A331512 a(n) = Sum_{k=0..n} n^(n-k) * (n+k+1) * binomial(n,k) * binomial(n+k,k).

1, 8, 90, 1328, 24150, 520272, 12926004, 363233600, 11376760230, 392615960600, 14791582824876, 603743206301424, 26528443526357500, 1248071683342913184, 62576263671773466600, 3330116426356595493120, 187430800395881065513734

Offset

0,2

Links

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

Formula

a(n) = [x^n] (1 - n*x)/(1 - 2*(n+2)*x + (n*x)^2)^(3/2).

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

Mathematica

a[n_] := Sum[If[n == n-k == 0, 1, n^(n-k)] * (n+k+1) * Binomial[n, k] * Binomial[n + k, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, May 05 2021 *)

Prog

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

Crossrefs

Cf. A001850, A108666, A331511, A331513.

Sequence in context: A166769 A323960 A187667 * A092956 A345876 A295623

Adjacent sequences: A331509 A331510 A331511 * A331513 A331514 A331515

Keyword

nonn

Author

Seiichi Manyama, Jan 19 2020

Status

approved