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A100822 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n with k cells in the first column. (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 A100822 #5 Mar 30 2012 17:36:00
%S A100822 1,1,1,2,3,1,6,8,9,1,24,30,32,33,1,120,144,150,152,153,1,720,840,864,
%T A100822 870,872,873,1,5040,5760,5880,5904,5910,5912,5913,1,40320,45360,46080,
%U A100822 46200,46224,46230,46232,46233,1,362880,403200,408240,408960,409080,409104,409110,409112,409113,1
%N A100822 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n with k cells in the first column. (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 A100822 Row n has n terms. Rows are circular permutations of the rows of A054115. Column 1 and row sums yield A000142 (the factorial numbers). Column 2 yields A059171.
%C A100822 T(n+1,n)=A007489(n).
%D A100822 E. Barcucci, A. del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
%F A100822 T(n, k)=sum((n-j)!, j=1..k) for 1<=k<n; T(n, n)=1.
%F A100822 T(n,k)=T(n-1,k-1)+(n-1)! for k<n; T(n,n)=1.
%e A100822 Triangle begins:
%e A100822 1;
%e A100822 1,1;
%e A100822 2,3,1;
%e A100822 6,8,9,1;
%e A100822 24,30,32,33,1;
%e A100822 T(2,1)=T(2,2)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having, respectively, 1 and 2 cells in their first columns.
%p A100822 T:=proc(n,k) if k=n then 1 elif k<n then sum((n-j)!,j=1..k) else 0 fi end: for n from 1 to 10 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%Y A100822 Cf. A000142, A054115, A059171, A007489.
%K A100822 nonn,tabl
%O A100822 1,4
%A A100822 _Emeric Deutsch_, Jan 06 2005, Aug 09 2006