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A325925 Number of Motzkin meanders of length n with an even number of humps and an odd number of peaks. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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..28.

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.: ( 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

EXAMPLE

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}.

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 *)

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.

Sequence in context: A109869 A197777 A197608 * A084132 A271235 A084770

Adjacent sequences:  A325922 A325923 A325924 * A325926 A325927 A325928

KEYWORD

nonn

AUTHOR

Andrei Asinowski, Jul 14 2019

STATUS

approved

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Last modified May 19 13:24 EDT 2022. Contains 353833 sequences. (Running on oeis4.)