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 1-5 of 5 results.

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

A098535 Expansion of (1+x)^(1/3)/(1+x-9*x^4)^(1/3).

Original entry on oeis.org

1, 0, 0, 0, 3, -3, 3, -3, 21, -39, 57, -75, 219, -489, 885, -1407, 3000, -6609, 13179, -23655, 46353, -96960, 198534, -381504, 742638, -1504011, 3071973, -6096117, 12008415, -24042522, 48733248, -97896198, 195048966, -390235269
Offset: 0

Views

Author

Paul Barry, Sep 13 2004

Keywords

Comments

Binomial transform is A098536.

Links

Programs

  • Magma
    Q:=Rationals(); R:=PowerSeriesRing(Q,30); Coefficients(R!((1+x)^(1/3)/(1+x-9*x^4)^(1/3))); // G. C. Greubel, Jan 17 2018
  • Mathematica
    CoefficientList[Series[(1+x)^(1/3)/(1+x-9*x^4)^(1/3), {x,0,50}], x] (* G. C. Greubel, Jan 17 2018 *)
  • PARI
    x='x+O('x^30); Vec((1+x)^(1/3)/(1+x-9*x^4)^(1/3)) \\ G. C. Greubel, Jan 17 2018

A098538 Expansion of 1/((1-x)^3 - 18*x^4)^(1/3).

Original entry on oeis.org

1, 1, 1, 1, 7, 25, 61, 121, 283, 841, 2521, 6769, 17119, 44665, 123685, 347497, 954787, 2578297, 7001617, 19307089, 53601175, 148305817, 408681997, 1127558041, 3124344427, 8681565865, 24127128841, 67020060721, 186282015823
Offset: 0

Views

Author

Paul Barry, Sep 13 2004

Keywords

Comments

Binomial transform of A098537.

Links

Crossrefs

Cf. A098536.

Programs

  • Magma
    Q:=Rationals(); R:=PowerSeriesRing(Q,30); Coefficients(R!(1/((1-x)^3-18*x^4)^(1/3))); // G. C. Greubel, Jan 17 2018
  • Mathematica
    CoefficientList[Series[1/((1-x)^3-18*x^4)^(1/3), {x,0,50}], x] (* G. C. Greubel, Jan 17 2018 *)
  • PARI
    x='x+O('x^30); Vec(1/((1-x)^3-18*x^4)^(1/3)) \\ G. C. Greubel, Jan 17 2018

A376803 Expansion of 1/((1 - x)^3 - 9*x^2)^(1/3).

Original entry on oeis.org

1, 1, 4, 13, 49, 187, 736, 2941, 11908, 48682, 200584, 831712, 3466867, 14515411, 61005634, 257238349, 1087792225, 4611606373, 19594364860, 83421726877, 355801896895, 1519998686401, 6503081167372, 27859917707863, 119502218725576, 513173645933326
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2024

Keywords

Crossrefs

Cf. A098536, A376802, A376804.

Programs

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

Formula

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

A376804 Expansion of 1/((1 - x)^3 - 9*x^3)^(1/3).

Original entry on oeis.org

1, 1, 1, 4, 13, 31, 79, 232, 673, 1891, 5401, 15742, 45958, 134122, 393394, 1159432, 3425101, 10137985, 30079405, 89437960, 266389615, 794667325, 2374097485, 7102303240, 21272892055, 63788000461, 191471030791, 575287348546, 1730027151334, 5206918491298
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2024

Keywords

Crossrefs

Cf. A098536, A376802, A376803.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/3)} (-9)^k * binomial(-1/3,k) * binomial(n,n-3*k).
Showing 1-5 of 5 results.

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