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.

A369479 Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^2)^3) ).

Original entry on oeis.org

1, 4, 25, 185, 1503, 12958, 116410, 1077872, 10213954, 98574454, 965545161, 9574235477, 95920415338, 969467658540, 9872949735243, 101211280459929, 1043597450013094, 10816134194658976, 112617367970103163, 1177413807406659659, 12355753915291229596
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)*(1+x+x^2)^3))/x)
    
  • PARI
    a(n, s=2, t=3, u=1) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((t+u)*(n+1)-k, n-s*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+3,k) * binomial(4*n-k+4,n-2*k).

A370172 Coefficient of x^n in the expansion of ( (1+x)^2 * (1+x+x^2)^3 )^n.

Original entry on oeis.org

1, 5, 51, 581, 6963, 85905, 1079943, 13756216, 176939187, 2292988919, 29892396451, 391576960230, 5150057095527, 67962810381653, 899458144305408, 11933576896320981, 158672857603511987, 2113800649819533735, 28207266176359605705, 376976971371883606824
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, s=2, t=3, u=2) = sum(k=0, n\s, binomial(t*n, k)*binomial((t+u)*n-k, n-s*k));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n,k) * binomial(5*n-k,n-2*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 * (1+x+x^2)^3) ). See A369480.

A369478 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, 93197, 823484, 7445184, 68545882, 640446224, 6057249180, 57878746750, 557903174040, 5418441862824, 52971933934834, 520869559359424, 5147999004530720, 51113415228327827, 509583784051748692, 5099262428810825568
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 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, binomial(t*(n+1), k)*binomial((t+u)*(n+1)-k, n-s*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+2,k) * binomial(4*n-k+4,n-2*k).
Showing 1-3 of 3 results.