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.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
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))/(1-3*t) - sqrt((1-t)*(1-2*t)*(1+t+2*t^2))/((1-t)*(1-2*t))) / (4*t).
EXAMPLE
For n = 3 the a(3) = 4 paths are UDH, HUD, UDU, UUD.
MAPLE
b:= proc(x, y, t, c) option remember; `if`(y<0, 0, `if`(x=0, c,
b(x-1, y-1, 0, irem(c+t, 2))+b(x-1, y, 0, c)+b(x-1, y+1, 1, c)))
end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..35); # Alois P. Heinz, Apr 16 2019
MATHEMATICA
b[x_, y_, t_, c_] := b[x, y, t, c] = If[y<0, 0, If[x==0, c, b[x-1, y-1, 0, Mod[c+t, 2]] + b[x-1, y, 0, c] + b[x-1, y+1, 1, c]]];
a[n_] := b[n, 0, 0, 0];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Apr 29 2019, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrei Asinowski, Apr 15 2019
STATUS
approved