

A083483


Number of forests with two connected components in the complete graph K_{n}.


7



0, 1, 3, 15, 110, 1080, 13377, 200704, 3542940, 72000000, 1656409535, 42568187904, 1208912928522, 37603105146880, 1271514111328125, 46443371157258240, 1822442358054692408, 76461926986744528896, 3415753581721829617275
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OFFSET

1,3


COMMENTS

Note that the above sequence is dominated by the sequence n^{n2} (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).


REFERENCES

W. Kook, Categories of acyclic graphs and automorphisms of free groups, Ph.D. thesis (G. Carlsson, advisor), Stanford University, 1996.


LINKS



FORMULA

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^(i2)*x^i/i!.
E.g.f.: (1/8)*LambertW(x)^2*(2+LambertW(x))^2.  Vladeta Jovovic, Jul 08 2003


MAPLE

f:=n>(n1)!*n^(n4)*(n+6)/(2*(n2)!); [seq(f(n), n=2..30)]; # N. J. A. Sloane, Apr 09 2014


MATHEMATICA

(* 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


PROG

(PARI) for(n=1, 30, print1(n^(n4)*(n1)*(n+6)/2, ", ")) \\ G. C. Greubel, Nov 14 2017


CROSSREFS



KEYWORD

nonn


AUTHOR

Woong Kook (andrewk(AT)math.uri.edu), Jun 08 2003


EXTENSIONS



STATUS

approved



