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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)).
1

%I #20 Nov 09 2024 04:20:32

%S 1,2,8,35,160,752,3605,17544,86400,429605,2153008,10860720,55086421,

%T 280692440,1435868960,7369703660,37934443008,195748568256,

%U 1012292239955,5244933087000,27220980100160,141486701601630,736387364237280,3837221866576800,20016901815607125

%N 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 G. C. Greubel, <a href="/A243204/b243204.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} binomial(2*k-1,k)*binomial(2*n-k-1,n-k).

%F 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 a(n) ~ 2^(4*n-3/2) / (sqrt(Pi*n) * 3^(n-1/2)). - _Vaclav Kotesovec_, Jun 02 2014

%t 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 (Maxima)

%o a(n):=sum(binomial(2*k-1,k)*binomial(2*n-k-1,n-k),k,0,n);

%o (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 Cf. A000108, A121988.

%K nonn,changed

%O 0,2

%A _Vladimir Kruchinin_, Jun 01 2014