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A325927 Number of Motzkin meanders of length n with an odd number of humps and an odd number of peaks.

%I A325927 #13 Oct 12 2020 19:33:52
%S A325927 0,0,1,4,13,38,105,280,737,1942,5183,14100,39151,110642,316751,914248,
%T A325927 2650655,7701562,22400559,65203428,189970159,554165922,1619018259,
%U A325927 4737859512,13887657307,40769959314,119849273449,352716050428,1039027117929
%N A325927 Number of Motzkin meanders of length n with an odd number of humps and an odd number of peaks.
%C A325927 A Motzkin meander is a lattice path with steps from the set {D=-1, H=0, U=1} that starts at (0,0), and never goes below the x-axis.
%C A325927 A peak is an occurrence of the pattern UD.
%C A325927 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 A325927 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 A325927 G.f.: ( sqrt((1+t)/(1-3*t)) - sqrt((1+t+2*t^2)/((1-2*t)*(1-t))) - sqrt((1+t^2)/(1-4*t+5*t^2)) + sqrt((1-t^2+2*t^3)/((1-2*t)*(1-t^2-2*t))) ) / (8*t).
%e A325927 For n=3, the a(3)=4 paths are UDH, UDU, UUD, HUD (1 hump, 1 peak).
%o A325927 (PARI) seq(n)={my(t='x + O('x*'x^n)); Vec(( sqrt((1+t)/(1-3*t)) - sqrt((1+t+2*t^2)/((1-2*t)*(1-t))) - sqrt((1+t^2)/(1-4*t+5*t^2)) + sqrt((1-t^2+2*t^3)/((1-2*t)*(1-t^2-2*t))) ) / (8*t), -n)} \\ _Andrew Howroyd_, Aug 12 2019
%Y A325927 Motzkin meanders and excursions with parity restrictions on the number of humps and peaks:
%Y A325927 A325921: Meanders, #humps=EVEN, #peaks=EVEN.
%Y A325927 A325922: Excursions, #humps=EVEN, #peaks=EVEN.
%Y A325927 A325923: Meanders, #humps=ODD, #peaks=EVEN.
%Y A325927 A325924: Excursions, #humps=ODD, #peaks=EVEN.
%Y A325927 A325925: Meanders, #humps=EVEN, #peaks=ODD.
%Y A325927 A325926: Excursions, #humps=EVEN, #peaks=ODD.
%Y A325927 A325927  (this sequence): Meanders, #humps=ODD, #peaks=ODD.
%Y A325927 A325928: Excursions, #humps=ODD, #peaks=ODD.
%K A325927 nonn
%O A325927 0,4
%A A325927 _Andrei Asinowski_, Aug 10 2019