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A274488 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having least column-height k (n>=2, k>=1).

%I A274488 #25 Nov 16 2018 17:03:25
%S A274488 1,1,1,3,1,1,8,3,1,1,22,8,3,1,1,62,22,8,3,1,1,178,62,22,8,3,1,1,519,
%T A274488 178,62,22,8,3,1,1,1533,519,178,62,22,8,3,1,1,4578,1533,519,178,62,22,
%U A274488 8,3,1,1,13800,4578,1533,519,178,62,22,8,3,1,1,41937,13800,4578,1533,519,178,62,22,8,3,1,1
%N A274488 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having least column-height k (n>=2, k>=1).
%C A274488 T(n,k) = number of bargraphs of semiperimeter n for which the width of the leftmost horizontal segment is k. A horizontal segment is a maximal sequence of adjacent horizontal steps (1,0). Example: T(4,1)=3 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 widths of their leftmost horizontal segments are 3, 1, 1, 2, 1.
%C A274488 Number of entries in row n is n-1.
%H A274488 G. C. Greubel, <a href="/A274488/b274488.txt">Rows n=2..102 of triangle, flattened</a>
%H A274488 M. Bousquet-Mélou and A. Rechnitzer, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00553-5">The site-perimeter of bargraphs</a>, Adv. in Appl. Math. 31 (2003), 86-112.
%H A274488 Emeric Deutsch, S Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv:1609.00088 [math.CO], 2016/2018.
%F A274488 G.f.: t(1-z)(1-2z-z^2-sqrt((1-z)(1-3z-z^2-z^3)))/(2z(1-tz)).
%e A274488 Row 4 is 3,1,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and, clearly, their least column-heights are 1,1,1,2,3.
%e A274488 Triangle starts
%e A274488 1;
%e A274488 1,1;
%e A274488 3,1,1;
%e A274488 8,3,1,1;
%e A274488 22,8,3,1,1
%p A274488 G:=(1/2)*t*(1-z)*(1-2*z-z^2-sqrt((1-z)*(1-3*z-z^2-z^3)))/(z*(1-t*z)): Gser:= simplify(series(G,z=0,28)):for n from 2 to 20 do P[n]:= sort(coeff(Gser,z,n)) end do: for n from 2 to 15 do seq(coeff(P[n],t,k),k=1..n-1) end do; # yields sequence in triangular form
%t A274488 gf = t(1-z)((1 - 2z - z^2 - Sqrt[(1-z)(1 - 3z - z^2 - z^3)])/(2z(1 - t z)));
%t A274488 Rest[CoefficientList[#, t]]& /@ Drop[CoefficientList[gf + O[z]^14, z], 2] // Flatten (* _Jean-François Alcover_, Nov 16 2018 *)
%Y A274488 Sum of entries in row n = A082582(n).
%Y A274488 T(n,1) = A188464(n-3)(n>=3).
%Y A274488 Sum(k*T(n,k),k>=1)= A008909(n).
%Y A274488 Cf. A273350, A274490.
%K A274488 nonn,tabl
%O A274488 2,4
%A A274488 _Emeric Deutsch_, Jul 01 2016