%I #23 Nov 09 2024 16:54:23
%S 0,1,1,4,11,40,138,512,1903,7256,27910,108704,426702,1688080,6719252,
%T 26895360,108171319,436935544,1771673454,7208637920,29422782282,
%U 120436341360,494277785356,2033457590656,8384379887334,34642507651952,143413034719036,594776592764224
%N Expansion of (sqrt(8*x+4*sqrt(1-4*x)-3)-1)/(2*sqrt(1-4*x)-2).
%H G. C. Greubel, <a href="/A243760/b243760.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = (Sum_{k=0..floor((n-1)/2)} 2^k*binomial(2*k+1,k)*binomial(2*n-2*k-2,n-1))/n, n>0, a(0)=0.
%F G.f. A(x) = x*C(x)*C(2*x^2*C(x)^2), where C(x) is the g.f. of A000108.
%F G.f. A(x) satisfies A(x)= x*(4*A(x)^4+4*A(x)^2+1)/(2*A(x)^2-A(x)+1).
%F a(n) ~ sqrt(2+3*sqrt(2)) * 2^(3*n-7/4) * ((1+2*sqrt(2))/7)^n / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jun 15 2014
%F Conjecture D-finite with recurrence: 245*n*(n-1)*(n+1)*a(n) -140*n*(n-1)*(28*n-53)*a(n-1) +4*(n-1)*(5812*n^2-29864*n+36585)*a(n-2) +16*(-3368*n^3+35040*n^2-111784*n+110667)*a(n-3) +128*(-152*n^3-1224*n^2+14756*n-29655)*a(n-4) +2048*(2*n-9)*(74*n^2-567*n+988)*a(n-5) -98304*(n-6)*(2*n-9)*(2*n-11)*a(n-6)=0. - _R. J. Mathar_, Jun 07 2016
%t CoefficientList[Series[(-1 + Sqrt[-3 + 4*Sqrt[1-4*x] + 8*x])/(-2 + 2*Sqrt[1-4*x]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 15 2014 *)
%o (Maxima)
%o a(n):=sum(2^k*binomial(2*k+1,k)*binomial(2*n-2*k-2,n-1),k,0,(n-1)/2)/n;
%o (PARI) my(x='x+O('x^50)); concat([0], Vec((sqrt(8*x+4*sqrt(1-4*x)-3)-1)/(2*sqrt(1-4*x)-2))) \\ _G. C. Greubel_, Jun 02 2017
%Y Cf. A000108.
%K nonn
%O 0,4
%A _Vladimir Kruchinin_, Jun 10 2014