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.

A377428 Expansion of e.g.f. (1/x) * Series_Reversion( x*(2 - exp(x))^4 ).

Original entry on oeis.org

1, 4, 56, 1432, 54184, 2734104, 173032680, 13192623448, 1177932112040, 120610734752920, 13935516914366824, 1793837540679492312, 254604546529825454376, 39504947952102355425304, 6652925600854130108675048, 1208610940763303680263653464, 235601431979292206398224418216
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(2-exp(x))^4)/x))
    
  • PARI
    a(n) = 4*sum(k=0, n, (4*n+k+3)!*stirling(n, k, 2))/(4*n+4)!;

Formula

E.g.f. A(x) satisfies A(x) = 1/(2 - exp(x*A(x)))^4.
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377424.
a(n) = (4/(4*n+4)!) * Sum_{k=0..n} (4*n+k+3)! * Stirling2(n,k).

A376391 Expansion of e.g.f. ( (1/x) * Series_Reversion( x*(2 - exp(x))^3 ) )^(2/3).

Original entry on oeis.org

1, 2, 20, 386, 11252, 441722, 21867764, 1308580226, 91904288420, 7413237414602, 675503178005108, 68631619821747842, 7693344955213551428, 943236099444038389082, 125565496331888560573172, 18037220418654308659836674, 2780985275750966018759898212, 458079154394191702424821932842
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*(2-exp(x))^3)/x)^(2/3)))
    
  • PARI
    a(n) = 2*sum(k=0, n, (3*n+k+1)!*stirling(n, k, 2))/(3*n+2)!;

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A367135.
a(n) = (2/(3*n+2)!) * Sum_{k=0..n} (3*n+k+1)! * Stirling2(n,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.