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 A325003 #19 Oct 25 2024 02:25:05
%S A325003 1,0,1,2,0,1,3,3,0,1,4,6,4,0,1,5,10,10,5,0,1,6,15,20,15,6,0,1,7,21,35,
%T A325003 35,21,7,0,1,8,28,56,70,56,28,8,0,1,9,36,84,126,126,84,36,9,0,1,10,45,
%U A325003 120,210,252,210,120,45,10,0,1,11,55,165,330,462,462,330,165,55,11,0
%N A325003 Triangle read by rows: T(n,k) is the number of achiral colorings of the facets (or vertices) of a regular n-dimensional simplex using exactly k colors.
%C A325003 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 triangular 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. An achiral coloring is the same as its reflection. For k <= n all the colorings are achiral.
%C A325003 The final zero in each row indicates no achiral colorings when each facet has a different color.
%H A325003 Robert A. Russell, <a href="/A325003/b325003.txt">Table of n, a(n) for n = 1..1325</a>
%F A325003 T(n,k) = binomial(n,k-1) - [k==n+1] = A007318(n,k-1) - [k==n+1].
%F A325003 T(n,k) = A325002(n,k) - 2*[k==n+1] = 2*A007318(n,k-1) - A325002(n,k).
%F A325003 G.f. for row n: x * (1+x)^n - x^(n+1).
%F A325003 G.f. for column k>1: x^(k-1)/(1-x)^k - x^(k-1).
%e A325003 Triangle begins with T(1,1):
%e A325003 1 0
%e A325003 1 2 0
%e A325003 1 3 3 0
%e A325003 1 4 6 4 0
%e A325003 1 5 10 10 5 0
%e A325003 1 6 15 20 15 6 0
%e A325003 1 7 21 35 35 21 7 0
%e A325003 1 8 28 56 70 56 28 8 0
%e A325003 1 9 36 84 126 126 84 36 9 0
%e A325003 1 10 45 120 210 252 210 120 45 10 0
%e A325003 1 11 55 165 330 462 462 330 165 55 11 0
%e A325003 1 12 66 220 495 792 924 792 495 220 66 12 0
%e A325003 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 0
%e A325003 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 0
%e A325003 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 0
%e A325003 For T(3,2)=3, the tetrahedron may have one, two, or three faces of one color.
%t A325003 Table[Binomial[n, k-1] - Boole[k==n+1], {n,1,15}, {k,1,n+1}] \\ Flatten
%Y A325003 Cf. A325002 (oriented), A007318(n,k-1) (unoriented), A325001 (up to k colors).
%Y A325003 Other n-dimensional polytopes: A325011 (orthotope), A325019 (orthoplex).
%Y A325003 Cf. A198321.
%K A325003 nonn,tabf
%O A325003 1,4
%A A325003 _Robert A. Russell_, Mar 23 2019