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

A369262 Expansion of (1/x) * Series_Reversion( x * (1-x) / (1+x^2)^3 ).

Original entry on oeis.org

1, 1, 5, 17, 80, 363, 1792, 8969, 46319, 242994, 1296046, 6996163, 38175142, 210162728, 1166020560, 6512854409, 36593709385, 206686641555, 1172856064443, 6683348391034, 38228129813288, 219411037878578, 1263245957786120, 7293833100110787, 42224142505632305
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^2)^3)/x)
    
  • PARI
    a(n, s=2, 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/2)} binomial(3*n+3,k) * binomial(2*n-2*k,n-2*k).
a(n) = (1/(n+1)) * [x^n] ( 1/(1-x) * (1+x^2)^3 )^(n+1). - Seiichi Manyama, Feb 14 2024

A369263 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 / (1+x^2)^3 ).

Original entry on oeis.org

1, 2, 10, 54, 329, 2126, 14356, 100030, 713956, 5193064, 38354066, 286860714, 2168308302, 16537766036, 127114940840, 983657456878, 7657060437148, 59917814944376, 471062428422152, 3718952705982232, 29471640802526185, 234356062245289566, 1869405604537134116
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^2)^3)/x)
    
  • PARI
    a(n, s=2, 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/2)} binomial(3*n+3,k) * binomial(3*n-2*k+1,n-2*k).
a(n) = (1/(n+1)) * [x^n] ( 1/(1-x)^2 * (1+x^2)^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
Showing 1-4 of 4 results.