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A161936 The number of direct isometries that are derangements of the (n-1)-dimensional facets of an n-cube. 3


%S 0,3,14,117,1164,13975,195642,3130281,56345048,1126900971,24791821350,

%T 595003712413,15470096522724,433162702636287,12994881079088594,

%U 415836194530835025,14138430614048390832,508983502105742069971,19341373080018198658878,773654923200727946355141

%N The number of direct isometries that are derangements of the (n-1)-dimensional facets of an n-cube.

%C a(n) plays the same role as A003221 plays for the derangement numbers A000166.

%H Gary Gordon, Elizabeth McMahon, <a href="http://arxiv.org/abs/0906.4253">Moving faces to other places: Facet derangements</a>, arXiv:0906.4253 [math.CO], 2009.

%H Gary Gordon and Elizabeth 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.

%F a(n) = (b(n) + (-1)^n)/2, where b(n) is sequence A000354, i.e., the number of (n-1)-dimensional facet derangements of an n-cube.

%F From _Peter Luschny_, May 09 2017: (Start)

%F a(n) = (-1)^n*(1-n*hypergeom([1,1-n],[],2)).

%F a(n) = (2^n*Gamma(n+1,-1/2)/exp(1/2)+(-1)^n)/2. (End)

%e For a square, the 3 rotations are direct edge derangements. For a 3-cube, the 6 edge-centered rotations and the 8 vertex-centered rotations are direct face derangements.

%p A161936 := n -> (2^n*GAMMA(n+1,-1/2)/exp(1/2)+(-1)^n)/2:

%p seq(A161936(n), n=1..20); # _Peter Luschny_, May 09 2017

%t a[n_] := (-1)^n*(1 - n*HypergeometricPFQ[{1, 1 - n}, {}, 2]);

%t Array[a, 20] (* _Jean-Fran├žois Alcover_, Jul 14 2018, after _Peter Luschny_ *)

%Y Cf. A000354, A161937.

%K easy,nonn

%O 1,2

%A Elizabeth McMahon, Gary Gordon (mcmahone(AT)lafayette.edu), Jun 29 2009

%E More terms from _Peter Luschny_, May 09 2017

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Last modified September 17 17:56 EDT 2019. Contains 327136 sequences. (Running on oeis4.)