A273713 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k doublerises (n>=2, k>=0). A doublerise in a bargraph is any pair of adjacent up steps.

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 8, 13, 9, 4, 1, 17, 32, 28, 14, 5, 1, 37, 80, 81, 50, 20, 6, 1, 82, 201, 231, 165, 80, 27, 7, 1, 185, 505, 653, 526, 295, 119, 35, 8, 1, 423, 1273, 1824, 1644, 1036, 483, 168, 44, 9, 1, 978, 3217, 5058, 5034, 3535, 1848, 742, 228, 54, 10, 1

Offset: 2

Author

Emeric Deutsch, May 28 2016

Keywords

• nonn
• tabl

Comments

Number of entries in row n is n-1.

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

T(n,0) = A004148(n-1) (the 2ndary structure numbers).

T(n,1) = A110320(n-2).

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

Examples

			Row 4 is 2,2,1 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 0, 0, 1, 1, 2 doublerises.
Triangle starts
1;
1,1;
2,2,1;
4,5,3,1;
8,13,9,4,1
		

Links

• Alois P. Heinz, Rows n = 2..150, 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. A004148, A082582, A110320, A273714.

Programs

Maple

eq := z*G^2-(1-z-t*z-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 .. n-2) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t) option remember; expand(`if`(n=0, (1-t),
      `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`(y<1, 0, b(n-1, y, 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n-2))(b(n, 0$2)):
seq(T(n), n=2..16);  # Alois P. Heinz, Jun 06 2016

Mathematica

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 == 1, z, 1]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1]] + If[y < 1, 0, b[n - 1, y, 0]]]];
T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 0, n - 2}]][b[n, 0, 0]];
Table[T[n], {n, 2, 16}] // Flatten (* Jean-François Alcover, Jul 29 2016, after Alois P. Heinz *)

Formula

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