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.

A360271 a(n) = Sum_{k=0..floor(n/4)} (-1)^k * binomial(n-3*k,k) * Catalan(n-3*k).

Original entry on oeis.org

1, 1, 2, 5, 13, 38, 117, 373, 1222, 4085, 13877, 47766, 166229, 583893, 2067414, 7371093, 26440789, 95355990, 345538389, 1257486165, 4593933398, 16841578325, 61938532181, 228454719830, 844882459989, 3132258655573, 11638656376150, 43337083401557
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2023

Keywords

Crossrefs

Cf. A157003.
Cf. A000108, A360219, A360272.

Programs

  • Maple
    A360271 := proc(n)
    add((-1)^k*binomial(n-3*k,k)*A000108(n-3*k),k=0..n/3) ;
end proc:
seq(A360271(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • PARI
    a(n) = sum(k=0, n\4, (-1)^k*binomial(n-3*k, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/(1+(sqrt(1-4*x*(1-x^3)))))

Formula

G.f.: c(x * (1-x^3)), where c(x) is the g.f. of A000108.
a(n) ~ 2 * sqrt(1-3*r) / (sqrt(Pi) * n^(3/2) * r^n), where r = 0.2541737124933... is the smallest positive root of the equation 1 - 4*r + 4*r^4 = 0. - Vaclav Kotesovec, Feb 01 2023
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(-n-1)*a(n-3) +2*(4*n-11)*a(n-4) +4*(-n+5)*a(n-7)=0. - R. J. Mathar, Mar 12 2023

A376574 G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)/(1 - x^3)).

Original entry on oeis.org

1, 1, 2, 5, 15, 46, 147, 486, 1646, 5684, 19940, 70864, 254592, 923153, 3374046, 12417246, 45975677, 171141378, 640105278, 2404375805, 9066188052, 34305301482, 130219435385, 495735347502, 1892254721982, 7240580768021, 27768359445128, 106718055778871
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2024

Keywords

Crossrefs

Cf. A002212, A085139.
Cf. A000108, A360272.

Programs

  • Maple
    A376574 := proc(n)
    add(A000108(n-3*k)*binomial(n-2*k-1,k),k=0..floor(n/3)) ;
end proc:
seq(A376574(n),n=0..80) ;
# R. J. Mathar, Oct 24 2024
  • PARI
    a(n) = sum(k=0, n\3, binomial(n-2*k-1, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x/(1-x^3))))

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k) * Catalan(n-3*k).
G.f.: 2/(1 + sqrt(1 - 4*x/(1 - x^3))).
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(-2*n+7)*a(n-3) +4*(n-5)*a(n-4) +(n-8)*a(n-6)=0. - R. J. Mathar, Oct 24 2024
Showing 1-2 of 2 results.

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

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