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A121745 Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns of odd length (0<=k<=n). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. 3
1, 0, 1, 1, 0, 1, 1, 3, 1, 1, 3, 6, 9, 4, 2, 9, 28, 36, 30, 13, 4, 27, 107, 195, 198, 130, 51, 12, 117, 532, 1106, 1371, 1096, 585, 197, 36, 459, 2573, 6547, 9914, 9886, 6733, 3132, 932, 144, 2421, 15139, 43606, 76017, 88793, 72689, 42200, 17027, 4412, 576, 11979 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Row sums are the factorials (A000142). T(n,0)=A121746 Sum(k*T(n,k), k=0..n)=A121747(n).

REFERENCES

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

LINKS

Table of n, a(n) for n=0..55.

FORMULA

The row generating polynomials P[n](t) are given by P[n](t)=Q[n](t,1,t,1), where Q[n](t,s,x,y) are defined by Q[n](t,s,x,y)=Q[n-1](t,s,y,x)+[floor(n/2)*x+floor((n-1)/2)*y]Q[n-1](t,s,t,s) for n>=2 and Q[0](t,s,x,y)=1, Q[1](t,s,x,y]=x.

EXAMPLE

T(2,0)=1, T(2,1)=0 and T(2,2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having 0 and 2 columns of odd length, respectively.

Triangle starts:

1;

0,1;

1,0,1;

1,3,1,1;

3,6,9,4,2;

9,28,36,30,13,4;

MAPLE

Q[0]:=1: Q[1]:=x: for n from 2 to 10 do Q[n]:=expand(subs({x=y, y=x}, Q[n-1])+(floor(n/2)*x+floor((n-1)/2)*y)*subs({x=t, y=s}, Q[n-1])) od: for n from 0 to 10 do P[n]:=sort(subs({x=t, y=1, s=1}, Q[n])) od: for n from 0 to 10 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form

CROSSREFS

Cf. A121746, A121747, A121748.

Sequence in context: A156710 A114588 A253223 * A252983 A089312 A246674

Adjacent sequences:  A121742 A121743 A121744 * A121746 A121747 A121748

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Aug 20 2006

STATUS

approved

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Last modified May 8 15:54 EDT 2021. Contains 343666 sequences. (Running on oeis4.)