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. An achiral coloring is identical to its reflection.
Also the number of achiral 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..2*n} A325011(n,j) * binomial(k,j).
A(n,k) = 2*A325005(n,k) - A325004(n,k) = (A325004(n,k) - 2*A325006(n,k)) / 2 = A325005(n,k) + A325006(n,k).
G.f. for row n: Sum{j=1..2*n} A325011(n,j) * x^j / (1-x)^(j+1).
Linear recurrence for row n: T(n,k) = Sum_{j=0..2*n} binomial(-2-j,2*n-j) * T(n,k-1-j).
G.f. for column k: 1/(1-x)^binomial(k+1,2) - (1+x)^binomial(k,2).
EXAMPLE
Array begins with A(1,1):
1 2 3 4 5 6 7 8 9 10 ...
1 6 18 40 75 126 196 288 405 550 ...
1 10 55 200 560 1316 2730 5160 9075 15070 ...
1 15 126 700 2850 9261 25480 61776 135675 275275 ...
1 21 252 1996 11376 50127 181027 559728 1529892 3784627 ...
1 28 462 5004 38550 225225 1053304 4119648 13942908 41918800 ...
1 36 792 11440 116160 881595 5263336 25794288 107427420 390891160 ...
For a(2,2)=6, all colorings are achiral: two with just one of the colors, two with one color on just one edge, one with opposite colors the same, and one with opposite colors different.
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
PROG
(PARI) a(n, k) = binomial(binomial(k+1, 2)+n-1, n) - binomial(binomial(k, 2), n)
array(rows, cols) = for(x=1, rows, for(y=1, cols, print1(a(x, y), ", ")); print(""))
/* Print initial 6 rows and 8 columns of array as follows: */
array(6, 8) \\ Felix Fröhlich, May 30 2019
CROSSREFS
KEYWORD
AUTHOR
Robert A. Russell, May 27 2019
STATUS
approved