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 A187071 #32 Feb 14 2025 01:09:45
%S A187071 1,2,8,40,206,1084,5802,31440,171946,947132,5247010,29203928,
%T A187071 163176586,914744612,5142354178,28978786976,163652047834,925925993132,
%U A187071 5247514156418,29783577676840,169270380108906,963186164033652,5486768119272258,31286597202864240
%N A187071 Expansion of d/dx arctan(x*A001003(x)).
%H A187071 G. C. Greubel, <a href="/A187071/b187071.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..199 from Nathaniel Johnston)
%F A187071 a(n) = Sum_{k=0..n} ( Sum_{j=0..n-k} (-1)^j*2^(-j)*binomial(n+1, j) * binomial(2*n-k-j, n) ) * (2^(n-k-1))*(1-(-1)^(k+1))*(-1)^(k/2).
%F A187071 G.f.: d/dx arctan(x*2/(1+x+sqrt(1-6*x+x^2))) = (sqrt(x^2-6*x+1)-x+3) / (4*sqrt(x^2-6*x+1) * ((-sqrt(x^2-6*x+1)+x+1)^2/16+1)).
%F A187071 Recurrence: 5*n*(17*n-24)*a(n) = (544*n^2 - 1023*n + 385)*a(n-1) - (323*n^2 - 643*n + 224)*a(n-2) + 2*(119*n^2 - 236*n + 91)*a(n-3) - 2*(n-2)*(17*n-7)*a(n-4). - _Vaclav Kotesovec_, Oct 24 2012
%F A187071 a(n) ~ sqrt(252+179*sqrt(2))*(3+2*sqrt(2))^n/(34*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 24 2012
%t A187071 CoefficientList[Series[(Sqrt[x^2-6*x+1]-x+3)/(4*Sqrt[x^2-6*x+1]*((-Sqrt[x^2-6*x+1]+x+1)^2/16+1)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 24 2012 *)
%o A187071 (Maxima)
%o A187071 a(n):=sum((sum((-1)^j*2^(-j)*binomial(n+1, j)*binomial(2*n-k-j, n), j, 0, n-k))*(2^(n-k-1))*(1-(-1)^(k+1))*(-1)^(k/2), k, 0, n);
%o A187071 (PARI) my(x='x+O('x^50)); Vec((sqrt(x^2-6*x+1)-x+3) / (4*sqrt(x^2-6*x+1)*((-sqrt(x^2-6*x+1)+x+1)^2/16+1))) \\ _G. C. Greubel_, Mar 26 2017
%Y A187071 Cf. A001003.
%K A187071 nonn
%O A187071 0,2
%A A187071 _Vladimir Kruchinin_, Apr 10 2011