A121583 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having k cells in the first two columns (n>=1, k>=1). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

1, 0, 2, 0, 1, 4, 1, 0, 2, 6, 10, 5, 1, 0, 6, 16, 29, 34, 23, 11, 1, 0, 24, 60, 102, 148, 154, 119, 77, 35, 1, 0, 120, 288, 474, 668, 867, 874, 719, 533, 341, 155, 1, 0, 720, 1680, 2712, 3768, 4834, 5906, 5914, 5039, 4013, 2957, 1901, 875, 1, 0, 5040, 11520, 18360

Offset: 1

Author

Emeric Deutsch, Aug 11 2006

Keywords

• nonn
• tabf

Comments

Row n has 2n-2 terms (n>=2). Row sums are the factorials (A000142). Sum(k*T(n,k), k=0..n)=A121584(n)

Examples

			T(2,2)=2 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, each having 2 cells in their first two columns.
Triangle starts:
1;
0,2;
0,1,4,1;
0,2,6,10,5,1;
0,6,16,29,34,23,11,1;
		

References

• E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

Crossrefs

Cf. A000142, A100822, A121581, A121584.

Programs

Maple

Q[1]:=t: for n from 2 to 9 do Q[n]:=expand(simplify(t*Q[n-1]+(t^n-t)/(t-1)*subs({t=s,s=1},Q[n-1]))) od: for n from 1 to 9 do P[n]:=sort(subs(s=t,Q[n])): od: 1; for n from 1 to 9 do seq(coeff(P[n],t,j),j=1..2*n-2) od; # yields sequence in triangular form

Formula

The generating polynomial of row n is P(n,t)=Q(n,t,t), where Q(1,t,s)=t and Q(n,t,s)=tQ(n-1,t,s)+(t^n-t)Q(n-1,s,1)/(t-1) for n>=2.