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

0,2

Links

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

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

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 *)

Prog

(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)}

Crossrefs

Cf. A331791, A331795.

Sequence in context: A293193 A295256 A269926 * A156132 A215364 A370930

Adjacent sequences: A331791 A331792 A331793 * A331795 A331796 A331797

Keyword

nonn

Author

Seiichi Manyama, Jan 26 2020

Status

approved