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Search: id:A101428
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| A101428 |
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Triangle read by rows: T(n,k) is the number of diagonally convex directed polyominoes with n diagonals and having k diagonals of length 1. |
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1
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| 1, 1, 2, 3, 5, 4, 12, 19, 16, 8, 55, 85, 73, 44, 16, 273, 416, 361, 234, 112, 32, 1428, 2156, 1883, 1269, 680, 272, 64, 7752, 11628, 10200, 7043, 4016, 1856, 640, 128, 43263, 64581, 56829, 39897, 23665, 11864, 4848, 1472, 256
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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Row n has n terms. Column 1 and row sums yield the ternary numbers (A001764).
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REFERENCES
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M. Bousqet-Melou, Percolation models and animals, Europ. J. Combinatorics, 17, 1996, 343-369 (Prop. 2.4).
E. Deutsch, S. Feretic, and M. Noy, Diagonally convex directed polyominoes and even trees: a bijection and related issues, Discrete Math., 256, 2002, 645-654.
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FORMULA
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G.f.=(1-tzg^2)/(1-tzg-tzg^2), where g=1+zg^3 is the g.f. of the ternary numbers (A001764). (The explicit expression of g is given at the Maple program).
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EXAMPLE
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T(2,2)=2 because the horizontal domino and the vertical domino have 2 diagonals of length 1 each.
Triangle begins:
1;
1,2;
3,5,4;
12,19,16,8;
55,85,73,44,16;
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MAPLE
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G:=t*z*g/(1-t*z*g-t*z*g^2):g:=2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z):Gser:=simplify(series(G, z=0, 12)):Gser:=simplify(series(G, z=0, 14)):for n from 1 to 10 do P[n]:=sort(coeff(Gser, z^n)) od:for n from 1 to 10 do seq(coeff(P[n], t^k), k=1..n) od;
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CROSSREFS
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Cf. A001764.
Adjacent sequences: A101425 A101426 A101427 this_sequence A101429 A101430 A101431
Sequence in context: A084933 A138153 A023395 this_sequence A101409 A131401 A061446
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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), Jan 17 2005
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