A325924 Number of Motzkin excursions of length n with an odd number of humps and an even number of peaks.

0, 0, 0, 1, 3, 7, 15, 34, 78, 191, 493, 1324, 3626, 10032, 27808, 77045, 213273, 590475, 1637117, 4550836, 12692866, 35532414, 99830094, 281412535, 795601139, 2254966896, 6405076658, 18227600051, 51960277037, 148352016215, 424186720927, 1214602291322

Offset: 0

Author

Andrei Asinowski, Jul 04 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).

Thus every peak is also a hump.

Examples

			For n = 5 the a(5) = 7 paths are UHHHD, UHHDH, HUHHD, HHUHD, HUHDH, UHDHH, UUHDD. In all these paths, 0 peaks and 1 hump.
For n = 0..6, we have only paths with 0 peaks and 1 hump.
For n=7, we have a(n)=34. Among them, 31 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. A325923.

Programs

Maple

b:= proc(x, y, t, p, h) option remember; `if`(y>x, 0, `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[-(4 x^3 + 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, 33}], x] (* Vincenzo Librandi, Jul 09 2019 *)

Formula

G.f.: -( 4*t^3 + 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, Aug 09 2019