login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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 x-axis, 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((1-t)^2*(1+t)*(1-3*t)) + sqrt((1-2*t)*(1+t+2*t^2)*(1-t)^3) - sqrt((1+t^2)*(1-4*t+5*t^2)) + sqrt((1-2*t)*(1-2*t-t^2)*(1-t^2+2*t^3)) ) / (8*t^2*(1-t)).

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 31 20:24 EDT 2021. Contains 346377 sequences. (Running on oeis4.)