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%I #27 Sep 08 2022 08:46:08
%S 0,1,2,6,20,73,283,1149,4833,20901,92406,415878,1899153,8778193,
%T 40988834,193056506,916091727,4375298013,21015823762,101454449878,
%U 491976470648,2395332765741,11704830457167
%N Expansion of 1/2 - (1/2)*(sqrt((2*sqrt(-3*x^2 - 2*x + 1) + 3*x - 2)/x)).
%H G. C. Greubel, <a href="/A243255/b243255.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..n-1} (binomial(2*k,k)*Sum_{j=0..n} (binomial(j,-n-k+2*j-1)*binomial(n,j)))/n, n>0, a(0)=0.
%F G.f.: A(x) = x*M(x)*C(x*M(x)), where M(x) is the g.f. of A001006, and C(x) is the g.f. of A000108.
%F A(x) satisfies A(x)=x*(1+A(x)-2*A(x)^3+A(x)^4)/(1-A(x)).
%F a(n) ~ 3^n * 7^(n+1/2) / (sqrt(5*Pi) * n^(3/2) * 4^(n+1)). - _Vaclav Kotesovec_, Jun 02 2014
%F Conjecture D-finite with recurrence: 2*n*(2*n+1)*a(n) + (-65*n^2+119*n-42)*a(n-1) + 3*(97*n^2-373*n+342)*a(n-2) - 9*(23*n-68)*(n-3)*a(n-3) - 567*(n-3)*(n-4)*a(n-4) = 0. - _R. J. Mathar_, Jul 15 2017
%t CoefficientList[Series[1/2 - (Sqrt[(2*Sqrt[-3*x^2-2*x+1]+3*x-2)/x])/2, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 02 2014 *)
%o (Maxima)
%o a(n):=sum(binomial(2*k,k)*sum(binomial(j,-n-k+2*j-1)*binomial(n,j),j,0,n),k,0,n-1)/n;
%o (PARI) x='x+O('x^30); concat([0], Vec(1/2-1/2*(sqrt((2*sqrt(-3*x^2 -2*x+1)+3*x-2)/x)))) \\ _G. C. Greubel_, Oct 06 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!(1/2-1/2*(Sqrt((2*Sqrt(-3*x^2-2*x+1)+3*x-2)/x)))); // _G. C. Greubel_, Oct 06 2018
%Y Cf. A000108, A001006.
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, Jun 01 2014
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