A382802 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 / (1 - log(1-x) * log(1-y))^3 - 1).

1, 1, 1, 2, 9, 2, 6, 26, 26, 6, 24, 94, 196, 94, 24, 120, 424, 996, 996, 424, 120, 720, 2312, 5448, 8204, 5448, 2312, 720, 5040, 14832, 33816, 58544, 58544, 33816, 14832, 5040, 40320, 109584, 238656, 431632, 556376, 431632, 238656, 109584, 40320

Offset: 1

Author

Seiichi Manyama, Apr 05 2025

Keywords

• nonn
• tabl

Examples

			Square array begins:
    1,    1,     2,      6,      24,      120, ...
    1,    9,    26,     94,     424,     2312, ...
    2,   26,   196,    996,    5448,    33816, ...
    6,   94,   996,   8204,   58544,   431632, ...
   24,  424,  5448,  58544,  556376,  5017480, ...
  120, 2312, 33816, 431632, 5017480, 55016408, ...
		

Crossrefs

Cf. A382741, A382800.

Programs

PARI

a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*abs(stirling(n, j, 1)*stirling(k, j, 1)))/3;

Formula

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

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

A(n,k) = (1/3) * A382800(n,k).