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%I #27 Nov 29 2024 18:00:16
%S 1,2,4,9,20,47,110,264,634,1541,3754,9204,22622,55817,138026,342203,
%T 849984,2115245,5271970,13158944,32886338,82285031,206101422,
%U 516728937,1296664512,3256472235,8184526438,20584627358,51805243138,130456806425,328703655114
%N Number of Motzkin meanders of length n with no level steps at odd level.
%C 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.
%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.: ((1+t)/sqrt((t-1)*(4*t^2+t-1)) -1) / (2*t).
%F D-finite with recurrence (n+1)*a(n) +(-n-2)*a(n-1) +(-5*n+3)*a(n-2) +(n+4)*a(n-3) +2*(2*n-5)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2023
%F a(n) ~ sqrt(13 + 53/sqrt(17)) * (1 + sqrt(17))^n / (sqrt(Pi*n) * 2^(n + 3/2)). - _Vaclav Kotesovec_, Jun 24 2023
%F a(n) = (A026569(n) + A026569(n+1))/2. - _Mark van Hoeij_, Nov 29 2024
%e For n = 3 the a(3) = 9 paths are UUU, UUH, UUD, UDU, UDH, HUU, HUD, HHU, HHH.
%Y Cf. A307555.
%K nonn
%O 0,2
%A _Andrei Asinowski_, Apr 14 2019