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Search: id:A121748
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| A121748 |
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Triangle read by rows: T(n,k) is the number of deco polyominoes of height n and having k columns of even length (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, 1, 1, 2, 3, 1, 6, 11, 6, 1, 16, 44, 42, 16, 2, 66, 209, 254, 147, 40, 4, 246, 1005, 1647, 1377, 615, 138, 12, 1248, 5792, 11246, 11799, 7192, 2533, 474, 36, 5976, 33164, 78760, 104276, 83932, 41912, 12628, 2088, 144, 36120, 223676, 605854, 940399
(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)=A121749 Sum(k*T(n,k), k=0..n)=A121750(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,1,s), where Q[n](t,s,x,y) are defined by Q[n](t,s,x,y)=Q[n-1](t,s,y,x)+[floor(n/2)*x+floor((n-1)/2)*y]Q[n-1](t,s,t,s) for n>=2 and Q[1](t,s,x,y]=x.
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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 of even length, respectively.
Triangle starts:
1;
1,1;
2,3,1;
6,11,6,1;
16,44,42,16,2;
66,209,254,147,40,4;
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MAPLE
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Q[1]:=x: for n from 2 to 11 do Q[n]:=expand(subs({x=y, y=x}, Q[n-1])+(floor(n/2)*x+floor((n-1)/2)*y)*subs({x=t, y=s}, Q[n-1])) od: for n from 1 to 11 do P[n]:=sort(subs({y=s, x=1, t=1}, Q[n])) od: for n from 0 to 11 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, A121745, A121749, A121750.
Adjacent sequences: A121745 A121746 A121747 this_sequence A121749 A121750 A121751
Sequence in context: A103136 A086960 A138771 this_sequence A008275 A130534 A107416
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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 20 2006
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