A384491 a(n) = n!^2 * Sum_{k=0..n} Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) / binomial(n,k)^2.

1, 2, 57, 6536, 1966816, 1226860992, 1373652478656, 2507498281198080, 6966291361870181376, 27969794062091821670400, 155875927262331497576140800, 1167389777699203314381963264000, 11441270265465265986005655905894400, 143525982910350708912088976768630784000

Offset: 0

Author

Vaclav Kotesovec, May 31 2025

Keywords

• nonn

Crossrefs

Cf. A187655, A384470, A384492.

Cf. A187657, A384471, A384472.

Programs

Mathematica

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

PARI

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

Formula

a(n) ~ sqrt(Pi) * 2^(2*n + 3/2) * n^(3*n + 1/2) / (sqrt(1-w) * exp(3*n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...