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%I #14 May 19 2016 02:51:27
%S 1,2,2,3,3,5,5,10,10,24,24,66,66,198,198,627,627,2057,2057,6919,6919,
%T 23715,23715,82501,82501,290513,290513,1033413,1033413,3707853,
%U 3707853,13402698,13402698,48760368,48760368,178405158,178405158,656043858
%N Partial sums of A097331; binomial transform of A166587.
%C Hankel transform is A131713. The Hankel transform of the sequence 1,1,2,2,... is A128017(n+3). A155587 doubled.
%H G. C. Greubel, <a href="/A166588/b166588.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: (1+2x-sqrt(1-4x^2))/(2x(1-x))=((1+x^2*c(x^2))/(1-x)-1)/x, c(x) the g.f. of A000108.
%F a(n) = Sum_{k=0..n} C(n,k)*A166587(k).
%F Conjecture: (-n-1)*a(n) + (n+1)*a(n-1) + 4*(n-2)*a(n-2) + 4*(-n+2)*a(n-3) = 0. - _R. J. Mathar_, Nov 15 2012
%F a(n) ~ 2^(n+1/2) * (3-(-1)^n) / (3 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 08 2014
%t CoefficientList[Series[(1+2*x-Sqrt[1-4*x^2])/(2*x*(1-x)), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Feb 08 2014 *)
%K easy,nonn
%O 0,2
%A _Paul Barry_, Oct 17 2009
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