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.
FORMULA
G.f.: (1+t)*(1-t^2-2*t^3-(1+t)*sqrt(1-2*t+t^2-4*t^3+4*t^4))/(2*t^4).
D-finite with recurrence: (n+4)*a(n) +(-n-4)*a(n-1) +(-n+2)*a(n-2) -3*n*a(n-3) +6*a(n-4) +4*(n-5)*a(n-5)=0. - R. J. Mathar, Jan 09 2020
EXAMPLE
a(4)=4 since we have 4 excursions of length 4, namely UHDH, UDUD, HUHD and HUDH.
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Valerie Roitner, Nov 26 2019
STATUS
approved