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

0, 3, 14, 117, 1164, 13975, 195642, 3130281, 56345048, 1126900971, 24791821350, 595003712413, 15470096522724, 433162702636287, 12994881079088594, 415836194530835025, 14138430614048390832, 508983502105742069971, 19341373080018198658878, 773654923200727946355141

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..20.

Gary Gordon, Elizabeth McMahon, Moving faces to other places: Facet derangements, arXiv:0906.4253 [math.CO], 2009.

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

Formula

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.

From Peter Luschny, May 09 2017: (Start)

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

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

Example

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.

Maple

A161936 := n -> (2^n*GAMMA(n+1, -1/2)/exp(1/2)+(-1)^n)/2:
seq(A161936(n), n=1..20); # Peter Luschny, May 09 2017

Mathematica

a[n_] := (-1)^n*(1 - n*HypergeometricPFQ[{1, 1 - n}, {}, 2]);
Array[a, 20] (* Jean-François Alcover, Jul 14 2018, after Peter Luschny *)

Crossrefs

Cf. A000354, A161937.

Sequence in context: A256159 A122081 A007140 * A304983 A349013 A260887

Adjacent sequences: A161933 A161934 A161935 * A161937 A161938 A161939

Keyword

easy,nonn

Author

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

Extensions

More terms from Peter Luschny, May 09 2017

Status

approved