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.

A357335 E.g.f. satisfies A(x) = (exp(x) - 1) * exp(2 * A(x)).

Original entry on oeis.org

0, 1, 5, 49, 757, 16081, 435477, 14345297, 556857973, 24894290257, 1259621627349, 71165987957329, 4440821632449077, 303338709537825105, 22512353926895739797, 1803812930088064925265, 155195078834104237961717, 14270228623788585753803089
Offset: 0

Views

Author

Seiichi Manyama, Sep 24 2022

Keywords

Links

Crossrefs

Cf. A048802, A357336.
Cf. A349524, A356000.
Cf. A357347.

Programs

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

Formula

E.g.f.: -LambertW(2 * (1 - exp(x)))/2.
a(n) = Sum_{k=1..n} (2 * k)^(k-1) * Stirling2(n,k).
a(n) ~ sqrt(1 + 2*exp(1)) * n^(n-1) / (2 * exp(n) * log(1 + exp(-1)/2)^(n - 1/2)). - Vaclav Kotesovec, Nov 14 2022
E.g.f.: Series_Reversion( log(1 + x * exp(-2*x)) ). - Seiichi Manyama, Sep 09 2024

A356001 Expansion of e.g.f. -LambertW((1 - exp(3*x))/3).

Original entry on oeis.org

0, 1, 5, 36, 379, 5461, 100476, 2250613, 59432141, 1807959042, 62262816157, 2394551966401, 101724440338494, 4730814590128615, 239057921691911861, 13042779411190737420, 764136645388807739239, 47846833035272035228849, 3188740106752561252031364
Offset: 0

Views

Author

Seiichi Manyama, Sep 24 2022

Keywords

Links

Crossrefs

Cf. A048802, A356000.
Cf. A357336.

Programs

  • Mathematica
    With[{m = 20}, Range[0, m]! * CoefficientList[Series[-ProductLog[(1 - Exp[3*x])/3], {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw((1-exp(3*x))/3))))
  • PARI
    a(n) = sum(k=1, n, 3^(n-k)*k^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} 3^(n-k) * k^(k-1) * Stirling2(n,k).
a(n) ~ 3^(n - 1/2) * sqrt(exp(1) + 3) * n^(n-1) / (exp(n) * (log(exp(1) + 3) - 1)^(n - 1/2)). - Vaclav Kotesovec, Oct 04 2022
E.g.f.: Series_Reversion( (log(1 + 3 * x * exp(-x)))/3 ). - Seiichi Manyama, Sep 11 2024

A357395 E.g.f. satisfies A(x) = exp(x * exp(3 * A(x))) - 1.

Original entry on oeis.org

0, 1, 7, 109, 2677, 90226, 3873007, 202134997, 12427851625, 879806921041, 70486590597331, 6304879010400202, 622838214328334077, 67347956304168803173, 7911963620634266270071, 1003477119181096373029261, 136658009168055564212000209, 19889317400287888238121299854
Offset: 0

Views

Author

Seiichi Manyama, Sep 26 2022

Keywords

Links

Crossrefs

Cf. A052888, A357394.
Cf. A357336.

Programs

  • Mathematica
    Table[Sum[(3*n)^(k-1) * StirlingS2[n, k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 14 2022 *)
  • PARI
    a(n) = sum(k=1, n, (3*n)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} (3 * n)^(k-1) * Stirling2(n,k).
a(n) ~ n^(n-1) / (3 * sqrt(1 + LambertW(1/3)) * LambertW(1/3)^n * exp(n*(4 - 1/LambertW(1/3)))). - Vaclav Kotesovec, Nov 14 2022
E.g.f.: Series_Reversion( exp(-3*x) * log(1 + x) ). - Seiichi Manyama, Sep 10 2024
Showing 1-3 of 3 results.

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