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A264869
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Triangular array: For n >= 2 and 0 <= k <= n - 2, T(n, k) equals the number of rooted duplication trees on n gene segments whose leftmost visible duplication event is (k, r), for 1 <= r <= (n - k)/2.
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5
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1, 1, 1, 2, 2, 2, 4, 6, 6, 6, 10, 16, 22, 22, 22, 26, 48, 70, 92, 92, 92, 74, 144, 236, 328, 420, 420, 420, 218, 454, 782, 1202, 1622, 2042, 2042, 2042, 672, 1454, 2656, 4278, 6320, 8362, 10404, 10404, 10404, 2126, 4782, 9060, 15380, 23742, 34146, 44550, 54954, 54954, 54954
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OFFSET
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2,4
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COMMENTS
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See Figure 3(a) in Gascuel et al. (2003).
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REFERENCES
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O. Gascuel (Ed.), Mathematics of Evolution and Phylogeny, Oxford University Press, 2005
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LINKS
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FORMULA
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T(n,k) = Sum_{j = 0.. k+1} T(n-1,j) for n >= 3, 0 <= k <= n - 2, with T(2,0) = 1 and T(n,k) = 0 for k >= n - 1.
T(n,k) = T(n,k-1) + T(n-1,k+1) for n >= 3, 1 <= k <= n - 2.
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EXAMPLE
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Triangle begins
n\k| 0 1 2 3 4 5 6 7
---+---------------------------------------
2 | 1
3 | 1 1
4 | 2 2 2
5 | 4 6 6 6
6 | 10 16 22 22 22
7 | 26 48 70 92 92 92
8 | 74 144 236 328 420 420 420
9 | 218 454 782 1202 1622 2042 2042 2042
...
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MAPLE
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A264869 := proc (n, k) option remember;
`if`(n <= 2, 1, add(A264869(n - 1, j), j = 0 .. min(k + 1, n - 3))) end proc:
seq(seq(A264869(n, k), k = 0..n - 2), n = 2..11);
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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