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).

A369485 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x+x^3)^2) ).

Original entry on oeis.org

1, 4, 22, 142, 1007, 7590, 59683, 484112, 4021061, 34029532, 292373296, 2543542676, 22360917140, 198341377680, 1772860026933, 15952960500612, 144397901220980, 1313835276189792, 12009823111155481, 110240431974732436, 1015727265740887873
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)^2*(1+x+x^3)^2))/x)
    
  • PARI
    a(n, s=3, t=2, u=2) = 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(4*n-k+4,n-3*k).

A370186 Coefficient of x^n in the expansion of ( (1+x) * (1+x+x^3)^2 )^n.

Original entry on oeis.org

1, 3, 15, 90, 583, 3913, 26790, 185839, 1301575, 9183681, 65181645, 464858661, 3328503814, 23913207750, 172295708971, 1244484142765, 9008351053031, 65332552755149, 474622993450725, 3453219378684621, 25158758123093013, 183521479226172667, 1340195580366321837
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, s=3, t=2, u=1) = 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(3*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) * (1+x+x^3)^2) ). See A369484.

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.