A325925 - OEIS

%I #14 Oct 12 2020 14:58:39

%S 0,0,0,0,0,2,14,68,274,986,3288,10416,31872,95382,281762,827084,

%T 2423078,7102598,20852296,61323328,180581128,532199414,1569071842,

%U 4626551740,13641716894,40223795038,118614194080,349847093824,1032173428200

%N Number of Motzkin meanders of length n with an even number of humps and an odd number of peaks.

%C 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.

%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)/(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).

%F a(n) ~ 3^(n + 1/2) / (4*sqrt(Pi*n)). - _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)=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}.

%t 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 *)

%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 (this sequence): Meanders, #humps=EVEN, #peaks=ODD.

%Y A325926: 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