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Search: id:A122104
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| A122104 |
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Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and such that the sum of the bottom levels of all columns is k (n>=1, k>=0; informally, the number of the "missing" cells in the right bottom corner of the polyomino). 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
2
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| 1, 2, 5, 1, 16, 5, 3, 65, 23, 20, 10, 2, 326, 119, 115, 84, 57, 11, 8, 1957, 719, 714, 582, 526, 310, 137, 55, 34, 6, 13700, 5039, 5033, 4222, 4173, 3291, 2506, 972, 748, 348, 220, 38, 30, 109601, 40319, 40312, 34026, 34454, 29792, 28055, 18723, 10613, 6745
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Row n has 1+floor((n-1)^2/4) terms. Row sums are the factorials (A000142). T(n,0)=A000522(n-1). T(n,1)=(n-1)!-1=A033312(n-1). T(n,2)=(n-1)!-n+1=A005096(n-1) for n>=2. Sum(k*T(n,k), k>=0)=A122105(n).
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REFERENCES
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E. Barcucci, S. Brunetti and F. Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.
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 row generating polynomials P[n](t) are given by P[n](t)=Q[n](t,1), where Q[1](t,x)=x and Q[n](t,x) = (1/t)Q[n-1](t,tx)+(n-1)xQ[n-1](t,x) for n>=2.
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EXAMPLE
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Triangle starts:
1;
2;
5,1;
16,5,3;
65,23,20,10,2;
326,119,115,84,57,11,8;
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MAPLE
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Q[1]:=x: for n from 2 to 10 do Q[n]:=simplify(subs(x=t*x, Q[n-1])/t+(n-1)*x*Q[n-1]) od: for n from 1 to 10 do P[n]:=sort(subs(x=1, Q[n])) od: for n from 1 to 10 do seq(coeff(P[n], t, j), j=0..floor((n-1)^2/4)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000142, A000522, A033312, A005096, A122105.
Sequence in context: A110220 A119518 A111797 this_sequence A104546 A121632 A121579
Adjacent sequences: A122101 A122102 A122103 this_sequence A122105 A122106 A122107
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 24 2006
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