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A361882 Expansion of 1/(1 - 9*x/(1 + x)^2)^(1/3).

%I A361882 #18 Mar 30 2023 09:39:14
%S A361882 1,3,12,63,357,2112,12834,79446,498504,3160566,20202882,129998400,
%T A361882 841084065,5466859635,35672889180,233564188167,1533744021741,
%U A361882 10097724827904,66633102118296,440600483618184,2918753549183712,19367330685385032,128704927930928088
%N A361882 Expansion of 1/(1 - 9*x/(1 + x)^2)^(1/3).
%H A361882 Winston de Greef, <a href="/A361882/b361882.txt">Table of n, a(n) for n = 0..1191</a>
%F A361882 a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-1/3,k) * binomial(n+k-1,n-k).
%F A361882 a(0) = 1; a(n) = (3/n) * Sum_{k=0..n-1}  (-1)^(n-1-k) * (n+2*k) * (n-k) * a(k).
%F A361882 (n-1)*n*a(n) = (7*n-6)*(n-1)*a(n-1) + 6*(n-2)*a(n-2) - (7*n-22)*(n-3)*a(n-3) + (n-3)*(n-4)*a(n-4) for n > 3.
%F A361882 a(n) ~ 3^(1/3) * phi^(4*n) / (Gamma(1/3) * 5^(1/6) * n^(2/3)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Mar 28 2023
%F A361882 a(n) = (-1)^(n - 1)*3*n*hypergeom([1 - n, 1 + n, 4/3], [3/2, 2], 9/4) for n >= 1. - _Peter Luschny_, Mar 30 2023
%p A361882 a := n -> if n = 0 then 1 else (-1)^(n - 1)*3*n*hypergeom([1 - n, 1 + n, 4/3], [3/2, 2], 9/4) fi: seq(simplify(a(n)), n = 0..22); # _Peter Luschny_, Mar 30 2023
%o A361882 (PARI) my(N=30, x='x+O('x^N)); Vec(1/(1-9*x/(1+x)^2)^(1/3))
%Y A361882 Cf. A004987, A180400, A361841, A361842, A361881.
%Y A361882 Cf. A361880.
%K A361882 nonn
%O A361882 0,2
%A A361882 _Seiichi Manyama_, Mar 28 2023