A336537 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).

1, 2, 10, 134, 3298, 122762, 6208970, 399606286, 31331798914, 2902190030354, 310441644900682, 37685712807847062, 5120833751373831138, 770270980249401539482, 127088854993223378639498, 22824507222500649365932062, 4432992797251355031727570434, 925899965014326913556521154594

Offset: 0

Author

Seiichi Manyama, Jul 25 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..311

Crossrefs

Main diagonal of A336534.

Cf. A336522, A336577.

Programs

Mathematica

a[0] = 1; a[n_] := Sum[2^k * Binomial[n, k] * Binomial[n^2, k - 1], {k, 1, n}] / n; Array[a, 18, 0] (* Amiram Eldar, Jul 25 2020 *)

PARI

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

PARI

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

PARI

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

Formula

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

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

a(n) ~ 2^(n - 1/2) * exp(n) * n^(n - 5/2) / sqrt(Pi). - Vaclav Kotesovec, Jul 31 2021

a(n) = 2*hypergeom([1-n, -n^2], [2], 2) for n > 0. - Stefano Spezia, Aug 09 2025