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A274486 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k horizontal segments (n>=2, k>=1). A horizontal segment is a maximal sequence of adjacent horizontal steps (1,0). 1
1, 2, 3, 2, 4, 8, 1, 5, 20, 10, 6, 40, 45, 6, 7, 70, 140, 56, 2, 8, 112, 350, 280, 44, 9, 168, 756, 1008, 366, 20, 10, 240, 1470, 2940, 1920, 320, 5, 11, 330, 2640, 7392, 7590, 2552, 190, 12, 440, 4455, 16632, 24684, 13904, 2445, 70 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

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

Sum(k*T(n,k), k>=0) = A273345(n+1).

LINKS

Alois P. Heinz, Rows n = 2..200, 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

Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

FORMULA

G.f.:  G = (1-2z+z^2-2tz^2-sqrt((1-z)((1-z)^3-4tz^2*(1-z+tz))))/(2tz).

EXAMPLE

Row 4 is 3,2 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and, clearly, they have 1,2,2,1,1 horizontal segments.

Triangle starts

1;

2;

3,2;

4,8,1;

5,20,10;

6,40,45,6.

MAPLE

G := ((1-2*z+z^2-2*t*z^2-sqrt((1-z)*((1-z)^3-4*t*z^2*(1-z+t*z))))*(1/2))/(t*z): Gser := simplify(series(G, z = 0, 23)): for n from 2 to 18 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 18 do seq(coeff(P[n], t, j), j = 1 .. degree(P[n])) 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>0 or y<2, 0, b(n, y-1, -1))+

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

    end:

T:= n-> (p-> seq(coeff(p, z, i), i=1..degree(p)))(b(n, 0$2)):

seq(T(n), n=2..20);  # Alois P. Heinz, Jun 27 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 > 0 || y < 2, 0, b[n, y - 1, -1]] + If[y < 1, 0, b[n - 1, y, 0]*If[t == 0, 1, z]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 1, Exponent[p, z]}]][b[n, 0, 0]]; Table[T[n], {n, 2, 20}] // Flatten (* Jean-François Alcover, Dec 02 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A082582, A273345.

Sequence in context: A303845 A132439 A116217 * A227961 A108838 A318176

Adjacent sequences:  A274483 A274484 A274485 * A274487 A274488 A274489

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch and Sergi Elizalde, Jun 27 2016

STATUS

approved

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Last modified January 28 13:29 EST 2020. Contains 331321 sequences. (Running on oeis4.)