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.

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, 8, 1, 0, 1, 13, 34, 43, 48, 55, 18, 35, 16, 1, 10, 1, 0, 1, 26, 72, 105, 148, 109, 116, 103, 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

Offset: 2

Author

Emeric Deutsch and Sergi Elizalde, Aug 25 2016

Keywords

• nonn
• tabf

Comments

Number of entries in row n is 2n-3.

Sum of entries in row n = A082582(n).

T(n,0) = A023431(n-2) = A025246(n+1).

Sum(k*T(n,k),k>=0) = 2*A273714(n).

Examples

			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.
Triangle starts:
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.
		

Links

• Alois P. Heinz, Rows n = 2..140, flattened
• M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
• Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Crossrefs

Cf. A023431, A025246, A082582, A273714.

Programs

Maple

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.
# second Maple program:
b:= proc(n, y, t) option remember; 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 or y<2, 0, b(n, y-1, -1)*`if`(t=-1, z, 1))+
     `if`(y<1, 0, b(n-1, y, 0))))
    end:
T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=2..12);  # Alois P. Heinz, Aug 25 2016

Mathematica

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 *)

Formula

G.f.: G = G(t,z) satisfies zG^2 - (1-z - t^2*z - 2tz^2+t^2*z^2)G + z^2 = 0.

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.