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.

A379933 Expansion of e.g.f. 1/( exp(-x) - x )^2.

Original entry on oeis.org

1, 4, 22, 158, 1408, 15002, 186100, 2634998, 41937136, 741170834, 14402727484, 305225470046, 7005711916840, 173134991854970, 4583675648417044, 129424786945875398, 3882446011526729440, 123304773913531035170, 4133369745467043807340, 145840627118145774415214
Offset: 0

Views

Author

Seiichi Manyama, Jan 06 2025

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(exp(-x)-x)^2))
    
  • PARI
    a(n) = n!*sum(k=0, n, (k+1)*(k+2)^(n-k)/(n-k)!);

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A072597.
a(n) = n! * Sum_{k=0..n} (k+1) * (k+2)^(n-k)/(n-k)!.

A379943 Expansion of e.g.f. 1/( exp(-x) - x )^4.

Original entry on oeis.org

1, 8, 76, 844, 10776, 155844, 2520856, 45125924, 886037216, 18938440324, 437820992136, 10886467502244, 289738784758096, 8218731027307844, 247539834718198136, 7889896358130120484, 265325716114102815936, 9388476560982511842564, 348703400008471862936296
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(exp(-x)-x)^4))
    
  • PARI
    a(n) = n!*sum(k=0, n, (k+4)^(n-k)*binomial(k+3, 3)/(n-k)!);

Formula

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A072597.
a(n) = n! * Sum_{k=0..n} (k+4)^(n-k) * binomial(k+3,3)/(n-k)!.

A379990 Expansion of e.g.f. exp(-2*x)/(exp(-x) - x)^3.

Original entry on oeis.org

1, 4, 25, 205, 2065, 24601, 337837, 5249581, 91006657, 1740663937, 36402220141, 826159146253, 20220201899377, 530828186303377, 14878044338021677, 443397290411503021, 14000282854007503105, 466866129420834410881, 16395362179348570608205, 604794784980600986425645
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+2, 2)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * binomial(k+2,2)/(n-k)!.
a(n) ~ n! * n^2 / (2 * (LambertW(1) + 1)^3 * LambertW(1)^(n+1)). - Vaclav Kotesovec, Jan 08 2025

A379991 Expansion of e.g.f. exp(-x)/(exp(-x) - x)^3.

Original entry on oeis.org

1, 5, 34, 293, 3052, 37247, 520918, 8211065, 143997160, 2780465147, 58611713434, 1339236707525, 32969142927580, 869913580853735, 24490615055384062, 732751463209179953, 23217990692393685712, 776689510407852177011, 27353286267329098887202
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (k+2)^(n-k)*binomial(k+2, 2)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (k+2)^(n-k) * binomial(k+2,2)/(n-k)!.
Showing 1-4 of 4 results.