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Number of levels in all bargraphs having semiperimeter n (n>=2). A level in a bargraph is a maximal sequence of two or more adjacent horizontal steps; it is preceded and followed by either an up step or a down step.
4

%I #23 Jul 26 2022 14:42:29

%S 0,1,2,7,23,75,245,801,2622,8595,28215,92751,305304,1006207,3320071,

%T 10966741,36261414,120010103,397528422,1317860989,4372180109,

%U 14515485973,48222552640,160300772873,533176676911,1774359032599,5907894024527,19680307851415,65588436120988,218679463049627

%N Number of levels in all bargraphs having semiperimeter n (n>=2). A level in a bargraph is a maximal sequence of two or more adjacent horizontal steps; it is preceded and followed by either an up step or a down step.

%H A. Blecher, C. Brennan, and A. Knopfmacher, <a href="http://amc-journal.eu/index.php/amc/article/view/600/835">Levels in bargraphs</a>, Ars Math. Contemp., 9, 2015, 297-310.

%H A. Blecher, C. Brennan, and A. Knopfmacher, <a href="http://dx.doi.org/10.1080/0035919X.2015.1059905">Peaks in bargraphs</a>, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.

%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.

%F a(n) = Sum(k*A273344(n,k), k>=0).

%F G.f. g(z) = (1-z)^2 (1-2z-z^2-sqrt((1-z)(1-3z-z^2-z^3)))/(2 sqrt((1-z)(1-3z-z^2-z^3))).

%F D-finite with recurrence n*a(n) +2*(-3*n+4)*a(n-1) +(9*n-28)*a(n-2) +2*a(n-3) +(-n+16)*a(n-4) +2*(-n+7)*a(n-5) +(-n+8)*a(n-6)=0. - _R. J. Mathar_, Jun 02 2016

%e a(4) = 2 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3]; they have 1, 0, 0, 1, 0 levels, respectively.

%p g := (1/2)*(1-z)^2*(1-2*z-z^2-sqrt((1-z)*(1-3*z-z^2-z^3)))/sqrt((1-z)*(1-3*z-z^2-z^3)): gser := series(g,z = 0,45): seq(coeff(gser, z, n), n = 2 .. 42);

%Y Cf. A082582, A273344.

%K nonn

%O 2,3

%A _Emeric Deutsch_, May 21 2016