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A121579
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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). A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Row n contains ceil(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*T(n,k), k=0..n)=A002538(n-2) for n>=3.
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REFERENCES
| E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
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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.
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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;
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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
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CROSSREFS
| Cf. A000142, A000522, A001147, A002538.
Sequence in context: A121632 A186361 A197365 * A106852 A162975 A187244
Adjacent sequences: A121576 A121577 A121578 * A121580 A121581 A121582
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 08 2006
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