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%I A199406 #25 Oct 18 2020 09:54:54
%S A199406 1,144,12111,358120,5131650,45528756,288936634,1433251296,5887880415,
%T A199406 20842168600,65402344161,185788177224,485443851256,1181242399260,
%U A199406 2703252560100,5864398969216,12138503871789,24101498435616,46112016365155,85335258695400,153249227870046
%N A199406 The number of inequivalent ways to color the edges of a cube using at most n colors.
%C A199406 Two edge colorings are equivalent if one is the mirror image of the other or the cube can be picked up and rotated in any manner to obtain the other.
%C A199406 The group here has order 48 (compare A060530). - _N. J. A. Sloane_, Aug 14 2012
%C A199406 Also the number of unoriented colorings of the 12 edges of a regular octahedron with n or fewer colors. The Schläfli symbols of the cube and octahedron are {4,3} and {3,4} respectively. They are mutually dual. For an unoriented coloring, chiral pairs are counted as one. - _Robert A. Russell_, Oct 17 2020
%H A199406 T. D. Noe, <a href="/A199406/b199406.txt">Table of n, a(n) for n = 1..1000</a>
%H A199406 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
%F A199406 a(n) = n^12/48 + n^8/16 + n^7/4 + n^6/12 + n^4/6 + n^3/4 + n^2/6.
%F A199406 Cycle index = 1/48(s_1^12+3s_1^4s_2^4+12s_1^2s_2^5+4s_2^6+8s_3^4+12s_4^3+8s_6^2).
%F A199406 G.f.: -x*(76*x^10 +10016*x^9 +212772*x^8 +1380453*x^7 +3384939*x^6 +3388593*x^5 +1380279*x^4 +211623*x^3 +10317*x^2 +131*x +1)/(x -1)^13. [_Colin Barker_, Aug 13 2012]
%F A199406 From _Robert A. Russell_, Oct 17 2020: (Start)
%F A199406 a(n) = A060530(n) - A337406(n) = (A060530(n) + A331351(n)) / 2 = A337406(n) + A331351(n).
%F A199406 a(n) = 1*C(n,1) + 142*C(n,2) + 11682*C(n,3) + 310536*C(n,4) + 3460725*C(n,5) + 19870590*C(n,6) + 65886660*C(n,7) + 133585200*C(n,8) + 168399000*C(n,9) + 128898000*C(n,10) + 54885600*C(n,11) + 9979200*C(n,12), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors. (End)
%t A199406 Table[CycleIndex[KSubsetGroup[Automorphisms[CubicalGraph], Edges[CubicalGraph]],s] /. Table[s[i]->n, {i,1,6}], {n,1,15}]
%t A199406 Table[(8n^2+12n^3+8n^4+4n^6+12n^7+3n^8+n^12)/48, {n,20}] (* _Robert A. Russell_, Oct 17 2020 *)
%Y A199406 Cf. A060530 (oriented), A337406 (chiral), A331351 (achiral), A128766 (cube vertices, octahedron faces), A198833 (cube faces, octahedron vertices), A063842(n-1) (tetrahedron), A337963 (dodecahedron, icosahedron).
%Y A199406 Row 3 of A337408 (orthotope edges, orthoplex ridges) and A337412 (orthoplex edges, orthotope ridges).
%K A199406 nonn,easy
%O A199406 1,2
%A A199406 _Geoffrey Critzer_, Nov 05 2011

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