A383121 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(n*k,k).

1, 0, 3, 47, 1093, 33029, 1236781, 55325416, 2879987209, 171061709417, 11418368571721, 846230146390001, 68949300160035373, 6126085419697733567, 589470974371501065845, 61068847238080533844679, 6777270943578364524130321, 802138434294752321142680145

Offset: 0

Author

Ilya Gutkovskiy, Apr 17 2025

Keywords

• nonn

Crossrefs

Cf. A002426, A349471, A378409, A383118, A383119, A383120.

Programs

Mathematica

Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[n k, k], {k, 0, n}], {n, 0, 17}]

PARI

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(n*k,k)); \\ Michel Marcus, Apr 17 2025

Formula

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

a(n) ~ exp(n - exp(-1) - 1/2) * n^n / sqrt(2*Pi*n). - Vaclav Kotesovec, Apr 17 2025