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 1-4 of 4 results.

A368975 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, 93199, 823548, 7446480, 68567202, 640757920, 6061477500, 57933260067, 558580920160, 5426644737984, 53069206438226, 522004849765080, 5161083186971000, 51262685633583970, 511272660117154692, 5118240198221249088
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 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, (-1)^k*binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

Formula

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

A368969 Expansion of (1/x) * Series_Reversion( x * (1-x+x^2)^2 ).

Original entry on oeis.org

1, 2, 5, 12, 22, 0, -284, -1938, -9367, -36938, -118105, -260130, 56637, 4890560, 35945616, 186674620, 782890326, 2632462236, 5987222046, -2241224328, -129137211280, -967479390360, -5145272296080, -22060975744080, -75535676951124, -172915138783080
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Crossrefs

Programs

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

Formula

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

A369076 Expansion of (1/x) * Series_Reversion( x * (1+x^2/(1-x))^2 ).

Original entry on oeis.org

1, 0, -2, -2, 9, 24, -37, -240, -2, 2126, 2919, -16052, -50663, 86940, 631995, 19094, -6491463, -9595434, 54443985, 181532910, -317331187, -2426618056, -133151895, 26332109928, 40544827703, -230619508548, -793966990358, 1384746844832, 10960715925621, 881359815524
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1+x^2/(1-x))^2)/x)
    
  • PARI
    a(n, s=2, t=2, u=-2) = sum(k=0, n\s, (-1)^k*binomial(t*(n+1)+k-1, k)*binomial((t+u+1)*(n+1)-(s-1)*k-2, n-s*k))/(n+1);

Formula

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

A372462 Coefficient of x^n in the expansion of 1 / ( (1-x) * (1-x+x^2)^2 )^n.

Original entry on oeis.org

1, 3, 17, 105, 673, 4403, 29183, 195170, 1313889, 8889963, 60392717, 411615867, 2813115487, 19270525316, 132273530462, 909530996780, 6263834506593, 43198661550219, 298296958413785, 2062180461738075, 14271253423675773, 98859742466265935
Offset: 0

Views

Author

Seiichi Manyama, May 01 2024

Keywords

Crossrefs

Programs

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

Formula

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n+k-1,k) * binomial(4*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) * (1-x+x^2)^2 ). See A368973.
Showing 1-4 of 4 results.