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Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having k cells in the second column (n>=1, k>=0). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.
2

%I #5 Apr 09 2013 09:38:12

%S 1,1,1,1,3,2,1,9,11,3,1,33,43,39,4,1,153,193,199,169,5,1,873,1057,

%T 1099,1081,923,6,1,5913,6937,7147,7171,7027,6117,7,1,46233,53017,

%U 54187,54403,54307,53413,47311,8,1,409113,461257,468907,470203,470323,469483,463399

%N Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having k cells in the second column (n>=1, k>=0). 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)=1; Sum(k*T(n,k), k=0..n)=A121582

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

%F The generating polynomial of row n is P(n,s)=Q(n,1,s), where Q(1,t,s)=t and Q(n,t,s)=tQ(n-1,t,s)+(t^n-t)Q(n-1,s,1)/(t-1) for n>=2.

%e T(2,0)=1 and T(2,1)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 1 cells in their second columns.

%e Triangle starts:

%e 1;

%e 1,1;

%e 1,3,2;

%e 1,9,11,3;

%e 1,33,43,39,4;

%p Q[1]:=t: for n from 2 to 11 do Q[n]:=expand(simplify(t*Q[n-1]+(t^n-t)/(t-1)*subs({t=s,s=1},Q[n-1]))): P[1]:=1: P[n]:=subs(t=1,Q[n]): od: for n from 1 to 11 do seq(coeff(P[n],s,j),j=0..n-1) od; # yields sequence in triangular form

%Y Cf. A000142, A121582, A100822, A121583.

%K nonn,tabl

%O 1,5

%A _Emeric Deutsch_, Aug 11 2006

%E Keyword tabf changed to tabl by _Michel Marcus_, Apr 09 2013