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.

A375662 Expansion of e.g.f. 1 / (1 - x^2 * (exp(x) - 1))^2.

Original entry on oeis.org

1, 0, 0, 12, 24, 40, 2220, 15204, 70672, 1723824, 22710420, 202577980, 3841065624, 71221859592, 994632663388, 19005155049300, 421055077585440, 8033764197776224, 172109549363348772, 4285658639255113836, 101794836650015825320, 2516190299149752959160
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^2*(exp(x)-1))^2))
    
  • PARI
    a(n) = n!*sum(k=0, n\3, (k+1)!*stirling(n-2*k, k, 2)/(n-2*k)!);

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A358013.
a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)! * Stirling2(n-2*k,k)/(n-2*k)!.

A375679 Expansion of e.g.f. 1 / (1 + x^2 * log(1 - x))^3.

Original entry on oeis.org

1, 0, 0, 18, 36, 120, 4860, 33264, 241920, 5598720, 72364320, 879500160, 18172978560, 331463508480, 5726430597888, 126134466796800, 2836325702246400, 62773403361177600, 1562890149787392000, 41009994647421972480, 1090182759179092992000
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x^2*log(1-x))^3))
    
  • PARI
    a(n) = n!*sum(k=0, n\3, (k+2)!*abs(stirling(n-2*k, k, 1))/(n-2*k)!)/2;

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A351503.
a(n) = (n!/2) * Sum_{k=0..floor(n/3)} (k+2)! * |Stirling1(n-2*k,k)|/(n-2*k)!.

A376439 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^2*(exp(x) - 1))^3 ).

Original entry on oeis.org

1, 0, 0, 18, 36, 60, 23850, 189126, 988008, 184207176, 3254640750, 35132272890, 4418970811596, 134653558474188, 2463781708180338, 246532610826062190, 11098269938629561680, 305828547775319369616, 27016544700449293891158
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^2*(exp(x)-1))^3)/x))
    
  • PARI
    a(n) = 3*n!*sum(k=0, n\3, (3*n+k+2)!*stirling(n-2*k, k, 2)/(n-2*k)!)/(3*n+3)!;

Formula

E.g.f. A(x) satisfies A(x) = 1/(1 - x^2*A(x)^2 * (exp(x*A(x)) - 1))^3.
a(n) = (3 * n!/(3*n+3)!) * Sum_{k=0..floor(n/3)} (3*n+k+2)! * Stirling2(n-2*k,k)/(n-2*k)!.
Showing 1-3 of 3 results.