%I #20 Jun 11 2019 06:09:08
%S 1,2,1,4,9,6,1,21,267,1718,5250,7980,5880,1680,1,494,228591,21539424,
%T 685479375,10257064650,86151316860,449772354360,1551283253100,
%U 3661969537800,6015983173200,6878457986400,5371454088000,2733402672000,817296480000,108972864000
%N Triangle read by rows: T(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.
%C Also called cross polytope and hyperoctahedron. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is an octahedron with eight triangular faces. For n=4, the figure is a 16-cell with sixteen tetrahedral facets. The Schläfli symbol, {3,...,3,4}, of the regular n-dimensional orthoplex (n>1) consists of n-2 threes followed by a four. Each of its 2^n facets is an (n-1)-dimensional simplex. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.
%C Also the number of oriented colorings of the vertices of a regular n-dimensional orthotope (cube) using exactly k colors.
%H Robert A. Russell, <a href="/A325016/b325016.txt">Table of n, a(n) for n = 1..510</a>, rows 1..8, flattened.
%H 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.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cross-polytope">Cross-polytope</a>
%F A325012(n,k) = Sum_{j=1..2^n} T(n,j) * binomial(k,j).
%F T(n,k) = A325017(n,k) + A325018(n,k) = 2*A325017(n,k) - A325019(n,k) = 2*A325018(n,k) + A325019(n,k).
%e Triangle begins with T(1,1):
%e 1 2
%e 1 4 9 6
%e 1 21 267 1718 5250 7980 5880 1680
%e For T(2,2)=4, two squares have three edges the same color, one has opposite edges the same color, and one has opposite edges different colors.
%t a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&, n, EvenQ], MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)
%t a37[n_] := a37[n] = DivisorSum[n, MoebiusMu[n/#]2^#&]/n; (* A001037 *)
%t CI0[{n_Integer}] := CI0[{n}] = CI[Transpose[If[EvenQ[n], p2 = IntegerExponent[n, 2]; sub = Divisors[n/2^p2]; {2^(p2+1) sub, a48 /@ (2^p2 sub) }, sub = Divisors[n]; {sub, a37 /@ sub}]]] 2^(n-1); (* even perm. *)
%t CI1[{n_Integer}] := CI1[{n}] = CI[sub = Divisors[n]; Transpose[If[EvenQ[n], {sub, a37 /@ sub}, {2 sub, (a37 /@ sub)/2}]]] 2^(n-1); (* odd perm. *)
%t 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 cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};
%t Unprotect[Times]; Times[CI[a_List], CI[b_List]] := (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];
%t CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]
%t CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]
%t pc[p_List] := Module[{ci,mb},mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; n!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
%t row[n_Integer] := row[n] = Factor[(Total[(CI0[#] pc[#]) & /@ IntegerPartitions[n]])/(n! 2^(n - 1))] /. CI[l_List] :> j^(Total[l][[2]])
%t array[n_, k_] := row[n] /. j -> k (* A325012 *)
%t Table[LinearSolve[Table[Binomial[i,j],{i,1,2^n},{j,1,2^n}],Table[array[n,k],{k,1,2^n}]],{n,1,6}] // Flatten
%Y Cf. A325017 (unoriented), A325018 (chiral), A325019 (achiral), A325012 (up to k colors).
%Y Other n-dimensional polytopes: A325002 (simplex), A325008 (orthotope).
%Y Cf. A000048, A001037.
%K nonn,tabf
%O 1,2
%A _Robert A. Russell_, May 28 2019