OFFSET
2,4
COMMENTS
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
FORMULA
G.f.: G(t,z) = t(1-2z)(1-2z-z^2-Q)/(z(1-z-tz)), where Q = sqrt((1-z)(1-3z-z^2-z^3)).
T(n,k)= T(n-1,k)+T(n-1,k-1) (n>=3, k>=2).
EXAMPLE
Row 4 is 2,2,1 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 lengths of their first descents are 1,2,1,2,3, respectively.
Triangle starts
1;
1,1;
2,2,1;
5,4,3,1;
14,9,7,4,1.
MAPLE
G := (1/2)*t*(1-2*z)*(1-2*z-z^2-sqrt((1-z)*(1-3*z-z^2-z^3)))/(z*(1-z-t*z)): Gser := simplify(series(G, z = 0, 20)): for n from 2 to 17 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 17 do seq(coeff(P[n], t, j), j = 1 .. n-1) end do; # yields sequence in triangular form
MATHEMATICA
m = maxExponent = 13;
G = ((1/2) t (1 - 2z)(1 - 2z - z^2 - Sqrt[(1 - z)(1 - 3z - z^2 - z^3)])/ (z(1 - z - t z)) + O[z]^m) + O[t]^m;
Drop[CoefficientList[#/t, t]& /@ CoefficientList[G, z], 2] // Flatten (* Jean-François Alcover, Aug 07 2018 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Aug 25 2016
STATUS
approved