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A325925
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Number of Motzkin meanders of length n with an even number of humps and an odd number of peaks.
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5
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0, 0, 0, 0, 0, 2, 14, 68, 274, 986, 3288, 10416, 31872, 95382, 281762, 827084, 2423078, 7102598, 20852296, 61323328, 180581128, 532199414, 1569071842, 4626551740, 13641716894, 40223795038, 118614194080, 349847093824, 1032173428200
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OFFSET
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0,6
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COMMENTS
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A Motzkin meander is a lattice path with steps from the set {D=-1, H=0, U=1} that starts at (0,0), and never goes below the x-axis.
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)/(1-3*t)) - sqrt((1+t+2*t^2)/((1-2*t)*(1-t))) + sqrt((1+t^2)/(1-4*t+5*t^2)) - sqrt((1-t^2+2*t^3)/((1-2*t)*(1-t^2-2*t))) ) / (8*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)=14 paths: 6 paths obtained by a permutation of {UD, UHD, H}, 6 paths obtained by a permutation of {UD, UHD, U}, and 2 paths obtained by a permutation of {UD, UHHD}.
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MATHEMATICA
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CoefficientList[Series[(Sqrt[(1 + x)/(1 - 3*x)] - Sqrt[(1 + x + 2*x^2)/((1 - 2*x)*(1 - x))] + Sqrt[(1 + x^2)/(1 - 4*x + 5*x^2)] - Sqrt[(1 - x^2 + 2*x^3)/((1 - 2*x)*(1 - 2*x - x^2))])/(8*x), {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 (this sequence): Meanders, #humps=EVEN, #peaks=ODD.
A325926: 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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