A361895 Expansion of 1/(1 - 9*x/(1 - x)^3)^(1/3).

1, 3, 27, 252, 2487, 25434, 266364, 2837082, 30601233, 333302931, 3658565127, 40413860334, 448778693844, 5005642415907, 56044616215041, 629552293867800, 7092072533703567, 80095810435943526, 906605837653876254, 10282430320166723448, 116829834042508121682

Offset: 0

Author

Seiichi Manyama, Mar 28 2023

Keywords

• nonn

Links

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

Crossrefs

Cf. A004987, A361375, A361843, A361844, A361845, A361880, A361896.

Programs

Mathematica

a[0]=1; a[n_]:=3*n*(1 + n)*HypergeometricPFQ[{1-n, 1+n/2, (3+n)/2}, {5/3, 2}, -4/3]/2; Array[a,21,0] (* Stefano Spezia, May 02 2024 *)

PARI

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

Formula

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

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

a(n) = 3*n*(1 + n)*hypergeom([1-n, 1+n/2, (3+n)/2], [5/3, 2], -4/3)/2 for n > 0. - Stefano Spezia, May 02 2024

a(n) ~ ((7 - sqrt(21))^(1/3) + (7 + sqrt(21))^(1/3))^(1/3) * (4 + (3*((39 - sqrt(21))/2))^(1/3) + (3*((39 + sqrt(21))/2))^(1/3))^n / (Gamma(1/3) * 2^(1/9) * 7^(2/9) * n^(2/3)). - Vaclav Kotesovec, Jul 11 2025