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%I #13 Mar 10 2024 13:37:16
%S 12232,241146903,243616903380,51700252145825,4112117375683170,
%T 166286156041490247,4099088542944703728,69240138924298950135,
%U 868045130573811864300,8550057218442459279340,69007402402972868503812
%N Number of chiral pairs of colorings of the 24 octahedral facets (or 24 vertices) 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 is self-dual.
%H Robert A. Russell, <a href="/A338950/b338950.txt">Table of n, a(n) for n = 2..30</a>
%H <a href="/index/Rec#order_25">Index entries for linear recurrences with constant coefficients</a>, signature (25, -300, 2300, -12650, 53130, -177100, 480700, -1081575, 2042975, -3268760, 4457400, -5200300, 5200300, -4457400, 3268760, -2042975, 1081575, -480700, 177100, -53130, 12650, -2300, 300, -25, 1).
%F a(n) = (96*n^2 + 144*n^3 - 48*n^4 - 52*n^6 - 228*n^7 - 24*n^8 + 36*n^9 + 21*n^12 + 60*n^13 + 18*n^14 - 12*n^15 - 12*n^18 + n^24) / 1152.
%F a(n) = 12232*C(n,2) + 241110207*C(n,3) + 242652389160*C(n,4) + 50484578975635*C(n,5) + 3805565293604340*C(n,6) + 138578521555036815*C(n,7) + 2881060406691096840*C(n,8) + 37995709352029326765*C(n,9) + 340998954354320550750*C(n,10) + 2186417251809922893300*C(n,11) + 10365972799754686653000*C(n,12) + 37236906263669699386800*C(n,13) + 103077047681129825503200*C(n,14) + 222282209861028829512000*C(n,15) + 375541963275099452832000*C(n,16) + 497391179860822639392000*C(n,17) + 513995707264282955712000*C(n,18) + 409785508676334510720000*C(n,19) + 247034122336026305280000*C(n,20) + 108861226736398456320000*C(n,21) + 33078014473191367680000*C(n,22) + 6193712343691192320000*C(n,23) + 538583682060103680000*C(n,24), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
%F a(n) = A338948(n) - A338949(n) = (A338948(n) - A338951(n)) / 2 = A338949(n) - A338951(n).
%t Table[(96n^2+144n^3-48n^4-52n^6-228n^7-24n^8+36n^9+21n^12+60n^13+18n^14-12n^15-12n^18+n^24)/1152,{n,2,15}]
%Y Cf. A338948 (oriented), A338949 (unoriented), A338951 (achiral), A338954 (edges, faces), A000389 (5-cell), A337954 (8-cell vertices, 16-cell facets), A234249(16-cell vertices, 8-cell facets), A338966 (120-cell, 600-cell).
%K nonn,easy
%O 2,1
%A _Robert A. Russell_, Nov 17 2020