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A060313
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Number of homeomorphically irreducible rooted trees (also known as series-reduced rooted trees, or rooted trees without nodes of degree 2) on n labeled nodes.
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2
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1, 2, 0, 16, 25, 576, 2989, 51584, 512649, 8927200, 130956001, 2533847328, 48008533885, 1059817074512, 24196291364925, 609350187214336, 16135860325700881, 459434230368302016, 13788624945433889593
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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REFERENCES
| I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
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FORMULA
| a(n)=n*(n - 2)!*Sum_{k=0..n - 2} ( - 1)^k*binomial(n, k)*(n - k)^(n - 2 - k)/(n - 2 - k)!, n>1. E.g.f.: x*(exp( - LambertW( - x/(1 + x))) - 1/2*(LambertW( - x/(1 + x)))^2).
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MATHEMATICA
| f[n_] := If[n < 2, 1, n(n - 2)!Sum[(-1)^k*Binomial[n, k](n - k)^(n - 2 - k)/(n - 2 - k)!, {k, 0, n - 2}]]; Table[ f[n], {n, 19}] (from Robert G. Wilson v Feb 12 2005)
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CROSSREFS
| Unlabeled case: A059123. Cf. A005512.
Sequence in context: A155585 A057375 A009045 * A015154 A009266 A009246
Adjacent sequences: A060310 A060311 A060312 * A060314 A060315 A060316
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KEYWORD
| easy,nonn
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AUTHOR
| Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 27 2001
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