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A276068 Sum of the lengths of the first descents in all bargraphs having semiperimeter n (n>=2). A descent is a maximal sequence of consecutive down steps. 1
1, 3, 9, 26, 74, 210, 598, 1715, 4963, 14504, 42808, 127553, 383451, 1162134, 3548060, 10904023, 33708595, 104756233, 327086895, 1025603074, 3228082910, 10195295005, 32300276271, 102622734570, 326893843104, 1043767139218, 3340051490096 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

LINKS

Table of n, a(n) for n=2..28.

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)(1-2z-z^2-Q)/(2z(1-2z)), where Q = sqrt((1-z)(1-3z-z^2-z^3)).

a(n) = Sum(k*A276067(n,k), k>=1).

EXAMPLE

a(4) = 9 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 the sum of the lengths of their first descents is 1+2+1+2+3.

MAPLE

g := (1/2)*(1-z)*(1-2*z-z^2-sqrt((1-z)*(1-3*z-z^2-z^3)))/(z*(1-2*z)): gser := series(g, z = 0, 38): seq(coeff(gser, z, n), n = 2 .. 35);

MATHEMATICA

G = (1/2)(1-z)(1 - 2z - z^2 - Sqrt[(1-z)(1 - 3z - z^2 - z^3)])/(z(1-2z)) + O[z]^29;

Drop[CoefficientList[G, z], 2] // Flatten (* Jean-François Alcover, Aug 07 2018 *)

CROSSREFS

Cf. A082582, A276067.

Sequence in context: A116423 A077845 A291000 * A171277 A289806 A303976

Adjacent sequences:  A276065 A276066 A276067 * A276069 A276070 A276071

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Aug 25 2016

STATUS

approved

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Last modified December 1 22:12 EST 2021. Contains 349435 sequences. (Running on oeis4.)