%I #18 Feb 24 2020 14:56:08
%S 1,1,3,8,23,67,199,600,1834,5674,17743,56011,178301,571812,1845913,
%T 5993985,19565770,64168531,211343740,698753053,2318315786,7716186315,
%U 25757105801,86208990248,289253059765,972729789813
%N 1-Fibonacci lattice paths.
%C a(n) is the number of lattice paths, never going below the x-axis, from (0,0) to (n,0) consisting of up steps U = (1,1), down steps D = (1,-1) and horizontal steps H(s) = (s,0) for every positive integer s, such that H(s) is colored by means of F(s) colors, where F(s) is the s-th Fibonacci number.
%H R. De Castro, A. Ramírez, and J. Ramírez, <a href="http://arxiv.org/abs/1310.2449">Applications in enumerative combinatorics of infinite weighted automata and graphs</a>, arXiv:1310.2449, (2013).
%F G.f.: (1-2*z-z^2-sqrt((1-2*z-z^2)^2-4*z^2*(1-z-z^2)^2))/(2*z^2*(1-z-z^2)).
%F D-finite with recurrence: n*(n+2)*a(n) = (7*n^2 + 5*n - 1)*a(n-1) - (14*n^2 - 8*n + 1)*a(n-2) + (3*n^2 - 12*n + 23)*a(n-3) + (n+1)*(13*n - 38)*a(n-4) - 4*(n-4)*n*a(n-5) - 4*(n-5)*(n+1)*a(n-6). - _Vaclav Kotesovec_, Mar 15 2014
%F a(n) ~ sqrt(663+161*sqrt(17)) * ((3+sqrt(17))/2)^(n-3/2) / (sqrt(2*Pi) * n^(3/2)). - _Vaclav Kotesovec_, Mar 15 2014
%t Rest[CoefficientList[Series[(1-2*x-x^2-Sqrt[(1-2*x-x^2)^2-4*x^2*(1-x-x^2)^2])/(2*x^2*(1-x-x^2)),{x,0,20}],x]] (* _Vaclav Kotesovec_, Mar 15 2014 *)
%Y Cf. A000045.
%K nonn
%O 1,3
%A _José Luis Ramírez Ramírez_, Oct 10 2013