A382735 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))^3.

1, 0, 0, 0, 3, 0, 0, 3, 3, 0, 0, 3, 27, 3, 0, 0, 3, 75, 75, 3, 0, 0, 3, 171, 579, 171, 3, 0, 0, 3, 363, 2667, 2667, 363, 3, 0, 0, 3, 747, 10083, 22779, 10083, 747, 3, 0, 0, 3, 1515, 34635, 142923, 142923, 34635, 1515, 3, 0, 0, 3, 3051, 112899, 761211, 1396803, 761211, 112899, 3051, 3, 0

Offset: 0

Author

Seiichi Manyama, Apr 04 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1, 0,   0,     0,      0,       0, ...
  0, 3,   3,     3,      3,       3, ...
  0, 3,  27,    75,    171,     363, ...
  0, 3,  75,   579,   2667,   10083, ...
  0, 3, 171,  2667,  22779,  142923, ...
  0, 3, 363, 10083, 142923, 1396803, ...
		

Crossrefs

Main diagonal gives A382738.

Cf. A371761, A382734, A382736.

Cf. A382673, A382741.

Programs

PARI

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

Formula

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