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A338955 Number of achiral colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

%I A338955 #15 Mar 13 2024 13:45:23
%S A338955 1,24124751133507584,883287060208158070437496209,
%T A338955 27692675763559261523047959805034496,
%U A338955 18070082615414169898334284655914306640625,1018202231744161700740376040914469837333037056
%N A338955 Number of achiral colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.
%C A338955 An achiral coloring is identical to its reflection. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. There are 576 elements in the automorphism group of the 24-cell that are not in its rotation group. They divide into 10 conjugacy classes. The first formula is obtained by averaging the edge (or face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C A338955   Count    Odd Cycle Indices     Count    Odd Cycle Indices
%C A338955      12    x_1^24x_2^36             96    x_1^2x_2^2x_3^2x_6^14
%C A338955      12    x_1^8x_2^44              96    x_3^8x_6^12
%C A338955   12+12    x_3^48                   96    x_2^3x_6^15
%C A338955   72+72    x_4^24                   96    x_6^16
%H A338955 Robert A. Russell, <a href="/A338955/b338955.txt">Table of n, a(n) for n = 1..30</a>
%H A338955 <a href="/index/Rec#order_61">Index entries for linear recurrences with constant coefficients</a>, order 61.
%F A338955 a(n) = (8*n^16 + 8*n^18 + 16*n^20 + 12*n^24 + 2*n^48 + n^52 + n^60) / 48.
%F A338955 a(n) = Sum_{j=1..Min(n,60)} A338959(n) * binomial(n,j).
%F A338955 a(n) = 2*A338953(n) - A338952(n) = A338952(n) - 2*A338954(n) = A338953(n) - A338954(n).
%t A338955 Table[(8n^16+8n^18+16n^20+12n^24+2n^48+n^52+n^60)/48,{n,15}]
%Y A338955 Cf. A338952 (oriented), A338953 (unoriented), A338954 (chiral), A338959 (exactly n colors), A338951 (vertices, facets), A331353 (5-cell), A331361 (8-cell edges, 16-cell faces), A331357 (16-cell edges, 8-cell faces), A338967 (120-cell, 600-cell).
%K A338955 nonn,easy
%O A338955 1,2
%A A338955 _Robert A. Russell_, Nov 17 2020