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A361375 Expansion of 1/(1 - 9*x/(1 - x))^(1/3).

%I A361375 #28 May 04 2025 03:36:41
%S A361375 1,3,21,165,1380,11982,106626,965442,8854725,82022115,765787773,
%T A361375 7195638909,67973370618,644991134880,6143707229880,58714212503784,
%U A361375 562741793028282,5407273475087934,52074626299010130,502513862912425650,4857975310180620720
%N A361375 Expansion of 1/(1 - 9*x/(1 - x))^(1/3).
%H A361375 Winston de Greef, <a href="/A361375/b361375.txt">Table of n, a(n) for n = 0..997</a>
%F A361375 a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(n-1,n-k).
%F A361375 a(0) = 1; a(n) = (3/n) * Sum_{k=0..n-1} (n+2*k) * a(k).
%F A361375 n*a(n) = (11*n-8)*a(n-1) - 10*(n-2)*a(n-2) for n > 1.
%F A361375 a(n) ~ 3^(2/3) * 10^(n - 1/3) / (Gamma(1/3) * n^(2/3)). - _Vaclav Kotesovec_, Mar 28 2023
%F A361375 a(n) = 3*hypergeom([1 - n, 4/3], [2], -9) for n >= 1. - _Peter Luschny_, Mar 30 2023
%p A361375 a := n -> if n = 0 then 1 else 3*hypergeom([1 - n, 4/3], [2], -9) fi:
%p A361375 seq(simplify(a(n)), n = 0..20); # _Peter Luschny_, Mar 30 2023
%o A361375 (PARI) my(N=30, x='x+O('x^N)); Vec(1/(1-9*x/(1-x))^(1/3))
%Y A361375 Cf. A004987, A361843, A361844, A361845, A361880, A361895, A361896.
%Y A361375 Cf. A085362.
%K A361375 nonn
%O A361375 0,2
%A A361375 _Seiichi Manyama_, Mar 28 2023