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A005804
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Number of phylogenetic rooted trees with n labels.
(Formerly M1890)
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4
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1, 2, 8, 58, 612, 8374, 140408, 2785906, 63830764, 1658336270, 48169385024, 1546832023114, 54413083601268, 2080827594898342, 85948745163598088, 3813417859420469410, 180876816831806597500, 9133309115320844870078
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| These are series-reduced rooted trees where each leaf is a non-empty subset of the set of n labels.
See A141268 for phylogenetic rooted trees with n unlabeled objects. - Thomas Wieder (thomas.wieder(AT)t-online.de), Jun 20 2008
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REFERENCES
| Foulds, L. R.; Robinson, R. W. Enumeration of phylogenetic trees without points of degree two. Ars Combin. 17 (1984), A, 169-183. Math. Rev. 85f:05045
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..100
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees
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FORMULA
| Stirling transform of [ 1, 1, 4, 26, 236, ... ] = A000311 [ Foulds and Robinson ].
G.f.: -LambertW(-1/2*exp(1/2*exp(z)-1))+1/2*exp(z)-1 series(-LambertW(-1/2*exp(1/2*exp(z)-1))+1/2*exp(z)-1,z=0,10). - Thomas Wieder (thomas.wieder(AT)t-online.de), Jun 20 2008
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EXAMPLE
| a(3)=8 because we have:
Set(Set(Z[3]),Set(Z[1]),Set(Z[2])),
Set(Z[3],Z[2],Z[1]),
Set(Set(Z[3],Z[1]),Set(Z[2])),
Set(Set(Set(Z[3]),Set(Z[2])),Set(Z[1])),
Set(Set(Set(Z[3]),Set(Z[1])),Set(Z[2])),
Set(Set(Z[3]),Set(Set(Z[1]),Set(Z[2]))),
Set(Set(Z[3]),Set(Z[2],Z[1])),
Set(Set(Z[3],Z[2]),Set(Z[1]))
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MAPLE
| combstruct command: A005804 := [H, {H=Union(Set(Z, card>=1), Set(H, card>=2))}, labelled]; seq(count(A00584, size=j), j=1..20); - Thomas Wieder (thomas.wieder(AT)t-online.de), Jun 20 2008
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CROSSREFS
| Cf. A000311, A005805.
Cf. A141268.
Sequence in context: A007347 A185898 A063074 * A162067 A179534 A086907
Adjacent sequences: A005801 A005802 A005803 * A005805 A005806 A005807
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms, comment from Christian G. Bower (bowerc(AT)usa.net), Dec 15 1999.
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