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
Keywords
Examples
For n=3, the a(3)=9 such meanders are UUU, UUH, UHU, UHH, UHD, HUU, HUH, HHU, HHH.
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.
- Helmut Prodinger, Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs, arXiv:2501.13645 [math.CO], 2025. See p. 8.
Crossrefs
Cf. A004149.
Programs
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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 *) -
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
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
Comments