A325923 Number of Motzkin meanders of length n with an odd number of humps and an even number of peaks.

0, 0, 0, 1, 5, 18, 56, 163, 459, 1286, 3640, 10479, 30659, 90738, 270092, 804833, 2393929, 7098790, 20984188, 61872587, 182130495, 535698422, 1575478728, 4635125097, 13645054833, 40196623234, 118493318904, 349506908369, 1031426887149

Offset: 0

Author

Andrei Asinowski, Jul 04 2019

Keywords

• nonn

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).

Examples

			For n = 4 the a(4) = 5 paths are UHDU, UHDH, UUHD, HUHD, UHHD: in all these paths, 0 peaks, 1 hump.
For n=0..6 we have only paths with 0 peaks and 1 hump.
For n=7, we have a(7)=163. Among them, 160 paths with 0 peaks and 1 hump, and 3 walks with 2 peaks and 3 humps: UDUDUHD, UDUHDUD, UHDUDUD.
		

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).

Crossrefs

Cf. A325921.

Programs

Maple

b:= proc(x, y, t, p, h) option remember; `if`(x=0, `if`(p+1=h, 1, 0),
      `if`(y>0, b(x-1, y-1, 0, irem(p+`if`(t=1, 1, 0), 2), irem(h+
      `if`(t=2, 1, 0), 2)), 0)+b(x-1, y, `if`(t>0, 2, 0), p, h)+
         b(x-1, y+1, 1, p, h))
    end:
a:= n-> b(n, 0$4):
seq(a(n), n=0..35);  # Alois P. Heinz, Jul 04 2019

Mathematica

CoefficientList[Series[((-1 + 4*x - 3*x^2 + Sqrt[(-(-1 + x)^2)*(-1 + 2*x + 3*x^2)])/ (1 - 4*x + 3*x^2) - (-1 + 4*x - 5*x^2 + 2*x^3 + Sqrt[(-1 + x)^3*(-1 + x + 4*x^3)])/ ((-1 + x)^2*(-1 + 2*x)) + (1 - 4*x + 5*x^2 - Sqrt[1 - 4*x + 6*x^2 - 4*x^3 + 5*x^4])/(1 - 4*x + 5*x^2) + (1 - 4*x + 3*x^2 + 2*x^3 - Sqrt[1 - 4*x + 2*x^2 + 8*x^3 - 11*x^4 + 4*x^5 + 4*x^6])/(1 - 4*x + 3*x^2 + 2*x^3)) / (8*x), {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 09 2019 *)

Formula

G.f.: ( (-3*t^2+4*t+sqrt(-3*t^4+4*t^3+2*t^2-4*t+1)-1)/(3*t^2-4*t+1) + (2*t^3-5*t^2+4*t+sqrt(4*t^6-12*t^5+13*t^4-8*t^3+6*t^2-4*t+1)-1)/(-2*t^3+5*t^2-4*t+1) - (-5*t^2+4*t+sqrt(5*t^4-4*t^3+6*t^2-4*t+1)-1)/(5*t^2-4*t+1) - (-2*t^3-3*t^2+4*t+sqrt(4*t^6+4*t^5-11*t^4+8*t^3+2*t^2-4*t+1)-1)/(2*t^3+3*t^2-4*t+1) ) / (8*t).

a(n) ~ 3^(n + 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 09 2019