A382740 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] (1/2) * (1 / (exp(x) + exp(y) - exp(x+y))^2 - 1).
1, 1, 1, 1, 7, 1, 1, 19, 19, 1, 1, 43, 127, 43, 1, 1, 91, 559, 559, 91, 1, 1, 187, 2071, 4327, 2071, 187, 1, 1, 379, 7039, 25831, 25831, 7039, 379, 1, 1, 763, 22807, 133783, 233551, 133783, 22807, 763, 1, 1, 1531, 71839, 636679, 1748791, 1748791, 636679, 71839, 1531, 1
Offset: 1
Examples
Square array begins: 1, 1, 1, 1, 1, 1, ... 1, 7, 19, 43, 91, 187, ... 1, 19, 127, 559, 2071, 7039, ... 1, 43, 559, 4327, 25831, 133783, ... 1, 91, 2071, 25831, 233551, 1748791, ... 1, 187, 7039, 133783, 1748791, 18207367, ...
Programs
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PARI
a(n, k) = sum(j=0, min(n, k), j!*(j+1)!*stirling(n, j, 2)*stirling(k, j, 2))/2;
Formula
E.g.f.: (1/2) * (1 / (exp(x) + exp(y) - exp(x+y))^2 - 1).
A(n,k) = A(k,n).
A(n,k) = (1/2) * A382734(n,k).