A307946 Coefficient of x^n in 1/(n+1) * (1 - n*x - n*x^2)^(n+1).

1, -1, 2, 0, -96, 1875, -32184, 554631, -9773056, 172718325, -2874200000, 35973317666, 218394869760, -46968959184459, 2890848443624064, -147665402789062500, 7121567693920010240, -337669517265832692843, 15985827659730523364352, -759295252512454596032456

Offset: 0

Author

Seiichi Manyama, May 07 2019

Keywords

• sign

Comments

Also coefficient of x^n in the expansion of 2/(1 + n*x + sqrt(1 + 2*n*x + n*(n+4)*x^2)).

Links

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

Crossrefs

Cf. A000108, A292716, A307911.

Programs

Mathematica

a[0] = 1; a[n_] := Sum[(-n)^(n-k) * Binomial[n, 2*k] * CatalanNumber[k], {k, 0, Floor[n/2]}]; Array[a, 20, 0] // Flatten (* Amiram Eldar, May 12 2021 *)
Join[{1}, Table[(-n)^n * Hypergeometric2F1[1/2 - n/2, -n/2, 2, -4/n], {n, 1, 20}]] (* Vaclav Kotesovec, May 12 2021 *)

PARI

{a(n) = polcoef((1-n*x-n*x^2)^(n+1)/(n+1), n)}

PARI

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

PARI

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

Formula

a(n) = Sum_{k=0..floor(n/2)} (-n)^(n-k) * binomial(n,k) * binomial(n-k,k)/(k+1) = Sum_{k=0..floor(n/2)} (-n)^(n-k) * binomial(n,2*k) * A000108(k).

For n>0, a(n) = (-n)^n * Hypergeometric2F1(1/2 - n/2, -n/2, 2, -4/n). - Vaclav Kotesovec, May 12 2021