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A264870
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Triangular array: For n >= 2 and 0 < k <= n - 2, T(n, k) equals the number of (unrooted) duplication trees on n gene segments that are canonical and whose leftmost visible duplication event is (k, r), for 1 <= r <= (n - k)/2.
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3
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1, 0, 1, 1, 1, 1, 2, 3, 3, 3, 5, 8, 11, 11, 11, 13, 24, 35, 46, 46, 46, 37, 72, 118, 164, 210, 210, 210, 109, 227, 391, 601, 811, 1021, 1021, 1021, 336, 727, 1328, 2139, 3160, 4181, 5202, 5202, 5202, 1063, 2391, 4530, 7690, 11871, 17073, 22275, 27477, 27477, 27477
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OFFSET
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0,7
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COMMENTS
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See Figure 3(b) in Gascuel et al. (2003).
From row 4 onwards, the entries are one-half the corresponding entries in A264879.
Row sums give the number of unrooted duplication trees on n gene segments, A086521.
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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 >= 4, 0 <= k <= n - 2, with T(2,0) = T(3,1) = 1, T(3,0) = 0 and T(n,k) = 0 for k >= n - 1.
T(n,k) = T(n,k-1) + T(n-1,k+1) for n >= 4, 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.| 0 1
.4.| 1 1 1
.5.| 2 3 3 3
.6.| 5 8 11 11 11
.7.| 13 24 35 46 46 46
.8.| 37 72 118 164 210 210 210
.9.| 109 227 391 601 811 1021 1021 1021
...
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MAPLE
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A264870 := proc (n, k) option remember;
`if`(n = 3 and k = 0, 0, `if`(n <= 4 and k <= n-2, 1, `if`(k > n - 2, 0, add(A264870(n-1, j), j = 0..min(k+1, n))))) end proc:
seq(seq(A264870(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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