A121697 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns ending at an odd level (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, 4, 8, 7, 3, 2, 14, 32, 37, 23, 10, 4, 44, 142, 207, 180, 97, 38, 12, 194, 730, 1267, 1327, 911, 425, 150, 36, 812, 3810, 8104, 10387, 8876, 5257, 2222, 708, 144, 4362, 23284, 56987, 84792, 85317, 60814, 31368, 11972, 3408, 576, 22716

Offset

0,7

Comments

Row sums are the factorials (A000142). T(n,0)=A121751 T(n,n)=A010551(n-1) for n>=1. Sum(k*T(n,k), k=0..n)=A121752(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=0..55.

Formula

The row generating polynomials P[n](t) are given by P[n](t)=Q[n](t,1), 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[0](t,s)=1, Q[1](t,s]=t.

Example

T(2,0)=1, T(2,1)=0 and T(2,2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having 0 and 2 columns ending at an odd level, respectively.
Triangle starts:
1;
0,1;
1,0,1;
2,2,1,1;
4,8,7,3,2;
14,32,37,23,10,4;

Maple

Q[0]:=1: 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 0 to 10 do P[n]:=sort(subs(s=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

Crossrefs

Cf. A000142, A010551, A121751, A121752, A121698.

Sequence in context: A199802 A297347 A342623 * A225201 A124976 A176663

Adjacent sequences: A121694 A121695 A121696 * A121698 A121699 A121700

Keyword

nonn,tabl

Author

Emeric Deutsch, Aug 23 2006

Status

approved