A382800 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] 1 / (1 - log(1-x) * log(1-y))^3.
1, 0, 0, 0, 3, 0, 0, 3, 3, 0, 0, 6, 27, 6, 0, 0, 18, 78, 78, 18, 0, 0, 72, 282, 588, 282, 72, 0, 0, 360, 1272, 2988, 2988, 1272, 360, 0, 0, 2160, 6936, 16344, 24612, 16344, 6936, 2160, 0, 0, 15120, 44496, 101448, 175632, 175632, 101448, 44496, 15120, 0
Offset: 0
Examples
Square array begins: 1, 0, 0, 0, 0, 0, ... 0, 3, 3, 6, 18, 72, ... 0, 3, 27, 78, 282, 1272, ... 0, 6, 78, 588, 2988, 16344, ... 0, 18, 282, 2988, 24612, 175632, ... 0, 72, 1272, 16344, 175632, 1669128, ...
Programs
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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)));
Formula
E.g.f.: 1 / (1 - log(1-x) * log(1-y))^3.
A(n,k) = A(k,n).
A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * binomial(j+2,2) * |Stirling1(n,j)| * |Stirling1(k,j)|.