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%I #14 May 02 2024 09:44:51
%S 1,2,8,36,176,912,4928,27472,156864,912832,5394176,32282240,195264000,
%T 1191825920,7331457024,45406194944,282896763904,1771868302336,
%U 11150040870912,70461597988864,446971590516736,2845144452292608
%N Expansion of (1-2x-sqrt(1-8x+8x^2))/(2x).
%C Binomial transform of A166228. Hankel transform is A166231.
%H Vincenzo Librandi, <a href="/A166229/b166229.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) = 0^n + Sum_{k = 0..n} C(n-1,k-1)*A006318(k). - _Paul Barry_, Nov 04 2009
%F G.f.: 1/(1-2x/(1-x-x/(1-2x/(1-x-x/(1-2x/(1-x-x/(1-... (continued fraction). - _Paul Barry_, Dec 10 2009
%F Recurrence: (n+1)*a(n) = 4*(2*n-1)*a(n-1) - 8*(n-2)*a(n-2). - _Vaclav Kotesovec_, Oct 20 2012
%F a(n) ~ sqrt(1+sqrt(2))*(4+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 20 2012
%F From _Peter Bala_, May 01 2024: (Start)
%F O.g.f.: A(x) = x*S(x/(1 - x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. for the large Schröder numbers A006318.
%F a(n) = A174347(n+1) - A174347(n).
%F The g.f. satisfies x^2*A(x)^2 - x*(1 - 2*x)*A(x) + x*(1 - x) = 0.
%F A(x) = (1 - x)/(1 - 2*x - x*(1 - x)/(1 - 2*x - x*(1 - x)/(1 - 2*x - ...))). (End)
%t CoefficientList[Series[(1-2*x-Sqrt[1-8*x+8*x^2])/(2*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%Y Cf. A006318, A174347.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Oct 09 2009
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