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A184120 Expansion of (1/(1+4x+2x^2))*c(x/(1+4x+2x^2)), c(x) the g.f. of A000108.

%I A184120 #26 Sep 08 2022 08:45:55
%S A184120 1,-3,8,-23,70,-218,688,-2195,7062,-22866,74416,-243206,797660,
%T A184120 -2624004,8654304,-28607171,94748774,-314361682,1044625200,
%U A184120 -3476135186,11581870900,-38632753228,128998096032,-431144781486,1442252806012
%N A184120 Expansion of (1/(1+4x+2x^2))*c(x/(1+4x+2x^2)), c(x) the g.f. of A000108.
%C A184120 Hankel transform is the (4,-3) Somos-4 sequence A184121.
%H A184120 Vincenzo Librandi, <a href="/A184120/b184120.txt">Table of n, a(n) for n = 0..1000</a>
%F A184120 G.f.: (sqrt(2*x^2+4*x+1)-sqrt(2*x^2+1))/(2*x*sqrt(2*x^2+4*x+1)).
%F A184120 G.f.: 1/(1+4x+2x^2-x/(1-x/(1+4x+2x^2-x/(1-x/(1+4x+2x^2-x/(1-x/(1-... (continued fraction).
%F A184120 Conjecture: (n+1)*a(n) +2*(2n+1)*a(n-1) +4*(n-1)*a(n-2) +4*(2n-5)*a(n-3) +4*(n-3)*a(n-4)=0. - _R. J. Mathar_, Nov 17 2011
%F A184120 G.f.: 1/(2*x) - G(0)/(2*x), where G(k)= 1 - 4*x*(4*k+1)/( (1+2*x^2)*(4*k+2) - x*(1+2*x^2)*(4*k+2)*(4*k+3)/(x*(4*k+3) - (1+2*x^2)*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 26 2013
%F A184120 a(n) ~ (-1)^n * (2+sqrt(2))^n / (sqrt(3*sqrt(2)-4) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 04 2014
%t A184120 CoefficientList[Series[(Sqrt[2x^2+4x+1]-Sqrt[2x^2+1])/(2x Sqrt[2x^2+4x+1]),{x,0,30}],x] (* _Harvey P. Dale_, Mar 09 2012 *)
%o A184120 (PARI) x='x+O('x^30); Vec((sqrt(2*x^2+4*x+1)-sqrt(2*x^2+1))/( 2*x*sqrt(2*x^2+4*x+1))) \\ _G. C. Greubel_, Aug 14 2018
%o A184120 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((Sqrt(2*x^2+4*x+1)-Sqrt(2*x^2+1))/( 2*x*Sqrt(2*x^2+4*x+1)))); // _G. C. Greubel_, Aug 14 2018
%K A184120 sign,easy
%O A184120 0,2
%A A184120 _Paul Barry_, Jan 09 2011