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A273349 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k level steps (n>=2,k>=0). A level step in a bargraph is any pair of adjacent horizontal steps at the same height. 1
1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 14, 12, 7, 1, 1, 33, 34, 19, 9, 1, 1, 79, 95, 61, 27, 11, 1, 1, 194, 261, 193, 95, 36, 13, 1, 1, 482, 728, 585, 333, 136, 46, 15, 1, 1, 1214, 2022, 1797, 1091, 521, 184, 57, 17, 1, 1, 3090, 5634, 5439, 3629, 1821, 763, 239, 69, 19, 1, 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) = A025243(n+1).

Sum(k*T(n,k),k>=0) = A271943(n-1). This implies that the number of level steps in all bargraphs of semiperimeter n is equal to the sum of the widths of all bargraphs of semiperimeter n-1.

REFERENCES

A. Blecher, C. Brennan, and A. Knopfmacher, Combinatorial parameters in bargraphs (preprint).

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.

FORMULA

G.f.:  G(t,z) = (1-tz-z-2z^2+tz^2-sqrt((1-z)(1-z-2tz-4z^2+t^2z^2+2tz^2-4z^3-t^2z^3+4tz^3)))/(2z) (z marks semiperimeter, t marks level steps; obtained from the expression for F in the Blecher et al. reference (Section 7.1) by setting x=z, y=z, w=t).

EXAMPLE

Row 4 is 3,1,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] which, clearly, have 2,0,0,1,0 level steps.

Triangle starts

1;

1,1;

3,1,1;

6,5,1,1;

14,12,7,1,1

MAPLE

G:=((1-t*z-z-2*z^2+t*z^2-sqrt((1-t*z-z-2*z^2+t*z^2)^2-4*z^3))*(1/2))/z: Gser:=simplify(series(G, z=0, 21)): 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 .. n-2) end do; # yields sequence in triangular form

# second Maple program:

b:= proc(n, y, t, w) option remember; expand(

      `if`(n=0, (1-t), `if`(t<0, 0, b(n-1, y+1, 1, 0))+

      `if`(t>0 or y<2, 0, b(n, y-1, -1, 0))+ `if`(y<1, 0,

      `if`(w=1, z, 1)*b(n-1, y, 0, min(w+1, 1)))))

    end:

T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):

seq(T(n), n=2..18);  # Alois P. Heinz, Jun 04 2016

MATHEMATICA

b[n_, y_, t_, w_] := b[n, y, t, w] = Expand[If[n == 0, 1 - t, If[t < 0, 0, b[n - 1, y + 1, 1, 0]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]] + If[y < 1, 0, If[w == 1, z, 1]*b[n - 1, y, 0, Min[w + 1, 1]]]]];

T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][ b[n, 0, 0, 0]];

Table[T[n], {n, 2, 18}] // Flatten (* Jean-François Alcover, Jul 29 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A025243, A082582, A271943.

Sequence in context: A273350 A278132 A203950 * A159572 A190907 A035582

Adjacent sequences:  A273346 A273347 A273348 * A273350 A273351 A273352

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jun 03 2016

STATUS

approved

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Last modified September 19 04:30 EDT 2019. Contains 327187 sequences. (Running on oeis4.)