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A003221 Number of even permutations of length n with no fixed points.
(Formerly M0922)
9
1, 0, 0, 2, 3, 24, 130, 930, 7413, 66752, 667476, 7342290, 88107415, 1145396472, 16035550518, 240533257874, 3848532125865, 65425046139840, 1177650830516968, 22375365779822562, 447507315596451051, 9397653627525472280, 206748379805560389930 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Bashir Ali and A. Umar, Some combinatorial properties of the alternating group, Southeast Asian Bulletin Math. 32 (2008), 823-830. [From Abdullahi Umar, Oct 09 2008]

L. Carlitz and R. A. Scoville, Problem E2354, Amer. Math. Monthly, 79 (1972), 394.

G. Gordon and E. McMahon, Moving faces to other places: facet derangements, Amer. Math. Monthly, 117 (2010), 865-88.

G. Gordon and E. McMahon, Moving faces to other places: facet derangements, arXiv:0906.4253 [math.CO], 2009.

Piotr Miska, Arithmetic Properties of the Sequence of Derangements and its Generalizations, arXiv:1508.01987 [math.NT], 2015. (see Chapter 5 p. 44)

J. M. Thomas, The number of even and odd absolute permutations of n letters, Bull. Amer. Math. Soc. 31 (1925), 303-304.

FORMULA

a(n) = (A000166(n)-(-1)^n*(n-1))/2.

From Abdullahi Umar, Oct 09 2008: (Start)

a(n) = (n!/2)*sum(((-1)^i)/i!, i=0..n-2)+((-1)^(n-1))*(n-1) for n>1, a(0)=1, a(1)=0.

a(n) = (n-1)*(a(n-1)+a(n-2))+((-1)^(n-1))*(n-1) for n>1, a(0)=1, a(1)=0.

a(n) = n*a(n-1)+((-1)^(n-1))*(n-2)*(n+1)/2 for n>0, a(0)=1.

E.g.f.: (1-x^2/2)*exp(-x)/(1-x). (End)

MAPLE

A003221 := n -> ((-1)^n*hypergeom([-n, 1], [], 1)-(-1)^n*(n-1))/2:

seq(simplify(A003221(n)), n=0..22); # Peter Luschny, May 09 2017

MATHEMATICA

a[n_] := (Round[n!/E] - (-1)^n*(n - 1))/2; a[0] = 1; Table[a[n], {n, 0, 22}] (* Jean-Fran├žois Alcover, Dec 13 2011, after Simon Plouffe *)

Range[0, 25]! CoefficientList[Series[(1 - x^2 / 2) E^(-x) / (1 - x), {x, 0, 25}], x] (* Vincenzo Librandi, Aug 11 2015 *)

PROG

(Python)

from __future__ import division

A003221_list, m, x = [], -1, 0

for n in range(10*2):

....x, m = x*n + m*(n*(n-1)//2-1), -m

....A003221_list.append(x) # Chai Wah Wu, Nov 03 2014

(PARI) a(n) = ( n!*sum(r=2, n, (-1)^r/r!) + (-1)^(n-1)*(n-1))/2; \\ Michel Marcus, Apr 22 2016

CROSSREFS

Cf. A000166, A000387.

Sequence in context: A009231 A012304 A047157 * A013312 A013318 A193338

Adjacent sequences:  A003218 A003219 A003220 * A003222 A003223 A003224

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Typo in second formula fixed by Josh Swanson, Nov 10 2013

STATUS

approved

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Last modified October 16 09:29 EDT 2019. Contains 328056 sequences. (Running on oeis4.)