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 1-2 of 2 results.

A360266 a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k,k) * binomial(2*(n-2*k),n-2*k).

Original entry on oeis.org

1, 2, 6, 22, 82, 312, 1210, 4752, 18834, 75184, 301856, 1217604, 4930626, 20032052, 81615072, 333328532, 1364264250, 5594210292, 22977466864, 94517423444, 389316529512, 1605533230256, 6628467569292, 27393187077144, 113310732332274, 469101108803052
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2023

Keywords

Comments

Diagonal of rational function 1/(1 - (x + y + x^3*y^2)). - Seiichi Manyama, Mar 23 2023

Links

Crossrefs

Cf. A006139, A360267.
Cf. A157004, A374598.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n-2*k, k)*binomial(2*(n-2*k), n-2*k));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1+x^2)))

Formula

G.f.: 1/sqrt(1 - 4*x*(1 + x^2)).
n*a(n) = 2*(2*n-1)*a(n-1) + 2*(2*n-3)*a(n-3).

A374599 Expansion of 1/sqrt(1 - 4*x - 8*x^4).

Original entry on oeis.org

1, 2, 6, 20, 74, 276, 1044, 3992, 15414, 59948, 234484, 921432, 3634916, 14386248, 57097704, 227166384, 905714150, 3617851980, 14475452484, 58004111160, 232737175404, 934969613528, 3760157234584, 15137340947280, 60994657996476, 245980435701752
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2024

Keywords

Links

Crossrefs

Cf. A360219, A360267.
Cf. A084609, A374598.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x-8*x^4))
  • PARI
    a(n) = sum(k=0, n\4, 2^k*binomial(n-3*k, k)*binomial(2*(n-3*k), n-3*k));

Formula

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.