A325921 Number of Motzkin meanders of length n with an even number of humps and an even number of peaks.

1, 2, 4, 8, 17, 38, 92, 239, 653, 1832, 5192, 14726, 41683, 117822, 333312, 945952, 2698117, 7740920, 22337788, 64788768, 188683267, 551179370, 1613612996, 4731245903, 13888157307, 40804653640, 119984904744, 353085202434, 1039830559085, 3064566227434

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.

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..2100

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

a(n) ~ 3^(n + 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Jul 03 2019

a(n) + A325923(n) = A307575(n). - R. J. Mathar, Jan 25 2023

a(n) + A325925(n) = A307555(n). - R. J. Mathar, Jan 25 2023

Example

For n=0, 1, 2, 3 there are 2^n paths: all the paths without D (0 humps, 0 peaks).
For example, for n=3: UUU, UUH, UHU, UHH, HUU, HUH, HHU, HHH.
For n=4, the "extra" path is UDUD (2 humps, 2 peaks).
The smallest example with #(humps) <> #(peaks) is UHDUHD (2 humps, 0 peaks).

Maple

b:= proc(x, y, t, p, h) option remember; `if`(x=0, `if`(p+h=0, 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 03 2019

Mathematica

CoefficientList[Series[(1/(8*x))*((-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)), {x, 0, 30}], x] (* Vaclav Kotesovec, Jul 03 2019 *)

Crossrefs

Cf. A307572, A325922.

Sequence in context: A090901 A101516 A118928 * A049312 A132043 A055545

Adjacent sequences: A325918 A325919 A325920 * A325922 A325923 A325924

Keyword

nonn

Author

Andrei Asinowski, Jun 27 2019

Status

approved