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A275448 The number of weakly alternating bargraphs of semiperimeter n. A bargraph is said to be weakly alternating if its ascents and descents alternate. An ascent (descent) is a maximal sequence of consecutive U (D) steps. 1

%I

%S 1,2,3,4,6,12,28,65,146,327,749,1756,4165,9913,23652,56687,136627,

%T 330969,804915,1963830,4805523,11793046,29019930,71589861,177006752,

%U 438561959,1088714711,2707615555,6745272783,16830750107,42058592797,105248042792

%N The number of weakly alternating bargraphs of semiperimeter n. A bargraph is said to be weakly alternating if its ascents and descents alternate. An ascent (descent) is a maximal sequence of consecutive U (D) steps.

%H M. Bousquet-Mélou and A. Rechnitzer, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00553-5">The site-perimeter of bargraphs</a>, Adv. in Appl. Math. 31 (2003), 86-112.

%H Emeric Deutsch, S Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088, 2016

%F G.f.: g(z) = (1-3z+3z^2 - Q)/(2z(1-z)), where Q = sqrt((1-3z+z^2)(1-3z+5z^2-4z^3)).

%e a(4)=3 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and the corresponding drawings show that only [1,1,1],[2,2], and [3] lead to weakly alternating bargraphs.

%p g := ((1-3*z+3*z^2-sqrt((1-3*z+z^2)*(1-3*z+5*z^2-4*z^3)))*(1/2))/(z*(1-z)): gser:= series(g,z=0,43): seq(coeff(gser,z,n), n=2..40);

%t terms = 32;

%t g[z_] = ((1 - 3z + 3z^2 - Sqrt[(1 - 3z + z^2)(1 - 3z + 5z^2 - 4z^3)])*(1/2) )/(z(1-z));

%t Drop[CoefficientList[g[z] + O[z]^(terms+2), z], 2] (* _Jean-François Alcover_, Aug 07 2018 *)

%Y Cf. A082582, A023432.

%K nonn

%O 2,2

%A _Emeric Deutsch_, _Sergi Elizalde_, Aug 26 2016

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Last modified May 24 06:53 EDT 2019. Contains 323529 sequences. (Running on oeis4.)