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 1-4 of 4 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

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

Links

Crossrefs

Cf. A052888, A357347, A357348, A357424.
Cf. A048802, A349557.

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

A357348 E.g.f. satisfies A(x) = (exp(x * exp(A(x))) - 1) * exp(3 * A(x)).

Original entry on oeis.org

0, 1, 9, 172, 5181, 214196, 11279542, 722242795, 54482959375, 4732518179422, 465226448603533, 51061919634063284, 6189640391474229790, 821277806639279795053, 118394082630978607655201, 18426248367244130561233924, 3079294928622816257125500821
Offset: 0

Views

Author

Seiichi Manyama, Sep 25 2022

Keywords

Links

Crossrefs

Cf. A052888, A357346, A357347, A357424.
Cf. A357345.

Programs

  • PARI
    a(n) = sum(k=1, n, (n+3*k)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} (n+3*k)^(k-1) * Stirling2(n,k).
E.g.f.: Series_Reversion( exp(-x) * log(1 + x * exp(-3*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

Links

Crossrefs

Cf. A052888, A357346, A357347, A357348.
Cf. A349588.

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 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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