A182661 Expansion of x^3*exp(-x)/(3*(1-x)).

2, 0, 20, 80, 630, 4928, 44520, 444960, 4894890, 58738240, 763597692, 10690366960, 160355505310, 2565688083840, 43616697426640, 785100553677888, 14916910519881810, 298338210397633920, 6265102418350314980, 137832253203706926480

Offset: 3

Author

Geoffrey Critzer, Feb 01 2011

Keywords

• nonn

Comments

a(n) is the number of 3-cycles in all derangements of {1,2,...n}.

Crossrefs

Cf. A000387.

Programs

Maple

egf:= x^3 * exp(-x)/(3*(1-x)):
a:= n-> n! * coeff (series (egf, x, n+1), x, n):
seq (a(n), n=3..25);

Mathematica

Table[Count[Flatten[Map[Length,Map[ToCycles,Derangements[n]],{2}]],3],{n,0,8}]
Range[0,20]! CoefficientList[Series[x^3/3 Exp[-x]/(1-x),{x,0,20}],x]

Formula

E.g.f.: x^3 * exp(-x)/(3*(1-x)).

In general, E.g.f. for the number of k cycles in the derangements of [n] is: x^k * exp(-x)/(k*(1-x)).