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 A129368 #10 Jan 31 2024 15:09:00
%S A129368 1,2,8,26,96,342,1266,4678,17548,66098,250854,956034,3660190,14059866,
%T A129368 54176466,209290554,810370944,3143964294,12219099594,47564314774,
%U A129368 185410843594,723668533278,2827767496998,11061197519166
%N A129368 a(n) = Sum_{k=floor((n+1)/2)..n} binomial(2*k,k).
%C A129368 Partial sums of A129369.
%H A129368 G. C. Greubel, <a href="/A129368/b129368.txt">Table of n, a(n) for n = 0..1000</a>
%F A129368 G.f.: (1/(1-x))*( 1/sqrt(1-4*x) - x/sqrt(1-4*x^2) ).
%F A129368 a(n) = Sum_{k=0..floor(n/2)} C(2*(n-k), n-k).
%t A129368 Table[Sum[Binomial[2k,k],{k,Floor[(n+1)/2],n}],{n,0,30}] (* _Harvey P. Dale_, Aug 13 2012 *)
%o A129368 (Magma) [(&+[Binomial(2*(n-k),n-k): k in [0..Floor(n/2)]]): n in [0..60]]; // _G. C. Greubel_, Jan 31 2024
%o A129368 (SageMath) [sum(binomial(2*(n-k),n-k) for k in range(1+(n//2))) for n in range(61)] # _G. C. Greubel_, Jan 31 2024
%Y A129368 Cf. A129369.
%K A129368 easy,nonn
%O A129368 0,2
%A A129368 _Paul Barry_, Apr 11 2007