A309976 Vacation Dyck paths. Discrete analog for vacation M/M/1 queue embedded chain.
1, 0, 0, 1, 0, 2, 1, 5, 4, 15, 14, 48, 49, 159, 173, 540, 616, 1869, 2211, 6565, 7994, 23335, 29092, 83756, 106489, 303093, 391815, 1104490, 1448313, 4049108, 5375784, 14922313, 20028144, 55248554, 74869485, 205394737, 280737471, 766396430, 1055627409, 2869157740, 3979545798, 10773488687, 15037617603
Offset: 0
Keywords
Examples
For n=0, the only path is the empty path, so a(0)=1. For n=1 and n=2, it is impossible to construct such a path, so a(1)=a(2)=0.
Programs
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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
Formula
G.f: (sqrt(-4*z^2 + 1) + 1)/(z*sqrt(-4*z^2 + 1) + sqrt(-4*z^2 + 1) - z + 1).
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
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