

A121579


Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having along the lower contour exactly k reentrant corners, i.e., a vertical step that is followed by a horizontal step (n>=1, k>=0).


2



1, 2, 5, 1, 16, 8, 65, 52, 3, 326, 344, 50, 1957, 2473, 595, 15, 13700, 19676, 6524, 420, 109601, 173472, 71862, 7840, 105, 986410, 1686912, 823836, 127232, 4410, 9864101, 17981193, 9976686, 1975750, 118125, 945, 108505112, 208769296, 128350992
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OFFSET

1,2


COMMENTS

A deco polyomino is a directed columnconvex 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) = (2n1)!! = A001147(n) (the double factorials).
Sum_{k=0..n} k*T(n,k) = A002538(n2) for n >= 3.


LINKS

Table of n, a(n) for n=1..39.
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 2942.


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(n1,t,t) + (n1)xQ(n1,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[n1])+(n1)*x*subs(x=1, Q[n1]))) 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

Cf. A000142, A000522, A001147, A002538.
Sequence in context: A121632 A186361 A197365 * A214733 A106852 A162975
Adjacent sequences: A121576 A121577 A121578 * A121580 A121581 A121582


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Aug 08 2006


STATUS

approved



