A382806 a(n) = Sum_{k=0..n} (k!)^2 * binomial(k+2,2) * Stirling1(n,k)^2.

1, 3, 27, 588, 24612, 1669128, 165049224, 22269896064, 3918921022656, 870149951146944, 237662482188210624, 78249086559726140160, 30547324837444471084800, 13946361918619108837939200, 7359961832428044552536217600, 4444946383758589481684168540160

Offset: 0

Author

Seiichi Manyama, Apr 05 2025

Keywords

• nonn

Crossrefs

Main diagonal of A382800.

Cf. A382792, A382804.

Cf. A382738.

Programs

PARI

a(n) = sum(k=0, n, k!^2*binomial(k+2, 2)*stirling(n, k, 1)^2);

Formula

a(n) == 0 (mod 3) for n > 0.

a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1-x) * log(1-y))^3.

a(n) = (n!)^2 * [(x*y)^n] 1 / (1 - log(1+x) * log(1+y))^3.

a(n) ~ sqrt(Pi) * n^(2*n + 5/2) / (2 * (exp(1) - 1)^(2*n+3)). - Vaclav Kotesovec, Apr 05 2025