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 A129366 #13 Feb 01 2024 01:52:52
%S A129366 1,1,3,7,21,61,193,617,2047,6895,23691,82435,290447,1033215,3707655,
%T A129366 13402071,48759741,178403101,656041801,2423300129,8987420549,
%U A129366 33453670773,124936234413,467995789277,1757899936601
%N A129366 a(n) = Sum_{k=0..floor(n/2)} A000108(n-k).
%C A129366 Partial sums of A129367 (prefixed by an initial 1).
%H A129366 G. C. Greubel, <a href="/A129366/b129366.txt">Table of n, a(n) for n = 0..1000</a>
%F A129366 G.f.: (1/(1-x))*(c(x) - x*c(x^2)), where c(x) is the g.f. of A000108(n).
%F A129366 G.f.: (sqrt(1-4*x^2) - sqrt(1-4*x))/(2*x*(1-x)).
%F A129366 a(n) = Sum_{k=floor((n+1)/2)..n} C(k), where C(n) = A000108(n).
%F A129366 Conjecture: n*(12*n+35)*(n-1)*a(n) + (n-1)*(12*n^2-701*n+1236)*a(n-1) + 2*(6*n^3-385*n^2+2285*n-3432)*a(n-2) + 4*(-405*n^3+5313*n^2-19970*n +23175)*a(n-3) + 8*(156*n^3-1724*n^2+5498*n-5175)*a(n-4) + 16*(393*n^3-4981*n^2+20393*n-26820)*a(n-5) - 32*(n-5)*(93*n-268)*(2*n-9)*a(n-6) = 0. - _R. J. Mathar_, Feb 05 2015
%t A129366 Table[Sum[CatalanNumber[k], {k, Floor[(n + 1)/2], n}], {n, 0, 30}] (* _Wesley Ivan Hurt_, Jun 18 2022 *)
%o A129366 (Magma) [(&+[Catalan(n-j): j in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, Jan 31 2024
%o A129366 (SageMath) [sum(catalan_number(n-j) for j in range(1+int(n//2))) for n in range(31)] # _G. C. Greubel_, Jan 31 2024
%Y A129366 Cf. A000108, A129367.
%K A129366 easy,nonn
%O A129366 0,3
%A A129366 _Paul Barry_, Apr 11 2007