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 A309976 Vacation Dyck paths. Discrete analog for vacation M/M/1 queue embedded chain. 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS 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. LINKS 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 EXAMPLE 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. PROG (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 CROSSREFS Sequence in context: A079285 A240607 A304298 * A257516 A124660 A217104 Adjacent sequences:  A309973 A309974 A309975 * A309977 A309978 A309979 KEYWORD nonn AUTHOR Igor Kleiner, Aug 25 2019 STATUS approved

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Last modified May 16 15:55 EDT 2021. Contains 343949 sequences. (Running on oeis4.)