login
A161937
The number of indirect isometries that are derangements of the (n-1)-dimensional facets of an n-cube.
2
1, 2, 15, 116, 1165, 13974, 195643, 3130280, 56345049, 1126900970, 24791821351, 595003712412, 15470096522725, 433162702636286, 12994881079088595, 415836194530835024, 14138430614048390833, 508983502105742069970, 19341373080018198658879, 773654923200727946355140
OFFSET
1,2
COMMENTS
a(n) plays the same role as A000387 plays for the derangement numbers A000166.
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.
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
EXAMPLE
For a square, the 2 diagonal reflections are indirect edge derangements. For a 3-cube, the 15 rotary reflections are indirect face derangements.
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 *)
CROSSREFS
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