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)] , be defined T by the initial conditions:, a(0) = [1, 0, 0, 0, 0] and a(n+1)=A(n)a(n), where A(n) is the, 5, by , 5, matrix 2 5 (n + 1) 5 (n + 1) (21 n + 28 n + 10) [[-----------, -----------------------------, 2 (6 n + 5) 12 (3 n + 2) (6 n + 5) 3 2 5 (n + 1) (21 n + 42 n + 28 n + 6) n --------------------------------------, 12 (2 n + 1) (3 n + 2) (6 n + 5) 5 4 3 2 2 (357 n + 1043 n + 1204 n + 672 n + 184 n + 20) (n + 1) - -----------------------------------------------------------, - n (n + 1) 24 (1 + 3 n) (2 n + 1) (3 n + 2) (6 n + 5) 7 6 5 4 3 2 (5103 n + 19026 n + 29946 n + 25690 n + 12915 n + 3778 n + 586 n + 36 14 )/(24 (6 n + 5) (3 n + 2) (2 n + 1) (1 + 3 n) (6 n + 1))], [- -------, 6 n + 5 2 4 3 2 45 n + 58 n + 20 105 n + 175 n + 72 n - 15 n - 10 - -------------------, - -----------------------------------, (3 n + 2) (6 n + 5) 3 (2 n + 1) (3 n + 2) (6 n + 5) 4 3 2 (11 n + 5) (n + 1) (21 n + 49 n + 42 n + 15 n + 2) 8 7 -----------------------------------------------------, (7119 n + 26397 n 2 (1 + 3 n) (2 n + 1) (3 n + 2) (6 n + 5) 4 2 6 5 3 - 2 + 16953 n + 21 n + 642 n + 41146 n + 34713 n + 4699 n )/(6 63 (6 n + 5) (3 n + 2) (2 n + 1) (1 + 3 n) (6 n + 1))], [-------------------, 2 (n + 1) (6 n + 5) 2 4 3 2 7 (51 n + 62 n + 20) 165 n + 112 n - 198 n - 226 n - 60 -----------------------------, ---------------------------------------, 4 (6 n + 5) (3 n + 2) (n + 1) 4 (6 n + 5) (3 n + 2) (2 n + 1) (n + 1) 5 4 3 2 2625 n + 6965 n + 7228 n + 3594 n + 856 n + 80 7 - --------------------------------------------------, - (72324 n 8 (1 + 3 n) (2 n + 1) (3 n + 2) (6 n + 5) 8 2 4 3 5 6 + 19971 n + 700 n - 104 n + 39167 n + 9164 n + 87330 n + 108976 n - 16)/(8 (6 n + 5) (3 n + 2) (2 n + 1) (1 + 3 n) (6 n + 1) (n + 1))], [ 2 35 7 (69 n + 74 n + 20) - ------------------, - ------------------------------, 2 2 (6 n + 5) (n + 1) 6 (3 n + 2) (6 n + 5) (n + 1) 4 3 2 7 (3 n + 84 n + 150 n + 94 n + 20) ----------------------------------------, 2 6 (2 n + 1) (3 n + 2) (6 n + 5) (n + 1) 5 4 3 2 5055 n + 12527 n + 11980 n + 5370 n + 1096 n + 80 8 7 -----------------------------------------------------, (27405 n + 93240 n 12 (6 n + 5) (3 n + 2) (2 n + 1) (1 + 3 n) (n + 1) 3 5 6 2 4 / + 2000 n + 88922 n + 128376 n - 1644 n - 476 n + 29729 n - 40) / ( / 2 12 (6 n + 5) (3 n + 2) (2 n + 1) (1 + 3 n) (6 n + 1) (n + 1) )], [ 35 35 n --------------------, --------------------, 3 3 2 (6 n + 5) (n + 1) 4 (6 n + 5) (n + 1) 3 2 2 7 (17 n + 54 n + 42 n + 10) 7 (27 n + 32 n + 10) n - -------------------------------, - ------------------------------, 3 2 12 (6 n + 5) (2 n + 1) (n + 1) 8 (6 n + 5) (2 n + 1) (n + 1) 6 5 4 3 2 1765 n + 3027 n + 843 n - 1223 n - 942 n - 228 n - 20 - ----------------------------------------------------------]] 3 24 (6 n + 5) (2 n + 1) (6 n + 1) (n + 1) T In other words, a(n)=A(n-1)A(n-2)...A(0) times, [1, 0, 0, 0, 0] Let 6 F(n, k) = binomial(n, k) 2 3 4 (a[0](n) + a[1](n) k + a[2](n) k + a[3](n) k + a[4](n) k ) We have, for every non-negative integer n ----- \ ) F(n, k) = 1 / ----- k Proof: We cleverly construct 6 / 6 3 G(n, k) = binomial(n, k) |-1/24 a[0](n) k (168 + 2268 n - 48 k + 23940 n \ 5 2 4 2 3 + 9072 n + 10920 n + 24192 n - 600 n k - 2520 n k - 4320 n k 4 / - 2592 n k) / ((6 n + 5) (3 n + 2) (2 n + 1) (1 + 3 n) (6 n + 1) / 6 6 3 4 2 4 (n + 1 - k) ) - 1/24 a[1](n) k (29556 n k - 2592 n k + 27216 n k 2 2 2 3 2 6 5 3 - 732 n k - 2952 n k - 4752 n k - 7560 n - 34272 n - 47684 n - 280 2 2 4 2 - 19754 n + 14436 n k - 3892 n + 240 k - 58254 n - 60 k + 3144 n k 5 / + 9072 n k) / ((6 n + 5) (3 n + 2) (2 n + 1) (1 + 3 n) (6 n + 1) / 6 6 7 3 (n + 1 - k) ) - 1/24 a[2](n) k (280 + 3780 n - 560 k - 3780 n + 40740 n 5 6 2 2 2 + 14532 n - 5670 n + 18284 n - 7416 n k + 360 k - 34992 n k 2 3 2 2 3 4 + 4512 n k - 76648 n k + 19356 n k - 936 n k - 83076 n k 3 2 2 3 5 4 2 3 3 + 36252 n k - 3528 n k - 42084 n k + 30240 n k - 5184 n k 4 3 6 3 4 2 5 / + 42378 n - 80 k - 7560 k n - 2592 k n + 9072 k n ) / ((6 n + 5) / 6 6 (3 n + 2) (2 n + 1) (1 + 3 n) (6 n + 1) (n + 1 - k) ) - 1/24 a[3](n) k ( 8 7 3 5 6 -280 - 3556 n + 960 k + 2142 n + 14679 n - 29764 n + 15113 n + 29309 n 2 2 2 2 3 - 15638 n + 12192 n k - 1140 k + 54732 n k - 13908 n k + 115404 n k 2 2 3 4 3 2 2 3 - 58698 n k + 6900 n k + 120330 n k - 114960 n k + 26256 n k 4 5 4 2 3 3 2 4 - 1284 n k + 54096 n k - 111426 n k + 44028 n k - 4248 n k 4 3 4 7 3 5 6 - 19061 n + 600 k - 120 k - 3780 k n + 9072 k n + 1890 k n 3 4 2 5 2 6 4 3 4 4 / + 33264 k n - 49896 k n - 7560 k n - 5616 k n - 2592 k n ) / ( / 6 (6 n + 5) (3 n + 2) (2 n + 1) (1 + 3 n) (6 n + 1) (n + 1 - k) ) - 1/24 6 8 9 7 a[4](n) k (392 + 4284 n - 1792 k + 21987 n + 5103 n + 36624 n 3 5 6 2 2 + 37772 n + 35329 n + 34874 n + 18004 n - 18768 n k + 3240 k 2 2 3 2 2 3 - 75360 n k + 32208 n k - 147712 n k + 123114 n k - 27084 n k 4 3 2 2 3 4 5 - 142506 n k + 232116 n k - 95736 n k + 11244 n k - 48300 n k 4 2 3 3 2 4 5 4 + 224952 n k - 164648 n k + 35712 n k - 1848 n k + 43855 n 3 4 5 7 3 5 6 - 2920 k + 1320 k - 240 k + 16128 k n - 57708 k n + 18872 k n 8 3 4 2 5 2 6 3 6 + 2142 k n - 143304 k n + 100212 k n + 9450 k n - 7560 k n 2 7 4 3 4 4 4 5 5 4 - 3780 k n + 52884 k n + 36288 k n + 9072 k n - 2592 k n 5 3 5 2 / - 6048 k n - 5112 k n ) / ((6 n + 5) (3 n + 2) (2 n + 1) (1 + 3 n) / 6 \ (6 n + 1) (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