A047865 Number of derangements of n where minimal cycle size is at least 4.

1, 0, 0, 0, 6, 24, 120, 720, 6300, 58464, 586656, 6384960, 76471560, 994831200, 13939507296, 209097854784, 3345235180560, 56866395720960, 1023601917024000, 19448577603454464, 388972171805410656, 8168409582839579520, 179704944537482689920

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 147, Eq. 5.2.9 (q=3).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 176, Eq. 5.2.9 (q=3).

Crossrefs

Cf. A038205, A000166.

Programs

Maple

with(combstruct): ZL3:=[S,{S=Set(Cycle(Z,card>3))},labeled]:
seq (count (ZL3, size=n), n=0..21); # Zerinvary Lajos, Sep 26 2007

Mathematica

nn=20;Range[0,nn]!CoefficientList[Series[Exp[-x-x^2/2-x^3/3]/(1-x),{x,0,nn}],x]  (* Geoffrey Critzer, Nov 11 2012 *)

Formula

a(n) = (n-1)*a(n-1) + (n-1)*(n-2)*(n-3)*a(n-4).

E.g.f.: A(x) = 1/(1-x)*exp(-x-x^2/2-x^3/3) = 1 + 6*x^4/4! + 24*x^5/5! + ... satisfies the differential equation A'(x) = x^3/(1-x)*A(x). - Peter Bala, Apr 18 2012

a(n) ~ n! * exp(-11/6). - Vaclav Kotesovec, Aug 13 2013

Extensions

Definition adjusted by Steven Finch, Mar 10 2022