A361375 Expansion of 1/(1 - 9*x/(1 - x))^(1/3).

1, 3, 21, 165, 1380, 11982, 106626, 965442, 8854725, 82022115, 765787773, 7195638909, 67973370618, 644991134880, 6143707229880, 58714212503784, 562741793028282, 5407273475087934, 52074626299010130, 502513862912425650, 4857975310180620720

Offset: 0

Author

Seiichi Manyama, Mar 28 2023

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 0..997

Crossrefs

Cf. A004987, A361843, A361844, A361845, A361880, A361895, A361896.

Cf. A085362.

Programs

Maple

a := n -> if n = 0 then 1 else 3*hypergeom([1 - n, 4/3], [2], -9) fi:
seq(simplify(a(n)), n = 0..20); # Peter Luschny, Mar 30 2023

PARI

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

Formula

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

a(0) = 1; a(n) = (3/n) * Sum_{k=0..n-1} (n+2*k) * a(k).

n*a(n) = (11*n-8)*a(n-1) - 10*(n-2)*a(n-2) for n > 1.

a(n) ~ 3^(2/3) * 10^(n - 1/3) / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Mar 28 2023

a(n) = 3*hypergeom([1 - n, 4/3], [2], -9) for n >= 1. - Peter Luschny, Mar 30 2023