A382673 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] exp(x+y) / (exp(x) + exp(y) - exp(x+y))^3.

1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 22, 52, 22, 1, 1, 46, 208, 208, 46, 1, 1, 94, 736, 1372, 736, 94, 1, 1, 190, 2440, 7516, 7516, 2440, 190, 1, 1, 382, 7792, 37012, 60316, 37012, 7792, 382, 1, 1, 766, 24328, 170668, 418996, 418996, 170668, 24328, 766, 1, 1, 1534, 74896, 754132, 2653036, 3964684, 2653036, 754132, 74896, 1534, 1

Offset: 0

Author

Seiichi Manyama, Apr 03 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,  1,    1,     1,      1,       1, ...
  1,  4,   10,    22,     46,      94, ...
  1, 10,   52,   208,    736,    2440, ...
  1, 22,  208,  1372,   7516,   37012, ...
  1, 46,  736,  7516,  60316,  418996, ...
  1, 94, 2440, 37012, 418996, 3964684, ...
  ...
		

Crossrefs

Columns k=0..2 give A000012, A033484, A382675.

Main diagonal gives A382676.

Cf. A099594, A136126, A382674.

Cf. A382735.

Programs

PARI

a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*stirling(n+1, j+1, 2)*stirling(k+1, j+1, 2));

Formula

E.g.f.: exp(x+y) / (exp(x) + exp(y) - exp(x+y))^3.

A(n,k) = A(k,n).

A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * binomial(j+2,2) * Stirling2(n+1,j+1) * Stirling2(k+1,j+1).