A382674 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))^4.

1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 29, 77, 29, 1, 1, 61, 325, 325, 61, 1, 1, 125, 1181, 2357, 1181, 125, 1, 1, 253, 3973, 13621, 13621, 3973, 253, 1, 1, 509, 12797, 69269, 118061, 69269, 12797, 509, 1, 1, 1021, 40165, 326005, 862261, 862261, 326005, 40165, 1021, 1

Offset: 0

Author

Seiichi Manyama, Apr 03 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,   1,    1,     1,      1,       1, ...
  1,   5,   13,    29,     61,     125, ...
  1,  13,   77,   325,   1181,    3973, ...
  1,  29,  325,  2357,  13621,   69269, ...
  1,  61, 1181, 13621, 118061,  862261, ...
  1, 125, 3973, 69269, 862261, 8712245, ...
  ...
		

Crossrefs

Columns k=0..2 give A000012, A036563(n+2), A382677.

Main diagonal gives A382678.

Cf. A099594, A136126, A382673.

Cf. A382736.

Programs

PARI

a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+3, 3)*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))^4.

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

A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * binomial(j+3,3) * Stirling2(n+1,j+1) * Stirling2(k+1,j+1).