A383132 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k) * n^k.

1, 2, 33, 2701, 524993, 181752001, 97735073905, 75179269556672, 78240951854025217, 105806762566689176353, 180297512864534759056001, 377878889913778527874694227, 955217573424445946022789385537, 2865620569274978738097814056365899, 10064763360358683666070320479027168465

Offset: 0

Author

Ilya Gutkovskiy, Apr 17 2025

Keywords

• nonn

Crossrefs

Cf. A084771, A187021, A331656, A340971, A383120, A383133.

Programs

Mathematica

Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k] Binomial[n k, k] n^k, {k, 0, n}], {n, 0, 14}]

Formula

a(n) = [x^n] ((1 + n*x)^n + x)^n.

a(n) ~ exp(n - 1/2) * n^(2*n - 1/2) / sqrt(2*Pi). - Vaclav Kotesovec, Apr 19 2025