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 A325005 #16 Aug 06 2019 01:13:11
%S A325005 1,3,1,6,6,1,10,21,10,1,15,55,56,15,1,21,120,220,126,21,1,28,231,680,
%T A325005 715,252,28,1,36,406,1771,3060,2002,462,36,1,45,666,4060,10626,11628,
%U A325005 5005,792,45,1,55,1035,8436,31465,53130,38760,11440,1287,55,1
%N A325005 Array read by descending antidiagonals: A(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthotope using up to k colors.
%C A325005 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.
%C A325005 Also the number of unoriented colorings of the vertices of a regular n-dimensional orthoplex using up to k colors.
%H A325005 Robert A. Russell, <a href="/A325005/b325005.txt">Table of n, a(n) for n = 1..325</a>
%H A325005 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 A325005 A(n,k) = binomial(n + binomial(k+1,2) - 1, n).
%F A325005 A(n,k) = Sum_{j=1..2n} A325009(n,j) * binomial(k,j).
%F A325005 A(n,k) = A325004(n,k) - A325006(n,k) = (A325004(n,k) + A325007(n,k)) / 2 = A325006(n,k) + A325007(n,k).
%F A325005 G.f. for row n: Sum_{j=1..2n} A325009(n,j) * x^j / (1-x)^(j+1).
%F A325005 Linear recurrence for row n: T(n,k) = Sum_{j=0..2n} binomial(-2-j,2n-j) * T(n,k-1-j).
%F A325005 G.f. for column k: 1/(1-x)^binomial(k+1,2) - 1.
%e A325005 Array begins with A(1,1):
%e A325005 1 3 6 10 15 21 28 36 45 55 ...
%e A325005 1 6 21 55 120 231 406 666 1035 1540 ...
%e A325005 1 10 56 220 680 1771 4060 8436 16215 29260 ...
%e A325005 1 15 126 715 3060 10626 31465 82251 194580 424270 ...
%e A325005 1 21 252 2002 11628 53130 201376 658008 1906884 5006386 ...
%e A325005 1 28 462 5005 38760 230230 1107568 4496388 15890700 50063860 ...
%e A325005 1 36 792 11440 116280 888030 5379616 26978328 115775100 436270780 ...
%e A325005 1 45 1287 24310 319770 3108105 23535820 145008513 752538150 3381098545 ...
%e A325005 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.
%t A325005 Table[Binomial[Binomial[d-n+2,2]+n-1,n],{d,1,11},{n,1,d}] // Flatten
%Y A325005 Cf. A325004 (oriented), A325006 (chiral), A325007 (achiral), A325009 (exactly k colors).
%Y A325005 Other n-dimensional polytopes: A325000 (simplex), A325013 (orthoplex).
%Y A325005 Rows 1-3 are A000217, A002817, A198833.
%K A325005 nonn,tabl,easy
%O A325005 1,2
%A A325005 _Robert A. Russell_, Mar 23 2019