240106

PARI


A292915

Fubini number に関係する数列を出力してみた。

N=15;

a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
A(n, k) = 2^k*a000670(n)-sum(j=0, k-1, 2^j*(k-1-j)^n);
for(n=0, N, for(k=0, n, print1(A(k, n-k), ", ")))

\\ 別の計算方法と一致することを確認
B(n, k) = k^n+sum(j=1, n, binomial(n, j)*B(n-j, k));
for(n=0, N, for(k=0, n, print1(A(k, n-k)-B(k, n-k), ", ")))

出力結果

1, 1, 1, 1, 2, 3, 1, 3, 6, 13, 1, 4, 11, 26, 75, 1, 5, 18, 51, 150, 541, 1, 6, 27, 94, 299, 1082, 4683, 1, 7, 38, 161, 582, 2163, 9366, 47293, 1, 8, 51, 258, 1083, 4294, 18731, 94586, 545835, 1, 9, 66, 391, 1910, 8345, 37398, 189171, 1091670, 7087261, 1, 10, 83, 566, 3195, 15666, 74067, 378214, 2183339, 14174522, 102247563, 1, 11, 102, 789, 5094, 28207, 144038, 754241, 4366422, 28349043, 204495126, 1622632573, 1, 12, 123, 1066, 7787, 48638, 272451, 1492098, 8726283, 56697574, 408990251, 3245265146, 28091567595, 1, 13, 146, 1403, 11478, 80469, 498246, 2906071, 17387030, 113375465, 817979478, 6490530291, 56183135190, 526858348381, 1, 14, 171, 1806, 16395, 128170, 878843, 5532206, 34383435, 226488786, 1635899907, 12981058534, 112366270379, 1053716696762, 10641342970443, 1, 15, 198, 2281, 22790, 197291, 1495542, 10240869, 67087254, 451024447, 3270751238, 25961939921, 224732536662, 2107433393523, 21282685940886, 230283190977853, 

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,