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 A337886 #4 Sep 28 2020 21:41:18
%S A337886 1,2,1,3,5,1,4,15,28,1,5,34,387,768,1,6,65,2784,202203,302032,1,7,111,
%T A337886 13125,11230976,7109211078,3098988832,1,8,175,46836,254729375,
%U A337886 9393953524224,50669807706182691,1831011525739328,1
%N A337886 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the triangular faces of a regular n-dimensional simplex using k or fewer colors.
%C A337886 An achiral arrangement is identical to its reflection. An n-simplex has n+1 vertices. For n=2, the figure is a triangle with one triangular face. For n=3, the figure is a tetrahedron with 4 triangular faces. For higher n, the number of triangular faces is C(n+1,3).
%C A337886 Also the number of achiral colorings of the peaks of a regular n-dimensional simplex. A peak of an n-simplex is an (n-3)-dimensional simplex.
%H A337886 E. M. Palmer and R. W. Robinson, <a href="https://doi.org/10.1007/BF02392038">Enumeration under two representations of the wreath product</a>, Acta Math., 131 (1973), 123-143.
%F A337886 The algorithm used in the Mathematica program below assigns each permutation of the vertices to a partition of n+1. It then determines the number of permutations for each partition and the cycle index for each partition using a formula for binary Lyndon words. If the value of m is increased, one can enumerate colorings of higher-dimensional elements beginning with T(m,1).
%F A337886 T(n,k) = A337884(n,k) - A337883(n,k) = A337883(n,k) - 2*A337885(n,k) = A337884(n,k) - A337885(n,k).
%e A337886 Table begins with T(2,1):
%e A337886 1 2 3 4 5 6 7 8 ...
%e A337886 1 5 15 34 65 111 175 260 ...
%e A337886 1 28 387 2784 13125 46836 137543 349952 ...
%e A337886 1 768 202203 11230976 254729375 3267720576 28271133933 183296831488 ...
%e A337886 For T(3,4)=34, the 34 achiral arrangements are AAAA, AAAB, AAAC, AAAD, AABB, AABC, AABD, AACC, AACD, AADD, ABBB, ABBC, ABBD, ABCC, ABDD, ACCC, ACCD, ACDD, ADDD, BBBB, BBBC, BBBD, BBCC, BBCD, BBDD, BCCC, BCCD, BCDD, BDDD, CCCC, CCCD, CCDD, CDDD, and DDDD.
%t A337886 m=2; (* dimension of color element, here a triangular face *)
%t A337886 lw[n_,k_]:=lw[n, k]=DivisorSum[GCD[n,k],MoebiusMu[#]Binomial[n/#,k/#]&]/n (*A051168*)
%t A337886 cxx[{a_, b_},{c_, d_}]:={LCM[a, c], GCD[a, c] b d}
%t A337886 compress[x:{{_, _} ...}] := (s=Sort[x];For[i=Length[s],i>1,i-=1,If[s[[i,1]]==s[[i-1,1]], s[[i-1,2]]+=s[[i,2]]; s=Delete[s,i], Null]]; s)
%t A337886 combine[a : {{_, _} ...}, b : {{_, _} ...}] := Outer[cxx, a, b, 1]
%t A337886 CX[p_List, 0] := {{1, 1}} (* cycle index for partition p, m vertices *)
%t A337886 CX[{n_Integer}, m_] := If[2m>n, CX[{n}, n-m], CX[{n},m] = Table[{n/k, lw[n/k, m/k]}, {k, Reverse[Divisors[GCD[n, m]]]}]]
%t A337886 CX[p_List, m_Integer] := CX[p, m] = Module[{v = Total[p], q, r}, If[2 m > v, CX[p, v - m], q = Drop[p, -1]; r = Last[p]; compress[Flatten[Join[{{CX[q, m]}}, Table[combine[CX[q, m - j], CX[{r}, j]], {j, Min[m, r]}]], 2]]]]
%t A337886 pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
%t A337886 row[n_Integer] := row[n] = Factor[Total[If[OddQ[Total[1-Mod[#, 2]]], pc[#] j^Total[CX[#, m+1]][[2]], 0] & /@ IntegerPartitions[n+1]]/((n+1)!/2)]
%t A337886 array[n_, k_] := row[n] /. j -> k
%t A337886 Table[array[n,d+m-n], {d,8}, {n,m,d+m-1}] // Flatten
%Y A337886 Cf. A337883 (oriented), A337884 (unoriented), A337885 (chiral), A051168 (binary Lyndon words).
%Y A337886 Other elements: A325001 (vertices), A327086 (edges).
%Y A337886 Other polytopes: A337890 (orthotope), A337894 (orthoplex).
%Y A337886 Rows 2-4 are A000027, A006003, A331353.
%K A337886 nonn,tabl
%O A337886 2,2
%A A337886 _Robert A. Russell_, Sep 28 2020