A361844 Expansion of 1/(1 - 9*x*(1-x)^2)^(1/3).

1, 3, 12, 57, 297, 1629, 9216, 53217, 311796, 1846818, 11032416, 66356712, 401364531, 2439135585, 14882263002, 91116281565, 559528781697, 3445002647847, 21260140172244, 131474746842345, 814564464082263, 5055177167348463, 31420067723814780

Offset: 0

Author

Seiichi Manyama, Mar 26 2023

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=2 of A361840.

Cf. A361815, A004987.

Programs

Maple

A361844 := n -> (-9)^n*binomial(-1/3, n)*hypergeom([1/3 - n*2/3, 2/3 - n*2/3,
-n*2/3], [1/2 - n, 2/3 - n], 3/4):
seq(simplify(A361844(n)), n = 0..22); # Peter Luschny, Mar 27 2023

PARI

my(N=30, x='x+O('x^N)); Vec(1/(1-9*x*(1-x)^2)^(1/3))

Formula

n*a(n) = 3 * ( (3*n-2)*a(n-1) - 2*(3*n-4)*a(n-2) + (3*n-6)*a(n-3) ) for n > 2.

a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-1/3,k) * binomial(2*k,n-k).

a(n) = (-9)^n*binomial(-1/3, n)*hypergeom([1/3 - n*2/3, 2/3 - n*2/3, -n*2/3], [1/2 - n, 2/3 - n], 3/4). - Peter Luschny, Mar 27 2023