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

%I #29 Dec 13 2019 05:38:22

%S 0,0,0,2,8,40,200,1400,11200,103040,1030400,11334400,135766400,

%T 1764963200,24709484800,370687116800,5930993868800,100826895769600,

%U 1814871926067200,34482566595276800,689651331905536000,14482682605174784000,318619017313845248000

%N Number of permutations of n elements containing a 3-cycle.

%C a(n)/n! is asymptotic to 1-e^(-1/3) = 1 - A092615. - _Michel Marcus_, Aug 08 2013

%H Alois P. Heinz, <a href="/A027617/b027617.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = n! * ( 1 - Sum_{k=0..floor(n/3)} (-1)^k / (3^k * k!) ).

%F E.g.f.: 1/(1-x) - exp(-x^3/3)/(1-x). - _Geoffrey Critzer_, Jan 23 2013

%F 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 Conjectures from _Stéphane Rézel_, Dec 11 2019: (Start)

%F Recurrence: a(n) = n*a(n-1), for n > 3 and n !== 0 (mod 3);

%F 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 (End)

%t 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 (PARI) a(n) = n! * (1 - sum(k=0, floor(n/3), (-1)^k/(k!*3^k) ) ); \\ _Stéphane Rézel_, Dec 11 2019

%Y Column k=3 of A293211.

%K nonn

%O 0,4

%A Joe Keane (jgk(AT)jgk.org)

%E More terms from _Geoffrey Critzer_, Jan 23 2013