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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,4

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).

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

FORMULA

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

EXAMPLE

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

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 *)

CROSSREFS

Cf. A004148, A082582, A110320, A273714.

Sequence in context: A202193 A105306 A183191 * A064189 A273897 A063415

Adjacent sequences:  A273710 A273711 A273712 * A273714 A273715 A273716

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, May 28 2016

STATUS

approved

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Last modified February 23 13:32 EST 2018. Contains 299581 sequences. (Running on oeis4.)