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Search: id:A121583
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| A121583 |
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Triangle read by rows: T(n,k) is the number of deco polyominoes of height n having k cells in the first two columns (n>=1, k>=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. |
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+0
3
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| 1, 0, 2, 0, 1, 4, 1, 0, 2, 6, 10, 5, 1, 0, 6, 16, 29, 34, 23, 11, 1, 0, 24, 60, 102, 148, 154, 119, 77, 35, 1, 0, 120, 288, 474, 668, 867, 874, 719, 533, 341, 155, 1, 0, 720, 1680, 2712, 3768, 4834, 5906, 5914, 5039, 4013, 2957, 1901, 875, 1, 0, 5040, 11520, 18360
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OFFSET
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1,3
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COMMENT
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Row n has 2n-2 terms (n>=2). Row sums are the factorials (A000142). Sum(k*T(n,k), k=0..n)=A121584(n)
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REFERENCES
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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
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The generating polynomial of row n is P(n,t)=Q(n,t,t), where Q(1,t,s)=t and Q(n,t,s)=tQ(n-1,t,s)+(t^n-t)Q(n-1,s,1)/(t-1) for n>=2.
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EXAMPLE
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T(2,2)=2 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, each having 2 cells in their first two columns.
Triangle starts:
1;
0,2;
0,1,4,1;
0,2,6,10,5,1;
0,6,16,29,34,23,11,1;
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MAPLE
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Q[1]:=t: for n from 2 to 9 do Q[n]:=expand(simplify(t*Q[n-1]+(t^n-t)/(t-1)*subs({t=s, s=1}, Q[n-1]))) od: for n from 1 to 9 do P[n]:=sort(subs(s=t, Q[n])): od: 1; for n from 1 to 9 do seq(coeff(P[n], t, j), j=1..2*n-2) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000142, A100822, A121581, A121584.
Sequence in context: A081265 A108643 A133838 this_sequence A124915 A158239 A159819
Adjacent sequences: A121580 A121581 A121582 this_sequence A121584 A121585 A121586
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 11 2006
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