A307577 Number of Motzkin meanders of length n with an odd number of peaks.

0, 0, 1, 4, 13, 40, 119, 348, 1011, 2928, 8471, 24516, 71023, 206024, 598513, 1741332, 5073733, 14804160, 43252855, 126526756, 370551287, 1086365336, 3188090101, 9364411252, 27529374201, 80993754352, 238463467529, 702563144252, 2071200546129, 6109619428824

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

Cf. A001006.

Sequence in context: A191132 A360606 A119915 * A137744 A027130 A027121

Adjacent sequences: A307574 A307575 A307576 * A307578 A307579 A307580

Keyword

nonn

Author

Andrei Asinowski, Apr 15 2019

Status

approved