A336714 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-2)^(n-k) * binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.

1, 1, 0, 2, 36, 766, 20910, 707472, 28740656, 1367040950, 74645106114, 4606416653654, 317237242964840, 24130334401571972, 2009783477119978508, 181958565624827141256, 17796032244661580019904, 1870078875109869688744870, 210155525478346375059816234, 25151873422906866362758095642

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 A336709.

Cf. A336578, A335871, A336712, A336713.

Programs

Mathematica

a[0] = 1; a[n_] := Sum[(-2)^(n - k) * Binomial[n, k] * Binomial[n + (n - 1)*k, k - 1], {k, 1, n}] / n; Array[a, 20, 0] (* Amiram Eldar, Aug 01 2020 *)

PARI

{a(n) = if(n==0, 1, sum(k=1, n, (-2)^(n-k)*binomial(n, k)*binomial(n+(n-1)*k, k-1))/n)}

Formula

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