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 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(binomial(k+1,2) + n-1, n) + binomial(binomial(k,2),n).
A(n,k) = Sum_{j=1..2n} A325008(n,j) * binomial(k,j).
A(n,k) = A325005(n,k) + A325006(n,k) = 2*A325005(n,k) - A325007(n,k) = 2*A325006(n,k) + A325007(n,k).
G.f. for row n: Sum{j=1..2n} A325008(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+x)^binomial(k,2) - 2.
EXAMPLE
Array begins with A(1,1):
1 4 9 16 25 36 49 64 81 100 ...
1 6 24 70 165 336 616 1044 1665 2530 ...
1 10 57 240 800 2226 5390 11712 23355 43450 ...
1 15 126 730 3270 11991 37450 102726 253485 573265 ...
1 21 252 2008 11880 56133 221725 756288 2283876 6228145 ...
1 28 462 5006 38970 235235 1161832 4873128 17838492 58208920 ...
1 36 792 11440 116400 894465 5495896 28162368 124122780 481650400 ...
1 45 1287 24310 319815 3114540 23739310 148116618 782798490 3596651740 ...
For A(1,2) = 4, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses two colors.
MATHEMATICA
Table[Binomial[Binomial[d-n+2, 2]+n-1, n]+Binomial[Binomial[d-n+1, 2], n], {d, 1, 11}, {n, 1, d}] // Flatten
CROSSREFS
KEYWORD
AUTHOR
Robert A. Russell, Mar 23 2019
STATUS
approved