A307969 Coefficient of x^n in 1/(n+1) * (1 - 2*x - 2*x^2)^(n+1).

1, -2, 2, 4, -24, 48, 24, -464, 1376, -704, -9920, 41600, -55040, -201216, 1266048, -2628864, -3021312, 37696512, -108659712, 15857664, 1067003904, -4155138048, 4378226688, 27416125440, -149814263808, 273526325248, 569660309504, -5103546466304, 13241145229312, 4914079858688

Offset

0,2

Comments

Also coefficient of x^n in the expansion of 2/(1 + 2*x + sqrt(1 + 4*x + 12*x^2)).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

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

(n+2) * a(n) = -2 * (2*n+1) * a(n-1) - 12 * (n-1) * a(n-2).

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

Mathematica

a[n_] := Sum[(-2)^(n-k) * Binomial[n, 2*k] * CatalanNumber[k], {k, 0, Floor[n/2]}]; Array[a, 30, 0] // Flatten (* Amiram Eldar, May 12 2021 *)
Table[(-2)^n * Hypergeometric2F1[1/2 - n/2, -n/2, 2, -2], {n, 0, 30}] (* Vaclav Kotesovec, May 12 2021 *)

Prog

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

Crossrefs

Column 2 of A307968.

Cf. A000108, A071356, A116093.

Sequence in context: A232275 A257612 A009541 * A212672 A362766 A176161

Adjacent sequences: A307966 A307967 A307968 * A307970 A307971 A307972

Keyword

sign

Author

Seiichi Manyama, May 08 2019

Status

approved