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A083483 Number of forests with two connected components in the complete graph K_{n}. 6
0, 1, 3, 15, 110, 1080, 13377, 200704, 3542940, 72000000, 1656409535, 42568187904, 1208912928522, 37603105146880, 1271514111328125, 46443371157258240, 1822442358054692408, 76461926986744528896, 3415753581721829617275 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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).

REFERENCES

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

N. Eaton, W. Kook, L. Thoma, Monotonicity for complete graphs, preprint

C. J. Liu and Yutze Chow, On operator and formal sum methods for graph enumeration problems, SIAM J. Algebraic Discrete Methods, 5 (1984), no. 3, 384--406. MR0752043 (86d:05059). See Eq. (47). - From N. J. A. Sloane, Apr 09 2014

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^{i-2}*x^{i}/i!.

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

a(n) = n^(n-4)*(n-1)*(n+6)/2. - Vaclav Kotesovec, Oct 18 2013

MAPLE

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

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

Table[n^(n - 4) (n - 1) (n + 6)/2, {n, 1, 40}] (* Vincenzo Librandi, Apr 10 2014 *)

PROG

(MAGMA) [n^(n-4)*(n-1)*(n+6)/2 : n in [1..20]]; // Vincenzo Librandi, Apr 10 2014

(PARI) for(n=1, 30, print1(n^(n-4)*(n-1)*(n+6)/2, ", ")) \\ G. C. Greubel, Nov 14 2017

CROSSREFS

Cf. A000272, A239910.

Column m=2 of A105599. A diagonal of A138464. - Alois P. Heinz, Apr 10 2014

Sequence in context: A217061 A054201 A090355 * A089468 A109498 A142967

Adjacent sequences:  A083480 A083481 A083482 * A083484 A083485 A083486

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

Edited by N. J. A. Sloane, Apr 09 2014

STATUS

approved

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Last modified October 15 15:20 EDT 2018. Contains 316236 sequences. (Running on oeis4.)