A320079 - OEIS

%I #18 Sep 10 2023 10:18:43

%S 1,1,0,1,1,0,1,2,3,0,1,3,10,14,0,1,4,21,76,88,0,1,5,36,222,772,694,0,

%T 1,6,55,488,3132,9808,6578,0,1,7,78,910,8824,55242,149552,72792,0,1,8,

%U 105,1524,20080,199456,1169262,2660544,920904,0,1,9,136,2366,39708,553870,5410208,28873800,54093696,13109088,0

%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k*log(1 - x)).

%F E.g.f. of column k: 1/(1 + k*log(1 - x)).

%F A(n,k) = Sum_{j=0..n} |Stirling1(n,j)|*j!*k^j.

%F A(0,k) = 1; A(n,k) = k * Sum_{j=1..n} (j-1)! * binomial(n,j) * A(n-j,k). - _Seiichi Manyama_, May 22 2022

%e E.g.f. of column k: A_k(x) = 1 + k*x/1! + k*(2*k + 1)*x^2/2! + 2*k*(3*k^2 + 3*k + 1)*x^3/3! + 2*k*(12*k^3 + 18*k^2 + 11*k + 3)*x^4/4! + ...

%e Square array begins:

%e   1,    1,     1,      1,       1,       1,  ...

%e   0,    1,     2,      3,       4,       5,  ...

%e   0,    3,    10,     21,      36,      55,  ...

%e   0,   14,    76,    222,     488,     910,  ...

%e   0,   88,   772,   3132,    8824,   20080,  ...

%e   0,  694,  9808,  55242,  199456,  553870,  ...

%t Table[Function[k, n! SeriesCoefficient[1/(1 + k Log[1 - x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

%Y Columns k=0..5 give A000007, A007840, A088500, A354263, A354264, A365588.

%Y Main diagonal gives A317171.

%Y Cf. A048594, A048994, A094416, A320080.

%K nonn,tabl

%O 0,8

%A _Ilya Gutkovskiy_, Oct 05 2018