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.

A375664 Expansion of e.g.f. 1 / (1 - x * (exp(x^2) - 1))^2.

Original entry on oeis.org

1, 0, 0, 12, 0, 120, 2160, 1680, 120960, 1481760, 6350400, 240166080, 2754259200, 31152401280, 894303970560, 11769588230400, 228232766361600, 5845147711603200, 98290727395660800, 2502848611354291200, 63417766359467520000, 1376904298716724377600
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2024

Keywords

Crossrefs

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A375588.
a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k+1)! * Stirling2(k,n-2*k)/k!.

A375665 Expansion of e.g.f. 1 / (1 - x * (exp(x^2) - 1))^3.

Original entry on oeis.org

1, 0, 0, 18, 0, 180, 4320, 2520, 241920, 3674160, 12700800, 599749920, 7903526400, 77863705920, 2660390853120, 37269497865600, 683129089843200, 19948163057222400, 339567800885913600, 8690022866119795200, 241333983478748160000, 5217670959100125926400
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2024

Keywords

Crossrefs

Programs

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

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A375588.
a(n) = (n!/2) * Sum_{k=0..floor(n/2)} (n-2*k+2)! * Stirling2(k,n-2*k)/k!.

A376345 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*(exp(x^2) - 1)) ).

Original entry on oeis.org

1, 0, 0, 6, 0, 60, 2880, 840, 201600, 7998480, 12700800, 1816547040, 67898476800, 311359688640, 35628798965760, 1317155266627200, 12924530383564800, 1308998905659244800, 49463008450023168000, 863080350836537433600, 81264621182097120768000, 3227330594664084337228800, 87828327888763088096870400
Offset: 0

Views

Author

Seiichi Manyama, Sep 21 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (2*n-2*k)! * Stirling2(k,n-2*k)/k!.

A375589 Expansion of e.g.f. 1 / (1 + x - x * exp(x^3)).

Original entry on oeis.org

1, 0, 0, 0, 24, 0, 0, 2520, 40320, 0, 604800, 39916800, 479001600, 259459200, 50854003200, 1961511552000, 21097146470400, 88921857024000, 8002967132160000, 243459152346009600, 2642401903325184000, 38318206628782080000, 2435557926202232832000
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k)! * Stirling2(k,n-3*k)/k!.
Showing 1-4 of 4 results.