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%I #20 Jul 22 2022 10:25:47
%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)).
%F D-finite with recurrence (n+1)*a(n) +(-7*n+2)*a(n-1) +3*(7*n-11)*a(n-2) +(-37*n+107)*a(n-3) +3*(13*n-54)*a(n-4) +3*(-7*n+37)*a(n-5) +2*(2*n-13)*a(n-6)=0. - _R. J. Mathar_, Jul 22 2022
%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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