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Search: id:A121698
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| A121698 |
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Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns ending at an even level (1<=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. |
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+0
4
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| 1, 1, 1, 2, 2, 2, 6, 8, 7, 3, 16, 36, 37, 23, 8, 62, 172, 220, 166, 80, 20, 230, 844, 1383, 1338, 835, 338, 72, 1114, 4796, 9331, 10828, 8265, 4282, 1452, 252, 5268, 27450, 64612, 91023, 85248, 55445, 25158, 7524, 1152, 30702, 181606, 489847, 798355
(list; table; graph; listen)
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OFFSET
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1,4
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COMMENT
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Row sums are the factorials (A000142). T(n,0)=A121753 Sum(k*T(n,k), k=0..n-1)=A121754(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](s) are given by P[n](s)=Q[n](1,s), where Q[n](t,s) are defined by Q[n](t,s)=Q[n-1](s,t)+[floor(n/2)*t+floor((n-1)/2)*s]Q[n-1](t,s) for n>=2 and Q[1](t,s]=t.
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EXAMPLE
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T(2,0)=1 and T(2,1)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes, having 0 and 1 columns ending at an even level, respectively.
Triangle starts:
1;
1,1;
2,2,2;
6,8,7,3;
16,36,37,23,8;
62,172,220,166,80,20;
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MAPLE
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Q[1]:=t: for n from 2 to 10 do Q[n]:=expand(subs({t=s, s=t}, Q[n-1])+(t*floor(n/2)+s*floor((n-1)/2))*Q[n-1]) od: for n from 1 to 10 do P[n]:=sort(subs(t=1, Q[n])) od: for n from 0 to 10 do seq(coeff(P[n], s, j), j=0..n-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000142, A121753, A121754, A121697.
Sequence in context: A081478 A105341 A151694 this_sequence A087482 A137227 A052537
Adjacent sequences: A121695 A121696 A121697 this_sequence A121699 A121700 A121701
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 23 2006
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