A065888 -id:A065888 - cp's OEIS frontend

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

N. J. A. Sloane, Jan 15 2000

nonn

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

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.

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

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

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

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

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

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

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

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

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

A065889 a(n) = number of unicyclic connected simple graphs whose cycle has length 4.

Original entry on oeis.org

3, 60, 1080, 20580, 430080, 9920232, 252000000, 7015381560, 212840939520, 6998969586180, 248180493969408, 9445533398437500, 384213343210045440, 16639691095281974160, 764619269867445288960, 37163398969133506235952, 1905131520000000000000000

Offset: 4

Len Smiley, Nov 27 2001

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..150

A065888 ( = 2*A065889) counts sagittal graphs with one cycle (length 4).
A column of A098909, A053507.
Main diagonal of A144209.
Cf. A053508.

List([4..25], n-> 12*Binomial(n,4)*n^(n-5)); # G. C. Greubel, May 16 2019

[12*Binomial(n,4)*n^(n-5) : n in [4..25]]; // G. C. Greubel, May 16 2019

Table[12*Binomial[n,4]*n^(n-5), {n,4,25}] (* G. C. Greubel, May 16 2019 *)

{a(n) = 12*binomial(n,4)*n^(n-5)}; \\ G. C. Greubel, May 16 2019

[12*binomial(n,4)*n^(n-5) for n in (4..25)] # G. C. Greubel, May 16 2019

E.g.f.: T^4/8, where T = T(x) is Euler's tree function (see A000169).
a(n) = (n-1)*(n-2)*(n-3)*n^(n-4)/2. - Vladeta Jovovic, Oct 26 2004
a(n) = 3 * A053508(n). - Alois P. Heinz, Jan 09 2025

cp's OEIS Frontend

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

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