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.

A357347 E.g.f. satisfies A(x) = (exp(x * exp(A(x))) - 1) * exp(2 * A(x)).

Original entry on oeis.org

0, 1, 7, 103, 2385, 75756, 3064239, 150689953, 8729691693, 582299930167, 43956280309659, 3704637865439380, 344825037782332457, 35131983926187957173, 3888817094785288023367, 464724955485177444101895, 59631976064836824117227621, 8177487264101392841050876136
Offset: 0

Views

Author

Seiichi Manyama, Sep 25 2022

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=1, n, (n+2*k)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} (n+2*k)^(k-1) * Stirling2(n,k).
E.g.f.: Series_Reversion( exp(-x) * log(1 + x * exp(-2*x)) ). - Seiichi Manyama, Sep 09 2024

A357346 E.g.f. satisfies A(x) = (exp(x * exp(A(x))) - 1) * exp(A(x)).

Original entry on oeis.org

0, 1, 5, 52, 849, 18996, 540986, 18726247, 763480675, 35837071558, 1903538106065, 112880374866172, 7392418912962210, 529898419942327801, 41266682731537698181, 3469461853041348996044, 313200848521114144611273, 30215925892728362737156556
Offset: 0

Views

Author

Seiichi Manyama, Sep 25 2022

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=1, n, (n+k)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} (n+k)^(k-1) * Stirling2(n,k).
E.g.f.: Series_Reversion( exp(-x) * log(1 + x * exp(-x)) ). - Seiichi Manyama, Sep 09 2024

A357424 E.g.f. satisfies A(x) * exp(A(x)) = exp(x * exp(A(x))) - 1.

Original entry on oeis.org

0, 1, 1, 4, 21, 156, 1470, 16843, 227367, 3533974, 62163477, 1220852524, 26480355110, 628693388909, 16216901961481, 451609382251836, 13504072800481613, 431544662700594212, 14677503631085378170, 529370720888418692643, 20180856622352239827687
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2022

Keywords

Crossrefs

Programs

  • Mathematica
    Join[{0,1},Table[Sum[(n-k)^(k-1) * StirlingS2[n,k], {k,1,n}], {n,2,20}]] (* Vaclav Kotesovec, Nov 14 2022 *)
  • PARI
    a(n) = sum(k=1, n, (n-k)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} (n-k)^(k-1) * Stirling2(n,k).
a(n) ~ n^(n-1) * (1 + exp(s)*s)^(n + 1/2) / (sqrt(exp(s)*(1 + s + s^2) - 1) * exp(n) * (1 + s)^(n - 1/2)), where s = 1.104072744884035178291292242554731... is the root of the equation 1 + s = (exp(-s) + s) * log(1 + exp(s)*s). - Vaclav Kotesovec, Nov 14 2022
E.g.f.: Series_Reversion( exp(-x) * log(1 + x * exp(x)) ). - Seiichi Manyama, Sep 09 2024
Showing 1-3 of 3 results.