OFFSET
0,7
COMMENTS
A Motzkin excursion is a lattice path with steps from the set {D=-1, H=0, U=1} that starts at (0,0), never goes below the x-axis, and terminates at the altitude 0.
A peak is an occurrence of the pattern UD.
A hump is an occurrence of the pattern UHH...HD (the number of Hs in the pattern is not fixed, and can be 0).
LINKS
Andrei Asinowski, Axel Bacher, Cyril Banderier, 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, 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).
FORMULA
G.f.: (1/4)*(t^3 - 2*t^2 + 2*t - 1 + sqrt(t^6 - 4*t^5 + 4*t^4 - 2*t^3 + 4*t^2 - 4*t + 1))/((t^2-t)*t)+(1/4)*(-t^3 - 2*t^2 - 1 + sqrt(t^6 + 4*t^5 - 4*t^4 + 2*t^3 + 4*t^2 - 4*t + 1) + 2*t)/((t^2-t)*t).
EXAMPLE
For n=0..5 we have a(n)=1 because for these values we have only the humpless paths HH...H. For n=6, the only "extra" path is UHDUHD. For n=7, the five "extra" paths are UHDUHHD, UHHDUHD, HUHDUHD, UHDHUHD, UHDUHDH.
MATHEMATICA
CoefficientList[Series[(1/4)*(x^3 - 2*x^2 + 2*x - 1 + Sqrt[x^6 - 4*x^5 + 4*x^4 - 2*x^3 + 4*x^2 - 4*x + 1])/((x^2-x)*x)+(1/4)*(-x^3 - 2*x^2 - 1 + Sqrt[x^6 + 4*x^5 - 4*x^4 + 2*x^3 + 4*x^2 - 4*x + 1] + 2*x)/((x^2-x)*x), {x, 0, 40}], x] (* Vaclav Kotesovec, Jun 05 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrei Asinowski, May 28 2019
STATUS
approved