cp's OEIS Frontend

A325928 Number of Motzkin excursions of length n with an odd number of humps and an odd number of peaks.

%I A325928 #11 Oct 12 2020 15:18:15
%S A325928 0,0,1,2,4,8,17,36,83,202,519,1382,3766,10352,28551,78756,217224,
%T A325928 599542,1657983,4598766,12803044,35785664,100412731,282753476,
%U A325928 798690091,2262087814,6421507153,18265543282,52047980674,148554917816,424656556001,1215691192244
%N A325928 Number of Motzkin excursions of length n with an odd number of humps and an odd number of peaks.
%C A325928 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 A325928 A peak is an occurrence of the pattern UD.
%C A325928 A hump is an occurrence of the pattern UHH...HD (the number of Hs in the pattern is not fixed, and can be 0).
%H A325928 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 A325928 G.f.: -1/2 + ( -sqrt((1-t)^2*(1+t)*(1-3*t)) + sqrt((1-2*t)*(1+t+2*t^2)*(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))
%e A325928 For n=4, the a(4)=4 paths are UDHH, HUDH, HHUD, and UUDD (1 hump, 1 peak).
%o A325928 (PARI) seq(n)={my(t='x + O('x*'x^n)); Vec(-1/2 + ( -sqrt((1-t)^2*(1+t)*(1-3*t)) + sqrt((1-2*t)*(1+t+2*t^2)*(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)), -n)} \\ _Andrew Howroyd_, Aug 12 2019
%Y A325928 Motzkin meanders and excursions with parity restrictions on the number of humps and peaks:
%Y A325928 A325921: Meanders, #humps=EVEN, #peaks=EVEN.
%Y A325928 A325922: Excursions, #humps=EVEN, #peaks=EVEN.
%Y A325928 A325923: Meanders, #humps=ODD, #peaks=EVEN.
%Y A325928 A325924: Excursions, #humps=ODD, #peaks=EVEN.
%Y A325928 A325925: Meanders, #humps=EVEN, #peaks=ODD.
%Y A325928 A325926: Excursions, #humps=EVEN, #peaks=ODD.
%Y A325928 A325927: Meanders, #humps=ODD, #peaks=ODD.
%Y A325928 A325928 (this sequence): Excursions, #humps=ODD, #peaks=ODD.
%K A325928 nonn
%O A325928 0,4
%A A325928 _Andrei Asinowski_, Aug 10 2019