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%I #16 Aug 19 2017 23:20:20
%S 1,1,1,2,2,1,4,6,2,1,8,14,10,2,1,16,35,28,15,2,1,33,84,88,46,21,2,1,
%T 69,203,247,176,68,28,2,1,146,486,693,558,311,94,36,2,1,312,1163,1882,
%U 1786,1083,507,124,45,2,1,673,2776,5052,5410,3898,1900,780,158,55,2,1
%N Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k even-length columns (n>=2,k>=0).
%C Number of entries in row n is n - 1.
%C Sum of entries in row n = A082582(n).
%C T(n,0) = A004149(n).
%C Sum(k*T(n,k),k>=0) = A273904(n).
%H 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 Emeric Deutsch, S Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088, 2016
%F G.f.: G = G(t,z) satisfies aG^2 + bG + c = 0, where a = z(tz^2-tz+z+t), b = tz^4+(1+t)z^3+(1-t)z^2+(1+t)z-1, c =tz^4+z^2.
%F The trivariate g.f. G(t,s,z), where t (s) marks number of odd-length (even-length) columns and z marks semiperimeter, satisfies AG^2 + BG + C = 0, where A = z(tsz^2-tsz+tz+s), B = tsz^4+(t+s)z^3+(1-ts)z^2+(t+s)z-1, C = tsz^4+s(1-t)z^3+tz^2.
%e 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, clearly, they have 0,1,1,2,0 columns of even length.
%e Triangle starts
%e 1;
%e 1,1;
%e 2,2,1;
%e 4,6,2,1;
%e 8,14,10,2,1
%p eq := z*(t*z^2-t*z+z+t)*f^2-(1-(t+1)*z-(1-t)*z^2-(t+1)*z^3-t*z^4)*f+z^2+t*z^4 = 0: f := RootOf(eq,f): fser := simplify(series(f, z = 0, 15)): for n from 2 to 12 do P[n]:=sort(coeff(fser,z,n)) end do: for n from 2 to 12 do seq(coeff(P[n],t,k),k=0..n-2) end do; # yields sequence in triangular form
%p # second Maple program:
%p b:= proc(n, y, t) option remember; expand(`if`(n=0, 1-t,
%p `if`(t<0, 0, b(n-1, y+1, 1))+ `if`(t>0 or y<2, 0,
%p b(n, y-1, -1))+ `if`(y<1, 0, b(n-1, y, 0)*z^
%p `if`(y::even, 1, 0))))
%p end:
%p T:= n-> (p-> seq(coeff(p, z, i), i=0..n-2))(b(n, 0$2)):
%p seq(T(n), n=2..15); # _Alois P. Heinz_, Jun 24 2016
%t b[n_, y_, t_] := b[n, y, t] = Expand[If[n == 0, 1 - t, If[t < 0, 0, b[n - 1, y + 1, 1]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1]] + If[y < 1, 0, b[n - 1, y, 0]*z^If[EvenQ[y], 1, 0]]]];
%t T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, n-2}]][b[n, 0, 0] ];
%t Table[T[n], {n, 2, 15}] // Flatten (* _Jean-François Alcover_, Jul 21 2016, after _Alois P. Heinz_ *)
%Y Cf. A004149, A082582, A273901, A273902, A273904.
%K nonn,tabl
%O 2,4
%A _Emeric Deutsch_, Jun 23 2016
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