A278135 Number 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
Offset: 2
Keywords
Examples
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.
Links
- A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.
Programs
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Maple
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);
Formula
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).
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