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Number of permutations of an n-set containing a 5-cycle.
2

%I #26 Aug 06 2024 09:56:18

%S 0,0,0,0,0,24,144,1008,8064,72576,653184,7185024,86220288,1120863744,

%T 15692092416,237124952064,3793999233024,64497986961408,

%U 1160963765305344,22058311540801536,441004037348818944,9261084784325197824,203743865255154352128,4686108900868550098944

%N Number of permutations of an n-set containing a 5-cycle.

%F a(n) = n!*(1 - Sum_{k=0..floor(n/5)} (-1)^k/(k!*5^k)).

%F a(n)/n! is asymptotic to 1-e^(-1/5) = 1 - A092618.

%F E.g.f.: (1-exp(-x^5/5))/(1-x) - _Geoffrey Critzer_, Jun 01 2013

%t nn = 20; a = Log[1/(1 - x)] - x^5/5; Range[0, nn]! CoefficientList[Series[1/(1 - x) - Exp[a], {x, 0, nn}],x] (* _Geoffrey Critzer_, Jun 01 2013 *)

%o (PARI) my(x='x+O('x^66)); concat([0,0,0,0,0], Vec(serlaplace((1-exp(-x^5/5))/(1-x)))) \\ _Joerg Arndt_, Jun 01 2013

%Y Column k=5 of A293211.

%K nonn

%O 0,6

%A _Rob Pratt_