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.

A377425 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^2))^2.

Original entry on oeis.org

1, 2, 24, 572, 20788, 1021892, 63498116, 4776128772, 422019084132, 42854861672612, 4918270207805188, 629575456637707076, 88938171122678982692, 13744507646644260776292, 2306659049841490720035780, 417774877069420589127228164, 81222489094387608969950071780
Offset: 0

Views

Author

Seiichi Manyama, Oct 28 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = 2*sum(k=0, n, (4*n+k+1)!*stirling(n, k, 2))/(4*n+2)!;

Formula

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

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).

A376438 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^2*(exp(x) - 1))^2 ).

Original entry on oeis.org

1, 0, 0, 12, 24, 40, 10860, 85764, 446992, 57788784, 1008736020, 10835748220, 965748698904, 28637803537512, 519426455756572, 37968161216666100, 1626852405783259680, 44177643556314690784, 2957776991432290423332, 163869985958022692795628, 6132727345895339422510120, 405409522521171206216078040
Offset: 0

Views

Author

Seiichi Manyama, Sep 22 2024

Keywords

Crossrefs

Programs

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

Formula

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