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

%I A027617 #29 Dec 13 2019 05:38:22
%S A027617 0,0,0,2,8,40,200,1400,11200,103040,1030400,11334400,135766400,
%T A027617 1764963200,24709484800,370687116800,5930993868800,100826895769600,
%U A027617 1814871926067200,34482566595276800,689651331905536000,14482682605174784000,318619017313845248000
%N A027617 Number of permutations of n elements containing a 3-cycle.
%C A027617 a(n)/n! is asymptotic to 1-e^(-1/3) = 1 - A092615. - _Michel Marcus_, Aug 08 2013
%H A027617 Alois P. Heinz, <a href="/A027617/b027617.txt">Table of n, a(n) for n = 0..200</a>
%F A027617 a(n) = n! * ( 1 - Sum_{k=0..floor(n/3)} (-1)^k / (3^k * k!) ).
%F A027617 E.g.f.: 1/(1-x) - exp(-x^3/3)/(1-x). - _Geoffrey Critzer_, Jan 23 2013
%F A027617 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
%F A027617 Conjectures from _Stéphane Rézel_, Dec 11 2019: (Start)
%F A027617 Recurrence: a(n) = n*a(n-1), for n > 3 and n !== 0 (mod 3);
%F A027617 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.
%F A027617 (End)
%t A027617 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 *)
%o A027617 (PARI) a(n) = n! * (1 - sum(k=0, floor(n/3), (-1)^k/(k!*3^k) ) ); \\ _Stéphane Rézel_, Dec 11 2019
%Y A027617 Column k=3 of A293211.
%K A027617 nonn
%O A027617 0,4
%A A027617 Joe Keane (jgk(AT)jgk.org)
%E A027617 More terms from _Geoffrey Critzer_, Jan 23 2013