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.

A360272 a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * Catalan(n-3*k).

Original entry on oeis.org

1, 1, 2, 5, 15, 46, 147, 485, 1642, 5669, 19883, 70646, 253755, 919925, 3361546, 12368661, 45786219, 170400470, 637200555, 2392962645, 9021255722, 34128098389, 129519490219, 492966689110, 1881289209003, 7197100511317, 27595769836714, 106032318322517
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2023

Keywords

Crossrefs

Cf. A052709, A125305.
Cf. A000108, A360267, A360271, A360274.

Programs

  • Maple
    A360272 := proc(n)
    add(binomial(n-3*k,k)*A000108(n-3*k),k=0..n/3) ;
end proc:
seq(A360272(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • PARI
    a(n) = sum(k=0, n\4, 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 = 0.2463187933841... is the smallest positive root of the equation1 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

A360273 a(n) = Sum_{k=0..floor(n/2)} Catalan(n-2*k).

Original entry on oeis.org

1, 1, 3, 6, 17, 48, 149, 477, 1579, 5339, 18375, 64125, 226387, 807025, 2900827, 10501870, 38258497, 140146660, 515897197, 1907409850, 7080017617, 26373676870, 98562581257, 369433290520, 1388466728581, 5231379691972
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2023

Keywords

Links

Crossrefs

Cf. A014137, A360274.
Cf. A000108.

Programs

  • Maple
    A360273 := proc(n)
    add(A000108(n-2*k),k=0..n/2) ;
end proc:
seq(A360273(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • Mathematica
    Table[Sum[CatalanNumber[n-2k],{k,0,Floor[n/2]}],{n,0,30}] (* Harvey P. Dale, Sep 08 2024 *)
  • PARI
    a(n) = sum(k=0, n\2, binomial(2*(n-2*k), n-2*k)/(n-2*k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/((1-x^2)*(1+sqrt(1-4*x))))

Formula

G.f.: c(x)/(1-x^2), where c(x) is the g.f. of A000108.
a(n) ~ 2^(2*n+4) / (15*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 01 2023
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(-n-1)*a(n-2) +2*(2*n-1)*a(n-3)=0. - R. J. Mathar, Mar 12 2023
Showing 1-2 of 2 results.

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

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