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%I A329679 #8 Jul 20 2024 12:30:38
%S A329679 1,1,1,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A329679 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A329679 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A329679 Number of excursions of length n with Motzkin-steps consisting only of consecutive steps UH, UD, HD and DH.
%C A329679 The Motzkin step set is U=(1,1), H=(1,0) and D=(1,-1). An excursion is a path starting at (0,0), ending on the x-axis and never crossing the x-axis, i.e., staying at nonnegative altitude.
%H A329679 Andrei Asinowski, Cyril Banderier, and Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, preprint, 2019.
%H A329679 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F A329679 G.f.: 1 + t + t^2 + 2t^3 + t^4.
%e A329679 We only have the following six excursions of this type: the empty walk, H, UD, UDH, UHD and UHDH.
%Y A329679 Cf. A329670, A329677, A329678 (other Motzkin excursions avoiding certain consecutive steps such that the sequence counting them has growth rate zero).
%K A329679 nonn,walk,easy
%O A329679 0,4
%A A329679 _Valerie Roitner_, Dec 16 2019

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