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.

A365729 G.f. satisfies A(x) = 1 + x^4*A(x)^3*(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 3, 7, 4, 0, 12, 45, 55, 22, 55, 286, 546, 455, 413, 1820, 4760, 6120, 5304, 12597, 38760, 67830, 73587, 108262, 309925, 672980, 928763, 1171390, 2598310, 6270030, 10607025, 14033565, 24293880, 57256875, 112293597, 167416275, 255805056
Offset: 0

Views

Author

Seiichi Manyama, Sep 17 2023

Keywords

Crossrefs

Cf. A365727, A365728, A365730, A365731.
Cf. A365697.

Programs

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

Formula

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

A365696 G.f. satisfies A(x) = 1 + x^4*A(x)^2 / (1 - x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 3, 6, 10, 15, 26, 49, 92, 165, 294, 535, 994, 1852, 3437, 6379, 11905, 22344, 42058, 79260, 149601, 283038, 536806, 1020066, 1941317, 3699922, 7062308, 13500402, 25842489, 49528164, 95031920, 182545222, 351023451, 675678911, 1301838177
Offset: 0

Views

Author

Seiichi Manyama, Sep 16 2023

Keywords

Crossrefs

Cf. A023427, A215341, A215342, A357308, A365697.
Cf. A112805.

Programs

  • Mathematica
    CoefficientList[Series[(1 + x - Sqrt[1 - 2*x + x^2 - 4*x^4])/(2*x*(1 + x^3)), {x, 0, 40}], x] (* Vaclav Kotesovec, Sep 26 2023 *)
  • PARI
    a(n) = sum(k=0, n\4, binomial(n-3*k-1, n-4*k)*binomial(n-2*k+1, k)/(n-2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k) * binomial(n-2*k+1,k) / (n-2*k+1).
From Vaclav Kotesovec, Sep 26 2023: (Start)
G.f.: (1 + x - sqrt(1 - 2*x + x^2 - 4*x^4)) / (2*x*(1 + x^3)).
a(n) ~ 2^(n + 3/2) / (sqrt(Pi) * 3^(3/2) * n^(3/2)). (End)
Showing 1-2 of 2 results.

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

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