A382736 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / (exp(x) + exp(y) - exp(x+y))^4.

1, 0, 0, 0, 4, 0, 0, 4, 4, 0, 0, 4, 44, 4, 0, 0, 4, 124, 124, 4, 0, 0, 4, 284, 1084, 284, 4, 0, 0, 4, 604, 5164, 5164, 604, 4, 0, 0, 4, 1244, 19804, 48044, 19804, 1244, 4, 0, 0, 4, 2524, 68524, 313804, 313804, 68524, 2524, 4, 0, 0, 4, 5084, 224284, 1707884, 3281404, 1707884, 224284, 5084, 4, 0

Offset: 0

Author

Seiichi Manyama, Apr 04 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1, 0,   0,     0,      0,       0, ...
  0, 4,   4,     4,      4,       4, ...
  0, 4,  44,   124,    284,     604, ...
  0, 4, 124,  1084,   5164,   19804, ...
  0, 4, 284,  5164,  48044,  313804, ...
  0, 4, 604, 19804, 313804, 3281404, ...
		

Crossrefs

Main diagonal gives A382739.

Cf. A371761, A382734, A382735.

Cf. A382674, A382742.

Programs

PARI

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

Formula

E.g.f.: 1 / (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,j) * Stirling2(k,j).