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 1-3 of 3 results.

A304446 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(n^2).

Original entry on oeis.org

1, 1, 14, 255, 6460, 209405, 8287038, 387605491, 20930373880, 1281932464680, 87828985857380, 6656774777650459, 553068813860022264, 49988877225605011590, 4883606791114233989450, 512829418039842285746460, 57607740718731604241384432, 6893420862444517638234527039
Offset: 0

Views

Author

Vaclav Kotesovec, May 12 2018

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 20; Table[SeriesCoefficient[Product[1/(1-x^k)^(n^2), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
    nmax = 20; Table[SeriesCoefficient[1/QPochhammer[x]^(n^2), {x, 0, n}], {n, 0, nmax}]

Formula

a(n) ~ exp(n + 3/2) * n^(n - 1/2) / sqrt(2*Pi).

A304448 Coefficient of x^n in Product_{k>=1} ((1+x^k)/(1-x^k))^(n^2).

Original entry on oeis.org

1, 2, 40, 1320, 61984, 3797560, 287566368, 25957422400, 2721948311680, 325260627848442, 43635601119149040, 6494550360714973304, 1062063969900788407680, 189301256401392643093560, 36526821128512112807216192, 7585918627122817713267856320
Offset: 0

Views

Author

Vaclav Kotesovec, May 12 2018

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 20; Table[SeriesCoefficient[Product[((1+x^k)/(1-x^k))^(n^2), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
    nmax = 20; Table[SeriesCoefficient[(QPochhammer[-1, x]/2/QPochhammer[x])^(n^2), {x, 0, n}], {n, 0, nmax}]

Formula

a(n) ~ 2^(n - 1/2) * exp(n + 1/2) * n^(n - 1/2) / sqrt(Pi).

A304459 Coefficient of x^n in Product_{k>=1} (1+x^k)^(n^3).

Original entry on oeis.org

1, 1, 36, 3681, 770576, 276218900, 151479085752, 117975860569973, 123825991870849088, 168480096257782525419, 288418999876101261408100, 606652152400218992684850772, 1537897976017806908644807294656, 4624364862288125600795358272563097
Offset: 0

Views

Author

Vaclav Kotesovec, May 13 2018

Keywords

Comments

In general, for m>=3, coefficient of x^n in Product_{k>=1} (1+x^k)^(n^m) is asymptotic to n^(m*n)/n!.

Crossrefs

Programs

  • Mathematica
    nmax = 20; Table[SeriesCoefficient[Product[(1+x^k)^(n^3), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
    nmax = 20; Table[SeriesCoefficient[(QPochhammer[-1, x]/2)^(n^3), {x, 0, n}], {n, 0, nmax}]

Formula

a(n) ~ exp(n) * n^(2*n - 1/2) / sqrt(2*Pi).
Showing 1-3 of 3 results.