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Number of chiral pairs of colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.
9

%I #13 Mar 13 2024 13:49:21

%S 68774446614978208476646592,

%T 5523164445430505077912054084256733211946217,

%U 5448873034189827051926943172520863487560602391778344960,10956401461402941741829554371669666304159415287557559324930859375

%N Number of chiral pairs of colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

%C Each member of a chiral pair is a reflection but not a rotation of the other. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual.

%H Robert A. Russell, <a href="/A338954/b338954.txt">Table of n, a(n) for n = 2..30</a>

%H <a href="/index/Rec#order_97">Index entries for linear recurrences with constant coefficients</a>, order 97.

%F a(n) = (96*n^8 + 144*n^12 - 48*n^16 - 64*n^18 - 192*n^20 - 60*n^24 +

%F 48*n^32 + 32*n^36 - 5*n^48 + 72*n^50 - 12*n^52 - 12*n^60 + n^96) / 1152.

%F a(n) = Sum_{j=2..Min(n,96)} A338958(n) * binomial(n,j).

%F a(n) = A338952(n) - A338953(n) = (A338952(n) - A338955(n)) / 2 = A338953(n) - A338955(n).

%t Table[(96n^8+144n^12-48n^16-64n^18-192n^20-60n^24+48n^32+32n^36-5n^48+72n^50-12n^52-12n^60+n^96)/1152,{n,2,15}]

%Y Cf. A338952 (oriented), A338953 (unoriented), A338955 (achiral), A338958 (exactly n colors), A338950 (vertices, facets), A331352 (5-cell), A331360 (8-cell edges, 16-cell faces), A331356 (16-cell edges, 8-cell faces), A338966 (120-cell, 600-cell).

%K nonn,easy

%O 2,1

%A _Robert A. Russell_, Nov 17 2020