A273722 The number of valleys of width 1 (i.e., DHU configurations, where U=(0,1), H(1,0), D=(0,-1)) in all bargraphs of semiperimeter n (n>=2).
0, 0, 0, 0, 1, 7, 34, 143, 558, 2083, 7559, 26913, 94547, 328943, 1136218, 3903245, 13352270, 45524764, 154811018, 525345268, 1779722313, 6020903806, 20346143381, 68691126090, 231732871764, 781268589267, 2632605033729, 8867115559325, 29855369535397
Offset: 2
Keywords
Examples
a(4)=0 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 they have no 1-width valleys. a(6)=1 because there is only one bargraph of semiperimeter 6 having a 1-width valley (it corresponds to the composition [2,1,2]).
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
Programs
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Maple
Q:=sqrt(1-4*z+2*z^2+z^4): g:=((1-5*z+6*z^2-z^3+z^4-(1-3*z+z^2)*Q)*(1/2))/(z*Q): gser:= series(g,z=0,40): seq(coeff(gser, z, n), n = 2 .. 35);
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Mathematica
terms = 29; g[z_] = (1 - 5z + 6z^2 - z^3 + z^4 - (1 - 3z + z^2) Q)/(2z Q) /. Q -> Sqrt[1 - 4z + 2z^2 + z^4]; Drop[CoefficientList[g[z] + O[z]^(terms+2), z], 2] (* Jean-François Alcover, Aug 07 2018 *)
Formula
G.f.: g(z)=(1-5z+6z^2-z^3+z^4-(1-3z+z^2)Q)/(2zQ), where Q = sqrt(1-4z+2z^2+z^4).
a(n) = Sum_{k >= 1} k*A273721(n,k).
Conjecture: -(n-6) *(2*n-7) *(2*n-9) *(n+1)*a(n) +2*(n-3) *(2*n-9) *(4*n^2-24*n+21)*a(n-1) +2*(-4*n^4+56*n^3-289*n^2+651*n-547) *a(n-2) +4*(2*n-5) *(n-4)*a(n-3) -(n-4) *(n-5) *(2*n-5) *(2*n-7) *a(n-4)=0. - R. J. Mathar, Jun 02 2016