OFFSET
0,4
LINKS
Harry J. Smith, Table of n, a(n) for n=0..100
FORMULA
E.g.f.: sinh(x) * exp(-x)/(1-x). Asymptotic expression: a(n) ~ n! * (1 - 1/e^2)/2 i.e. as n goes to infinity the fraction for permutations that has an odd number of fixed points is about (1 - 1/e^2)/2 = 0.432332...
a(n) = n! - A062282(n) = n! - sum k=0 ... [n/2] sum l=0...n-2k (-1)^l * n!/((2k)! * l!)
Recurrence: a(n) = n*a(n-1)+(-2)^(n-1). - Vladeta Jovovic, Apr 11 2003
More generally, e.g.f. for number of degree-n permutations with an odd number of k-cycles is sinh(x^k/k)*exp(-x^k/k)/(1-x). - Vladeta Jovovic, Jan 31 2006
a(n) = (Gamma(n+1) - Gamma(n+1,-2)*exp(-2))/2, where Gamma(a,x) is the incomplete gamma function. - Ilya Gutkovskiy, May 06 2016
MATHEMATICA
nn = 20; d = Exp[-x]/(1 - x); Range[0, nn]! CoefficientList[Series[Sinh[x] d, {x, 0, nn}], x] (* Geoffrey Critzer, Jan 14 2012 *)
a[n_] := -n!/2 Sum[(-2)^i/i!, {i, 1, n}]
Table[a[n], {n, 0, 20}] (* Gerry Martens , May 06 2016 *)
PROG
(PARI) { for (n=0, 100, if (n, a=n*a + (-2)^(n-1), a=0); write("b063083.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 17 2009
CROSSREFS
KEYWORD
nonn
AUTHOR
Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 05 2001
EXTENSIONS
More terms from Wouter Meeussen, Aug 09 2001
STATUS
approved