A360273 - OEIS

%I #16 Sep 08 2024 15:49:31

%S 1,1,3,6,17,48,149,477,1579,5339,18375,64125,226387,807025,2900827,

%T 10501870,38258497,140146660,515897197,1907409850,7080017617,

%U 26373676870,98562581257,369433290520,1388466728581,5231379691972

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

%H Harvey P. Dale, <a href="/A360273/b360273.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: c(x)/(1-x^2), where c(x) is the g.f. of A000108.

%F a(n) ~ 2^(2*n+4) / (15*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Feb 01 2023

%F 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

%p A360273 := proc(n)

%p     add(A000108(n-2*k),k=0..n/2) ;

%p end proc:

%p seq(A360273(n),n=0..70) ; # _R. J. Mathar_, Mar 12 2023

%t Table[Sum[CatalanNumber[n-2k],{k,0,Floor[n/2]}],{n,0,30}] (* _Harvey P. Dale_, Sep 08 2024 *)

%o (PARI) a(n) = sum(k=0, n\2, binomial(2*(n-2*k), n-2*k)/(n-2*k+1));

%o (PARI) my(N=30, x='x+O('x^N)); Vec(2/((1-x^2)*(1+sqrt(1-4*x))))

%Y Cf. A014137, A360274.

%Y Cf. A000108.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Jan 31 2023