A307575 Number of Motzkin meanders of length n with an even number of peaks.

1, 2, 4, 9, 22, 56, 148, 402, 1112, 3118, 8832, 25205, 72342, 208560, 603404, 1750785, 5092046, 14839710, 43321976, 126661355, 370813762, 1086877792, 3189091724, 9366371000, 27533212140, 81001276874, 238478223648, 702592110803, 2071257446234, 6109731270056

Offset

0,2

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

Table of n, a(n) for n=0..29.

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)) -2) / (4*t).

D-finite with recurrence -3*(n+1)*(n-2)*a(n) +4*(4*n^2-7*n-3)*a(n-1) +3*(-7*n^2+17*n-2)*a(n-2) +4*n*(n-3)*a(n-3) -(n-3)*(25*n-82)*a(n-4) +4*(n-3)*(6*n-19)*a(n-5) +(61*n^2-575*n+1302)*a(n-6) -4*(11*n-37)*(n-6)*a(n-7) -12*(n-6)*(n-7)*a(n-8)=0. - R. J. Mathar, Mar 06 2022

a(n) + A307577(n) = A005773(n+1). - R. J. Mathar, Jan 25 2023

Example

For n = 3 the a(3) = 9 paths are UUU, UUH, UHU, UHH, UHD, HUU, HUH, HHU, HHH.

Maple

b:= proc(x, y, t, c) option remember; `if`(y<0, 0, `if`(x=0, 1-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, 1-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];
a /@ Range[0, 35] (* Jean-François Alcover, May 12 2020, after Maple *)

Crossrefs

Cf. A001006.

Sequence in context: A037245 A244886 A143017 * A301362 A130018 A352176

Adjacent sequences: A307572 A307573 A307574 * A307576 A307577 A307578

Keyword

nonn

Author

Andrei Asinowski, Apr 15 2019

Status

approved