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 A325009 #5 May 27 2019 18:02:07
%S A325009 1,1,1,4,6,3,1,8,29,52,45,15,1,13,84,297,600,690,420,105,1,19,192,
%T A325009 1116,3933,8661,11970,10080,4725,945,1,26,381,3321,18080,63919,150332,
%U A325009 236978,247275,163800,62370,10395,1,34,687,8484,66645,346644,1231857,3052008,5316885,6483330,5415795,2952180,945945,135135
%N A325009 Triangle read by rows: T(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthotope using exactly k colors. Row n has 2n columns.
%C A325009 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 A325009 Also the number of unoriented colorings of the vertices of a regular n-dimensional orthoplex using exactly k colors.
%H A325009 Robert A. Russell, <a href="/A325009/b325009.txt">Table of n, a(n) for n = 1..132</a>
%F A325009 T(n,k) = Sum{j=0..k-1} binomial(-j-2, k-j-1) * binomial(n+binomial(j+2, 2)-1, n).
%F A325009 T(n,k) = A325009(n,k) + A325010(n,k) = 2*A325009(n,k) - A325011(n,k) = 2*A325010(n,k) + A325011(n,k).
%e A325009 The triangle begins with T(1,1):
%e A325009 1 1
%e A325009 1 4 6 3
%e A325009 1 8 29 52 45 15
%e A325009 1 13 84 297 600 690 420 105
%e A325009 1 19 192 1116 3933 8661 11970 10080 4725 945
%e A325009 1 26 381 3321 18080 63919 150332 236978 247275 163800 62370 10395
%e A325009 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 A325009 Table[Sum[Binomial[-j-2,k-j-1]Binomial[n+Binomial[j+2,2]-1,n],{j,0,k-1}],{n,1,10},{k,1,2n}] // Flatten
%Y A325009 Cf. A325008 (oriented), A325010 (chiral), A325011 (achiral), A325005 (up to k colors).
%Y A325009 Other n-dimensional polytopes: A007318(n,k-1) (simplex), A325017 (orthoplex).
%K A325009 nonn,tabf,easy
%O A325009 1,4
%A A325009 _Robert A. Russell_, May 27 2019