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.

A369231 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1-x+x^3)^2 ).

Original entry on oeis.org

1, 1, 2, 7, 26, 98, 385, 1569, 6556, 27908, 120624, 528030, 2336202, 10430155, 46930285, 212597901, 968833424, 4438398734, 20428750419, 94424634294, 438104297376, 2039690282940, 9526029685218, 44617396906698, 209526541600978, 986339358246758, 4653571637230839
Offset: 0

Views

Author

Seiichi Manyama, Jan 17 2024

Keywords

Crossrefs

Programs

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

Formula

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

A369230 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 / (1-x+x^2)^3 ).

Original entry on oeis.org

1, 0, 3, 3, 24, 54, 283, 900, 4098, 15286, 66555, 268173, 1156951, 4852722, 21007605, 90167059, 393152058, 1712432070, 7524092134, 33112353060, 146518404963, 649861681966, 2893369443183, 12913307575722, 57800647230933, 259298148600504, 1165967972216967
Offset: 0

Views

Author

Seiichi Manyama, Jan 17 2024

Keywords

Crossrefs

Programs

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

Formula

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

A372411 Coefficient of x^n in the expansion of ( (1-x+x^2)^2 / (1-x)^3 )^n.

Original entry on oeis.org

1, 1, 7, 34, 183, 1001, 5578, 31459, 179063, 1026493, 5918007, 34277728, 199309146, 1162682314, 6801575641, 39885002534, 234384591991, 1379936226605, 8137835460115, 48062073927739, 284233390132183, 1682950066882489, 9975692904121556, 59190095764321975
Offset: 0

Views

Author

Seiichi Manyama, Apr 29 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k) * binomial(2*n-k-1,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)^3 / (1-x+x^2)^2 ). See A369229.
Showing 1-3 of 3 results.