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
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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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 *) -
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))}
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
a(n) = Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k). - Seiichi Manyama, Aug 24 2025
From Seiichi Manyama, Aug 27 2025: (Start)
a(n) = [x^n] (1+4*x+3*x^2)^(n+1).
E.g.f.: exp(4*x) * BesselI(1, 2*sqrt(3)*x) / sqrt(3), with offset 1. (End)