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.

A368079 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 * (1-x^2)^3 ).

Original entry on oeis.org

1, 3, 18, 127, 996, 8322, 72644, 654615, 6043455, 56866028, 543368586, 5258196762, 51426990112, 507537393600, 5048033356128, 50549237164615, 509197913456922, 5156339940802941, 52460340305220466, 535976129228082972, 5496745175387480976
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^2)^3)/x)
    
  • PARI
    a(n, s=2, t=3, u=3) = 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/2)} binomial(3*n+k+2,k) * binomial(4*n-2*k+2,n-2*k).
a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(3*n+k+2,k) * binomial(7*n-k+5,n-k).
a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^3 * (1-x)^6 )^(n+1). - Seiichi Manyama, Feb 16 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

A369300 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, 5589, 33546, 206359, 1294096, 8242375, 53173095, 346724250, 2281555440, 15131448440, 101038950441, 678724811604, 4583483218340, 31098830566098, 211898222878937, 1449322361547669, 9947227335902244, 68486384818253877
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-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(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

A369403 Expansion of (1/x) * Series_Reversion( x / (1+x)^3 * (1-x^3)^3 ).

Original entry on oeis.org

1, 3, 12, 58, 318, 1887, 11775, 76041, 503607, 3401326, 23337339, 162214074, 1139835938, 8083530360, 57783277608, 415904602938, 3011669994078, 21924967877547, 160374157346266, 1178091991206162, 8687419007293458, 64285383562018856, 477208235856114384
Offset: 0

Views

Author

Seiichi Manyama, Jan 22 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-1, k)*binomial(u*(n+1), n-s*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(3*n+k+2,k) * binomial(3*n+3,n-3*k).
a(n) = (1/(n+1)) * [x^n] ( (1+x)^3 / (1-x^3)^3 )^(n+1). - Seiichi Manyama, Feb 16 2024
Showing 1-4 of 4 results.