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A101409
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Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges in which the leftmost leaf is at level k.
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0
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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, 246675, 366850, 323587, 229936, 140161, 74050, 33360, 12256, 3328, 512
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| T(n,k) is also the number of diagonally convex directed polyominoes with n diagonals and having k diagonals of length 1. Proof: the two triangles have the same g.f.
Row n has n terms. Column 1 and row sums yield the ternary numbers (A001764).
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REFERENCES
| 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
| T(n, k)=sum([(k+i)/(2*n-k+i)] binomial(k-1, i) binomial(3n-2k+i-1, n-k), i=0..k-1).
G.f.=(1-tzg^2)/(1-tzg-tzg^2), where g=1+zg^3 is the g.f. of the ternary numbers (A001764). (An explicit expression for g is given in the Maple program).
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EXAMPLE
| T(2,1)=1 and T(2,2)=2 because the noncrossing trees with 2 edges are /\, /_ and _\.
Or, 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
| 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;
T:=proc(n, k) if k=1 then binomial(3*n-3, n-1)/(2*n-1) elif k<=n then sum(((k+i)/(2*n-k+i))*binomial(k-1, i)*binomial(3*n-2*k+i-1, n-k), i=0..k-1) else 0 fi end: for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A001764.
Sequence in context: A138153 A023395 A193798 * A131401 A061446 A107476
Adjacent sequences: A101406 A101407 A101408 * A101410 A101411 A101412
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 15 2005 and Jan 17 2005
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Jan 25 2009 at the suggestion of R. J. Mathar and Max Alekseyev.
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