A325922 Number of Motzkin excursions of length n with an even number of humps and an even number of peaks.

1, 1, 1, 1, 2, 4, 11, 31, 86, 230, 608, 1588, 4151, 10925, 29083, 78373, 213702, 588366, 1631906, 4550346, 12736029, 35746763, 100561622, 283486702, 800798659, 2266802139, 6429960961, 18276530005, 52051825058, 148520257620, 424507695627

Offset: 0

Author

Andrei Asinowski, Jun 27 2019

Keywords

• nonn

Comments

A Motzkin excursion is a lattice path with steps from the set {D=-1, H=0, U=1} that starts at (0,0), never goes below the x-axis, and terminates at the altitude 0.

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=3 the a(5)=4 paths are HHHHH, UDUDH, UDHUD, HUDUD.
		

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

Mathematica

CoefficientList[Series[(4 (1 - 2 x + 2 x^2) - Sqrt[(1 - 2 x - 3 x^2) (1 - x)^2] - Sqrt[(1 - x - 4 x^3) (1 - x)^3] - Sqrt[(1 + x^2) (1 - 4 x + 5  x^2)] - Sqrt[(1 - 2 x) (1 - 2 x - x^2) (1 - x^2 + 2 x^3)]) / (8  x^2 (1 - x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 30 2019 *)

Formula

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

a(n) ~ 3^(n + 3/2) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 03 2019

conjecture: a(n)+A325924(n) = A307557(n). - R. J. Mathar, Jan 25 2023