A308036 Coefficient of x^n in 1/(n+1) * (1 + x - 3*x^2)^(n+1).

1, 1, -2, -8, 1, 61, 91, -377, -1469, 1027, 16120, 18250, -132065, -427517, 620062, 5707648, 3746683, -55581941, -144227438, 351490672, 2274331579, 278638399, -25348074713, -50842003745, 195685236757, 957135268261, -459626168864, -12027281377922

Offset

0,3

Comments

Also coefficient of x^n in the expansion of 2/(1 - x + sqrt(1 - 2*x + 13*x^2)).

Links

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

Formula

a(n) = Sum_{k=0..floor(n/2)} (-3)^k * binomial(n,k) * binomial(n-k,k)/(k+1) = Sum_{k=0..floor(n/2)} (-3)^k * binomial(n,2*k) * A000108(k).

(n+2) * a(n) = (2*n+1) * a(n-1) - 13 * (n-1) * a(n-2).

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

Mathematica

a[n_] := Sum[(-3)^k * Binomial[n, 2*k] * CatalanNumber[k], {k, 0, Floor[n/2]}]; Array[a, 28, 0] // Flatten (* Amiram Eldar, May 12 2021 *)
Table[Hypergeometric2F1[1/2 - n/2, -n/2, 2, -12], {n, 0, 30}] (* Vaclav Kotesovec, May 12 2021 *)

Prog

(PARI) {a(n) = polcoef((1+x-3*x^2)^(n+1)/(n+1), n)}
(PARI) {a(n) = sum(k=0, n\2, (-3)^k*binomial(n, k)*binomial(n-k, k)/(k+1))}
(PARI) {a(n) = sum(k=0, n\2, (-3)^k*binomial(n, 2*k)*binomial(2*k, k)/(k+1))}

Crossrefs

Column 3 of A308035.

Sequence in context: A065249 A062038 A318106 * A305677 A296569 A021082

Adjacent sequences: A308033 A308034 A308035 * A308037 A308038 A308039

Keyword

sign

Author

Seiichi Manyama, May 10 2019

Status

approved