A273899 Triangle read by rows: for n>=1, 0<=k<=n-1, T(n,k) is the number of bargraphs of semiperimeter n having k columns of length 1.

0, 0, 1, 1, 0, 1, 2, 2, 0, 1, 5, 4, 3, 0, 1, 13, 11, 6, 4, 0, 1, 35, 30, 18, 8, 5, 0, 1, 97, 84, 51, 26, 10, 6, 0, 1, 275, 240, 148, 76, 35, 12, 7, 0, 1, 794, 697, 435, 228, 105, 45, 14, 8, 0, 1, 2327, 2052, 1293, 688, 325, 138, 56, 16, 9, 0, 1, 6905, 6111, 3881, 2091, 1005, 440, 175, 68, 18, 10, 0, 1

Offset

1,7

Comments

Number of entries in row n is n.

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

T(n,0) = A082582(n-1).

Sum(k*T(n,k), k>=0) = A273900(n).

Links

Alois P. Heinz, Rows n = 1..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(t,z) = (1/2)(1-2z-z^2+2tz^2-2t^2z^2-2tz^3+2t^2z^3 - sqrt(1-4z+2z^2+z^4))/(1-t+t^2z+tz^2-t^2z^2), where z marks semiperimeter and t marks columns of length 1.

Example

Row 4 is 2,2,0,1 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 pictures give the values 3,1,1,0,0 for the number of columns of length 1.
Triangle starts
0;
0,1;
1,0,1;
2,2,0,1;
5,4,3,0,1;
13,11,6,4,0,1;
35,30,18,8,5,0,1.

Maple

G:=((1-2*z-z^2+2*t*z^2-2*t^2*z^2-2*t*z^3+2*t^2*z^3-sqrt(1-4*z+2*z^2+z^4))*(1/2))/(1-t+t^2*z+t*z^2-t^2*z^2): Gser:= simplify(series(G, z=0, 22)): for n from 1 to 18 do P[n]:= sort(coeff(Gser, z, n)) end do: for n from 1 to 18 do seq(coeff(P[n], t, j), j=0..n-1)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, `if`(y=1, z, 1)*b(n-1, y, 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n-1))(b(n, 0$2)):
seq(T(n), n=1..18);  # Alois P. Heinz, Jun 07 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, If[y==1, z, 1]*b[n-1, y, 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, n-1}]][ b[n, 0, 0]]; Table[T[n], {n, 1, 18}] // Flatten (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)

Crossrefs

Cf. A082582, A273900.

Sequence in context: A267617 A158852 A188285 * A102404 A089246 A291684

Adjacent sequences: A273896 A273897 A273898 * A273900 A273901 A273902

Keyword

nonn,tabl

Author

Emeric Deutsch, Jun 07 2016

Status

approved