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.

A369267 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^3)^2 ).

Original entry on oeis.org

1, 2, 7, 32, 163, 884, 5009, 29310, 175750, 1074264, 6668825, 41929970, 266464579, 1708829584, 11044663663, 71871779008, 470495357634, 3096311833496, 20472771422946, 135937759368388, 906056228361095, 6059922934991008, 40657629626645463
Offset: 0

Views

Author

Seiichi Manyama, Jan 18 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^2/(1+x^3)^2)/x)
    
  • PARI
    a(n, s=3, t=2, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((u+1)*(n+1)-s*k-2, 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-3*k+1,n-3*k).
a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^2 * (1+x^3)^2 )^(n+1). - Seiichi Manyama, Feb 14 2024

A369268 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^3)^3 ).

Original entry on oeis.org

1, 1, 2, 8, 29, 105, 414, 1695, 7046, 29853, 128644, 561262, 2474142, 11006108, 49343508, 222715440, 1011217425, 4615519083, 21165513228, 97467424198, 450541090701, 2089777230606, 9723511785608, 45371996501895, 212271904284993, 995513843930049, 4679212044797252
Offset: 0

Views

Author

Seiichi Manyama, Jan 18 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(3*n+3,k) * binomial(2*n-3*k,n-3*k).
a(n) = (1/(n+1)) * [x^n] ( 1/(1-x) * (1+x^3)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A370248 Coefficient of x^n in the expansion of ( 1/(1-x) * (1+x^3)^2 )^n.

Original entry on oeis.org

1, 1, 3, 16, 67, 276, 1200, 5293, 23427, 104425, 468428, 2110725, 9546256, 43315546, 197088195, 898910916, 4108495491, 18812770011, 86285313327, 396332663094, 1822878714492, 8394131895424, 38696042930251, 178561943852670, 824720550229584, 3812313399877776
Offset: 0

Views

Author

Seiichi Manyama, Feb 13 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, s=3, t=2, u=1) = sum(k=0, n\s, binomial(t*n, k)*binomial((u+1)*n-s*k-1, n-s*k));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(2*n,k) * binomial(2*n-3*k-1,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^3)^2 ). See A369266.
Showing 1-3 of 3 results.