

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
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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 xaxis.
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)/(13*t))  sqrt((1+t+2*t^2)/((12*t)*(1t))) + sqrt((1+t^2)/(14*t+5*t^2))  sqrt((1t^2+2*t^3)/((12*t)*(1t^22*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



