A161937 The number of indirect isometries that are derangements of the (n-1)-dimensional facets of an n-cube.

1, 2, 15, 116, 1165, 13974, 195643, 3130280, 56345049, 1126900970, 24791821351, 595003712412, 15470096522725, 433162702636286, 12994881079088595, 415836194530835024, 14138430614048390833, 508983502105742069970, 19341373080018198658879, 773654923200727946355140

Offset: 1

Author

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

Keywords

• easy
• nonn

Comments

a(n) plays the same role as A000387 plays for the derangement numbers A000166.

Examples

			For a square, the 2 diagonal reflections are indirect edge derangements. For a 3-cube, the 15 rotary reflections are indirect face derangements.
		

Links

• G. Gordon and E. 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.

Crossrefs

Cf. A000354, A000387, A161936.

Programs

Maple

a := n -> (-1)^(n+1)*n*hypergeom([1,1-n],[],2):
seq(simplify(a(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 *)

Formula

a(n) = (b(n) + (-1)^(n+1))/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)

E.g.f.: x*exp(-x) / (1 - 2*x). - Ilya Gutkovskiy, May 23 2020

Extensions

More terms from Peter Luschny, May 09 2017