A361842 Expansion of 1/(1 - 9*x*(1+x)^3)^(1/3).

1, 3, 27, 243, 2352, 23607, 242757, 2539431, 26904492, 287858421, 3104029755, 33684914907, 367483636746, 4026930734223, 44295829667055, 488855016668727, 5410588668898995, 60035381850523284, 667643481187840206, 7439651232903588528, 83050643822779921347

Offset: 0

Author

Seiichi Manyama, Mar 26 2023

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 0..932

Crossrefs

Column k=3 of A361839.

Cf. A099234, A361812, A361845.

Programs

Mathematica

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,21,0] (* Stefano Spezia, Jul 11 2024 *)

PARI

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

Formula

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.

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

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