

A325926


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


4



0, 0, 0, 0, 0, 2, 8, 26, 76, 212, 568, 1504, 3968, 10526, 28192, 76398, 209268, 578396, 1609376, 4499336, 12620080, 35482718, 99958776, 282107702, 797637908, 2259545652, 6413273704, 18238099464, 51963195440, 148315593178, 424034498656, 1214186436154
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OFFSET

0,6


COMMENTS

A Motzkin excursion is a lattice path with steps from the set {D=1, H=0, U=1} that starts at (0,0), never goes below the xaxis, and terminates at the altitude 0.
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..31.
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((1t)^2*(1+t)*(13*t)) + sqrt((12*t)*(1+t+2*t^2)*(1t)^3)  sqrt((1+t^2)*(14*t+5*t^2)) + sqrt((12*t)*(12*tt^2)*(1t^2+2*t^3)) ) / (8*t^2*(1t)).
a(n) ~ 3^(n + 3/2) / (8*sqrt(Pi)*n^(3/2)).  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)=8 paths: 6 paths obtained by a permutation of {UD, UHD, H}, and 2 paths obtained by a permutation of {UD, UHHD}.


MATHEMATICA

CoefficientList[Series[(1/(8*(1  x)*x^2))* (Sqrt[(1  3*x)*(1  x)^2*(1 + x)] + Sqrt[(1  2*x)*(1  x)^3*(1 + x + 2*x^2)]  Sqrt[(1 + x^2)*(1  4*x + 5*x^2)] + Sqrt[(1  2*x)*(1  2*x  x^2)*(1  x^2 + 2*x^3)]), {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: Meanders, #humps=EVEN, #peaks=ODD.
A325926 (this sequence): Excursions, #humps=EVEN, #peaks=ODD.
A325927: Meanders, #humps=ODD, #peaks=ODD.
A325928: Excursions, #humps=ODD, #peaks=ODD.
Sequence in context: A268502 A167826 A301995 * A097040 A302237 A224289
Adjacent sequences: A325923 A325924 A325925 * A325927 A325928 A325929


KEYWORD

nonn


AUTHOR

Andrei Asinowski, Jul 14 2019


STATUS

approved



