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A238113
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Expansion of (3 - 5*x - 3*sqrt(x^2-6*x+1))/(4*x).
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3
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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
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OFFSET
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0,2
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COMMENTS
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Number of associate averaging words of degree n.
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LINKS
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FORMULA
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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
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
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MATHEMATICA
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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 *)
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PROG
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(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<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((3-5*x-3*Sqrt(x^2-6*x+1))/(4*x))); // Vincenzo Librandi, Jan 27 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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