A336713 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.

1, 1, 1, 6, 76, 1447, 37206, 1212194, 47975271, 2238595055, 120453255172, 7347494056729, 501273291296174, 37833413358907566, 3130557361463956074, 281854137496597897755, 27433898122963009937892, 2870816347095046227070383, 321430790732030793454519088

Offset: 0

Author

Seiichi Manyama, Aug 01 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..339

Crossrefs

Main diagonal of A336708.

Cf. A336578, A335871, A336712, A336714.

Programs

Mathematica

a[0] = 1; a[n_] := Sum[(-1)^(n - k) * Binomial[n, k] * Binomial[n + (n - 1)*k, k - 1], {k, 1, n}] / n; Array[a, 19, 0] (* Amiram Eldar, Aug 01 2020 *)

PARI

{a(n) = if(n==0, 1, sum(k=1, n, (-1)^(n-k)*binomial(n, k)*binomial(n+(n-1)*k, k-1))/n)}

Formula

a(n) ~ exp(n - 1/2 - exp(-1)) * n^(n - 5/2) / sqrt(2*Pi). - Vaclav Kotesovec, Aug 04 2025