A325924 - OEIS

%I #15 Aug 09 2019 13:06:48

%S 0,0,0,1,3,7,15,34,78,191,493,1324,3626,10032,27808,77045,213273,

%T 590475,1637117,4550836,12692866,35532414,99830094,281412535,

%U 795601139,2254966896,6405076658,18227600051,51960277037,148352016215,424186720927,1214602291322

%N Number of Motzkin excursions of length n with an odd number of humps and an even number of peaks.

%C 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.

%C A peak is an occurrence of the pattern UD.

%C A hump is an occurrence of the pattern UHH...HD (the number of Hs in the pattern is not fixed, and can be 0).

%C Thus every peak is also a hump.

%H Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/patterns2019.pdf">Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata</a>, Algorithmica (2019).

%F G.f.: -( 4*t^3 + 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)).

%F a(n) ~ 3^(n + 3/2) / (8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 09 2019

%e For n = 5 the a(5) = 7 paths are UHHHD, UHHDH, HUHHD, HHUHD, HUHDH, UHDHH, UUHDD. In all these paths, 0 peaks and 1 hump.

%e For n = 0..6, we have only paths with 0 peaks and 1 hump.

%e For n=7, we have a(n)=34. Among them, 31 paths with 0 peaks and 1 hump, and 3 walks with 2 peaks and 3 humps: UDUDUHD, UDUHDUD, UHDUDUD.

%p b:= proc(x, y, t, p, h) option remember; `if`(y>x, 0, `if`(x=0,

%p       `if`(p+1=h, 1, 0), `if`(y>0, b(x-1, y-1, 0, irem(p+

%p       `if`(t=1, 1, 0), 2), irem(h+`if`(t=2, 1, 0), 2)), 0)+

%p        b(x-1, y, `if`(t>0, 2, 0), p, h)+b(x-1, y+1, 1, p, h)))

%p     end:

%p a:= n-> b(n, 0$4):

%p seq(a(n), n=0..35);  # _Alois P. Heinz_, Jul 04 2019

%t CoefficientList[Series[-(4 x^3 + 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, 33}], x] (* _Vincenzo Librandi_, Jul 09 2019 *)

%Y Cf. A325923.

%K nonn

%O 0,5

%A _Andrei Asinowski_, Jul 04 2019