A027617 Number of permutations of n elements containing a 3-cycle.

0, 0, 0, 2, 8, 40, 200, 1400, 11200, 103040, 1030400, 11334400, 135766400, 1764963200, 24709484800, 370687116800, 5930993868800, 100826895769600, 1814871926067200, 34482566595276800, 689651331905536000, 14482682605174784000, 318619017313845248000

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn

Comments

a(n)/n! is asymptotic to 1-e^(-1/3) = 1 - A092615. - Michel Marcus, Aug 08 2013

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Column k=3 of A293211.

Programs

Mathematica

nn=20;Range[0,nn]!CoefficientList[Series[1/(1-x)-Exp[-x^3/3]/(1-x), {x,0,nn}],x]  (* Geoffrey Critzer, Jan 23 2013 *)

PARI

a(n) = n! * (1 - sum(k=0, floor(n/3), (-1)^k/(k!*3^k) ) ); \\ Stéphane Rézel, Dec 11 2019

Formula

a(n) = n! * ( 1 - Sum_{k=0..floor(n/3)} (-1)^k / (3^k * k!) ).

E.g.f.: 1/(1-x) - exp(-x^3/3)/(1-x). - Geoffrey Critzer, Jan 23 2013

Recurrence: a(n) = n*a(n-1) - (n-2)*(n-1)*a(n-3) + (n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Aug 13 2013

Conjectures from Stéphane Rézel, Dec 11 2019: (Start)

Recurrence: a(n) = n*a(n-1), for n > 3 and n !== 0 (mod 3);

for k > 1, a(3*k) = a(3*k-1)*S(k)/S(k-1) where S(k) = 3*k*S(k-1) - (-1)^k with S(1) = 1.

(End)

Extensions

More terms from Geoffrey Critzer, Jan 23 2013