A053506 a(n) = (n-1)*n^(n-2).

0, 1, 6, 48, 500, 6480, 100842, 1835008, 38263752, 900000000, 23579476910, 681091006464, 21505924728444, 737020860878848, 27246730957031250, 1080863910568919040, 45798768824157052688, 2064472028642102280192, 98646963440126439346902, 4980736000000000000000000

Offset: 1

Author

N. J. A. Sloane, Jan 15 2000

Keywords

• nonn

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

Crossrefs

Cf. A000169, A000312, A053507, A053508, A053509, A083483, A195242.

Cf. A001865 which is the sum of A000169 + A053506 + A065513 + A065888 + ...

Programs

GAP

List([1..20], n-> (n-1)*n^(n-2)) # G. C. Greubel, May 15 2019

Magma

[(n-1)*n^(n-2): n in [1..20]]; // G. C. Greubel, May 15 2019

Mathematica

Table[(n-1)*n^(n-2), {n,20}]

PARI

vector(20, n, (n-1)*n^(n-2)) \\ G. C. Greubel, Jan 18 2017

Sage

[(n-1)*n^(n-2) for n in (1..20)] # G. C. Greubel, May 15 2019

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