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.

A368975 Expansion of (1/x) * Series_Reversion( x * (1-x)^2 * (1-x+x^2)^2 ).

Original entry on oeis.org

1, 4, 24, 170, 1320, 10868, 93199, 823548, 7446480, 68567202, 640757920, 6061477500, 57933260067, 558580920160, 5426644737984, 53069206438226, 522004849765080, 5161083186971000, 51262685633583970, 511272660117154692, 5118240198221249088
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n+k+1,k) * binomial(5*n-k+3,n-2*k).

A368973 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x+x^2)^2 ).

Original entry on oeis.org

1, 3, 13, 65, 351, 1989, 11650, 69903, 427225, 2649229, 16622079, 105310673, 672687322, 4327037010, 28002409452, 182179075689, 1190778886791, 7815755146095, 51491064226095, 340374137775879, 2256891800364421, 15006481967365535, 100037043223408890
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n+k+1,k) * binomial(4*n-k+2,n-2*k).

A367027 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3 - x^2*A(x)^5.

Original entry on oeis.org

1, 1, 2, 4, 5, -13, -147, -816, -3534, -12650, -35420, -53040, 199056, 2391340, 14555740, 68264112, 261045693, 769660569, 1167906402, -5145668100, -61758940705, -385813067255, -1857144860445, -7266981925560, -21793022441775, -32643056947527, 161919845140752
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2023

Keywords

Crossrefs

Programs

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

Formula

a(n) = (1/(2*n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-k,k) * binomial(3*n-2*k,n-2*k).
G.f.: ( (1/x) * Series_Reversion( x * (1-x+x^2)^2 ) )^(1/2). - Seiichi Manyama, Mar 08 2025

A369076 Expansion of (1/x) * Series_Reversion( x * (1+x^2/(1-x))^2 ).

Original entry on oeis.org

1, 0, -2, -2, 9, 24, -37, -240, -2, 2126, 2919, -16052, -50663, 86940, 631995, 19094, -6491463, -9595434, 54443985, 181532910, -317331187, -2426618056, -133151895, 26332109928, 40544827703, -230619508548, -793966990358, 1384746844832, 10960715925621, 881359815524
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1+x^2/(1-x))^2)/x)
    
  • PARI
    a(n, s=2, t=2, u=-2) = sum(k=0, n\s, (-1)^k*binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n+k+1,k) * binomial(n-k-1,n-2*k).
Showing 1-4 of 4 results.