A384492 a(n) = n!^3 * Sum_{k=0..n} Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) / binomial(n,k)^3.

1, 2, 113, 38992, 47071264, 147015606528, 988250901343488, 12631667044878213120, 280790763724247161061376, 10147405862241529912885248000, 565550513462476798468573003776000, 46592777163703224212146175606784000000, 5479872142880875751798643810680954683392000

Offset: 0

Author

Vaclav Kotesovec, May 31 2025

Keywords

• nonn

Crossrefs

Cf. A187655, A384470, A384491.

Cf. A187657, A384471, A384472.

Programs

Mathematica

Table[n!^3 * Sum[StirlingS2[2*k, k] * StirlingS2[2*n-2*k, n-k] / Binomial[n, k]^3, {k, 0, n}], {n, 0, 15}]

PARI

a(n) = n!^3 * sum(k=0, n, stirling(2*k,k,2) * stirling(2*n-2*k,n-k,2) / binomial(n,k)^3); \\ Michel Marcus, May 31 2025

Formula

a(n) ~ Pi * 2^(2*n+2) * n^(4*n+1) / (sqrt(1-w) * exp(4*n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...