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.

A383582 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * binomial(2*(n-4*k),n-4*k).

Original entry on oeis.org

1, 2, 6, 20, 71, 256, 942, 3512, 13221, 50138, 191260, 733088, 2821037, 10892100, 42174848, 163706656, 636816019, 2481902842, 9689155902, 37882580356, 148313102097, 581365577564, 2281393560802, 8961689897248, 35235582858441, 138657185501870, 546064549476476
Offset: 0

Views

Author

Seiichi Manyama, Apr 30 2025

Keywords

Links

Crossrefs

Cf. A026375, A383573, A383581.
Cf. A376792, A383572.
Cf. A360292.

Programs

  • Magma
    [&+[Binomial(n-3*k,k) * Binomial(2*(n-4*k),n-4*k): k in [0..n div 4]]: n in [0..45]]; // Vincenzo Librandi, May 02 2025
  • Mathematica
    Table[Sum[Binomial[n-3*k,k]* Binomial[2*(n-4*k),n-4*k],{k,0,Floor[n/4]}],{n,0,30}] (* Vincenzo Librandi, May 02 2025 *)
  • PARI
    a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*(n-4*k), n-4*k));

Formula

G.f.: 1/sqrt((1 - x^4) * (1 - x^4 - 4*x)).
a(n) ~ (2 + sqrt(2) + sqrt(10 + 8*sqrt(2)))^n / (sqrt((sqrt(5 + 32*sqrt(2)) - 7)*Pi*n) * 2^(n + 7/4)). - Vaclav Kotesovec, May 01 2025

A383571 Expansion of 1/sqrt((1-x^3)^2 - 4*x^4).

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 1, 6, 6, 1, 12, 30, 21, 20, 90, 141, 100, 210, 561, 672, 672, 1681, 3206, 3528, 5125, 11622, 17892, 21253, 38172, 74052, 102565, 141680, 268092, 454741, 622182, 979836, 1790361, 2784366, 3993132, 6741593, 11587758, 17380116, 26551097, 45489082, 74098518
Offset: 0

Views

Author

Seiichi Manyama, Apr 30 2025

Keywords

Comments

Number of lattice paths from (0,0) to (n,n) using steps (4,0),(0,4),(3,3).
Diagonal of the rational function 1 / (1 - x^4 - y^4 - x^3*y^3).

Crossrefs

Cf. A053442, A376791.
Cf. A002426, A182883, A383572.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n-2*k, k)*binomial(k, n-3*k));

Formula

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

A383584 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k-1,k) * binomial(k,n-4*k).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 3, 0, 0, 3, 12, 10, 0, 4, 30, 60, 35, 5, 60, 210, 280, 132, 105, 560, 1260, 1267, 630, 1260, 4200, 6938, 5796, 4236, 11550, 27729, 36396, 28644, 34155, 90100, 168663, 188100, 163020, 276573, 631290, 973830, 995280, 1068222, 2111252, 4104100
Offset: 0

Views

Author

Seiichi Manyama, May 01 2025

Keywords

Links

Crossrefs

Cf. A005717, A383583.
Cf. A383572.

Programs

  • Magma
    [&+[Binomial(n-3*k-1,k) * Binomial(k,n-4*k): k in [0..Floor(n div 4)]]: n in [0..45]]; // Vincenzo Librandi, May 02 2025
  • Mathematica
    Table[Sum[Binomial[n-3*k-1,k]* Binomial[k,n-4*k],{k,0,Floor[n/4]}],{n,0,40}] (* Vincenzo Librandi, May 02 2025 *)
  • PARI
    a(n) = sum(k=0, n\4, binomial(n-3*k-1, k)*binomial(k, n-4*k));

Formula

G.f.: (1/2) * ( 1 + 1/sqrt(1 - 4*x^5/(1-x^4)^2) ).
Showing 1-3 of 3 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.