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A063083 Number of permutations of n elements with an odd number of fixed points.

%I A063083 #26 Jan 28 2022 06:25:49
%S A063083 0,1,0,4,8,56,304,2192,17408,156928,1568768,17257472,207087616,
%T A063083 2692143104,37689995264,565349945344,9045599092736,153775184642048,
%U A063083 2767953323425792,52591113145352192,1051822262906519552,22088267521037959168,485941885462833004544
%N A063083 Number of permutations of n elements with an odd number of fixed points.
%H A063083 Harry J. Smith, <a href="/A063083/b063083.txt">Table of n, a(n) for n=0..100</a>
%F A063083 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...
%F A063083 a(n) = n! - A062282(n) = n! - sum k=0 ... [n/2] sum l=0...n-2k (-1)^l * n!/((2k)! * l!)
%F A063083 Recurrence: a(n) = n*a(n-1)+(-2)^(n-1). - _Vladeta Jovovic_, Apr 11 2003
%F A063083 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
%F A063083 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
%t A063083 nn = 20; d = Exp[-x]/(1 - x); Range[0, nn]! CoefficientList[Series[Sinh[x]  d, {x, 0, nn}], x] (* _Geoffrey Critzer_, Jan 14 2012 *)
%t A063083 a[n_] := -n!/2 Sum[(-2)^i/i!, {i, 1, n}]
%t A063083 Table[a[n], {n, 0, 20}] (* _Gerry Martens_ , May 06 2016 *)
%o A063083 (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
%Y A063083 Cf. A062282.
%K A063083 nonn
%O A063083 0,4
%A A063083 Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 05 2001
%E A063083 More terms from _Wouter Meeussen_, Aug 09 2001