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%I #19 Mar 30 2023 09:14:43
%S 1,3,15,93,618,4278,30390,219810,1611105,11929395,89045079,669018837,
%T 5053759440,38350056072,292147584072,2233020788184,17117923408746,
%U 131560216858110,1013413369611606,7822237588031586,60487791859818348,468511159492134516
%N Expansion of 1/(1 - 9*x/(1 + x))^(1/3).
%H Winston de Greef, <a href="/A361881/b361881.txt">Table of n, a(n) for n = 0..1103</a>
%F a(n) = (-1)^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} (-1)^(n-1-k) * (n+2*k) * a(k).
%F n*a(n) = (7*n-4)*a(n-1) + 8*(n-2)*a(n-2) for n > 1.
%F a(n) ~ 3^(2/3) * 2^(3*n-1) / (Gamma(1/3) * n^(2/3)). - _Vaclav Kotesovec_, Mar 28 2023
%F a(n) = (-1)^(1 - 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 (-1)^(1-n)*3*hypergeom([1 - n, 4/3], [2], 9) fi:
%p seq(simplify(a(n)), n = 0..21); # _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, A180400, A361841, A361842, A361882.
%Y Cf. A361375.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Mar 28 2023
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