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A308435
Peak- and valleyless Motzkin meanders.
2
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
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).
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
KEYWORD
nonn
AUTHOR
Andrei Asinowski, May 27 2019
STATUS
approved