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.

A370171 Coefficient of x^n in the expansion of ( (1+x) * (1+x+x^2)^3 )^n.

Original entry on oeis.org

1, 4, 34, 319, 3146, 31929, 330145, 3458620, 36585194, 389893576, 4179819559, 45025583343, 486961123577, 5284324727023, 57508473997848, 627410367071169, 6859805605391466, 75144918246760324, 824558759018846116, 9061483047671168437, 99716283188165243471
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, s=2, t=3, 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/2)} binomial(3*n,k) * binomial(4*n-k,n-2*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^2)^3) ). See A369479.

A369477 Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^2)^2) ).

Original entry on oeis.org

1, 3, 14, 77, 464, 2964, 19717, 135131, 947549, 6765642, 49022225, 359545750, 2664127354, 19913283809, 149968276974, 1136856855549, 8668000962927, 66428474900907, 511414514214628, 3953420853213504, 30674783555852576, 238808419235022293, 1864869207177530320
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^2)^2))/x)
    
  • PARI
    a(n, s=2, 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/2)} binomial(2*n+2,k) * binomial(3*n-k+3,n-2*k).

A369480 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x+x^2)^3) ).

Original entry on oeis.org

1, 5, 38, 342, 3379, 35427, 387038, 4358119, 50222276, 589439699, 7021368716, 84669873678, 1031603223880, 12679812357672, 157038146685360, 1957792379658934, 24549963008189965, 309435808369427643, 3918185776941808956, 49818464846052855850, 635788103792527271239
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^2)^3))/x)
    
  • PARI
    a(n, s=2, t=3, 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/2)} binomial(3*n+3,k) * binomial(5*n-k+5,n-2*k).
Showing 1-3 of 3 results.