A121581 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having k cells in the second column (n>=1, k>=0). 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, 1, 1, 1, 3, 2, 1, 9, 11, 3, 1, 33, 43, 39, 4, 1, 153, 193, 199, 169, 5, 1, 873, 1057, 1099, 1081, 923, 6, 1, 5913, 6937, 7147, 7171, 7027, 6117, 7, 1, 46233, 53017, 54187, 54403, 54307, 53413, 47311, 8, 1, 409113, 461257, 468907, 470203, 470323, 469483, 463399

Offset: 1

Author

Emeric Deutsch, Aug 11 2006

Keywords

• nonn
• tabl

Comments

Row sums are the factorials (A000142). T(n,0)=1; Sum(k*T(n,k), k=0..n)=A121582

Examples

			T(2,0)=1 and T(2,1)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 1 cells in their second columns.
Triangle starts:
1;
1,1;
1,3,2;
1,9,11,3;
1,33,43,39,4;
		

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, A121582, A100822, A121583.

Programs

Maple

Q[1]:=t: for n from 2 to 11 do Q[n]:=expand(simplify(t*Q[n-1]+(t^n-t)/(t-1)*subs({t=s,s=1},Q[n-1]))): P[1]:=1: P[n]:=subs(t=1,Q[n]): od: for n from 1 to 11 do seq(coeff(P[n],s,j),j=0..n-1) od; # yields sequence in triangular form

Formula

The generating polynomial of row n is P(n,s)=Q(n,1,s), 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.

Extensions

Keyword tabf changed to tabl by Michel Marcus, Apr 09 2013