 z 62 76 - 21833674609911819508632559569 z + 35975053011032307812548020 z 74 72 - 138270680856055679975557548 z + 459298131435845957303975358 z 104 + 3063483222 z )) And in Maple-input format, it is: -(1+8303917256824372169006*z^28-1054339089665487349671*z^26-575*z^2+ 113241999547978841108*z^24-10214174560705559693*z^22+129134*z^4-16125253*z^6-\ 73081599075*z^102+1297992896*z^8-73081599075*z^10+3031509417506*z^12-\ 96039935380929*z^14-47438629899203595*z^18+2386024648654896*z^16-\ 3210485219769185009871495375*z^50+2006200880250947229061829572*z^48+ 766961660460281770*z^20+6680054134968534174269594*z^36-\ 1572504425982331277880007*z^34-1094873226930323286747317785*z^66+ 319185301197313832210096*z^80+3031509417506*z^100-10214174560705559693*z^90+ 113241999547978841108*z^88+8303917256824372169006*z^84-47438629899203595*z^94-\ 1054339089665487349671*z^86+2386024648654896*z^96-96039935380929*z^98+ 766961660460281770*z^92-55662520362725866193109*z^82+ 2006200880250947229061829572*z^64+z^112-575*z^110-16125253*z^106+129134*z^108-\ 55662520362725866193109*z^30-216413303232100863993113139*z^42+ 521389292345010662197390306*z^44-1094873226930323286747317785*z^46-\ 5489780744504534873163727533*z^58+5870159196546392486480518634*z^56-\ 5489780744504534873163727533*z^54+4489862520146358608966169494*z^52+ 4489862520146358608966169494*z^60-216413303232100863993113139*z^70+ 521389292345010662197390306*z^68-1572504425982331277880007*z^78+ 319185301197313832210096*z^32-24542759339520352922220769*z^38+ 78184536942549757805522836*z^40-3210485219769185009871495375*z^62+ 6680054134968534174269594*z^76-24542759339520352922220769*z^74+ 78184536942549757805522836*z^72+1297992896*z^104)/(-1+z^2)/(1+ 36416323354672055333356*z^28-4372410059115572598104*z^26-915*z^2+ 443546489165233891318*z^24-37746999547210015766*z^22+248280*z^4-34846196*z^6-\ 185865921226*z^102+3063483222*z^8-185865921226*z^10+8251124970324*z^12-\ 278717519475435*z^14-155664434871793689*z^18+7367822163167288*z^16-\ 21833674609911819508632559569*z^50+13377774469031592207346168928*z^48+ 2671904041864732332*z^20+35975053011032307812548020*z^36-\ 8072000967465999594363871*z^34-7121852809538400854697033515*z^66+ 1557822871956543001576896*z^80+8251124970324*z^100-37746999547210015766*z^90+ 443546489165233891318*z^88+36416323354672055333356*z^84-155664434871793689*z^94 -4372410059115572598104*z^86+7367822163167288*z^96-278717519475435*z^98+ 2671904041864732332*z^92-257749474522995803759405*z^82+ 13377774469031592207346168928*z^64+z^112-915*z^110-34846196*z^106+248280*z^108-\ 257749474522995803759405*z^30-1320785229289628445495005810*z^42+ 3292443268220263122767036364*z^44-7121852809538400854697033515*z^46-\ 38199753882026178160696834722*z^58+40965021457633153704748526908*z^56-\ 38199753882026178160696834722*z^54+30973075036460716714954910344*z^52+ 30973075036460716714954910344*z^60-1320785229289628445495005810*z^70+ 3292443268220263122767036364*z^68-8072000967465999594363871*z^78+ 1557822871956543001576896*z^32-138270680856055679975557548*z^38+ 459298131435845957303975358*z^40-21833674609911819508632559569*z^62+ 35975053011032307812548020*z^76-138270680856055679975557548*z^74+ 459298131435845957303975358*z^72+3063483222*z^104) The first , 40, terms are: [0, 341, 0, 192295, 0, 110135948, 0, 63132455637, 0, 36191811537887, 0, 20747796660073939, 0, 11894170360593394547, 0, 6818618502171812864339, 0, 3908936714556416544245777, 0, 2240891798546463207139787676, 0, 1284645012506280214587599119779, 0, 736453589282946422470582190749625, 0, 422189697458111722594669126838372465, 0, 242030378062883417285713556618644370017, 0, 138749723780459375610124415704976966056945, 0, 79541609624524932010834350463849472567033003, 0, 45599136987625233983624626081277151869454696636, 0, 26140799813222674589030272633731275178683156691705, 0, 14985840961429355949804905057979685575284203697940491, 0, 8590993042518080074871451842924951848103139093442465443] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {3, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 13036535 z - 98167678 z - 356 z + 442303835 z 22 4 6 8 10 - 1259180906 z + 30405 z - 938398 z + 13036535 z - 98167678 z 12 14 18 16 + 442303835 z - 1259180906 z - 2864081548 z + 2334931992 z 20 36 34 30 32 / 2 + 2334931992 z + z - 356 z - 938398 z + 30405 z ) / ((-1 + z ) / 36 34 32 30 28 26 (z - 687 z + 74559 z - 2815278 z + 48617243 z - 451110811 z 24 22 20 18 + 2446781247 z - 8083958486 z + 16542460918 z - 21018135572 z 16 14 12 10 + 16542460918 z - 8083958486 z + 2446781247 z - 451110811 z 8 6 4 2 + 48617243 z - 2815278 z + 74559 z - 687 z + 1)) And in Maple-input format, it is: -(1+13036535*z^28-98167678*z^26-356*z^2+442303835*z^24-1259180906*z^22+30405*z^ 4-938398*z^6+13036535*z^8-98167678*z^10+442303835*z^12-1259180906*z^14-\ 2864081548*z^18+2334931992*z^16+2334931992*z^20+z^36-356*z^34-938398*z^30+30405 *z^32)/(-1+z^2)/(z^36-687*z^34+74559*z^32-2815278*z^30+48617243*z^28-451110811* z^26+2446781247*z^24-8083958486*z^22+16542460918*z^20-21018135572*z^18+ 16542460918*z^16-8083958486*z^14+2446781247*z^12-451110811*z^10+48617243*z^8-\ 2815278*z^6+74559*z^4-687*z^2+1) The first , 40, terms are: [0, 332, 0, 183575, 0, 103269367, 0, 58157069944, 0, 32755285122869, 0, 18448702646333957, 0, 10390847069720145184, 0, 5852429052907026370111, 0, 3296259331792253198773343, 0, 1856549736080142648319469156, 0, 1045663152322835127623418743521, 0, 588948093834413770672935889980673, 0, 331712804895656006557499698377907508, 0, 186830360915901604615031666953841615855, 0, 105228327772721398830660294854389675040719, 0, 59267674223611418329670948736169785611167696, 0, 33381288881288603908969739894706494713856231237, 0, 18801318964733671520075513547383861009751904337333, 0, 10589453153553860429480451290970744174547216888594888, 0, 5964289968254376673095744907092382839674242084458987175] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {4, 6}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 298792219605078000 z - 129872314506190514 z - 479 z 24 22 4 6 + 44166468211338422 z - 11683469236593824 z + 79520 z - 6731719 z 8 10 12 14 + 338924645 z - 11031227000 z + 244775489821 z - 3843505511207 z 18 16 50 - 372488718790095 z + 43901121412208 z - 372488718790095 z 48 20 36 + 2385944914853833 z + 2385944914853833 z + 769684223912280830 z 34 66 64 30 - 865938949731230800 z - 479 z + 79520 z - 540171401690666354 z 42 44 46 - 129872314506190514 z + 44166468211338422 z - 11683469236593824 z 58 56 54 - 11031227000 z + 244775489821 z - 3843505511207 z 52 60 68 32 + 43901121412208 z + 338924645 z + z + 769684223912280830 z 38 40 62 / - 540171401690666354 z + 298792219605078000 z - 6731719 z ) / (-1 / 28 26 2 - 2730694714077962114 z + 1042267626284150294 z + 813 z 24 22 4 6 - 310966189948867686 z + 72194257626110809 z - 165775 z + 16103503 z 8 10 12 14 - 910636273 z + 33101642421 z - 820908410349 z + 14454016078721 z 18 16 50 + 1783214081078159 z - 185893558695375 z + 12959554784551133 z 48 20 36 - 72194257626110809 z - 12959554784551133 z - 11503756280149923486 z 34 66 64 + 11503756280149923486 z + 165775 z - 16103503 z 30 42 + 5610037038881398018 z + 2730694714077962114 z 44 46 58 - 1042267626284150294 z + 310966189948867686 z + 820908410349 z 56 54 52 - 14454016078721 z + 185893558695375 z - 1783214081078159 z 60 70 68 32 - 33101642421 z + z - 813 z - 9056685982293473454 z 38 40 62 + 9056685982293473454 z - 5610037038881398018 z + 910636273 z ) And in Maple-input format, it is: -(1+298792219605078000*z^28-129872314506190514*z^26-479*z^2+44166468211338422*z ^24-11683469236593824*z^22+79520*z^4-6731719*z^6+338924645*z^8-11031227000*z^10 +244775489821*z^12-3843505511207*z^14-372488718790095*z^18+43901121412208*z^16-\ 372488718790095*z^50+2385944914853833*z^48+2385944914853833*z^20+ 769684223912280830*z^36-865938949731230800*z^34-479*z^66+79520*z^64-\ 540171401690666354*z^30-129872314506190514*z^42+44166468211338422*z^44-\ 11683469236593824*z^46-11031227000*z^58+244775489821*z^56-3843505511207*z^54+ 43901121412208*z^52+338924645*z^60+z^68+769684223912280830*z^32-\ 540171401690666354*z^38+298792219605078000*z^40-6731719*z^62)/(-1-\ 2730694714077962114*z^28+1042267626284150294*z^26+813*z^2-310966189948867686*z^ 24+72194257626110809*z^22-165775*z^4+16103503*z^6-910636273*z^8+33101642421*z^ 10-820908410349*z^12+14454016078721*z^14+1783214081078159*z^18-185893558695375* z^16+12959554784551133*z^50-72194257626110809*z^48-12959554784551133*z^20-\ 11503756280149923486*z^36+11503756280149923486*z^34+165775*z^66-16103503*z^64+ 5610037038881398018*z^30+2730694714077962114*z^42-1042267626284150294*z^44+ 310966189948867686*z^46+820908410349*z^58-14454016078721*z^56+185893558695375*z ^54-1783214081078159*z^52-33101642421*z^60+z^70-813*z^68-9056685982293473454*z^ 32+9056685982293473454*z^38-5610037038881398018*z^40+910636273*z^62) The first , 40, terms are: [0, 334, 0, 185287, 0, 104641265, 0, 59164254394, 0, 33455320777547, 0, 18918063197364391, 0, 10697660925245103570, 0, 6049243527801017475013, 0, 3420686857609883506875291, 0, 1934307745915183588827662502, 0, 1093799758075802545103040307133, 0, 618514770128706515120861830169957, 0, 349753707701147600188644935762572598, 0, 197776450876566679304322861858718583827, 0, 111837340562969910223458728615817709779709, 0, 63241051645748849011510161446318383279743202, 0, 35761138391952404666141736363678855886381559951, 0, 20221975849674839089834787341578243718157088187363, 0, 11434991324461209629783344609942169712602309160951978, 0, 6466184489712249361507578712603627509754568759119055929] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 5921554188309778 z - 3275866314578662 z - 409 z 24 22 4 6 + 1424797062956622 z - 485217136426472 z + 50695 z - 3097912 z 8 10 12 14 + 111690858 z - 2611364048 z + 42017228775 z - 484423020289 z 18 16 50 - 26332532810776 z + 4117320001905 z - 484423020289 z 48 20 36 + 4117320001905 z + 128631148999264 z + 5921554188309778 z 34 64 30 42 - 8438347704466104 z + z - 8438347704466104 z - 485217136426472 z 44 46 58 56 + 128631148999264 z - 26332532810776 z - 3097912 z + 111690858 z 54 52 60 32 - 2611364048 z + 42017228775 z + 50695 z + 9494217843158380 z 38 40 62 / 2 - 3275866314578662 z + 1424797062956622 z - 409 z ) / ((-1 + z ) (1 / 28 26 2 + 40825513926946112 z - 21535696323386932 z - 764 z 24 22 4 6 + 8792517141435998 z - 2774291990660628 z + 113688 z - 7858108 z 8 10 12 14 + 316040897 z - 8220141208 z + 147085520936 z - 1884575657936 z 18 16 50 - 125599720880364 z + 17767670382753 z - 1884575657936 z 48 20 36 + 17767670382753 z + 674383860777760 z + 40825513926946112 z 34 64 30 - 59926140821736748 z + z - 59926140821736748 z 42 44 46 - 2774291990660628 z + 674383860777760 z - 125599720880364 z 58 56 54 52 - 7858108 z + 316040897 z - 8220141208 z + 147085520936 z 60 32 38 + 113688 z + 68106626540346942 z - 21535696323386932 z 40 62 + 8792517141435998 z - 764 z )) And in Maple-input format, it is: -(1+5921554188309778*z^28-3275866314578662*z^26-409*z^2+1424797062956622*z^24-\ 485217136426472*z^22+50695*z^4-3097912*z^6+111690858*z^8-2611364048*z^10+ 42017228775*z^12-484423020289*z^14-26332532810776*z^18+4117320001905*z^16-\ 484423020289*z^50+4117320001905*z^48+128631148999264*z^20+5921554188309778*z^36 -8438347704466104*z^34+z^64-8438347704466104*z^30-485217136426472*z^42+ 128631148999264*z^44-26332532810776*z^46-3097912*z^58+111690858*z^56-2611364048 *z^54+42017228775*z^52+50695*z^60+9494217843158380*z^32-3275866314578662*z^38+ 1424797062956622*z^40-409*z^62)/(-1+z^2)/(1+40825513926946112*z^28-\ 21535696323386932*z^26-764*z^2+8792517141435998*z^24-2774291990660628*z^22+ 113688*z^4-7858108*z^6+316040897*z^8-8220141208*z^10+147085520936*z^12-\ 1884575657936*z^14-125599720880364*z^18+17767670382753*z^16-1884575657936*z^50+ 17767670382753*z^48+674383860777760*z^20+40825513926946112*z^36-\ 59926140821736748*z^34+z^64-59926140821736748*z^30-2774291990660628*z^42+ 674383860777760*z^44-125599720880364*z^46-7858108*z^58+316040897*z^56-\ 8220141208*z^54+147085520936*z^52+113688*z^60+68106626540346942*z^32-\ 21535696323386932*z^38+8792517141435998*z^40-764*z^62) The first , 40, terms are: [0, 356, 0, 208583, 0, 123694967, 0, 73379659468, 0, 43531701027281, 0, 25824739309825257, 0, 15320265086029955820, 0, 9088592165137657048703, 0, 5391715294084507950259647, 0, 3198580515563443113545699076, 0, 1897525510256702478027409872521, 0, 1125687799496086126874844613931353, 0, 667802891231188020632612096786845508, 0, 396167304768132637988742867249533337967, 0, 235022242982341595637672067839726006319119, 0, 139424566418419663182439833715873114055785772, 0, 82712212573109013240777147376930630574962645913, 0, 49068182777833285294744252079559302549404564227297, 0, 29109202694713431064388260073906384480018129482828172, 0, 17268739813708831812080024853981547710655957050339249959] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 1799362 z + 31453326 z + 403 z - 271130219 z 22 4 6 8 10 + 1243411561 z - 44817 z + 1799362 z - 31453326 z + 271130219 z 12 14 18 16 - 1243411561 z + 3284274667 z + 5270676024 z - 5270676024 z 20 34 30 32 / 36 34 - 3284274667 z + z + 44817 z - 403 z ) / (z - 774 z / 32 30 28 26 + 108063 z - 5296260 z + 118168733 z - 1343147158 z 24 22 20 18 + 8326557163 z - 29393051764 z + 61530617193 z - 78476257066 z 16 14 12 10 + 61530617193 z - 29393051764 z + 8326557163 z - 1343147158 z 8 6 4 2 + 118168733 z - 5296260 z + 108063 z - 774 z + 1) And in Maple-input format, it is: -(-1-1799362*z^28+31453326*z^26+403*z^2-271130219*z^24+1243411561*z^22-44817*z^ 4+1799362*z^6-31453326*z^8+271130219*z^10-1243411561*z^12+3284274667*z^14+ 5270676024*z^18-5270676024*z^16-3284274667*z^20+z^34+44817*z^30-403*z^32)/(z^36 -774*z^34+108063*z^32-5296260*z^30+118168733*z^28-1343147158*z^26+8326557163*z^ 24-29393051764*z^22+61530617193*z^20-78476257066*z^18+61530617193*z^16-\ 29393051764*z^14+8326557163*z^12-1343147158*z^10+118168733*z^8-5296260*z^6+ 108063*z^4-774*z^2+1) The first , 40, terms are: [0, 371, 0, 223908, 0, 136710317, 0, 83495812207, 0, 50995538063323, 0, 31145824189782833, 0, 19022495416916489940, 0, 11618101055896357718687, 0, 7095823615122503538685525, 0, 4333816045979492878554212077, 0, 2646903663237314444269621528583, 0, 1616611994634146367101854058931060, 0, 987355292711762912292221242839258857, 0, 603033057580760048961905851581884256835, 0, 368305989970888197412017490829037524076679, 0, 224945051590757142204388741260023834233103477, 0, 137386514509763924058109565445675154522300428804, 0, 83909622530754698528119905106536668109166444315019, 0, 51248295936304229375124345365651033820593613244692457, 0, 31300198441631511620828258574867824943530167532485207321] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 328662379328546218 z - 143578619334459286 z - 492 z 24 22 4 6 + 49127902834420516 z - 13084878990144856 z + 84543 z - 7375310 z 8 10 12 14 + 379636150 z - 12525477922 z + 279699861572 z - 4396279449420 z 18 16 50 - 422788752839510 z + 50081354441303 z - 422788752839510 z 48 20 36 + 2690811271327683 z + 2690811271327683 z + 841572313763809502 z 34 66 64 30 - 946097178200786444 z - 492 z + 84543 z - 591964483885060494 z 42 44 46 - 143578619334459286 z + 49127902834420516 z - 13084878990144856 z 58 56 54 - 12525477922 z + 279699861572 z - 4396279449420 z 52 60 68 32 + 50081354441303 z + 379636150 z + z + 841572313763809502 z 38 40 62 / - 591964483885060494 z + 328662379328546218 z - 7375310 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 2126402108196258630 z - 876836312387985802 z - 827 z 24 22 4 6 + 279663194924598794 z - 68717584314011808 z + 174381 z - 17389096 z 8 10 12 14 + 999750254 z - 36582096042 z + 904653290746 z - 15742410851312 z 18 16 50 - 1848300155359771 z + 198370877404429 z - 1848300155359771 z 48 20 36 + 12931597582988101 z + 12931597582988101 z + 5837929541341276616 z 34 66 64 30 - 6622497933425886444 z - 827 z + 174381 z - 3998459450666519368 z 42 44 46 - 876836312387985802 z + 279663194924598794 z - 68717584314011808 z 58 56 54 - 36582096042 z + 904653290746 z - 15742410851312 z 52 60 68 32 + 198370877404429 z + 999750254 z + z + 5837929541341276616 z 38 40 62 - 3998459450666519368 z + 2126402108196258630 z - 17389096 z )) And in Maple-input format, it is: -(1+328662379328546218*z^28-143578619334459286*z^26-492*z^2+49127902834420516*z ^24-13084878990144856*z^22+84543*z^4-7375310*z^6+379636150*z^8-12525477922*z^10 +279699861572*z^12-4396279449420*z^14-422788752839510*z^18+50081354441303*z^16-\ 422788752839510*z^50+2690811271327683*z^48+2690811271327683*z^20+ 841572313763809502*z^36-946097178200786444*z^34-492*z^66+84543*z^64-\ 591964483885060494*z^30-143578619334459286*z^42+49127902834420516*z^44-\ 13084878990144856*z^46-12525477922*z^58+279699861572*z^56-4396279449420*z^54+ 50081354441303*z^52+379636150*z^60+z^68+841572313763809502*z^32-\ 591964483885060494*z^38+328662379328546218*z^40-7375310*z^62)/(-1+z^2)/(1+ 2126402108196258630*z^28-876836312387985802*z^26-827*z^2+279663194924598794*z^ 24-68717584314011808*z^22+174381*z^4-17389096*z^6+999750254*z^8-36582096042*z^ 10+904653290746*z^12-15742410851312*z^14-1848300155359771*z^18+198370877404429* z^16-1848300155359771*z^50+12931597582988101*z^48+12931597582988101*z^20+ 5837929541341276616*z^36-6622497933425886444*z^34-827*z^66+174381*z^64-\ 3998459450666519368*z^30-876836312387985802*z^42+279663194924598794*z^44-\ 68717584314011808*z^46-36582096042*z^58+904653290746*z^56-15742410851312*z^54+ 198370877404429*z^52+999750254*z^60+z^68+5837929541341276616*z^32-\ 3998459450666519368*z^38+2126402108196258630*z^40-17389096*z^62) The first , 40, terms are: [0, 336, 0, 187543, 0, 106603883, 0, 60672806252, 0, 34536435157777, 0, 19659350719733321, 0, 11190821305580381924, 0, 6370226983978229324267, 0, 3626167618873146372769047, 0, 2064148063184560306244880056, 0, 1174989045918301457714601000265, 0, 668847009008789598368332097050073, 0, 380732333652250989217281968061645416, 0, 216726856719423842285767310341720990775, 0, 123368903221127387116960074532770509835723, 0, 70226120160495062118346092184683391460284532, 0, 39975292184908974632867370163445606271016530745, 0, 22755407555147793127415797658081475528802665225313, 0, 12953215466336344541041112715638649118533429877814236, 0, 7373446971261919115620355474529320589461712243519403723] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 7}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 22 4 6 8 10 f(z) = - (-1 + 302 z + z - 13777 z + 174853 z - 859945 z + 1856487 z 12 14 18 16 20 / - 1856487 z + 859945 z + 13777 z - 174853 z - 302 z ) / ( / 12 24 10 22 8 20 28406759 z + z - 18678592 z - 643 z + 5537873 z + 38544 z 6 18 4 16 2 14 - 730000 z - 730000 z + 38544 z + 5537873 z - 643 z - 18678592 z + 1) And in Maple-input format, it is: -(-1+302*z^2+z^22-13777*z^4+174853*z^6-859945*z^8+1856487*z^10-1856487*z^12+ 859945*z^14+13777*z^18-174853*z^16-302*z^20)/(28406759*z^12+z^24-18678592*z^10-\ 643*z^22+5537873*z^8+38544*z^20-730000*z^6-730000*z^18+38544*z^4+5537873*z^16-\ 643*z^2-18678592*z^14+1) The first , 40, terms are: [0, 341, 0, 194496, 0, 112472571, 0, 65067461401, 0, 37643345391631, 0, 21777745080111181, 0, 12599044992987556096, 0, 7288905926232040637611, 0, 4216839422377110732459951, 0, 2439561560304629455054791071, 0, 1411355759704334658974892199771, 0, 816509455167001040470450200244096, 0, 472373946677186112554708374324067181, 0, 273281765553765415614282436445592012751, 0, 158101275291586929698638645853668509683481, 0, 91466085189311482758731074659231620428643931, 0, 52915732175017877025833886922425808706823471936, 0, 30613256332363861797558284515584111992139089614661, 0, 17710639629275799009924532626538810595149276160902321, 0, 10246108831828868894115206403174725761829517700640776401] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {3, 7}, {4, 5}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 2515351 z + 50594746 z + 462 z - 507144267 z 22 4 6 8 10 + 2696136244 z - 55005 z + 2515351 z - 50594746 z + 507144267 z 12 14 18 16 - 2696136244 z + 7929416248 z + 13452806284 z - 13452806284 z 20 34 30 32 / 36 34 - 7929416248 z + z + 55005 z - 462 z ) / (z - 854 z / 32 30 28 26 + 139730 z - 7983386 z + 201566378 z - 2585522582 z 24 22 20 18 + 18134399379 z - 71889264172 z + 162967000444 z - 213822068404 z 16 14 12 10 + 162967000444 z - 71889264172 z + 18134399379 z - 2585522582 z 8 6 4 2 + 201566378 z - 7983386 z + 139730 z - 854 z + 1) And in Maple-input format, it is: -(-1-2515351*z^28+50594746*z^26+462*z^2-507144267*z^24+2696136244*z^22-55005*z^ 4+2515351*z^6-50594746*z^8+507144267*z^10-2696136244*z^12+7929416248*z^14+ 13452806284*z^18-13452806284*z^16-7929416248*z^20+z^34+55005*z^30-462*z^32)/(z^ 36-854*z^34+139730*z^32-7983386*z^30+201566378*z^28-2585522582*z^26+18134399379 *z^24-71889264172*z^22+162967000444*z^20-213822068404*z^18+162967000444*z^16-\ 71889264172*z^14+18134399379*z^12-2585522582*z^10+201566378*z^8-7983386*z^6+ 139730*z^4-854*z^2+1) The first , 40, terms are: [0, 392, 0, 250043, 0, 164230597, 0, 108292937128, 0, 71453481132239, 0, 47151214855632863, 0, 31115022966773459656, 0, 20532822983238115177525, 0, 13549629431235133396299947, 0, 8941414110500219869734140840, 0, 5900448261527967541107206057617, 0, 3893711815487036348512206128612209, 0, 2569464392548186694006872236485740904, 0, 1695592169598130049615055181208345982155, 0, 1118922999661921187922968259420738837870421, 0, 738378427090461127216132129922311160481830536, 0, 487256676069296360110210457993126897628501001151, 0, 321541176805026959494190110355925341584242577502447, 0, 212185350060670784802409959568131743193551638513698280, 0, 140021328614063803263682480387637021750163335977797085861] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 109971100871144 z - 127201346902872 z - 548 z 24 22 4 6 + 109971100871144 z - 70898879767626 z + 93495 z - 6794990 z 8 10 12 14 + 245874304 z - 5032271250 z + 63589476710 z - 526379779120 z 18 16 50 48 - 11828486419254 z + 2974758009722 z - 548 z + 93495 z 20 36 34 + 33849764142464 z + 2974758009722 z - 11828486419254 z 30 42 44 46 52 - 70898879767626 z - 5032271250 z + 245874304 z - 6794990 z + z 32 38 40 / 2 + 33849764142464 z - 526379779120 z + 63589476710 z ) / ((-1 + z ) / 28 26 2 (1 + 865535342016830 z - 1016708892608730 z - 951 z 24 22 4 6 + 865535342016830 z - 533113975536924 z + 216793 z - 18919620 z 8 10 12 14 + 794959940 z - 18770860628 z + 273233069388 z - 2597486999004 z 18 16 50 48 - 74861409049548 z + 16734395334300 z - 951 z + 216793 z 20 36 34 + 236402325699004 z + 16734395334300 z - 74861409049548 z 30 42 44 46 - 533113975536924 z - 18770860628 z + 794959940 z - 18919620 z 52 32 38 40 + z + 236402325699004 z - 2597486999004 z + 273233069388 z )) And in Maple-input format, it is: -(1+109971100871144*z^28-127201346902872*z^26-548*z^2+109971100871144*z^24-\ 70898879767626*z^22+93495*z^4-6794990*z^6+245874304*z^8-5032271250*z^10+ 63589476710*z^12-526379779120*z^14-11828486419254*z^18+2974758009722*z^16-548*z ^50+93495*z^48+33849764142464*z^20+2974758009722*z^36-11828486419254*z^34-\ 70898879767626*z^30-5032271250*z^42+245874304*z^44-6794990*z^46+z^52+ 33849764142464*z^32-526379779120*z^38+63589476710*z^40)/(-1+z^2)/(1+ 865535342016830*z^28-1016708892608730*z^26-951*z^2+865535342016830*z^24-\ 533113975536924*z^22+216793*z^4-18919620*z^6+794959940*z^8-18770860628*z^10+ 273233069388*z^12-2597486999004*z^14-74861409049548*z^18+16734395334300*z^16-\ 951*z^50+216793*z^48+236402325699004*z^20+16734395334300*z^36-74861409049548*z^ 34-533113975536924*z^30-18770860628*z^42+794959940*z^44-18919620*z^46+z^52+ 236402325699004*z^32-2597486999004*z^38+273233069388*z^40) The first , 40, terms are: [0, 404, 0, 260359, 0, 172234615, 0, 114438848980, 0, 76110793779745, 0, 50630830980246241, 0, 33682631769300392212, 0, 22407948956018775223767, 0, 14907313443461986202077095, 0, 9917379557701671883690927316, 0, 6597730174999330513807895388097, 0, 4389268827548873443535713658548289, 0, 2920046816322989713921667937626580308, 0, 1942618180866639147034028516789327344487, 0, 1292364689887014562843378847187572538643479, 0, 859770853727776397896705240323235716919016852, 0, 571979354373449754358909924438797028918734758753, 0, 380520437989610412743318119922386579353560670847393, 0, 253148654091881315538765146636206027876425218527333204, 0, 168412086896365017781178140834668010164476967937551954743] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 57926451543918 z - 67022962242616 z - 490 z 24 22 4 6 + 57926451543918 z - 37317895821041 z + 69114 z - 4350084 z 8 10 12 14 + 144199886 z - 2803458423 z + 34399086379 z - 280190274050 z 18 16 50 48 - 6225651327464 z + 1570734534376 z - 490 z + 69114 z 20 36 34 + 17806296903058 z + 1570734534376 z - 6225651327464 z 30 42 44 46 52 - 37317895821041 z - 2803458423 z + 144199886 z - 4350084 z + z 32 38 40 / 2 + 17806296903058 z - 280190274050 z + 34399086379 z ) / ((-1 + z ) / 28 26 2 (1 + 445474101506351 z - 523243894610327 z - 893 z 24 22 4 6 + 445474101506351 z - 274509844720342 z + 163193 z - 12199760 z 8 10 12 14 + 468035460 z - 10467130924 z + 147364199111 z - 1372083204983 z 18 16 50 48 - 38754797957674 z + 8727489921499 z - 893 z + 163193 z 20 36 34 + 121919705872902 z + 8727489921499 z - 38754797957674 z 30 42 44 46 - 274509844720342 z - 10467130924 z + 468035460 z - 12199760 z 52 32 38 40 + z + 121919705872902 z - 1372083204983 z + 147364199111 z )) And in Maple-input format, it is: -(1+57926451543918*z^28-67022962242616*z^26-490*z^2+57926451543918*z^24-\ 37317895821041*z^22+69114*z^4-4350084*z^6+144199886*z^8-2803458423*z^10+ 34399086379*z^12-280190274050*z^14-6225651327464*z^18+1570734534376*z^16-490*z^ 50+69114*z^48+17806296903058*z^20+1570734534376*z^36-6225651327464*z^34-\ 37317895821041*z^30-2803458423*z^42+144199886*z^44-4350084*z^46+z^52+ 17806296903058*z^32-280190274050*z^38+34399086379*z^40)/(-1+z^2)/(1+ 445474101506351*z^28-523243894610327*z^26-893*z^2+445474101506351*z^24-\ 274509844720342*z^22+163193*z^4-12199760*z^6+468035460*z^8-10467130924*z^10+ 147364199111*z^12-1372083204983*z^14-38754797957674*z^18+8727489921499*z^16-893 *z^50+163193*z^48+121919705872902*z^20+8727489921499*z^36-38754797957674*z^34-\ 274509844720342*z^30-10467130924*z^42+468035460*z^44-12199760*z^46+z^52+ 121919705872902*z^32-1372083204983*z^38+147364199111*z^40) The first , 40, terms are: [0, 404, 0, 266204, 0, 179708501, 0, 121637648028, 0, 82361592461439, 0, 55770410136238111, 0, 37764709035688175184, 0, 25572247906590963711271, 0, 17316165768224176544310027, 0, 11725586444350705649075032500, 0, 7939943503120307898573482159661, 0, 5376507448898123609314711280995556, 0, 3640684891339159773712169693629085372, 0, 2465278176237343661708177396974518720205, 0, 1669355263537934539086051600156527576056597, 0, 1130398598731616408327088593968040937937842284, 0, 765445810082593802396510464160806263848865792404, 0, 518319191858896558295243998594785316298459765598645, 0, 350978189586368164598970443141910211017484258706051940, 0, 237663762986533940591906781506451852011274412560765745795] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 436472242862061040 z - 191112398270822970 z - 555 z 24 22 4 6 + 65529972640362558 z - 17482671538775472 z + 102340 z - 9318579 z 8 10 12 14 + 492456113 z - 16505686340 z + 371881479553 z - 5871528925715 z 18 16 50 - 565848530894651 z + 67007761918252 z - 565848530894651 z 48 20 36 + 3599311731440289 z + 3599311731440289 z + 1113669830954973550 z 34 66 64 30 - 1251328911068890328 z - 555 z + 102340 z - 784507476484854682 z 42 44 46 - 191112398270822970 z + 65529972640362558 z - 17482671538775472 z 58 56 54 - 16505686340 z + 371881479553 z - 5871528925715 z 52 60 68 32 + 67007761918252 z + 492456113 z + z + 1113669830954973550 z 38 40 62 / - 784507476484854682 z + 436472242862061040 z - 9318579 z ) / (-1 / 28 26 2 - 4345505648820979722 z + 1676187693816558790 z + 967 z 24 22 4 - 505815175183447274 z + 118765545516472613 z - 221441 z 6 8 10 12 + 23437105 z - 1410069199 z + 53448968449 z - 1359546328933 z 14 18 16 + 24235299503819 z + 2987120036241093 z - 312561843998613 z 50 48 20 + 21538823597544995 z - 118765545516472613 z - 21538823597544995 z 36 34 66 - 17967635489201164202 z + 17967635489201164202 z + 221441 z 64 30 42 - 23437105 z + 8848723005348595722 z + 4345505648820979722 z 44 46 58 - 1676187693816558790 z + 505815175183447274 z + 1359546328933 z 56 54 52 - 24235299503819 z + 312561843998613 z - 2987120036241093 z 60 70 68 32 - 53448968449 z + z - 967 z - 14193541372594158614 z 38 40 62 + 14193541372594158614 z - 8848723005348595722 z + 1410069199 z ) And in Maple-input format, it is: -(1+436472242862061040*z^28-191112398270822970*z^26-555*z^2+65529972640362558*z ^24-17482671538775472*z^22+102340*z^4-9318579*z^6+492456113*z^8-16505686340*z^ 10+371881479553*z^12-5871528925715*z^14-565848530894651*z^18+67007761918252*z^ 16-565848530894651*z^50+3599311731440289*z^48+3599311731440289*z^20+ 1113669830954973550*z^36-1251328911068890328*z^34-555*z^66+102340*z^64-\ 784507476484854682*z^30-191112398270822970*z^42+65529972640362558*z^44-\ 17482671538775472*z^46-16505686340*z^58+371881479553*z^56-5871528925715*z^54+ 67007761918252*z^52+492456113*z^60+z^68+1113669830954973550*z^32-\ 784507476484854682*z^38+436472242862061040*z^40-9318579*z^62)/(-1-\ 4345505648820979722*z^28+1676187693816558790*z^26+967*z^2-505815175183447274*z^ 24+118765545516472613*z^22-221441*z^4+23437105*z^6-1410069199*z^8+53448968449*z ^10-1359546328933*z^12+24235299503819*z^14+2987120036241093*z^18-\ 312561843998613*z^16+21538823597544995*z^50-118765545516472613*z^48-\ 21538823597544995*z^20-17967635489201164202*z^36+17967635489201164202*z^34+ 221441*z^66-23437105*z^64+8848723005348595722*z^30+4345505648820979722*z^42-\ 1676187693816558790*z^44+505815175183447274*z^46+1359546328933*z^58-\ 24235299503819*z^56+312561843998613*z^54-2987120036241093*z^52-53448968449*z^60 +z^70-967*z^68-14193541372594158614*z^32+14193541372594158614*z^38-\ 8848723005348595722*z^40+1410069199*z^62) The first , 40, terms are: [0, 412, 0, 279303, 0, 192970835, 0, 133492135996, 0, 92357289344833, 0, 63898741183859261, 0, 44209345880174230204, 0, 30586934764339551137711, 0, 21162054800362711670340403, 0, 14641302509097423065502965116, 0, 10129816847536398015299302943757, 0, 7008474096206072338806812716745461, 0, 4848923716666577007888859995066688444, 0, 3354804610435699108435194506618924391195, 0, 2321074661479511471534002018729956709398007, 0, 1605872236911772988953403668369473129790501820, 0, 1111048120976952067144476572335539994496285778181, 0, 768696225485736437386802731889963041616614778123385, 0, 531834648670699533662349082354513110026980545471393852, 0, 367958218277910357053159246322240546366569134036742798731] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 447657241869187520 z - 198927795149617830 z - 573 z 24 22 4 6 + 69429943315500526 z - 18890440248686112 z + 110648 z - 10503953 z 8 10 12 14 + 569872021 z - 19315337828 z + 435038014277 z - 6811601091249 z 18 16 50 - 636262903749725 z + 76670831061576 z - 636262903749725 z 48 20 36 + 3968822919900369 z + 3968822919900369 z + 1121991979184350182 z 34 66 64 30 - 1257784645471726456 z - 573 z + 110648 z - 795771441820835710 z 42 44 46 - 198927795149617830 z + 69429943315500526 z - 18890440248686112 z 58 56 54 - 19315337828 z + 435038014277 z - 6811601091249 z 52 60 68 32 + 76670831061576 z + 569872021 z + z + 1121991979184350182 z 38 40 62 / - 795771441820835710 z + 447657241869187520 z - 10503953 z ) / (-1 / 28 26 2 - 4555304381098299978 z + 1784894781186124258 z + 997 z 24 22 4 - 548124940068606314 z + 131052927591902581 z - 241081 z 6 8 10 12 + 26635907 z - 1642820111 z + 62900780899 z - 1600082267679 z 14 18 16 + 28343373932491 z + 3409195547019741 z - 361677128737551 z 50 48 20 + 24189375712211353 z - 131052927591902581 z - 24189375712211353 z 36 34 66 - 18343124626423370590 z + 18343124626423370590 z + 241081 z 64 30 42 - 26635907 z + 9158047798950934606 z + 4555304381098299978 z 44 46 58 - 1784894781186124258 z + 548124940068606314 z + 1600082267679 z 56 54 52 - 28343373932491 z + 361677128737551 z - 3409195547019741 z 60 70 68 32 - 62900780899 z + z - 997 z - 14557801089150311302 z 38 40 62 + 14557801089150311302 z - 9158047798950934606 z + 1642820111 z ) And in Maple-input format, it is: -(1+447657241869187520*z^28-198927795149617830*z^26-573*z^2+69429943315500526*z ^24-18890440248686112*z^22+110648*z^4-10503953*z^6+569872021*z^8-19315337828*z^ 10+435038014277*z^12-6811601091249*z^14-636262903749725*z^18+76670831061576*z^ 16-636262903749725*z^50+3968822919900369*z^48+3968822919900369*z^20+ 1121991979184350182*z^36-1257784645471726456*z^34-573*z^66+110648*z^64-\ 795771441820835710*z^30-198927795149617830*z^42+69429943315500526*z^44-\ 18890440248686112*z^46-19315337828*z^58+435038014277*z^56-6811601091249*z^54+ 76670831061576*z^52+569872021*z^60+z^68+1121991979184350182*z^32-\ 795771441820835710*z^38+447657241869187520*z^40-10503953*z^62)/(-1-\ 4555304381098299978*z^28+1784894781186124258*z^26+997*z^2-548124940068606314*z^ 24+131052927591902581*z^22-241081*z^4+26635907*z^6-1642820111*z^8+62900780899*z ^10-1600082267679*z^12+28343373932491*z^14+3409195547019741*z^18-\ 361677128737551*z^16+24189375712211353*z^50-131052927591902581*z^48-\ 24189375712211353*z^20-18343124626423370590*z^36+18343124626423370590*z^34+ 241081*z^66-26635907*z^64+9158047798950934606*z^30+4555304381098299978*z^42-\ 1784894781186124258*z^44+548124940068606314*z^46+1600082267679*z^58-\ 28343373932491*z^56+361677128737551*z^54-3409195547019741*z^52-62900780899*z^60 +z^70-997*z^68-14557801089150311302*z^32+14557801089150311302*z^38-\ 9158047798950934606*z^40+1642820111*z^62) The first , 40, terms are: [0, 424, 0, 292295, 0, 205331725, 0, 144469635408, 0, 101667221059623, 0, 71547848736400687, 0, 50351655236355804608, 0, 35434893934425639951981, 0, 24937249585472580356274519, 0, 17549549433276957201191536856, 0, 12350467306328610487054598627513, 0, 8691621587917163592581133381251817, 0, 6116714773274206625946174888365259000, 0, 4304628226076325103642456837356834573719, 0, 3029375220454045808709208689062535583558605, 0, 2131917960001524956918621619565013667307270752, 0, 1500333850191025987185887072613708900735955877247, 0, 1055857544362270313190389267664756961078389534758295, 0, 743058055941869489352723896903914364538464240414328752, 0, 522925916898757249248586303172990803359273960434907280365] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 22 4 6 8 10 f(z) = - (-1 + 346 z + z - 13806 z + 166926 z - 805302 z + 1728095 z 12 14 18 16 20 / - 1728095 z + 805302 z + 13806 z - 166926 z - 346 z ) / ( / 12 24 10 22 8 20 28879053 z + z - 19067134 z - 778 z + 5755648 z + 43742 z 6 18 4 16 2 14 - 788803 z - 788803 z + 43742 z + 5755648 z - 778 z - 19067134 z + 1) And in Maple-input format, it is: -(-1+346*z^2+z^22-13806*z^4+166926*z^6-805302*z^8+1728095*z^10-1728095*z^12+ 805302*z^14+13806*z^18-166926*z^16-346*z^20)/(28879053*z^12+z^24-19067134*z^10-\ 778*z^22+5755648*z^8+43742*z^20-788803*z^6-788803*z^18+43742*z^4+5755648*z^16-\ 778*z^2-19067134*z^14+1) The first , 40, terms are: [0, 432, 0, 306160, 0, 219917813, 0, 158039820344, 0, 113574366076969, 0, 81619596877701723, 0, 58655478789935669280, 0, 42152440455423280012197, 0, 30292621816914154147363671, 0, 21769627728102302535322273576, 0, 15644624433122388012301158170507, 0, 11242924165280953801513693572632560, 0, 8079666234661449071455907664303386768, 0, 5806408146478588893500732349187093126559, 0, 4172743599093188093632348836368002056056159, 0, 2998719467272189069466707736909755605878237328, 0, 2155013417395545999277825702753706687685565615920, 0, 1548688658555766675609338891566107790724945548746507, 0, 1112956671999705930067967496344594618748450786787355496, 0, 799820252382934134629876149369809835654825027718928271191] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 14 12 10 8 6 f(z) = - (73982 z - 2156079 z + 16357952 z - 16357952 z + 2156079 z 18 4 16 2 / 14 12 + z - 73982 z - 530 z + 530 z - 1) / (-8009774 z + 99015657 z / 10 8 20 6 18 4 - 395576540 z + 99015657 z + z - 8009774 z - 962 z + 209491 z 16 2 + 209491 z - 962 z + 1) And in Maple-input format, it is: -(73982*z^14-2156079*z^12+16357952*z^10-16357952*z^8+2156079*z^6+z^18-73982*z^4 -530*z^16+530*z^2-1)/(-8009774*z^14+99015657*z^12-395576540*z^10+99015657*z^8+z ^20-8009774*z^6-962*z^18+209491*z^4+209491*z^16-962*z^2+1) The first , 40, terms are: [0, 432, 0, 280075, 0, 184785733, 0, 122468247984, 0, 81304448476519, 0, 54011414637820903, 0, 35889187721892453360, 0, 23849670377197139536117, 0, 15849538316374348256980411, 0, 10533113020360408161784288176, 0, 7000017329944588843410001487617, 0, 4652028212368617552923458919334145, 0, 3091618443008119065342676600912888368, 0, 2054610780983794290174159371198898924379, 0, 1365442080453199614611883138586940969561749, 0, 907438123669617839075326317248661802014417648, 0, 603060337016464325448585999740014633356110101735, 0, 400778590593487445206693775021407401883202946192487, 0, 266347277683647380593626084531074140484969920796489136, 0, 177007639768952102722316864656008006375184126379475602341] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 50960592737051021 z + 30698641433719587 z + 583 z 24 22 4 6 - 14252775587298553 z + 5057498004724372 z - 113173 z + 10238335 z 8 10 12 14 - 511912473 z + 15629642605 z - 310571137441 z + 4195844266723 z 18 16 50 + 271258448810695 z - 39783467122213 z + 310571137441 z 48 20 36 - 4195844266723 z - 1356486374749744 z - 30698641433719587 z 34 30 42 + 50960592737051021 z + 65566515359387801 z + 1356486374749744 z 44 46 58 56 - 271258448810695 z + 39783467122213 z + 113173 z - 10238335 z 54 52 60 32 + 511912473 z - 15629642605 z - 583 z - 65566515359387801 z 38 40 62 / + 14252775587298553 z - 5057498004724372 z + z ) / (1 / 28 26 2 + 629102346091204376 z - 329644647076283500 z - 1003 z 24 22 4 6 + 132404622249727438 z - 40475106864331302 z + 253341 z - 26979700 z 8 10 12 14 + 1539410992 z - 53281760053 z + 1203651293527 z - 18603761496213 z 18 16 50 - 1605226267284092 z + 203198561237296 z - 18603761496213 z 48 20 36 + 203198561237296 z + 9332368933958206 z + 629102346091204376 z 34 64 30 - 925125257079295013 z + z - 925125257079295013 z 42 44 46 - 40475106864331302 z + 9332368933958206 z - 1605226267284092 z 58 56 54 52 - 26979700 z + 1539410992 z - 53281760053 z + 1203651293527 z 60 32 38 + 253341 z + 1051650309199944279 z - 329644647076283500 z 40 62 + 132404622249727438 z - 1003 z ) And in Maple-input format, it is: -(-1-50960592737051021*z^28+30698641433719587*z^26+583*z^2-14252775587298553*z^ 24+5057498004724372*z^22-113173*z^4+10238335*z^6-511912473*z^8+15629642605*z^10 -310571137441*z^12+4195844266723*z^14+271258448810695*z^18-39783467122213*z^16+ 310571137441*z^50-4195844266723*z^48-1356486374749744*z^20-30698641433719587*z^ 36+50960592737051021*z^34+65566515359387801*z^30+1356486374749744*z^42-\ 271258448810695*z^44+39783467122213*z^46+113173*z^58-10238335*z^56+511912473*z^ 54-15629642605*z^52-583*z^60-65566515359387801*z^32+14252775587298553*z^38-\ 5057498004724372*z^40+z^62)/(1+629102346091204376*z^28-329644647076283500*z^26-\ 1003*z^2+132404622249727438*z^24-40475106864331302*z^22+253341*z^4-26979700*z^6 +1539410992*z^8-53281760053*z^10+1203651293527*z^12-18603761496213*z^14-\ 1605226267284092*z^18+203198561237296*z^16-18603761496213*z^50+203198561237296* z^48+9332368933958206*z^20+629102346091204376*z^36-925125257079295013*z^34+z^64 -925125257079295013*z^30-40475106864331302*z^42+9332368933958206*z^44-\ 1605226267284092*z^46-26979700*z^58+1539410992*z^56-53281760053*z^54+ 1203651293527*z^52+253341*z^60+1051650309199944279*z^32-329644647076283500*z^38 +132404622249727438*z^40-1003*z^62) The first , 40, terms are: [0, 420, 0, 281092, 0, 192273421, 0, 131942088372, 0, 90602051220763, 0, 62223765125231047, 0, 42735478257051391752, 0, 29351070851102540229511, 0, 20158583810839127416546627, 0, 13845104501562548875961856420, 0, 9508948409553292111814375786053, 0, 6530835602615139387854885800693156, 0, 4485439608314188975086892160325732588, 0, 3080642312999440067173947450084967289281, 0, 2115814254818420317835138873905525360796425, 0, 1453161226211092937118713914630942379965701948, 0, 998044863613829242943567171278241612482473864340, 0, 685466644596010516995984140389457235376803420989461, 0, 470784969678201730454588730906729816587224667947389636, 0, 323339566443167824193313708599235181962521286111755521963] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 7}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 2047093917 z - 8533279206 z - 426 z + 23235254379 z 22 4 6 8 10 - 42060796532 z + 41959 z - 1567192 z + 29303452 z - 312952004 z 12 14 18 16 + 2047093917 z - 8533279206 z - 42060796532 z + 23235254379 z 20 36 34 30 32 + 51193918504 z + 41959 z - 1567192 z - 312952004 z + 29303452 z 38 40 / 42 30 16 - 426 z + z ) / (z + 12345973257 z - 219259076582 z / 8 6 40 32 2 - 115897146 z + 5057431 z - 846 z - 1522740506 z + 846 z 38 34 20 18 + 109928 z + 115897146 z - 742237774164 z + 494763651551 z 4 28 10 12 - 109928 z - 64274735628 z + 1522740506 z - 12345973257 z 14 24 26 + 64274735628 z - 1 - 494763651551 z + 219259076582 z 22 36 + 742237774164 z - 5057431 z ) And in Maple-input format, it is: -(1+2047093917*z^28-8533279206*z^26-426*z^2+23235254379*z^24-42060796532*z^22+ 41959*z^4-1567192*z^6+29303452*z^8-312952004*z^10+2047093917*z^12-8533279206*z^ 14-42060796532*z^18+23235254379*z^16+51193918504*z^20+41959*z^36-1567192*z^34-\ 312952004*z^30+29303452*z^32-426*z^38+z^40)/(z^42+12345973257*z^30-219259076582 *z^16-115897146*z^8+5057431*z^6-846*z^40-1522740506*z^32+846*z^2+109928*z^38+ 115897146*z^34-742237774164*z^20+494763651551*z^18-109928*z^4-64274735628*z^28+ 1522740506*z^10-12345973257*z^12+64274735628*z^14-1-494763651551*z^24+ 219259076582*z^26+742237774164*z^22-5057431*z^36) The first , 40, terms are: [0, 420, 0, 287351, 0, 200419425, 0, 140004440148, 0, 97817840656271, 0, 68344418602946031, 0, 47751725738251290564, 0, 33363777256971785301969, 0, 23311024959640159295999847, 0, 16287241180058410740619622356, 0, 11379775271326025446015710118241, 0, 7950965040774417955087184821345889, 0, 5555280624859151714374887749029251700, 0, 3881433594877780704119444180584610599879, 0, 2711929021916743215906721678094969652062257, 0, 1894804803467459615895688447064411168040445412, 0, 1323886139433624789447604926993155937299122711279, 0, 924989480170782572379927520948910698202386670365455, 0, 646283326746402404115705190011984039767336531126437940, 0, 451553393183757813125519762192399088889370602379892317057] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 159530266366855817173 z - 34799508830436151781 z - 617 z 24 22 4 + 6217068350419801990 z - 902298721365928464 z + 136610 z 6 8 10 12 - 15729508 z + 1101585289 z - 51186032342 z + 1668832117587 z 14 18 16 - 39712953109912 z - 9774724463429334 z + 710423080845087 z 50 48 - 19686841873597623436966 z + 30258804600041648982796 z 20 36 + 105339377937405833 z + 10762065702356524785420 z 34 66 80 - 4932364325366325764736 z - 902298721365928464 z + 1101585289 z 88 84 86 82 64 + z + 136610 z - 617 z - 15729508 z + 6217068350419801990 z 30 42 - 602995325005865408890 z - 39137455333593784157198 z 44 46 + 42637884669123250620176 z - 39137455333593784157198 z 58 56 - 602995325005865408890 z + 1889702456458265862098 z 54 52 - 4932364325366325764736 z + 10762065702356524785420 z 60 70 + 159530266366855817173 z - 9774724463429334 z 68 78 32 + 105339377937405833 z - 51186032342 z + 1889702456458265862098 z 38 40 - 19686841873597623436966 z + 30258804600041648982796 z 62 76 74 - 34799508830436151781 z + 1668832117587 z - 39712953109912 z 72 / 2 28 + 710423080845087 z ) / ((-1 + z ) (1 + 932004283095812578336 z / 26 2 24 - 190227768175040323534 z - 1060 z + 31665262977643861663 z 22 4 6 8 - 4267306585064754295 z + 287746 z - 37661719 z + 2919440771 z 10 12 14 - 148534742326 z + 5277652654160 z - 136554408664901 z 18 16 - 39555908516715936 z + 2651986851670613 z 50 48 - 146175994866015681630214 z + 230117706864411227913681 z 20 36 + 461330590715285662 z + 77325442609325420437737 z 34 66 80 - 34009445774597115197622 z - 4267306585064754295 z + 2919440771 z 88 84 86 82 64 + z + 287746 z - 1060 z - 37661719 z + 31665262977643861663 z 30 42 - 3746020374017943134246 z - 302004452568785168492066 z 44 46 + 330626808737022616008157 z - 302004452568785168492066 z 58 56 - 3746020374017943134246 z + 12409760977917385227029 z 54 52 - 34009445774597115197622 z + 77325442609325420437737 z 60 70 + 932004283095812578336 z - 39555908516715936 z 68 78 32 + 461330590715285662 z - 148534742326 z + 12409760977917385227029 z 38 40 - 146175994866015681630214 z + 230117706864411227913681 z 62 76 74 - 190227768175040323534 z + 5277652654160 z - 136554408664901 z 72 + 2651986851670613 z )) And in Maple-input format, it is: -(1+159530266366855817173*z^28-34799508830436151781*z^26-617*z^2+ 6217068350419801990*z^24-902298721365928464*z^22+136610*z^4-15729508*z^6+ 1101585289*z^8-51186032342*z^10+1668832117587*z^12-39712953109912*z^14-\ 9774724463429334*z^18+710423080845087*z^16-19686841873597623436966*z^50+ 30258804600041648982796*z^48+105339377937405833*z^20+10762065702356524785420*z^ 36-4932364325366325764736*z^34-902298721365928464*z^66+1101585289*z^80+z^88+ 136610*z^84-617*z^86-15729508*z^82+6217068350419801990*z^64-\ 602995325005865408890*z^30-39137455333593784157198*z^42+42637884669123250620176 *z^44-39137455333593784157198*z^46-602995325005865408890*z^58+ 1889702456458265862098*z^56-4932364325366325764736*z^54+10762065702356524785420 *z^52+159530266366855817173*z^60-9774724463429334*z^70+105339377937405833*z^68-\ 51186032342*z^78+1889702456458265862098*z^32-19686841873597623436966*z^38+ 30258804600041648982796*z^40-34799508830436151781*z^62+1668832117587*z^76-\ 39712953109912*z^74+710423080845087*z^72)/(-1+z^2)/(1+932004283095812578336*z^ 28-190227768175040323534*z^26-1060*z^2+31665262977643861663*z^24-\ 4267306585064754295*z^22+287746*z^4-37661719*z^6+2919440771*z^8-148534742326*z^ 10+5277652654160*z^12-136554408664901*z^14-39555908516715936*z^18+ 2651986851670613*z^16-146175994866015681630214*z^50+230117706864411227913681*z^ 48+461330590715285662*z^20+77325442609325420437737*z^36-34009445774597115197622 *z^34-4267306585064754295*z^66+2919440771*z^80+z^88+287746*z^84-1060*z^86-\ 37661719*z^82+31665262977643861663*z^64-3746020374017943134246*z^30-\ 302004452568785168492066*z^42+330626808737022616008157*z^44-\ 302004452568785168492066*z^46-3746020374017943134246*z^58+ 12409760977917385227029*z^56-34009445774597115197622*z^54+ 77325442609325420437737*z^52+932004283095812578336*z^60-39555908516715936*z^70+ 461330590715285662*z^68-148534742326*z^78+12409760977917385227029*z^32-\ 146175994866015681630214*z^38+230117706864411227913681*z^40-\ 190227768175040323534*z^62+5277652654160*z^76-136554408664901*z^74+ 2651986851670613*z^72) The first , 40, terms are: [0, 444, 0, 318888, 0, 232330261, 0, 169399684452, 0, 123523635485321, 0, 90072244893526683, 0, 65679877578318567728, 0, 47893186943859880310101, 0, 34923289724697013191247863, 0, 25465755062968378275958310032, 0, 18569404149274291552706811310775, 0, 13540645844597771404643537838658424, 0, 9873719609736830629047413226391165832, 0, 7199829317644283185073300257768570921451, 0, 5250052082914774870288708445394971752720867, 0, 3828291707661829586183162740955209857892929360, 0, 2791556572676055273279253162604116686043111217864, 0, 2035578449482998124528291959988774335020051659930751, 0, 1484325864844454443272885067467401595352904240896143880, 0, 1082357338576569438029886565328679261150192842342516991927] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 2555744282281 z + 5160678174964 z + 508 z 24 22 4 6 - 7300084110459 z + 7300084110459 z - 75595 z + 4724105 z 8 10 12 14 - 150977972 z + 2719382939 z - 29371952937 z + 198629472476 z 18 16 20 + 2555744282281 z - 870643677635 z - 5160678174964 z 36 34 30 42 - 2719382939 z + 29371952937 z + 870643677635 z + 75595 z 44 46 32 38 40 / - 508 z + z - 198629472476 z + 150977972 z - 4724105 z ) / (1 / 28 26 2 24 + 57980757413056 z - 98958829338436 z - 988 z + 118118290246950 z 22 4 6 8 - 98958829338436 z + 183216 z - 13434680 z + 505391448 z 10 12 14 - 10905305356 z + 143293881068 z - 1193681788500 z 18 16 48 20 - 23506174208584 z + 6485927317688 z + z + 57980757413056 z 36 34 30 42 + 143293881068 z - 1193681788500 z - 23506174208584 z - 13434680 z 44 46 32 38 + 183216 z - 988 z + 6485927317688 z - 10905305356 z 40 + 505391448 z ) And in Maple-input format, it is: -(-1-2555744282281*z^28+5160678174964*z^26+508*z^2-7300084110459*z^24+ 7300084110459*z^22-75595*z^4+4724105*z^6-150977972*z^8+2719382939*z^10-\ 29371952937*z^12+198629472476*z^14+2555744282281*z^18-870643677635*z^16-\ 5160678174964*z^20-2719382939*z^36+29371952937*z^34+870643677635*z^30+75595*z^ 42-508*z^44+z^46-198629472476*z^32+150977972*z^38-4724105*z^40)/(1+ 57980757413056*z^28-98958829338436*z^26-988*z^2+118118290246950*z^24-\ 98958829338436*z^22+183216*z^4-13434680*z^6+505391448*z^8-10905305356*z^10+ 143293881068*z^12-1193681788500*z^14-23506174208584*z^18+6485927317688*z^16+z^ 48+57980757413056*z^20+143293881068*z^36-1193681788500*z^34-23506174208584*z^30 -13434680*z^42+183216*z^44-988*z^46+6485927317688*z^32-10905305356*z^38+ 505391448*z^40) The first , 40, terms are: [0, 480, 0, 366619, 0, 282986467, 0, 218514395616, 0, 168735581305033, 0, 130297087968035545, 0, 100615046994342735744, 0, 77694662480823750342931, 0, 59995605093469759136616043, 0, 46328441613549019898342100864, 0, 35774695484992875112791221819185, 0, 27625121685410452404799776852088657, 0, 21332043160394243691602223288776760000, 0, 16472545191988063024676174873502871786699, 0, 12720054195553582313893275558141073306311091, 0, 9822390945178149837024964463581793821952733888, 0, 7584823334608361330183551237705905919550695167289, 0, 5856979765752552158843777395948143185456477944116073, 0, 4522743703193464385636662024566103786088456291385031584, 0, 3492450276912964702124315813261094003923006317761890204995] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 352512465030688256 z - 136182471361832508 z - 528 z 24 22 4 6 + 42208256784497252 z - 10432111377059824 z + 88402 z - 7205128 z 8 10 12 14 + 342437049 z - 10479228788 z + 219996615104 z - 3311907025380 z 18 16 50 - 312287639173472 z + 36925174841108 z - 10432111377059824 z 48 20 36 + 42208256784497252 z + 2039840203713448 z + 1882777441230131852 z 34 66 64 - 1696544993375403736 z - 7205128 z + 342437049 z 30 42 44 - 735429546258408236 z - 735429546258408236 z + 352512465030688256 z 46 58 56 - 136182471361832508 z - 3311907025380 z + 36925174841108 z 54 52 60 70 - 312287639173472 z + 2039840203713448 z + 219996615104 z - 528 z 68 32 38 + 88402 z + 1240718304413356494 z - 1696544993375403736 z 40 62 72 / 2 + 1240718304413356494 z - 10479228788 z + z ) / ((-1 + z ) (1 / 28 26 2 + 2473433810383957144 z - 897510448711810944 z - 966 z 24 22 4 6 + 258722269928922744 z - 58999319490070788 z + 200102 z - 18667514 z 8 10 12 14 + 991014973 z - 33595808392 z + 779137165800 z - 12940571750128 z 18 16 50 - 1477883023809292 z + 158954054983408 z - 58999319490070788 z 48 20 + 258722269928922744 z + 10577672834231016 z 36 34 66 + 14892418309688993252 z - 13314610680847932016 z - 18667514 z 64 30 42 + 991014973 z - 5430268913223665864 z - 5430268913223665864 z 44 46 58 + 2473433810383957144 z - 897510448711810944 z - 12940571750128 z 56 54 52 + 158954054983408 z - 1477883023809292 z + 10577672834231016 z 60 70 68 32 + 779137165800 z - 966 z + 200102 z + 9513777681162814042 z 38 40 62 - 13314610680847932016 z + 9513777681162814042 z - 33595808392 z 72 + z )) And in Maple-input format, it is: -(1+352512465030688256*z^28-136182471361832508*z^26-528*z^2+42208256784497252*z ^24-10432111377059824*z^22+88402*z^4-7205128*z^6+342437049*z^8-10479228788*z^10 +219996615104*z^12-3311907025380*z^14-312287639173472*z^18+36925174841108*z^16-\ 10432111377059824*z^50+42208256784497252*z^48+2039840203713448*z^20+ 1882777441230131852*z^36-1696544993375403736*z^34-7205128*z^66+342437049*z^64-\ 735429546258408236*z^30-735429546258408236*z^42+352512465030688256*z^44-\ 136182471361832508*z^46-3311907025380*z^58+36925174841108*z^56-312287639173472* z^54+2039840203713448*z^52+219996615104*z^60-528*z^70+88402*z^68+ 1240718304413356494*z^32-1696544993375403736*z^38+1240718304413356494*z^40-\ 10479228788*z^62+z^72)/(-1+z^2)/(1+2473433810383957144*z^28-897510448711810944* z^26-966*z^2+258722269928922744*z^24-58999319490070788*z^22+200102*z^4-18667514 *z^6+991014973*z^8-33595808392*z^10+779137165800*z^12-12940571750128*z^14-\ 1477883023809292*z^18+158954054983408*z^16-58999319490070788*z^50+ 258722269928922744*z^48+10577672834231016*z^20+14892418309688993252*z^36-\ 13314610680847932016*z^34-18667514*z^66+991014973*z^64-5430268913223665864*z^30 -5430268913223665864*z^42+2473433810383957144*z^44-897510448711810944*z^46-\ 12940571750128*z^58+158954054983408*z^56-1477883023809292*z^54+ 10577672834231016*z^52+779137165800*z^60-966*z^70+200102*z^68+ 9513777681162814042*z^32-13314610680847932016*z^38+9513777681162814042*z^40-\ 33595808392*z^62+z^72) The first , 40, terms are: [0, 439, 0, 311847, 0, 224949685, 0, 162439530785, 0, 117313509435051, 0, 84724760748291555, 0, 61189009276314116597, 0, 44191280408925138452045, 0, 31915360941431334824590907, 0, 23049575799419566838721564499, 0, 16646621848745606236732849269609, 0, 12022347890534758937999168079751981, 0, 8682653460581994610372791458804205535, 0, 6270694526813131861271039760497696990975, 0, 4528754951135939748958020227168332358931305, 0, 3270709698860403558166095424021284281211217689, 0, 2362137507911812187932945690021747686837699886511, 0, 1705958070270782973319545997978357032615233381181871, 0, 1232059068438731369147448173688965715173533259938122045, 0, 889804722973742598593810808969302053709405376809277212121] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {5, 6}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 336856600391277836 z - 151984369556408080 z - 572 z 24 22 4 6 + 54059419401595740 z - 15037112265647672 z + 109241 z - 10178424 z 8 10 12 14 + 540666900 z - 17918433008 z + 394036496260 z - 6014993716264 z 18 16 50 - 532781936489200 z + 65939169931484 z - 532781936489200 z 48 20 36 + 3237729759081836 z + 3237729759081836 z + 829351387550976766 z 34 66 64 30 - 927624451622547880 z - 572 z + 109241 z - 592198948359103016 z 42 44 46 - 151984369556408080 z + 54059419401595740 z - 15037112265647672 z 58 56 54 - 17918433008 z + 394036496260 z - 6014993716264 z 52 60 68 32 + 65939169931484 z + 540666900 z + z + 829351387550976766 z 38 40 62 / - 592198948359103016 z + 336856600391277836 z - 10178424 z ) / ((1 / 28 26 2 + 2488766011369925980 z - 1058411879349875000 z - 1026 z 24 22 4 6 + 349889294278581164 z - 89377416006456876 z + 245065 z - 26240716 z 8 10 12 14 + 1558497476 z - 57337950408 z + 1399849873972 z - 23762830828740 z 18 16 50 - 2602736389123848 z + 289881877408812 z - 2602736389123848 z 48 20 36 + 17505905769434268 z + 17505905769434268 z + 6571170622736068078 z 34 66 64 - 7415615692632505836 z - 1026 z + 245065 z 30 42 - 4569399433985528388 z - 1058411879349875000 z 44 46 58 + 349889294278581164 z - 89377416006456876 z - 57337950408 z 56 54 52 + 1399849873972 z - 23762830828740 z + 289881877408812 z 60 68 32 38 + 1558497476 z + z + 6571170622736068078 z - 4569399433985528388 z 40 62 2 + 2488766011369925980 z - 26240716 z ) (-1 + z )) And in Maple-input format, it is: -(1+336856600391277836*z^28-151984369556408080*z^26-572*z^2+54059419401595740*z ^24-15037112265647672*z^22+109241*z^4-10178424*z^6+540666900*z^8-17918433008*z^ 10+394036496260*z^12-6014993716264*z^14-532781936489200*z^18+65939169931484*z^ 16-532781936489200*z^50+3237729759081836*z^48+3237729759081836*z^20+ 829351387550976766*z^36-927624451622547880*z^34-572*z^66+109241*z^64-\ 592198948359103016*z^30-151984369556408080*z^42+54059419401595740*z^44-\ 15037112265647672*z^46-17918433008*z^58+394036496260*z^56-6014993716264*z^54+ 65939169931484*z^52+540666900*z^60+z^68+829351387550976766*z^32-\ 592198948359103016*z^38+336856600391277836*z^40-10178424*z^62)/(1+ 2488766011369925980*z^28-1058411879349875000*z^26-1026*z^2+349889294278581164*z ^24-89377416006456876*z^22+245065*z^4-26240716*z^6+1558497476*z^8-57337950408*z ^10+1399849873972*z^12-23762830828740*z^14-2602736389123848*z^18+ 289881877408812*z^16-2602736389123848*z^50+17505905769434268*z^48+ 17505905769434268*z^20+6571170622736068078*z^36-7415615692632505836*z^34-1026*z ^66+245065*z^64-4569399433985528388*z^30-1058411879349875000*z^42+ 349889294278581164*z^44-89377416006456876*z^46-57337950408*z^58+1399849873972*z ^56-23762830828740*z^54+289881877408812*z^52+1558497476*z^60+z^68+ 6571170622736068078*z^32-4569399433985528388*z^38+2488766011369925980*z^40-\ 26240716*z^62)/(-1+z^2) The first , 40, terms are: [0, 455, 0, 330435, 0, 243692697, 0, 179962279297, 0, 132922432522843, 0, 98180714098508671, 0, 72519649005846683625, 0, 53565533086138421437913, 0, 39565367379865198439665679, 0, 29224357960735714262533414059, 0, 21586128378022958117928182830193, 0, 15944266049791265953146880294693417, 0, 11776990084699317945698505534673429107, 0, 8698894952119253399096343796128843373623, 0, 6425306707727483858090330328138663743253649, 0, 4745955263928560760677660648794372776490039921, 0, 3505527812411294384831698048463663735843803955031, 0, 2589304904955403927291851179192864239691089481347347, 0, 1912550762566733848358028371950812707997962763256037897, 0, 1412676588374822987341615763763139922315677683582589060497] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 8139081154559675 z - 4508585986135598 z - 474 z 24 22 4 6 + 1964022179503826 z - 669904262784952 z + 63470 z - 4054984 z 8 10 12 14 + 150019628 z - 3560110727 z + 57762902420 z - 668844389772 z 18 16 50 - 36431134827877 z + 5695069745424 z - 668844389772 z 48 20 36 + 5695069745424 z + 177826625142204 z + 8139081154559675 z 34 64 30 - 11588109130087180 z + z - 11588109130087180 z 42 44 46 - 669904262784952 z + 177826625142204 z - 36431134827877 z 58 56 54 52 - 4054984 z + 150019628 z - 3560110727 z + 57762902420 z 60 32 38 + 63470 z + 13034039988656776 z - 4508585986135598 z 40 62 / 28 + 1964022179503826 z - 474 z ) / (-1 - 94970853781974514 z / 26 2 24 + 46210534960950553 z + 935 z - 17630692910687529 z 22 4 6 8 + 5256912887878108 z - 152120 z + 11155392 z - 468428005 z 10 12 14 + 12609564933 z - 232524856500 z + 3067269511996 z 18 16 50 + 218127833051013 z - 29817206973453 z + 29817206973453 z 48 20 36 - 218127833051013 z - 1218827815416308 z - 153365513980404248 z 34 66 64 30 + 194839788235230217 z + z - 935 z + 153365513980404248 z 42 44 46 + 17630692910687529 z - 5256912887878108 z + 1218827815416308 z 58 56 54 52 + 468428005 z - 12609564933 z + 232524856500 z - 3067269511996 z 60 32 38 - 11155392 z - 194839788235230217 z + 94970853781974514 z 40 62 - 46210534960950553 z + 152120 z ) And in Maple-input format, it is: -(1+8139081154559675*z^28-4508585986135598*z^26-474*z^2+1964022179503826*z^24-\ 669904262784952*z^22+63470*z^4-4054984*z^6+150019628*z^8-3560110727*z^10+ 57762902420*z^12-668844389772*z^14-36431134827877*z^18+5695069745424*z^16-\ 668844389772*z^50+5695069745424*z^48+177826625142204*z^20+8139081154559675*z^36 -11588109130087180*z^34+z^64-11588109130087180*z^30-669904262784952*z^42+ 177826625142204*z^44-36431134827877*z^46-4054984*z^58+150019628*z^56-3560110727 *z^54+57762902420*z^52+63470*z^60+13034039988656776*z^32-4508585986135598*z^38+ 1964022179503826*z^40-474*z^62)/(-1-94970853781974514*z^28+46210534960950553*z^ 26+935*z^2-17630692910687529*z^24+5256912887878108*z^22-152120*z^4+11155392*z^6 -468428005*z^8+12609564933*z^10-232524856500*z^12+3067269511996*z^14+ 218127833051013*z^18-29817206973453*z^16+29817206973453*z^50-218127833051013*z^ 48-1218827815416308*z^20-153365513980404248*z^36+194839788235230217*z^34+z^66-\ 935*z^64+153365513980404248*z^30+17630692910687529*z^42-5256912887878108*z^44+ 1218827815416308*z^46+468428005*z^58-12609564933*z^56+232524856500*z^54-\ 3067269511996*z^52-11155392*z^60-194839788235230217*z^32+94970853781974514*z^38 -46210534960950553*z^40+152120*z^62) The first , 40, terms are: [0, 461, 0, 342385, 0, 257103063, 0, 193131985040, 0, 145080431102661, 0, 108984306493857039, 0, 81868932648754257439, 0, 61499883707398383325103, 0, 46198668719846359614288191, 0, 34704406950162457975260982063, 0, 26069925717336046752451711121527, 0, 19583709581425521104247766412394189, 0, 14711268652161523619160675677523981008, 0, 11051094505677259105900551984436282731855, 0, 8301574300695410807019946949425535135267849, 0, 6236136686240655699005105632560464963925465933, 0, 4684581425262781511729775673502102458160246780545, 0, 3519054221235553102562932258986034892146128048639521, 0, 2643511017913644077441729003049373941298407043910915533, 0, 1985803588833952343377431237662342794401577066640214045753] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 7}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 1333931 z + 20627050 z + 422 z - 164915523 z 22 4 6 8 10 + 739564236 z - 39761 z + 1333931 z - 20627050 z + 164915523 z 12 14 18 16 - 739564236 z + 1961583760 z + 3171074484 z - 3171074484 z 20 34 30 32 / 4 - 1961583760 z + z + 39761 z - 422 z ) / (105682 z / 28 18 14 20 + 85639754 z - 53245502628 z - 19301314892 z + 41314169756 z 22 36 26 16 32 - 19301314892 z + z - 900190682 z + 41314169756 z + 105682 z 12 10 6 8 + 5417373139 z - 900190682 z - 4373986 z + 85639754 z 24 2 34 30 + 5417373139 z + 1 - 882 z - 882 z - 4373986 z ) And in Maple-input format, it is: -(-1-1333931*z^28+20627050*z^26+422*z^2-164915523*z^24+739564236*z^22-39761*z^4 +1333931*z^6-20627050*z^8+164915523*z^10-739564236*z^12+1961583760*z^14+ 3171074484*z^18-3171074484*z^16-1961583760*z^20+z^34+39761*z^30-422*z^32)/( 105682*z^4+85639754*z^28-53245502628*z^18-19301314892*z^14+41314169756*z^20-\ 19301314892*z^22+z^36-900190682*z^26+41314169756*z^16+105682*z^32+5417373139*z^ 12-900190682*z^10-4373986*z^6+85639754*z^8+5417373139*z^24+1-882*z^2-882*z^34-\ 4373986*z^30) The first , 40, terms are: [0, 460, 0, 339799, 0, 254129053, 0, 190178207684, 0, 142327929655275, 0, 106517686638642691, 0, 79717476031027456484, 0, 59660292189448141335173, 0, 44649563176887204226334687, 0, 33415583795384683783099536748, 0, 25008111188470940682967148175609, 0, 18715986799685007140760284498850121, 0, 14006981944628216083725036739897031980, 0, 10482778455499509606618898732736417384847, 0, 7845276347288246330930962097633012621876981, 0, 5871378587900113318493823991481868119762437796, 0, 4394120104432994494894526513971859411274232734355, 0, 3288544794568902423032625231109541338960755996912347, 0, 2461135929119466402469650083080119513493573591389682948, 0, 1841905900630062824783049166208444907929125313362399415725] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 26 2 24 22 4 6 f(z) = - (-1 + z + 498 z - 498 z + 66817 z - 66817 z + 2756233 z 8 10 12 14 - 38750563 z + 188693851 z - 400519033 z + 400519033 z 18 16 20 / 20 + 38750563 z - 188693851 z - 2756233 z ) / (175480987 z / 4 8 14 10 2 + 176286 z + 175480987 z - 7172768024 z - 1394968824 z - 954 z 24 18 16 26 22 + 176286 z - 1394968824 z + 1 + 4743886303 z - 954 z - 9090140 z 6 28 12 - 9090140 z + z + 4743886303 z ) And in Maple-input format, it is: -(-1+z^26+498*z^2-498*z^24+66817*z^22-66817*z^4+2756233*z^6-38750563*z^8+ 188693851*z^10-400519033*z^12+400519033*z^14+38750563*z^18-188693851*z^16-\ 2756233*z^20)/(175480987*z^20+176286*z^4+175480987*z^8-7172768024*z^14-\ 1394968824*z^10-954*z^2+176286*z^24-1394968824*z^18+1+4743886303*z^16-954*z^26-\ 9090140*z^22-9090140*z^6+z^28+4743886303*z^12) The first , 40, terms are: [0, 456, 0, 325555, 0, 236526961, 0, 172264305480, 0, 125524283053675, 0, 91475946916316899, 0, 66664732247649339432, 0, 48583358388781614433945, 0, 35406206136519302024530363, 0, 25803068484010915548268807848, 0, 18804567027030920194328400628249, 0, 13704251690154324887063363716924585, 0, 9987282055106486145657811191130823400, 0, 7278456726668479516640802254503912777195, 0, 5304339262213199272640214868383654885908233, 0, 3865656699753291992993348163943609425035508840, 0, 2817184380885052842654445925110667961157356808243, 0, 2053086565192863618890340287151983756334776299496347, 0, 1496233073268767070442856254243278961652794502731719432, 0, 1090413549773100219334799533175090892289131688624566397057] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 4153138331313 z + 8537352036868 z + 524 z 24 22 4 6 - 12185904123343 z + 12185904123343 z - 83183 z + 5654317 z 8 10 12 14 - 195644476 z + 3772754243 z - 43044898557 z + 303787289012 z 18 16 20 + 4153138331313 z - 1378048112083 z - 8537352036868 z 36 34 30 42 - 3772754243 z + 43044898557 z + 1378048112083 z + 83183 z 44 46 32 38 40 / - 524 z + z - 303787289012 z + 195644476 z - 5654317 z ) / (1 / 28 26 2 24 + 95696443545128 z - 166067965288804 z - 1004 z + 199353509104502 z 22 4 6 8 - 166067965288804 z + 199240 z - 15841716 z + 641869880 z 10 12 14 - 14767196788 z + 204816659276 z - 1785434395196 z 18 16 48 20 - 37796994033676 z + 10086520403800 z + z + 95696443545128 z 36 34 30 42 + 204816659276 z - 1785434395196 z - 37796994033676 z - 15841716 z 44 46 32 38 + 199240 z - 1004 z + 10086520403800 z - 14767196788 z 40 + 641869880 z ) And in Maple-input format, it is: -(-1-4153138331313*z^28+8537352036868*z^26+524*z^2-12185904123343*z^24+ 12185904123343*z^22-83183*z^4+5654317*z^6-195644476*z^8+3772754243*z^10-\ 43044898557*z^12+303787289012*z^14+4153138331313*z^18-1378048112083*z^16-\ 8537352036868*z^20-3772754243*z^36+43044898557*z^34+1378048112083*z^30+83183*z^ 42-524*z^44+z^46-303787289012*z^32+195644476*z^38-5654317*z^40)/(1+ 95696443545128*z^28-166067965288804*z^26-1004*z^2+199353509104502*z^24-\ 166067965288804*z^22+199240*z^4-15841716*z^6+641869880*z^8-14767196788*z^10+ 204816659276*z^12-1785434395196*z^14-37796994033676*z^18+10086520403800*z^16+z^ 48+95696443545128*z^20+204816659276*z^36-1785434395196*z^34-37796994033676*z^30 -15841716*z^42+199240*z^44-1004*z^46+10086520403800*z^32-14767196788*z^38+ 641869880*z^40) The first , 40, terms are: [0, 480, 0, 365863, 0, 281878651, 0, 217269419760, 0, 167475789654853, 0, 129094465198886209, 0, 99509263696848720144, 0, 76704253176581202842527, 0, 59125576055421851396860267, 0, 45575487749860164581589926592, 0, 35130737369875858083024385230733, 0, 27079659903515983060656121278078613, 0, 20873685991097933686850087327586460800, 0, 16089964512396400853130145070558524644547, 0, 12402551141212726901611712959101270511890295, 0, 9560199756306969918324919702201649620373904720, 0, 7369243499975215739292659535427974614399003901289, 0, 5680399065521709575058913146503469929718058351783645, 0, 4378595108668675874570477994388155953354943787199954352, 0, 3375131730097278842524647068489940590249925085821042162403] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 500805454418369840 z - 221470187452547586 z - 571 z 24 22 4 6 + 76852807470960990 z - 20770655848244064 z + 109400 z - 10366203 z 8 10 12 14 + 567745253 z - 19577701544 z + 449531173061 z - 7164895454963 z 18 16 50 - 687503065289699 z + 81859741439224 z - 687503065289699 z 48 20 36 + 4329679061493441 z + 4329679061493441 z + 1262872753907036214 z 34 66 64 30 - 1416867355653320944 z - 571 z + 109400 z - 893573796340014322 z 42 44 46 - 221470187452547586 z + 76852807470960990 z - 20770655848244064 z 58 56 54 - 19577701544 z + 449531173061 z - 7164895454963 z 52 60 68 32 + 81859741439224 z + 567745253 z + z + 1262872753907036214 z 38 40 62 / - 893573796340014322 z + 500805454418369840 z - 10366203 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 3705410845442108202 z - 1536366130752196152 z - 1032 z 24 22 4 + 492782630760102602 z - 121675503831953128 z + 243849 z 6 8 10 12 - 26328126 z + 1603707765 z - 61168120354 z + 1556158280073 z 14 18 16 - 27571648282638 z - 3283456364562224 z + 350948494262301 z 50 48 20 - 3283456364562224 z + 22968657744978789 z + 22968657744978789 z 36 34 66 + 10095937230837940018 z - 11440620113851009932 z - 1032 z 64 30 42 + 243849 z - 6935789946776882980 z - 1536366130752196152 z 44 46 58 + 492782630760102602 z - 121675503831953128 z - 61168120354 z 56 54 52 + 1556158280073 z - 27571648282638 z + 350948494262301 z 60 68 32 + 1603707765 z + z + 10095937230837940018 z 38 40 62 - 6935789946776882980 z + 3705410845442108202 z - 26328126 z )) And in Maple-input format, it is: -(1+500805454418369840*z^28-221470187452547586*z^26-571*z^2+76852807470960990*z ^24-20770655848244064*z^22+109400*z^4-10366203*z^6+567745253*z^8-19577701544*z^ 10+449531173061*z^12-7164895454963*z^14-687503065289699*z^18+81859741439224*z^ 16-687503065289699*z^50+4329679061493441*z^48+4329679061493441*z^20+ 1262872753907036214*z^36-1416867355653320944*z^34-571*z^66+109400*z^64-\ 893573796340014322*z^30-221470187452547586*z^42+76852807470960990*z^44-\ 20770655848244064*z^46-19577701544*z^58+449531173061*z^56-7164895454963*z^54+ 81859741439224*z^52+567745253*z^60+z^68+1262872753907036214*z^32-\ 893573796340014322*z^38+500805454418369840*z^40-10366203*z^62)/(-1+z^2)/(1+ 3705410845442108202*z^28-1536366130752196152*z^26-1032*z^2+492782630760102602*z ^24-121675503831953128*z^22+243849*z^4-26328126*z^6+1603707765*z^8-61168120354* z^10+1556158280073*z^12-27571648282638*z^14-3283456364562224*z^18+ 350948494262301*z^16-3283456364562224*z^50+22968657744978789*z^48+ 22968657744978789*z^20+10095937230837940018*z^36-11440620113851009932*z^34-1032 *z^66+243849*z^64-6935789946776882980*z^30-1536366130752196152*z^42+ 492782630760102602*z^44-121675503831953128*z^46-61168120354*z^58+1556158280073* z^56-27571648282638*z^54+350948494262301*z^52+1603707765*z^60+z^68+ 10095937230837940018*z^32-6935789946776882980*z^38+3705410845442108202*z^40-\ 26328126*z^62) The first , 40, terms are: [0, 462, 0, 341765, 0, 256113995, 0, 192087963682, 0, 144080903854719, 0, 108073079292879007, 0, 81064224629369388706, 0, 60805241370495306168011, 0, 45609237336559281160546789, 0, 34210908293893454229686602894, 0, 25661166795616686696516780096961, 0, 19248114539540577436588039742866753, 0, 14437765682296002157746009988515597070, 0, 10829584241554170465566483257616063495333, 0, 8123133275998611554786308744218814514956107, 0, 6093058860601935059183892088718548238352974498, 0, 4570325885019512668883358929418834462315350216799, 0, 3428143264845446801350524226955914743688598174315711, 0, 2571406621752318323926108364174526963208859588003206690, 0, 1928779372261669269021845094480277918430165657903001180619] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 60681298606190 z - 70109606228200 z - 498 z 24 22 4 6 + 60681298606190 z - 39259917173921 z + 72068 z - 4594138 z 8 10 12 14 + 154939460 z - 3049982137 z + 37564810905 z - 304843078648 z 18 16 50 48 - 6654039157594 z + 1695074013690 z - 498 z + 72068 z 20 36 34 + 18862563549730 z + 1695074013690 z - 6654039157594 z 30 42 44 46 52 - 39259917173921 z - 3049982137 z + 154939460 z - 4594138 z + z 32 38 40 / 2 + 18862563549730 z - 304843078648 z + 37564810905 z ) / ((-1 + z ) / 28 26 2 (1 + 490152625092243 z - 575845221278897 z - 983 z 24 22 4 6 + 490152625092243 z - 301868684272186 z + 173297 z - 12651734 z 8 10 12 14 + 486862914 z - 11040844698 z + 157785934305 z - 1486660316823 z 18 16 50 48 - 42492583836690 z + 9529451383461 z - 983 z + 173297 z 20 36 34 + 133940226571914 z + 9529451383461 z - 42492583836690 z 30 42 44 46 - 301868684272186 z - 11040844698 z + 486862914 z - 12651734 z 52 32 38 40 + z + 133940226571914 z - 1486660316823 z + 157785934305 z )) And in Maple-input format, it is: -(1+60681298606190*z^28-70109606228200*z^26-498*z^2+60681298606190*z^24-\ 39259917173921*z^22+72068*z^4-4594138*z^6+154939460*z^8-3049982137*z^10+ 37564810905*z^12-304843078648*z^14-6654039157594*z^18+1695074013690*z^16-498*z^ 50+72068*z^48+18862563549730*z^20+1695074013690*z^36-6654039157594*z^34-\ 39259917173921*z^30-3049982137*z^42+154939460*z^44-4594138*z^46+z^52+ 18862563549730*z^32-304843078648*z^38+37564810905*z^40)/(-1+z^2)/(1+ 490152625092243*z^28-575845221278897*z^26-983*z^2+490152625092243*z^24-\ 301868684272186*z^22+173297*z^4-12651734*z^6+486862914*z^8-11040844698*z^10+ 157785934305*z^12-1486660316823*z^14-42492583836690*z^18+9529451383461*z^16-983 *z^50+173297*z^48+133940226571914*z^20+9529451383461*z^36-42492583836690*z^34-\ 301868684272186*z^30-11040844698*z^42+486862914*z^44-12651734*z^46+z^52+ 133940226571914*z^32-1486660316823*z^38+157785934305*z^40) The first , 40, terms are: [0, 486, 0, 376012, 0, 293526621, 0, 229187213582, 0, 178952477819727, 0, 139728626307298217, 0, 109102096570367930968, 0, 85188467735700648675783, 0, 66516366502713050422989453, 0, 51936924454259658814750706674, 0, 40553088865537804859672464405587, 0, 31664428223634213705443187371123964, 0, 24724035647549985547087743219165062610, 0, 19304878470695896678696311039808640997219, 0, 15073523516993833358962410160233109469934243, 0, 11769621423013068315096745386975327396096532370, 0, 9189887705079488931899414800481156117341744041852, 0, 7175594948775387046589547544495001367824194616289307, 0, 5602806532710019481058065162834179461369119690588001282, 0, 4374750981218003019746071774045939706588872969994912735373] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 4}, {3, 5}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 531968995940296 z - 462864605233032 z - 486 z 24 22 4 6 + 304619101973755 z - 151195561616506 z + 67558 z - 4376438 z 8 10 12 14 + 157010395 z - 3423875276 z + 48476944828 z - 467447194220 z 18 16 50 48 - 15599504267114 z + 3177137811753 z - 4376438 z + 157010395 z 20 36 34 + 56301195821690 z + 56301195821690 z - 151195561616506 z 30 42 44 - 462864605233032 z - 467447194220 z + 48476944828 z 46 56 54 52 32 - 3423875276 z + z - 486 z + 67558 z + 304619101973755 z 38 40 / 28 - 15599504267114 z + 3177137811753 z ) / (-1 - 8023550593100304 z / 26 2 24 + 6031552740405251 z + 959 z - 3406090248515517 z 22 4 6 8 + 1442299361166460 z - 162020 z + 12040484 z - 493183641 z 10 12 14 + 12393153831 z - 204244374312 z + 2309556574920 z 18 16 50 + 107151800260651 z - 18491109823637 z + 493183641 z 48 20 36 - 12393153831 z - 456277602834812 z - 1442299361166460 z 34 30 42 + 3406090248515517 z + 8023550593100304 z + 18491109823637 z 44 46 58 56 54 - 2309556574920 z + 204244374312 z + z - 959 z + 162020 z 52 32 38 - 12040484 z - 6031552740405251 z + 456277602834812 z 40 - 107151800260651 z ) And in Maple-input format, it is: -(1+531968995940296*z^28-462864605233032*z^26-486*z^2+304619101973755*z^24-\ 151195561616506*z^22+67558*z^4-4376438*z^6+157010395*z^8-3423875276*z^10+ 48476944828*z^12-467447194220*z^14-15599504267114*z^18+3177137811753*z^16-\ 4376438*z^50+157010395*z^48+56301195821690*z^20+56301195821690*z^36-\ 151195561616506*z^34-462864605233032*z^30-467447194220*z^42+48476944828*z^44-\ 3423875276*z^46+z^56-486*z^54+67558*z^52+304619101973755*z^32-15599504267114*z^ 38+3177137811753*z^40)/(-1-8023550593100304*z^28+6031552740405251*z^26+959*z^2-\ 3406090248515517*z^24+1442299361166460*z^22-162020*z^4+12040484*z^6-493183641*z ^8+12393153831*z^10-204244374312*z^12+2309556574920*z^14+107151800260651*z^18-\ 18491109823637*z^16+493183641*z^50-12393153831*z^48-456277602834812*z^20-\ 1442299361166460*z^36+3406090248515517*z^34+8023550593100304*z^30+ 18491109823637*z^42-2309556574920*z^44+204244374312*z^46+z^58-959*z^56+162020*z ^54-12040484*z^52-6031552740405251*z^32+456277602834812*z^38-107151800260651*z^ 40) The first , 40, terms are: [0, 473, 0, 359145, 0, 275448641, 0, 211325549505, 0, 162132986203017, 0, 124391684948261081, 0, 95435810748451806497, 0, 73220280435897395915233, 0, 56176077188471266062673113, 0, 43099420402328673436075354057, 0, 33066745347744078978811199076161, 0, 25369474523930158261908024394462657, 0, 19463972968971802242362269876925406889, 0, 14933152966156765966847306209026595842265, 0, 11457016399792953111536376885257430911985729, 0, 8790054255963797260683127813848371179135026113, 0, 6743907063289493987425785191684972907024136385241, 0, 5174061633058621051742866778781252456295512431881513, 0, 3969644529714370462578916310303710116360362641956012737, 0, 3045591415380167286555645134553496742788286688796171183681] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {3, 6}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 22 4 6 8 10 f(z) = - (-1 + 415 z + z - 34103 z + 551961 z - 2835494 z + 6088938 z 12 14 18 16 20 / - 6088938 z + 2835494 z + 34103 z - 551961 z - 415 z ) / ( / 22 20 10 18 8 -892 z + 94942 z - 70185656 z - 2341068 z + 20280383 z 16 6 14 4 12 + 20280383 z - 2341068 z - 70185656 z + 94942 z + 105862084 z 2 24 - 892 z + z + 1) And in Maple-input format, it is: -(-1+415*z^2+z^22-34103*z^4+551961*z^6-2835494*z^8+6088938*z^10-6088938*z^12+ 2835494*z^14+34103*z^18-551961*z^16-415*z^20)/(-892*z^22+94942*z^20-70185656*z^ 10-2341068*z^18+20280383*z^8+20280383*z^16-2341068*z^6-70185656*z^14+94942*z^4+ 105862084*z^12-892*z^2+z^24+1) The first , 40, terms are: [0, 477, 0, 364645, 0, 281765113, 0, 217813599753, 0, 168382436716117, 0, 130169744291113453, 0, 100629074287330166001, 0, 77792355530244814694161, 0, 60138192182199431863043533, 0, 46490457016177192318911241845, 0, 35939932932515534189780223602665, 0, 27783740193254297970460865096897305, 0, 21478510284823343034568642066956859461, 0, 16604186507881065758993682340667849047549, 0, 12836039647652415482100584966182987685459425, 0, 9923034396048301647402680342141256086562224673, 0, 7671105288551073034995101036132603001796445255101, 0, 5930227992706617506650417475990916187810446633474181, 0, 4584424632779829102527658548794933675021039578386704089, 0, 3544037301683255775940401288348090268305292725107309087785] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {3, 7}, {4, 5}, {4, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 8639977658798 z + 17924675477219 z + 624 z 24 22 4 6 - 25708628145577 z + 25708628145577 z - 115289 z + 9220677 z 8 10 12 14 - 358635683 z + 7315216965 z - 85673341354 z + 614928984035 z 18 16 20 + 8639977658798 z - 2830900125197 z - 17924675477219 z 36 34 30 42 - 7315216965 z + 85673341354 z + 2830900125197 z + 115289 z 44 46 32 38 40 / - 624 z + z - 614928984035 z + 358635683 z - 9220677 z ) / (1 / 28 26 2 + 221561877559166 z - 387750691356996 z - 1188 z 24 22 4 6 + 466782471290365 z - 387750691356996 z + 285054 z - 27283016 z 8 10 12 14 + 1249389545 z - 30578233572 z + 438736570434 z - 3918794601156 z 18 16 48 20 - 86305412526136 z + 22612493623886 z + z + 221561877559166 z 36 34 30 42 + 438736570434 z - 3918794601156 z - 86305412526136 z - 27283016 z 44 46 32 38 + 285054 z - 1188 z + 22612493623886 z - 30578233572 z 40 + 1249389545 z ) And in Maple-input format, it is: -(-1-8639977658798*z^28+17924675477219*z^26+624*z^2-25708628145577*z^24+ 25708628145577*z^22-115289*z^4+9220677*z^6-358635683*z^8+7315216965*z^10-\ 85673341354*z^12+614928984035*z^14+8639977658798*z^18-2830900125197*z^16-\ 17924675477219*z^20-7315216965*z^36+85673341354*z^34+2830900125197*z^30+115289* z^42-624*z^44+z^46-614928984035*z^32+358635683*z^38-9220677*z^40)/(1+ 221561877559166*z^28-387750691356996*z^26-1188*z^2+466782471290365*z^24-\ 387750691356996*z^22+285054*z^4-27283016*z^6+1249389545*z^8-30578233572*z^10+ 438736570434*z^12-3918794601156*z^14-86305412526136*z^18+22612493623886*z^16+z^ 48+221561877559166*z^20+438736570434*z^36-3918794601156*z^34-86305412526136*z^ 30-27283016*z^42+285054*z^44-1188*z^46+22612493623886*z^32-30578233572*z^38+ 1249389545*z^40) The first , 40, terms are: [0, 564, 0, 500267, 0, 451609079, 0, 408405343596, 0, 369419973665281, 0, 334166474330038921, 0, 302278477330602970524, 0, 273433566693791457507059, 0, 247341200092562887962555299, 0, 223738697010162209107307588052, 0, 202388460135415086777313312652941, 0, 183075566952759159145252502547915901, 0, 165605604161399019701028457671495199284, 0, 149802710357549535385506030242887316470499, 0, 135507805693634663880226824040783316174009603, 0, 122576990496886504343279530523396272468193175900, 0, 110880096702648802068415699061977947239881058343721, 0, 100299377517357537074675610456879473201378889177189025, 0, 90728322120313294415278885475668571907931517456209596972, 0, 82070583472392813872610205510224829736617172106254880381767] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {3, 7}, {4, 5}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 22 4 6 8 10 f(z) = - (-1 + 454 z + z - 31773 z + 537755 z - 3066810 z + 6743591 z 12 14 18 16 20 / 24 - 6743591 z + 3066810 z + 31773 z - 537755 z - 454 z ) / (z / 22 20 18 16 14 - 1022 z + 98614 z - 2449808 z + 22327130 z - 84605810 z 12 10 8 6 4 + 129448414 z - 84605810 z + 22327130 z - 2449808 z + 98614 z 2 - 1022 z + 1) And in Maple-input format, it is: -(-1+454*z^2+z^22-31773*z^4+537755*z^6-3066810*z^8+6743591*z^10-6743591*z^12+ 3066810*z^14+31773*z^18-537755*z^16-454*z^20)/(z^24-1022*z^22+98614*z^20-\ 2449808*z^18+22327130*z^16-84605810*z^14+129448414*z^12-84605810*z^10+22327130* z^8-2449808*z^6+98614*z^4-1022*z^2+1) The first , 40, terms are: [0, 568, 0, 513655, 0, 470854711, 0, 431932171096, 0, 396247564570177, 0, 363512534999540545, 0, 333481934875623917848, 0, 305932238734972671966007, 0, 280658485789836356992292983, 0, 257472654675863432020453491832, 0, 236202257416467114823114311879169, 0, 216689055693800810117724484484914177, 0, 198787883617412176837691141899411562488, 0, 182365567778975585918975435675477846154871, 0, 167299936526086634292776854474801485249845047, 0, 153478911082355218637935959020453149927625866648, 0, 140799671752131896804376568056669887079414130741057, 0, 129167893006945030631063102101176850148243310822927937, 0, 118497041763174248895692707008908041881880680454361738968, 0, 108707733630589480666313596341388153779326142398795168375863] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 26 2 24 22 4 6 f(z) = - (-1 + z + 577 z - 577 z + 82608 z - 82608 z + 2963560 z 8 10 12 14 - 35558040 z + 162230416 z - 332357438 z + 332357438 z 18 16 20 / 28 26 + 35558040 z - 162230416 z - 2963560 z ) / (z - 1145 z / 24 22 20 18 + 237709 z - 10883920 z + 184880476 z - 1378392952 z 16 14 12 10 + 4559395242 z - 6785881942 z + 4559395242 z - 1378392952 z 8 6 4 2 + 184880476 z - 10883920 z + 237709 z - 1145 z + 1) And in Maple-input format, it is: -(-1+z^26+577*z^2-577*z^24+82608*z^22-82608*z^4+2963560*z^6-35558040*z^8+ 162230416*z^10-332357438*z^12+332357438*z^14+35558040*z^18-162230416*z^16-\ 2963560*z^20)/(z^28-1145*z^26+237709*z^24-10883920*z^22+184880476*z^20-\ 1378392952*z^18+4559395242*z^16-6785881942*z^14+4559395242*z^12-1378392952*z^10 +184880476*z^8-10883920*z^6+237709*z^4-1145*z^2+1) The first , 40, terms are: [0, 568, 0, 495259, 0, 439973203, 0, 392074539928, 0, 349626321493081, 0, 311820339421924201, 0, 278111610931951733464, 0, 248048715640624699118371, 0, 221235873463522213027439947, 0, 197321438139009295319780251384, 0, 175992041613682452285545847015025, 0, 156968242020111860605598714789038225, 0, 140000814089171971164552007483356681592, 0, 124867474530797371969042350836023071077611, 0, 111369968097161579467294483500519369933655171, 0, 99331469953212047259515333374020633819743659096, 0, 88594269099104857543846957766645588881425909666889, 0, 79017702254059387329841127113069221960534372495012217, 0, 70476311086550284999093125661631052179558428419340888600, 0, 62858198640084507115850311121943309058689072349375925302771] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 675869128197905860 z - 303491164763561446 z - 667 z 24 22 4 6 + 107237885573154114 z - 29567622513118192 z + 143554 z - 14780577 z 8 10 12 14 + 852245035 z - 30132796476 z + 695562108767 z - 11002471747725 z 18 16 50 - 1020263384890703 z + 123817729334266 z - 1020263384890703 z 48 20 36 + 6293580352010605 z + 6293580352010605 z + 1672407415399148742 z 34 66 64 30 - 1871694284575474856 z - 667 z + 143554 z - 1191995298107942402 z 42 44 46 - 303491164763561446 z + 107237885573154114 z - 29567622513118192 z 58 56 54 - 30132796476 z + 695562108767 z - 11002471747725 z 52 60 68 32 + 123817729334266 z + 852245035 z + z + 1672407415399148742 z 38 40 62 / - 1191995298107942402 z + 675869128197905860 z - 14780577 z ) / ( / 2 28 26 2 (-1 + z ) (1 + 5559869636615031022 z - 2342926527945212772 z - 1246 z 24 22 4 + 765518956966935874 z - 192777166394673208 z + 336047 z 6 8 10 12 - 39913688 z + 2585350061 z - 102036550990 z + 2631373601081 z 14 18 16 - 46582936641408 z - 5405814816698446 z + 586937373272739 z 50 48 20 - 5405814816698446 z + 37117633883459869 z + 37117633883459869 z 36 34 66 + 14835415821134701882 z - 16764015779996940660 z - 1246 z 64 30 42 + 336047 z - 10275347552937703320 z - 2342926527945212772 z 44 46 58 + 765518956966935874 z - 192777166394673208 z - 102036550990 z 56 54 52 + 2631373601081 z - 46582936641408 z + 586937373272739 z 60 68 32 + 2585350061 z + z + 14835415821134701882 z 38 40 62 - 10275347552937703320 z + 5559869636615031022 z - 39913688 z )) And in Maple-input format, it is: -(1+675869128197905860*z^28-303491164763561446*z^26-667*z^2+107237885573154114* z^24-29567622513118192*z^22+143554*z^4-14780577*z^6+852245035*z^8-30132796476*z ^10+695562108767*z^12-11002471747725*z^14-1020263384890703*z^18+123817729334266 *z^16-1020263384890703*z^50+6293580352010605*z^48+6293580352010605*z^20+ 1672407415399148742*z^36-1871694284575474856*z^34-667*z^66+143554*z^64-\ 1191995298107942402*z^30-303491164763561446*z^42+107237885573154114*z^44-\ 29567622513118192*z^46-30132796476*z^58+695562108767*z^56-11002471747725*z^54+ 123817729334266*z^52+852245035*z^60+z^68+1672407415399148742*z^32-\ 1191995298107942402*z^38+675869128197905860*z^40-14780577*z^62)/(-1+z^2)/(1+ 5559869636615031022*z^28-2342926527945212772*z^26-1246*z^2+765518956966935874*z ^24-192777166394673208*z^22+336047*z^4-39913688*z^6+2585350061*z^8-102036550990 *z^10+2631373601081*z^12-46582936641408*z^14-5405814816698446*z^18+ 586937373272739*z^16-5405814816698446*z^50+37117633883459869*z^48+ 37117633883459869*z^20+14835415821134701882*z^36-16764015779996940660*z^34-1246 *z^66+336047*z^64-10275347552937703320*z^30-2342926527945212772*z^42+ 765518956966935874*z^44-192777166394673208*z^46-102036550990*z^58+2631373601081 *z^56-46582936641408*z^54+586937373272739*z^52+2585350061*z^60+z^68+ 14835415821134701882*z^32-10275347552937703320*z^38+5559869636615031022*z^40-\ 39913688*z^62) The first , 40, terms are: [0, 580, 0, 529521, 0, 490151905, 0, 454187526468, 0, 420911955069521, 0, 390079929461648785, 0, 361507017857971502532, 0, 335027115483003362864705, 0, 310486839199222770059054609, 0, 287744105508801469695802436804, 0, 266667245886390117633389477740289, 0, 247134237231576828459354358630890753, 0, 229031994572400424771080604946032592836, 0, 212255716267707293152219844813347995730513, 0, 196708277253731649165671199750249276867593985, 0, 182299666744091852662982837780277285719749283012, 0, 168946467118615188457509580497067434930109911638801, 0, 156571370982971788159805256272846024827713066655601361, 0, 145102733603041209141483591871180220617882477414544323140, 0, 134474158122847369181382706433004527662754960307497259629601] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 688712268798876 z - 600843977597622 z - 563 z 24 22 4 6 + 398421327256826 z - 199985540339352 z + 84970 z - 5809399 z 8 10 12 14 + 215014021 z - 4755414072 z + 67525630616 z - 648361017896 z 18 16 50 48 - 21189940935064 z + 4367540174680 z - 5809399 z + 215014021 z 20 36 34 + 75459840065608 z + 75459840065608 z - 199985540339352 z 30 42 44 - 600843977597622 z - 648361017896 z + 67525630616 z 46 56 54 52 32 - 4755414072 z + z - 563 z + 84970 z + 398421327256826 z 38 40 / 2 - 21189940935064 z + 4367540174680 z ) / ((-1 + z ) (1 / 28 26 2 + 6171432888150186 z - 5305806558972562 z - 1162 z 24 22 4 6 + 3369978653738530 z - 1578671409285594 z + 214579 z - 16902428 z 8 10 12 14 + 714553389 z - 18111107102 z + 295259901036 z - 3249674495402 z 18 16 50 48 - 136884039095374 z + 24963697198504 z - 16902428 z + 714553389 z 20 36 34 + 543679322286116 z + 543679322286116 z - 1578671409285594 z 30 42 44 - 5305806558972562 z - 3249674495402 z + 295259901036 z 46 56 54 52 32 - 18111107102 z + z - 1162 z + 214579 z + 3369978653738530 z 38 40 - 136884039095374 z + 24963697198504 z )) And in Maple-input format, it is: -(1+688712268798876*z^28-600843977597622*z^26-563*z^2+398421327256826*z^24-\ 199985540339352*z^22+84970*z^4-5809399*z^6+215014021*z^8-4755414072*z^10+ 67525630616*z^12-648361017896*z^14-21189940935064*z^18+4367540174680*z^16-\ 5809399*z^50+215014021*z^48+75459840065608*z^20+75459840065608*z^36-\ 199985540339352*z^34-600843977597622*z^30-648361017896*z^42+67525630616*z^44-\ 4755414072*z^46+z^56-563*z^54+84970*z^52+398421327256826*z^32-21189940935064*z^ 38+4367540174680*z^40)/(-1+z^2)/(1+6171432888150186*z^28-5305806558972562*z^26-\ 1162*z^2+3369978653738530*z^24-1578671409285594*z^22+214579*z^4-16902428*z^6+ 714553389*z^8-18111107102*z^10+295259901036*z^12-3249674495402*z^14-\ 136884039095374*z^18+24963697198504*z^16-16902428*z^50+714553389*z^48+ 543679322286116*z^20+543679322286116*z^36-1578671409285594*z^34-\ 5305806558972562*z^30-3249674495402*z^42+295259901036*z^44-18111107102*z^46+z^ 56-1162*z^54+214579*z^52+3369978653738530*z^32-136884039095374*z^38+ 24963697198504*z^40) The first , 40, terms are: [0, 600, 0, 567029, 0, 541317735, 0, 516974884720, 0, 493738397581147, 0, 471547110132174971, 0, 450353275735041721872, 0, 430112008990284189938871, 0, 410780492510993111293397525, 0, 392317837004783995409548599352, 0, 374684991227141450519022424042769, 0, 357844659123201581187753691502760625, 0, 341761220922178338101546958482203128696, 0, 326400657794937598318047664043977846420757, 0, 311730479899085327230054821196278293952574167, 0, 297719657658181407185980492084191129061380070736, 0, 284338556129636946126335230476411898851642405777531, 0, 271558872322467103456685402647442430369005849375154523, 0, 259353575332316818739681893374054546441775378971595916592, 0, 247696849167136213357503060479000303242375705770555238510919] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 2797751 z + 48934784 z + 536 z - 417160255 z 22 4 6 8 10 + 1904968408 z - 67169 z + 2797751 z - 48934784 z + 417160255 z 12 14 18 16 - 1904968408 z + 5033406632 z + 8086024832 z - 8086024832 z 20 34 30 32 / 28 - 5033406632 z + z + 67169 z - 536 z ) / (215835352 z / 10 30 16 4 - 2479652784 z - 9531046 z + 117771734848 z + 185416 z 14 22 18 6 - 55724377104 z - 55724377104 z - 150713707084 z - 9531046 z 24 12 20 + 15575187275 z + 1 + 15575187275 z + 117771734848 z 26 2 34 32 36 8 - 2479652784 z - 1152 z - 1152 z + 185416 z + z + 215835352 z ) And in Maple-input format, it is: -(-1-2797751*z^28+48934784*z^26+536*z^2-417160255*z^24+1904968408*z^22-67169*z^ 4+2797751*z^6-48934784*z^8+417160255*z^10-1904968408*z^12+5033406632*z^14+ 8086024832*z^18-8086024832*z^16-5033406632*z^20+z^34+67169*z^30-536*z^32)/( 215835352*z^28-2479652784*z^10-9531046*z^30+117771734848*z^16+185416*z^4-\ 55724377104*z^14-55724377104*z^22-150713707084*z^18-9531046*z^6+15575187275*z^ 24+1+15575187275*z^12+117771734848*z^20-2479652784*z^26-1152*z^2-1152*z^34+ 185416*z^32+z^36+215835352*z^8) The first , 40, terms are: [0, 616, 0, 591385, 0, 573792559, 0, 557061010576, 0, 540849588618415, 0, 525113417031994735, 0, 509835475381082770720, 0, 495002080785103826008447, 0, 480600260646632704239768937, 0, 466617454257917265131870625016, 0, 453041470165919436424824705940417, 0, 439860472042230350435012022105771073, 0, 427062967976575094322315897935992833496, 0, 414637800414774926338845507735947057884873, 0, 402574136426262587284829957658239719413499615, 0, 390861458258827194218362064187123708296858581504, 0, 379489554168613444241522756409866660355015897822703, 0, 368448509516966771602902014000630143067054779606012975, 0, 357728698126316505282545884831150689750438667051809954736, 0, 347320773887555568372753927186673594829740033513591522466767] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 4447530938550 z + 9001195986561 z + 600 z 24 22 4 6 - 12749931646665 z + 12749931646665 z - 101475 z + 7064493 z 8 10 12 14 - 243321927 z + 4579025027 z - 50482557942 z + 343710169443 z 18 16 20 + 4447530938550 z - 1511153596989 z - 9001195986561 z 36 34 30 42 - 4579025027 z + 50482557942 z + 1511153596989 z + 101475 z 44 46 32 38 40 / - 600 z + z - 343710169443 z + 243321927 z - 7064493 z ) / (1 / 28 26 2 + 114462453660822 z - 196936587161004 z - 1216 z 24 22 4 6 + 235726429190301 z - 196936587161004 z + 260574 z - 21240972 z 8 10 12 14 + 858148197 z - 19414947620 z + 263740009994 z - 2250749712648 z 18 16 48 20 - 45839761203744 z + 12456276884826 z + z + 114462453660822 z 36 34 30 42 + 263740009994 z - 2250749712648 z - 45839761203744 z - 21240972 z 44 46 32 38 + 260574 z - 1216 z + 12456276884826 z - 19414947620 z 40 + 858148197 z ) And in Maple-input format, it is: -(-1-4447530938550*z^28+9001195986561*z^26+600*z^2-12749931646665*z^24+ 12749931646665*z^22-101475*z^4+7064493*z^6-243321927*z^8+4579025027*z^10-\ 50482557942*z^12+343710169443*z^14+4447530938550*z^18-1511153596989*z^16-\ 9001195986561*z^20-4579025027*z^36+50482557942*z^34+1511153596989*z^30+101475*z ^42-600*z^44+z^46-343710169443*z^32+243321927*z^38-7064493*z^40)/(1+ 114462453660822*z^28-196936587161004*z^26-1216*z^2+235726429190301*z^24-\ 196936587161004*z^22+260574*z^4-21240972*z^6+858148197*z^8-19414947620*z^10+ 263740009994*z^12-2250749712648*z^14-45839761203744*z^18+12456276884826*z^16+z^ 48+114462453660822*z^20+263740009994*z^36-2250749712648*z^34-45839761203744*z^ 30-21240972*z^42+260574*z^44-1216*z^46+12456276884826*z^32-19414947620*z^38+ 858148197*z^40) The first , 40, terms are: [0, 616, 0, 589957, 0, 571050607, 0, 553139695276, 0, 535834405338643, 0, 519075959703235807, 0, 502842324820522862164, 0, 487116468324164268760243, 0, 471882432021029865883506361, 0, 457124824953289128558225477136, 0, 442828746081234023462567517183709, 0, 428979761459695312519035730627357093, 0, 415563888685459354874053113632865411904, 0, 402567582656818151092457443581724334334289, 0, 389977721882461044791655425596336647694037227, 0, 377781595231636206193300440586481108785616275972, 0, 365966889100335451766203963134524027366234727743047, 0, 354521674979050273957162697105513199662966629106696683, 0, 343434397409358854937372411871655084661425506382189315068, 0, 332693862317160940663056946729749521961212813492278139480311] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 7}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (1 + 2453074337 z - 10044414492 z - 484 z + 26782958731 z 22 4 6 8 10 - 47738995824 z + 45523 z - 1766064 z + 34460944 z - 376089312 z 12 14 18 16 + 2453074337 z - 10044414492 z - 47738995824 z + 26782958731 z 20 36 34 30 32 + 57781461472 z + 45523 z - 1766064 z - 376089312 z + 34460944 z 38 40 / 2 40 38 36 34 - 484 z + z ) / ((-1 + z ) (z - 1119 z + 127405 z - 5852434 z / 32 30 28 26 + 137548226 z - 1828107666 z + 14531901355 z - 71327375073 z 24 22 20 + 220068318631 z - 431033677148 z + 538961688828 z 18 16 14 12 - 431033677148 z + 220068318631 z - 71327375073 z + 14531901355 z 10 8 6 4 2 - 1828107666 z + 137548226 z - 5852434 z + 127405 z - 1119 z + 1)) And in Maple-input format, it is: -(1+2453074337*z^28-10044414492*z^26-484*z^2+26782958731*z^24-47738995824*z^22+ 45523*z^4-1766064*z^6+34460944*z^8-376089312*z^10+2453074337*z^12-10044414492*z ^14-47738995824*z^18+26782958731*z^16+57781461472*z^20+45523*z^36-1766064*z^34-\ 376089312*z^30+34460944*z^32-484*z^38+z^40)/(-1+z^2)/(z^40-1119*z^38+127405*z^ 36-5852434*z^34+137548226*z^32-1828107666*z^30+14531901355*z^28-71327375073*z^ 26+220068318631*z^24-431033677148*z^22+538961688828*z^20-431033677148*z^18+ 220068318631*z^16-71327375073*z^14+14531901355*z^12-1828107666*z^10+137548226*z ^8-5852434*z^6+127405*z^4-1119*z^2+1) The first , 40, terms are: [0, 636, 0, 629319, 0, 627309791, 0, 625398608652, 0, 623495691158217, 0, 621598646362379129, 0, 619707376763404469868, 0, 617821861660874832178415, 0, 615942083412799989689216695, 0, 614068024558377219163977079836, 0, 612199667695573778132517350426065, 0, 610336995475558321313241907651075889, 0, 608479990602296405425259289426066645724, 0, 606628635832378843225266903591151934629591, 0, 604782913974861085429654659296659099343431119, 0, 602942807891103569519844485936112102302289684908, 0, 601108300494612576101684066852902559094979623612185, 0, 599279374750881570468293973747383378824080991654125865, 0, 597456013677233026940796287540795675021474700237988979660, 0, 595638200342660734474700347744868894482460067228782030517759] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (1 + 77517873331666 z - 89589562226242 z - 568 z 24 22 4 6 + 77517873331666 z - 50109529625194 z + 86237 z - 5683485 z 8 10 12 14 + 195273573 z - 3870901322 z + 47748134029 z - 387521470046 z 18 16 50 48 - 8470865327878 z + 2155688991438 z - 568 z + 86237 z 20 36 34 + 24045067862302 z + 2155688991438 z - 8470865327878 z 30 42 44 46 52 - 50109529625194 z - 3870901322 z + 195273573 z - 5683485 z + z 32 38 40 / 2 + 24045067862302 z - 387521470046 z + 47748134029 z ) / ((-1 + z ) / 28 26 2 (1 + 702608151246827 z - 826405197554776 z - 1196 z 24 22 4 6 + 702608151246827 z - 431245212759064 z + 221670 z - 16753006 z 8 10 12 14 + 658536773 z - 15114537096 z + 217827431138 z - 2067767206320 z 18 16 50 48 - 59980643094220 z + 13353102096799 z - 1196 z + 221670 z 20 36 34 + 190330324089795 z + 13353102096799 z - 59980643094220 z 30 42 44 46 - 431245212759064 z - 15114537096 z + 658536773 z - 16753006 z 52 32 38 40 + z + 190330324089795 z - 2067767206320 z + 217827431138 z )) And in Maple-input format, it is: -(1+77517873331666*z^28-89589562226242*z^26-568*z^2+77517873331666*z^24-\ 50109529625194*z^22+86237*z^4-5683485*z^6+195273573*z^8-3870901322*z^10+ 47748134029*z^12-387521470046*z^14-8470865327878*z^18+2155688991438*z^16-568*z^ 50+86237*z^48+24045067862302*z^20+2155688991438*z^36-8470865327878*z^34-\ 50109529625194*z^30-3870901322*z^42+195273573*z^44-5683485*z^46+z^52+ 24045067862302*z^32-387521470046*z^38+47748134029*z^40)/(-1+z^2)/(1+ 702608151246827*z^28-826405197554776*z^26-1196*z^2+702608151246827*z^24-\ 431245212759064*z^22+221670*z^4-16753006*z^6+658536773*z^8-15114537096*z^10+ 217827431138*z^12-2067767206320*z^14-59980643094220*z^18+13353102096799*z^16-\ 1196*z^50+221670*z^48+190330324089795*z^20+13353102096799*z^36-59980643094220*z ^34-431245212759064*z^30-15114537096*z^42+658536773*z^44-16753006*z^46+z^52+ 190330324089795*z^32-2067767206320*z^38+217827431138*z^40) The first , 40, terms are: [0, 629, 0, 616284, 0, 608800425, 0, 601582413779, 0, 594461599900953, 0, 587426024598675587, 0, 580473798448463421892, 0, 573603859462152383555039, 0, 566815227197912709211283499, 0, 560106938812780594214423468211, 0, 553478043382512167126903209659863, 0, 546927601282872513928342068655566692, 0, 540454684014134247061988489722703167787, 0, 534058374065764148838416731413122616479777, 0, 527737764786045668596638545947054936758491355, 0, 521491960253536258276262588574309301812349575057, 0, 515320075150071558451944245090345833674853542080444, 0, 509221234635274822203150200594547012356320537207260525, 0, 503194574222551722318793869464669047695909951363128758153, 0, 497239239656552783687248531946302273129425361002359277980409] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 2500927 z + 41519902 z + 522 z - 335507223 z 22 4 6 8 10 + 1493204520 z - 61017 z + 2500927 z - 41519902 z + 335507223 z 12 14 18 16 - 1493204520 z + 3922022232 z + 6306723880 z - 6306723880 z 20 34 30 32 / 36 34 - 3922022232 z + z + 61017 z - 522 z ) / (z - 1172 z / 32 30 28 26 + 167616 z - 8307616 z + 180758352 z - 2004980484 z 24 22 20 18 + 12303188183 z - 43946196840 z + 93805838192 z - 120773192592 z 16 14 12 10 + 93805838192 z - 43946196840 z + 12303188183 z - 2004980484 z 8 6 4 2 + 180758352 z - 8307616 z + 167616 z - 1172 z + 1) And in Maple-input format, it is: -(-1-2500927*z^28+41519902*z^26+522*z^2-335507223*z^24+1493204520*z^22-61017*z^ 4+2500927*z^6-41519902*z^8+335507223*z^10-1493204520*z^12+3922022232*z^14+ 6306723880*z^18-6306723880*z^16-3922022232*z^20+z^34+61017*z^30-522*z^32)/(z^36 -1172*z^34+167616*z^32-8307616*z^30+180758352*z^28-2004980484*z^26+12303188183* z^24-43946196840*z^22+93805838192*z^20-120773192592*z^18+93805838192*z^16-\ 43946196840*z^14+12303188183*z^12-2004980484*z^10+180758352*z^8-8307616*z^6+ 167616*z^4-1172*z^2+1) The first , 40, terms are: [0, 650, 0, 655201, 0, 664751861, 0, 674527722226, 0, 684450777370773, 0, 694520034960728301, 0, 704737443029089014690, 0, 715105165541889212314509, 0, 725625412589632490538907401, 0, 736300427928898360130576882778, 0, 747132488419958630794908155530729, 0, 758123904426892225565292626752620953, 0, 769277020303278771334327298565701780410, 0, 780594214891635870644984910960526851220281, 0, 792077902030752220264686791931419584518725597, 0, 803730531070529998649183184320204187467929680194, 0, 815554587394405144554049964446385782679390401640893, 0, 827552592949453522085152703972943466676719180493883077, 0, 839727106784295752279228648396118892311971684580541885266, 0, 852080725594915416828687496789325093234415701498278878992805] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {3, 7}, {4, 5}, {4, 6}, {4, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 18 16 14 12 10 8 f(z) = - (z - 574 z + 54664 z - 680169 z + 1916294 z - 1916294 z 6 4 2 / 20 18 16 + 680169 z - 54664 z + 574 z - 1) / (z - 1358 z + 182867 z / 14 12 10 8 6 - 3790720 z + 25238717 z - 45315766 z + 25238717 z - 3790720 z 4 2 + 182867 z - 1358 z + 1) And in Maple-input format, it is: -(z^18-574*z^16+54664*z^14-680169*z^12+1916294*z^10-1916294*z^8+680169*z^6-\ 54664*z^4+574*z^2-1)/(z^20-1358*z^18+182867*z^16-3790720*z^14+25238717*z^12-\ 45315766*z^10+25238717*z^8-3790720*z^6+182867*z^4-1358*z^2+1) The first , 40, terms are: [0, 784, 0, 936469, 0, 1131467725, 0, 1368232495984, 0, 1654681769091721, 0, 2001118748144894713, 0, 2420090640822768487984, 0, 2926782462212976528340957, 0, 3539559849259907948037327109, 0, 4280633802050310185092882796368, 0, 5176865635702543615138078782776977, 0, 6260740593499784099220462102471413617, 0, 7571545320553774674947909365827428213968, 0, 9156791865921967565334328007714503309297957, 0, 11073939826815066670458838716906143271088291261, 0, 13392479056372395420130721045948734779241467369136, 0, 16196448425795526274645401175423076886989525157783577, 0, 19587481937082828918546398519511154768435022649004910953, 0, 23688492597209704269401430661114900193881909338536079777520, 0, 28648127581203041041388103358618462508161847497587505194732013] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {3, 7}, {4, 5}, {4, 6}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 22 f(z) = - (-1 - 648 z + 96287 z + 648 z - 4198419 z + 60847799 z 4 6 8 10 12 - 96287 z + 4198419 z - 60847799 z + 384035001 z - 1212872368 z 14 18 16 20 30 + 2107886613 z + 1212872368 z - 2107886613 z - 384035001 z + z ) / 32 30 28 26 24 / (z - 1432 z + 284846 z - 15457640 z + 312684233 z / 22 20 18 16 - 2941722092 z + 13927889566 z - 34452264628 z + 46341384996 z 14 12 10 8 - 34452264628 z + 13927889566 z - 2941722092 z + 312684233 z 6 4 2 - 15457640 z + 284846 z - 1432 z + 1) And in Maple-input format, it is: -(-1-648*z^28+96287*z^26+648*z^2-4198419*z^24+60847799*z^22-96287*z^4+4198419*z ^6-60847799*z^8+384035001*z^10-1212872368*z^12+2107886613*z^14+1212872368*z^18-\ 2107886613*z^16-384035001*z^20+z^30)/(z^32-1432*z^30+284846*z^28-15457640*z^26+ 312684233*z^24-2941722092*z^22+13927889566*z^20-34452264628*z^18+46341384996*z^ 16-34452264628*z^14+13927889566*z^12-2941722092*z^10+312684233*z^8-15457640*z^6 +284846*z^4-1432*z^2+1) The first , 40, terms are: [0, 784, 0, 934129, 0, 1125612685, 0, 1357661409112, 0, 1637741710020853, 0, 1975631228859323017, 0, 2383236761619627422920, 0, 2874938790329971888265857, 0, 3468087367667528429043828037, 0, 4183612561900312658987769614368, 0, 5046762731589182960030664261956197, 0, 6087995409176249136115975497464581405, 0, 7344052033686416349737781792857547539584, 0, 8859254425891378484671993789381685366413981, 0, 10687068749333149871942207666441458114834380121, 0, 12891992143174458887244642992359522673792476212712, 0, 15551828599403656485671687024652053705256952827794401, 0, 18760434392079557746773362425661862755708437361828345245, 0, 22631029935156137595812119154916238968678796990429038088504, 0, 27300194932701709363049176921503199028072839138898927974367413] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {3, 7}, {4, 5}, {4, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 24 f(z) = - (-1 - 3633850 z + 65920608 z + 613 z - 577970837 z 22 4 6 8 10 + 2723320870 z - 79638 z + 3633850 z - 65920608 z + 577970837 z 12 14 18 16 - 2723320870 z + 7335696481 z + 11882719944 z - 11882719944 z 20 34 30 32 / 36 34 - 7335696481 z + z + 79638 z - 613 z ) / (z - 1413 z / 32 30 28 26 + 232602 z - 12936561 z + 306447431 z - 3667900846 z 24 22 20 18 + 24107723419 z - 90278345107 z + 196173896322 z - 253273217371 z 16 14 12 10 + 196173896322 z - 90278345107 z + 24107723419 z - 3667900846 z 8 6 4 2 + 306447431 z - 12936561 z + 232602 z - 1413 z + 1) And in Maple-input format, it is: -(-1-3633850*z^28+65920608*z^26+613*z^2-577970837*z^24+2723320870*z^22-79638*z^ 4+3633850*z^6-65920608*z^8+577970837*z^10-2723320870*z^12+7335696481*z^14+ 11882719944*z^18-11882719944*z^16-7335696481*z^20+z^34+79638*z^30-613*z^32)/(z^ 36-1413*z^34+232602*z^32-12936561*z^30+306447431*z^28-3667900846*z^26+ 24107723419*z^24-90278345107*z^22+196173896322*z^20-253273217371*z^18+ 196173896322*z^16-90278345107*z^14+24107723419*z^12-3667900846*z^10+306447431*z ^8-12936561*z^6+232602*z^4-1413*z^2+1) The first , 40, terms are: [0, 800, 0, 977436, 0, 1204338179, 0, 1484485000432, 0, 1829848428921463, 0, 2255565024497912909, 0, 2780325605707319854176, 0, 3427172568144242926844993, 0, 4224509460425824462564280503, 0, 5207347989600667421784325418320, 0, 6418845392347050151293400280789567, 0, 7912199502157070049351895623639384228, 0, 9752984709146656582159705379991423531392, 0, 12022031384688473974780118324654472455841769, 0, 14818975208572882170985183848931520720798321041, 0, 18266632252514971509954648261984161922869368887776, 0, 22516391933471203021843380145630120644775193960480580, 0, 27754864645719484857462774038363126464790151811618317359, 0, 34212075974618741866657708734739359923998409516043474295504, 0, 42171567306619921442083909878390701459083743877548123517140319] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 28 26 2 f(z) = - (-1 - 85426563761518 z + 117120510347820 z + 691 z 24 22 4 6 - 117120510347820 z + 85426563761518 z - 127410 z + 9970246 z 8 10 12 14 - 390205802 z + 8326922222 z - 105820498918 z + 855686418106 z 18 16 50 48 + 17188904045174 z - 4616825876978 z + z - 691 z 20 36 34 - 45217089460826 z - 855686418106 z + 4616825876978 z 30 42 44 46 + 45217089460826 z + 390205802 z - 9970246 z + 127410 z 32 38 40 / - 17188904045174 z + 105820498918 z - 8326922222 z ) / (1 / 28 26 2 + 2065580287083982 z - 2432638692366874 z - 1527 z 24 22 4 6 + 2065580287083982 z - 1262421445026206 z + 344489 z - 31072362 z 8 10 12 14 + 1413136686 z - 36089514058 z + 560290634308 z - 5595368625412 z 18 16 50 48 - 171686692524862 z + 37380654839732 z - 1527 z + 344489 z 20 36 34 + 552535992391626 z + 37380654839732 z - 171686692524862 z 30 42 44 46 - 1262421445026206 z - 36089514058 z + 1413136686 z - 31072362 z 52 32 38 40 + z + 552535992391626 z - 5595368625412 z + 560290634308 z ) And in Maple-input format, it is: -(-1-85426563761518*z^28+117120510347820*z^26+691*z^2-117120510347820*z^24+ 85426563761518*z^22-127410*z^4+9970246*z^6-390205802*z^8+8326922222*z^10-\ 105820498918*z^12+855686418106*z^14+17188904045174*z^18-4616825876978*z^16+z^50 -691*z^48-45217089460826*z^20-855686418106*z^36+4616825876978*z^34+ 45217089460826*z^30+390205802*z^42-9970246*z^44+127410*z^46-17188904045174*z^32 +105820498918*z^38-8326922222*z^40)/(1+2065580287083982*z^28-2432638692366874*z ^26-1527*z^2+2065580287083982*z^24-1262421445026206*z^22+344489*z^4-31072362*z^ 6+1413136686*z^8-36089514058*z^10+560290634308*z^12-5595368625412*z^14-\ 171686692524862*z^18+37380654839732*z^16-1527*z^50+344489*z^48+552535992391626* z^20+37380654839732*z^36-171686692524862*z^34-1262421445026206*z^30-36089514058 *z^42+1413136686*z^44-31072362*z^46+z^52+552535992391626*z^32-5595368625412*z^ 38+560290634308*z^40) The first , 40, terms are: [0, 836, 0, 1059493, 0, 1350955123, 0, 1722878352492, 0, 2197213395242943, 0, 2802142088328724799, 0, 3573617712093490439084, 0, 4557493234238164503933379, 0, 5812245813960678318940932501, 0, 7412452343134604963581766611780, 0, 9453221955515165664628989781282529, 0, 12055848888259171505248078350596165345, 0, 15375021669912951173106843432990627110596, 0, 19608017116115942883307566409480400539783157, 0, 25006425583013347847457377089047483693494526179, 0, 31891104375099165124847517813849962201941177520556, 0, 40671248071309218053110308285759955939840206304066175, 0, 51868709224430244348772535638616593436271490929935105791, 0, 66149014947647059891958498762746987574202810894545308067692, 0, 84360922875695472336915418073907967143214277532814932337278163] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 7}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 18 16 14 12 10 8 f(z) = - (z - 501 z + 20620 z - 147796 z + 365098 z - 365098 z 6 4 2 / 20 18 16 + 147796 z - 20620 z + 501 z - 1) / (z - 1364 z + 74841 z / 14 12 10 8 6 - 1061296 z + 5081158 z - 8770744 z + 5081158 z - 1061296 z 4 2 + 74841 z - 1364 z + 1) And in Maple-input format, it is: -(z^18-501*z^16+20620*z^14-147796*z^12+365098*z^10-365098*z^8+147796*z^6-20620* z^4+501*z^2-1)/(z^20-1364*z^18+74841*z^16-1061296*z^14+5081158*z^12-8770744*z^ 10+5081158*z^8-1061296*z^6+74841*z^4-1364*z^2+1) The first , 40, terms are: [0, 863, 0, 1122911, 0, 1467976321, 0, 1919191102081, 0, 2509099211717471, 0, 3280329402785366111, 0, 4288615191959719942657, 0, 5606821147112515191249409, 0, 7330208463249615422105854559, 0, 9583319086675122872402282612063, 0, 12528975837109739841097534837732993, 0, 16380048927426599770234422065648135809, 0, 21414839197804994584798116443204770517087, 0, 27997189745872953171635143894148504025160031, 0, 36602779335684058512510241402320724916639901697, 0, 47853497699506556857324379065620363830937092007937, 0, 62562386890773604585850612708537944826530313289375071, 0, 81792396410580471906281032444337086354032433595326424159, 0, 106933198093376217286495577007928549353277500117866825250433, 0, 139801611840267245738334586045527648261877666410437251303538305] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {3, 7}, {4, 5}, {4, 6}, {4, 7}, {5, 6}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 26 2 24 22 4 6 f(z) = - (-1 + z + 739 z - 739 z + 113882 z - 113882 z + 5483338 z 8 10 12 14 - 73621814 z + 375442374 z - 812461276 z + 812461276 z 18 16 20 / 28 18 + 73621814 z - 375442374 z - 5483338 z ) / (z - 3622719496 z / 14 4 22 2 - 21262445354 z + 1 + 353401 z - 20891992 z - 1819 z 10 8 24 16 - 3622719496 z + 422140976 z + 353401 z + 13606539750 z 20 26 12 6 + 422140976 z - 1819 z + 13606539750 z - 20891992 z ) And in Maple-input format, it is: -(-1+z^26+739*z^2-739*z^24+113882*z^22-113882*z^4+5483338*z^6-73621814*z^8+ 375442374*z^10-812461276*z^12+812461276*z^14+73621814*z^18-375442374*z^16-\ 5483338*z^20)/(z^28-3622719496*z^18-21262445354*z^14+1+353401*z^4-20891992*z^22 -1819*z^2-3622719496*z^10+422140976*z^8+353401*z^24+13606539750*z^16+422140976* z^20-1819*z^26+13606539750*z^12-20891992*z^6) The first , 40, terms are: [0, 1080, 0, 1725001, 0, 2771512393, 0, 4453978796664, 0, 7157918222048257, 0, 11503390806065618305, 0, 18486941173604920797432, 0, 29710109160771969779586505, 0, 47746708261701664177813323721, 0, 76733078885576154805829396898744, 0, 123316676890243399511304491321963905, 0, 198180537261295856293676961468291764353, 0, 318493218756887576827442709098045663768760, 0, 511846076289416622508464853021049166984329929, 0, 822580797278610021809330877878188527672731994313, 0, 1321958298394607635071609257554333981936110447134712, 0, 2124501019809801377254257597946453580635406432529068161, 0, 3414256401774630276591035522426088732065179008814404064513, 0, 5487004557005421570432928145844728824018321449235383752239992, 0, 8818089641114654937320505820655144047655747378723758778032809289] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {3, 7}, {4, 5}, {4, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 2 22 4 6 8 10 f(z) = - (-1 + 628 z + z - 44408 z + 640929 z - 3189461 z + 6689880 z 12 14 18 16 20 / - 6689880 z + 3189461 z + 44408 z - 640929 z - 628 z ) / ( / 12 24 10 22 8 20 172798561 z + z - 113938640 z - 1744 z + 31255874 z + 155857 z 6 18 4 16 2 - 3554616 z - 3554616 z + 155857 z + 31255874 z - 1744 z 14 - 113938640 z + 1) And in Maple-input format, it is: -(-1+628*z^2+z^22-44408*z^4+640929*z^6-3189461*z^8+6689880*z^10-6689880*z^12+ 3189461*z^14+44408*z^18-640929*z^16-628*z^20)/(172798561*z^12+z^24-113938640*z^ 10-1744*z^22+31255874*z^8+155857*z^20-3554616*z^6-3554616*z^18+155857*z^4+ 31255874*z^16-1744*z^2-113938640*z^14+1) The first , 40, terms are: [0, 1116, 0, 1834855, 0, 3028964395, 0, 5000477794188, 0, 8255235399986413, 0, 13628480652302167765, 0, 22499114348192833829100, 0, 37143549556554276997959331, 0, 61319892521673677546994388591, 0, 101232361036063413761633456640636, 0, 167123432535596068914151122017842201, 0, 275902304526217226518518634096946442985, 0, 455484192060643922364180042546591748957500, 0, 751954027978854417022629919400262050868820063, 0, 1241392939753968090193142696065201707203775498163, 0, 2049402454845730087672542799345361034270967707817260, 0, 3383336804509387306845874216318329916619112129163504645, 0, 5585514892734657602695971509631859006146966925673207602621, 0, 9221067372121891459898807625858307651239124687536942793498956, 0, 15222962450930165743344757704141088216242693410533880625064217371] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {3, 7}, {4, 5}, {4, 6}, {4, 7}, {5, 6}, {5, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : f(z) = - ( 14 12 10 8 6 4 2 z - 771 z + 75128 z - 410268 z + 410268 z - 75128 z + 771 z - 1) / 16 14 12 10 8 6 / (z - 2227 z + 273809 z - 4388036 z + 15256956 z - 4388036 z / 4 2 + 273809 z - 2227 z + 1) And in Maple-input format, it is: -(z^14-771*z^12+75128*z^10-410268*z^8+410268*z^6-75128*z^4+771*z^2-1)/(z^16-\ 2227*z^14+273809*z^12-4388036*z^10+15256956*z^8-4388036*z^6+273809*z^4-2227*z^2 +1) The first , 40, terms are: [0, 1456, 0, 3043831, 0, 6383923501, 0, 13389943448176, 0, 28084762579393531, 0, 58906445691644474131, 0, 123553451966195720201776, 0, 259147455227818240153257301, 0, 543549390838032979386283816831, 0, 1140068846213160947346135757465456, 0, 2391239868932482130460767126219119801, 0, 5015511238435452620893592532772563515401, 0, 10519794902090855491241264140782732252726256, 0, 22064766585305893038659609415816691596162714031, 0, 46279792428963336658547390284935353906514015087301, 0, 97069650793146572775130361076210544954952196887283376, 0, 203598949143222953560451677281446277071721156869083479331, 0, 427039056528174589372021164831742707197760358602784086324331, 0, 895693993352537164432484824267565844416863361528453635992699376, 0, 1878675304901260207866239355844816391811062442995795203383556957501] -------------------------- Theorem: Let, A(n), be the number of perfect matchings in the Cartesian product of the graph, {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {2, 7}, {3, 4}, {3, 5}, {3, 6}, {3, 7}, {4, 5}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}} with the n-path, and let f(z) be its (ordinary) generating function, i.e. infinity ----- \ n f(z) = ) A(n) z / ----- n = 0 Then : 6 4 2 z - 783 z + 783 z - 1 f(z) = - -------------------------------------- 6 4 8 2 -2695 z + 39408 z + z - 2695 z + 1 And in Maple-input format, it is: -(z^6-783*z^4+783*z^2-1)/(-2695*z^6+39408*z^4+z^8-2695*z^2+1) The first , 40, terms are: [0, 1912, 0, 5114215, 0, 13707464023, 0, 36740079710104, 0, 98474344859319409, 0, 263940543276260456593, 0, 707439186161806982514520, 0, 1896147503164961601589311607, 0, 5082239468900972681893333775431, 0, 13621913894431739336433644549152696, 0, 36510782162619901570625974038756597217, 0, 97859759242142341289685035710988955530785, 0, 262293270965161947948674553572186081805078328, 0, 703024006255441410682874315087406438509609265799, 0, 1884313507368234146537032068152988257812818843446839, 0, 5050520839199143981157318291999971033882539576873187288, 0, 13536898529592760223115878065596479461056336192005852959825, 0, 36282915690245528268672613263422118087031850316918628642048177, 0, 97249009299109284517385803206939842089803311832627484391275347736, 0, 260656279401514812727992491452528165066666321807434096033158081130391] The whole thing took, 10474.669, seconds. Here is the list of all triples [Graph, GeneratingFunction,Sequence] [[{{1, 2}}, -1/(-1+z+z^2), [1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141]], [{{1, 2}, {1, 3 }}, -(-1+z^2)/(-4*z^2+1+z^4), [0, 3, 0, 11, 0, 41, 0, 153, 0, 571, 0, 2131, 0, 7953, 0, 29681, 0, 110771, 0, 413403, 0, 1542841, 0, 5757961, 0, 21489003, 0, 80198051, 0, 299303201, 0, 1117014753, 0, 4168755811, 0, 15558008491, 0, 58063278153, 0, 216695104121]], [{{1, 2}, {1, 3}, {2, 3}}, -(-1+z^2)/(-5*z^2+1+ z^4), [0, 4, 0, 19, 0, 91, 0, 436, 0, 2089, 0, 10009, 0, 47956, 0, 229771, 0, 1100899, 0, 5274724, 0, 25272721, 0, 121088881, 0, 580171684, 0, 2779769539, 0, 13318676011, 0, 63813610516, 0, 305749376569, 0, 1464933272329, 0, 7018916985076, 0, 33629651653051]], [{{1, 2}, {1, 3}, {1, 4}}, -(-1+z^2)/(-5*z^ 2+1+z^4), [0, 4, 0, 19, 0, 91, 0, 436, 0, 2089, 0, 10009, 0, 47956, 0, 229771, 0, 1100899, 0, 5274724, 0, 25272721, 0, 121088881, 0, 580171684, 0, 2779769539, 0, 13318676011, 0, 63813610516, 0, 305749376569, 0, 1464933272329, 0, 7018916985076, 0, 33629651653051]], [{{1, 2}, {1, 3}, {2, 4}}, -(-1+z^2)/(z^4-z ^3-5*z^2-z+1), [1, 5, 11, 36, 95, 281, 781, 2245, 6336, 18061, 51205, 145601, 413351, 1174500, 3335651, 9475901, 26915305, 76455961, 217172736, 616891945, 1752296281, 4977472781, 14138673395, 40161441636, 114079985111, 324048393905, 920471087701, 2614631600701, 7426955448000, 21096536145301, 59925473898301, 170220478472105, 483517428660911, 1373448758774436, 3901330906652795, 11081871650713781, 31478457514091281, 89415697915538545, 253988526230055936, 721463601671126161]], [{{1, 2}, {1, 3}, {1, 4}, {2, 3}}, -(-1+z^2)/(z^4-z^3-6*z ^2-z+1), [1, 6, 13, 49, 132, 433, 1261, 3942, 11809, 36289, 109824, 335425, 1018849, 3104934, 9443629, 28756657, 87504516, 266383153, 810723277, 2467770054 , 7510988353, 22861948801, 69584925696, 211799836801, 644660351425, 1962182349126, 5972359368781, 18178313978161, 55329992188548, 168410053077169, 512595960817837, 1560207957491238, 4748863783286881, 14454297435974977, 43995092132369664, 133909532574015169, 407585519020921249, 1240583509161406950, 3776011063728579949, 11493188105143927729]], [{{1, 2}, {1, 3}, {2, 4}, {3, 4}}, -(-1+z)/(1-3*z-3*z^2+z^3), [2, 9, 32, 121, 450, 1681, 6272, 23409, 87362, 326041, 1216800, 4541161, 16947842, 63250209, 236052992, 880961761, 3287794050, 12270214441, 45793063712, 170902040409, 637815097922, 2380358351281, 8883618307200, 33154114877521, 123732841202882, 461777249934009, 1723376158533152, 6431727384198601, 24003533378261250, 89582406128846401, 334326091137124352, 1247721958419651009, 4656561742541479682, 17378525011746267721, 64857538304443591200, 242051628206028097081, 903348974519668797122, 3371344269872647091409, 12582028104970919568512, 46956768150011031182641]], [{{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}}, -(-1+z^2) /(-2*z^3-7*z^2-2*z+z^4+1), [2, 10, 36, 145, 560, 2197, 8568, 33490, 130790, 510949, 1995840, 7796413, 30454814, 118965250, 464711184, 1815292333, 7091038640, 27699580729, 108202305420, 422668460890, 1651061182538, 6449506621417, 25193576136960, 98413152528025, 384429290075066, 1501688293498810, 5866014346442172, 22914292174998121, 89509632072014000, 349649649768400381, 1365829294044452832, 5335311108436738210, 20841216942649433006, 81411620582676538765, 318016552686728132160, 1242261572229054163477, 4852621037487908896598, 18955694565369976663090, 74046242984926695797160, 289245328430189991865669]], [{{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}, -(-1+z)/(1-4*z-4*z^2+z^3), [3, 16, 75, 361, 1728, 8281 , 39675, 190096, 910803, 4363921, 20908800, 100180081, 479991603, 2299777936, 11018898075, 52794712441, 252954664128, 1211978608201, 5806938376875, 27822713276176, 133306628004003, 638710426743841, 3060245505715200, 14662517101832161, 70252340003445603, 336599182915395856, 1612743574573533675, 7727118689952272521, 37022849875187828928, 177387130685986872121, 849912803554746531675, 4072176887087745786256, 19510971631883982399603, 93482681272332166211761, 447902434729776848659200, 2146029492376552077084241, 10282245027152983536762003, 49265195643388365606725776, 236043733189788844496866875, 1130953470305555856877608601]], [{{1, 2}, {1, 3}, {1, 4}, {1, 5}}, -(-1+z^2)/(-6*z^2+1+z^4), [0, 5, 0, 29, 0, 169, 0, 985, 0, 5741, 0, 33461, 0, 195025, 0, 1136689, 0, 6625109, 0, 38613965, 0, 225058681, 0 , 1311738121, 0, 7645370045, 0, 44560482149, 0, 259717522849, 0, 1513744654945, 0, 8822750406821, 0, 51422757785981, 0, 299713796309065, 0, 1746860020068409]], [{{1, 2}, {1, 3}, {1, 4}, {2, 5}}, -(-34*z^4-11*z^8+34*z^6-1+11*z^2+z^10)/(-18* z^10+89*z^8-152*z^6+89*z^4+z^12-18*z^2+1), [0, 7, 0, 71, 0, 773, 0, 8581, 0, 95847, 0, 1072839, 0, 12017505, 0, 134651297, 0, 1508860231, 0, 16908413479, 0, 189479536517, 0, 2123360889605, 0, 23795019256967, 0, 266654293034375, 0, 2988210579801025, 0, 33486815887487041, 0, 375263671216306823, 0, 4205321430942608391, 0, 47126140274266744581, 0, 528110190129944930181]], [{{1, 2}, {1, 3}, {2, 4}, {3, 5}}, -(z^6-7*z^4+7*z^2-1)/(z^8-15*z^6+32*z^4-15*z^2+1), [0, 8, 0, 95, 0, 1183, 0, 14824, 0, 185921, 0, 2332097, 0, 29253160, 0, 366944287, 0, 4602858719, 0, 57737128904, 0, 724240365697, 0, 9084693297025, 0, 113956161827912, 0, 1429438110270431, 0, 17930520634652959, 0, 224916047725262248, 0, 2821291671062267585, 0, 35389589910135145793, 0, 443918325373278904936, 0, 5568402462493067660191]], [{{1, 2}, {1, 3}, {1, 4}, { 1, 5}, {2, 3}}, -(94*z^10-210*z^8+17*z^2-17*z^12+210*z^6-94*z^4-1+z^14)/(-25*z^ 2+205*z^4-696*z^6+1044*z^8-696*z^10+205*z^12-25*z^14+z^16+1), [0, 8, 0, 89, 0, 1071, 0, 13264, 0, 166239, 0, 2094735, 0, 26463328, 0, 334744623, 0, 4237029785 , 0, 53647679768, 0, 679379618545, 0, 8604208653457, 0, 108975330035192, 0, 1380241523321753, 0, 17481830863039407, 0, 221422251196465216, 0, 2804509402019720367, 0, 35521655583318475839, 0, 449914340164057313776, 0, 5698579943625197681583]], [{{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 5}}, -(-17*z^12 +99*z^10-242*z^8+242*z^6-99*z^4+17*z^2-1+z^14)/(z^16-26*z^14+217*z^12-770*z^10+ 1203*z^8-770*z^6+217*z^4-26*z^2+1), [0, 9, 0, 116, 0, 1591, 0, 22163, 0, 310155 , 0, 4346911, 0, 60954980, 0, 854907185, 0, 11991084425, 0, 168193387801, 0, 2359192846593, 0, 33091730619076, 0, 464168946440111, 0, 6510778110086075, 0, 91325022076889379, 0, 1280992852374594279, 0, 17968161322019256468, 0, 252034836694020152825, 0, 3535228674445477014833, 0, 49587755252978412752721]], [{{1, 2}, {1, 3}, {1, 4}, {2, 3}, {4, 5}}, -(-1+22*z^2-184*z^4+737*z^6-1482*z^8 +1482*z^10-737*z^12+184*z^14-22*z^16+z^18)/(1-32*z^2+359*z^4-1888*z^6+5081*z^8-\ 7112*z^10+5081*z^12-1888*z^14+359*z^16-32*z^18+z^20), [0, 10, 0, 145, 0, 2201, 0, 33658, 0, 515477, 0, 7897561, 0, 121010866, 0, 1854262901, 0, 28413431085, 0 , 435389580706, 0, 6671648045989, 0, 102232389065885, 0, 1566549074783858, 0, 24004880231519541, 0, 367836736108241917, 0, 5636514965881852098, 0, 86370658655129449233, 0, 1323493460453613345757, 0, 20280439782633665595754, 0, 310765598967697148467345]], [{{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}}, -(z^10-\ 14*z^8+55*z^6-55*z^4+14*z^2-1)/(-26*z^10+154*z^8-268*z^6+154*z^4+z^12-26*z^2+1) , [0, 12, 0, 213, 0, 3903, 0, 71752, 0, 1319751, 0, 24277231, 0, 446600352, 0, 8215660303, 0, 151135631573, 0, 2780299176772, 0, 51146541729081, 0, 940894736640041, 0, 17308754179042772, 0, 318412847633477173, 0, 5857541253534036623, 0, 107755669416656143472, 0, 1982279558927628297791, 0, 36466130006078774450231, 0, 670833047559097274020472, 0, 12340683741962581448385023]], [{{1, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 5}}, -(-1-8 *z^4+8*z^2+z^6)/(1+41*z^4-19*z^2-19*z^6+z^8), [0, 11, 0, 176, 0, 2911, 0, 48301 , 0, 801701, 0, 13307111, 0, 220880176, 0, 3666315811, 0, 60855946601, 0, 1010127453401, 0, 16766766924211, 0, 278305942640176, 0, 4619507031938711, 0, 76677648402694901, 0, 1272746577484955101, 0, 21125893715367851311, 0, 350661626727725280176, 0, 5820514772820509871611, 0, 96612773221767455293201, 0 , 1603643030542052158306801]], [{{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4} }, -(-25*z^16+228*z^14-956*z^12+1930*z^10-1930*z^8+956*z^6-228*z^4+25*z^2-1+z^ 18)/(477*z^16+7358*z^12+z^20+7358*z^8-10300*z^10-2676*z^14-38*z^18+477*z^4-38*z ^2-2676*z^6+1), [0, 13, 0, 245, 0, 4829, 0, 95997, 0, 1912789, 0, 38142605, 0, 760805121, 0, 15176819777, 0, 302764305101, 0, 6039969963221, 0, 120494499836029, 0, 2403812274318557, 0, 47955034267442037, 0, 956682848980371533, 0, 19085424234544653633, 0, 380746277170430152641, 0, 7595730021342875759821, 0, 151531658625523988015477, 0, 3022993649336995071440861, 0, 60307467722963389108943997]], [{{1, 2}, {1, 3}, { 1, 4}, {1, 5}, {2, 3}, {4, 5}}, -(z^10-14*z^8+47*z^6-47*z^4+14*z^2-1)/(-3*z^2+1 +z^4)/(z^8-23*z^6+76*z^4-23*z^2+1), [0, 12, 0, 213, 0, 4013, 0, 76396, 0, 1457033, 0, 27797817, 0, 530368556, 0, 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