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%I #20 Jul 22 2022 10:18:49
%S 1,3,9,26,74,210,598,1715,4963,14504,42808,127553,383451,1162134,
%T 3548060,10904023,33708595,104756233,327086895,1025603074,3228082910,
%U 10195295005,32300276271,102622734570,326893843104,1043767139218,3340051490096
%N 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.
%H M. Bousquet-Mélou and A. Rechnitzer, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00553-5">The site-perimeter of bargraphs</a>, Adv. in Appl. Math. 31 (2003), 86-112.
%H Emeric Deutsch, S Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088, 2016
%F 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)).
%F a(n) = Sum(k*A276067(n,k), k>=1).
%F 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
%e 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.
%p 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);
%t G = (1/2)(1-z)(1 - 2z - z^2 - Sqrt[(1-z)(1 - 3z - z^2 - z^3)])/(z(1-2z)) + O[z]^29;
%t Drop[CoefficientList[G, z], 2] // Flatten (* _Jean-François Alcover_, Aug 07 2018 *)
%Y Cf. A082582, A276067.
%K nonn
%O 2,2
%A _Emeric Deutsch_, Aug 25 2016
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