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%I #25 Sep 08 2022 08:46:08
%S 0,1,2,4,10,29,86,259,806,2573,8332,27301,90498,302933,1022074,
%T 3472148,11868242,40788053,140851104,488485195,1700694884,5941906864,
%U 20826229564,73208454375,258031793698,911704863957,3228661787336
%N Expansion of -(x*(1-sqrt((2*(1-sqrt(4*x^2+1)))/x+1)))/(1-sqrt(4*x^2+1)) - 1.
%H G. C. Greubel, <a href="/A243814/b243814.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum(i=0..floor((n-1)/2), (-1)^i*binomial(n,i)*binomial(2*n-4*i,n -2*i +1))/n, n>0, a(0)=0.
%F G.f. A(x) satisfies A(x) = x*(A(x)^4 + 4*A(x)^3 + 5*A(x)^2 + 4*A(x) + 1) / (A(x)^2 + 2*A(x)+1).
%F a(n) ~ 15^(n+1/2) / (4^n * n^(3/2) * sqrt(17*Pi)). - _Vaclav Kotesovec_, Jun 15 2014
%F Conjecture D-finite with recurrence: 2*n*(n+1)*(2*n-1)*a(n) -n*(47*n^2-127*n+86)*a(n-1) +4*(38*n^3-186*n^2+291*n-145)*a(n-2) +4*(-94*n^3+724*n^2-1887*n+1650)*a(n-3) +16*(n-4)*(64*n^2-382*n+595)*a(n-4) -16*(n-5)*(47*n-174)*(n-4)*a(n-5) +1920*(n-5)*(n-6)*(n-4)*a(n-6)=0. - _R. J. Mathar_, Jan 25 2020
%t CoefficientList[Series[-1 + 2/(1 + Sqrt[(2 + x - 2*Sqrt[1 + 4*x^2])/x]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 15 2014 *)
%o (Maxima)
%o a(n):=if n=0 then 0 else sum((-1)^i*binomial(n,i)*binomial(2*n-4*i,n-2*i+1),i,0,(n-1)/2)/n;
%o (PARI) x='x+O('x^30); concat([0], Vec(-(x*(1-sqrt((2*(1-sqrt(4*x^2+1)))/x+1)))/(1-sqrt(4*x^2+1))-1)) \\ _G. C. Greubel_, Oct 06 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!(-(x*(1-Sqrt((2*(1-Sqrt(4*x^2+1)))/x+1)))/(1-Sqrt(4*x^2+1))-1)); // _G. C. Greubel_, Oct 06 2018
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, Jun 11 2014
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