OFFSET
1,5
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. The chiral colorings of its facets come in pairs, each the reflection of the other.
Also the number of chiral pairs of 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
The triangle begins with T(1,1):
0 1
0 0 3 3
0 0 1 16 30 15
0 0 0 15 135 330 315 105
0 0 0 6 222 1581 4410 5880 3780 945
0 0 0 1 205 3760 23604 71078 116550 107100 51975 10395
0 0 0 0 120 5715 73755 427260 1351980 2552130 2962575 2079000 810810 135135
For T(2,3)=3, the three squares have the two edges with the same color adjacent.
MATHEMATICA
Table[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