This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A240586 #17 May 16 2025 23:41:43
%S A240586 1,4,22,140,950,6692,48284,354216,2630310,19713188,148817524,
%T A240586 1130011896,8621650492,66043991080,507628779896,3913088587472,
%U A240586 30240258982662,234210742764964,1817484391184900,14128074297880536,109992814064010196,857525947713607096,6693820044841440008
%N A240586 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 A240586 G. C. Greubel, <a href="/A240586/b240586.txt">Table of n, a(n) for n = 1..1000</a>
%F A240586 a(n) = Sum_{m=1..n} binomial(n-1, m-1)*Sum_{j=0..m} j*(-1)^(j-m)*binomial(m, j)*Sum_{k=1..n} binomial(k, n-k) * binomial(2*k+j-1, k+j-1) / (k+j).
%F A240586 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 A240586 a(n) ~ 8^(n-1) * (sqrt(3)-1) / sqrt(Pi*n). - _Vaclav Kotesovec_, Apr 12 2014
%t A240586 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 A240586 (Maxima)
%o A240586 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 A240586 (PARI) my(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 A240586 nonn
%O A240586 1,2
%A A240586 _Vladimir Kruchinin_, Apr 08 2014