A361880 -id:A361880 - 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

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

A361896 Expansion of 1/(1 - 9*x/(1 - x)^4)^(1/3).

Original entry on oeis.org

1, 3, 30, 300, 3165, 34584, 386880, 4400928, 50692266, 589584042, 6910397886, 81507086634, 966408021984, 11509174498254, 137584249375308, 1650109151463594, 19847075122106145, 239316542492974317, 2892135259684291248, 35021199836282568456, 424837125616822551264

Offset: 0

Seiichi Manyama, Mar 28 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..907

Cf. A004987, A361375, A361843, A361844, A361845, A361880, A361895.

my(N=30, x='x+O('x^N)); Vec(1/(1-9*x/(1-x)^4)^(1/3))

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

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

A361882 Expansion of 1/(1 - 9*x/(1 + x)^2)^(1/3).

Original entry on oeis.org

1, 3, 12, 63, 357, 2112, 12834, 79446, 498504, 3160566, 20202882, 129998400, 841084065, 5466859635, 35672889180, 233564188167, 1533744021741, 10097724827904, 66633102118296, 440600483618184, 2918753549183712, 19367330685385032, 128704927930928088

Offset: 0

Seiichi Manyama, Mar 28 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..1191

Cf. A004987, A180400, A361841, A361842, A361881.

Cf. A361880.

a := n -> if n = 0 then 1 else (-1)^(n - 1)*3*n*hypergeom([1 - n, 1 + n, 4/3], [3/2, 2], 9/4) fi: seq(simplify(a(n)), n = 0..22); # Peter Luschny, Mar 30 2023

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

a(n) = (-1)^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}  (-1)^(n-1-k) * (n+2*k) * (n-k) * a(k).
(n-1)*n*a(n) = (7*n-6)*(n-1)*a(n-1) + 6*(n-2)*a(n-2) - (7*n-22)*(n-3)*a(n-3) + (n-3)*(n-4)*a(n-4) for n > 3.
a(n) ~ 3^(1/3) * phi^(4*n) / (Gamma(1/3) * 5^(1/6) * n^(2/3)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Mar 28 2023
a(n) = (-1)^(n - 1)*3*n*hypergeom([1 - n, 1 + n, 4/3], [3/2, 2], 9/4) for n >= 1. - Peter Luschny, Mar 30 2023

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

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

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A361896 Expansion of 1/(1 - 9*x/(1 - x)^4)^(1/3).

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A361882 Expansion of 1/(1 - 9*x/(1 + x)^2)^(1/3).

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