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A325920
Number of Motzkin excursions of length n with an odd number of humps and without peaks.
0
0, 0, 0, 1, 3, 7, 15, 31, 63, 128, 262, 545, 1161, 2547, 5767, 13456, 32202, 78544, 194016, 482726, 1204874, 3008782, 7505420, 18689551, 46454971, 115296751, 285886839, 708673484, 1757339598, 4361894604, 10841959912
OFFSET
0,5
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).
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 = 5 the a(5) = 7 paths are UHHHD, UHHDH, HUHHD, HHUHD, HUHDH, UHDHH, UUHDD.
CROSSREFS
Sequence in context: A178460 A351755 A057613 * A351754 A174743 A146686
KEYWORD
nonn
AUTHOR
Andrei Asinowski, Jun 25 2019
STATUS
approved