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

0,5

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).

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

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 *)

Crossrefs

Cf. A038205, A000166.

Sequence in context: A354074 A293300 A293487 * A355285 A182083 A293049

Adjacent sequences: A047862 A047863 A047864 * A047866 A047867 A047868

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Definition adjusted by Steven Finch, Mar 10 2022

Status

approved