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 A337960 #9 Mar 09 2024 11:13:26
%S A337960 1,1048,133875,4211872,61198135,545203800,3465030541,17197766272,
%T A337960 70665499413,250166670040,785039389519,2230057075104,5826818931739,
%U A337960 14178299017624,32446195329465,70387069393408,145689159233737
%N A337960 Number of achiral colorings of the 30 triangular faces of a regular icosahedron or the 30 vertices of a regular dodecahedron using n or fewer colors.
%C A337960 An achiral coloring is identical to its reflection. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.
%C A337960 There are 60 elements in the automorphism group of the regular dodecahedron/icosahedron that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C A337960 Conjugacy Class Count Odd Cycle Indices
%C A337960 Inversion 1 x_2^10
%C A337960 Edge rotation* 15 x_1^4x_2^8 Asterisk indicates that the
%C A337960 Vertex rotation* 20 x_2^1x_6^3 operation is followed by an
%C A337960 Small face rotation* 12 x_10^2 inversion.
%C A337960 Large face rotation* 12 x_10^2
%H A337960 <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 A337960 a(n) = n^2 * (15*n^10 + n^8 + 20*n^2 + 24) / 60.
%F A337960 a(n) = 1*C(n,1) + 1046*C(n,2) + 130734*C(n,3) + 3682656*C(n,4) + 41467050*C(n,5) + 238531284*C(n,6) + 791012880*C(n,7) + 1603496160*C(n,8) + 2021060160*C(n,9) + 1546836480*C(n,10) + 658627200*C(n,11) + 119750400*C(n,12), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
%F A337960 a(n) = 2*A252704(n) - A054472(n) = A054472(n) - 2*A337959(n) = A252704(n) - A337959(n).
%t A337960 Table[(15n^12+n^10+20n^4+24n^2)/60,{n,30}]
%Y A337960 Cf. A054472 (oriented), A252704 (unoriented), A337959 (chiral).
%Y A337960 Other elements: A337953 (edges), A337962 (dodecahedron faces, icosahedron vertices).
%Y A337960 Other polyhedra: A006003 (tetrahedron), A337898 (cube faces, octahedron vertices), A337897 (octahedron faces, cube vertices).
%K A337960 nonn,easy
%O A337960 1,2
%A A337960 _Robert A. Russell_, Oct 03 2020