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 A325001 #21 Oct 20 2020 08:17:16
%S A325001 1,2,1,3,4,1,4,9,5,1,5,16,15,6,1,6,25,34,21,7,1,7,36,65,56,28,8,1,8,
%T A325001 49,111,125,84,36,9,1,9,64,175,246,210,120,45,10,1,10,81,260,441,461,
%U A325001 330,165,55,11,1,11,100,369,736,917,792,495,220,66,12,1
%N A325001 Array read by descending antidiagonals: A(n,k) is the number of achiral colorings of the facets (or vertices) of a regular n-dimensional simplex using up to k colors.
%C A325001 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.
%H A325001 Robert A. Russell, <a href="/A325001/b325001.txt">Table of n, a(n) for n = 1..1275</a>
%F A325001 A(n,k) = binomial(n+k,n+1) - binomial(k,n+1).
%F A325001 A(n,k) = Sum_{j=1..n} A325003(n,j) * binomial(k,j).
%F A325001 A(n,k) = 2*A325000(n,k) - A324999(n,k) = A324999(n,k) - 2*A325000(n,k-n) = A325000(n,k) - A325000(n,k-n).
%F A325001 G.f. for row n: (x - x^(n+1)) / (1-x)^(n+2).
%F A325001 Linear recurrence for row n: A(n,k) = Sum_{j=1..n+1} -binomial(j-n-2,j) * A(n,k-j).
%F A325001 G.f. for column k: (1 - (1-x^2)^k) / (x*(1-x)^k).
%e A325001 The array begins with A(1,1):
%e A325001 1 2 3 4 5 6 7 8 9 10 11 12 13 ...
%e A325001 1 4 9 16 25 36 49 64 81 100 121 144 169 ...
%e A325001 1 5 15 34 65 111 175 260 369 505 671 870 1105 ...
%e A325001 1 6 21 56 125 246 441 736 1161 1750 2541 3576 4901 ...
%e A325001 1 7 28 84 210 461 917 1688 2919 4795 7546 11452 16848 ...
%e A325001 1 8 36 120 330 792 1715 3424 6399 11320 19118 31032 48672 ...
%e A325001 1 9 45 165 495 1287 3003 6434 12861 24265 43593 75087 124683 ...
%e A325001 1 10 55 220 715 2002 5005 11440 24309 48610 92323 167740 293215 ...
%e A325001 ...
%e A325001 For A(2,2)=4, the triangle may have 0, 1, 2, or 3 edges of one color.
%t A325001 Table[Binomial[d+1,n+1] - Binomial[d+1-n,n+1], {d,1,15}, {n,1,d}] // Flatten
%Y A325001 Cf. A324999 (oriented), A325000 (unoriented), A325000(n,k-n) (chiral), A325003 (exactly k colors), A327086 (edges, ridges), A337886 (faces, peaks), A325007 (orthotope facets, orthoplex vertices), A325015 (orthoplex facets, orthotope vertices).
%Y A325001 Rows 1-4 are A000027, A000290, A006003, A132366(n-1).
%Y A325001 Column 2 is A162880.
%K A325001 nonn,tabl
%O A325001 1,2
%A A325001 _Robert A. Russell_, Mar 23 2019