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A027617
Number of permutations of n elements containing a 3-cycle.
3
0, 0, 0, 2, 8, 40, 200, 1400, 11200, 103040, 1030400, 11334400, 135766400, 1764963200, 24709484800, 370687116800, 5930993868800, 100826895769600, 1814871926067200, 34482566595276800, 689651331905536000, 14482682605174784000, 318619017313845248000
OFFSET
0,4
COMMENTS
a(n)/n! is asymptotic to 1-e^(-1/3) = 1 - A092615. - Michel Marcus, Aug 08 2013
LINKS
FORMULA
a(n) = n! * ( 1 - Sum_{k=0..floor(n/3)} (-1)^k / (3^k * k!) ).
E.g.f.: 1/(1-x) - exp(-x^3/3)/(1-x). - Geoffrey Critzer, Jan 23 2013
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
Conjectures from Stéphane Rézel, Dec 11 2019: (Start)
Recurrence: a(n) = n*a(n-1), for n > 3 and n !== 0 (mod 3);
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.
(End)
MATHEMATICA
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 *)
PROG
(PARI) a(n) = n! * (1 - sum(k=0, floor(n/3), (-1)^k/(k!*3^k) ) ); \\ Stéphane Rézel, Dec 11 2019
CROSSREFS
Column k=3 of A293211.
Sequence in context: A221587 A186947 A071007 * A187071 A154626 A003305
KEYWORD
nonn
AUTHOR
Joe Keane (jgk(AT)jgk.org)
EXTENSIONS
More terms from Geoffrey Critzer, Jan 23 2013
STATUS
approved