A331793 Expansion of ((1 - 5*x)/sqrt(1 - 10*x + 9*x^2) - 1)/(8*x^2).
1, 10, 87, 740, 6285, 53550, 458115, 3934600, 33913881, 293244050, 2542684463, 22101612780, 192530903461, 1680415209270, 14692052109915, 128653303453200, 1128147127156785, 9905115333850650, 87066787614156807, 766127762539955700, 6747880819438628541
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
a[n_] := Sum[4^k * Binomial[n + 1, k] * Binomial[n + 1, k + 1], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, May 05 2021 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(((1-5*x)/sqrt(1-10*x+9*x^2)-1)/(8*x^2)) -
PARI
a(n) = sum(k=0, n, 4^k*binomial(n+1, k)*binomial(n+1, k+1));
Formula
a(n) = (2/(n+2)) * A331516(n) = Sum_{k=0..n} 4^k * binomial(n+1,k) * binomial(n+1,k+1).
n * (n+2) * a(n) = (n+1) * (5 * (2*n+1) * a(n-1) - 9 * n * a(n-2)) for n>1.
a(n) ~ 3^(2*n + 3) / (2^(5/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Jan 26 2020
From Seiichi Manyama, Aug 23 2025: (Start)
a(n) = Sum_{k=0..floor(n/2)} 4^k * 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+1,k+1) * binomial(2*k+2,k+2). (End)
From Seiichi Manyama, Aug 25 2025: (Start)
a(n) = [x^n] (1+5*x+4*x^2)^(n+1).
E.g.f.: exp(5*x) * BesselI(1, 4*x) / 2, with offset 1. (End)