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Search: id:A083483
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| A083483 |
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Number of forests with two connected components in complete graphs K_{n}. |
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2
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| 0, 1, 3, 15, 110, 1080, 13377, 200704, 3542940, 72000000, 1656409535, 42568187904, 1208912928522, 37603105146880, 1271514111328125, 46443371157258240, 1822442358054692408, 76461926986744528896, 3415753581721829617275
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Note that the above sequence is dominated by the sequence n^{n-2} (n >0), which enumerates the number of spanning trees in K_{n} : 1, 1, 3, 16, 125, 1296, 16807, 262144, ... This is a consequence of the result in [EKT] which shows that the sequence of independent set numbers of cycle matroid of K_{n} is (strictly) monotone increasing (when n > 3).
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REFERENCES
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[K] W. Kook, Categories of acyclic graphs and automorphisms of free groups, Ph.D. thesis (G. Carlsson, advisor), Stanford University, 1996
[EKT] N. Eaton, W. Kook, L. Thoma, Monotonicity for complete graphs, preprint
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FORMULA
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exponential generating function=T(x)^{2}/2!, where T(x) is the e.g.f. for the number of spanning trees in K_{n}, i.e. T(x)= sum_{i>= 1}i^{i-2}*x^{i}/i!
E.g.f.: 1/8*LambertW(-x)^2*(2+LambertW(-x))^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 08 2003
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MATHEMATICA
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(* first 20 terms starting with n=1 *) T := Sum[i^(i - 2)*(x^i)/i!, {i, 1, 20}]; T2 := Expand[(T^{2})/2! ]; C2[i_] := Coefficient[T2, x^{i}]*i!; M := MatrixForm[Table[C2[i], {i, 20}]]; M
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CROSSREFS
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Sequence in context: A110328 A054201 A090355 this_sequence A089468 A109498 A142967
Adjacent sequences: A083480 A083481 A083482 this_sequence A083484 A083485 A083486
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KEYWORD
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nonn
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AUTHOR
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Woong Kook (andrewk(AT)math.uri.edu), Jun 08 2003
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