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.

A292415 a(n) = [x^n] Product_{k>=1} (1 / (1 - 2^n*x^k)).

Original entry on oeis.org

1, 2, 20, 584, 70160, 34670624, 69827571776, 567417533153408, 18519367937794769152, 2422592506675732350501376, 1268890961666701371908767613952, 2659755334364276805914758224778627072, 22306192375831301664022382396371369078034432
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 16 2017

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 15; Table[SeriesCoefficient[Product[1/(1-2^n*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
  • PARI
    {a(n)= polcoef(prod(k=1, n, 1/(1-2^n*x^k +x*O(x^n))), n)};
    for(n=0,20, print1(a(n), ", ")) \\ G. C. Greubel, Feb 02 2019

Formula

a(n) ~ 2^(n^2).

A292416 a(n) = [x^n] Product_{k>=1} (1 + 2^n*x^k) / (1 - 2^n*x^k).

Original entry on oeis.org

1, 4, 40, 1296, 149024, 71573568, 141871849600, 1143771307901184, 37183988027710374400, 4854666820584582571623424, 2540262650941956832151944038400, 5322109355556594174041950822678401024, 44623279107562668799968801377926722975965184
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 16 2017

Keywords

Comments

Convolution of A292414 and A292415.

Crossrefs

Programs

  • Mathematica
    nmax = 15; Table[SeriesCoefficient[Product[(1+2^n*x^k)/(1-2^n*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
  • PARI
    {a(n)= polcoef(prod(k=1, n, ((1+2^n*x^k)/(1-2^n*x^k) +x*O(x^n))), n)};
    for(n=0,20, print1(a(n), ", ")) \\ G. C. Greubel, Feb 02 2019

Formula

a(n) ~ 2^(n^2 + 1).

A318246 a(n) = [x^n] Product_{k>=1} (1 + 3^n*x^k).

Original entry on oeis.org

1, 3, 9, 756, 6642, 118341, 388484100, 10474704297, 564988219686, 22878342156600, 12158489037532504050, 984798697643349485688, 159533936817604246934415, 19383278088136495245171156, 2616739259326831261950662430, 608267042060342812170824926328855679
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 22 2018

Keywords

Comments

Conjecture: In general, if m > 1 and a(n) = [x^n] Product_{k>=1} (1 + m^n * x^k), then log(a(n)) ~ log(m)*(sqrt(2)*n^(3/2) - n/2).

Crossrefs

Cf. A292414.

Programs

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

Formula

Conjecture: log(a(n)) ~ log(3)*sqrt(2)*n^(3/2).

A318247 a(n) = [x^n] Product_{k>=1} (1 + n!*x^k).

Original entry on oeis.org

1, 1, 2, 42, 600, 28920, 374285520, 128100273840, 131101518683520, 143354704247556480, 173401404266683545849388800, 2538767479410416957720411116800, 105287752487031026606448840363801600, 4510685217145833106538730603088118860800, 288804138719404983322786510403231912442931200
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 22 2018

Keywords

Crossrefs

Programs

  • Mathematica
    nmax = 15; Table[SeriesCoefficient[Product[(1+n!*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
Showing 1-4 of 4 results.