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A273349 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k level steps (n>=2,k>=0). A level step in a bargraph is any pair of adjacent horizontal steps at the same height.

%I A273349 #23 Nov 08 2016 20:01:58
%S A273349 1,1,1,3,1,1,6,5,1,1,14,12,7,1,1,33,34,19,9,1,1,79,95,61,27,11,1,1,
%T A273349 194,261,193,95,36,13,1,1,482,728,585,333,136,46,15,1,1,1214,2022,
%U A273349 1797,1091,521,184,57,17,1,1,3090,5634,5439,3629,1821,763,239,69,19,1,1
%N A273349 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k level steps (n>=2,k>=0). A level step in a bargraph is any pair of adjacent horizontal steps at the same height.
%C A273349 Number of entries in row n is n-1.
%C A273349 Sum of entries in row n = A082582(n).
%C A273349 T(n,0) = A025243(n+1).
%C A273349 Sum(k*T(n,k),k>=0) = A271943(n-1). This implies that the number of level steps in all bargraphs of semiperimeter n is equal to the sum of the widths of all bargraphs of semiperimeter n-1.
%D A273349 A. Blecher, C. Brennan, and A. Knopfmacher, Combinatorial parameters in bargraphs (preprint).
%H A273349 Alois P. Heinz, <a href="/A273349/b273349.txt">Rows n = 2..150, flattened</a>
%H A273349 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.
%F A273349 G.f.:  G(t,z) = (1-tz-z-2z^2+tz^2-sqrt((1-z)(1-z-2tz-4z^2+t^2z^2+2tz^2-4z^3-t^2z^3+4tz^3)))/(2z) (z marks semiperimeter, t marks level steps; obtained from the expression for F in the Blecher et al. reference (Section 7.1) by setting x=z, y=z, w=t).
%e A273349 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] which, clearly, have 2,0,0,1,0 level steps.
%e A273349 Triangle starts
%e A273349 1;
%e A273349 1,1;
%e A273349 3,1,1;
%e A273349 6,5,1,1;
%e A273349 14,12,7,1,1
%p A273349 G:=((1-t*z-z-2*z^2+t*z^2-sqrt((1-t*z-z-2*z^2+t*z^2)^2-4*z^3))*(1/2))/z: Gser:=simplify(series(G,z=0,21)): for n from 2 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 18 do seq(coeff(P[n], t, j), j = 0 .. n-2) end do; # yields sequence in triangular form
%p A273349 # second Maple program:
%p A273349 b:= proc(n, y, t, w) option remember; expand(
%p A273349       `if`(n=0, (1-t), `if`(t<0, 0, b(n-1, y+1, 1, 0))+
%p A273349       `if`(t>0 or y<2, 0, b(n, y-1, -1, 0))+ `if`(y<1, 0,
%p A273349       `if`(w=1, z, 1)*b(n-1, y, 0, min(w+1, 1)))))
%p A273349     end:
%p A273349 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):
%p A273349 seq(T(n), n=2..18);  # _Alois P. Heinz_, Jun 04 2016
%t A273349 b[n_, y_, t_, w_] := b[n, y, t, w] = Expand[If[n == 0, 1 - t, If[t < 0, 0, b[n - 1, y + 1, 1, 0]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]] + If[y < 1, 0, If[w == 1, z, 1]*b[n - 1, y, 0, Min[w + 1, 1]]]]];
%t A273349 T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][ b[n, 0, 0, 0]];
%t A273349 Table[T[n], {n, 2, 18}] // Flatten (* _Jean-François Alcover_, Jul 29 2016, after _Alois P. Heinz_ *)
%Y A273349 Cf. A025243, A082582, A271943.
%K A273349 nonn,tabl
%O A273349 2,4
%A A273349 _Emeric Deutsch_, Jun 03 2016