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 A325002 #13 May 25 2019 11:43:50
%S A325002 1,2,1,2,2,1,3,3,2,1,4,6,4,2,1,5,10,10,5,2,1,6,15,20,15,6,2,1,7,21,35,
%T A325002 35,21,7,2,1,8,28,56,70,56,28,8,2,1,9,36,84,126,126,84,36,9,2,1,10,45,
%U A325002 120,210,252,210,120,45,10,2,1,11,55,165,330,462,462,330,165,55,11,2
%N A325002 Triangle read by rows: T(n,k) is the number of oriented colorings of the facets (or vertices) of a regular n-dimensional simplex using exactly k colors.
%C A325002 For n=1, the figure is a line segment with two vertices. For n=2, the figure is a triangle with three edges. For n=3, the figure is a tetrahedron with four faces. The Schläfli symbol, {3,...,3}, of the regular n-dimensional simplex consists of n-1 threes. Each of its n+1 facets is a regular (n-1)-dimensional simplex. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two. The only chiral pair occurs when k=n+1; for k <= n all the colorings are achiral.
%H A325002 Robert A. Russell, <a href="/A325002/b325002.txt">Table of n, a(n) for n = 1..1325</a>
%F A325002 T(n,k) = binomial(n,k-1) + [k==n+1] = A007318(n,k-1) + [k==n+1].
%F A325002 T(n,k) = 2*A007318(n,k-1) - A325003(n,k) = [k==n+1] + A325003(n,k).
%F A325002 G.f. for row n: x * (1+x)^n + x^(n+1).
%F A325002 G.f. for column k>1: x^(k-1)/(1-x)^k + x^(k-1).
%e A325002 Triangle begins with T(1,1):
%e A325002 1 2
%e A325002 1 2 2
%e A325002 1 3 3 2
%e A325002 1 4 6 4 2
%e A325002 1 5 10 10 5 2
%e A325002 1 6 15 20 15 6 2
%e A325002 1 7 21 35 35 21 7 2
%e A325002 1 8 28 56 70 56 28 8 2
%e A325002 1 9 36 84 126 126 84 36 9 2
%e A325002 1 10 45 120 210 252 210 120 45 10 2
%e A325002 1 11 55 165 330 462 462 330 165 55 11 2
%e A325002 1 12 66 220 495 792 924 792 495 220 66 12 2
%e A325002 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 2
%e A325002 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 2
%e A325002 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 2
%e A325002 For T(3,2)=3, the tetrahedron may have one, two, or three faces of one color.
%t A325002 Table[Binomial[n,k-1] + Boole[k==n+1], {n,1,15}, {k,1,n+1}] \\ Flatten
%Y A325002 Cf. A007318(n,k-1) (unoriented), A325003 (achiral), A325001 (up to k colors).
%Y A325002 Other n-dimensional polytopes: A325008 (orthotope), A325016 (orthoplex).
%K A325002 nonn,tabf
%O A325002 1,2
%A A325002 _Robert A. Russell_, Mar 23 2019