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A325926
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Number of Motzkin excursions of length n with an even number of humps and an odd number of peaks.
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4
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0, 0, 0, 0, 0, 2, 8, 26, 76, 212, 568, 1504, 3968, 10526, 28192, 76398, 209268, 578396, 1609376, 4499336, 12620080, 35482718, 99958776, 282107702, 797637908, 2259545652, 6413273704, 18238099464, 51963195440, 148315593178, 424034498656, 1214186436154
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OFFSET
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0,6
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COMMENTS
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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.
A peak is an occurrence of the pattern UD.
A hump is an occurrence of the pattern UHH...HD (the number of Hs in the pattern is not fixed, and can be 0).
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LINKS
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FORMULA
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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)).
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EXAMPLE
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For n=5, the a(5)=2 paths are UDUHD and UHDUD (2 humps, 1 peak).
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}.
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MATHEMATICA
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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 *)
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CROSSREFS
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Motzkin meanders and excursions with restrictions on the number of humps and peaks:
A325921: Meanders, #humps=EVEN, #peaks=EVEN.
A325922: Excursions, #humps=EVEN, #peaks=EVEN.
A325923: Meanders, #humps=ODD, #peaks=EVEN.
A325924: Excursions, #humps=ODD, #peaks=EVEN.
A325925: Meanders, #humps=EVEN, #peaks=ODD.
A325926 (this sequence): Excursions, #humps=EVEN, #peaks=ODD.
A325927: Meanders, #humps=ODD, #peaks=ODD.
A325928: Excursions, #humps=ODD, #peaks=ODD.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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