OFFSET
2,2
LINKS
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).
Conjecture D-finite with recurrence (n+1)*a(n) +(-6*n+1)*a(n-1) +(10*n-13)*a(n-2) +(-4*n+13)*a(n-3) +(n-4)*a(n-4) +2*(-n+6)*a(n-5)=0. - R. J. Mathar, Jul 22 2022
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Aug 25 2016
STATUS
approved