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

0,5

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

Table of n, a(n) for n=0..30.

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).

Example

For n = 4 the a(4) = 5 paths are UUHD, UHHD, UHDU, UHDH, HUHD.

Crossrefs

Sequence in context: A344847 A145129 A001793 * A317849 A307572 A325923

Adjacent sequences: A325916 A325917 A325918 * A325920 A325921 A325922

Keyword

nonn

Author

Andrei Asinowski, Jun 25 2019

Status

approved