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A121585 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 1-cell columns starting at level 0 (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.

%I A121585 #3 Mar 30 2012 17:36:10
%S A121585 1,0,1,1,0,1,3,1,1,1,12,5,3,3,1,60,27,14,12,6,1,360,168,83,62,36,10,1,
%T A121585 2520,1200,583,393,242,86,15,1,20160,9720,4698,2941,1845,758,176,21,1,
%U A121585 181440,88200,42606,25285,15856,7151,1990,323,28,1,1814400,887040
%N A121585 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k 1-cell columns starting at level 0 (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 A121585 Row sums are the factorials (A000142). T(n,0)=n!/2=A001710(n) for n>=2 Sum(k*T(n,k), k=0..n)=n!=A000142(n) for n>=1.
%D A121585 E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
%F A121585 The row generating polynomials satisfy P(n,t)=(n-1)!+(t+n-2)P(n-1,t) for n>=1 and P(0,t)=1.
%e A121585 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 starting at level 0.
%e A121585 Triangle starts:
%e A121585 1;
%e A121585 0,1;
%e A121585 1,0,1;
%e A121585 3,1,1,1;
%e A121585 12,5,3,3,1;
%e A121585 60,27,14,12,6,1;
%p A121585 P[0]:=1: for n from 1 to 10 do P[n]:=sort(expand((n-1)!+(t+n-2)*P[n-1])) od: for n from 0 to 10 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
%Y A121585 Cf. A000142, A001710.
%K A121585 nonn,tabl
%O A121585 0,7
%A A121585 _Emeric Deutsch_, Aug 12 2006