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 1-5 of 5 results.

A369128 Expansion of (1/x) * Series_Reversion( x / ((1+x)^5+x^5) ).

Original entry on oeis.org

1, 5, 35, 285, 2530, 23752, 231910, 2331040, 23960235, 250692365, 2661086895, 28587333725, 310217791590, 3395464391870, 37442295427120, 415570885425280, 4638842010800025, 52044582325415025, 586553425250933055, 6637525235622842585, 75387741117556006435
Offset: 0

Views

Author

Seiichi Manyama, Jan 14 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^5+x^5))/x)
    
  • PARI
    a(n) = sum(k=0, n\5, binomial(n+1, k)*binomial(5*n-5*k+5, n-5*k))/(n+1);

Formula

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

A369158 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2+x^4) ).

Original entry on oeis.org

1, 2, 5, 14, 43, 142, 495, 1794, 6686, 25436, 98311, 384826, 1522283, 6075838, 24437937, 98956270, 403080170, 1650502292, 6790018182, 28050896964, 116322826479, 484029536374, 2020386475025, 8457397801150, 35495812337114, 149336478356692, 629685490668799
Offset: 0

Views

Author

Seiichi Manyama, Jan 15 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2+x^4))/x)
    
  • PARI
    a(n) = sum(k=0, n\4, binomial(n+1, k)*binomial(2*n-2*k+2, n-4*k))/(n+1);

Formula

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

A369156 Expansion of (1/x) * Series_Reversion( x / ((1+x)^4-x^4) ).

Original entry on oeis.org

1, 4, 22, 140, 968, 7064, 53544, 417456, 3326304, 26967040, 221733568, 1844667136, 15498804480, 131325820032, 1120928667264, 9628975973120, 83181462291968, 722175844640768, 6297942966129664, 55143987250677760, 484589284705202176, 4272491458636754944
Offset: 0

Views

Author

Seiichi Manyama, Jan 15 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^4-x^4))/x)
    
  • PARI
    a(n) = sum(k=0, n\4, (-1)^k*binomial(n+1, k)*binomial(4*n-4*k+4, n-4*k))/(n+1);

Formula

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

A369159 Expansion of (1/x) * Series_Reversion( x / ((1+x)^3+x^4) ).

Original entry on oeis.org

1, 3, 12, 55, 274, 1443, 7905, 44593, 257305, 1511553, 9010170, 54361486, 331336454, 2037132958, 12619056108, 78682008194, 493427982703, 3110202012353, 19693920616872, 125214061831251, 799059649687239, 5116372686471627, 32860439054510610
Offset: 0

Views

Author

Seiichi Manyama, Jan 15 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^3+x^4))/x)
    
  • PARI
    a(n) = sum(k=0, n\4, binomial(n+1, k)*binomial(3*n-3*k+3, n-4*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} binomial(n+1,k) * binomial(3*n-3*k+3,n-4*k).

A369213 Expansion of (1/x) * Series_Reversion( x / ((1+x)^4+x^2) ).

Original entry on oeis.org

1, 4, 23, 152, 1091, 8264, 65021, 526236, 4352942, 36637576, 312763225, 2701521420, 23567184019, 207343098824, 1837623853627, 16391011930424, 147029997389386, 1325506554640872, 12003342144724338, 109136630802023808, 995907341988015935
Offset: 0

Views

Author

Seiichi Manyama, Jan 16 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^4+x^2))/x)
    
  • PARI
    a(n) = sum(k=0, n\2, binomial(n+1, k)*binomial(4*n-4*k+4, n-2*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+1,k) * binomial(4*n-4*k+4,n-2*k).
D-finite with recurrence -3*(4813*n-632)*(3*n+2)*(3*n+4)*(n+1)*a(n) +2*(206141*n^4+1346849*n^3+118471*n^2-121301*n-7584)*a(n-1) +4*(1658281*n^4-3845638*n^3+4346111*n^2-2458136*n+406104)*a(n-2) +8*(n-2)*(2032705*n^3-6230304*n^2+5971619*n-935490)*a(n-3) +16*(n-2)*(n-3)*(958321*n^2-2152552*n+309963)*a(n-4) +544*(8765*n-1142)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 28 2024
Showing 1-5 of 5 results.