A361841 Expansion of 1/(1 - 9*x*(1+x)^2)^(1/3).

1, 3, 24, 201, 1809, 16893, 161676, 1574289, 15527052, 154662930, 1552725504, 15688410264, 159355067283, 1625899880673, 16652520666414, 171119405299005, 1763475423260049, 18219685282559559, 188664151412242368, 1957539823296458841, 20347733657193596127

Offset: 0

Author

Seiichi Manyama, Mar 26 2023

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 0..966

Crossrefs

Column k=2 of A361839.

Cf. A002478, A137635, A361844.

Programs

Maple

A361841 := 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(A361841(n)), n = 0..20); # 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) = 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