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Search: id:A121697
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| A121697 |
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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 odd level (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. |
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+0
4
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| 1, 0, 1, 1, 0, 1, 2, 2, 1, 1, 4, 8, 7, 3, 2, 14, 32, 37, 23, 10, 4, 44, 142, 207, 180, 97, 38, 12, 194, 730, 1267, 1327, 911, 425, 150, 36, 812, 3810, 8104, 10387, 8876, 5257, 2222, 708, 144, 4362, 23284, 56987, 84792, 85317, 60814, 31368, 11972, 3408, 576, 22716
(list; table; graph; listen)
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OFFSET
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0,7
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COMMENT
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Row sums are the factorials (A000142). T(n,0)=A121751 T(n,n)=A010551(n-1) for n>=1. Sum(k*T(n,k), k=0..n)=A121752(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[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[0](t,s)=1, Q[1](t,s]=t.
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EXAMPLE
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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 ending at an odd level, respectively.
Triangle starts:
1;
0,1;
1,0,1;
2,2,1,1;
4,8,7,3,2;
14,32,37,23,10,4;
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MAPLE
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Q[0]:=1: 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 0 to 10 do P[n]:=sort(subs(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
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CROSSREFS
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Cf. A000142, A010551, A121751, A121752, A121698.
Sequence in context: A108017 A133135 A132311 this_sequence A124976 A113021 A152937
Adjacent sequences: A121694 A121695 A121696 this_sequence A121698 A121699 A121700
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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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