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.

A375796 Expansion of e.g.f. 1/(1 - (exp(x^3) - 1)/x^2).

Original entry on oeis.org

1, 1, 2, 6, 36, 240, 1800, 15960, 164640, 1905120, 24343200, 342619200, 5269017600, 87749101440, 1573083832320, 30218175187200, 619256461824000, 13483023576422400, 310821905134540800, 7563477205380096000, 193736838233562624000, 5210638309494858240000
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x^3)-1)/x^2)))
    
  • PARI
    a(n) = n!*sum(k=0, n\3, (n-3*k)!*stirling(n-2*k, n-3*k, 2)/(n-2*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k)! * Stirling2(n-2*k,n-3*k)/(n-2*k)!.

A375810 Expansion of e.g.f. 1/(1 - (exp(x^2) - 1)/x)^2.

Original entry on oeis.org

1, 2, 6, 30, 192, 1480, 13500, 141540, 1676640, 22141728, 322388640, 5130084960, 88561408320, 1648294876800, 32898981155040, 700940855815200, 15877318955097600, 380996919471168000, 9654670629548904960, 257627854786123261440, 7220676423560766566400
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x^2)-1)/x)^2))
    
  • PARI
    a(n) = n!*sum(k=0, n\2, (n-2*k+1)!*stirling(n-k, n-2*k, 2)/(n-k)!);

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A375795.
a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k+1)! * Stirling2(n-k,n-2*k)/(n-k)!.

A375811 Expansion of e.g.f. 1/(1 - (exp(x^2) - 1)/x)^3.

Original entry on oeis.org

1, 3, 12, 69, 504, 4380, 44280, 509670, 6572160, 93813552, 1467910080, 24976440720, 459045195840, 9061616266560, 191187467190720, 4293103436622000, 102216550583347200, 2572022267758944000, 68195046359419499520, 1900236334204732043520
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x^2)-1)/x)^3))
    
  • PARI
    a(n) = n!*sum(k=0, n\2, (n-2*k+2)!*stirling(n-k, n-2*k, 2)/(n-k)!)/2;

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A375795.
a(n) = (n!/2) * Sum_{k=0..floor(n/2)} (n-2*k+2)! * Stirling2(n-k,n-2*k)/(n-k)!.
Showing 1-3 of 3 results.