A336522 a(n) is the coefficient of x^(n^2) in expansion of ( (1 + x)/(1 - x) )^n.

1, 2, 16, 326, 11008, 525002, 32497680, 2478629134, 224921989120, 23681262354194, 2838826197080080, 381825269929428822, 56949892477659339520, 9329658433405643973850, 1665421971238565711337488, 321771059958076157377283102, 66901218825369170336327860224, 14894388013750938445628478094370

Offset: 0

Author

Seiichi Manyama, Jul 24 2020

Keywords

• nonn

Links

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

Crossrefs

Main diagonal of A336521.

Cf. A336537.

Programs

Mathematica

a[n_] := Sum[Binomial[n, k] * Binomial[n^2 + k - 1, n - 1], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Jul 24 2020 *)

PARI

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

PARI

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

Formula

a(n) = (1/n) * [x^n] ( (1 + x)/(1 - x) )^(n^2) for n > 0.

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

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

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

a(n) ~ 2^(n - 1/2) * exp(n) * n^(n - 3/2) / sqrt(Pi). - Vaclav Kotesovec, Jul 31 2021

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