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.

A354230 Expansion of e.g.f. 1/(1 - log(1 + x)^5).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, -1800, 21000, -235200, 2693880, -28690200, 210447600, 1465952400, -123513355680, 4155643171680, -114924516470400, 2886135295680000, -66750668391381120, 1375830884058456960, -22036006671394705920, 70186623981895296000, 16180846322732941893120
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A006252, A354229.
Cf. A353200, A354135, A354232.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1-Log[1+x]^5),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 20 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x)^5)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i, j)*stirling(j, 5, 1)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 1));

Formula

a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n,k) * Stirling1(k,5) * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling1(n,5*k).

A354231 Expansion of e.g.f. exp(log(1 + x)^3).

Original entry on oeis.org

1, 0, 0, 6, -36, 210, -990, 2184, 37128, -863736, 13020480, -168384744, 1940801544, -18825129648, 107706637584, 1386022834944, -73429347222720, 2034345021802560, -46869707752067520, 976421492688165120, -18675350766042871680, 319467427583225518080
Offset: 0

Views

Author

Seiichi Manyama, May 20 2022

Keywords

Crossrefs

Cf. A009199, A354232.
Cf. A353344, A354136, A354229.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(log(1+x)^3)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^log(1+x)^2))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 3, 1)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 1)/k!);

Formula

E.g.f.: (1 + x)^(log(1 + x)^2).
a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling1(k,3) * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling1(n,3*k)/k!.

A354250 Expansion of e.g.f. Sum_{k>=0} (3*k)! * log(1+x)^k / k!.

Original entry on oeis.org

1, 6, 714, 360732, 476832204, 1302897016944, 6382799223892560, 50956720815425427360, 619019914356960664044960, 10866561174598537960652828160, 264763399994627082733034386813440, 8668743073576807048450006051943930880
Offset: 0

Views

Author

Seiichi Manyama, May 21 2022

Keywords

Crossrefs

Cf. A006252, A354243.
Cf. A316748, A354229, A354251.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (3*k)!*log(1+x)^k/k!)))
  • PARI
    a(n) = sum(k=0, n, (3*k)!*stirling(n, k, 1));

Formula

a(n) = Sum_{k=0..n} (3*k)! * Stirling1(n,k).
Showing 1-3 of 3 results.

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

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