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A006677
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Number of planted binary phylogenetic trees with n labels.
(Formerly M1806)
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1
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1, 2, 7, 41, 346, 3797, 51157, 816356, 15050581, 314726117, 7359554632, 190283748371, 5389914888541, 165983936096162, 5521346346543307, 197294173392918461, 7536892461493548226, 306520422583290179057
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Series-reduced planted binary trees where each leaf is a non-empty subset of the set of n labels.
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REFERENCES
| Foulds, L. R.; Robinson, R. W. Enumeration of binary phylogenetic trees. Combinatorial mathematics, VIII (Geelong, 1980), pp. 187-202, Lecture Notes in Math., 884, Springer, Berlin-New York, 1981. Math. Rev. 83a:05071.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Plouffe Simon, Master Thesis, uqam 1992.
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 1..101
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 160
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees
Plouffe Simon, Master Thesis, uqam 1992.
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FORMULA
| Stirling transform of A001147(n-1).
E.g.f.: E^x/(3 - 2E^x)^(1/2). This sequence is the Stirling transform of the sequence ( (2n-1)!! + n(2n-3)!! )_(n>=0) = (1, 2, 5, 24, 165, 1470, 16065, 207900,...) with egf (1+x)/(1-2x)^(1/2). (Both remarks assume offset 0.) - David Callan (callan(AT)stat.wisc.edu), Jul 22 2008
E.g.f.: 1-(3-2*exp(x))^(1/2). - Simon Plouffe, Feb 17 2011
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MAPLE
| stirtr:= proc(p) proc (n) add (p(k) *combinat[stirling2] (n, k), k=0..n) end end: f:= n-> `if`(n=0, 1, (2*n-2)!/ (n-1)!/ 2^(n-1)): a:= stirtr(f): seq (a(n), n=1..20); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 15 2008]
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MATHEMATICA
| max = 18; f[x_] := 1 - (3 - 2*Exp[x])^(1/2); Drop[ CoefficientList[ Series[ f[x], {x, 0, max}], x]*Range[0, max]!, 1] (* From Jean-François Alcover, Dec 20 2011, after Simon Plouffe *)
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CROSSREFS
| Sequence in context: A106871 A107376 A087804 * A101390 A113144 A006846
Adjacent sequences: A006674 A006675 A006676 * A006678 A006679 A006680
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| More terms, formula and comment from Christian G. Bower (bowerc(AT)usa.net), Dec 15 1999.
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