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.

A383581 a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k,k) * binomial(2*(n-3*k),n-3*k).

Original entry on oeis.org

1, 2, 6, 21, 74, 270, 1005, 3788, 14418, 55289, 213270, 826614, 3216629, 12558928, 49175136, 193023965, 759299438, 2992534344, 11813985377, 46709675040, 184928644350, 733047010709, 2908981549006, 11555513379450, 45945148281437, 182835149061920, 728149606630164
Offset: 0

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Author

Seiichi Manyama, Apr 30 2025

Keywords

Links

Crossrefs

Cf. A026375, A383573, A383582.
Cf. A376791, A383571.
Cf. A360291.

Programs

  • Magma
    [&+[Binomial(n-2*k,k) * Binomial(2*(n-3*k),n-3*k): k in [0..n div 3]]: n in [0..25]]; // Vincenzo Librandi, May 02 2025
  • Mathematica
    Table[Sum[Binomial[n-2*k,k]* Binomial[2*(n-3*k),n-3*k],{k,0,Floor[n/3]}],{n,0,30}] (* Vincenzo Librandi, May 02 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, binomial(n-2*k, k)*binomial(2*(n-3*k), n-3*k));

Formula

G.f.: 1/sqrt((1 - x^3) * (1 - x^3 - 4*x)).

A383572 Expansion of 1/sqrt((1-x^4)^2 - 4*x^5).

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 0, 0, 1, 6, 6, 0, 1, 12, 30, 20, 1, 20, 90, 140, 71, 30, 210, 560, 631, 294, 420, 1680, 3151, 2828, 1680, 4200, 11551, 16704, 13272, 12672, 34651, 72162, 86064, 69960, 102961, 252362, 423390, 446160, 429001, 805508, 1685970, 2393820, 2419561
Offset: 0

Views

Author

Seiichi Manyama, Apr 30 2025

Keywords

Comments

Number of lattice paths from (0,0) to (n,n) using steps (5,0),(0,5),(4,4).
Diagonal of the rational function 1 / (1 - x^5 - y^5 - x^4*y^4).

Crossrefs

Cf. A002426, A182883, A383571.
Cf. A376792.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(k, n-4*k));

Formula

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

A383583 a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k) * binomial(k,n-3*k).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 2, 3, 0, 3, 12, 10, 4, 30, 60, 40, 60, 210, 286, 231, 560, 1267, 1428, 1722, 4208, 7182, 8064, 13275, 28080, 40656, 51754, 97020, 176088, 240251, 355872, 667810, 1081092, 1506648, 2475616, 4401696, 6693492, 9904752, 16950662, 28359201
Offset: 0

Views

Author

Seiichi Manyama, May 01 2025

Keywords

Links

Crossrefs

Cf. A005717, A383584.
Cf. A383571.

Programs

  • Magma
    [&+[Binomial(n-2*k-1, k) * Binomial(k, n-3*k): k in [0..Floor(n div 3)]]: n in [0..45]]; // Vincenzo Librandi, May 03 2025
  • Mathematica
    Table[Sum[Binomial[n-2*k-1,k]* Binomial[k,n-3*k],{k,0,Floor[n/3]}],{n,0,45}] (* Vincenzo Librandi, May 03 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, binomial(n-2*k-1, k)*binomial(k, n-3*k));

Formula

G.f.: (1/2) * ( 1 + 1/sqrt(1 - 4*x^4/(1-x^3)^2) ).
a(n) ~ phi^(n-1) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 01 2025
Showing 1-3 of 3 results.

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

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