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.

A379993 Expansion of e.g.f. 1/(1 - x * exp(x))^4.

Original entry on oeis.org

1, 4, 28, 252, 2776, 35940, 533304, 8908228, 165247072, 3368072196, 74782987240, 1796037420804, 46379441090448, 1281203788073092, 37694510810334616, 1176606639075726660, 38833052393329645504, 1351066066253778043908, 49417629820950190273992
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Programs

  • Mathematica
    nmax=18;CoefficientList[Series[1/(1 - x * Exp[x])^4,{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Feb 05 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(k+3, 3)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(k+3,3)/(n-k)!.
a(n) == 0 (mod 4) for n>0.

A379994 Expansion of e.g.f. exp(-3*x)/(exp(-x) - x)^4.

Original entry on oeis.org

1, 5, 37, 349, 3969, 52641, 796069, 13502693, 253667297, 5225690017, 117090480021, 2834363683317, 73697918467105, 2048252470006913, 60587779740857573, 1900347489736371301, 62992469337321611073, 2200238756416513719489, 80765631192992760237205
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(n-k)*binomial(k+3, 3)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * binomial(k+3,3)/(n-k)!.
a(n) == 1 (mod 4).

A379996 Expansion of e.g.f. exp(-x)/(exp(-x) - x)^4.

Original entry on oeis.org

1, 7, 61, 639, 7825, 109683, 1731645, 30403495, 587595649, 12395233539, 283385424829, 6979650164391, 184235963026833, 5188528445210035, 155284012799863453, 4921569063327156807, 164672737994453759617, 5800532536265417597571, 214559605389429001486557
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[-x]/(Exp[-x]-x)^4,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 19 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, (k+3)^(n-k)*binomial(k+3, 3)/(n-k)!);

Formula

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