%I #26 Mar 30 2023 09:14:54

%S 1,3,21,165,1380,11982,106626,965442,8854725,82022115,765787773,

%T 7195638909,67973370618,644991134880,6143707229880,58714212503784,

%U 562741793028282,5407273475087934,52074626299010130,502513862912425650,4857975310180620720

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

%H Winston de Greef, <a href="/A361375/b361375.txt">Table of n, a(n) for n = 0..997</a>

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

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

%F n*a(n) = (11*n-8)*a(n-1) - 10*(n-2)*a(n-2) for n > 1.

%F a(n) ~ 3^(2/3) * 10^(n - 1/3) / (Gamma(1/3) * n^(2/3)). - _Vaclav Kotesovec_, Mar 28 2023

%F a(n) = 3*hypergeom([1 - n, 4/3], [2], -9) for n >= 1. - _Peter Luschny_, Mar 30 2023

%p a := n -> if n = 0 then 1 else 3*hypergeom([1 - n, 4/3], [2], -9) fi:

%p seq(simplify(a(n)), n = 0..20); # _Peter Luschny_, Mar 30 2023

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

%Y Cf. A004987, A361843, A361844, A361845, A361880, A361895, A361896.

%Y Cf. A085362, A361843.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 28 2023