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

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

A369297 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x^3) ).

Original entry on oeis.org

1, 2, 7, 31, 153, 806, 4440, 25266, 147364, 876282, 5292527, 32378125, 200218715, 1249456536, 7858638756, 49766595855, 317051378103, 2030589300596, 13066646029059, 84439101344619, 547746622599561, 3565472378360110, 23282050305073680, 152466688160732190
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))/x)
    
  • PARI
    a(n, s=3, t=1, u=2) = 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(n+k,k) * binomial(3*n-3*k+1,n-3*k).
a(n) = (1/(n+1)) * [x^n] 1/( (1-x)^2 * (1-x^3) )^(n+1). - Seiichi Manyama, Feb 14 2024

A370247 Coefficient of x^n in the expansion of ( 1/(1-x)^2 * (1+x^3) )^n.

Original entry on oeis.org

1, 2, 10, 59, 362, 2277, 14575, 94474, 618154, 4074197, 27008885, 179897720, 1202961215, 8070830588, 54302131642, 366252974259, 2475575739306, 16764524795037, 113719280941453, 772551326290528, 5255393538550837, 35794109754866998, 244060675562790316
Offset: 0

Views

Author

Seiichi Manyama, Feb 13 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, s=3, t=1, u=2) = 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(n,k) * binomial(3*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)^2 / (1+x^3) ). See A369265.
Showing 1-4 of 4 results.