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.

A291979 a(n) = (-1)^n*n!*[x^n] exp(-x)/(1 + log(1+x)).

Original entry on oeis.org

1, 2, 6, 27, 167, 1310, 12394, 137053, 1733325, 24670114, 390204086, 6789564639, 128884276179, 2650516064222, 58701784670138, 1392959655437473, 35257885037803417, 948208649740610466, 27000743345935785670, 811575543670852269347, 25677856392014665436799
Offset: 0

Views

Author

Peter Luschny, Sep 16 2017

Keywords

Comments

Row sums of A291978.

Links

Crossrefs

Cf. A132393, A291978, A343707, A343709, A343710.

Programs

  • Maple
    a_list := proc(n) exp(-x)/(1 + log(1+x)): series(%, x, n+1):
seq((-1)^k*k!*coeff(%, x, k), k=0..n) end: a_list(20);
  • Mathematica
    nmax = 20; CoefficientList[Series[E^(-x)/(1 + Log[1+x]), {x, 0, nmax}], x] * Range[0, nmax]! * (-1)^Range[0, nmax] (* Vaclav Kotesovec, Sep 18 2017 *)
  • PARI
    N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/(1+log(1-x)))) \\ Seiichi Manyama, Oct 20 2021

Formula

a(n) ~ sqrt(2*Pi) * n^(n+1/2) * exp(1 - exp(-1)) / (exp(1)-1)^(n+1). - Vaclav Kotesovec, Sep 18 2017
a(n) = 1 + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k). - Ilya Gutkovskiy, Apr 26 2021
a(n) = Sum_{k=0..n} Sum_{j=0..k} binomial(n,k)*j!*A132393(k,j). - Fabian Pereyra, Aug 29 2024

A343707 a(n) = 1 + 2 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 3, 15, 113, 1145, 14539, 221663, 3943281, 80173345, 1833831619, 46606646175, 1302954958689, 39737420405753, 1312901360002283, 46714233470065999, 1780859204826798401, 72416689888874547969, 3128792006916853876291, 143132514626658326870767, 6911638338982428907738641
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2021

Keywords

Links

Crossrefs

Cf. A088500, A201339, A291979, A343709, A343710.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + 2 Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Exp[x]/(1 + 2 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/(1+2*log(1-x)))) \\ Seiichi Manyama, Oct 20 2021

Formula

E.g.f.: exp(x) / (1 + 2 * log(1 - x)).
a(n) = Sum_{k=0..n} binomial(n,k) * A088500(k).

A343709 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 4, 28, 295, 4159, 73348, 1552468, 38336569, 1081926157, 34350646636, 1211796777748, 47023762576987, 1990643657768683, 91291802205304972, 4508735102829489580, 238583762726054522989, 13466532093135977880025, 807606110028529741369396, 51282242176105846536128236
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2021

Keywords

Links

Crossrefs

Cf. A201354, A291979, A343707, A343710.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + 3 Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[x]/(1 + 3 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/(1+3*log(1-x)))) \\ Seiichi Manyama, Oct 20 2021

Formula

E.g.f.: exp(x) / (1 + 3 * log(1 - x)).
Showing 1-3 of 3 results.

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