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.

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A354393 Expansion of e.g.f. 1/(1 + (exp(x) - 1)^4 / 24).

Original entry on oeis.org

1, 0, 0, 0, -1, -10, -65, -350, -1631, -5250, 18395, 685850, 10485739, 127737610, 1336804105, 11432407350, 54280609109, -712071643930, -29671691715185, -660215774400350, -11770593620859521, -176475952496559870, -2055362595355830475, -9749893741512339250
Offset: 0

Views

Author

Seiichi Manyama, May 25 2022

Keywords

Crossrefs

Cf. A354391, A354392, A354394.
Cf. A346895, A354390, A354397.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+(exp(x)-1)^4/24)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, binomial(i, j)*stirling(j, 4, 2)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/(-24)^k);

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * Stirling2(k,4) * a(n-k).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k)/(-24)^k.

A354389 Expansion of e.g.f. 1/(1 + log(1 + x)^2 / 2).

Original entry on oeis.org

1, 0, -1, 3, -5, -10, 146, -756, 1086, 23400, -300066, 1855590, 341826, -165915828, 2158958556, -10006622640, -172337345496, 4941605486016, -64365944851512, 339328464492456, 5510899593823176, -157099566384759600, 1059259019507498160, 41957473280879898720
Offset: 0

Views

Author

Seiichi Manyama, May 25 2022

Keywords

Crossrefs

Cf. A346921, A354317, A354390, A354391.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1+(Log[1+x]^2)/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 08 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1+x)^2/2)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, binomial(i, j)*stirling(j, 2, 1)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 1)/(-2)^k);

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * Stirling1(k,2) * a(n-k).
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling1(n,2*k)/(-2)^k.
Showing 1-2 of 2 results.

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