A029572 Number of permutations of an n-set containing a 5-cycle.

0, 0, 0, 0, 0, 24, 144, 1008, 8064, 72576, 653184, 7185024, 86220288, 1120863744, 15692092416, 237124952064, 3793999233024, 64497986961408, 1160963765305344, 22058311540801536, 441004037348818944, 9261084784325197824, 203743865255154352128, 4686108900868550098944

Offset: 0

Author

Rob Pratt

Keywords

• nonn

Crossrefs

Column k=5 of A293211.

Programs

Mathematica

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 *)

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

Formula

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

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

E.g.f.: (1-exp(-x^5/5))/(1-x) - Geoffrey Critzer, Jun 01 2013