A382741 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] (1/3) * (1 / (exp(x) + exp(y) - exp(x+y))^3 - 1).

1, 1, 1, 1, 9, 1, 1, 25, 25, 1, 1, 57, 193, 57, 1, 1, 121, 889, 889, 121, 1, 1, 249, 3361, 7593, 3361, 249, 1, 1, 505, 11545, 47641, 47641, 11545, 505, 1, 1, 1017, 37633, 253737, 465601, 253737, 37633, 1017, 1, 1, 2041, 118969, 1228249, 3657721, 3657721, 1228249, 118969, 2041, 1

Offset: 1

Author

Seiichi Manyama, Apr 04 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,   1,     1,      1,       1,        1, ...
  1,   9,    25,     57,     121,      249, ...
  1,  25,   193,    889,    3361,    11545, ...
  1,  57,   889,   7593,   47641,   253737, ...
  1, 121,  3361,  47641,  465601,  3657721, ...
  1, 249, 11545, 253737, 3657721, 40666089, ...
		

Crossrefs

Cf. A272644, A382740, A382742.

Cf. A382735.

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))/3;

Formula

E.g.f.: (1/3) * (1 / (exp(x) + exp(y) - exp(x+y))^3 - 1).

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

A(n,k) = (1/3) * A382735(n,k).