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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.
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%I #2 Mar 30 2012 17:36:11

%S 1,1,1,2,2,2,6,8,7,3,16,36,37,23,8,62,172,220,166,80,20,230,844,1383,

%T 1338,835,338,72,1114,4796,9331,10828,8265,4282,1452,252,5268,27450,

%U 64612,91023,85248,55445,25158,7524,1152,30702,181606,489847,798355

%N 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.

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

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

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

%F 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.

%e 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.

%e Triangle starts:

%e 1;

%e 1,1;

%e 2,2,2;

%e 6,8,7,3;

%e 16,36,37,23,8;

%e 62,172,220,166,80,20;

%p 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

%Y Cf. A000142, A121753, A121754, A121697.

%K nonn,tabl

%O 1,4

%A _Emeric Deutsch_, Aug 23 2006