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.

A380663 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x/(1 - x)) ).

Original entry on oeis.org

1, 2, 15, 208, 4285, 117936, 4075099, 169736960, 8282604537, 463604723200, 29287449579751, 2061571190059008, 160023548976361525, 13580237335641417728, 1250935473495646861875, 124307671411309327876096, 13255531892787507819759601, 1509841440567809574906101760
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(2*n, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = exp(x * A(x)/(1 - x*A(x)))/(1 - x*A(x)).
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n,n-k)/k!.

A380673 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x * (1 - x)) ).

Original entry on oeis.org

1, 2, 11, 106, 1501, 28416, 677839, 19566128, 663801849, 25897000960, 1142424023731, 56232973813248, 3055417111781269, 181644488496644096, 11728204122824976375, 817281148114199197696, 61136484485752079320561, 4886365932210442324672512, 415573028022035962921316059
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(2*n-2*k, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = exp(x * A(x) * (1 - x*A(x)))/(1 - x*A(x)).
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n-2*k,n-k)/k!.
a(n) = A277184(n+1)/(n+1).

A380762 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * (1 + x)^2) / (1 + x) ).

Original entry on oeis.org

1, 2, 15, 208, 4249, 115656, 3946879, 162225680, 7807264497, 430828353280, 26825288214031, 1860715287986688, 142304071119852745, 11897080341213068288, 1079508321205459768575, 105660694801273960216576, 11097101798773200862180321, 1244852059489783737208012800
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n, q=1, r=1, s=1, t=2, u=1) = q*n!*sum(k=0, n, (r*n+(s-r)*k+q)^(k-1)*binomial(r*u*n+((s-r)*u+t)*k+q*u, n-k)/k!);

Formula

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