A329677 Number of excursions of length n with Motzkin-steps consisting only of consecutive steps UH, HD, and DH.

1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0

Comments

The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.

Links

Table of n, a(n) for n=0..96.

Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, preprint, 2019.

Formula

G.f.: 1 + t + t^3 + t^4.

Example

We only have the following four excursions of this type: the empty walk, H, UHD and UHDH.

Mathematica

PadRight[#, 105] &@ CoefficientList[Series[1 + x + x^3 + x^4, {x, 0, 105}], x] (* Michael De Vlieger, Dec 16 2019 *)

Crossrefs

Cf. A329670, A329678, A329679 (other Motzkin excursions avoiding certain consecutive steps such that the sequence counting them has growth rate zero).

Sequence in context: A284793 A260397 A137161 * A077050 A128432 A195198

Adjacent sequences: A329674 A329675 A329676 * A329678 A329679 A329680

Keyword

nonn,walk,easy

Author

Valerie Roitner, Dec 16 2019

Status

approved