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 A325008 #5 May 27 2019 17:54:22
%S A325008 1,2,1,4,9,6,1,8,30,68,75,30,1,13,84,312,735,1020,735,210,1,19,192,
%T A325008 1122,4155,10242,16380,15960,8505,1890,1,26,381,3322,18285,67679,
%U A325008 173936,308056,363825,270900,114345,20790,1,34,687,8484,66765,352359,1305612,3479268,6668865,9035460,8378370,5031180,1756755,270270
%N A325008 Triangle read by rows: T(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthotope using exactly k colors. Row n has 2n columns.
%C A325008 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 A325008 Also the number of oriented colorings of the vertices of a regular n-dimensional orthoplex using exactly k colors.
%H A325008 Robert A. Russell, <a href="/A325008/b325008.txt">Table of n, a(n) for n = 1..132</a>
%F A325008 T(n,k) = Sum_{j=0..k-1} binomial(-j-2,k-j-1) * binomial(n + binomial(j+2,2)-1, n) + Sum_{j=0..k-2} binomial(j-k-1,j) * binomial(binomial(k-j,2),n).
%F A325008 T(n,k) = A325009(n,k) + A325010(n,k) = (A325009(n,k) + A325011(n,k)) / 2 = 2*A325010(n,k) + A325011(n,k).
%e A325008 Triangle begins with T(1,1):
%e A325008 1 2
%e A325008 1 4 9 6
%e A325008 1 8 30 68 75 30
%e A325008 1 13 84 312 735 1020 735 210
%e A325008 1 19 192 1122 4155 10242 16380 15960 8505 1890
%e A325008 1 26 381 3322 18285 67679 173936 308056 363825 270900 114345 20790
%e A325008 For T(2,2)=4, there are two squares with just one edge for one color, one square with opposite edges the same color, and one square with opposite edges different colors.
%t A325008 Table[Sum[Binomial[-j-2,k-j-1] Binomial[n + Binomial[j+2,2]-1, n], {j,0,k-1}] + Sum[Binomial[j-k-1,j] Binomial[Binomial[k-j,2],n],{j,0,k-2}], {n,1,10},{k,1,2n}] // Flatten
%Y A325008 Cf. A325009 (unoriented), A325010 (chiral), A325011 (achiral), A325004 (up to k colors).
%Y A325008 Other n-dimensional polytopes: A325002 (simplex), A325016 (orthoplex).
%K A325008 nonn,tabf,easy
%O A325008 1,2
%A A325008 _Robert A. Russell_, May 27 2019