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%I A243204 #20 Nov 09 2024 04:20:32
%S A243204 1,2,8,35,160,752,3605,17544,86400,429605,2153008,10860720,55086421,
%T A243204 280692440,1435868960,7369703660,37934443008,195748568256,
%U A243204 1012292239955,5244933087000,27220980100160,141486701601630,736387364237280,3837221866576800,20016901815607125
%N A243204 Expansion of 2*x/((1-sqrt(1-2*(1-sqrt(1-4*x))))*sqrt(1-2*(1-sqrt(1-4*x))) * sqrt(1-4*x)).
%H A243204 G. C. Greubel, <a href="/A243204/b243204.txt">Table of n, a(n) for n = 0..1000</a>
%F A243204 a(n) = Sum_{k=0..n} binomial(2*k-1,k)*binomial(2*n-k-1,n-k).
%F A243204 G.f.: A(x) = x*F'(x)/F(x), where F(x)=x*C(x)*C(x*C(x)), C(x) is g.f. of A000108.
%F A243204 a(n) ~ 2^(4*n-3/2) / (sqrt(Pi*n) * 3^(n-1/2)). - _Vaclav Kotesovec_, Jun 02 2014
%t A243204 CoefficientList[Series[2*x / (Sqrt[1-4*x] + Sqrt[-1+2*Sqrt[1-4*x]] *Sqrt[1-4*x] + 8*x-2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 02 2014 *)
%o A243204 (Maxima)
%o A243204 a(n):=sum(binomial(2*k-1,k)*binomial(2*n-k-1,n-k),k,0,n);
%o A243204 (PARI) my(x='x+O('x^50)); Vec(2*x/((1-sqrt(1-2*(1-sqrt(1-4*x))))*sqrt(1-2*(1-sqrt(1-4*x)))*sqrt(1-4*x))) \\ _G. C. Greubel_, Jun 01 2017
%Y A243204 Cf. A000108, A121988.
%K A243204 nonn
%O A243204 0,2
%A A243204 _Vladimir Kruchinin_, Jun 01 2014

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