A331792 Expansion of ((1 - 4*x)/sqrt(1 - 8*x + 4*x^2) - 1)/(6*x^2).

1, 8, 57, 400, 2810, 19824, 140497, 999968, 7143966, 51206320, 368094122, 2652720096, 19159794004, 138658606688, 1005231020865, 7299082678336, 53074479789878, 386419850997552, 2816685368479342, 20553133273532000, 150120362670452076

Offset

0,2

Links

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

Formula

a(n) = (2/(n+2)) * A331515(n) = Sum_{k=0..n} 3^k * binomial(n+1,k) * binomial(n+1,k+1).

n * (n+2) * a(n) = (n+1) * (4 * (2*n+1) * a(n-1) - 4 * n * a(n-2)) for n>1.

a(n) ~ 2^(n + 1/2) * (2 + sqrt(3))^(n + 3/2) / (3^(3/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Jan 26 2020

Mathematica

a[n_] := Sum[3^k * Binomial[n + 1, k] * Binomial[n + 1, k + 1], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, May 05 2021 *)

Prog

(PARI) N=20; x='x+O('x^N); Vec(((1-4*x)/sqrt(1-8*x+4*x^2)-1)/(6*x^2))
(PARI) {a(n) = sum(k=0, n, 3^k*binomial(n+1, k)*binomial(n+1, k+1))}

Crossrefs

Column 4 of A331791.

Cf. A331515.

Sequence in context: A164031 A297369 A023000 * A097114 A022038 A277671

Adjacent sequences: A331789 A331790 A331791 * A331793 A331794 A331795

Keyword

nonn

Author

Seiichi Manyama, Jan 26 2020

Status

approved