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.

A358080 Expansion of e.g.f. 1/(1 - x^2 * exp(x)).

Original entry on oeis.org

1, 0, 2, 6, 36, 260, 2190, 21882, 248696, 3181320, 45229050, 707208590, 12063902532, 222939837276, 4436813677478, 94605994108290, 2151763873634160, 51999544476324752, 1330540380342907506, 35936656483848501654, 1021700660649312689660
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2022

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} k^(n - 2*k)/(n - 2*k)!.
a(n) ~ n! / ((1 + LambertW(1/2)) * 2^(n+1) * LambertW(1/2)^n). - Vaclav Kotesovec, Oct 30 2022

A370985 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^3*exp(x)) ).

Original entry on oeis.org

1, 0, 0, 6, 24, 60, 3000, 45570, 403536, 10644984, 297562320, 5517833310, 142801022760, 5076208052916, 150282366476424, 4713707747551530, 189345734667052320, 7517503455423740400, 295622259241028433696, 13370535071068474177974, 642403497550155241197240
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Crossrefs

Programs

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

Formula

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

A366546 Expansion of e.g.f. -log(1 - x^3 * exp(x)).

Original entry on oeis.org

0, 0, 0, 6, 24, 60, 480, 5250, 40656, 363384, 4839120, 65198430, 859543080, 13311494196, 233478687624, 4190929145130, 79746180437280, 1667320408619760, 36965002127643936, 854734007793179574, 20962277675893792440, 544839141515795731500
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2023

Keywords

Crossrefs

Programs

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

Formula

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

A375630 Expansion of e.g.f. exp(2*x) / (1 - x^3 * exp(x)).

Original entry on oeis.org

1, 2, 4, 14, 88, 572, 4024, 37298, 404464, 4601528, 58426864, 846080798, 13209174136, 219868220756, 3981563464792, 77708414601098, 1606665377716576, 35246223612156272, 821962294211430496, 20227931586257247542, 522932344617513862696, 14204133017700173041292
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (k+2)^(n-3*k)/(n-3*k)!.
Showing 1-4 of 4 results.