Theorem. Let the column-vector of functions of,n T a(n):=, [a[0](n), a[1](n), a[2](n), a[3](n), a[4](n), a[5](n), a[6](n)] , be defined T by the initial conditions:, a(0) = [1, 0, 0, 0, 0, 0, 0] and a(n+1)=A(n)a(n), where A(n) is the, 7, by , 7, matrix 2 7 (n + 1) 7 (n + 1) (12 n + 18 n + 7) [[-----------, ----------------------------, 2 (7 + 8 n) 4 (4 n + 3) (7 + 8 n) 3 2 7 (n + 1) (48 n + 108 n + 81 n + 20) n ----------------------------------------, 4 (5 + 8 n) (4 n + 3) (7 + 8 n) 4 3 2 2 7 (144 n + 408 n + 432 n + 201 n + 35) (n + 1) - --------------------------------------------------, - 7 (n + 1) 8 (5 + 8 n) (4 n + 3) (7 + 8 n) 6 5 4 3 2 (1056 n + 4296 n + 7236 n + 6450 n + 3204 n + 839 n + 90) n/(4 8 7 6 (5 + 8 n) (7 + 8 n) (4 n + 3) (3 + 8 n)), - (960 n - 10944 n - 49428 n 5 4 3 2 2 - 82191 n - 71739 n - 36000 n - 10420 n - 1618 n - 105) (n + 1) /(4 (3 + 8 n) (4 n + 3) (7 + 8 n) (1 + 4 n) (5 + 8 n)), n (3 n + 2) (n + 1) ( 9 8 7 6 5 4 259584 n + 1223808 n + 2528448 n + 2994936 n + 2231880 n + 1079421 n 3 2 + 336644 n + 64767 n + 6886 n + 300)/(8 (3 + 8 n) (4 n + 3) (8 n + 1) 2 27 308 n + 456 n + 175 (7 + 8 n) (1 + 4 n) (5 + 8 n))], [- -------, - ---------------------, 7 + 8 n 2 (7 + 8 n) (4 n + 3) 4 3 2 3 (336 n + 672 n + 380 n - 35) - ---------------------------------, 2 (5 + 8 n) (7 + 8 n) (4 n + 3) 4 3 2 (13 n + 7) (n + 1) (336 n + 924 n + 945 n + 422 n + 70) 8 ----------------------------------------------------------, (55104 n 2 (5 + 8 n) (7 + 8 n) (1 + 2 n) (4 n + 3) 7 6 5 4 3 2 + 249312 n + 478548 n + 503580 n + 312020 n + 112388 n + 20902 n + 1162 n - 105)/(2 (5 + 8 n) (7 + 8 n) (1 + 2 n) (4 n + 3) (3 + 8 n)), - 9 8 7 6 5 (n + 1) (21888 n + 217440 n + 700776 n + 1144116 n + 1099530 n 4 3 2 + 659622 n + 249761 n + 57945 n + 7522 n + 420)/(2 (3 + 8 n) (4 n + 3) 12 9 (1 + 2 n) (7 + 8 n) (1 + 4 n) (5 + 8 n)), - (3122688 n + 76100736 n 10 11 3 6 7 + 48571104 n + 18365952 n + 327916 n + 27425530 n + 55418628 n 8 2 5 4 + 78310614 n + 27088 n + 9447496 n - 15 + 2204926 n + 802 n)/(2 (3 + 8 n) (4 n + 3) (8 n + 1) (1 + 2 n) (7 + 8 n) (1 + 4 n) (5 + 8 n))], [ 2 90 15 (64 n + 93 n + 35) -----------------, -----------------------------, (n + 1) (7 + 8 n) 2 (7 + 8 n) (4 n + 3) (n + 1) 4 3 2 5 (448 n + 652 n - 34 n - 403 n - 147) -----------------------------------------, 2 (5 + 8 n) (7 + 8 n) (4 n + 3) (n + 1) 5 4 3 2 3 (10752 n + 34776 n + 44572 n + 28174 n + 8786 n + 1085) - -------------------------------------------------------------, - ( 4 (5 + 8 n) (7 + 8 n) (1 + 2 n) (4 n + 3) 8 7 6 5 4 3 86016 n + 382536 n + 716898 n + 728095 n + 425685 n + 136906 n 2 + 18276 n - 1127 n - 420)/((3 + 8 n) (4 n + 3) (1 + 2 n) (7 + 8 n) 9 8 7 6 (5 + 8 n) (n + 1)), 3 (129024 n + 856128 n + 2306976 n + 3395112 n 5 4 3 2 + 3043300 n + 1733324 n + 629008 n + 140476 n + 17572 n + 945)/(4 (3 + 8 n) (4 n + 3) (1 + 2 n) (7 + 8 n) (1 + 4 n) (5 + 8 n)), ( 10 9 11 3 5 166017888 n + 258737016 n + 62985600 n + 969623 n + 30432293 n 6 7 8 2 4 + 90201275 n + 184980348 n + 264206748 n + 70499 n + 6883406 n 12 + 890 n + 10727424 n - 120)/(2 (3 + 8 n) (4 n + 3) (8 n + 1) (1 + 2 n) 168 (7 + 8 n) (1 + 4 n) (5 + 8 n) (n + 1))], [- ------------------, 2 (7 + 8 n) (n + 1) 2 12 (68 n + 96 n + 35) - ----------------------------, 2 (7 + 8 n) (4 n + 3) (n + 1) 4 3 2 3 (288 n - 228 n - 1449 n - 1318 n - 357) - --------------------------------------------, 2 (5 + 8 n) (7 + 8 n) (4 n + 3) (n + 1) 5 4 3 2 32704 n + 103496 n + 129402 n + 79474 n + 23939 n + 2835 8 ------------------------------------------------------------, (142464 n 2 (5 + 8 n) (7 + 8 n) (1 + 2 n) (4 n + 3) (n + 1) 7 6 5 4 3 2 + 613872 n + 1099244 n + 1038816 n + 531124 n + 119528 n - 8889 n / - 9722 n - 1365) / ((5 + 8 n) (7 + 8 n) (1 + 2 n) (4 n + 3) (3 + 8 n) / 2 9 8 7 6 5 (n + 1) ), - (317184 n + 1821792 n + 4455360 n + 6105314 n + 5171185 n 4 3 2 + 2806077 n + 973814 n + 207943 n + 24771 n + 1260)/((3 + 8 n) 9 (4 n + 3) (1 + 2 n) (7 + 8 n) (1 + 4 n) (5 + 8 n) (n + 1)), - (477874992 n 10 4 5 6 + 309685824 n + 10970768 n - 3196 n + 51296960 n + 157523906 n 7 8 2 3 11 + 331101072 n + 481493060 n + 71703 n + 1387394 n + 118341888 n 12 / - 399 + 20256768 n ) / (2 (3 + 8 n) (4 n + 3) (8 n + 1) (1 + 2 n) / 2 189 (7 + 8 n) (1 + 4 n) (5 + 8 n) (n + 1) )], [------------------, 3 (7 + 8 n) (n + 1) 2 21 (76 n + 102 n + 35) ------------------------------, 3 2 (7 + 8 n) (4 n + 3) (n + 1) 4 3 2 9 (144 n + 1236 n + 2243 n + 1536 n + 364) - ---------------------------------------------, 3 2 (5 + 8 n) (7 + 8 n) (4 n + 3) (n + 1) 5 4 3 2 3 (25824 n + 79200 n + 95460 n + 56075 n + 15951 n + 1750) - --------------------------------------------------------------, - ( 2 4 (5 + 8 n) (7 + 8 n) (1 + 2 n) (4 n + 3) (n + 1) 8 7 6 5 4 3 255808 n + 1031472 n + 1662136 n + 1284394 n + 360376 n - 143555 n 2 / - 143712 n - 42407 n - 4515) / (2 (5 + 8 n) (7 + 8 n) (1 + 2 n) / 3 9 8 7 (4 n + 3) (3 + 8 n) (n + 1) ), 3 (715008 n + 3787392 n + 8662768 n 6 5 4 3 2 + 11202572 n + 9000470 n + 4641140 n + 1527940 n + 307535 n + 34070 n / + 1575) / (4 (3 + 8 n) (4 n + 3) (1 + 2 n) (7 + 8 n) (1 + 4 n) (5 + 8 n) / 2 4 10 9 (n + 1) ), (-21316 n + 17040619 n + 660734592 n + 1000351088 n 12 2 8 3 7 + 44362752 n - 35278 n + 983200808 n + 1575962 n + 654049460 n 6 5 11 / + 297089851 n + 90194848 n + 256236288 n - 1344) / (4 (3 + 8 n) / 3 (4 n + 3) (8 n + 1) (1 + 2 n) (7 + 8 n) (1 + 4 n) (5 + 8 n) (n + 1) )], [ 126 21 (10 n + 7) (1 + 2 n) - ------------------, - ----------------------------, 4 4 (7 + 8 n) (n + 1) (7 + 8 n) (4 n + 3) (n + 1) 4 3 2 21 (80 n + 344 n + 490 n + 292 n + 63) -----------------------------------------, 4 (5 + 8 n) (7 + 8 n) (4 n + 3) (n + 1) 5 4 3 2 3 (4272 n + 12444 n + 14085 n + 7625 n + 1929 n + 175) 8 ----------------------------------------------------------, 3 (17344 n 3 (5 + 8 n) (7 + 8 n) (1 + 2 n) (4 n + 3) (n + 1) 7 6 5 4 3 2 + 56640 n + 55228 n - 17914 n - 80493 n - 71564 n - 31202 n - 6936 n / 4 - 630) / ((5 + 8 n) (7 + 8 n) (1 + 2 n) (4 n + 3) (3 + 8 n) (n + 1) ), - / 9 8 7 6 5 (969472 n + 4824384 n + 10397680 n + 12684232 n + 9601804 n 4 3 2 / + 4644754 n + 1421193 n + 260959 n + 25377 n + 945) / (2 (3 + 8 n) / 3 3 (4 n + 3) (1 + 2 n) (7 + 8 n) (1 + 4 n) (5 + 8 n) (n + 1) ), - 3 (-66392 n 10 4 6 5 2 + 62030656 n - 4002 n + 570879 n + 21106510 n + 5235262 n - 31022 n 8 7 11 12 + 84099516 n + 51976036 n - 189 + 24826368 n + 4408320 n 9 / + 90216768 n ) / (2 (3 + 8 n) (4 n + 3) (8 n + 1) (1 + 2 n) (7 + 8 n) / 4 42 21 n (1 + 4 n) (5 + 8 n) (n + 1) )], [------------------, ------------------, 5 5 (7 + 8 n) (n + 1) (7 + 8 n) (n + 1) 3 2 2 21 (12 n + 33 n + 27 n + 7) 21 (44 n + 60 n + 21) n - -----------------------------, - ------------------------------, 5 4 (7 + 8 n) (5 + 8 n) (n + 1) 2 (7 + 8 n) (5 + 8 n) (n + 1) 6 5 4 3 2 3 (40 n - 1082 n - 3468 n - 4142 n - 2373 n - 654 n - 70) - --------------------------------------------------------------, 5 (5 + 8 n) (7 + 8 n) (3 + 8 n) (n + 1) 5 4 3 2 3 (4056 n + 12750 n + 15825 n + 9680 n + 2920 n + 350) n 4 ------------------------------------------------------------, (-17368 n 4 2 (5 + 8 n) (7 + 8 n) (3 + 8 n) (n + 1) 5 6 7 3 8 - 1187 n + 132609 n + 418711 n + 560260 n - 27796 n + 369504 n - 63 2 9 / - 8772 n + 98336 n ) / (2 (5 + 8 n) (7 + 8 n) (8 n + 1) (3 + 8 n) / 5 (n + 1) )]] T In other words, a(n)=A(n-1)A(n-2)...A(0) times, [1, 0, 0, 0, 0, 0, 0] Let 8 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 + a[4](n) k + a[5](n) k + a[6](n) k ) We have, for every non-negative integer n ----- \ ) F(n, k) = 1 / ----- k Proof: We cleverly construct 8 / 8 6 5 G(n, k) = binomial(n, k) |1/8 a[0](n) k (7056 n k + 131072 n k + 344064 n k \ 4 3 2 4 + 358400 n k + 188160 n k + 51968 n k - 33372 n - 2459520 n 3 6 7 2 5 - 1080576 n - 2138112 n - 589824 n - 265608 n - 3161088 n + 360 k / - 1620) / ((8 n + 1) (1 + 4 n) (3 + 8 n) (1 + 2 n) (5 + 8 n) (4 n + 3) / 8 8 7 6 (7 + 8 n) (n + 1 - k) ) + 1/8 a[1](n) k (3600384 n + 7727616 n 5 8 2 2 2 4 + 8884800 n + 688128 n + 3150 + 59168 k n + 8152 k n - 2901120 n k 2 3 3 5 6 - 42680 n k + 209920 k n - 1319456 n k - 3568640 n k - 2285568 n k 7 2 2 6 2 5 2 4 - 589824 k n - 333072 n k + 131072 k n + 360448 k n + 389120 k n 2 3 2 4 / - 2100 k + 420 k + 2390256 n + 552504 n + 66630 n + 5970672 n ) / ( / (8 n + 1) (1 + 4 n) (3 + 8 n) (1 + 2 n) (5 + 8 n) (4 n + 3) (7 + 8 n) 8 8 9 (n + 1 - k) ) + 1/8 a[2](n) k (-3024 - 63000 n + 5292 k + 344064 n 3 5 6 7 8 2 - 2123634 n - 6591744 n - 4375296 n - 573696 n + 860160 n - 509934 n 4 6 5 3 - 4979916 n + 109608 n k + 9739776 n k + 12130432 n k + 3664448 n k 4 2 2 3 3 3 4 + 8694080 n k + 881824 n k - 2772 k + 504 k + 9648 k n + 423936 k n 3 5 3 3 3 2 2 7 3 6 + 376832 k n + 236800 k n + 68608 k n - 589824 k n + 131072 k n 8 2 2 2 2 3 7 + 688128 k n - 422256 k n - 55416 k n - 1620576 k n + 4104192 k n 2 6 2 5 2 4 / - 2433024 k n - 4015104 k n - 3424128 k n ) / ((8 n + 1) (1 + 4 n) / 8 (3 + 8 n) (1 + 2 n) (5 + 8 n) (4 n + 3) (7 + 8 n) (n + 1 - k) ) + 1/8 8 10 9 3 a[3](n) k (2205 + 42987 n - 8400 k - 516096 n - 3483648 n + 1002093 n 5 6 7 8 2 - 2048160 n - 8602464 n - 12442272 n - 9249408 n + 308157 n 4 6 5 3 + 1128606 n - 172952 n k - 13554432 n k - 18027392 n k - 5674664 n k 4 2 2 3 4 3 7 - 13265892 n k - 1380108 n k + 8610 k - 3780 k + 630 k - 589824 k n 3 3 4 3 5 3 3 - 73872 k n - 4043904 k n - 4500480 k n - 2007744 k n 3 2 2 7 3 6 8 - 545016 k n + 4608000 k n - 2580480 k n + 172032 k n 2 2 2 2 3 7 + 1353828 k n + 174088 k n + 5386936 k n - 4505856 k n 2 6 2 5 2 4 4 6 + 11917824 k n + 15929536 k n + 12130800 k n + 131072 k n 4 5 4 4 4 3 4 2 4 + 393216 k n + 462848 k n + 270336 k n + 81440 k n + 11808 k n 9 2 8 / + 344064 k n + 688128 k n ) / ((8 n + 1) (1 + 4 n) (3 + 8 n) (1 + 2 n) / 8 8 (5 + 8 n) (4 n + 3) (7 + 8 n) (n + 1 - k) ) + 1/8 a[4](n) k (-2520 10 9 3 5 - 50988 n + 11340 k - 5107200 n - 11531520 n - 1615038 n - 6256476 n 6 7 8 2 4 - 8610288 n - 12070272 n - 14480928 n - 398454 n - 3871080 n 11 6 5 3 - 946176 n + 228312 n k + 7572624 n k + 17528464 n k + 6926768 n k 4 2 2 3 4 5 + 15051488 n k + 1763656 n k - 18480 k + 14280 k - 5460 k + 840 k 3 7 3 3 4 3 5 + 5111808 k n + 278624 k n + 16481056 k n + 20351488 k n 3 3 3 2 2 7 3 6 + 7767360 k n + 2064896 k n - 8855808 k n + 14261760 k n 8 2 2 2 2 3 - 8058624 k n - 2837952 k n - 369496 k n - 11187568 k n 7 2 6 2 5 2 4 - 5310912 k n - 24662400 k n - 33094192 k n - 25128072 k n 4 7 5 6 5 5 5 4 5 2 - 589824 k n + 131072 k n + 409600 k n + 505856 k n + 99584 k n 5 5 3 3 8 10 2 9 + 15184 k n + 312064 k n + 688128 k n - 516096 k n + 344064 k n 4 6 4 5 4 4 4 3 - 2727936 k n - 5024768 k n - 4775808 k n - 2512832 k n 4 2 4 9 2 8 / - 722376 k n - 102896 k n - 3548160 k n - 516096 k n ) / ((8 n + 1) / 8 (1 + 4 n) (3 + 8 n) (1 + 2 n) (5 + 8 n) (4 n + 3) (7 + 8 n) (n + 1 - k) ) 8 10 9 + 1/8 a[5](n) k (3150 + 63660 n - 15960 k + 7257600 n + 17947248 n 3 5 6 7 8 + 2201064 n + 11942214 n + 18939174 n + 25078050 n + 25720116 n 2 4 12 11 + 507642 n + 6051018 n - 30720 n + 1215744 n - 310616 n k 6 5 3 4 - 17896200 n k - 25095344 n k - 8938520 n k - 19657860 n k 2 2 3 4 5 6 - 2318508 n k + 35070 k - 40320 k + 25830 k - 8820 k + 1260 k 3 7 3 3 4 3 5 - 13623552 k n - 756552 k n - 41916168 k n - 52588848 k n 3 3 3 2 2 7 3 6 - 19808592 k n - 5386944 k n + 5624064 k n - 37885824 k n 8 2 2 2 2 3 - 5836320 k n + 4932080 k n + 672712 k n + 18694784 k n 7 2 6 2 5 2 4 - 7635984 k n + 33553152 k n + 50915472 k n + 40539544 k n 6 6 6 5 6 4 6 3 6 + 131072 k n + 425984 k n + 552960 k n + 363520 k n + 21096 k n 11 6 2 2 10 3 9 - 946176 k n + 125728 k n - 516096 k n + 344064 k n 4 8 5 7 4 7 5 6 + 688128 k n - 589824 k n + 5615616 k n - 2875392 k n 5 5 5 4 5 2 5 - 5587968 k n - 5635200 k n - 989808 k n - 154392 k n 5 3 3 8 10 2 9 - 3175776 k n - 1204224 k n - 4311552 k n - 3612672 k n 4 6 4 5 4 4 4 3 + 16771584 k n + 25465664 k n + 21983536 k n + 11114152 k n 4 2 4 9 2 8 / + 3197252 k n + 469848 k n - 7109760 k n - 6249600 k n ) / ( / (8 n + 1) (1 + 4 n) (3 + 8 n) (1 + 2 n) (5 + 8 n) (4 n + 3) (7 + 8 n) 8 8 10 (n + 1 - k) ) + 1/8 a[6](n) k (-4176 - 71160 n + 27108 k + 58057104 n 9 3 5 6 7 + 67740936 n - 1783488 n - 1804443 n + 8698683 n + 29919609 n 8 2 4 12 11 + 54632454 n - 492096 n - 3426579 n + 10748160 n + 32654592 n 6 5 3 + 451272 n k + 29224176 n k + 33646384 n k + 11359472 n k 4 2 2 3 4 + 24898952 n k + 3080608 n k - 75768 k + 117516 k - 109200 k 5 6 3 7 3 3 4 + 60900 k - 18900 k + 20703168 k n + 1834744 k n + 83510368 k n 3 5 3 3 3 2 2 7 + 102695664 k n + 40767728 k n + 11768504 k n - 15675216 k n 3 6 8 2 2 2 + 71265744 k n + 18137640 k n - 8113788 k n - 1224008 k n 2 3 7 2 6 2 5 - 29002280 k n + 20461056 k n - 53389272 k n - 76641296 k n 2 4 4 9 3 10 2 11 - 61090980 k n + 344064 k n - 516096 k n - 946176 k n 12 5 8 6 7 7 6 7 5 - 30720 k n + 688128 k n - 589824 k n + 131072 k n + 442368 k n 7 4 7 3 7 2 7 7 + 604160 k n + 426240 k n + 163328 k n + 32112 k n + 2520 k 6 6 6 5 6 4 6 3 - 3022848 k n - 6190080 k n - 6637440 k n - 4044576 k n 6 11 6 2 2 10 - 255960 k n + 1708032 k n - 1402512 k n - 3515904 k n 3 9 4 8 5 7 4 7 - 3677184 k n - 1892352 k n + 6119424 k n - 18809088 k n 5 6 5 5 5 4 5 2 + 19447296 k n + 31341440 k n + 28915392 k n + 5082144 k n 5 5 3 3 8 10 + 870680 k n + 15861056 k n - 3822336 k n + 8347776 k n 2 9 4 6 4 5 4 4 - 2937984 k n - 53411328 k n - 77418112 k n - 65426364 k n 4 3 4 2 4 9 - 33388744 k n - 10069716 k n - 1637080 k n + 16059264 k n 2 8 13 / - 241248 k n + 1557504 n ) / ((8 n + 1) (1 + 4 n) (3 + 8 n) (1 + 2 n) / 8 \ (5 + 8 n) (4 n + 3) (7 + 8 n) (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