A Linear Recurrence Equation for The Diagonal Coefficients of The Power Ser\ ies Of 1 ------------------------------------- 2 2 2 a ((1 - x) + (1 - y) + (1 - z) - 2) By Shalosh B. Ekhad n n n Let , A(n), be the coefficient of, x y z , in the Maclaurin Expansion of the Formal Power series, 1 ------------------------------------- 2 2 2 a ((1 - x) + (1 - y) + (1 - z) - 2) then , A(n), satisfies the following linear recurrence equation with polynomi\ al coefficients (2 a + 1 + 3 n) (2 a + 4 + 3 n) (2 a + 2 + 3 n) (2 a + 3 n) (2 a + 5 + 3 n) 5 4 4 3 2 3 (2 a + 3 + 3 n) (32 a + 400 a n + 928 a + 10936 a + 2048 n a 3 2 2 2 2 3 2 + 9504 n a + 64520 a + 37128 n a + 5344 a n + 85204 n a 2 3 4 3 + 224768 n a + 65466 n a + 339502 n a + 190164 a + 7084 n a + 200102 n 4 5 2 / 3 + 507177 n + 43769 n + 3795 n + 224550 + 452939 n ) A(n) / ((n + 3) / %1 (2 + n) (2 a + 4 n + 11) (2 a + 4 n + 9)) - (2 a + 5 + 3 n) 6 2 5 2 (2 a + 3 + 3 n) (2 a + 4 + 3 n) (28569744 + 43456 a n + 1201696 a n 5 3 2 3 3 3 7 + 220512 a n + 82928768 n a + 48257776 n a + 4992 a n 3 4 7 6 8 7 + 13652816 a n + 8052484 n + 140352 a + 554070 n + 9088 a 5 4 4 5 5 + 1222688 a + 6562336 a + 15749120 a n + 2129248 n a + 172549733 n 2 2 2 3 2 4 + 497634562 n a + 180032752 n a + 223189956 a n + 13736640 n a 3 4 3 3 2 5 + 5176912 n a + 469825139 n + 22248592 a + 22041592 a n 6 2 4 6 3 + 20162278 n a + 97380516 a n + 49971265 n + 512946602 n a 4 2 2 + 361665430 n + 258669238 n a + 368178661 n + 46628608 a 5 4 4 3 2 2 + 106998808 n a + 713344 a n + 69084632 n a + 279115896 n a 4 2 6 7 3 5 8 + 306995210 n a + 2027008 a n + 1588334 n a + 1506296 a n + 256 a 6 / 3 + 158080 a n + 158511814 n + 55350740 a) A(1 + n) / ((n + 3) %1 / 6 2 (2 + n) (2 a + 4 n + 11) (2 a + 4 n + 9)) - (153797120 a n 3 6 5 2 8 5 3 + 891715752 a n + 1379820224 a n + 627792318 a n + 766696608 a n 4 5 5 4 2 3 + 527499664 a n + 209748288 a n + 26998012088 n a 3 3 2 7 7 3 4 + 25915415928 n a + 976377780 a n + 17790464 a n + 14652435352 a n 9 7 6 8 7 + 180406710 n + 7116183465 n + 81715776 a + 1483249968 n + 10679040 a 10 9 5 4 4 + 1024 a + 45568 a + 424503072 a + 1512088832 a + 5329847088 a n 5 5 2 9 + 1221128448 n a + 45932380392 n + 71628907476 n a + 37638810 a n 2 2 3 2 4 + 27828140816 n a + 68936854496 a n + 7670683952 n a 3 4 3 3 2 5 + 5780789840 n a + 61955294781 n + 3635401856 a + 22455877100 a n 6 2 4 6 10 + 19252449198 n a + 49639353548 a n + 22033808133 n + 9734175 n 6 3 3 4 + 56089472 a n + 100555295310 n a + 65158595358 n + 28987808392 n a 2 2 5 4 4 + 37669738758 n + 5628424064 a + 51074823900 n a + 2409758400 a n 3 2 2 2 8 4 + 15308605632 n a + 58616504336 n a + 65850660 a n + 88676252984 n a 2 6 7 3 5 + 6242713680 a n + 4587339804 n a + 4887470128 a n + 2000016360 8 7 2 6 + 908800 a + 9730688 a n + 184582272 a n + 13170877404 n 5 5 3 7 7 3 + 5055641328 a + 22624736 a n + 68723376 a n + 1749120 a n 8 2 9 6 4 4 6 8 + 268032 a n + 24576 a n + 7561984 a n + 47422368 a n + 994048 a n / 3 ) A(2 + n) / ((n + 3) (2 a + 4 n + 11) (2 a + 4 n + 9) %1) + A(n + 3) = / 0 5 4 4 2 3 3 3 2 3 %1 := 32 a + 400 a n + 528 a + 2048 n a + 5408 n a + 3480 a + 5344 a n 2 2 2 2 4 3 + 21096 n a + 26980 n a + 11100 a + 7084 n a + 37130 n a 2 5 4 3 + 70874 n a + 58028 n a + 17048 a + 3795 n + 24794 n + 62976 n 2 + 77297 n + 45504 n + 10184 Proof: Thanks to Cauchy's Integral Formula, A(n) equals to a constant (independent of n) times the contour-integral with respect to the complex variables, x, y, z around any poly-circle around the origin, of the function F(n, x, y, z) = 1 ---------------------------------------------------------------- 2 2 2 a (1 + n) (1 + n) (1 + n) ((1 - x) + (1 - y) + (1 - z) - 2) x y z Let's cleverly constuct rational functions 8 6 2 3 6 -1/1824 z (-107595941483520 z y[1] - 51778953216 a n + 10331711126656 a n 5 2 8 5 3 - 383273938944 a n + 14644672511968 a n - 532953188352 a n 4 5 5 4 2 3 + 1721534591744 a n - 92096123904 a n - 4050797188608 n a 3 3 2 7 7 - 8934831270144 n a + 21177935305088 a n - 1336393728 a n 3 4 2 9 2 - 8294345307264 a n - 3303161856 y[1] y[2] a n 2 9 2 2 9 3 - 3303161856 y[1] y[2] a n - 21323776 y[1] y[2] a n 2 9 3 2 4 6 - 21323776 y[1] y[2] a n + 157513492172032 y[1] y[2] a n 2 4 6 2 10 2 + 157513492172032 y[1] y[2] a n - 67239936 y[1] y[2] a n 2 10 2 2 6 4 - 67239936 y[1] y[2] a n - 171852980224 y[1] y[2] a n 2 6 4 2 10 - 171852980224 y[1] y[2] a n + 6686821882352 y[1] y[2] a n 2 10 6 2 5 + 6686821882352 y[1] y[2] a n + 1823355622400 a n y[2] 3 6 5 3 6 5 + 655287954501504 a n y[1] + 655287954501504 a n y[2] 5 2 5 5 2 5 - 27199693751808 a n y[1] - 27199693751808 a n y[2] 8 5 8 5 + 724677204078704 a n y[1] + 724677204078704 a n y[2] 5 3 5 5 3 5 - 76413605152512 a n y[1] - 76413605152512 a n y[2] 4 5 5 4 5 5 + 52347535635456 a n y[1] + 52347535635456 a n y[2] 5 4 5 5 4 5 - 74436014384640 a n y[1] - 74436014384640 a n y[2] 2 3 5 2 3 5 - 368317693908608 n a y[1] - 368317693908608 n a y[2] 3 3 5 3 3 5 - 650289722395968 n a y[1] - 650289722395968 n a y[2] 2 7 5 2 7 5 + 1141926311870464 a n y[1] + 1141926311870464 a n y[2] 7 5 7 5 + 3731773722624 a n y[1] + 3731773722624 a n y[2] 3 4 5 3 4 5 - 627393935981568 a n y[1] - 627393935981568 a n y[2] 4 5 4 5 - 56058388180992 a n y[1] - 56058388180992 a n y[2] 5 5 5 5 - 2498122957824 n a y[1] - 2498122957824 n a y[2] 2 5 2 5 + 219784293423744 n a y[1] + 219784293423744 n a y[2] 9 5 9 5 + 610528622354688 a n y[1] + 610528622354688 a n y[2] 2 5 2 5 - 62481608446464 n a y[1] - 62481608446464 n a y[2] 2 3 5 2 3 5 - 692480351147264 a n y[1] - 692480351147264 a n y[2] 2 4 5 2 4 5 - 161230544897024 n a y[1] - 161230544897024 n a y[2] 3 4 5 3 4 5 - 275026726609920 n a y[1] - 275026726609920 n a y[2] 2 5 5 2 5 5 - 603235325074336 a n y[1] - 603235325074336 a n y[2] 6 5 6 5 - 689536676781520 n a y[1] - 689536676781520 n a y[2] 2 4 5 2 4 5 - 971052122762912 a n y[1] - 971052122762912 a n y[2] 6 3 5 6 3 5 - 19722736164352 a n y[1] - 19722736164352 a n y[2] 3 5 3 5 + 137292746842304 n a y[1] + 137292746842304 n a y[2] 5 5 + 95161046658048 n a y[1] + 95161046658048 n a y[2] 5 5 5 5 - 873539232558256 n a y[1] - 873539232558256 n a y[2] 4 4 5 4 4 5 - 234028314698112 a n y[1] - 234028314698112 a n y[2] 3 5 3 5 - 119488395062016 n a y[1] - 119488395062016 n a y[2] 2 2 5 2 2 5 - 282394958338304 n a y[1] - 282394958338304 n a y[2] 2 8 5 2 8 5 + 1029639576402528 a n y[1] + 1029639576402528 a n y[2] 4 5 4 5 - 361029463454816 n a y[1] - 361029463454816 n a y[2] 2 6 5 2 6 5 + 378490241263072 a n y[1] + 378490241263072 a n y[2] 7 5 7 5 + 131896052586352 n a y[1] + 131896052586352 n a y[2] 3 5 5 3 5 5 - 74781883883776 a n y[1] - 74781883883776 a n y[2] 7 2 5 7 2 5 + 1343055109120 a n y[1] + 1343055109120 a n y[2] 6 5 6 5 + 7880204851200 a n y[1] + 7880204851200 a n y[2] 7 5 5 7 5 5 + 935134584832 a n y[1] + 935134584832 a n y[2] 5 5 5 5 5 5 + 22762529245696 a n y[1] + 22762529245696 a n y[2] 3 8 5 3 8 5 + 503939775753728 a n y[1] + 503939775753728 a n y[2] 10 5 10 5 + 9391472640 a n y[1] + 9391472640 a n y[2] 5 6 5 5 6 5 + 104506814099456 a n y[1] + 104506814099456 a n y[2] 2 9 5 2 9 5 + 378052527467424 a n y[1] + 378052527467424 a n y[2] 8 3 5 8 3 5 - 1117730080768 a n y[1] - 1117730080768 a n y[2] 2 10 5 2 10 5 - 87805201388096 a n y[1] - 87805201388096 a n y[2] 3 9 5 3 9 5 + 31737805006080 a n y[1] + 31737805006080 a n y[2] 6 6 5 6 6 5 + 21206896541696 a n y[1] + 21206896541696 a n y[2] 3 7 5 3 7 5 + 889135343151040 a n y[1] + 889135343151040 a n y[2] 8 4 5 8 4 5 - 784555239424 a n y[1] - 784555239424 a n y[2] 7 3 5 7 3 5 - 5243352679424 a n y[1] - 5243352679424 a n y[2] 6 5 5 6 5 5 + 4653532063232 a n y[1] + 4653532063232 a n y[2] 8 2 5 8 2 5 + 113590151168 a n y[1] + 113590151168 a n y[2] 11 5 11 5 + 357433344 a n y[1] + 357433344 a n y[2] 5 7 5 5 7 5 + 76300114216704 a n y[1] + 76300114216704 a n y[2] 4 7 5 4 7 5 + 339536944027392 a n y[1] + 339536944027392 a n y[2] 9 5 9 5 + 119620583424 a n y[1] + 119620583424 a n y[2] 4 8 5 4 8 5 + 112239079276928 a n y[1] + 112239079276928 a n y[2] 7 4 5 7 4 5 - 5252481291264 a n y[1] - 5252481291264 a n y[2] 10 2 5 10 2 5 - 6916308992 a n y[1] - 6916308992 a n y[2] 6 4 5 6 4 5 - 22500186279936 a n y[1] - 22500186279936 a n y[2] 10 5 10 5 + 176710931670912 a n y[1] + 176710931670912 a n y[2] 9 2 5 9 2 5 - 26959609856 a n y[1] - 26959609856 a n y[2] 9 3 5 9 3 5 - 156346388480 a n y[1] - 156346388480 a n y[2] 4 6 5 4 6 5 + 345758101551360 a n y[1] + 345758101551360 a n y[2] 8 5 8 5 + 873313542144 a n y[1] + 873313542144 a n y[2] 11 5 11 5 - 74454529872288 a n y[1] - 74454529872288 a n y[2] 8 5 5 8 5 5 + 221164441600 a n y[1] + 221164441600 a n y[2] 5 8 5 5 8 5 + 6321572096512 a n y[1] + 6321572096512 a n y[2] 9 4 5 9 4 5 - 62961459200 a n y[1] - 62961459200 a n y[2] 11 2 5 11 2 5 - 589283328 a n y[1] - 589283328 a n y[2] 7 6 5 7 6 5 + 3117385739264 a n y[1] + 3117385739264 a n y[2] 12 5 12 5 - 85179642475136 a n y[1] - 85179642475136 a n y[2] 4 9 5 4 9 5 - 45955327404288 a n y[1] - 45955327404288 a n y[2] 2 11 5 2 11 5 - 156683346145024 a n y[1] - 156683346145024 a n y[2] 10 3 5 10 3 5 - 12651896832 a n y[1] - 12651896832 a n y[2] 6 7 5 6 7 5 + 10426072049152 a n y[1] + 10426072049152 a n y[2] 3 10 5 3 10 5 - 133275829553152 a n y[1] - 133275829553152 a n y[2] 8 6 5 8 6 5 + 340641329152 a n y[1] + 340641329152 a n y[2] 3 11 5 3 11 5 - 80091651823296 a n y[1] - 80091651823296 a n y[2] 10 4 5 10 4 5 - 1398677504 a n y[1] - 1398677504 a n y[2] 6 8 5 6 8 5 - 2253246240256 a n y[1] - 2253246240256 a n y[2] 9 5 5 9 5 5 + 37712023552 a n y[1] + 37712023552 a n y[2] 11 3 5 11 3 5 - 486506496 a n y[1] - 486506496 a n y[2] 13 5 13 5 - 29248358185072 a n y[1] - 29248358185072 a n y[2] 4 10 5 4 10 5 - 54356117542016 a n y[1] - 54356117542016 a n y[2] 7 7 5 7 7 5 + 838867386368 a n y[1] + 838867386368 a n y[2] 12 5 12 5 5 9 5 + 3538944 a n y[1] + 3538944 a n y[2] - 19515354456320 a n y[1] 5 9 5 12 2 5 - 19515354456320 a n y[2] - 17661952 a n y[1] 12 2 5 2 12 5 - 17661952 a n y[2] - 65875943269600 a n y[1] 2 12 5 5 18 - 65875943269600 a n y[2] - 49499284032 z n 2 5 7 2 5 7 + 21454022178304 y[1] y[2] a n + 21454022178304 y[1] y[2] a n 2 4 7 2 4 7 + 128021776856704 y[1] y[2] a n + 128021776856704 y[1] y[2] a n 2 9 2 9 - 1585741824 y[1] y[2] a n - 1585741824 y[1] y[2] a n 2 4 8 2 4 8 + 28755744846976 y[1] y[2] a n + 28755744846976 y[1] y[2] a n 2 7 4 2 7 4 + 390853296128 y[1] y[2] a n + 390853296128 y[1] y[2] a n 5 2 5 2 5 4 + 67575900049920 z a + 1336046417587584 z n + 3594258705466848 z n 5 13 5 14 5 + 77189649922992 z n + 1820916401136 z n + 84027236364288 z a 5 5 8 + 343635870306816 z n - 223864750080 z a 2 7 3 2 7 3 - 656248545280 y[1] y[2] a n - 656248545280 y[1] y[2] a n 2 6 5 2 6 5 + 7352417876480 y[1] y[2] a n + 7352417876480 y[1] y[2] a n 2 8 2 2 8 2 - 66364416000 y[1] y[2] a n - 66364416000 y[1] y[2] a n 2 6 6 2 6 6 + 6754171130368 y[1] y[2] a n + 6754171130368 y[1] y[2] a n 2 3 7 2 3 7 + 378874688310848 y[1] y[2] a n + 378874688310848 y[1] y[2] a n 2 8 4 2 8 4 + 62784784384 y[1] y[2] a n + 62784784384 y[1] y[2] a n 4 2 2 4 + 26000752619520 y[1] y[2] + 26000752619520 y[1] y[2] 5 14 5 5 14 5 + 21586523136 a n y[1] + 21586523136 a n y[2] 5 13 5 5 13 5 + 230302690048 a n y[1] + 230302690048 a n y[2] 5 12 5 5 12 5 + 710676660224 a n y[1] + 710676660224 a n y[2] 4 7 15 7 15 - 337530820608 a n + 155713536 z a y[2] + 155713536 z a y[1] 7 20 7 20 + 4128929640 z n y[2] + 4128929640 z n y[1] 4 4 13 6 4 4 12 7 - 196608 y[1] y[2] a n - 4456448 y[1] y[2] a n 4 4 11 8 4 4 10 9 - 40910848 y[1] y[2] a n - 191979520 y[1] y[2] a n 4 4 9 10 4 4 8 11 - 442437632 y[1] y[2] a n - 199401472 y[1] y[2] a n 4 4 3 15 4 4 4 14 - 801810021824 y[1] y[2] a n - 568599684992 y[1] y[2] a n 4 4 7 10 4 4 9 9 - 327491722240 y[1] y[2] a n - 8784613376 y[1] y[2] a n 4 4 4 13 4 4 13 4 - 3019818113792 y[1] y[2] a n + 3145728 y[1] y[2] a n 4 4 4 11 4 4 11 4 + 14610853552768 y[1] y[2] a n + 1033617408 y[1] y[2] a n 4 4 7 9 4 4 7 8 - 552211866624 y[1] y[2] a n + 2774585462784 y[1] y[2] a n 4 4 10 5 4 4 12 3 + 49731444736 y[1] y[2] a n - 260636672 y[1] y[2] a n 4 4 3 14 4 4 9 8 - 4694946198272 y[1] y[2] a n - 25160572928 y[1] y[2] a n 4 4 2 15 4 4 6 11 - 3978789743936 y[1] y[2] a n - 631415311360 y[1] y[2] a n 4 4 2 13 4 4 4 15 + 1901991904544 y[1] y[2] a n - 41166194432 y[1] y[2] a n 4 4 3 16 4 4 7 12 - 53283392832 y[1] y[2] a n + 950012928 y[1] y[2] a n 4 4 3 13 4 4 13 5 - 10718681779648 y[1] y[2] a n - 131072 y[1] y[2] a n 4 4 5 10 4 4 5 11 + 6370957328896 y[1] y[2] a n - 2895410118912 y[1] y[2] a n 4 4 6 12 4 4 8 7 - 69166319104 y[1] y[2] a n + 1466379042816 y[1] y[2] a n 4 4 9 7 4 4 5 12 + 77489676288 y[1] y[2] a n - 1345064806912 y[1] y[2] a n 4 4 13 4 4 10 7 - 524288 y[1] y[2] a n - 1402912768 y[1] y[2] a n 4 4 6 9 4 4 2 17 + 3327027407360 y[1] y[2] a n - 35562513888 y[1] y[2] a n 4 4 6 13 4 4 5 14 - 1078290944 y[1] y[2] a n - 15612207872 y[1] y[2] a n 4 4 11 5 4 4 11 7 + 3416883200 y[1] y[2] a n - 256819200 y[1] y[2] a n 4 4 13 2 4 4 8 8 - 11599872 y[1] y[2] a n + 16926558208 y[1] y[2] a n 4 4 12 5 4 4 3 12 + 70844416 y[1] y[2] a n + 15295065221952 y[1] y[2] a n 4 4 11 6 4 4 6 10 + 394002432 y[1] y[2] a n - 1657967201280 y[1] y[2] a n 3 3 3 6 3 3 7 3 - 80919818113024 y[1] y[2] n a + 3610761517165568 y[1] y[2] n a 3 3 3 5 3 3 9 - 394495850588160 y[1] y[2] n a + 3356475991407168 y[1] y[2] n a 3 3 4 5 3 3 6 4 - 178859093259264 y[1] y[2] n a + 1496736107512832 y[1] y[2] n a 3 3 5 5 3 3 3 3 + 291464679186432 y[1] y[2] n a - 2557203337197568 y[1] y[2] n a 3 3 4 3 3 3 8 2 - 2971048615232512 y[1] y[2] n a + 4781165796789888 y[1] y[2] n a 3 3 2 7 3 3 5 3 - 11712184713216 y[1] y[2] n a - 623523585144320 y[1] y[2] n a 3 3 6 3 3 7 - 16393727950848 y[1] y[2] n a - 3631956295680 y[1] y[2] n a 3 3 10 3 3 11 - 3003383808 y[1] y[2] n a - 75497472 y[1] y[2] n a 3 3 9 3 3 5 - 52993327104 y[1] y[2] n a - 51212471132160 y[1] y[2] n a 3 3 4 3 3 2 4 - 111069244366848 y[1] y[2] n a - 597546626531328 y[1] y[2] n a 3 3 2 5 3 3 3 - 236575219249152 y[1] y[2] n a - 2682013470019584 y[1] y[2] n a 3 3 10 3 3 2 2 + 365683353347904 y[1] y[2] n a - 1149910117152768 y[1] y[2] n a 3 3 4 2 3 3 3 4 - 5060102236834816 y[1] y[2] n a - 1240981623451648 y[1] y[2] n a 3 3 4 4 3 3 3 - 1020250419249152 y[1] y[2] n a - 165149335535616 y[1] y[2] n a 3 3 6 2 3 3 7 + 2236224261587072 y[1] y[2] n a + 4841259697797184 y[1] y[2] n a 3 3 5 2 3 3 - 3027937878274176 y[1] y[2] n a - 27646369873920 y[1] y[2] n 2 8 3 2 8 3 - 31419494400 y[1] y[2] a n - 31419494400 y[1] y[2] a n 2 2 10 2 2 10 - 58670802091040 y[1] y[2] a n - 58670802091040 y[1] y[2] a n 2 3 9 2 3 9 - 7196668259008 y[1] y[2] a n - 7196668259008 y[1] y[2] a n 5 6 5 2 - 92929794048 n a + 2406679031808 a y[2] - 2956489030656 n a 2 10 2 10 - 44826624 y[1] y[2] a n - 44826624 y[1] y[2] a n 2 5 6 2 5 6 + 44051212805120 y[1] y[2] a n + 44051212805120 y[1] y[2] a n 2 2 9 2 2 9 + 110956367524672 y[1] y[2] a n + 110956367524672 y[1] y[2] a n 8 7 5 8 7 5 + 33127983104 a n y[1] + 33127983104 a n y[2] 5 10 5 5 10 5 - 10030951913984 a n y[1] - 10030951913984 a n y[2] 9 6 5 9 6 5 + 25646792704 a n y[1] + 25646792704 a n y[2] 11 4 5 11 4 5 + 115720192 a n y[1] + 115720192 a n y[2] 9 5 2 4 + 158757227889672 n y[2] - 283928677007232 y[1] y[2] n a 2 4 2 2 6 - 283928677007232 y[1] y[2] n a + 238657604147520 y[1] y[2] a n 2 2 6 2 7 + 238657604147520 y[1] y[2] a n + 293129864194192 y[1] y[2] n a 2 7 2 3 5 + 293129864194192 y[1] y[2] n a - 53706425973248 y[1] y[2] a n 2 3 5 2 7 2 - 53706425973248 y[1] y[2] a n - 735055773696 y[1] y[2] a n 2 7 2 2 6 - 735055773696 y[1] y[2] a n - 1254244737024 y[1] y[2] a n 2 6 2 7 5 - 1254244737024 y[1] y[2] a n + 1025975602176 y[1] y[2] a n 2 7 5 2 5 5 + 1025975602176 y[1] y[2] a n + 27552249809152 y[1] y[2] a n 2 5 5 2 3 8 + 27552249809152 y[1] y[2] a n + 185383639989312 y[1] y[2] a n 2 3 8 4 3 16 + 185383639989312 y[1] y[2] a n + 4564016969972 y[1] y[2] a n 4 3 16 4 2 18 + 4564016969972 y[1] y[2] a n + 165956952594 y[1] y[2] a n 4 2 18 4 2 17 + 165956952594 y[1] y[2] a n + 1892718483060 y[1] y[2] a n 4 2 17 4 19 + 1892718483060 y[1] y[2] a n + 24511991526 y[1] y[2] a n 4 19 4 18 + 24511991526 y[1] y[2] a n + 221862730134 y[1] y[2] a n 4 18 3 3 7 + 221862730134 y[1] y[2] a n + 1443970035822816 y[1] y[2] n 3 3 11 3 3 16 - 158208483032640 y[1] y[2] n + 4929713209248 y[1] y[2] n 3 3 19 3 3 14 + 8138180160 y[1] y[2] n - 33060778718400 y[1] y[2] n 3 3 5 3 3 3 - 2654365322596608 y[1] y[2] n - 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169631034036224 z y[1] y[2] a n 7 9 5 7 11 3 - 169631034036224 z y[1] y[2] a n - 8040601026560 z y[1] y[2] a n 7 11 3 7 2 14 - 8040601026560 z y[1] y[2] a n + 162612309262544 z y[1] y[2] a n 7 2 14 7 13 + 162612309262544 z y[1] y[2] a n + 42519724032 z y[1] y[2] a n 7 13 7 13 2 + 42519724032 z y[1] y[2] a n - 53717139456 z y[1] y[2] a n 7 13 2 7 13 3 - 53717139456 z y[1] y[2] a n - 50117902336 z y[1] y[2] a n 7 13 3 7 8 8 - 50117902336 z y[1] y[2] a n + 142778333622272 z y[1] y[2] a n 7 8 8 7 15 + 142778333622272 z y[1] y[2] a n + 63564518702280 z y[1] y[2] a n 7 15 7 5 11 + 63564518702280 z y[1] y[2] a n + 188750304901632 z y[1] y[2] a n 7 5 11 7 13 + 188750304901632 z y[1] y[2] a n - 13211985733784 z y[1] y[2] a n 7 13 7 3 15 - 13211985733784 z y[1] y[2] a n - 41168538180832 z y[1] y[2] a n 7 3 15 7 2 16 - 41168538180832 z y[1] y[2] a n - 10993458105568 z y[1] y[2] a n 7 2 16 7 6 12 - 10993458105568 z y[1] y[2] a n - 77059580318464 z y[1] y[2] a n 7 6 12 - 77059580318464 z y[1] y[2] a n 7 2 13 + 1032280040702272 z y[1] y[2] a n 7 2 13 7 12 5 + 1032280040702272 z y[1] y[2] a n + 64650838016 z y[1] y[2] a n 7 12 5 7 4 10 + 64650838016 z y[1] y[2] a n + 7828832852446208 z y[1] y[2] a n 7 4 10 + 7828832852446208 z y[1] y[2] a n 7 4 11 + 3196996644119872 z y[1] y[2] a n 7 4 11 + 3196996644119872 z y[1] y[2] a n 7 5 12 - 302747848323072 z y[1] y[2] a n 7 5 12 - 302747848323072 z y[1] y[2] a n 7 7 7 + 1213630453339648 z y[1] y[2] a n 7 7 7 7 8 7 + 1213630453339648 z y[1] y[2] a n + 316675178967040 z y[1] y[2] a n 7 8 7 7 4 12 + 316675178967040 z y[1] y[2] a n + 114529671627008 z y[1] y[2] a n 7 4 12 7 12 + 114529671627008 z y[1] y[2] a n + 854710697984 z y[1] y[2] a n 7 12 7 9 7 + 854710697984 z y[1] y[2] a n + 42624168316928 z y[1] y[2] a n 7 9 7 7 5 9 + 42624168316928 z y[1] y[2] a n + 7539745734800640 z y[1] y[2] a n 7 5 9 7 17 + 7539745734800640 z y[1] y[2] a n - 810336776184 z y[1] y[2] a n 7 17 7 5 13 - 810336776184 z y[1] y[2] a n - 97279612013824 z y[1] y[2] a n 7 5 13 7 4 14 - 97279612013824 z y[1] y[2] a n - 80674465672576 z y[1] y[2] a n 7 4 14 7 10 5 - 80674465672576 z y[1] y[2] a n - 18008660291584 z y[1] y[2] a n 7 10 5 7 10 7 - 18008660291584 z y[1] y[2] a n + 3482064195584 z y[1] y[2] a n 7 10 7 7 12 2 + 3482064195584 z y[1] y[2] a n - 291126460416 z y[1] y[2] a n 7 12 2 7 7 8 - 291126460416 z y[1] y[2] a n + 1172402604665344 z y[1] y[2] a n 7 7 8 7 11 5 + 1172402604665344 z y[1] y[2] a n - 583646412800 z y[1] y[2] a n 7 11 5 7 2 12 - 583646412800 z y[1] y[2] a n + 1589624586043792 z y[1] y[2] a n 7 2 12 7 10 6 + 1589624586043792 z y[1] y[2] a n + 5549275594752 z y[1] y[2] a n 7 10 6 7 5 10 + 5549275594752 z y[1] y[2] a n + 3340359917359360 z y[1] y[2] a n 7 5 10 7 15 3 + 3340359917359360 z y[1] y[2] a n + 5898240 z y[1] y[2] a n 7 15 3 7 15 2 + 5898240 z y[1] y[2] a n - 43646976 z y[1] y[2] a n 7 15 2 7 15 - 43646976 z y[1] y[2] a n - 43646976 z y[1] y[2] a n 7 15 7 14 5 - 43646976 z y[1] y[2] a n + 80609280 z y[1] y[2] a n 7 14 5 7 13 6 + 80609280 z y[1] y[2] a n + 720306176 z y[1] y[2] a n 7 13 6 7 12 7 + 720306176 z y[1] y[2] a n + 2466824192 z y[1] y[2] a n 7 12 7 7 11 8 + 2466824192 z y[1] y[2] a n - 7937966080 z y[1] y[2] a n 7 11 8 7 10 9 - 7937966080 z y[1] y[2] a n - 128886247424 z y[1] y[2] a n 7 10 9 7 9 10 - 128886247424 z y[1] y[2] a n - 700165136384 z y[1] y[2] a n 7 9 10 7 8 11 - 700165136384 z y[1] y[2] a n - 2331379569664 z y[1] y[2] a n 7 8 11 7 7 12 - 2331379569664 z y[1] y[2] a n - 5302486193152 z y[1] y[2] a n 7 7 12 7 3 16 - 5302486193152 z y[1] y[2] a n - 3664345600096 z y[1] y[2] a n 7 3 16 7 2 17 - 3664345600096 z y[1] y[2] a n - 1038518216784 z y[1] y[2] a n 7 2 17 7 6 13 - 1038518216784 z y[1] y[2] a n - 8493228408576 z y[1] y[2] a n 7 6 13 7 15 4 - 8493228408576 z y[1] y[2] a n + 3538944 z y[1] y[2] a n 7 15 4 7 18 + 3538944 z y[1] y[2] a n - 120324695040 z y[1] y[2] a n 7 18 7 5 14 - 120324695040 z y[1] y[2] a n - 9563983716096 z y[1] y[2] a n 7 5 14 7 4 15 - 9563983716096 z y[1] y[2] a n - 7371234153152 z y[1] y[2] a n 7 4 15 5 / - 7371234153152 z y[1] y[2] a n + 420948893675520 z y[1] y[2] ) / (a / 5 4 4 2 3 3 3 2 2 (16 n + 176 n + 16 n a + 4 a n + 755 n + 136 n a + 24 n a 2 2 2 2 + 1570 n + 416 n a + 44 n a + 1569 n + 536 n a + 24 a + 594 + 240 a) 6 5 4 3 2 2 5 4 (n + 6 n + 4 n - 30 n - 41 n + 24 n + 36) (-9 + n ) (32 a + 400 a n 4 2 3 3 3 2 3 2 2 + 528 a + 2048 n a + 5408 n a + 3480 a + 5344 a n + 21096 n a 2 2 4 3 2 + 26980 n a + 11100 a + 7084 n a + 37130 n a + 70874 n a + 58028 n a 5 4 3 2 + 17048 a + 3795 n + 24794 n + 62976 n + 77297 n + 45504 n + 10184) n) that certifies it (Check!) and the theorem follows upon contour-integrating with respect to, x, y, z QED! This took, 11450.699, seconds . This took, 11450.775, seconds.