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%I #31 Jan 30 2020 21:29:18
%S 1,0,0,1,0,2,1,5,4,15,14,48,49,159,173,540,616,1869,2211,6565,7994,
%T 23335,29092,83756,106489,303093,391815,1104490,1448313,4049108,
%U 5375784,14922313,20028144,55248554,74869485,205394737,280737471,766396430,1055627409,2869157740,3979545798,10773488687,15037617603
%N Vacation Dyck paths. Discrete analog for vacation M/M/1 queue embedded chain.
%C The Dyck path starts in red. At any point at any height > 0, the path can take a horizontal step and its color will change to blue. The color remains blue until the first time the path visits the y=0 line, at which point it changes to red again.
%F G.f: (sqrt(-4*z^2 + 1) + 1)/(z*sqrt(-4*z^2 + 1) + sqrt(-4*z^2 + 1) - z + 1).
%F D-finite with recurrence: n*a(n) -n*a(n-1) +6*(-n+2)*a(n-2) +3*(n-4)*a(n-3) +8*(n-3)*a(n-4) +4*(n-3)*a(n-5)=0. - _R. J. Mathar_, Jan 27 2020
%e For n=0, the only path is the empty path, so a(0)=1.
%e For n=1 and n=2, it is impossible to construct such a path, so a(1)=a(2)=0.
%o (PARI) a(n) = my(z='z+O('z^(n+1))); Vec((sqrt(-4*z^2 + 1) + 1)/(z*sqrt(-4*z^2 + 1) + sqrt(-4*z^2 + 1) - z + 1))[n+1] \\ _Jianing Song_, Nov 21 2019
%K nonn
%O 0,6
%A _Igor Kleiner_, Aug 25 2019
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