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%I #18 Jul 12 2024 10:17:21
%S 1,3,9,27,78,207,459,567,-1926,-20763,-120123,-569349,-2410200,
%T -9379449,-33818715,-112292001,-335018295,-837341388,-1317232530,
%U 2358000072,35974607355,228270292803,1148026536963,5094839173779,20667058966044,77501033284779
%N Expansion of 1/(1 - 9*x*(1-x)^3)^(1/3).
%H Winston de Greef, <a href="/A361845/b361845.txt">Table of n, a(n) for n = 0..1669</a>
%F n*a(n) = 3 * ( (3*n-2)*a(n-1) - 3*(3*n-4)*a(n-2) + 3*(3*n-6)*a(n-3) - (3*n-8)*a(n-4) ) for n > 3.
%F a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-1/3,k) * binomial(3*k,n-k).
%F a(n) = (-9)^n*binomial(-1/3, n)*hypergeom([(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4], [1/3-n, 2/3-n, 2/3-n], 2^8/3^5). - _Stefano Spezia_, Jul 11 2024
%t a[n_]:=(-9)^n*Binomial[-1/3, n]HypergeometricPFQ[{(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4}, {1/3-n, 2/3-n, 2/3-n}, 2^8/3^5]; Array[a,26,0] (* _Stefano Spezia_, Jul 11 2024 *)
%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/(1-9*x*(1-x)^3)^(1/3))
%Y Column k=3 of A361840.
%Y Cf. A361816.
%K sign
%O 0,2
%A _Seiichi Manyama_, Mar 26 2023
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