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

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

Author

Seiichi Manyama, Jan 31 2023

Keywords

• nonn
• easy

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

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