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 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 up to k colors.
LINKS
Robert A. Russell, Table of n, a(n) for n = 1..325
Robin Chapman, answer to Coloring the faces of a hypercube, Math StackExchange, September 30, 2010.
FORMULA
A(n,k) = binomial(n + binomial(k+1,2) - 1, n).
A(n,k) = Sum_{j=1..2n} A325009(n,j) * binomial(k,j).
A(n,k) = A325004(n,k) - A325006(n,k) = (A325004(n,k) + A325007(n,k)) / 2 = A325006(n,k) + A325007(n,k).
G.f. for row n: Sum_{j=1..2n} A325009(n,j) * x^j / (1-x)^(j+1).
Linear recurrence for row n: T(n,k) = Sum_{j=0..2n} binomial(-2-j,2n-j) * T(n,k-1-j).
G.f. for column k: 1/(1-x)^binomial(k+1,2) - 1.
EXAMPLE
Array begins with A(1,1):
1 3 6 10 15 21 28 36 45 55 ...
1 6 21 55 120 231 406 666 1035 1540 ...
1 10 56 220 680 1771 4060 8436 16215 29260 ...
1 15 126 715 3060 10626 31465 82251 194580 424270 ...
1 21 252 2002 11628 53130 201376 658008 1906884 5006386 ...
1 28 462 5005 38760 230230 1107568 4496388 15890700 50063860 ...
1 36 792 11440 116280 888030 5379616 26978328 115775100 436270780 ...
1 45 1287 24310 319770 3108105 23535820 145008513 752538150 3381098545 ...
For A(1,2) = 3, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses one color for each vertex.
MATHEMATICA
Table[Binomial[Binomial[d-n+2, 2]+n-1, n], {d, 1, 11}, {n, 1, d}] // Flatten
CROSSREFS
KEYWORD
AUTHOR
Robert A. Russell, Mar 23 2019
STATUS
approved