A273345 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.

0, 1, 2, 7, 23, 75, 245, 801, 2622, 8595, 28215, 92751, 305304, 1006207, 3320071, 10966741, 36261414, 120010103, 397528422, 1317860989, 4372180109, 14515485973, 48222552640, 160300772873, 533176676911, 1774359032599, 5907894024527, 19680307851415, 65588436120988, 218679463049627

Offset: 2

Author

Emeric Deutsch, May 21 2016

Keywords

• nonn

Examples

			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.
		

Links

• A. Blecher, C. Brennan, and A. Knopfmacher, Levels in bargraphs, Ars Math. Contemp., 9, 2015, 297-310.
• A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.
• M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Crossrefs

Cf. A082582, A273344.

Programs

Maple

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);

Formula

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

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))).

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