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 A327086 #21 Jun 09 2021 02:31:42
%S A327086 1,2,1,3,4,1,4,9,10,1,5,16,45,28,1,6,25,136,387,128,1,7,36,325,2784,
%T A327086 8352,792,1,8,49,666,13125,186304,382563,7620,1,9,64,1225,46836,
%U A327086 2117750,36507008,44526672,124344
%N A327086 Array read by descending antidiagonals: A(n,k) is the number of achiral colorings of the edges of a regular n-dimensional simplex using up to k colors.
%C A327086 An n-dimensional simplex has n+1 vertices and (n+1)*n/2 edges. For n=1, the figure is a line segment with one edge. For n-2, the figure is a triangle with three edges. For n=3, the figure is a tetrahedron with six edges. The Schläfli symbol, {3,...,3}, of the regular n-dimensional simplex consists of n-1 threes. An achiral coloring is identical to its reflection.
%C A327086 A(n,k) is also the number of achiral colorings of (n-2)-dimensional regular simplices in an n-dimensional simplex using up to k colors. Thus, A(2,k) is also the number of achiral colorings of the vertices (0-dimensional simplices) of an equilateral triangle.
%H A327086 Robert A. Russell, <a href="/A327086/b327086.txt">Table of n, a(n) for n = 1..325</a> First 25 antidiagonals.
%H A327086 Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/html/book/hyl00_42.html">The cycle type of the induced action on 2-subsets</a>
%H A327086 E. M. Palmer and R. W. Robinson, <a href="https://projecteuclid.org/euclid.acta/1485889789">Enumeration under two representations of the wreath product</a>, Acta Math., 131 (1973), 123-143.
%F A327086 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.
%F A327086 A(n,k) = Sum_{j=1..(n+1)*n/2} A327090(n,j) * binomial(k,j).
%F A327086 A(n,k) = 2*A327084(n,k) - A327083(n,k) = A327083(n,k) - 2*A327085(n,k) = A327084(n,k) - A327085(n,k).
%e A327086 Array begins with A(1,1):
%e A327086 1 2 3 4 5 6 7 8 9 10 11 12 ...
%e A327086 1 4 9 16 25 36 49 64 81 100 121 144 ...
%e A327086 1 10 45 136 325 666 1225 2080 3321 5050 7381 10440 ...
%e A327086 1 28 387 2784 13125 46836 137543 349952 797769 1667500 3248971 5973408 ...
%e A327086 ...
%e A327086 For A(2,3) = 9, the colorings are AAA, AAB, AAC, ABB, ACC, BBB, BBC, BCC, and CCC.
%t A327086 CycleX[{2}] = {{1,1}}; (* cycle index for permutation with given cycle structure *)
%t A327086 CycleX[{n_Integer}] := CycleX[n] = If[EvenQ[n], {{n/2,1}, {n,(n-2)/2}}, {{n,(n-1)/2}}]
%t A327086 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 A327086 CycleX[p_List] := CycleX[p] = compress[Join[CycleX[Drop[p, -1]], If[Last[p] > 1, CycleX[{Last[p]}], ## &[]], If[# == Last[p], {#, Last[p]}, {LCM[#, Last[p]], GCD[#, Last[p]]}] & /@ Drop[p, -1]]]
%t A327086 pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
%t A327086 row[n_Integer] := row[n] = Factor[Total[If[EvenQ[Total[1-Mod[#,2]]], 0, pc[#] j^Total[CycleX[#]][[2]]] & /@ IntegerPartitions[n+1]]/((n+1)!/2)]
%t A327086 array[n_, k_] := row[n] /. j -> k
%t A327086 Table[array[n,d-n+1], {d,1,10}, {n,1,d}] // Flatten
%t A327086 (* Using Fripertinger's exponent per Andrew Howroyd's code in A063841: *)
%t A327086 pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))]
%t A327086 ex[v_] := Sum[GCD[v[[i]], v[[j]]], {i,2,Length[v]}, {j,i-1}] + Total[Quotient[v,2]]
%t A327086 array[n_,k_] := Total[If[OddQ[Total[1-Mod[#,2]]], pc[#]k^ex[#], 0] &/@ IntegerPartitions[n+1]]/((n+1)!/2)
%t A327086 Table[array[n,d-n+1], {d,10}, {n,d}] // Flatten
%Y A327086 Cf. A327083 (oriented), A327084 (unoriented), A327085 (chiral), A327090 (exactly k colors), A325001 (vertices, facets), A337886 (faces, peaks), A337410 (orthotope edges, orthoplex ridges), A337414 (orthoplex edges, orthotope ridges).
%Y A327086 Rows 1-4 are A000027, A000290, A037270, A331353.
%K A327086 nonn,tabl
%O A327086 1,2
%A A327086 _Robert A. Russell_, Aug 19 2019