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A276066 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having a total of k double rises and double falls (n>=2,k>=0). A double rise (fall) in a bargraph is any pair of adjacent up (down) steps.

%I A276066 #18 Aug 19 2017 23:08:41
%S A276066 1,1,0,1,1,2,1,0,1,2,4,1,4,1,0,1,4,6,8,8,1,6,1,0,1,7,14,22,12,19,12,1,
%T A276066 8,1,0,1,13,34,43,48,55,18,35,16,1,10,1,0,1,26,72,105,148,109,116,103,
%U A276066 24,56,20,1,12,1,0,1,52,154,276,344,347,398,205,232,166,30,82,24,1,14,1,0,1
%N A276066 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having a total of k double rises and double falls (n>=2,k>=0). A double rise (fall) in a bargraph is any pair of adjacent up (down) steps.
%C A276066 Number of entries in row n is 2n-3.
%C A276066 Sum of entries in row n = A082582(n).
%C A276066 T(n,0) = A023431(n-2) = A025246(n+1).
%C A276066 Sum(k*T(n,k),k>=0) = 2*A273714(n).
%H A276066 Alois P. Heinz, <a href="/A276066/b276066.txt">Rows n = 2..140, flattened</a>
%H A276066 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 A276066 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 A276066 G.f.: G = G(t,z) satisfies zG^2 - (1-z - t^2*z - 2tz^2+t^2*z^2)G + z^2 = 0.
%F A276066 The g.f. B(t,s,z) of bargraphs, where t(s) marks double rises (falls) and z marks semiperimeter, satisfies zB^2 - (1-(1+ts)z +(ts- t-s)z^2)B + z^2 = 0.
%e A276066 Row 4 is 1,2,1,0,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and the corresponding drawings show that they have a total of  0, 1, 1, 2, 4 double rises and double falls, respectively.
%e A276066 Triangle starts:
%e A276066 1;
%e A276066 1,0,1;
%e A276066 1,2,1,0,1;
%e A276066 2,4,1,4,1,0,1;
%e A276066 4,6,8,8,1,6,1,0,1.
%p A276066 eq := z*G^2-(1-z-t^2*z-2*t*z^2+t^2*z^2)*G+z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 22)): for n from 2 to 20 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 20 do seq(coeff(P[n], t, j), j = 0 .. 2*n-4) end do; # yields sequence in triangular form.
%p A276066 # second Maple program:
%p A276066 b:= proc(n, y, t) option remember; expand(`if`(n=0, (1-t)*
%p A276066       z^(y-1), `if`(t<0, 0, b(n-1, y+1, 1)*`if`(t=1, z, 1))+
%p A276066      `if`(t>0 or y<2, 0, b(n, y-1, -1)*`if`(t=-1, z, 1))+
%p A276066      `if`(y<1, 0, b(n-1, y, 0))))
%p A276066     end:
%p A276066 T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
%p A276066 seq(T(n), n=2..12);  # _Alois P. Heinz_, Aug 25 2016
%t A276066 b[n_, y_, t_] := b[n, y, t] = Expand[If[n == 0, (1 - t)*z^(y - 1), If[t < 0, 0, b[n - 1, y + 1, 1]*If[t == 1, z, 1]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1]*If[t == -1, z, 1]] + If[y < 1, 0, b[n - 1, y, 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0]]; Table[T[n], {n, 2, 12}] // Flatten (* _Jean-François Alcover_, Dec 02 2016 after _Alois P. Heinz_ *)
%Y A276066 Cf. A023431, A025246, A082582, A273714.
%K A276066 nonn,tabf
%O A276066 2,6
%A A276066 _Emeric Deutsch_ and _Sergi Elizalde_, Aug 25 2016