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%I #25 Feb 28 2021 09:53:53
%S 0,0,0,0,24,600,7560,61320,351120,1515024,5266800,15531120,40308840,
%T 94534440,204228024,412284600,786283680,1428742560,2490276960,
%U 4186173024,6816915000,10793253240,16666437480,25164280680,37233759024,54090894000,77278702800
%N Number of n-colorings of the vertices of the 4-dimensional cross polytope such that no two adjacent vertices have the same color.
%C The 4-dimensional cross-polytope is sometimes called the 16-cell. It is one of the six convex regular 4-polytopes.
%H Colin Barker, <a href="/A334281/b334281.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChromaticPolynomial.html">Chromatic Polynomial</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cross-polytope">Cross-polytope</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tur%C3%A1n_graph">Turán graph</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F a(n) = n*(n - 1)*(n - 2)*(n - 3)*(465 - 392n + 125n^2 - 18n^3 + n^4).
%F a(n) = -2790n + 7467n^2 - 7852n^3 + 4300n^4 - 1346n^5 + 244n^6 - 24n^7 + n^8.
%F From _Colin Barker_, Apr 22 2020: (Start)
%F G.f.: 24*x^4*(1 + 16*x + 126*x^2 + 536*x^3 + 1001*x^4) / (1 - x)^9.
%F a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>8.
%F (End)
%o (PARI) concat([0,0,0,0], Vec(24*x^4*(1 + 16*x + 126*x^2 + 536*x^3 + 1001*x^4) / (1 - x)^9 + O(x^30))) \\ _Colin Barker_, Apr 22 2020
%Y Cf. A091940 (2-dimensional), A115400 (3-dimensional).
%Y Cf. A334279.
%K nonn,easy
%O 0,5
%A _Peter Kagey_, Apr 21 2020
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