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Expansion of (((8-8 / sqrt(1-8*x)) / (2*sqrt(8*x+2*sqrt(1-8*x)+2))+4 / sqrt(1-8*x))*((x*(sqrt(8*x+2*sqrt(1-8*x)+2)-sqrt(1-8*x)-1))-4*x^2)) / (sqrt(8*x+2*sqrt(1-8*x)+2)-sqrt(1-8*x)-1)^2.
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%I #14 Apr 06 2017 02:27:25

%S 1,4,22,140,950,6692,48284,354216,2630310,19713188,148817524,

%T 1130011896,8621650492,66043991080,507628779896,3913088587472,

%U 30240258982662,234210742764964,1817484391184900,14128074297880536

%N Expansion of (((8-8 / sqrt(1-8*x)) / (2*sqrt(8*x+2*sqrt(1-8*x)+2))+4 / sqrt(1-8*x))*((x*(sqrt(8*x+2*sqrt(1-8*x)+2)-sqrt(1-8*x)-1))-4*x^2)) / (sqrt(8*x+2*sqrt(1-8*x)+2)-sqrt(1-8*x)-1)^2.

%H G. C. Greubel, <a href="/A240586/b240586.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = Sum_{m=1..n} (Sum_{j=0..m} j*(Sum_{k=1..n} (binomial(k, n-k) * binomial(2*k+j-1, k+j-1)) / (k+j))) * (-1)^(j-m) * binomial(m, j))) * binomial(n-1, m-1)).

%F A(x) = B'(x) * (x*B(x)-x^2) / B(x)^2, where B(x) = (-1-sqrt(1-8*x)+sqrt(2+2*sqrt(1-8*x)+8*x))/4, B(x)/x is g.f. of A186997.

%F a(n) ~ 8^(n-1) * (sqrt(3)-1) / sqrt(Pi*n). - _Vaclav Kotesovec_, Apr 12 2014

%t Rest[CoefficientList[Series[(((8-8 / Sqrt[1-8*x]) / (2*Sqrt[8*x+2*Sqrt[1-8*x]+2])+4 / Sqrt[1-8*x])*((x*(Sqrt[8*x+2*Sqrt[1-8*x]+2]-Sqrt[1-8*x]-1))-4*x^2)) / (Sqrt[8*x+2*Sqrt[1-8*x]+2]-Sqrt[1-8*x]-1)^2, {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Apr 12 2014 *)

%o (Maxima)

%o a(n):=sum((sum(j*(sum((binomial(k,n-k)*binomial(2*k+j-1,k+j-1)) / (k+j),k,1,n))*(-1)^(j-m)*binomial(m,j),j,0,m))*binomial(n-1,m-1),m,1,n);

%o (PARI) x='x+O('x^50); Vec((((8-8 / sqrt(1-8*x)) / (2*sqrt(8*x+2*sqrt(1-8*x)+2))+4 / sqrt(1-8*x))*((x*(sqrt(8*x+2*sqrt(1-8*x)+2)-sqrt(1-8*x)-1))-4*x^2)) / (sqrt(8*x+2*sqrt(1-8*x)+2)-sqrt(1-8*x)-1)^2) \\ _G. C. Greubel_, Apr 05 2017

%K nonn

%O 1,2

%A _Vladimir Kruchinin_, Apr 08 2014