A331795 a(n) = Sum_{k=0..n} (-n)^k * binomial(n+1,k) * binomial(n+1,k+1).

1, 0, -3, 40, -515, 7056, -102935, 1554288, -22775319, 265497760, 586651461, -230587852560, 13426823564869, -637734419560224, 28594259589697425, -1264238490602458784, 56015489395280490385, -2499557487903373341888, 112150411888789509887053

Offset: 0

Author

Seiichi Manyama, Jan 26 2020

Keywords

• sign
• changed

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..386

Crossrefs

Cf. A331791, A331794.

Programs

Mathematica

Flatten[{1, Table[Sum[(-1)^k * 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