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%I #11 Jan 31 2023 14:49:48
%S 1,1,2,4,9,20,47,112,274,679,1708,4341,11143,28831,75135,197013,
%T 519447,1376256,3662327,9784106,26232033,70558313,190348160,514904151,
%U 1396328313,3795324358,10338000693,28215285901,77149545999,211314835549,579730469034
%N Number of UFU-free Motzkin paths of length n.
%C a(n) = number of Motzkin paths (A001006) of length n that contain no consecutive UFU.
%H Andrew Howroyd, <a href="/A095980/b095980.txt">Table of n, a(n) for n = 0..200</a>
%H Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2301.10449">Partial Motzkin paths with air pockets of the first kind avoiding peaks, valleys or double rises</a>, arXiv:2301.10449 [math.CO], 2023.
%F G.f.: (1 - x - x^3 - (1 - 2*x - 3*x^2 + 2*x^3 - 2*x^4 + x^6)^(1/2))/(2*x^2*(1 - x + x^2)).
%e a(5) = 20 because, of the 21 Motzkin paths of length 5, only UFUDD contains an occurrence of UFU.
%t CoefficientList[Series[(1 - x - x^3 - (1 - 2*x - 3*x^2 + 2*x^3 - 2*x^4 + x^6)^(1/2))/(2*x^2*(1 - x + x^2)), {x, 0, 30}], x] (* _Michael De Vlieger_, Jan 31 2023 *)
%o (PARI) seq(n)={Vec((1 - x - x^3 - (1 - 2*x - 3*x^2 + 2*x^3 - 2*x^4 + x^6 + O(x^3*x^n))^(1/2))/(2*x^2*(1 - x + x^2)))} \\ _Andrew Howroyd_, Nov 05 2019
%Y Cf. A001006.
%K nonn
%O 0,3
%A _David Callan_, Jul 16 2004
%E Terms a(23) and beyond from _Andrew Howroyd_, Nov 05 2019
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