A325919 Number of Motzkin meanders of length n with an odd number of humps and without peaks.

0, 0, 0, 1, 5, 18, 56, 160, 432, 1121, 2827, 6988, 17052, 41334, 100082, 243205, 595313, 1471278, 3674756, 9272410, 23605202, 60513201, 155893167, 402819550, 1042358942, 2697994240, 6979913196, 18041181065, 46583002021, 120161923640, 309719942306

Offset: 0

Author

Andrei Asinowski, Jun 25 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 UUHD, UHHD, UHDU, UHDH, HUHD.
		

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).

Formula

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