This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A325007 #12 Oct 01 2019 07:16:06
%S A325007 1,2,1,3,6,1,4,18,10,1,5,40,55,15,1,6,75,200,126,21,1,7,126,560,700,
%T A325007 252,28,1,8,196,1316,2850,1996,462,36,1,9,288,2730,9261,11376,5004,
%U A325007 792,45,1,10,405,5160,25480,50127,38550,11440,1287,55,1,11,550,9075,61776,181027,225225,116160,24310,2002,66,1
%N A325007 Array read by descending antidiagonals: A(n,k) is the number of achiral colorings of the facets of a regular n-dimensional orthotope using up to k colors.
%C A325007 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.
%C A325007 Also the number of achiral colorings of the vertices of a regular n-dimensional orthoplex using up to k colors.
%H A325007 Robert A. Russell, <a href="/A325007/b325007.txt">Table of n, a(n) for n = 1..325</a>
%H A325007 Robin Chapman, answer to <a href="https://math.stackexchange.com/q/5732/">Coloring the faces of a hypercube</a>, Math StackExchange, September 30, 2010.
%F A325007 A(n,k) = binomial(binomial(k+1,2) + n-1, n) - binomial(binomial(k,2),n).
%F A325007 A(n,k) = Sum_{j=1..2*n} A325011(n,j) * binomial(k,j).
%F A325007 A(n,k) = 2*A325005(n,k) - A325004(n,k) = (A325004(n,k) - 2*A325006(n,k)) / 2 = A325005(n,k) + A325006(n,k).
%F A325007 G.f. for row n: Sum{j=1..2*n} A325011(n,j) * x^j / (1-x)^(j+1).
%F A325007 Linear recurrence for row n: T(n,k) = Sum_{j=0..2*n} binomial(-2-j,2*n-j) * T(n,k-1-j).
%F A325007 G.f. for column k: 1/(1-x)^binomial(k+1,2) - (1+x)^binomial(k,2).
%e A325007 Array begins with A(1,1):
%e A325007 1 2 3 4 5 6 7 8 9 10 ...
%e A325007 1 6 18 40 75 126 196 288 405 550 ...
%e A325007 1 10 55 200 560 1316 2730 5160 9075 15070 ...
%e A325007 1 15 126 700 2850 9261 25480 61776 135675 275275 ...
%e A325007 1 21 252 1996 11376 50127 181027 559728 1529892 3784627 ...
%e A325007 1 28 462 5004 38550 225225 1053304 4119648 13942908 41918800 ...
%e A325007 1 36 792 11440 116160 881595 5263336 25794288 107427420 390891160 ...
%e A325007 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.
%t A325007 Table[Binomial[Binomial[d-n+2,2]+n-1,n]-Binomial[Binomial[d-n+1,2],n],{d,1,11},{n,1,d}] // Flatten
%o A325007 (PARI) a(n, k) = binomial(binomial(k+1, 2)+n-1, n) - binomial(binomial(k, 2), n)
%o A325007 array(rows, cols) = for(x=1, rows, for(y=1, cols, print1(a(x, y), ", ")); print(""))
%o A325007 /* Print initial 6 rows and 8 columns of array as follows: */
%o A325007 array(6, 8) \\ _Felix Fröhlich_, May 30 2019
%Y A325007 Cf. A325004 (oriented), A325005 (unoriented), A325006 (chiral), A325011 (exactly k colors).
%Y A325007 Other n-dimensional polytopes: A325001 (simplex), A325015 (orthoplex).
%Y A325007 Rows 1-2 are A000027, A002411; column 2 is A186783(n+2).
%K A325007 nonn,tabl,easy
%O A325007 1,2
%A A325007 _Robert A. Russell_, May 27 2019