A382738 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling2(n,k)^2.
1, 3, 27, 579, 22779, 1396803, 121998267, 14333812419, 2175860165499, 414000255441603, 96422983358827707, 26970211126038920259, 8918364340126714711419, 3440770498298077165166403, 1531504734740033368269820347, 778873986278207207346380124099
Offset: 0
Programs
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Mathematica
Table[Sum[k! * (k+2)! * StirlingS2[n,k]^2/2, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 30 2025 *) -
PARI
a(n) = sum(k=0, n, k!^2*binomial(k+2, 2)*stirling(n, k, 2)^2);
Formula
a(n) == 0 (mod 3) for n > 0.
a(n) = (n!)^2 * [(x*y)^n] 1 / (exp(x) + exp(y) - exp(x+y))^3.
a(n) ~ sqrt(Pi) * n^(2*n + 5/2) / (16 * sqrt(1 - log(2)) * exp(2*n) * log(2)^(2*n+3)). - Vaclav Kotesovec, Aug 30 2025