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The number of ways to color the faces of a regular dodecahedron with n colors, counting mirror images as one.
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%I #22 Jun 13 2015 00:55:20

%S 1,82,5379,148648,2085655,18356514,116081245,574795936,2359033605,

%T 8345970370,26180606287,74354990568,194253329803,472634761522,

%U 1081541381145,2346163937920,4856060529001,9641643580530,18446420258299,34136541925480,61303301959263

%N The number of ways to color the faces of a regular dodecahedron with n colors, counting mirror images as one.

%C The cycle index using the full automorphism group for faces of a dodecahedron is (x1^12+15*x2^6+20*x3^4+24*x1^2*x5^2+15*x1^4*x2^4+x2^6+20*x6^2+24*x2*x10)/120.

%C Also the number of ways to color the vertices of a regular icosahedron with n colors, counting mirror images as one.

%D F. S. Roberts and B. Tesman, Applied Combinatorics, 2d Ed., Pearson Prentice Hall, 2005, pages 439-488.

%D J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge University Press, 1992, pages 461-474.

%H Colin Barker, <a href="/A252705/b252705.txt">Table of n, a(n) for n = 1..1000</a>

%H <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 a(n) = n^2*(n^2+1)*(n^8-n^6+16*n^4+44)/120.

%F G.f.: x*(x+1)*(x^10+68*x^9+4323*x^8+80508*x^7+469548*x^6+886944*x^5+469548*x^4 +80508*x^3+4323*x^2+68*x+1)/(1-x)^13.

%F a(n) = C(n,1)+80*C(n,2)+5136*C(n,3)+127620*C(n,4)+1395390*C(n,5)+7965948*C(n,6) +26368272*C(n,7)+53438112*C(n,8)+67359600*C(n,9)+51559200*C(n,10)+21954240*C(n,11)+3991680*C(n,12). Each term indicates the number of ways to use n colors to color the dodecahedron with exactly 1, 2, 3, ..., 10, 11, or 12 colors.

%e For n=2, a(2)=82, the number of ways to color the faces of a regular dodecahedron with two colors, counting mirror images as the same. Of these, two use the same color for all faces, and 80 use both colors.

%t Table[n^2(n^2+1)(n^8-n^6+16n^4+44)/120,{n,1,30}]

%o (PARI) vector(60, n, n^2*(n^2+1)*(n^8-n^6+16*n^4+44)/120) \\ _Michel Marcus_, Dec 21 2014

%Y Cf. A000545 (number when mirror images are counted separately).

%Y Cf. A000332 (tetrahedron), A198833 (cube), A128766 (octahedron), A252704 (icosahedron).

%K nonn,easy

%O 1,2

%A _Robert A. Russell_, Dec 20 2014