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%I #30 Dec 23 2021 03:02:20
%S 1,3,9,32,126,538,2429,11412,55201,272993,1373784,7011297,36201841,
%T 188761743,992491049,5256244537,28013213196,150128293038,808543940999,
%U 4373798584407,23753913152691,129469596050953,707969244301884,3882857013894482,21353585584100401
%N Pseudo-involution companion for the Fibonacci generating function.
%C a(n) is the number of colored Schröder paths of semilength n with steps U=(1,0) and D=(1,-1) of 1 color and H=(2,0) of 2 colors, red and blue, where H does not follow D, and no two red H steps are consecutive.
%H Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.
%F G.f.: A(x) satisfies A(-x*A(x)) = 1/A(x) and F(-x*A(x)) = 1/F(x), where x*F(x) is g.f. of A000045. I.e., the Riordan array (F(x), x*A(x)) is a pseudo-involution.
%F G.f.: A(x) = (F(x) - 1)*C(F(x) - 1)/x, where C(x) is the g.f. of A000108 and x*F(x) is g.f. of A000045.
%F G.f.: A(x) = (1 - sqrt((1 - 5*x - 5*x^2)/(1 - x - x^2)))/(2x).
%F G.f.: Let B(x) = 2 + g.f.(A200031(n)), then A(x) = 1 + x*A(x)*B(x^2*A(x)).
%F a(n) ~ sqrt(3) * 5^(n/2 + 1) * phi^(2*n + 1) / (8*sqrt(Pi)*n^(3/2)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, May 25 2021
%Y Cf. A000045, A000108, A200031.
%K nonn
%O 0,2
%A _Alexander Burstein_, May 24 2021
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