A331794 a(n) = Sum_{k=0..n} n^k * binomial(n+1,k) * binomial(n+1,k+1).

1, 4, 33, 400, 6285, 120456, 2714173, 70129984, 2040655401, 65956468600, 2342384363561, 90607200956064, 3789863084012629, 170370561866229648, 8188781210421259365, 418938023982360898816, 22724122083014879989905, 1302374806940392958470104

Offset: 0

Author

Seiichi Manyama, Jan 26 2020

Keywords

• nonn
• changed

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..380

Crossrefs

Cf. A331791, A331795.

Programs

Mathematica

Flatten[{1, Table[Sum[n^k * Binomial[n+1,k] * Binomial[n+1,k+1], {k,0,n}], {n,1,20}]}] (* Vaclav Kotesovec, Jan 26 2020 *)
Table[(n+1) * Hypergeometric2F1[-1 - n, -n, 2, n], {n, 0, 20}] (* Vaclav Kotesovec, Jan 26 2020 *)

PARI

a(n) = sum(k=0, n, n^k*binomial(n+1, k)*binomial(n+1, k+1));

PARI

a(n) = polcoef(2/(1-2*(n+1)*x+((n-1)*x)^2+(1-(n+1)*x)*sqrt(1-2*(n+1)*x+((n-1)*x)^2)), n);

Formula

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

a(n) = (n+1) * 2F1(-1 - n, -n; 2; n), where 2F1 is the hypergeometric function. - Vaclav Kotesovec, Jan 26 2020

a(n) = Sum_{k=0..floor(n/2)} n^k * (n+1)^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k). - Seiichi Manyama, Aug 24 2025