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 A337953 #9 Mar 08 2024 12:09:08
%S A337953 1,33328,32524281,4312863360,191243490675,4239501280272,
%T A337953 58236754527707,563536359633920,4172726943804861,25016666666700400,
%U A337953 126431377927701253,554909560378102656,2163457078062360639,7625429483925609552,24638829565429941975
%N A337953 Number of achiral colorings of the 30 edges of a regular dodecahedron or icosahedron using n or fewer colors.
%C A337953 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 A337953 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 edge cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C A337953 Conjugacy Class Count Odd Cycle Indices
%C A337953 Inversion 1 x_2^15
%C A337953 Edge rotation* 15 x_1^4x_2^13 Asterisk indicates that the
%C A337953 Vertex rotation* 20 x_6^5 operation is followed by an
%C A337953 Small face rotation* 12 x_10^3 inversion.
%C A337953 Large face rotation* 12 x_10^3
%H A337953 <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (18, -153, 816, -3060, 8568, -18564, 31824, -43758, 48620, -43758, 31824, -18564, 8568, -3060, 816, -153, 18, -1).
%F A337953 a(n) = n^3 * (15*n^14 + n^12 + 20*n^2 + 24) / 60.
%F A337953 a(n) = 1*C(n,1) + 33326*C(n,2) + 32424300*C(n,3) + 4182966200*C(n,4) + 170004083410*C(n,5) + 3156083300916*C(n,6) + 32426546302332*C(n,7) + 205938803790720*C(n,8) + 864860752435680*C(n,9) + 2503126577952000*C(n,10) + 5110943178781440*C(n,11) + 7428048096268800*C(n,12) + 7644417350169600*C(n,13) + 5446616304729600*C(n,14) + 2556525184012800*C(n,15) + 711374856192000*C(n,16) + 88921857024000*C(n,17), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
%F A337953 a(n) = 2*A337963(n) - A282670(n) = A282670(n) - 2*A337964(n) = A337963(n) - A337964(n).
%t A337953 Table[(15n^17+n^15+20n^5+24n^3)/60,{n,30}]
%Y A337953 Cf. A282670 (oriented), A337963 (unoriented), A337964 (chiral).
%Y A337953 Other elements: A337960 (dodecahedron vertices, icosahedron faces), A337962 (dodecahedron faces, icosahedron vertices).
%Y A337953 Cf. A037270 (tetrahedron), A331351 (cube/octahedron).
%K A337953 nonn,easy
%O A337953 1,2
%A A337953 _Robert A. Russell_, Oct 03 2020