A325925 Number of Motzkin meanders of length n with an even number of humps and an odd number of peaks.

0, 0, 0, 0, 0, 2, 14, 68, 274, 986, 3288, 10416, 31872, 95382, 281762, 827084, 2423078, 7102598, 20852296, 61323328, 180581128, 532199414, 1569071842, 4626551740, 13641716894, 40223795038, 118614194080, 349847093824, 1032173428200

Offset: 0

Author

Andrei Asinowski, Jul 14 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=5, the a(5)=2 paths are UDUHD and UHDUD (2 humps, 1 peak).
For n=6, we have a(6)=14 paths: 6 paths obtained by a permutation of {UD, UHD, H}, 6 paths obtained by a permutation of {UD, UHD, U}, and 2 paths obtained by a permutation of {UD, UHHD}.
		

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

Motzkin meanders and excursions with restrictions on the number of humps and peaks:

A325921: Meanders, #humps=EVEN, #peaks=EVEN.

A325922: Excursions, #humps=EVEN, #peaks=EVEN.

A325923: Meanders, #humps=ODD, #peaks=EVEN.

A325924: Excursions, #humps=ODD, #peaks=EVEN.

A325925 (this sequence): Meanders, #humps=EVEN, #peaks=ODD.

A325926: Excursions, #humps=EVEN, #peaks=ODD.

A325927: Meanders, #humps=ODD, #peaks=ODD.

A325928: Excursions, #humps=ODD, #peaks=ODD.

Programs

Mathematica

CoefficientList[Series[(Sqrt[(1 + x)/(1 - 3*x)] - Sqrt[(1 + x + 2*x^2)/((1 - 2*x)*(1 - x))] + Sqrt[(1 + x^2)/(1 - 4*x + 5*x^2)] - Sqrt[(1 - x^2 + 2*x^3)/((1 - 2*x)*(1 - 2*x - x^2))])/(8*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 09 2019 *)

Formula

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

a(n) ~ 3^(n + 1/2) / (4*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 09 2019