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%I #6 Jul 22 2022 10:16:04
%S 0,0,0,0,1,9,51,236,979,3805,14190,51488,183333,644121,2241127,
%T 7741378,26593899,90971184,310159487,1054693058,3578948942,
%U 12124108632,41015411703,138597840864,467913141789,1578497031981,5321685955902,17931990439148,60397664457791,203355625940891
%N Number of horizontal steps in the valleys of all bargraphs having semiperimeter n (n >=2).
%H A. Blecher, C. Brennan, and A. Knopfmacher, <a href="http://dx.doi.org/10.1080/0035919X.2015.1059905">Peaks in bargraphs</a>, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.
%F G.f.: g(z) = 2z^6/(Q(R + (1-3z+z^2)(1-z)^2*Q)), where Q = sqrt((1-z)(1-3z-z^2-z^3)) and R = 1 - 7z + 17z^2 - 18z^3 + 9z^4 - 3z^5 + z^6.
%F a(n) = Sum(k*A278134(n,k), k>=0).
%F Conjecture D-finite with recurrence -7*(n+1)*(n-6)*a(n) +3*(13*n^2-69*n+14)*a(n-1) +(-61*n^2+331*n-256)*a(n-2) +3*(11*n^2-59*n+68)
%F *a(n-3) -(n-2)*(9*n-25)*a(n-4) +(9*n^2-55*n+80)*a(n-5) -(3*n-4)*(n-5)*a(n-6) -(n-5)*(n-6)*a(n-7)=0. - _R. J. Mathar_, Jul 22 2022
%e a(6) = 1 because among the 35 (=A082582(6)) bargraphs of semiperimeter 6 only one has a valley; it corresponds to the composition [2,1,2] and its width is 1.
%p Q := sqrt((1-z)*(1-3*z-z^2-z^3)): R := 1-7*z+17*z^2-18*z^3+9*z^4-3*z^5+z^6: g := 2*z^6/(Q*(R+(1-3*z+z^2)*(1-z)^2*Q)): gser := series(g, z = 0, 35): seq(coeff(gser, z, j), j = 2 .. 33);
%Y Cf. A082582, A273719, A273720, A278134
%K nonn
%O 2,6
%A _Emeric Deutsch_, Jan 06 2017
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