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 A325004 #12 May 25 2019 11:44:14
%S A325004 1,4,1,9,6,1,16,24,10,1,25,70,57,15,1,36,165,240,126,21,1,49,336,800,
%T A325004 730,252,28,1,64,616,2226,3270,2008,462,36,1,81,1044,5390,11991,11880,
%U A325004 5006,792,45,1,100,1665,11712,37450,56133,38970,11440,1287,55,1
%N A325004 Array read by descending antidiagonals: A(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthotope using up to k colors.
%C A325004 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.
%C A325004 Also the number of oriented colorings of the vertices of a regular n-dimensional orthoplex using up to k colors.
%H A325004 Robert A. Russell, <a href="/A325004/b325004.txt">Table of n, a(n) for n = 1..325</a>
%H A325004 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 A325004 A(n,k) = binomial(binomial(k+1,2) + n-1, n) + binomial(binomial(k,2),n).
%F A325004 A(n,k) = Sum_{j=1..2n} A325008(n,j) * binomial(k,j).
%F A325004 A(n,k) = A325005(n,k) + A325006(n,k) = 2*A325005(n,k) - A325007(n,k) = 2*A325006(n,k) + A325007(n,k).
%F A325004 G.f. for row n: Sum{j=1..2n} A325008(n,j) * x^j / (1-x)^(j+1).
%F A325004 Linear recurrence for row n: T(n,k) = Sum_{j=0..2n} binomial(-2-j,2n-j) * T(n,k-1-j).
%F A325004 G.f. for column k: 1/(1-x)^binomial(k+1,2) + (1+x)^binomial(k,2) - 2.
%e A325004 Array begins with A(1,1):
%e A325004 1 4 9 16 25 36 49 64 81 100 ...
%e A325004 1 6 24 70 165 336 616 1044 1665 2530 ...
%e A325004 1 10 57 240 800 2226 5390 11712 23355 43450 ...
%e A325004 1 15 126 730 3270 11991 37450 102726 253485 573265 ...
%e A325004 1 21 252 2008 11880 56133 221725 756288 2283876 6228145 ...
%e A325004 1 28 462 5006 38970 235235 1161832 4873128 17838492 58208920 ...
%e A325004 1 36 792 11440 116400 894465 5495896 28162368 124122780 481650400 ...
%e A325004 1 45 1287 24310 319815 3114540 23739310 148116618 782798490 3596651740 ...
%e A325004 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.
%t A325004 Table[Binomial[Binomial[d-n+2,2]+n-1,n]+Binomial[Binomial[d-n+1,2],n],{d,1,11},{n,1,d}] // Flatten
%Y A325004 Cf. A325005 (unoriented), A325006 (chiral), A325007 (achiral), A325008 (exactly k colors)
%Y A325004 Other n-dimensional polytopes: A324999 (simplex), A325012 (orthoplex)
%Y A325004 Rows 1-3 are A000290, A006528, A047780.
%K A325004 nonn,tabl,easy
%O A325004 1,2
%A A325004 _Robert A. Russell_, Mar 23 2019