OFFSET
0,4
COMMENTS
A Motzkin meander is a lattice path with steps from the set {D=-1, H=0, U=1} that starts at (0,0), and never goes below the x-axis.
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.: ( sqrt((1+t)/(1-3*t)) - sqrt((1+t+2*t^2)/((1-2*t)*(1-t))) - sqrt((1+t^2)/(1-4*t+5*t^2)) + sqrt((1-t^2+2*t^3)/((1-2*t)*(1-t^2-2*t))) ) / (8*t).
EXAMPLE
For n=3, the a(3)=4 paths are UDH, UDU, UUD, HUD (1 hump, 1 peak).
PROG
(PARI) seq(n)={my(t='x + O('x*'x^n)); Vec(( sqrt((1+t)/(1-3*t)) - sqrt((1+t+2*t^2)/((1-2*t)*(1-t))) - sqrt((1+t^2)/(1-4*t+5*t^2)) + sqrt((1-t^2+2*t^3)/((1-2*t)*(1-t^2-2*t))) ) / (8*t), -n)} \\ Andrew Howroyd, Aug 12 2019
CROSSREFS
Motzkin meanders and excursions with parity restrictions on the number of humps and peaks:
A325921: Meanders, #humps=EVEN, #peaks=EVEN.
A325922: Excursions, #humps=EVEN, #peaks=EVEN.
A325923: Meanders, #humps=ODD, #peaks=EVEN.
A325924: Excursions, #humps=ODD, #peaks=EVEN.
A325925: Meanders, #humps=EVEN, #peaks=ODD.
A325926: Excursions, #humps=EVEN, #peaks=ODD.
A325927 (this sequence): Meanders, #humps=ODD, #peaks=ODD.
A325928: Excursions, #humps=ODD, #peaks=ODD.
KEYWORD
nonn
AUTHOR
Andrei Asinowski, Aug 10 2019
STATUS
approved