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.

A369269 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^3)^3 ).

Original entry on oeis.org

1, 2, 7, 33, 173, 962, 5586, 33498, 205846, 1289386, 8202247, 52845855, 344129832, 2261377872, 14976646685, 99863119809, 669860309538, 4517037850220, 30603008068997, 208211448723097, 1421986458302466, 9745007758311114, 66993247112160800
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)^3)/x)
    
  • PARI
    a(n, s=3, t=3, 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(3*n+3,k) * binomial(3*n-3*k+1,n-3*k).
a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^2 * (1+x^3)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A369299 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x^3)^3 ).

Original entry on oeis.org

1, 1, 2, 8, 29, 105, 417, 1719, 7181, 30603, 132736, 582790, 2585352, 11575613, 52237278, 237328704, 1084701387, 4983867447, 23007263941, 106658256768, 496336303014, 2317687534865, 10856677523580, 51001805706435, 240225121539000, 1134240896062656, 5367428039668751
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-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+k+2,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

A369270 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1+x^3)^3 ).

Original entry on oeis.org

1, 3, 15, 94, 657, 4902, 38233, 307953, 2541831, 21386810, 182754162, 1581699162, 13836248406, 122139271098, 1086638457429, 9733419373534, 87707244737511, 794505072627735, 7231017033165776, 66089527981542462, 606340568510978940, 5582088822346925210
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)^3/(1+x^3)^3)/x)
    
  • PARI
    a(n, s=3, t=3, u=3) = 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(4*n-3*k+2,n-3*k).
a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^3 * (1+x^3)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A369266 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^3)^2 ).

Original entry on oeis.org

1, 1, 2, 7, 24, 84, 313, 1209, 4769, 19166, 78253, 323570, 1352122, 5701467, 24229122, 103663575, 446163435, 1930390329, 8391341664, 36630504952, 160509484616, 705750073063, 3112865367660, 13769327908980, 61066953746400, 271488240652950, 1209671359828154
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)^2)/x)
    
  • PARI
    a(n, s=3, t=2, 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(2*n+2,k) * binomial(2*n-3*k,n-3*k).
a(n) = (1/(n+1)) * [x^n] ( 1/(1-x) * (1+x^3)^2 )^(n+1). - Seiichi Manyama, Feb 14 2024
Showing 1-4 of 4 results.