OFFSET
0,5
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).
FORMULA
G.f.: (4*(1-2*t+2*t^2) - sqrt((1-2*t-3*t^2)*(1-t)^2) - sqrt((1-t-4*t^3)*(1-t)^3) - sqrt((1+t^2)*(1-4*t+5*t^2)) - sqrt((1-2*t)*(1-2*t-t^2)*(1-t^2+2*t^3)) ) / (8*t^2*(1-t)).
a(n) ~ 3^(n + 3/2) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 03 2019
EXAMPLE
For n=3 the a(5)=4 paths are HHHHH, UDUDH, UDHUD, HUDUD.
MATHEMATICA
CoefficientList[Series[(4 (1 - 2 x + 2 x^2) - Sqrt[(1 - 2 x - 3 x^2) (1 - x)^2] - Sqrt[(1 - x - 4 x^3) (1 - x)^3] - Sqrt[(1 + x^2) (1 - 4 x + 5 x^2)] - Sqrt[(1 - 2 x) (1 - 2 x - x^2) (1 - x^2 + 2 x^3)]) / (8 x^2 (1 - x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 30 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrei Asinowski, Jun 27 2019
STATUS
approved