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%I #29 Dec 09 2020 19:58:44
%S 1,2,8,36,172,860,4460,23820,130268,726236,4112972,23599724,136906748,
%T 801671996,4732110828,28128179276,168222049052,1011509012636,
%U 6111445499532,37084090264364,225899543897916,1380918157453052,8468524718133804,52085281291575052
%N Expansion of 1/(1-2x/(1-2x/(1-x/(1-2x/(1-2x/(1-x/(1-2x/(1-... (continued fraction).
%H Vincenzo Librandi, <a href="/A186338/b186338.txt">Table of n, a(n) for n = 0..200</a>
%H Xiaomei Chen, Yuan Xiang, <a href="https://arxiv.org/abs/2009.04900">Counting generalized Schröder paths</a>, arXiv:2009.04900 [math.CO], 2020.
%F G.f.: (sqrt(1-10x+25x^2-16x^3)+3x-1)/(2x(2x-1)).
%F Conjecture: (n+1)*a(n) +3*(1-4n)*a(n-1) +15*(3n-4)*a(n-2) +6*(26-11n)*a(n-3) +16*(2n-7)*a(n-4)=0. - _R. J. Mathar_, Nov 17 2011
%F a(n) = Sum_{k, 0<=k<=n} A091866(n,k)*2^k. - _Philippe Deléham_, Nov 27 2011
%F a(n) ~ sqrt(7*sqrt(17)-17)*((9+sqrt(17))/2)^n/(2*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2012
%F From _Vladimir Kruchinin_, Jan 25 2020: (Start)
%F a(n) = Sum_{j=0..n} Sum_{i=0..j} C(j+1, i)*C(2*j-i, j-i)*C(n-j+i-1,n-j) /(j+1)*2^(n-j).
%F a(n) = Sum_{i=0..n-1} a(i)*(2^(n-i-1)+a(n-i-1)). (End)
%t CoefficientList[Series[(Sqrt[1-10*x+25*x^2-16*x^3]+3*x-1)/(2*x*(2*x-1)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 24 2012 *)
%o (Maxima)
%o a(n):=sum(sum(binomial(j+1, i)*binomial(2*j-i, j-i)*binomial(n-j+i-1, n-j), i, 0, j)/(j+1)*2^(n-j), j, 0, n); /* _Vladimir Kruchinin_, Jan 25 2020 */
%Y Hankel transform is A186339.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Feb 18 2011
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