OFFSET
1,3
COMMENTS
a(n) is the number of endofunctions f of [n] which interchange a pair a<->b and for all x in [n] some iterate f^k(x) = a. E.g., a(3) = 6: 1<->2<-3; 3->1<->2; 2<->3<-1; 1->2<->3; 1<->3<-2; 2->1<->3. - Len Smiley, Nov 27 2001
If offset is 0: right side of the binomial sum n-> sum( i^(i-1) * (n-i+1)^(n-i)*binomial(n, i), i=1..n) - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
a(n) is the number of birooted labeled trees on n nodes in which the two root nodes are adjacent. - N. J. A. Sloane, May 01 2018
a(n) is the number of ways to partition the complete graph K_n into two components and choose an arborescence on each component. - Harry Richman, May 11 2022
REFERENCES
A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.36)
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Prop. 5.3.2.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..250
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Eric Weisstein's World of Mathematics, Graph Edge
FORMULA
E.g.f.: LambertW(-x)^2/2. - Vladeta Jovovic, Apr 07 2001
E.g.f. if offset 0: W(-x)^2/((1+W(-x))*x), W(x) Lambert's function (principal branch).
The sequence 1, 1, 6, 48, ... satisfies a(n) = (n*(n+1)^n + 0^n)/(n+1); it is the main diagonal of A085388. - Paul Barry, Jun 30 2003
a(n) = Sum_{i=1..n-1} binomial(n-1,i-1)*i^(i-2)*(n-i)^(n-i). - Dmitry Kruchinin, Oct 28 2013
If offset = 0 and a(0) = 1 then a(n) = Sum_{k=0..n} (-1)^(n-k)* binomial(-k,-n)*n^k (cf. A195242). - Peter Luschny, Apr 11 2016
MATHEMATICA
Table[(n-1)*n^(n-2), {n, 20}]
PROG
(PARI) vector(20, n, (n-1)*n^(n-2)) \\ G. C. Greubel, Jan 18 2017
(Magma) [(n-1)*n^(n-2): n in [1..20]]; // G. C. Greubel, May 15 2019
(Sage) [(n-1)*n^(n-2) for n in (1..20)] # G. C. Greubel, May 15 2019
(GAP) List([1..20], n-> (n-1)*n^(n-2)) # G. C. Greubel, May 15 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 15 2000
STATUS
approved