|
%I #3 Mar 30 2012 17:36:15
%S 1,1,1,1,3,2,1,7,10,6,1,16,37,42,24,1,44,125,214,216,120,1,169,465,
%T 959,1406,1320,720,1,895,2199,4469,7880,10476,9360,5040,1,5942,13504,
%U 24902,44203,70676,87732,75600,40320
%N Triangle read by rows: T(n,k) is the number of deco polyominoes of height n with k cells in the second row (0<=k<=n-1; 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). Sum(k*T(n,k),k=0..n-1)=A134437(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 generating polynomial of row n is P(n,t)=Q(n,t,1), where Q(1,t,x)=x and Q(n,t,x)=xQ(n-1,1,t)+x(1+(n-2)t)Q(n-1,t,x) for n>=2.
%e T(2,0)=T(2,1)=1 because the horizontal domino has no cells in the 2nd row and the vertical domino has 1 cell in the 2nd row.
%e Triangle starts:
%e 1;
%e 1,1;
%e 1,3,2;
%e 1,7,10,6;
%e 1,16,37,42,24;
%p Q[1]:=x: for n from 2 to 10 do Q[n]:=expand(x*subs({t=1,x=t}, Q[n-1])+x*Q[n-1]+(n-2)*t*x*Q[n-1]) end do: for n to 9 do P[n]:=sort(subs(x=1,Q[n])) end do: for n to 9 do seq(coeff(P[n],t,j),j=0..n-1) end do; # yields sequence in triangular form
%Y Cf. A000142, A134437.
%K nonn,tabl
%O 1,5
%A _Emeric Deutsch_, Nov 30 2007
|
