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A325922
Number of Motzkin excursions of length n with an even number of humps and an even number of peaks.
5
1, 1, 1, 1, 2, 4, 11, 31, 86, 230, 608, 1588, 4151, 10925, 29083, 78373, 213702, 588366, 1631906, 4550346, 12736029, 35746763, 100561622, 283486702, 800798659, 2266802139, 6429960961, 18276530005, 52051825058, 148520257620, 424507695627
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.: (4*(1-2*t+2*t^2) - sqrt((1-2*t-3*t^2)*(1-t)^2) - sqrt((1-t-4*t^3)*(1-t)^3) - sqrt((1+t^2)*(1-4*t+5*t^2)) - sqrt((1-2*t)*(1-2*t-t^2)*(1-t^2+2*t^3)) ) / (8*t^2*(1-t)).
a(n) ~ 3^(n + 3/2) / (8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 03 2019
conjecture: a(n)+A325924(n) = A307557(n). - R. J. Mathar, Jan 25 2023
EXAMPLE
For n=3 the a(5)=4 paths are HHHHH, UDUDH, UDHUD, HUDUD.
MATHEMATICA
CoefficientList[Series[(4 (1 - 2 x + 2 x^2) - Sqrt[(1 - 2 x - 3 x^2) (1 - x)^2] - Sqrt[(1 - x - 4 x^3) (1 - x)^3] - Sqrt[(1 + x^2) (1 - 4 x + 5 x^2)] - Sqrt[(1 - 2 x) (1 - 2 x - x^2) (1 - x^2 + 2 x^3)]) / (8 x^2 (1 - x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 30 2019 *)
CROSSREFS
Cf. A325921.
Sequence in context: A276687 A298891 A002387 * A148160 A148161 A263375
KEYWORD
nonn
AUTHOR
Andrei Asinowski, Jun 27 2019
STATUS
approved