OFFSET
1,2
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. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.
Also the number of oriented 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
Triangle begins with T(1,1):
1 2
1 4 9 6
1 8 30 68 75 30
1 13 84 312 735 1020 735 210
1 19 192 1122 4155 10242 16380 15960 8505 1890
1 26 381 3322 18285 67679 173936 308056 363825 270900 114345 20790
For T(2,2)=4, there are two squares with just one edge for one color, one square with opposite edges the same color, and one square with opposite edges different colors.
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