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A273721 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k valleys of width 1 (i.e., DHU configurations, where U=(0,1), H=(1,0), D=(0,-1)), (n>=2, k>=0).

%I A273721 #20 Aug 19 2017 23:09:09
%S A273721 1,2,5,13,34,1,90,7,241,34,652,141,1,1780,536,11,4899,1929,77,13581,
%T A273721 6690,433,1,37893,22601,2132,16,106340,74898,9598,151,299978,244632,
%U A273721 40517,1091,1,850187,790144,162988,6670,22,2419788,2529873,631655,36330,268
%N A273721 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k valleys of width 1 (i.e., DHU configurations, where U=(0,1), H=(1,0), D=(0,-1)), (n>=2, k>=0).
%C A273721 Sum of entries in row n = A082582(n).
%C A273721 T(n,0) = A023425(n+2).
%C A273721 Sum_{k>=0} k*T(n,k) = A273722(n).
%H A273721 Alois P. Heinz, <a href="/A273721/b273721.txt">Rows n = 2..250, flattened</a>
%H A273721 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 A273721 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 A273721 G.f.: G=G(t,z), where t marks number of 1-width valleys and z marks semiperimeter, satisfies z(t+z-tz)G^2-(1-2z-z^2-z^3+tz^3)G+z^2 = 0.
%F A273721 For p-width valleys, the corresponding g.f. satisfies aG^2 - bG + c = 0, where a = z-(1-t)(1-z)z^p, b = 1-2z-z^2(1+z^p-tz^p), c=z^2.
%e A273721 Row 4 is 5 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.
%e A273721 T(6,1) = 1 because there is only one bargraph of semiperimeter 6 having a 1-width valley (it corresponds to the composition [2,1,2]).
%e A273721 Triangle starts
%e A273721 1;
%e A273721 2;
%e A273721 5;
%e A273721 13;
%e A273721 34, 1;
%p A273721 eq := z*(t+z-t*z)*G^2-(1-2*z-z^2-z^3+t*z^3)*G+z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 25)): for n from 2 to 22 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 22 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
%p A273721 # second Maple program:
%p A273721 b:= proc(n, y, t, h) option remember; expand(
%p A273721       `if`(n=0, (1-t), `if`(t<0, 0, b(n-1, y+1, 1, 0)*z^h)+
%p A273721       `if`(t>0 or y<2, 0, b(n, y-1, -1, 0))+
%p A273721       `if`(y<1, 0, b(n-1, y, 0, `if`(t<0, 1, 0)))))
%p A273721     end:
%p A273721 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):
%p A273721 seq(T(n), n=2..22);  # _Alois P. Heinz_, Jun 06 2016
%t A273721 b[n_, y_, t_, h_] := b[n, y, t, h] = Expand[If[n == 0, (1 - t), If[t < 0, 0, b[n - 1, y + 1, 1, 0]*z^h] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]] + If[y < 1, 0, b[n - 1, y, 0, If[t < 0, 1, 0]]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0, 0]]; Table[T[n], {n, 2, 22}] // Flatten (* _Jean-François Alcover_, Nov 29 2016 after _Alois P. Heinz_ *)
%Y A273721 Cf. A023425, A082582, A273722.
%K A273721 nonn,tabf
%O A273721 2,2
%A A273721 _Emeric Deutsch_, Jun 01 2016