A361880 - OEIS

%I #17 Mar 29 2023 12:09:13

%S 1,3,24,207,1893,17952,174402,1723494,17250000,174354822,1776119970,

%T 18208500000,187659221409,1942674634371,20187543581880,

%U 210472842939975,2200677521078253,23068297001178240,242353695578011416,2551260130246575048,26905595698893121728

%N Expansion of 1/(1 - 9*x/(1 - x)^2)^(1/3).

%H Winston de Greef, <a href="/A361880/b361880.txt">Table of n, a(n) for n = 0..961</a>

%F a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(n+k-1,n-k).

%F a(0) = 1; a(n) = (3/n) * Sum_{k=0..n-1} (n+2*k) * (n-k) * a(k).

%F (n-1)*n*a(n) = (11*n-6)*(n-1)*a(n-1) - 18*(n-2)*a(n-2) - (11*n-38)*(n-3)*a(n-3) + (n-3)*(n-4)*a(n-4) for n > 3.

%F a(n) ~ 3^(1/3) * ((11 + 3*sqrt(13))/2)^n / (Gamma(1/3) * 13^(1/6) * n^(2/3)). - _Vaclav Kotesovec_, Mar 28 2023

%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/(1-9*x/(1-x)^2)^(1/3))

%Y Cf. A004987, A361375, A361843, A361844, A361845, A361895, A361896.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 28 2023