OFFSET
1,2
COMMENTS
A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.
Row n contains ceiling(n/2) terms.
Row sums are the factorials (A000142).
T(n,0) = A000522(n).
T(2n+1,n) = (2n-1)!! = A001147(n) (the double factorials).
Sum_{k=0..n} k*T(n,k) = A002538(n-2) for n >= 3.
LINKS
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
FORMULA
The row generating polynomials are P(n,t) = Q(n,t,1), where Q(1,t,x) = 1 and Q(n,t,x) = Q(n-1,t,t) + (n-1)xQ(n-1,t,1) for n >= 2.
EXAMPLE
T(2,0)=2 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having no reentrant corners along the lower contour.
Triangle starts:
1;
2;
5, 1;
16, 8;
65, 52, 3;
326, 344, 50;
MAPLE
Q[1]:=1: for n from 2 to 13 do Q[n]:=sort(expand(subs(x=t, Q[n-1])+(n-1)*x*subs(x=1, Q[n-1]))) od: for n from 1 to 13 do P[n]:=subs(x=1, Q[n]) od: for n from 1 to 13 do seq(coeff(P[n], t, j), j=0..ceil(n/2)-1) od; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Aug 08 2006
STATUS
approved