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.

A358264 Expansion of e.g.f. 1/(1 - x * exp(x^2/2)).

Original entry on oeis.org

1, 1, 2, 9, 48, 315, 2520, 23415, 248640, 2972025, 39463200, 576413145, 9184855680, 158550787395, 2947473809280, 58707685211175, 1247293022976000, 28156003910859825, 672972205556851200, 16978695795089253225, 450907982644863744000, 12573634144960773960075
Offset: 0

Views

Author

Seiichi Manyama, Nov 06 2022

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^k/(2^k * k!).
a(n) ~ n! / ((1 + LambertW(1)) * LambertW(1)^(n/2)). - Vaclav Kotesovec, Nov 13 2022

A370931 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x^3/6)) ).

Original entry on oeis.org

1, 1, 4, 30, 340, 5180, 99360, 2300830, 62473600, 1946941920, 68507714800, 2686816932800, 116225776497600, 5497681373384200, 282305750023897600, 15640212734095950000, 929908726447266966400, 59061538103044360083200, 3990922849835432102592000
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x*exp(x^3/6)))/x))
    
  • PARI
    a(n) = sum(k=0, n\3, (n-3*k)^k*(2*n-3*k)!/(6^k*k!*(n-3*k)!))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (n-3*k)^k * (2*n-3*k)!/(6^k * k! * (n-3*k)!).

A375634 Expansion of e.g.f. exp(x^3) / (1 - x * exp(x^3/6)).

Original entry on oeis.org

1, 1, 2, 12, 52, 280, 2160, 16870, 150080, 1557360, 17491600, 215908000, 2923905600, 42734091400, 672515043200, 11347480544400, 204142655516800, 3902009862150400, 78976503077472000, 1687212801048174400, 37941749192547200000
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3)/(1-x*exp(x^3/6))))
    
  • PARI
    a(n) = n!*sum(k=0, n\3, ((n-3*k+6)/6)^k/k!);

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} ((n-3*k+6)/6)^k/k!.
Showing 1-3 of 3 results.