A307572 Number of Motzkin meanders of length n with an odd number of humps.

0, 0, 1, 5, 18, 56, 161, 443, 1196, 3228, 8823, 24579, 69810, 201380, 586843, 1719081, 5044584, 14800352, 43384747, 127076015, 372100654, 1089864344, 3194496987, 9372984609, 27532712140, 80966582548, 238342592353, 702222958797, 2070454005078, 6108341367004

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 hump is an occurrence of the pattern UHH...HD (the number of Hs in the pattern is not fixed, and can be 0).

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^2)/(1-4*t+3*t^2)) - sqrt((1+t^2)/(1-4*t+5*t^2))) / (4*t).

Conjecture: D-finite with recurrence -3*(n+1)*(n-2)*a(n) +12*(2*n^2-4*n-1)*a(n-1) +2*(-35*n^2+107*n-48)*a(n-2) +4*(21*n^2-89*n+80)*a(n-3) +4*(-5*n^2+32*n-43)*a(n-4) +4*(-8*n^2+62*n-115)*a(n-5) +2*(31*n^2-283*n+616)*a(n-6) -4*(23*n-97)*(n-6)*a(n-7) +15*(n-6)*(n-7)*a(n-8)=0. - R. J. Mathar, May 06 2020

a(n) ~ 3^(n + 1/2) / (2*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 08 2023

Example

For n = 3 the a(3) = 5 paths are UDH, HUD, UHD, UUD, UDU.

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, t, 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, t, 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: A001793 A325919 A317849 * A325923 A335720 A093374

Adjacent sequences: A307569 A307570 A307571 * A307573 A307574 A307575

Keyword

nonn

Author

Andrei Asinowski, Apr 15 2019

Status

approved