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 A238113 Expansion of (3-5*x-3*sqrt(x^2-6*x+1))/(4*x). 1
 1, 3, 9, 33, 135, 591, 2709, 12837, 62379, 309147, 1556577, 7940169, 40946607, 213118119, 1118080557, 5906404557, 31390735059, 167727039027, 900478280889, 4855086475761, 26277928981335, 142724482802943, 777647813128389, 4249385026394613, 23282201473312635, 127874913883456971, 703929221807756049 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of associate averaging words of degree n. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Li Guo and Jun Pei, Averaging algebras, Schroeder numbers and rooted trees, arXiv:1401.7386 [math.RA], 2014. FORMULA 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 a(n) = -3/4*GegenbauerC(n+1,-1/2,3), n>0. - Benedict W. J. Irwin, Sep 26 2016 From Alexander Burstein, Apr 19 2018: (Start) 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. Series reversion of x*A(x) is x*A(-x). (End) 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 MATHEMATICA CoefficientList[Series[(3-5x-3*Sqrt[x^2-6x+1])/(4x), {x, 0, 30}], x] (* Harvey P. Dale, Mar 29 2016 *) Join[{1}, Table[-(3/4) GegenbauerC[n+1, -(1/2), 3], {n, 1, 30}]] (* Benedict W. J. Irwin, Sep 26 2016 *) PROG (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 (MAGMA) m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((3-5*x-3*Sqrt(x^2-6*x+1))/(4*x))); // Vincenzo Librandi, Jan 27 2020 CROSSREFS Sequence in context: A236408 A217617 A320181 * A098742 A320182 A320183 Adjacent sequences:  A238110 A238111 A238112 * A238114 A238115 A238116 KEYWORD nonn AUTHOR N. J. A. Sloane, Mar 04 2014 STATUS approved

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Last modified August 2 21:30 EDT 2021. Contains 346429 sequences. (Running on oeis4.)