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A306295
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Maximal number of coalescent histories among non-matching pairs consisting of a caterpillar gene tree and a caterpillar species tree with n+2 leaves.
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1
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1, 3, 10, 32, 107, 359, 1234, 4274, 15032, 53242, 190588, 686272, 2490399, 9081375, 33312770, 122692130, 453999656, 1685601038, 6282014804, 23478897364, 88026769844, 330831420218, 1246635155180, 4707414286652, 17815452662152, 67546709440004, 256595322436760
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = C(n+1) - C(floor((n+1)/2))*C(ceiling((n+1)/2)), where C(n) is the n-th term in the Catalan sequence A000108.
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EXAMPLE
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For n=1, a non-matching caterpillar gene tree and species tree with n+2=3 leaves have only one coalescent history: all coalescences must take place above the root of the species tree. Hence, a(1)=1.
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MATHEMATICA
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b[n_] :=
Binomial[2 n - 2, n - 1]/
n - (2 Floor[(n - 1)/2])!/(Floor[(n - 1)/2]! Floor[(n + 1)/
2]!) (2 Ceiling[(n - 1)/2])!/(Ceiling[(n - 1)/
2]! Ceiling[(n + 1)/2]!)
a[n_] := b[n+2]
Table[a[n], {n, 1, 30}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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