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A274492 Number of horizontal segments of length 1 in all bargraphs of semiperimeter n (n>=2). By a horizontal segment of length 1 we mean a horizontal step that is not adjacent to any other horizontal step. 1
1, 1, 5, 16, 52, 170, 556, 1821, 5973, 19620, 64536, 212553, 700903, 2313864, 7646670, 25294673, 83748689, 277518319, 920332567, 3054319120, 10143305864, 33707066667, 112078220233, 372875904038, 1241182355688, 4133534991928, 13772413826888, 45908128269573 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,3

LINKS

Table of n, a(n) for n=2..29.

M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

FORMULA

G.f.: g(z)=(1-z)^3*(1-2z-z^2-Q)/(2zQ), where Q = sqrt((1-z)(1-3z-z^2-z^3)).

a(n) = Sum(k*A274491(n,k), k>=0).

EXAMPLE

a(4)=5 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 pictures give the values 0,2,2,0,1 for the number of horizontal segments of length 1.

MAPLE

g:=(1/2)*(1-z)^3*(1-2*z-z^2-Q)/(z*Q): Q:=sqrt((1-z)*(1-3*z-z^2-z^3)): gser:= series(g, z=0, 30): seq(coeff(gser, z, n), n=2..27);

MATHEMATICA

g = (1-z)^3 (1-2z-z^2-Q)/(2z Q) /. Q -> Sqrt[(1-z)(1-3z-z^2-z^3)];

a[n_] := SeriesCoefficient[g, {z, 0, n}];

Table[a[n], {n, 2, 29}] (* Jean-François Alcover, Jul 25 2018 *)

CROSSREFS

Cf. A082582, A274491.

Sequence in context: A317817 A077840 A007343 * A147536 A173871 A108300

Adjacent sequences:  A274489 A274490 A274491 * A274493 A274494 A274495

KEYWORD

nonn

AUTHOR

Emeric Deutsch and Sergi Elizalde, Jun 27 2016

STATUS

approved

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Last modified April 18 22:08 EDT 2019. Contains 322237 sequences. (Running on oeis4.)