A338949 Number of unoriented colorings of the 24 octahedral facets (or 24 vertices) of the 4-D 24-cell using subsets of a set of n colors.

1, 18736, 249563343, 245072692820, 51780391393325, 4114243321427946, 166320182540310771, 4099464588809407728, 69243270244113372390, 868065984969662449300, 8550173137863803682016, 69007957379144017626756

Offset: 1

Author

Robert A. Russell, Nov 17 2020

Keywords

• nonn
• easy

Comments

Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It is self-dual.

Links

• Robert A. Russell, Table of n, a(n) for n = 1..30
• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

Cf. A338948 (oriented), A338950 (chiral), A338951 (achiral), A338953 (edges, faces), A000389 (5-cell), A128767 (8-cell vertices, 16-cell facets), A337957 (16-cell vertices, 8-cell facets), A338965 (120-cell, 600-cell).

Programs

Mathematica

Table[(96n^2+144n^3+144n^4+140n^6+300n^7+120n^8+36n^9+45n^12+84n^13+18n^14+12n^15+12n^18+n^24)/1152,{n,15}]
LinearRecurrence[{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},{1,18736,249563343,245072692820,51780391393325,4114243321427946,166320182540310771,4099464588809407728,69243270244113372390,868065984969662449300,8550173137863803682016,69007957379144017626756,471182396311499869193288,2790108355121570273031710,14612960014479438426745050,68774495831757984888966336,294660451484256436406752191,1161683435155207577365494648,4252399462403852518286044405,14563558286595288907896687700,46968928774940328123724865031,143447144215320073513164583826,416884377543198363455158598933,1157756823443195554136397711600,3083952997773835021725260467500},20] (* Harvey P. Dale, Mar 24 2024 *)

Formula

a(n) = (96*n^2 + 144*n^3 + 144*n^4 + 140*n^6 + 300*n^7 + 120*n^8 + 36*n^9 + 45*n^12 + 84*n^13 + 18*n^14 + 12*n^15 + 12*n^18 + n^24) / 1152.

a(n) = 1*C(n,1) + 18734*C(n,2) + 249507138*C(n,3) + 244074551860*C(n,4) + 50557523375300*C(n,5) + 3807232072474470*C(n,6) + 138599298699649830*C(n,7) + 2881219380682352640*C(n,8) + 37996512548398853085*C(n,9) + 341001760994302265550*C(n,10) + 2186424231002014796100*C(n,11) + 10365985337974980021000*C(n,12) + 37236922591331944681200*C(n,13) + 103077062953464218018400*C(n,14) + 222282219864764987928000*C(n,15) + 375541967632270447008000*C(n,16) + 497391180994576316448000*C(n,17) + 513995707397665741248000*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 unoriented colorings using exactly k colors.

a(n) = A338948(n) - A338950(n) = (A338948(n) + A338951(n)) / 2 = A338950(n) + A338951(n).