A308435 Peak- and valleyless Motzkin meanders.

1, 2, 4, 9, 20, 45, 102, 233, 535, 1234, 2857, 6636, 15456, 36085, 84424, 197883, 464585, 1092348, 2571770, 6062109, 14305022, 33789777, 79887365, 189031914, 447639473, 1060798484, 2515512091, 5968826698, 14171068794, 33662866431, 80005478832, 190237068767, 452548530595

Offset

0,2

Comments

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.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..2616

Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Algorithmica (2019).

Andrei Asinowski, Axel Bacher, Cyril Banderier, and Bernhard Gittenberger, Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Asymptotic Aspects and Borges's Theorem, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018).

Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).

Helmut Prodinger, Motzkin paths of bounded height with two forbidden contiguous subwords of length two, arXiv:2310.12497 [math.CO], 2023.

Formula

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

D-finite with recurrence (n+2)*a(n) +(-2*n-3)*a(n-1) +(-n-1)*a(n-2) +(-n+4)*a(n-4) +(2*n-9)*a(n-5) +(n-5)*a(n-6)=0. - R. J. Mathar, Jan 25 2023

Example

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

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A004149.

Sequence in context: A167750 A343756 A329276 * A352018 A188460 A111099

Adjacent sequences: A308432 A308433 A308434 * A308436 A308437 A308438

Keyword

nonn

Author

Andrei Asinowski, May 27 2019

Status

approved