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 A338116 #12 Oct 14 2020 10:38:31
%S A338116 1,1,3,3,0,1,26,306,1400,2800,2520,840,0,0,0,1,766,199902,10426768,
%T A338116 200588850,1903776420,10360383600,35133957600,77643846000,
%U A338116 113816253600,109880971200,67199932800,23610787200,3632428800,0,0,0,0,0,0
%N A338116 Triangle read by rows: T(n,k) is the number of achiral colorings of the faces (and peaks) of a regular n-dimensional simplex using exactly k colors. Row n has C(n+1,3) columns.
%C A338116 An n-dimensional simplex has n+1 vertices, C(n+1,3) faces, and C(n+1,3) peaks, which are (n-3)-dimensional simplexes. For n=2, the figure is a triangle with one face. For n=3, the figure is a tetrahedron with four triangular faces and four peaks (vertices). For n=4, the figure is a 4-simplex with ten triangular faces and ten peaks (edges). The Schläfli symbol {3,...,3}, of the regular n-dimensional simplex consists of n-1 3's. An achiral coloring is identical to its reflection.
%C A338116 The algorithm used in the Mathematica program below assigns each permutation of the vertices to a cycle-structure partition of n+1. It then determines the number of permutations for each partition and the cycle index for each partition. If the value of m is increased, one can enumerate colorings of higher-dimensional elements beginning with T(m,1).
%H A338116 G. Royle, <a href="http://teaching.csse.uwa.edu.au/units/CITS7209/partition.pdf">Partitions and Permutations</a>
%F A338116 A337886(n,k) = Sum_{j=1..C(n+1,3)} T(n,j) * binomial(k,j).
%F A338116 T(n,k) = 2*A338114(n,k) - A338113(n,k) = A338113(n,k) - 2*A338115(n,k) = A338114(n,k) - A338115(n,k).
%F A338116 T(3,k) = A325003(3,k); T(4,k) = A327090(4,k).
%e A338116 Triangle begins with T(2,1):
%e A338116 1
%e A338116 1 3 3 0
%e A338116 1 26 306 1400 2800 2520 840 0 0 0
%e A338116 1 766 199902 10426768 200588850 1903776420 10360383600 35133957600 ...
%e A338116 ...
%e A338116 For T(3,3)=3, one of the three colors appears on two faces (vertices) of the tetrahedron.
%t A338116 m=2; (* dimension of color element, here a triangular face *)
%t A338116 lw[n_, k_]:=lw[n, k]=DivisorSum[GCD[n, k], MoebiusMu[#]Binomial[n/#, k/#]&]/n (*A051168*)
%t A338116 cxx[{a_, b_}, {c_, d_}]:={LCM[a, c], GCD[a, c] b d}
%t A338116 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 A338116 combine[a : {{_, _} ...}, b : {{_, _} ...}] := Outer[cxx, a, b, 1]
%t A338116 CX[p_List, 0] := {{1, 1}} (* cycle index for partition p, m vertices *)
%t A338116 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 A338116 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 A338116 pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
%t A338116 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 A338116 array[n_, k_] := row[n] /. j -> k
%t A338116 Table[LinearSolve[Table[Binomial[i,j],{i,Binomial[n+1,m+1]},{j,Binomial[n+1,m+1]}], Table[array[n,k],{k,Binomial[n+1,m+1]}]], {n,m,m+4}] // Flatten
%Y A338116 Cf. A338113 (oriented), A338114 (unoriented), A338115 (chiral), A337886 (k or fewer colors), A325003 (vertices and facets), A327090 (edges and ridges).
%K A338116 nonn,tabf
%O A338116 2,3
%A A338116 _Robert A. Russell_, Oct 10 2020