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A063083 Number of permutations of n elements with an odd number of fixed points. 8
0, 1, 0, 4, 8, 56, 304, 2192, 17408, 156928, 1568768, 17257472, 207087616, 2692143104, 37689995264, 565349945344, 9045599092736, 153775184642048, 2767953323425792, 52591113145352192, 1051822262906519552 (list; graph; refs; listen; history; text; internal format)
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

Cf. A062282.

Sequence in context: A123106 A123288 A192200 * A270399 A269998 A335527

Adjacent sequences:  A063080 A063081 A063082 * A063084 A063085 A063086

KEYWORD

nonn

AUTHOR

Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 05 2001

EXTENSIONS

More terms from Wouter Meeussen, Aug 09 2001

STATUS

approved

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Last modified October 23 16:20 EDT 2021. Contains 348215 sequences. (Running on oeis4.)