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A273717 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k L-shaped corners (n>=2, k>=0).

%I A273717 #41 Mar 07 2023 11:19:22
%S A273717 1,2,4,1,8,5,16,18,1,32,56,9,64,160,50,1,128,432,220,14,256,1120,840,
%T A273717 110,1,512,2816,2912,645,20,1024,6912,9408,3150,210,1,2048,16640,
%U A273717 28800,13552,1575,27,4096,39424,84480,53088,9534,364,1,8192,92160,239360,193440,49644,3388,35,16384,212992,658944,665280,231000,24822,588
%N A273717 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k L-shaped corners (n>=2, k>=0).
%C A273717 Each L-shaped corner can be viewed as a descent (as defined in Sec. 5.1 of the Blecher et al. reference). - _Emeric Deutsch_, Jul 02 2016
%D A273717 A. Blecher, C. Brennan, and A. Knopfmacher, Combinatorial parameters in bargraphs (preprint).
%H A273717 Alois P. Heinz, <a href="/A273717/b273717.txt">Rows n = 2..200, flattened</a>
%H A273717 Jean-Luc Baril and José Luis Ramírez, <a href="https://arxiv.org/abs/2302.12741">Descent distribution on Catalan words avoiding ordered pairs of Relations</a>, arXiv:2302.12741 [math.CO], 2023.
%H A273717 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. Appl. Math., 31, 2003, 86-112.
%H A273717 Emeric Deutsch and S. Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088 [math.CO], 2016.
%F A273717 G.f.: G = G(t,z) satisfies t*z*G^2 - (1 - 2*z - t*z^2)*G + z^2 = 0.
%F A273717 Sum_{k>=1} k*T(n,k) = A273718(n).
%F A273717 T(n,0) = 2^(n-2).
%F A273717 T(n,1) = n*(n-3)*2^(n-6) = A001793(n-3) for n>=4.
%e A273717 Row 4 is 4,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] of which only [2,1] yields a |_ - shaped corner.
%e A273717 Triangle starts:
%e A273717   1;
%e A273717   2;
%e A273717   4,1;
%e A273717   8,5;
%e A273717   16,18,1.
%p A273717 eq := t*z*G^2-(1-2*z-t*z^2)*G+z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 22)): 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 .. degree(P[n])) end do; # yields sequence in triangular form
%p A273717 # second Maple program:
%p A273717 b:= proc(n, y, t) option remember; expand(
%p A273717       `if`(n=0, (1-t), `if`(t<0, 0, b(n-1, y+1, 1))+
%p A273717       `if`(t>0 or y<2, 0, b(n, y-1, -1))+
%p A273717       `if`(y<1, 0, b(n-1, y, 0)*`if`(t<0, z, 1))))
%p A273717     end:
%p A273717 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
%p A273717 seq(T(n), n=2..18);  # _Alois P. Heinz_, Jun 06 2016
%t A273717 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]*If[t < 0, z, 1]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0]]; Table[T[n], {n, 2, 18}] // Flatten (* _Jean-François Alcover_, Dec 02 2016 after _Alois P. Heinz_ *)
%Y A273717 Sum of entries in row n = A082582(n).
%Y A273717 Cf. A001793, A273718.
%K A273717 nonn,tabf
%O A273717 2,2
%A A273717 _Emeric Deutsch_, May 29 2016