OFFSET
0,4
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.
a(n) also counts excursions avoiding the consecutive steps HH and DD. This can easily be seen by time reversal.
a(n) also counts excursions avoiding the consecutive steps HH and DU.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..2857
Helmut Prodinger, Motzkin paths of bounded height with two forbidden contiguous subwords of length two, arXiv:2310.12497 [math.CO], 2023.
FORMULA
G.f.: (1/2)*(1 - t^3 - t^2 - sqrt(t^6 + 2*t^5 - 3*t^4 - 6*t^3 - 2*t^2 + 1))/t^3.
a(0) = a(1) = a(2) = 1; a(n) = a(n-2) + a(n-3) + Sum_{k=0..n-3} a(k) * a(n-k-3). - Ilya Gutkovskiy, Nov 09 2021
D-finite with recurrence (n+3)*a(n) +2*-n*a(n-2) +3*(-2*n+3)*a(n-3) +3*(-n+3)*a(n-4) +(2*n-9)*a(n-5) +(n-6)*a(n-6)=0. - R. J. Mathar, Jan 25 2023
EXAMPLE
a(3)=3 as there are 3 excursions of length 3, namely: UDH, UHD and HUD.
MATHEMATICA
CoefficientList[Series[(1/2)*(1 - x^3 - x^2 - Sqrt[x^6 + 2*x^5 - 3*x^4 - 6*x^3 - 2*x^2 + 1])/x^3, {x, 0, 40}], x] (* Michael De Vlieger, Oct 24 2023 *)
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Valerie Roitner, Nov 19 2019
STATUS
approved