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A083483
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Number of forests with two connected components in the complete graph K_{n}.
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7
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0, 1, 3, 15, 110, 1080, 13377, 200704, 3542940, 72000000, 1656409535, 42568187904, 1208912928522, 37603105146880, 1271514111328125, 46443371157258240, 1822442358054692408, 76461926986744528896, 3415753581721829617275
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OFFSET
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1,3
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COMMENTS
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Note that the above sequence is dominated by the sequence n^{n-2} (n > 0), A000272, 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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W. Kook, Categories of acyclic graphs and automorphisms of free groups, Ph.D. thesis (G. Carlsson, advisor), Stanford University, 1996.
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LINKS
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FORMULA
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E.g.f.: 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, Jul 08 2003
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MAPLE
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f:=n->(n-1)!*n^(n-4)*(n+6)/(2*(n-2)!); [seq(f(n), n=2..30)]; # N. J. A. Sloane, Apr 09 2014
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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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PROG
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(PARI) for(n=1, 30, print1(n^(n-4)*(n-1)*(n+6)/2, ", ")) \\ G. C. Greubel, Nov 14 2017
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CROSSREFS
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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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EXTENSIONS
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STATUS
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approved
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