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.

A369441 Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x^2)^2) ).

Original entry on oeis.org

1, 2, 7, 30, 141, 704, 3666, 19686, 108222, 606062, 3445308, 19829680, 115323955, 676659960, 4000719012, 23811922678, 142557391306, 857894530348, 5186614665121, 31487226410770, 191871141682557, 1173163962971056, 7195329233469552, 44255915928488880
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2024

Keywords

Crossrefs

Cf. A369440.

Programs

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

Formula

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

A369477 Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^2)^2) ).

Original entry on oeis.org

1, 3, 14, 77, 464, 2964, 19717, 135131, 947549, 6765642, 49022225, 359545750, 2664127354, 19913283809, 149968276974, 1136856855549, 8668000962927, 66428474900907, 511414514214628, 3953420853213504, 30674783555852576, 238808419235022293, 1864869207177530320
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)^2))/x)
    
  • PARI
    a(n, s=2, t=2, 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(2*n+2,k) * binomial(3*n-k+3,n-2*k).

A370194 Coefficient of x^n in the expansion of ( (1+x) * (1+x^2)^2 )^n.

Original entry on oeis.org

1, 1, 5, 19, 77, 326, 1391, 6028, 26349, 116011, 513730, 2285570, 10208111, 45742724, 205550840, 925918544, 4179740909, 18903381337, 85635147983, 388517336189, 1765019420602, 8028115465732, 36555667019338, 166621503161184, 760161934681647, 3470945792364701
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2024

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_]:=SeriesCoefficient[((1+x)*(1+x^2)^2)^n,{x,0,n}]; Array[a,26,0] (* Stefano Spezia, Apr 30 2024 *)
  • PARI
    a(n, s=2, t=2, u=1) = sum(k=0, n\s, binomial(t*n, k)*binomial(u*n, n-s*k));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k) * binomial(n,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) * (1+x^2)^2) ). See A369440.
Showing 1-3 of 3 results.