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.

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A278668 Expansion of Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^3 in powers of x.

Original entry on oeis.org

1, 3, 9, 22, 51, 107, 218, 420, 788, 1428, 2531, 4375, 7430, 12377, 20313, 32833, 52402, 82585, 128750, 198588, 303428, 459375, 689710, 1027243, 1518709, 2229375, 3251022, 4710777, 6785378, 9717677, 13841991, 19614182, 27656250, 38810312, 54216128, 75406438
Offset: 0

Views

Author

Seiichi Manyama, Nov 25 2016

Keywords

Examples

			G.f.: 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 107*x^5 + 218*x^6 + ...

Links

Crossrefs

Cf. Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^k: A035959 (k=1), A160461 (k=2), this sequence (k=3), A278680 (k=4), A277212 (k=5), A182821 (k=6).

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

Formula

G.f.: Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^3.
a(n) ~ exp(2*Pi*sqrt(7*n/15)) * 7^(3/4) / (20 * 3^(3/4) * 5^(1/4) * n^(5/4)). - Vaclav Kotesovec, Nov 10 2017

A278680 Expansion of Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^4 in powers of x.

Original entry on oeis.org

1, 4, 14, 40, 105, 251, 570, 1226, 2540, 5075, 9855, 18630, 34439, 62340, 110805, 193624, 333235, 565415, 947040, 1567130, 2564425, 4152535, 6658711, 10579380, 16663755, 26033200, 40357641, 62106290, 94912385, 144088840, 217368655, 325945320, 485950150, 720515475
Offset: 0

Views

Author

Seiichi Manyama, Nov 25 2016

Keywords

Examples

			G.f.: 1 + 4*x + 14*x^2 + 40*x^3 + 105*x^4 + 251*x^5 + 570*x^6 + ...

Links

Crossrefs

Cf. Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^k: A035959 (k=1), A160461 (k=2), A278668 (k=3), this sequence (k=4), A277212 (k=5), A182821 (k=6).

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

Formula

G.f.: Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^4.
a(n) ~ 19 * exp(Pi*sqrt(38*n/15)) / (120 * sqrt(10) * n^(3/2)). - Vaclav Kotesovec, Nov 10 2017

A319362 a(n) = [x^n] exp(Sum_{k>=1} sigma(n*k)*x^k/k).

Original entry on oeis.org

1, 1, 8, 39, 385, 917, 31247, 22527, 1081986, 2464860, 50099635, 14931071, 19684696065, 394805109, 82267017929, 496514888157, 11386442827781, 284625019799, 3469798073972537, 7725084195239, 136470024990370842, 28400489198168457, 241211623942678951, 5776331152550399
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 17 2018

Keywords

Links

Crossrefs

Cf. A000041, A182818, A182819, A182820, A182821, A283077, A283119, A283120, A283121, A319363.

Programs

  • Mathematica
    Table[SeriesCoefficient[Exp[Sum[DivisorSigma[1, n k] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Previous Showing 11-13 of 13 results.

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