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 1-3 of 3 results.

A369511 Expansion of (1/x) * Series_Reversion( x * ((1-x)^2-x^3)^2 ).

Original entry on oeis.org

1, 4, 26, 206, 1815, 17082, 168159, 1710234, 17828973, 189504744, 2045971440, 22374997320, 247344411792, 2759394009008, 31027178033064, 351270123392892, 4000793799046578, 45809545263096832, 527010005799822844, 6088666065809281348, 70612995488695876634
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^2-x^3)^2)/x)
    
  • PARI
    a(n) = sum(k=0, n\3, binomial(2*n+k+1, k)*binomial(5*n-k+3, n-3*k))/(n+1);

Formula

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

A370281 Coefficient of x^n in the expansion of 1/( (1-x)^2 - x^3 )^n.

Original entry on oeis.org

1, 2, 10, 59, 370, 2392, 15757, 105156, 708546, 4809695, 32841380, 225321967, 1552063981, 10726766624, 74348039020, 516586596484, 3597106344450, 25095046641861, 175369603836301, 1227366066102925, 8601637753421020, 60355768595163030, 423972992316330225
Offset: 0

Views

Author

Seiichi Manyama, Feb 13 2024

Keywords

Crossrefs

Cf. A369214.

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n+k-1,k) * binomial(3*n-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 - x^3) ). See A369214.

A371431 Expansion of (1/x) * Series_Reversion( x * ((1-x)^2 + x^3) ).

Original entry on oeis.org

1, 2, 7, 29, 131, 623, 3064, 15423, 78936, 408958, 2137993, 11252163, 59508232, 315786764, 1679410076, 8941421014, 47613443433, 253359512287, 1346009853489, 7133000408765, 37669665812955, 198034693198875, 1035095172883710, 5371011415598595, 27615259784888724
Offset: 0

Views

Author

Seiichi Manyama, Mar 23 2024

Keywords

Crossrefs

Cf. A369214.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^2+x^3))/x)
    
  • PARI
    a(n) = sum(k=0, n\3, (-1)^k*binomial(n+k, k)*binomial(3*n-k+1, n-3*k))/(n+1);

Formula

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