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%I #44 Sep 08 2022 08:45:51
%S 1,3,11,47,223,1135,6063,33535,190399,1103231,6497407,38779647,
%T 234043647,1425869567,8757326591,54163521535,337060285439,
%U 2108928587775,13258969458687,83720567447551,530692157964287,3375836610256895
%N Expansion of (1 - 2*x - sqrt(1 - 8*x + 8*x^2))/(2*x*(1-x)).
%C Binomial transform of large Schroeder numbers A006318.
%C Hankel transform is 2^binomial(n+1,2).
%C Series reversion of (-1)^(n+1)*A001333(n). - _Vladimir Reshetnikov_, Nov 08 2015
%C Series reversion of x + 3*x^2 + 11*x^3 + ... is x - 3*x^2 + 7*x^3 - ... - _Michael Somos_, Nov 09 2015
%H Vincenzo Librandi, <a href="/A174347/b174347.txt">Table of n, a(n) for n = 0..200</a>
%H Paul Barry, <a href="http://arxiv.org/abs/1311.2292">Laurent Biorthogonal Polynomials and Riordan Arrays</a>, arXiv preprint arXiv:1311.2292 [math.CA], 2013.
%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%F G.f.: 1/(1-x-2x/(1-x/(1-x-2x/(1-x/(1-x-2x/(1-x/(1-... (continued fraction);
%F a(n) = Sum_{k=0..n} binomial(n,k)*A006318(k).
%F D-finite with recurrence: (n+1)*a(n) + 3*(1-3n)*a(n-1) + 4*(4n-5)*a(n-2) + 8(2-n)*a(n-3) = 0. - _R. J. Mathar_, Dec 08 2011
%F a(n) ~ 2*sqrt(2*sqrt(2)-2)*(4+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 20 2012
%F 0 = a(n)*(+64*a(n+1) - 224*a(n+2) + 192*a(n+3) - 32*a(n+4)) + a(n+1)*(-32*a(n+1) + 208*a(n+2) - 260*a(n+3) + 52*a(n+4)) + a(n+2)*(-12*a(n+2) + 61*a(n+3) - 21*a(n+4)) + a(n+3)*(+3*a(n+3) + a(n+4)) for all n>=0. - _Michael Somos_, Nov 09 2015
%e G.f. = 1 + 3*x + 11*x^2 + 47*x^3 + 223*x^4 + 1135*x^5 + 6063*x^6 + 33535*x^7 + ...
%t CoefficientList[Series[(1-2*x-Sqrt[1-8*x+8*x^2])/(2*x*(1-x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%o (PARI) x='x+O('x^35); Vec((1-2*x-sqrt(1-8*x+8*x^2))/(2*x*(1-x))) \\ _Altug Alkan_, Nov 08 2015
%o (Magma) m:=35; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-2*x-Sqrt(1-8*x+8*x^2))/(2*x*(1-x)))); // _G. C. Greubel_, Sep 22 2018
%Y Cf. A006318, A001333.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Mar 16 2010