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A308435 Peak- and valleyless Motzkin meanders. 1

%I

%S 1,2,4,9,20,45,102,233,535,1234,2857,6636,15456,36085,84424,197883,

%T 464585,1092348,2571770,6062109,14305022,33789777,79887365,189031914,

%U 447639473,1060798484,2515512091,5968826698,14171068794,33662866431,80005478832,190237068767,452548530595

%N Peak- and valleyless Motzkin meanders.

%C a(n) is the number of Motzkin meanders that avoid UD and DU. A Motzkin meander is a lattice paths that starts at (0,0), uses steps U=1, H=0, D=-1, and never goes below the x-axis.

%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).

%H Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, <a href="https://doi.org/10.4230/LIPIcs.AofA.2018.10">Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Asymptotic Aspects and Borges's Theorem</a>, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018).

%H Andrei Asinowski, Cyril Banderier, Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019).

%F G.f.: -(1+t-sqrt((1-t^4)/(1-2*t-t^2)))/(2*t^2).

%e For n=3, the a(3)=9 such meanders are UUU, UUH, UHU, UHH, UHD, HUU, HUH, HHU, HHH.

%t CoefficientList[Series[-(1+x-Sqrt[(1-x^4)/(1-2*x-x^2)])/(2*x^2), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Jun 05 2019 *)

%o (PARI) my(t='t + O('t^40)); Vec(-(1+t-sqrt((1-t^4)/(1-2*t-t^2)))/(2*t^2)) \\ _Michel Marcus_, May 27 2019

%Y Cf. A004149.

%K nonn

%O 0,2

%A _Andrei Asinowski_, May 27 2019

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Last modified January 20 14:17 EST 2021. Contains 340302 sequences. (Running on oeis4.)