cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

1, 3, 21, 165, 1380, 11982, 106626, 965442, 8854725, 82022115, 765787773, 7195638909, 67973370618, 644991134880, 6143707229880, 58714212503784, 562741793028282, 5407273475087934, 52074626299010130, 502513862912425650, 4857975310180620720
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2023

Keywords

Links

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

A376802 Expansion of 1/((1 - x)^3 - 9*x)^(1/3).

Original entry on oeis.org

1, 4, 31, 283, 2770, 28204, 294568, 3131650, 33732883, 367035814, 4025600941, 44439461275, 493218155119, 5498860571026, 61543476786067, 691095770653867, 7783168304357434, 87878978740300960, 994484816394177214, 11276915136560900662, 128106749179069022344
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2024

Keywords

Links

Crossrefs

Partial sums of A361895.
Cf. A098536, A376803, A376804.
Cf. A376805, A376806.
Cf. A004987.

Programs

  • Mathematica
    CoefficientList[Series[1/Surd[((1-x)^3-9x),3],{x,0,30}],x] (* Harvey P. Dale, Dec 11 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/((1-x)^3-9*x)^(1/3))

Formula

a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(n+2*k,n-k).
a(n) = hypergeom([(1+n)/2, 1+n/2, -n], [2/3, 1], -4/3). - Stefano Spezia, May 04 2025

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

Views

Author

Seiichi Manyama, Mar 28 2023

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

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

Views

Author

Seiichi Manyama, Mar 28 2023

Keywords

Links

Crossrefs

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

Programs

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

Formula

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).
Showing 1-4 of 4 results.

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