A362837 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = (-1)^n * n! * Sum_{j=0..floor(n/2)} k^(n-j) * Stirling1(n-j,j)/(n-j)!.

1, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 3, 0, 1, 0, 6, 12, 20, 0, 1, 0, 8, 27, 112, 90, 0, 1, 0, 10, 48, 324, 960, 594, 0, 1, 0, 12, 75, 704, 4050, 10848, 4200, 0, 1, 0, 14, 108, 1300, 11520, 64962, 141120, 34544, 0, 1, 0, 16, 147, 2160, 26250, 239616, 1224720, 2122496, 316008, 0

Offset: 0

Author

Seiichi Manyama, May 05 2023

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,  1,   1,    1,     1,     1, ...
  0,  0,   0,    0,     0,     0, ...
  0,  2,   4,    6,     8,    10, ...
  0,  3,  12,   27,    48,    75, ...
  0, 20, 112,  324,   704,  1300, ...
  0, 90, 960, 4050, 11520, 26250, ...
		

Crossrefs

Columns k=0..3 give: A000007, A066166, A053491, A351735.

Main diagonal gives A362838.

Cf. A362834.

Programs

PARI

T(n, k) = (-1)^n*n!*sum(j=0, n\2, k^(n-j)*stirling(n-j, j, 1)/(n-j)!);

Formula

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