A361896 -id:A361896 - cp's OEIS frontend

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

Seiichi Manyama, Mar 28 2023

nonn

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

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

Cf. A085362.

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

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

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

A361880 Expansion of 1/(1 - 9*x/(1 - x)^2)^(1/3).

Original entry on oeis.org

1, 3, 24, 207, 1893, 17952, 174402, 1723494, 17250000, 174354822, 1776119970, 18208500000, 187659221409, 1942674634371, 20187543581880, 210472842939975, 2200677521078253, 23068297001178240, 242353695578011416, 2551260130246575048, 26905595698893121728

Offset: 0

Seiichi Manyama, Mar 28 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..961

Cf. A004987, A361375, A361843, A361844, A361845, A361895, A361896.

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

a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(n+k-1,n-k).
a(0) = 1; a(n) = (3/n) * Sum_{k=0..n-1} (n+2*k) * (n-k) * a(k).
(n-1)*n*a(n) = (11*n-6)*(n-1)*a(n-1) - 18*(n-2)*a(n-2) - (11*n-38)*(n-3)*a(n-3) + (n-3)*(n-4)*a(n-4) for n > 3.
a(n) ~ 3^(1/3) * ((11 + 3*sqrt(13))/2)^n / (Gamma(1/3) * 13^(1/6) * n^(2/3)). - Vaclav Kotesovec, Mar 28 2023

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

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Mar 28 2023

nonn

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

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

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 *)

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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

A361880 Expansion of 1/(1 - 9*x/(1 - x)^2)^(1/3).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula