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A329687
Number of excursions of length n with Motzkin-steps avoiding the consecutive steps UU, UD, HH and DH.
0
1, 1, 0, 1, 1, 0, 2, 2, 0, 5, 5, 0, 14, 14, 0, 42, 42, 0, 132, 132, 0, 429, 429, 0, 1430, 1430, 0, 4862, 4862, 0, 16796, 16796, 0, 58786, 58786, 0, 208012, 208012, 0, 742900, 742900, 0, 2674440, 2674440, 0, 9694845, 9694845, 0, 35357670, 35357670, 0, 129644790, 129644790, 0, 477638700
OFFSET
0,7
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-sqrt(1-4t^3))/(2t^3).
D-finite with recurrence: +(n+3)*a(n) +(n+1)*a(n-1) +2*(-2*n+3)*a(n-3) +2*(-2*n+7)*a(n-4)=0. - R. J. Mathar, Feb 21 2020
EXAMPLE
a(6)=2 since we have the following two excursions of length 6: UHDUHD and UHUHDD.
CROSSREFS
Cf. A000108.
Sequence in context: A166299 A182508 A213626 * A356035 A321127 A222128
KEYWORD
nonn,walk
AUTHOR
Valerie Roitner, Nov 29 2019
STATUS
approved