A121554 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 1-cell columns (0<=k<=n). 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, 1, 1, 0, 1, 2, 2, 1, 1, 7, 7, 6, 3, 1, 30, 35, 30, 18, 6, 1, 157, 205, 184, 117, 46, 10, 1, 972, 1392, 1304, 874, 381, 101, 15, 1, 6961, 10764, 10499, 7355, 3470, 1052, 197, 21, 1, 56660, 93493, 94668, 68909, 34622, 11606, 2542, 351, 28, 1, 516901, 901900

Offset: 0

Author

Emeric Deutsch, Aug 08 2006

Keywords

• nonn
• tabl

Comments

Row sums are the factorials (A000142). T(n,0)=A001053(n). Sum(k*T(n,k), k=0..n)=A121555(n).

Examples

			T(2,0)=1, T(2,1)=0, T(2,2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 2 columns with exactly 1 cell.
Triangle starts:
1;
0,1;
1,0,1;
2,2,1,1;
7,7,6,3,1;
30,35,30,18,6,1;
		

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, A001053, A121555.

Programs

Maple

Q[0]:=1: for n from 1 to 10 do Q[n]:=sort(expand(subs(x=1/t,Q[n-1])+(t*x+n-2)*subs(x=1,Q[n-1]))) od: for n from 0 to 10 do P[n]:=subs(x=1,Q[n]) od: for n from 0 to 10 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

Formula

The row generating polynomials are P(n,t)=Q(n,t,1), where Q(0,t,x)=1 and Q(n,t,x)=Q(n-1,t,1/t)+(tx+n-2)Q(n-1,t,1) for n>=1.