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%I #14 Oct 12 2020 14:58:54
%S 0,0,0,0,0,2,8,26,76,212,568,1504,3968,10526,28192,76398,209268,
%T 578396,1609376,4499336,12620080,35482718,99958776,282107702,
%U 797637908,2259545652,6413273704,18238099464,51963195440,148315593178,424034498656,1214186436154
%N Number of Motzkin excursions of length n with an even number of humps and an odd number of peaks.
%C A Motzkin excursion is a lattice path with steps from the set {D=-1, H=0, U=1} that starts at (0,0), never goes below the x-axis, and terminates at the altitude 0.
%C A peak is an occurrence of the pattern UD.
%C A hump is an occurrence of the pattern UHH...HD (the number of Hs in the pattern is not fixed, and can be 0).
%H Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/patterns2019.pdf">Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata</a>, Algorithmica (2019).
%F G.f.: ( -sqrt((1-t)^2*(1+t)*(1-3*t)) + sqrt((1-2*t)*(1+t+2*t^2)*(1-t)^3) - sqrt((1+t^2)*(1-4*t+5*t^2)) + sqrt((1-2*t)*(1-2*t-t^2)*(1-t^2+2*t^3)) ) / (8*t^2*(1-t)).
%F a(n) ~ 3^(n + 3/2) / (8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 09 2019
%e For n=5, the a(5)=2 paths are UDUHD and UHDUD (2 humps, 1 peak).
%e For n=6, we have a(6)=8 paths: 6 paths obtained by a permutation of {UD, UHD, H}, and 2 paths obtained by a permutation of {UD, UHHD}.
%t CoefficientList[Series[(1/(8*(1 - x)*x^2))* (-Sqrt[(1 - 3*x)*(1 - x)^2*(1 + x)] + Sqrt[(1 - 2*x)*(1 - x)^3*(1 + x + 2*x^2)] - Sqrt[(1 + x^2)*(1 - 4*x + 5*x^2)] + Sqrt[(1 - 2*x)*(1 - 2*x - x^2)*(1 - x^2 + 2*x^3)]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Aug 09 2019 *)
%Y Motzkin meanders and excursions with restrictions on the number of humps and peaks:
%Y A325921: Meanders, #humps=EVEN, #peaks=EVEN.
%Y A325922: Excursions, #humps=EVEN, #peaks=EVEN.
%Y A325923: Meanders, #humps=ODD, #peaks=EVEN.
%Y A325924: Excursions, #humps=ODD, #peaks=EVEN.
%Y A325925: Meanders, #humps=EVEN, #peaks=ODD.
%Y A325926 (this sequence): Excursions, #humps=EVEN, #peaks=ODD.
%Y A325927: Meanders, #humps=ODD, #peaks=ODD.
%Y A325928: Excursions, #humps=ODD, #peaks=ODD.
%K nonn
%O 0,6
%A _Andrei Asinowski_, Jul 14 2019