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Number of derangements of n where minimal cycle size is at least 4.
7

%I #36 Mar 10 2022 16:54:33

%S 1,0,0,0,6,24,120,720,6300,58464,586656,6384960,76471560,994831200,

%T 13939507296,209097854784,3345235180560,56866395720960,

%U 1023601917024000,19448577603454464,388972171805410656,8168409582839579520,179704944537482689920

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

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

%H Alois P. Heinz, <a href="/A047865/b047865.txt">Table of n, a(n) for n = 0..200</a>

%H H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, 2nd edn., Academic Press, NY, 1994, p. 176, Eq. 5.2.9 (q=3).

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

%F 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

%F a(n) ~ n! * exp(-11/6). - _Vaclav Kotesovec_, Aug 13 2013

%p with(combstruct): ZL3:=[S,{S=Set(Cycle(Z,card>3))},labeled]:

%p seq (count (ZL3, size=n), n=0..21); # _Zerinvary Lajos_, Sep 26 2007

%t 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 *)

%Y Cf. A038205, A000166.

%K nonn

%O 0,5

%A _N. J. A. Sloane_

%E Definition adjusted by _Steven Finch_, Mar 10 2022