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A003221 Number of even permutations of length n with no fixed points.

%I A003221 M0922 #59 May 21 2025 18:43:12
%S A003221 1,0,0,2,3,24,130,930,7413,66752,667476,7342290,88107415,1145396472,
%T A003221 16035550518,240533257874,3848532125865,65425046139840,
%U A003221 1177650830516968,22375365779822562,447507315596451051,9397653627525472280,206748379805560389930
%N A003221 Number of even permutations of length n with no fixed points.
%D A003221 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003221 Vincenzo Librandi, <a href="/A003221/b003221.txt">Table of n, a(n) for n = 0..100</a>
%H A003221 Bashir Ali and A. Umar, <a href="http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_200805&amp;filename=Some Combinatorial Properties of the Alternating Group.pdf">Some combinatorial properties of the alternating group</a>, Southeast Asian Bulletin Math. 32 (2008), 823-830. [From _Abdullahi Umar_, Oct 09 2008]
%H A003221 L. Carlitz and R. A. Scoville, <a href="http://www.jstor.org/stable/2978096">Problem E2354</a>, Amer. Math. Monthly, 79 (1972), 394.
%H A003221 G. Gordon and E. McMahon, <a href="http://www.jstor.org/stable/10.4169/000298910X523353">Moving faces to other places: facet derangements</a>, Amer. Math. Monthly, 117 (2010), 865-88.
%H A003221 G. Gordon and E. McMahon, <a href="http://arxiv.org/abs/0906.4253">Moving faces to other places: facet derangements</a>, arXiv:0906.4253 [math.CO], 2009.
%H A003221 Karen Meagher, Peter Sin, <a href="https://arxiv.org/abs/1911.11252">All 2-transitive groups have the EKR-module property</a>, arXiv:1911.11252 [math.CO], 2019.
%H A003221 Piotr Miska, <a href="http://arxiv.org/abs/1508.01987">Arithmetic Properties of the Sequence of Derangements and its Generalizations</a>, arXiv:1508.01987 [math.NT], 2015. (see Chapter 5 p. 44)
%H A003221 J. M. Thomas, <a href="http://dx.doi.org/10.1090/S0002-9904-1925-04036-7">The number of even and odd absolute permutations of n letters</a>, Bull. Amer. Math. Soc. 31 (1925), 303-304.
%F A003221 a(n) = (A000166(n)-(-1)^n*(n-1))/2.
%F A003221 From _Abdullahi Umar_, Oct 09 2008: (Start)
%F A003221 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.
%F A003221 a(n) = (n-1)*(a(n-1)+a(n-2))+((-1)^(n-1))*(n-1) for n>1, a(0)=1, a(1)=0.
%F A003221 a(n) = n*a(n-1)+((-1)^(n-1))*(n-2)*(n+1)/2 for n>0, a(0)=1.
%F A003221 E.g.f.: (1-x^2/2)*exp(-x)/(1-x). (End)
%p A003221 A003221 := n -> ((-1)^n*hypergeom([-n,1],[],1)-(-1)^n*(n-1))/2:
%p A003221 seq(simplify(A003221(n)), n=0..22); # _Peter Luschny_, May 09 2017
%t A003221 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_ *)
%t A003221 Range[0, 25]! CoefficientList[Series[(1 - x^2 / 2) E^(-x) / (1 - x), {x, 0, 25}], x] (* _Vincenzo Librandi_, Aug 11 2015 *)
%o A003221 (Python)
%o A003221 A003221_list, m, x = [], -1, 0
%o A003221 for n in range(10*2):
%o A003221     x, m = x*n + m*(n*(n-1)//2-1), -m
%o A003221     A003221_list.append(x) # _Chai Wah Wu_, Nov 03 2014
%o A003221 (PARI) a(n) = ( n!*sum(r=2, n, (-1)^r/r!) + (-1)^(n-1)*(n-1))/2; \\ _Michel Marcus_, Apr 22 2016
%Y A003221 Cf. A000166, A000387.
%K A003221 nonn,easy,nice
%O A003221 0,4
%A A003221 _N. J. A. Sloane_
%E A003221 Typo in second formula fixed by _Josh Swanson_, Nov 10 2013