A383133 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(n*k,k) * n^k.

1, 0, 17, 1889, 412225, 151448249, 84430503361, 66535567456546, 70456680210155009, 96530372235620300465, 166169585125820280654001, 351113456811120647774884511, 893491183170443755035588745153, 2695374684029443253628238600963667, 9511442599320236554084097413617603681

Offset: 0

Author

Ilya Gutkovskiy, Apr 17 2025

Keywords

• nonn

Crossrefs

Cf. A307885, A331657, A340972, A383121, A383132.

Programs

Mathematica

Unprotect[Power]; 0^0 = 1; Table[Sum[(-1)^(n - k) 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