A027616 Number of permutations of n elements containing a 2-cycle.

0, 0, 1, 3, 9, 45, 285, 1995, 15855, 142695, 1427895, 15706845, 188471745, 2450132685, 34301992725, 514529890875, 8232476226975, 139952095858575, 2519137759913775, 47863617438361725, 957272348112505425, 20102719310362613925, 442259824841726816925, 10171975971359716789275

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..449
• Larry Carter and Stan Wagon, The Mensa Correctional Institute, The American Mathematical Monthly 125.4 (2018): 306-319.

Crossrefs

Cf. A000266, A088436, A114320.

Column k=2 of A293211.

Programs

Magma

A027616:= func< n | Factorial(n)*(1- (&+[(-1/2)^j/Factorial(j): j in [0..Floor(n/2)]]) ) >;
[A027616(n): n in [0..30]]; // G. C. Greubel, Aug 05 2022

Maple

S:= series((1-exp(-x^2/2))/(1-x), x, 101):
seq(coeff(S,x,j)*j!,j=0..100); # Robert Israel, May 12 2016

Mathematica

nn=30; Table[n!,{n,0,nn}]-Range[0,nn]!CoefficientList[Series[Exp[-x^2/2]/(1-x),{x,0,nn}],x]  (* Geoffrey Critzer, Oct 20 2012 *)

PARI

a(n) = n! * (1 - sum(k=0,floor(n/2), (-1)^k / (2^k * k!) ) );
/* Joerg Arndt, Oct 20 2012 */

PARI

N=33; x='x+O('x^N);
v=Vec( 'a0 + serlaplace( (1-exp(-x^2/2))/(1-x) ) );
v[1]-='a0;  v
/* Joerg Arndt, Oct 20 2012 */

SageMath

def A027616(n): return factorial(n)*(1-sum((-1/2)^k/factorial(k) for k in (0..(n//2))))
[A027616(n) for n in (0..30)] # G. C. Greubel, Aug 05 2022

Formula

E.g.f.: (1 - exp(-x^2/2)) / (1-x).

a(n) = n! * ( 1 - Sum_{k=0..floor(n/2)} (-1)^k / (2^k * k!) ).

a(n) + A000266(n) = n!. - Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 09 2003

Limit_{n -> oo} a(n)/n! = 1 - e^(-1/2) = 1 - A092605. - Michel Marcus, Aug 08 2013

Extensions

Added more terms, Geoffrey Critzer, Oct 20 2012