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.

A369483 Expansion of (1/x) * Series_Reversion( x / (1+x+x^3)^2 ).

Original entry on oeis.org

1, 2, 5, 16, 60, 242, 1014, 4370, 19278, 86678, 395751, 1829742, 8549100, 40302810, 191469165, 915751966, 4405727502, 21307102900, 103526683797, 505118705078, 2473833623696, 12157124607612, 59929746189165, 296271556144028, 1468494529164194, 7296261411708962
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^3)^2)/x)
    
  • PARI
    a(n, s=3, t=2, u=0) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((t+u)*(n+1)-k, n-s*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+2,k) * binomial(2*n-k+2,n-3*k).

A369484 Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^3)^2) ).

Original entry on oeis.org

1, 3, 12, 57, 301, 1700, 10045, 61303, 383335, 2443113, 15811317, 103627692, 686402602, 4587643765, 30900426417, 209539509967, 1429344492215, 9801262309209, 67523359213569, 467136798336153, 3243948604314619, 22604271635042853, 158001453530915361
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)*(1+x+x^3)^2))/x)
    
  • PARI
    a(n, s=3, t=2, u=1) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((t+u)*(n+1)-k, n-s*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(2*n+2,k) * binomial(3*n-k+3,n-3*k).

A370187 Coefficient of x^n in the expansion of ( (1+x)^2 * (1+x+x^3)^2 )^n.

Original entry on oeis.org

1, 4, 28, 226, 1940, 17214, 155914, 1432106, 13289076, 124276528, 1169346298, 11057293526, 104986087178, 1000248093420, 9557756114130, 91559051752596, 879027678226452, 8455595252761536, 81476137225450096, 786286875175380088, 7598503022428758570
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, s=3, t=2, u=2) = sum(k=0, n\s, binomial(t*n, k)*binomial((t+u)*n-k, n-s*k));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n,k) * binomial(4*n-k,n-3*k).
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x+x^3)^2) ). See A369485.

A372375 Expansion of (1/x) * Series_Reversion( x * (1+x) / (1+x+x^3)^2 ).

Original entry on oeis.org

1, 1, 1, 3, 9, 21, 54, 161, 470, 1347, 4007, 12199, 37141, 113802, 352905, 1101969, 3455220, 10891968, 34515825, 109814395, 350616323, 1123368287, 3610647348, 11637410625, 37605280548, 121812321775, 395455199269, 1286446544052, 4192913001804, 13690359696969
Offset: 0

Views

Author

Seiichi Manyama, Apr 29 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1+x)/(1+x+x^3)^2)/x)
    
  • PARI
    a(n, s=3, t=2, u=-1) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((t+u)*(n+1)-k, n-s*k))/(n+1);

Formula

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