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.

A293493 Expansion of e.g.f. exp(x/(1 - x^3)).

Original entry on oeis.org

1, 1, 1, 1, 25, 121, 361, 5881, 62161, 365905, 5145841, 84831121, 812318761, 11450816521, 243450591385, 3371056121161, 51784222133281, 1275605799044641, 23531635543402081, 410559590525890465, 11089633716053137081, 256375957896260034841, 5161258224477109736521
Offset: 0

Views

Author

Seiichi Manyama, Oct 10 2017

Keywords

Crossrefs

Cf. A293494.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(1-x^3))))
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(3*k-2)))))

Formula

E.g.f.: Product_{k>0} exp(x^(3*k-2)).
a(n) ~ exp(2*sqrt(3*n)/3 - n + 1/6) * n^(n-1/4) / (sqrt(2) * 3^(1/4)). - Vaclav Kotesovec, Oct 10 2017
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * (3*k+1)! * a(n-3*k-1). - Ilya Gutkovskiy, Feb 24 2022
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k)/(n-3*k)!. - Seiichi Manyama, Jun 08 2024

A293526 E.g.f.: exp(x^3/(1 - x^4)).

Original entry on oeis.org

1, 0, 0, 6, 0, 0, 360, 5040, 0, 60480, 3628800, 39916800, 19958400, 3113510400, 130767436800, 1318571654400, 3487131648000, 355687428096000, 12813639597158400, 126713646259200000, 1013709170073600000, 85161707377883136000, 2819368492175499264000
Offset: 0

Views

Author

Seiichi Manyama, Oct 11 2017

Keywords

Crossrefs

E.g.f.: Product_{k>0} exp(x^(m*k-1)): A088009 (m=2), A293494 (m=3), this sequence (m=4).
Cf. A293507.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(x^3/(1-x^4))))
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(4*k-1)))))

Formula

E.g.f.: Product_{k>0} exp(x^(4*k-1)).
a(n) ~ exp(sqrt(n) - n - 1/4) * n^(n - 1/4) / 2. - Vaclav Kotesovec, Oct 15 2017

A293567 Expansion of e.g.f.: exp(x^2/(x^3 - 1)).

Original entry on oeis.org

1, 0, -2, 0, 12, -120, -120, 5040, -38640, -181440, 5412960, -33264000, -478336320, 12194582400, -50871300480, -2168559993600, 49692144902400, -59775248332800, -15819216007795200, 329479616206540800, 1101564635255884800, -174845824790757120000
Offset: 0

Views

Author

Seiichi Manyama, Oct 12 2017

Keywords

Crossrefs

E.g.f.: Product_{k>0} exp(x^(-(m*k-1))): A293532 (m=2), this sequence (m=3), A293568 (m=4).
Cf. A293494.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x^2/(x^3-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 31 2024 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(x^2/(x^3-1))))
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, exp(x^(3*k-1)))))

Formula

E.g.f.: Product_{k>0} exp(x^(-(3*k-1))).

Extensions

Definition clarified by Harvey P. Dale, Mar 31 2024
Showing 1-3 of 3 results.

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