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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)

*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

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proposed

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proposed

allocated for Emeric DeutschNumber of horizontal steps in the valleys of all bargraphs having semiperimeter n (n >=2).

0, 0, 0, 0, 1, 9, 51, 236, 979, 3805, 14190, 51488, 183333, 644121, 2241127, 7741378, 26593899, 90971184, 310159487, 1054693058, 3578948942, 12124108632, 41015411703, 138597840864, 467913141789, 1578497031981, 5321685955902, 17931990439148, 60397664457791, 203355625940891

2,6

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.

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.

a(n) = Sum(k*A278134(n,k), k>=0).

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.

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);

Cf. A082582, A273719, A273720, A278134

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nonn

Emeric Deutsch, Jan 06 2017

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allocated for Emeric Deutsch

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