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%I #2 Mar 30 2012 17:36:10
%S 1,0,1,1,0,1,2,2,1,1,7,7,6,3,1,30,35,30,18,6,1,157,205,184,117,46,10,
%T 1,972,1392,1304,874,381,101,15,1,6961,10764,10499,7355,3470,1052,197,
%U 21,1,56660,93493,94668,68909,34622,11606,2542,351,28,1,516901,901900
%N Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 1-cell columns (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)=A001053(n). Sum(k*T(n,k), k=0..n)=A121555(n).
%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 are P(n,t)=Q(n,t,1), where Q(0,t,x)=1 and Q(n,t,x)=Q(n-1,t,1/t)+(tx+n-2)Q(n-1,t,1) for n>=1.
%e T(2,0)=1, T(2,1)=0, T(2,2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 2 columns with exactly 1 cell.
%e Triangle starts:
%e 1;
%e 0,1;
%e 1,0,1;
%e 2,2,1,1;
%e 7,7,6,3,1;
%e 30,35,30,18,6,1;
%p Q[0]:=1: for n from 1 to 10 do Q[n]:=sort(expand(subs(x=1/t,Q[n-1])+(t*x+n-2)*subs(x=1,Q[n-1]))) od: for n from 0 to 10 do P[n]:=subs(x=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
%Y Cf. A000142, A001053, A121555.
%K nonn,tabl
%O 0,7
%A _Emeric Deutsch_, Aug 08 2006
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