A121698 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns ending at an even level (1<=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, 1, 1, 2, 2, 2, 6, 8, 7, 3, 16, 36, 37, 23, 8, 62, 172, 220, 166, 80, 20, 230, 844, 1383, 1338, 835, 338, 72, 1114, 4796, 9331, 10828, 8265, 4282, 1452, 252, 5268, 27450, 64612, 91023, 85248, 55445, 25158, 7524, 1152, 30702, 181606, 489847, 798355

Offset

1,4

Comments

Row sums are the factorials (A000142). T(n,0)=A121753 Sum(k*T(n,k), k=0..n-1)=A121754(n).

References

E. Barcucci, S. Brunetti and F. Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.

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=1..49.

Formula

The row generating polynomials P[n](s) are given by P[n](s)=Q[n](1,s), where Q[n](t,s) are defined by Q[n](t,s)=Q[n-1](s,t)+[floor(n/2)*t+floor((n-1)/2)*s]Q[n-1](t,s) for n>=2 and Q[1](t,s]=t.

Example

T(2,0)=1 and T(2,1)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having 0 and 1 columns ending at an even level, respectively.
Triangle starts:
1;
1,1;
2,2,2;
6,8,7,3;
16,36,37,23,8;
62,172,220,166,80,20;

Maple

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

Crossrefs

Cf. A000142, A121753, A121754, A121697.

Sequence in context: A361424 A298745 A323860 * A087482 A137227 A323862

Adjacent sequences: A121695 A121696 A121697 * A121699 A121700 A121701

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 23 2006

Status

approved