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, and L. Thoma, Monotonicity for complete graphs, preprint
A. Kassel, R. Kenyon, and W. Wu, Random two-component spanning forests, Ann. Inst. H. Poincaré Probab. Statist., 51 (2015), 1457-1464.
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
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