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%I #16 Jan 30 2020 21:29:16
%S 1,1,2,6,14,38,94,248,628,1638,4190,10872,27940,72316,186260,481488,
%T 1241512,3207302,8274646,21369496,55148068,142396436,367537484,
%U 948920560,2449445432,6323741404,16324167564,42143003504
%N Expansion of ((x-1)*sqrt(1-4*x^2))/((x-1)*sqrt(1-4*x^2)+x).
%F a(n)=sum(k=1..n, sum(i=0..(n-k)/2, binomial((2*i+k-2)/2,i)*4^i*binomial(n-2*i-1,k-1))), n>0, a(0)=1.
%F Conjecture D-finite with recurrence: (-n+1)*a(n) +(3*n-5)*a(n-1) +2*(3*n-7)*a(n-2) +4*(-6*n+19)*a(n-3) +4*(n-3)*a(n-4) +4*(11*n-53)*a(n-5) +16*(-3*n+16)*a(n-6) +16*(n-6)*a(n-7)=0. - _R. J. Mathar_, Nov 28 2013
%t CoefficientList[Series[((x-1)Sqrt[1-4x^2])/((x-1)Sqrt[1-4x^2]+x), {x,0,60}],x] (* _Harvey P. Dale_, May 24 2011 *)
%o (Maxima)
%o a(n):=sum(sum(binomial((2*i+k-2)/2,i)*4^i*binomial(n-2*i-1,k-1),i,0,(n-k)/2),k,1,n);
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, May 19 2011