Theorem. Let the column-vector of functions of,n a(n):=, [a[0](n), a[1](n), a[2](n), a[3](n), a[4](n), a[5](n), a[6](n), a[7](n), T a[8](n)] , be defined T by the initial conditions:, a(0) = [1, 0, 0, 0, 0, 0, 0, 0, 0] and a(n+1)=A(n)a(n), where A(n) is the, 9, by , 9, matrix 2 9 (n + 1) 3 (n + 1) (55 n + 88 n + 36) [[------------, -----------------------------, 2 (10 n + 9) 4 (10 n + 9) (5 n + 4) 3 2 3 (n + 1) (275 n + 660 n + 528 n + 140) n -------------------------------------------, - 3 4 (10 n + 7) (5 n + 4) (10 n + 9) 5 4 3 2 2 (37125 n + 134475 n + 194040 n + 139216 n + 49672 n + 7056) (n + 1) /( 7 40 (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9)), - 3 (n + 1) (125125 n 6 5 4 3 2 + 632500 n + 1365320 n + 1630970 n + 1164141 n + 496356 n + 117012 n + 11760) n/(40 (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9)), 3 ( 9 8 7 6 5 4 138875 n + 950125 n + 2775465 n + 4576869 n + 4716646 n + 3159068 n 3 2 2 + 1377656 n + 377688 n + 59128 n + 4032) (n + 1) /(16 (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9)), 3 (n + 1) ( 11 10 9 8 7 7734375 n + 53872500 n + 169623300 n + 318489710 n + 395986591 n 6 5 4 3 2 + 342092388 n + 209392590 n + 90746310 n + 27268260 n + 5406592 n + 636056 n + 33600) n/(16 (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) 12 (10 n + 7) (5 n + 4) (10 n + 9)), (-108864 - 2904336 n + 3190501875 n 13 11 3 5 + 588878125 n + 7173751750 n - 236708528 n - 3132583148 n 6 7 2 4 - 6260781142 n - 7896570440 n - 34350000 n - 1051238640 n 8 9 10 2 - 4663089541 n + 2641771825 n + 7954674200 n ) (n + 1) /(160 (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9)), 15 14 13 - n (n + 1) (24511609375 n + 195702787500 n + 723951662500 n 12 11 10 + 1644511687500 n + 2562568190050 n + 2897726115240 n 9 8 7 6 + 2452703212740 n + 1579612870968 n + 778888377663 n + 293389536196 n 5 4 3 2 + 83518485000 n + 17591642716 n + 2644046688 n + 266374888 n + 15963216 n + 423360)/(160 (1 + 10 n) (5 n + 1) (3 + 10 n) (5 n + 2) 44 (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9))], [- --------, 10 n + 9 2 4 3 2 2 (195 n + 310 n + 126) 2 (825 n + 1815 n + 1190 n + 105 n - 84) - ------------------------, - -------------------------------------------, (10 n + 9) (5 n + 4) (10 n + 7) (5 n + 4) (10 n + 9) 4 3 2 7 (n + 1) (825 n + 2475 n + 2772 n + 1369 n + 252) 7 -----------------------------------------------------, (88275 n (10 n + 7) (5 n + 4) (10 n + 9) 6 5 4 3 2 + 389400 n + 710094 n + 682605 n + 361760 n + 97615 n + 8925 n - 588) 8 /(5 (2 n + 1) (10 n + 7) (5 n + 4) (10 n + 9)), - (n + 1) (122925 n 7 6 5 4 3 + 737055 n + 1862949 n + 2604030 n + 2207370 n + 1163575 n 2 + 372750 n + 66392 n + 5040)/(2 (5 n + 2) (2 n + 1) (10 n + 7) (5 n + 4) 11 10 9 8 (10 n + 9)), - (5696625 n + 36161400 n + 102516810 n + 170739789 n 7 6 5 4 3 + 184763436 n + 135524499 n + 68091401 n + 23059620 n + 5002732 n 2 + 615348 n + 29112 n - 672)/(2 (3 + 10 n) (5 n + 2) (2 n + 1) (10 n + 7) 12 11 (5 n + 4) (10 n + 9)), - (n + 1) (23643125 n + 106734375 n 10 9 8 7 6 + 177782000 n + 78351625 n - 174062779 n - 351174131 n - 321986896 n 5 4 3 2 - 181929289 n - 67608800 n - 16578292 n - 2581278 n - 231108 n - 9072) /(4 (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) (10 n + 7) (5 n + 4) 13 14 11 (10 n + 9)), (-1008 + 157420562750 n + 46338380000 n + 457900655795 n 3 10 9 + 95098300 n + 140060 n + 461896277513 n + 344071587585 n 8 4 2 15 + 191827560705 n + 904160990 n + 5777690 n + 6275121875 n 12 7 6 5 + 325714575725 n + 80217083090 n + 24957374105 n + 5663922793 n )/(20 (1 + 10 n) (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) (10 n + 7) (5 n + 4) 2 385 77 (85 n + 134 n + 54) (10 n + 9))], [--------------------, ------------------------------, 2 (n + 1) (10 n + 9) 4 (5 n + 4) (10 n + 9) (n + 1) 4 3 2 7 (3075 n + 5630 n + 1730 n - 1768 n - 888) ----------------------------------------------, 4 (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) 5 4 3 2 7 (152625 n + 545325 n + 775200 n + 547166 n + 191752 n + 26712) - --------------------------------------------------------------------, - 7 8 (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) 8 7 6 5 4 3 (416625 n + 2060850 n + 4318710 n + 4942510 n + 3297113 n + 1242658 n 2 + 216326 n - 1320 n - 4032)/(8 (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) 9 8 7 (10 n + 9) (n + 1)), (25810125 n + 164981025 n + 457542855 n 6 5 4 3 2 + 723759531 n + 720364958 n + 468076670 n + 198582860 n + 53038600 n + 8091368 n + 537264)/(16 (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) 12 11 10 (5 n + 4) (10 n + 9)), 5 (198994125 n + 1377580050 n + 4303428690 n 9 8 7 6 + 7999382622 n + 9818355261 n + 8340757362 n + 4991674660 n 5 4 3 2 + 2095921878 n + 600275564 n + 109533516 n + 10706976 n + 225744 n - 32256)/(16 (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) 13 12 11 (10 n + 9) (n + 1)), (3021253125 n + 13560050625 n + 18747126750 n 10 9 8 7 - 11628749550 n - 77298139875 n - 128357373639 n - 124283985560 n 6 5 4 3 - 80109403904 n - 35808514880 n - 11162494460 n - 2379889832 n 2 - 330236240 n - 26824992 n - 967680)/(32 (5 n + 1) (3 + 10 n) (5 n + 2) 14 (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9)), - (6781931543750 n 13 11 12 + 15356300905000 n + 26756380371160 n + 23814518484350 n 15 16 7 + 1836414868750 n + 230061734375 n - 48384 + 2589620206410 n 6 9 8 4 + 723821256150 n + 14322704059940 n + 6977105425115 n + 21653294256 n 2 10 3 + 111307552 n + 22463187073140 n + 1942080 n + 2069613856 n 5 + 148897171716 n )/(32 (1 + 10 n) (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) (n + 1))], [ 2 495 55 (145 n + 226 n + 90) - -------------------, - -------------------------------, 2 2 (10 n + 9) (n + 1) 2 (10 n + 9) (5 n + 4) (n + 1) 4 3 2 11 (1525 n + 1380 n - 2502 n - 3554 n - 1152) - ------------------------------------------------, 2 2 (10 n + 9) (5 n + 4) (10 n + 7) (n + 1) 5 4 3 2 1444125 n + 5110875 n + 7188930 n + 5014556 n + 1733894 n + 237888 ----------------------------------------------------------------------, ( 4 (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) 8 7 6 5 4 3436125 n + 16655100 n + 33910890 n + 37125904 n + 22912573 n 3 2 / + 7244024 n + 525722 n - 302106 n - 61992) / (4 (10 n + 9) (5 n + 4) / 2 9 8 (10 n + 7) (5 n + 3) (2 n + 1) (n + 1) ), - 5 (7915875 n + 49213725 n 7 6 5 4 3 + 133335015 n + 206678761 n + 201991318 n + 129036460 n + 53849106 n 2 + 14144668 n + 2120264 n + 138096)/(8 (5 n + 2) (2 n + 1) (5 n + 3) 12 (10 n + 7) (5 n + 4) (10 n + 9) (n + 1)), - (1263136875 n 11 10 9 8 + 8697265500 n + 26986345650 n + 49737139320 n + 60383066187 n 7 6 5 4 + 50563398480 n + 29669735144 n + 12103564576 n + 3307600920 n 3 2 / + 550347056 n + 40579900 n - 1618968 n - 356832) / (8 (3 + 10 n) / 2 (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) ), - 13 12 11 10 (2091684375 n + 5252630625 n - 13546962000 n - 84899334300 n 9 8 7 6 - 186317166915 n - 241109819667 n - 207818531170 n - 125007234414 n 5 4 3 2 - 53299277366 n - 16030667540 n - 3317995836 n - 448284400 n - 35479632 n - 1245888)/(16 (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) 14 (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) (n + 1)), (9093221238750 n 11 15 16 + 35494538984520 n + 2466396487500 n + 309245578125 n 12 3 7 6 + 31736433139950 n + 2395556976 n + 3327189748092 n + 917442895026 n 5 2 8 4 + 185325937074 n + 115366592 n + 9060364130221 n + 26241988888 n 10 9 + 823008 n + 29629649540048 n + 18759955854540 n - 107136 13 / + 20536909181250 n ) / (16 (1 + 10 n) (5 n + 1) (3 + 10 n) (5 n + 2) / 2 (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) )], [ 2 825 825 (15 n + 23 n + 9) -------------------, -------------------------------, 3 3 (10 n + 9) (n + 1) 2 (10 n + 9) (5 n + 4) (n + 1) 4 3 2 275 (25 n - 155 n - 463 n - 394 n - 108) -------------------------------------------, 3 2 (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) 5 4 3 2 11 (227875 n + 796400 n + 1104510 n + 758119 n + 257258 n + 34524) - ----------------------------------------------------------------------, - 2 4 (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) 8 7 6 5 4 (4994625 n + 23388725 n + 45216015 n + 45337626 n + 23220025 n 3 2 / + 3500173 n - 2157571 n - 1109826 n - 155736) / (4 (2 n + 1) (5 n + 3) / 3 8 7 (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) ), 5 (15621375 n + 79042425 n 6 5 4 3 2 + 171638770 n + 208878843 n + 155710209 n + 72720836 n + 20744132 n + 3297964 n + 223272)/(8 (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) 12 11 (5 n + 4) (10 n + 9) (n + 1)), 5 (408106875 n + 2786631375 n 10 9 8 7 + 8557204425 n + 15566009170 n + 18580371487 n + 15209421431 n 6 5 4 3 2 + 8642014393 n + 3354310866 n + 838294784 n + 112163058 n + 859040 n / - 1870296 n - 181440) / (8 (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) / 3 13 (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) ), - (264928125 n 12 11 10 9 + 16614510000 n + 100492326000 n + 285048379275 n + 487013279525 n 8 7 6 5 + 554142773048 n + 440750195890 n + 250553383951 n + 102286274464 n 4 3 2 + 29673622030 n + 5946381524 n + 778662920 n + 59648928 n + 2020032) / / (16 (10 n + 9) (5 n + 4) (10 n + 7) (5 n + 3) (2 n + 1) (5 n + 2) / 2 16 15 (3 + 10 n) (5 n + 1) (n + 1) ), - (535660640625 n + 4264286503125 n 14 13 10 + 15677763151875 n + 35271967500000 n + 49927544207352 n 3 5 6 7 + 3302796312 n + 287687783174 n + 1462662646893 n + 5407630782783 n 2 4 8 - 2364576 n + 125892032 n + 39095932944 n + 14945172022891 n 9 11 12 + 31311303896610 n + 60276905725420 n - 273024 + 54235060397100 n ) / / (16 (1 + 10 n) (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) / 3 924 (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) )], [- -------------------, 4 (10 n + 9) (n + 1) 2 66 (95 n + 142 n + 54) - -----------------------------, 4 (10 n + 9) (5 n + 4) (n + 1) 4 3 2 33 (300 n + 2045 n + 3716 n + 2645 n + 660) ----------------------------------------------, 4 (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) 5 4 3 2 11 (653500 n + 2244275 n + 3050835 n + 2045509 n + 674753 n + 87444) ------------------------------------------------------------------------, 3 10 (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) 8 7 6 5 4 11 (1010500 n + 4418375 n + 7600580 n + 5919845 n + 923876 n 3 2 / - 1901263 n - 1511660 n - 469017 n - 55188) / (10 (2 n + 1) (5 n + 3) / 4 9 8 (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) ), - (51220500 n + 302675375 n 7 6 5 4 + 782690985 n + 1161037486 n + 1087207426 n + 665278357 n 3 2 / + 265414273 n + 66373814 n + 9405184 n + 572544) / (4 (5 n + 2) / 3 (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) ), - ( 12 11 10 9 1071262500 n + 7215429375 n + 21782648700 n + 38770404065 n 8 7 6 5 + 44964287004 n + 35357956829 n + 18907473716 n + 6607366289 n 4 3 2 / + 1302610838 n + 42255990 n - 44051566 n - 9909132 n - 716688) / (4 / (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) 4 13 12 11 (n + 1) ), (13914037500 n + 134811538125 n + 549494430375 n 10 9 8 + 1285125136025 n + 1951636827275 n + 2047347314983 n 7 6 5 4 + 1532080224735 n + 829306365471 n + 324673499349 n + 90669465820 n 3 2 / + 17509392214 n + 2205812840 n + 161751768 n + 5195232) / (40 / (10 n + 9) (5 n + 4) (10 n + 7) (5 n + 3) (2 n + 1) (5 n + 2) (3 + 10 n) 3 12 16 (5 n + 1) (n + 1) ), (152270250279650 n + 1537608187500 n 15 11 13 + 12196137628125 n + 167661764397520 n + 99785049106875 n 8 10 9 + 39827294509510 n + 137300163387510 n - 20188032 n + 84908796639548 n 6 7 2 4 + 3699273632926 n + 14092628091387 n + 91992440 n + 88017859236 n 3 5 14 / + 6321813472 n + 696603527297 n - 1081728 + 44624587837500 n ) / (40 / (1 + 10 n) (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) 4 693 (5 n + 4) (10 n + 9) (n + 1) )], [-------------------, 5 (10 n + 9) (n + 1) 2 231 (35 n + 50 n + 18) -------------------------------, 5 2 (10 n + 9) (5 n + 4) (n + 1) 4 3 2 33 (1175 n + 5310 n + 8130 n + 5240 n + 1224) - ------------------------------------------------, 5 2 (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) 5 4 3 2 33 (64875 n + 216975 n + 285930 n + 184606 n + 58042 n + 7056) - ------------------------------------------------------------------, - 11 4 4 (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) 8 7 6 5 4 3 (969125 n + 3469850 n + 3528460 n - 2168004 n - 8142355 n - 8011190 n 2 / - 3975080 n - 1018110 n - 107352) / (20 (2 n + 1) (5 n + 3) (10 n + 7) / 5 9 8 7 (5 n + 4) (10 n + 9) (n + 1) ), 11 (8018875 n + 46114775 n + 116056155 n 6 5 4 3 2 + 167451761 n + 152308978 n + 90306666 n + 34768834 n + 8336276 n / + 1120392 n + 63504) / (8 (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) / 4 12 11 (5 n + 4) (10 n + 9) (n + 1) ), (1411918125 n + 9277661250 n 10 9 8 7 + 27140973000 n + 46339017800 n + 50702503861 n + 36479142414 n 6 5 4 3 + 16666283974 n + 3980145008 n - 183232080 n - 468259852 n 2 / - 152311508 n - 22997064 n - 1409184) / (8 (3 + 10 n) (5 n + 2) / 5 (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) ), - 7 ( 13 12 11 10 1294115625 n + 10377613125 n + 37235266500 n + 79307075350 n 9 8 7 6 + 111961529955 n + 110622492505 n + 78627281062 n + 40635392116 n 5 4 3 2 + 15229550230 n + 4072578072 n + 751225828 n + 89825568 n + 6173136 n / + 181440) / (16 (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) / 4 14 (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) ), - (65590238505000 n 13 16 15 + 145153298453750 n + 2295438234375 n + 18080228718750 n 12 11 4 + 218747050868950 n + 237243999022400 n - 2132352 + 81772996360 n 9 10 + 115301479554280 n + 190738461575912 n - 54829920 n 8 3 5 6 + 52542869934875 n + 3139675520 n + 768472554082 n + 4449069086960 n 7 2 / + 17893945771810 n - 381766880 n ) / (80 (1 + 10 n) (5 n + 1) / (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) 2 5 330 33 (45 n + 58 n + 18) (n + 1) )], [- -------------------, - -----------------------------, 6 6 (10 n + 9) (n + 1) (10 n + 9) (5 n + 4) (n + 1) 4 3 2 33 (475 n + 1840 n + 2550 n + 1526 n + 336) ----------------------------------------------, 6 (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) 5 4 3 2 33 (14375 n + 45925 n + 57210 n + 34336 n + 9742 n + 1008) 8 --------------------------------------------------------------, 33 (3375 n 5 2 (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) 7 6 5 4 3 2 - 12400 n - 106860 n - 254346 n - 310269 n - 219616 n - 91444 n / - 20856 n - 2016) / (2 (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) / 6 9 8 7 (10 n + 9) (n + 1) ), - 11 (2105375 n + 11705325 n + 28430165 n 6 5 4 3 2 + 39482573 n + 34425254 n + 19444220 n + 7061040 n + 1570400 n / + 189864 n + 9072) / (4 (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) / 5 12 11 (5 n + 4) (10 n + 9) (n + 1) ), - 11 (23186875 n + 144177000 n 10 9 8 7 6 + 391237600 n + 598086830 n + 544178755 n + 263396744 n + 4591186 n 5 4 3 2 - 86053258 n - 62688620 n - 23801880 n - 5331776 n - 668064 n - 36288) / / (4 (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) / 6 13 12 11 (10 n + 9) (n + 1) ), (3526321875 n + 26029240625 n + 86990955250 n 10 9 8 7 + 174047635100 n + 232195283175 n + 217658136561 n + 147095952240 n 6 5 4 3 + 72311782138 n + 25736949716 n + 6506814160 n + 1124582208 n 2 / + 123809400 n + 7558128 n + 181440) / (8 (5 n + 1) (3 + 10 n) (5 n + 2) / 5 (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) ), (-3810144 n 3 12 15 14 - 251885408 n + 9011381280650 n + 790439100000 n + 2820427360000 n 13 16 11 + 6120777662500 n + 101775609375 n + 9501552785040 n - 120960 5 6 7 2 + 16057777740 n + 123993254428 n + 572936643776 n - 47757296 n 4 8 9 10 + 432256688 n + 1832107536631 n + 4266063711488 n + 7377502964644 n ) / / (8 (1 + 10 n) (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) / 6 165 (10 n + 7) (5 n + 4) (10 n + 9) (n + 1) )], [---------------------, 7 2 (10 n + 9) (n + 1) 3 2 165 n 33 (135 n + 364 n + 308 n + 84) ---------------------, - ---------------------------------, 7 7 4 (10 n + 9) (n + 1) 4 (10 n + 9) (10 n + 7) (n + 1) 2 231 (65 n + 96 n + 36) n - --------------------------------, 6 8 (10 n + 9) (10 n + 7) (n + 1) 6 5 4 3 2 33 (505 n + 3213 n + 7313 n + 8123 n + 4764 n + 1416 n + 168) ------------------------------------------------------------------, 7 8 (2 n + 1) (10 n + 7) (10 n + 9) (n + 1) 5 4 3 2 165 (1125 n + 3910 n + 5391 n + 3684 n + 1248 n + 168) n 9 ------------------------------------------------------------, 11 (85655 n 6 16 (2 n + 1) (10 n + 7) (10 n + 9) (n + 1) 8 7 6 5 4 3 + 260037 n + 112465 n - 561881 n - 1119894 n - 996640 n - 501514 n 2 / - 146400 n - 23048 n - 1512) / (16 (3 + 10 n) (2 n + 1) (10 n + 7) / 7 8 7 6 (10 n + 9) (n + 1) ), - 11 (713065 n + 3531281 n + 7566657 n 5 4 3 2 + 9150365 n + 6821374 n + 3206070 n + 926968 n + 150752 n + 10584) n / 6 / (32 (3 + 10 n) (2 n + 1) (10 n + 7) (10 n + 9) (n + 1) ), - ( / 12 9 10 127282575 n - 5040 + 2354526618 n + 1696146151 n - 136496 n 3 5 6 7 2 - 8053788 n + 39264318 n + 359133909 n + 1122166002 n - 1510936 n 4 8 11 / - 16672392 n + 2044354749 n + 700945730 n ) / (32 (1 + 10 n) / 7 (3 + 10 n) (2 n + 1) (10 n + 7) (10 n + 9) (n + 1) )]] T In other words, a(n)=A(n-1)A(n-2)...A(0) times, [1, 0, 0, 0, 0, 0, 0, 0, 0] Let 10 2 3 F(n, k) = binomial(n, k) (a[0](n) + a[1](n) k + a[2](n) k + a[3](n) k 4 5 6 7 8 + a[4](n) k + a[5](n) k + a[6](n) k + a[7](n) k + a[8](n) k ) We have, for every non-negative integer n ----- \ ) F(n, k) = 1 / ----- k Proof: We cleverly construct 10 / 10 6 G(n, k) = binomial(n, k) |-1/160 a[0](n) k (443520 + 10995600000 n \ 7 2 4 8 + 9966000000 n + 141990640 n + 3209514000 n + 5060000000 n 9 8 7 6 + 1100000000 n - 200000000 n k - 720000000 n k - 1092000000 n k 5 4 3 2 - 907200000 n k - 448980000 n k - 134568000 n k - 23624800 n k 3 5 / - 2191680 n k + 12497760 n - 80640 k + 870060400 n + 7458990000 n ) / ( / (1 + 10 n) (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) 10 10 (5 n + 4) (10 n + 9) (n + 1 - k) ) - 1/160 a[1](n) k (-997920 3 5 6 - 28757520 n + 544320 k - 2162680520 n - 21451969000 n - 35643685000 n 7 2 2 4 - 38585800000 n - 337479120 n - 90720 k - 8482045000 n 8 9 10 8 - 26125000000 n - 10010000000 n - 1650000000 n + 5280000000 n k 7 6 5 4 + 10782000000 n k + 12264000000 n k + 8534190000 n k + 3750432000 n k 3 2 2 7 + 1034330800 n k + 171129600 n k + 15222240 n k - 740000000 k n 9 2 2 2 2 8 + 1100000000 k n - 26251200 k n - 2453040 k n - 200000000 k n 2 5 2 4 2 3 2 6 - 971600000 k n - 488180000 k n - 148106000 k n - 1148000000 k n ) / / ((1 + 10 n) (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) / 10 10 (10 n + 7) (5 n + 4) (10 n + 9) (n + 1 - k) ) - 1/160 a[2](n) k (950400 11 3 5 + 27086400 n - 1520640 k - 825000000 n + 1955049360 n + 17671260200 n 6 7 2 2 + 26711135000 n + 24147695000 n + 312623520 n + 673920 k 4 8 9 10 + 7393657040 n + 10391975000 n - 1053250000 n - 2887500000 n 3 8 7 6 - 103680 k - 30695500000 n k - 48092750000 n k - 46650760000 n k 5 4 3 2 - 29237558000 n k - 11953475200 n k - 3132061600 n k - 499524000 n k 2 7 9 2 2 - 43289280 n k + 11644000000 k n - 10945000000 k n + 207576000 k n 2 2 8 2 5 2 4 + 18685440 k n + 5500000000 k n + 9773890000 k n + 4395160000 k n 2 3 2 6 3 8 3 7 + 1235471600 k n + 13669000000 k n - 200000000 k n - 760000000 k n 3 6 3 5 10 3 4 - 1208000000 k n - 1044400000 k n - 1650000000 k n - 534380000 k n 3 3 3 2 3 2 9 / - 164584000 k n - 29527200 k n - 2784960 k n + 1100000000 k n ) / / ((1 + 10 n) (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) 10 10 (5 n + 4) (10 n + 9) (n + 1 - k) ) - 1/160 a[3](n) k (-399168 12 11 3 - 9633888 n + 2419200 k + 2227500000 n + 16689750000 n - 196308728 n 5 6 7 2 + 10115170604 n + 37059199760 n + 79801165400 n - 79864752 n 2 4 8 9 - 2237760 k + 1091864048 n + 109609841000 n + 97081022500 n 10 3 4 8 + 53573025000 n + 846720 k - 120960 k + 39650875000 n k 7 6 5 + 69609945000 n k + 70710885000 n k + 45292756200 n k 4 3 2 + 18730676400 n k + 4940320800 n k + 791164400 n k + 68748000 n k 2 7 9 2 2 2 - 58605500000 k n + 9561750000 k n - 714200000 k n - 62934720 k n 2 8 2 5 2 4 - 35508000000 k n - 38629577000 k n - 16297206200 k n 2 3 2 6 3 8 - 4383585800 k n - 59390135000 k n + 5720000000 k n 3 7 3 6 3 5 + 12552000000 k n + 15225000000 k n + 11211690000 k n 10 3 4 3 3 3 2 - 1237500000 k n + 5173938000 k n + 1486922800 k n + 254424400 k n 3 2 9 4 2 4 + 23233440 k n - 11880000000 k n - 33722800 k n - 3220320 k n 4 4 3 9 4 3 4 7 - 589380000 k n + 1100000000 k n - 185022000 k n - 780000000 k n 4 6 4 5 4 8 11 - 1272000000 k n - 1126800000 k n - 200000000 k n - 825000000 k n 2 10 / - 1650000000 k n ) / ((1 + 10 n) (5 n + 1) (3 + 10 n) (5 n + 2) / 10 (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) (n + 1 - k) ) - 1/160 10 12 a[4](n) k (532224 + 14636160 n - 2927232 k + 24255000000 n 13 11 3 5 + 3753750000 n + 68120662500 n + 961263336 n + 9264282236 n 6 7 2 2 + 20203655120 n + 42190384940 n + 161295024 n + 4717440 k 4 8 9 10 + 3556780304 n + 78694913000 n + 111959721500 n + 109392525000 n 3 4 5 8 7 - 3265920 k + 1088640 k - 145152 k + 51267876000 n k - 6993036400 n k 6 5 4 3 - 44228037240 n k - 39689843944 n k - 18996720480 n k - 5422202720 n k 2 2 7 9 - 910743120 n k - 81583200 n k + 124502995000 k n + 74673115000 k n 2 2 2 2 8 + 1506626880 k n + 132685920 k n + 73040275000 k n 2 5 2 4 2 3 + 82142705200 k n + 34524475960 k n + 9261069160 k n 2 6 3 8 3 7 + 126672610000 k n - 40562500000 k n - 70203450000 k n 3 6 3 5 10 - 74146810000 k n - 50036016000 k n + 49811025000 k n 3 4 3 3 3 2 3 - 21812368000 k n - 6037433680 k n - 1007740560 k n - 90542880 k n 2 9 4 2 4 4 4 + 20946750000 k n + 316749360 k n + 29466720 k n + 6127806000 k n 3 9 4 3 4 7 - 12815000000 k n + 1809024400 k n + 13506000000 k n 4 6 4 5 4 8 + 16946400000 k n + 12887190000 k n + 5940000000 k n 11 2 10 5 8 5 7 + 16681500000 k n + 412500000 k n - 200000000 k n - 800000000 k n 5 3 5 2 5 6 5 5 - 210920000 k n - 39276000 k n - 1340000000 k n - 1220000000 k n 5 4 12 5 4 9 - 655460000 k n + 2227500000 k n - 3816000 k n + 1100000000 k n 3 10 2 11 / - 1650000000 k n - 825000000 k n ) / ((1 + 10 n) (5 n + 1) / (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) 10 10 (n + 1 - k) ) - 1/160 a[5](n) k (-997920 - 30309840 n + 3689280 k 12 13 14 11 - 106920618750 n - 23421750000 n - 2083125000 n - 275326425000 n 3 5 6 7 - 2876898200 n - 50178519160 n - 134654036770 n - 278594386840 n 2 2 4 8 - 388629120 n - 7953120 k - 14147232220 n - 442277963710 n 9 10 3 4 - 527505585200 n - 457461270750 n + 8467200 k - 4838400 k 5 6 8 7 + 1451520 k - 181440 k + 38559435200 n k + 64301443780 n k 6 5 4 3 + 75878122540 n k + 54625722880 n k + 24499627600 n k + 6846475800 n k 2 2 7 9 + 1143961040 n k + 102559200 n k - 142471618200 k n + 38749084000 k n 2 2 2 2 8 - 2479444400 k n - 221450640 k n - 36381141500 k n 2 5 2 4 2 3 - 123810345540 k n - 54287398300 k n - 14955472000 k n 2 6 3 8 3 7 - 176181026080 k n + 110826375000 k n + 190124245000 k n 3 6 3 5 10 + 197224960000 k n + 131087345200 k n + 55039462500 k n 3 4 3 3 3 2 + 56594229400 k n + 15596017400 k n + 2602018000 k n 3 2 9 4 2 4 + 234192000 k n + 42241787500 k n - 1426510000 k n - 131534400 k n 4 4 3 9 4 3 - 28926584600 k n + 33101750000 k n - 8286376200 k n 4 7 4 6 4 5 - 82966000000 k n - 91240385000 k n - 63973105000 k n 4 8 11 2 10 - 45859000000 k n + 47939512500 k n + 44565675000 k n 5 8 5 7 5 3 + 6160000000 k n + 14506000000 k n + 2233288400 k n 5 2 5 6 5 5 + 403388640 k n + 18847600000 k n + 14845990000 k n 5 4 12 5 4 9 + 7311724000 k n + 21194250000 k n + 38522880 k n - 13750000000 k n 3 10 2 11 6 6 7 + 2062500000 k n + 16673250000 k n - 4679280 k n - 820000000 k n 6 6 6 5 6 4 6 3 - 1412000000 k n - 1325200000 k n - 735380000 k n - 244498000 k n 6 2 4 10 3 11 6 8 - 46936800 k n - 1650000000 k n - 825000000 k n - 200000000 k n 13 2 12 5 9 / + 3753750000 k n + 2227500000 k n + 1100000000 k n ) / ((1 + 10 n) / (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) 10 10 (10 n + 9) (n + 1 - k) ) - 1/160 a[6](n) k (887040 + 24842400 n 12 13 15 - 4878720 k - 746934168750 n - 342320137500 n - 11601562500 n 14 11 3 5 - 93807656250 n - 1087830892500 n + 1636621800 n + 5673654140 n 6 7 2 2 - 26124973600 n - 153779024080 n + 278314080 n + 12579840 k 4 8 9 10 + 5217973640 n - 435588228130 n - 824544508480 n - 1115783154750 n 3 4 5 6 7 - 18023040 k + 15120000 k - 7499520 k + 2056320 k - 241920 k 8 7 6 - 332342508740 n k - 250751273760 n k - 169244573120 n k 5 4 3 2 - 91654845600 n k - 35869928000 n k - 9406145520 n k - 1529216880 n k 2 7 9 - 135823200 n k + 227006969500 k n - 395864327100 k n 2 2 2 2 8 + 3806164000 k n + 345461280 k n + 96513476400 k n 2 5 2 4 2 3 + 183924394240 k n + 81090116800 k n + 22607423800 k n 2 6 3 8 3 7 + 264046825780 k n - 157159574000 k n - 336738026000 k n 3 6 3 5 10 - 373774315560 k n - 255853361480 k n - 378115452000 k n 3 4 3 3 3 2 - 112560579600 k n - 31556332000 k n - 5360294800 k n 3 2 9 4 2 4 - 491088480 k n + 8004414000 k n + 4355966800 k n + 406370400 k n 4 4 3 9 4 3 + 87701220000 k n - 834625000 k n + 25113091200 k n 4 7 4 6 4 5 + 267892995000 k n + 285544095000 k n + 195990070200 k n 4 8 11 2 10 + 153275375000 k n - 254762475000 k n + 10722937500 k n 5 8 5 7 5 3 - 51397500000 k n - 96972550000 k n - 11447362720 k n 5 2 5 6 5 5 - 2058837280 k n - 111025060000 k n - 81084094000 k n 5 4 12 5 - 38248036000 k n - 108017662500 k n - 197563200 k n 4 9 3 10 2 11 + 46026750000 k n + 37836975000 k n + 28683187500 k n 6 6 7 6 6 + 52830240 k n + 15552000000 k n + 20943000000 k n 6 5 6 4 6 3 + 17139690000 k n + 8797692000 k n + 2808614800 k n 6 2 4 10 3 11 + 530749600 k n + 3712500000 k n + 16665000000 k n 6 8 13 2 12 + 6380000000 k n - 24632437500 k n + 18133500000 k n 5 9 2 13 14 - 14685000000 k n + 3753750000 k n - 2083125000 k n 6 9 7 4 7 3 7 2 + 1100000000 k n - 832380000 k n - 288936000 k n - 58055200 k n 7 7 8 7 7 7 6 - 6037440 k n - 200000000 k n - 840000000 k n - 1488000000 k n 7 5 4 11 3 12 - 1443600000 k n - 825000000 k n + 2227500000 k n 5 10 / - 1650000000 k n ) / ((1 + 10 n) (5 n + 1) (3 + 10 n) (5 n + 2) / 10 (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) (10 n + 9) (n + 1 - k) ) - 1/160 10 16 a[7](n) k (-798336 - 20344896 n + 6410880 k - 5888781250 n 12 13 15 14 + 337848350675 n + 32414683125 n - 32669896875 n - 57916313125 n 11 3 5 6 + 744910474935 n - 817898136 n + 14091482416 n + 81373801388 n 7 2 2 4 + 260136487072 n - 194287104 n - 20200320 k - 304799044 n 8 9 10 3 + 563852947823 n + 874957823047 n + 974232689661 n + 36408960 k 4 5 6 7 8 - 40763520 k + 29030400 k - 12882240 k + 3265920 k - 362880 k 8 7 6 - 77168471490 n k + 87081032480 n k + 129609628160 n k 5 4 3 2 + 90815767580 n k + 39538114640 n k + 10977124680 n k + 1862915840 n k 2 7 9 + 172289760 n k - 429729014600 k n - 329546266720 k n 2 2 2 2 8 - 5714939600 k n - 536432640 k n - 338604564650 k n 2 5 2 4 2 3 - 271631802800 k n - 117988152100 k n - 33163653800 k n 2 6 3 8 3 7 - 414547555150 k n + 279451387600 k n + 580624950500 k n 3 6 3 5 10 + 644944926920 k n + 445050255860 k n - 570099642750 k n 3 4 3 3 3 2 + 198726042200 k n + 57016292200 k n + 10004545600 k n 3 2 9 4 2 + 954498720 k n - 281762052000 k n - 10816864000 k n 4 4 4 3 9 - 1052286240 k n - 206464635400 k n + 28077626500 k n 4 3 4 7 4 6 - 60395637400 k n - 601622823800 k n - 656298335680 k n 4 5 4 8 11 - 455359204340 k n - 315182604000 k n - 661960942500 k n 2 10 5 8 5 7 - 277828823250 k n + 200653475000 k n + 359350445000 k n 5 3 5 2 5 6 + 39877761680 k n + 7357964640 k n + 395423045000 k n 5 5 5 4 12 + 282003870200 k n + 132037586240 k n - 531402093750 k n 5 4 9 3 10 + 734774400 k n - 55177787500 k n - 22407000000 k n 2 11 6 6 7 - 220609125000 k n - 318318480 k n - 112302500000 k n 6 6 6 5 6 4 - 133889635000 k n - 102158033000 k n - 50629712200 k n 6 3 6 2 4 10 - 16043838600 k n - 3082884000 k n + 29624925000 k n 3 11 6 8 13 + 10351687500 k n - 57178000000 k n - 279395737500 k n 2 12 5 9 2 13 - 106487906250 k n + 59721750000 k n - 25843125000 k n 14 6 9 7 4 - 85805156250 k n - 15620000000 k n + 10677870000 k n 7 3 7 2 7 7 8 + 3609451600 k n + 730050000 k n + 78481440 k n + 6600000000 k n 7 7 7 6 7 5 + 16644000000 k n + 23247000000 k n + 19825890000 k n 4 11 3 12 5 10 + 16656750000 k n + 15072750000 k n + 5362500000 k n 5 11 8 8 8 7 8 6 - 825000000 k n - 200000000 k n - 860000000 k n - 1568000000 k n 6 10 7 9 15 - 1650000000 k n + 1100000000 k n - 11601562500 k n 2 14 8 5 8 4 8 3 - 2083125000 k n - 1576400000 k n - 950180000 k n - 348614000 k n 8 2 8 3 13 4 12 - 75013200 k n - 8451360 k n + 3753750000 k n + 2227500000 k n ) / / ((1 + 10 n) (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) / 10 10 (10 n + 7) (5 n + 4) (10 n + 9) (n + 1 - k) ) - 1/160 a[8](n) k (1178496 17 16 + 27968160 n - 10188288 k + 24511609375 n + 226103178125 n 12 13 15 + 5461985903990 n + 4306046987550 n + 955163137500 n 14 11 3 5 + 2455921448750 n + 5183368535180 n + 1691678384 n + 25194419932 n 6 7 2 2 + 89781659066 n + 308314090216 n + 284417936 n + 38638080 k 4 8 9 10 + 7059911936 n + 900900668349 n + 2087196652871 n + 3749476712520 n 3 4 5 6 7 - 85294080 k + 120960000 k - 114283008 k + 71971200 k - 29151360 k 8 8 7 6 + 6894720 k + 217082234270 n k - 20303489160 n k - 113425858980 n k 5 4 3 2 - 93330466932 n k - 43658505360 n k - 12837863840 n k - 2330634080 n k 2 7 9 - 236893440 n k + 487707618240 k n + 513143640680 k n 2 2 2 2 8 + 8559511520 k n + 882405120 k n + 231687377680 k n 2 5 2 4 2 3 + 358588160340 k n + 160174298640 k n + 46787502640 k n 2 6 3 8 3 7 + 527262258150 k n - 577177456240 k n - 989437064280 k n 3 6 3 5 10 - 1050200478960 k n - 720417482820 k n + 710393426688 k n 3 4 3 3 3 2 - 326678833200 k n - 97118261120 k n - 18110702240 k n 3 2 9 4 2 - 1905640320 k n - 50832857020 k n + 24471124640 k n 4 4 4 3 9 + 2636607360 k n + 422029818000 k n - 236155869400 k n 4 3 4 7 4 6 + 128255264800 k n + 1187459676780 k n + 1300849052880 k n 4 5 4 8 11 + 911401996380 k n + 617996948800 k n + 680468522800 k n 2 10 5 8 5 7 - 239830453500 k n - 514858234000 k n - 950859255600 k n 5 3 5 2 5 6 - 111447018560 k n - 21867555200 k n - 1048362064440 k n 5 5 5 4 12 - 750206646696 k n - 356650820640 k n + 432486101300 k n 5 4 9 3 10 - 2422485120 k n + 107827296500 k n - 169815327000 k n 2 11 6 6 7 - 347781967500 k n + 1478316000 k n + 466159695000 k n 6 6 6 5 6 4 + 531341070000 k n + 395910871200 k n + 195905693400 k n 6 3 6 2 4 10 + 63562487800 k n + 12915010400 k n - 43200300000 k n 3 11 6 8 13 - 174394275000 k n + 253226875000 k n + 144079127500 k n 2 12 5 9 2 13 - 346085850000 k n - 121409365000 k n - 219746175000 k n 14 6 9 7 4 - 7572317500 k n + 74186750000 k n - 67245859200 k n 7 3 7 2 7 - 22927998800 k n - 4862390000 k n - 578125920 k n 7 8 7 7 7 6 - 63200500000 k n - 129035250000 k n - 160257510000 k n 7 5 4 11 3 12 - 128148552000 k n - 7054987500 k n - 102331350000 k n 5 10 5 11 8 8 + 19929525000 k n + 16648500000 k n + 6820000000 k n 8 7 8 6 6 10 + 17782000000 k n + 25774000000 k n + 7012500000 k n 7 9 15 2 14 - 16555000000 k n - 23907125000 k n - 77802656250 k n 8 5 8 4 8 3 + 22968190000 k n + 13067698000 k n + 4745898800 k n 8 2 8 3 13 + 1059404400 k n + 131545440 k n - 27053812500 k n 4 12 7 10 3 14 + 12012000000 k n - 1650000000 k n - 2083125000 k n 4 13 2 15 9 9 4 + 3753750000 k n - 11601562500 k n - 725760 k - 1092980000 k n 9 9 8 9 7 9 6 - 13273920 k n - 200000000 k n - 880000000 k n - 1652000000 k n 9 5 9 3 9 2 6 11 - 1724800000 k n - 429352000 k n - 101800800 k n - 825000000 k n 8 9 5 12 16 / + 1100000000 k n + 2227500000 k n - 5888781250 k n ) / ((1 + 10 n) / (5 n + 1) (3 + 10 n) (5 n + 2) (2 n + 1) (5 n + 3) (10 n + 7) (5 n + 4) 10 \ (10 n + 9) (n + 1 - k) )| / with the motive that F(n + 1, k) - F(n, k) = G(n, k + 1) - G(n, k) (check!) and the identity follows upon summing with respect to k. QED