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A102892
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Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges in which the number of edges from the root to the first branch point is k.
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2
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1, 0, 1, 1, 0, 2, 5, 3, 0, 4, 25, 16, 6, 0, 8, 130, 83, 32, 12, 0, 16, 700, 442, 166, 64, 24, 0, 32, 3876, 2420, 884, 332, 128, 48, 0, 64, 21945, 13566, 4840, 1768, 664, 256, 96, 0, 128, 126500, 77539, 27132, 9680, 3536, 1328, 512, 192, 0, 256
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| The statistic "number of edges from the root to the first branchpoint" is equal to 0 if root is a branchpoint and it is equal to the total number of edges if there is no branchpoint. Row n contains n+1 terms. Row sums yield the ternary numbers (A001764). Column 0 yields A102893. Column 1 yields A030983.
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REFERENCES
| M. Noy, Enumeration of noncrossing trees on a circle, Discr. Math. 180 (1998), 301-313.
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FORMULA
| T(n, 0)=5binomial(3n-1, n-2)/(3n-1) for n>0; T(n, k)=[2^(k-1)/(n-k+1)]binomial(3n-3k+1, n-k)-[2^k/(n-k)]binomial(3n-3k-2, n-k-1) for 0<k<n; T(n, n)=2^(n-1) (n>0). G.f.=(1/2)g(2-g)+g^2*(1-2z)/[2(1-2tz)], where g=1+zg^3 is the g.f. of the ternary numbers (A001764).
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EXAMPLE
| T(2,0)=1 because we have /\ and T(2,2)=2 because we have /_ and _\.
Triangle starts:
1;
0,1;
1,0,2;
5,3,0,4;
25,16,6,0,8;
130,83,32,12,0,16;
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MAPLE
| T:=proc(n, k) if n=0 and k=0 then 1 elif k=0 then 5*binomial(3*n-1, n-2)/(3*n-1) elif k<n then (2^(k-1)/(n-k+1))*binomial(3*n-3*k+1, n-k)-(2^k/(n-k))*binomial(3*n-3*k-2, n-k-1) elif k=n then 2^(n-1) else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A001764, A102893, A030983.
Sequence in context: A160127 A130280 A011035 * A132898 A062706 A059217
Adjacent sequences: A102889 A102890 A102891 * A102893 A102894 A102895
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 16 2005
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