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%I #18 Jan 30 2020 21:29:17
%S 1,1,5,7,33,51,233,379,1697,2851,12585,21627,94449,165075,714873,
%T 1266027,5445441,9746883,41687369,75275227,320420753,582881971,
%U 2471008281,4523575371,19108837601,35174066851
%N Expansion of -(2*x*sqrt(1-8*x^2)-2*x) / (16*x^3+sqrt(1-8*x^2)*(4*x^2+2*x-1)-8*x^2-2*x+1).
%F a(n) = sum(i=0..floor(n/2), 2^i*binomial(n,i)).
%F G.f. A(x) = (x*C'(2*x^2))/(C(2*x^2)*(1-x*C(2*x^2))), where C(x) is g.f. of A000108.
%F a(n) ~ 2^(3*n/2) * (2+sqrt(2) + (-1)^n*(2-sqrt(2))) / sqrt(2*Pi*n). - _Vaclav Kotesovec_, May 29 2014
%F D-finite with recurrence: n^2*a(n) = (3*n^2-4)*a(n-1) + 4*(2*n^2 - 2*n - 1)*a(n-2) - 24*(n-2)*(n+1)*a(n-3). - _Vaclav Kotesovec_, May 29 2014
%t CoefficientList[Series[-(2*x*Sqrt[1-8*x^2]-2*x)/(16*x^3+Sqrt[1-8*x^2]*(4*x^2+2*x-1)-8*x^2-2*x+1),{x,0,20}],x] (* _Vaclav Kotesovec_, May 29 2014 *)
%o (Maxima)
%o a(n):=sum(2^(i)*binomial(n,i),i,0,floor((n)/2));
%Y Cf. A000108, A128418.
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, May 29 2014