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 A325926 Number of Motzkin excursions of length n with an even number of humps and an odd number of peaks. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS 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). LINKS Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Algorithmica (2019). FORMULA 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)). a(n) ~ 3^(n + 3/2) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 09 2019 EXAMPLE 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}. MATHEMATICA 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 *) CROSSREFS 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. Sequence in context: A268502 A167826 A301995 * A097040 A302237 A224289 Adjacent sequences:  A325923 A325924 A325925 * A325927 A325928 A325929 KEYWORD nonn AUTHOR Andrei Asinowski, Jul 14 2019 STATUS approved

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Last modified July 31 20:24 EDT 2021. Contains 346377 sequences. (Running on oeis4.)