A361845 Expansion of 1/(1 - 9*x*(1-x)^3)^(1/3).

1, 3, 9, 27, 78, 207, 459, 567, -1926, -20763, -120123, -569349, -2410200, -9379449, -33818715, -112292001, -335018295, -837341388, -1317232530, 2358000072, 35974607355, 228270292803, 1148026536963, 5094839173779, 20667058966044, 77501033284779

Offset: 0

Author

Seiichi Manyama, Mar 26 2023

Keywords

• sign

Links

• Winston de Greef, Table of n, a(n) for n = 0..1669

Crossrefs

Column k=3 of A361840.

Cf. A361816.

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,26,0] (* 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) = (-1)^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