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

Author

Emeric Deutsch, Aug 23 2006

Keywords

• nonn
• tabl

Comments

Row sums are the factorials (A000142).

T(n,0) = A121753(n).

Sum_{k=0..n-1} k*T(n,k) = A121754(n).

Examples

			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;
		

Links

• Elena Barcucci, Sara Brunetti, and Francesco Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.
• Elena Barcucci, Alberto Del Lungo, and Renzo Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

Crossrefs

Cf. A000142, A121753, A121754, A121697.

Programs

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

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.