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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-François 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 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.)