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.

A364399 G.f. satisfies A(x) = 1 + x/A(x)^3*(1 + 1/A(x)^2).

Original entry on oeis.org

1, 2, -16, 212, -3400, 60384, -1142960, 22598832, -461250208, 9644611008, -205537131008, 4447969973888, -97482797466624, 2159242220999936, -48260706692535552, 1087076798266594048, -24652590023639251456, 562396337623786449920
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(4*n+2*k-2, n-1))/n);

Formula

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A363380.
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(4*n+2*k-2,n-1) for n > 0.
a(n) ~ c*(-1)^(n-1)*256^n*27^(-n)*3F2([-n, 2*n, 2*n-1/2], [3*n/2, (3*n+1)/2], -1)*n^(-3/2), with c = (1/8)*sqrt(3/(2*Pi)). - Stefano Spezia, Oct 21 2023

A379245 G.f. A(x) satisfies A(x) = ( (1 + x*A(x)^2)/(1 - x*A(x)) )^3.

Original entry on oeis.org

1, 6, 72, 1100, 18984, 352608, 6879152, 139012368, 2884353888, 61091682368, 1315450042368, 28709737064064, 633684940733696, 14120739728984832, 317243001537462528, 7178031348934793472, 163423203504309020160, 3741114809852278047744
Offset: 0

Views

Author

Seiichi Manyama, Dec 18 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*n+3*k+3, k)*binomial(4*n+2*k+2, n-k)/(n+k+1));

Formula

G.f.: B(x)^3 where B(x) is the g.f. of A363380.
a(n) = Sum_{k=0..n} binomial(3*n+3*k+3,k) * binomial(4*n+2*k+2,n-k)/(n+k+1).

A379247 G.f. A(x) satisfies A(x) = 1 + x * A(x)^4 * (1 + A(x)^5).

Original entry on oeis.org

1, 2, 26, 506, 11650, 294338, 7889658, 220337562, 6341770050, 186793134530, 5603256962842, 170587626013306, 5257389708399426, 163705194058656258, 5142396822771086970, 162763301041914082970, 5185766155796261822338, 166183971861135163491458
Offset: 0

Views

Author

Seiichi Manyama, Dec 18 2024

Keywords

Crossrefs

Cf. A363380.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*binomial(4*n+5*k+1, n)/(4*n+5*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*n+5*k+1,n)/(4*n+5*k+1).

A371680 G.f. satisfies A(x) = ( 1 + x * A(x)^2 * (1 + A(x)) )^2.

Original entry on oeis.org

1, 4, 44, 648, 10960, 200992, 3886928, 78043488, 1611405504, 33998715264, 729793915264, 15886841223936, 349900041893376, 7782694227059712, 174573007616191744, 3944500600180286976, 89696369377912622080, 2051147782339517224960
Offset: 0

Views

Author

Seiichi Manyama, Apr 02 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n, r=2, t=4, u=2) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

Formula

G.f.: B(x)^2 where B(x) is the g.f. of A363380.
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*n+2*k+2,n)/(2*n+k+1).
Showing 1-4 of 4 results.