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%I A338965 #21 Mar 13 2024 13:51:15
%S A338965 1,92307499707443390526727850063504,
%T A338965 124792381938502167392338612231208163827413085862945471,
%U A338965 122697712831832245109951221276235414511846772206539032522116543043328
%N A338965 Number of unoriented colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using subsets of a set of n colors.
%C A338965 Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual.
%C A338965 Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide formulas here.
%C A338965 For the 600 facets of the 600-cell (vertices of the 120-cell), the formula is (960*n^20 + 1440*n^30 + 960*n^40 + 1200*n^50 + 2064*n^60 + 1440*n^66 + 40*n^100 + 1600*n^104 + 1200*n^114 + 624*n^120 + 60*n^150 + 1800*n^152 + 40*n^200 + 400*n^208 + 61*n^300 + 450*n^302 + 60*n^330 + n^600) / 14400.
%C A338965 For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the formula is (960 n^24 + 1440 n^36 + 960 n^48 + 1200 n^60 + 336 n^72 + 1728 n^76 + 1440 n^84 + 1640 n^120 + 1200 n^132 + 336 n^144 + 288 n^152 + 60 n^180 + 1800 n^182 + 440 n^240 + 61 n^360 + 450 n^364 + 60 n^396 + n^720) / 14400.
%C A338965 For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the formula is (960*n^40 + 1440*n^60 + 960*n^80 + 1200*n^100 + 2064*n^120 + 1440*n^128 + 40*n^200 + 1600*n^202 + 1200*n^216 + 624*n^240 + 60*n^300 + 1800*n^302 + 40*n^400 + 400*n^404 + 61*n^600 + 450*n^604 + 60*n^640 + n^1200) / 14400.
%H A338965 Robert A. Russell, <a href="/A338965/b338965.txt">Table of n, a(n) for n = 1..30</a>
%H A338965 <a href="/index/Rec#order_121">Index entries for linear recurrences with constant coefficients</a>, order 121.
%F A338965 a(n) = (960*n^4 + 1440*n^6 + 960*n^8 + 1200*n^10 + 336*n^12 + 288*n^16 + 1440*n^17 + 1440*n^19 + 40*n^20 + 400*n^22 + 1200*n^23 + 336*n^24 + 1200*n^27 + 60*n^30 + 1800*n^31 + 288*n^32 + 40*n^40 + 400*n^44 + n^60 + 60*n^61 + 450*n^62 + 60*n^75 +*n^120) / 14400.
%F A338965 a(n) = Sum_{j=1..Min(n,120)} A338981(n) * binomial(n,j).
%F A338965 a(n) = A338964(n) - A338966(n) =(A338964(n) + A338967(n)) / 2 = A338966(n) + A338967(n).
%t A338965 Table[(960n^4+1440n^6+960n^8+1200n^10+336n^12+288n^16+1440n^17+1440n^19+40n^20+400n^22+1200n^23+336n^24+1200n^27+60n^30+1800n^31+288n^32+40n^40+400n^44+n^60+60n^61+450n^62+60n^75+n^120)/14400,{n,10}]
%Y A338965 Cf. A338964 (oriented), A338966 (chiral), A338967 (achiral), A338981 (exactly n colors), A000389 (5-cell), A128767 (8-cell vertices, 16-cell facets), A337957(16-cell vertices, 8-cell facets), A338949 (24-cell).
%K A338965 nonn,easy
%O A338965 1,2
%A A338965 _Robert A. Russell_, Dec 04 2020

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