A121585 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 1-cell columns starting at level 0 (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, 3, 1, 1, 1, 12, 5, 3, 3, 1, 60, 27, 14, 12, 6, 1, 360, 168, 83, 62, 36, 10, 1, 2520, 1200, 583, 393, 242, 86, 15, 1, 20160, 9720, 4698, 2941, 1845, 758, 176, 21, 1, 181440, 88200, 42606, 25285, 15856, 7151, 1990, 323, 28, 1, 1814400, 887040

Offset

0,7

Comments

Row sums are the factorials (A000142). T(n,0)=n!/2=A001710(n) for n>=2 Sum(k*T(n,k), k=0..n)=n!=A000142(n) for n>=1.

References

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

Links

Table of n, a(n) for n=0..56.

Formula

The row generating polynomials satisfy P(n,t)=(n-1)!+(t+n-2)P(n-1,t) for n>=1 and P(0,t)=1.

Example

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 starting at level 0.
Triangle starts:
1;
0,1;
1,0,1;
3,1,1,1;
12,5,3,3,1;
60,27,14,12,6,1;

Maple

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

Crossrefs

Cf. A000142, A001710.

Sequence in context: A362783 A152795 A338817 * A261959 A348988 A257565

Adjacent sequences: A121582 A121583 A121584 * A121586 A121587 A121588

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 12 2006

Status

approved