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A342073 Number of n-colorings of the vertices of the 5-dimensional cross polytope such that no two adjacent vertices have the same color.

%I A342073 #20 Jan 20 2024 03:08:14
%S A342073 0,0,0,0,0,120,4320,78120,913920,7575120,46751040,224587440,881591040,
%T A342073 2946869640,8659691040,22915652760,55611279360,125508233760,
%U A342073 266320172160,535945217760,1030028705280,1901347885080,3386866301280,5844714201480,9803816225280
%N A342073 Number of n-colorings of the vertices of the 5-dimensional cross polytope such that no two adjacent vertices have the same color.
%H A342073 Peter Kagey, <a href="/A342073/b342073.txt">Table of n, a(n) for n = 0..1000</a>
%H A342073 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F A342073 a(n) = -205056*n + 593016*n^2 - 698250*n^3 + 448015*n^4 - 175004*n^5 + 43608*n^6 - 6990*n^7 + 700*n^8 - 40*n^9 + n^10.
%F A342073 a(n) = (n - 4)*(n - 3)*(n - 2)*(n - 1)*n*(-8544 + 6909*n - 2240*n^2 + 365*n^3 - 30*n^4 + n^5).
%F A342073 a(n) = Sum_{i=1..10} A334279(5,i)*n^i.
%F A342073 From _Chai Wah Wu_, Jan 19 2024: (Start)
%F A342073 a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n > 10.
%F A342073 G.f.: x^5*(-2170680*x^5 - 1145400*x^4 - 272400*x^3 - 37200*x^2 - 3000*x - 120)/(x - 1)^11. (End)
%t A342073 p = ChromaticPolynomial[CompleteGraph[Table[2, 5]], x];
%t A342073 Table[p /. x -> n, {n, 0, 50}]
%Y A342073 Analogous for k-dimensional cross polytope: A091940 (k=2), A115400 (k=3), A334281 (k=4), A342074 (k=6), A342075 (k=7)
%Y A342073 Cf. A334279, A342088.
%K A342073 nonn
%O A342073 0,6
%A A342073 _Peter Kagey_, Feb 27 2021