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%I #3 Mar 30 2012 17:36:11
%S 1,1,1,2,3,1,6,11,6,1,16,44,42,16,2,66,209,254,147,40,4,246,1005,1647,
%T 1377,615,138,12,1248,5792,11246,11799,7192,2533,474,36,5976,33164,
%U 78760,104276,83932,41912,12628,2088,144,36120,223676,605854,940399
%N Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns of even length (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.
%C Row sums are the factorials (A000142). T(n,0)=A121749 Sum(k*T(n,k), k=0..n)=A121750(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,1,s), where Q[n](t,s,x,y) are defined by Q[n](t,s,x,y)=Q[n-1](t,s,y,x)+[floor(n/2)*x+floor((n-1)/2)*y]Q[n-1](t,s,t,s) for n>=2 and Q[1](t,s,x,y]=x.
%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 of even length, respectively.
%e Triangle starts:
%e 1;
%e 1,1;
%e 2,3,1;
%e 6,11,6,1;
%e 16,44,42,16,2;
%e 66,209,254,147,40,4;
%p Q[1]:=x: for n from 2 to 11 do Q[n]:=expand(subs({x=y,y=x},Q[n-1])+(floor(n/2)*x+floor((n-1)/2)*y)*subs({x=t,y=s},Q[n-1])) od: for n from 1 to 11 do P[n]:=sort(subs({y=s,x=1,t=1},Q[n])) od: for n from 0 to 11 do seq(coeff(P[n],s,j),j=0..n-1) od; # yields sequence in triangular form
%Y Cf. A000142, A121745, A121749, A121750.
%K nonn,tabl
%O 1,4
%A _Emeric Deutsch_, Aug 20 2006
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