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%I #45 Sep 08 2022 08:46:07
%S 1,3,9,33,135,591,2709,12837,62379,309147,1556577,7940169,40946607,
%T 213118119,1118080557,5906404557,31390735059,167727039027,
%U 900478280889,4855086475761,26277928981335,142724482802943,777647813128389,4249385026394613,23282201473312635,127874913883456971,703929221807756049
%N Expansion of (3 - 5*x - 3*sqrt(x^2-6*x+1))/(4*x).
%C Number of associate averaging words of degree n.
%H G. C. Greubel, <a href="/A238113/b238113.txt">Table of n, a(n) for n = 0..1000</a>
%H Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.
%H Li Guo and Jun Pei, <a href="http://arxiv.org/abs/1401.7386">Averaging algebras, Schroeder numbers and rooted trees</a>, arXiv:1401.7386 [math.RA], 2014.
%F a(n) ~ 3*sqrt(3*sqrt(2)-4) * (3+2*sqrt(2))^(n+1) / (4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 05 2014
%F a(n) = -3/4*GegenbauerC(n+1,-1/2,3), n>0. - _Benedict W. J. Irwin_, Sep 26 2016
%F From _Alexander Burstein_, Apr 19 2018: (Start)
%F G.f.: A(x) = r+s-1 = (3*r-1)/2 = 3*s-2 = 1+3*x*r*s, where r=r(x) is g.f. of A006318 and s=s(x) is g.f. of A001003.
%F Series reversion of x*A(x) is x*A(-x). (End)
%F D-finite with recurrence: (n+1)*a(n) +3*(-2*n+1)*a(n-1) +(n-2)*a(n-2)=0. - _R. J. Mathar_, Jan 25 2020
%t CoefficientList[Series[(3-5x-3*Sqrt[x^2-6x+1])/(4x),{x,0,30}],x] (* _Harvey P. Dale_, Mar 29 2016 *)
%t Join[{1},Table[-(3/4) GegenbauerC[n+1,-(1/2),3],{n,1,30}]] (* _Benedict W. J. Irwin_, Sep 26 2016 *)
%o (PARI) x='x+O('x^50); Vec((3-5*x-3*sqrt(x^2-6*x+1))/(4*x)) \\ _G. C. Greubel_, Jun 01 2017
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((3-5*x-3*Sqrt(x^2-6*x+1))/(4*x))); // _Vincenzo Librandi_, Jan 27 2020
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Mar 04 2014
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