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.
%I A360273 #16 Sep 08 2024 15:49:31
%S A360273 1,1,3,6,17,48,149,477,1579,5339,18375,64125,226387,807025,2900827,
%T A360273 10501870,38258497,140146660,515897197,1907409850,7080017617,
%U A360273 26373676870,98562581257,369433290520,1388466728581,5231379691972
%N A360273 a(n) = Sum_{k=0..floor(n/2)} Catalan(n-2*k).
%H A360273 Harvey P. Dale, <a href="/A360273/b360273.txt">Table of n, a(n) for n = 0..1000</a>
%F A360273 G.f.: c(x)/(1-x^2), where c(x) is the g.f. of A000108.
%F A360273 a(n) ~ 2^(2*n+4) / (15*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Feb 01 2023
%F A360273 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 A360273 := proc(n)
%p A360273 add(A000108(n-2*k),k=0..n/2) ;
%p A360273 end proc:
%p A360273 seq(A360273(n),n=0..70) ; # _R. J. Mathar_, Mar 12 2023
%t A360273 Table[Sum[CatalanNumber[n-2k],{k,0,Floor[n/2]}],{n,0,30}] (* _Harvey P. Dale_, Sep 08 2024 *)
%o A360273 (PARI) a(n) = sum(k=0, n\2, binomial(2*(n-2*k), n-2*k)/(n-2*k+1));
%o A360273 (PARI) my(N=30, x='x+O('x^N)); Vec(2/((1-x^2)*(1+sqrt(1-4*x))))
%Y A360273 Cf. A014137, A360274.
%Y A360273 Cf. A000108.
%K A360273 nonn,easy
%O A360273 0,3
%A A360273 _Seiichi Manyama_, Jan 31 2023