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Search: id:A073663
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| A073663 |
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Total number of branches of length k (k=1,2,3,...) in all ordered trees with n+k edges (it is independent of k). |
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+0
2
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| 1, 2, 8, 30, 113, 428, 1629, 6226, 23881, 91884, 354484, 1370812, 5312058, 20622904, 80196055, 312319530, 1217938665, 4755296460, 18586968840, 72723903780, 284804791230, 1116315593640, 4378929921210, 17189573707956
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OFFSET
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0,2
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REFERENCES
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J. Riordan, Enumeration of plane trees by branches and endpoints, J. Combinat. Theory, A, 19, 214-222, 1975.
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FORMULA
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a(n)=binom(2n+2, n)-2binom(2n, n-1)+binom(2n-2, n-2) (n>0); a(n)=3(3n^3+2n^2+n-2)binomial(2n, n)/[2(n+1)(n+2)(2n-1)] (n>0); g.f.=(1-z)^2C^2/sqrt(1-4z), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
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EXAMPLE
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a(2)=8 because for n=2 and k=1 (for example), the five ordered trees with n+k=3 edges have alltogether 0+3+1+1+3=8 branches of length 1.
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CROSSREFS
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First differences of A076540.
Sequence in context: A077839 A052530 A162551 this_sequence A155116 A133915 A150759
Adjacent sequences: A073660 A073661 A073662 this_sequence A073664 A073665 A073666
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 01 2002
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