OFFSET
1,4
COMMENTS
Also called hypercube, n-dimensional cube, and measure polytope. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is a cube with six square faces. For n=4, the figure is a tesseract with eight cubic facets. The Schläfli symbol, {4,3,...,3}, of the regular n-dimensional orthotope (n>1) consists of a four followed by n-2 threes. Each of its 2n facets is an (n-1)-dimensional orthotope. An achiral coloring is identical to its reflection.
Also the number of achiral colorings of the vertices of a regular n-dimensional orthoplex using exactly k colors.
LINKS
Robert A. Russell, Table of n, a(n) for n = 1..132
FORMULA
EXAMPLE
Table begins with T(1,1):
1 0
1 4 3 0
1 8 28 36 15 0
1 13 84 282 465 360 105 0
1 19 192 1110 3711 7080 7560 4200 945 0
1 26 381 3320 17875 60159 126728 165900 130725 56700 10395 0
For T(2,3)=3, each of the three chiral pairs has two opposite edges with the same color.
MATHEMATICA
Table[Sum[Binomial[-j-2, k-j-1] Binomial[n + Binomial[j+2, 2]-1, n], {j, 0, k-1}] - Sum[Binomial[j-k-1, j] Binomial[Binomial[k-j, 2], n], {j, 0, k-2}], {n, 1, 10}, {k, 1, 2n}] // Flatten
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Robert A. Russell, May 27 2019
STATUS
approved