A361881 Expansion of 1/(1 - 9*x/(1 + x))^(1/3).
1, 3, 15, 93, 618, 4278, 30390, 219810, 1611105, 11929395, 89045079, 669018837, 5053759440, 38350056072, 292147584072, 2233020788184, 17117923408746, 131560216858110, 1013413369611606, 7822237588031586, 60487791859818348, 468511159492134516
Offset: 0
Keywords
Links
- Winston de Greef, Table of n, a(n) for n = 0..1103
Programs
-
Maple
a := n -> if n = 0 then 1 else (-1)^(1-n)*3*hypergeom([1 - n, 4/3], [2], 9) fi: seq(simplify(a(n)), n = 0..21); # Peter Luschny, Mar 30 2023
-
Mathematica
CoefficientList[Series[1/CubeRoot[(1-9x/(1+x))],{x,0,30}],x] (* Harvey P. Dale, Apr 15 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(1/(1-9*x/(1+x))^(1/3))
Formula
a(n) = (-1)^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} (-1)^(n-1-k) * (n+2*k) * a(k).
n*a(n) = (7*n-4)*a(n-1) + 8*(n-2)*a(n-2) for n > 1.
a(n) ~ 3^(2/3) * 2^(3*n-1) / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Mar 28 2023
a(n) = (-1)^(1 - n)*3*hypergeom([1 - n, 4/3], [2], 9) for n >= 1. - Peter Luschny, Mar 30 2023