A383049 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the n-th term of the inverse Stirling transform of j-> (j+1)^k.

1, 1, 1, 1, 2, 0, 1, 4, 1, 0, 1, 8, 5, -1, 0, 1, 16, 19, -3, 2, 0, 1, 32, 65, -1, 4, -6, 0, 1, 64, 211, 45, -10, -8, 24, 0, 1, 128, 665, 359, -116, 48, 20, -120, 0, 1, 256, 2059, 2037, -538, 340, -234, -52, 720, 0, 1, 512, 6305, 10079, -1316, 984, -1240, 1302, 72, -5040, 0

Offset: 0

Author

Seiichi Manyama, Apr 14 2025

Keywords

• sign
• tabl

Examples

			Square array begins:
  1,  1,  1,    1,     1,     1,     1, ...
  1,  2,  4,    8,    16,    32,    64, ...
  0,  1,  5,   19,    65,   211,   665, ...
  0, -1, -3,   -1,    45,   359,  2037, ...
  0,  2,  4,  -10,  -116,  -538, -1316, ...
  0, -6, -8,   48,   340,   984, -1148, ...
  0, 24, 20, -234, -1240, -1866, 16400, ...
		

Links

• Christian G. Bower, PARI programs for transforms, 2007.
• N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Crossrefs

Columns k=0..6 give A019590(n+1), A302190 (for n > 0), A222627, A222636, A222748, A223023, A383050.

Main diagonal gives A383051.

Cf. A362924, A362925.

Programs

PARI

a(n, k) = sum(j=0, n, (j+1)^k*stirling(n, j, 1));

Formula

A(n,k) = Sum_{j=0..n} (j+1)^k * Stirling1(n,j).

E.g.f. of column k: Sum_{j>=0} (j+1)^k * log(1+x)^j / j!.

E.g.f. of column k: (1+x) * Sum_{j=0..k} Stirling2(k+1,j+1) * log(1+x)^j.