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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.
4
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
2,3
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.
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
EXAMPLE
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.
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);
CROSSREFS
Sequence in context: A077832 A030282 A291015 * A042575 A256030 A052924
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 21 2016
STATUS
approved