A379820 Array read by ascending antidiagonals: A(n, k) = (-1)^(n + k) * Sum_{j=0..k} j! * Stirling1(n, j) * Stirling1(k, j).

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 3, 2, 0, 0, 6, 8, 8, 6, 0, 0, 24, 28, 28, 28, 24, 0, 0, 120, 124, 114, 114, 124, 120, 0, 0, 720, 668, 558, 518, 558, 668, 720, 0, 0, 5040, 4248, 3234, 2744, 2744, 3234, 4248, 5040, 0, 0, 40320, 31176, 21768, 16888, 15446, 16888, 21768, 31176, 40320, 0

Offset: 0

Author

Peter Luschny, Jan 03 2025

Keywords

• nonn
• tabl

Examples

			Array begins:
  [0] 1,   0,   0,    0,     0,     0,      0,       0,        0, ...
  [1] 0,   1,   1,    2,     6,    24,    120,     720,     5040, ...
  [2] 0,   1,   3,    8,    28,   124,    668,    4248,    31176, ...
  [3] 0,   2,   8,   28,   114,   558,   3234,   21768,   167280, ...
  [4] 0,   6,  28,  114,   518,  2744,  16888,  119232,   952944, ...
  [5] 0,  24, 124,  558,  2744, 15446,  99730,  732120,  6045240, ...
  [6] 0, 120, 668, 3234, 16888, 99730, 669422, 5074992, 43062864, ...
.
Triangle T(n, k) = A(n - k, k) starts:
  [0] 1;
  [1] 0,  0;
  [2] 0,  1,    0;
  [3] 0,  1,    1,   0;
  [4] 0,  2,    3,   2,   0;
  [5] 0,  6,    8,   8,   6,   0;
  [6] 0,  24,  28,  28,  28,  24,   0;
  [7] 0, 120, 124, 114, 114, 124, 120,   0;
  [8] 0, 720, 668, 558, 518, 558, 668, 720, 0;
		

Crossrefs

The corresponding array with Stirling2 numbers is A108470.

Programs

Maple

A := (n, k) -> local j; (-1)^(n + k)*add(j!*Stirling1(n, j)*Stirling1(k, j), j = 0..k):
seq(lprint(seq(A(n, k), k = 0..8)), n = 0..8);