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. Two unoriented colorings are the same if congruent; chiral pairs are counted as one.
Also the number of unoriented 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):
1 1
1 4 6 3
1 8 29 52 45 15
1 13 84 297 600 690 420 105
1 19 192 1116 3933 8661 11970 10080 4725 945
1 26 381 3321 18080 63919 150332 236978 247275 163800 62370 10395
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}], {n, 1, 10}, {k, 1, 2n}] // Flatten
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Robert A. Russell, May 27 2019
STATUS
approved