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%I #9 Feb 23 2017 04:35:00
%S 0,0,0,6,72,375,1320,3675,8736,18522,36000,65340,112200,184041,290472,
%T 443625,658560,953700,1351296,1877922,2565000,3449355,4573800,5987751,
%U 7747872,9918750,12573600,15795000,19675656,24319197,29841000,36369045,44044800,53024136
%N Number of inequivalent ways to color the faces of a cube using at most n colors so that no color appears more than twice.
%C Also the number of inequivalent ways to color the corners of an octahedron using at most n colors so that no color appears more than twice.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = (n-2)^2*(n-1)*n^2*(n+5)/24.
%F G.f.: 3*x^3*(-2-10*x+x^2+x^3)/(x-1)^7 . - _R. J. Mathar_, Feb 23 2017
%e For n=3 we get a(3)=6 ways to color the faces of a cube with three colors so that no color appears more than twice.
%t Table[(3 n (n - 1) (n - 2)^2 + 6 n (n - 1) (n - 2) + n (n - 1) (n - 2) (n - 3) (n - 4) (n - 5) + 15 n (n - 1) (n - 2) (n - 3) (n - 4) + 45 n (n - 1) (n - 2) (n - 3) + 15 n (n - 1) (n - 2))/24, {n, 0, 16}]
%Y Cf. A249460, A282816. A047780 (face colorings without restriction).
%K nonn,easy
%O 0,4
%A _David Nacin_, Feb 21 2017
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