A320079 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)).

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, 1, 6, 55, 488, 3132, 9808, 6578, 0, 1, 7, 78, 910, 8824, 55242, 149552, 72792, 0, 1, 8, 105, 1524, 20080, 199456, 1169262, 2660544, 920904, 0, 1, 9, 136, 2366, 39708, 553870, 5410208, 28873800, 54093696, 13109088, 0

Offset: 0

Author

Ilya Gutkovskiy, Oct 05 2018

Keywords

• nonn
• tabl

Examples

			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! + ...
Square array begins:
  1,    1,     1,      1,       1,       1,  ...
  0,    1,     2,      3,       4,       5,  ...
  0,    3,    10,     21,      36,      55,  ...
  0,   14,    76,    222,     488,     910,  ...
  0,   88,   772,   3132,    8824,   20080,  ...
  0,  694,  9808,  55242,  199456,  553870,  ...
		

Crossrefs

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

Main diagonal gives A317171.

Cf. A048594, A048994, A094416, A320080.

Programs

Mathematica

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

Formula

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

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

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