A325918 Number of Motzkin excursions of length n with an even number of humps and without peaks.

1, 1, 1, 1, 1, 1, 2, 6, 19, 57, 161, 433, 1122, 2826, 6968, 16916, 40630, 96958, 230732, 549278, 1311473, 3146659, 7596281, 18460921, 45163078, 111164142, 275067208, 683577528, 1704485046, 4260677154, 10669252349

Offset

0,7

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..30.

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).

Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Asymptotic Aspects and Borges's Theorem, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018).

Formula

G.f.: (1/4)*(t^3 - 2*t^2 + 2*t - 1 + sqrt(t^6 - 4*t^5 + 4*t^4 - 2*t^3 + 4*t^2 - 4*t + 1))/((t^2-t)*t)+(1/4)*(-t^3 - 2*t^2 - 1 + sqrt(t^6 + 4*t^5 - 4*t^4 + 2*t^3 + 4*t^2 - 4*t + 1) + 2*t)/((t^2-t)*t).

Example

For n=0..5 we have a(n)=1 because for these values we have only the humpless paths HH...H. For n=6, the only "extra" path is UHDUHD. For n=7, the five "extra" paths are UHDUHHD, UHHDUHD, HUHDUHD, UHDHUHD, UHDUHDH.

Mathematica

CoefficientList[Series[(1/4)*(x^3 - 2*x^2 + 2*x - 1 + Sqrt[x^6 - 4*x^5 + 4*x^4 - 2*x^3 + 4*x^2 - 4*x + 1])/((x^2-x)*x)+(1/4)*(-x^3 - 2*x^2 - 1 + Sqrt[x^6 + 4*x^5 - 4*x^4 + 2*x^3 + 4*x^2 - 4*x + 1] + 2*x)/((x^2-x)*x), {x, 0, 40}], x] (* Vaclav Kotesovec, Jun 05 2019 *)

Crossrefs

Sequence in context: A226433 A121483 A077834 * A307564 A308240 A328658

Adjacent sequences: A325915 A325916 A325917 * A325919 A325920 A325921

Keyword

nonn

Author

Andrei Asinowski, May 28 2019

Status

approved