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A342074 Number of n-colorings of the vertices of the 6-dimensional cross polytope such that no two adjacent vertices have the same color.

%I A342074 #23 Jan 19 2024 12:42:54
%S A342074 0,0,0,0,0,0,720,35280,866880,13849920,158004000,1347524640,
%T A342074 8866186560,46496324160,201705744240,748737990000,2444976293760,
%U A342074 7178449299840,19276199691840,47983899216960,111920569776000,246727594270080,517702915311120,1039979954779920
%N A342074 Number of n-colorings of the vertices of the 6-dimensional cross polytope such that no two adjacent vertices have the same color.
%H A342074 Peter Kagey, <a href="/A342074/b342074.txt">Table of n, a(n) for n = 0..1000</a>
%H A342074 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
%F A342074 a(n) = -22852200*n + 70164670*n^2 - 89812001*n^3 + 64407806*n^4 - 29113410*n^5 + 8790285*n^6 - 1822164*n^7 + 260868*n^8 - 25405*n^9 + 1610*n^10 - 60*n^11 + n^12.
%F A342074 a(n) = (n - 5)*(n - 4)*(n - 3)*(n - 2)*(n - 1)*n*(190435 - 149879*n + 49144*n^2 - 8605*n^3 + 850*n^4 - 45*n^5 + n^6).
%F A342074 a(n) = Sum_{i=1..12} A334279(6,i)*n^i.
%F A342074 From _Chai Wah Wu_, Jan 19 2024: (Start)
%F A342074 a(n) = 13*a(n-1) - 78*a(n-2) + 286*a(n-3) - 715*a(n-4) + 1287*a(n-5) - 1716*a(n-6) + 1716*a(n-7) - 1287*a(n-8) + 715*a(n-9) - 286*a(n-10) + 78*a(n-11) - 13*a(n-12) + a(n-13) for n > 12.
%F A342074 G.f.: x^6*(-287250480*x^6 - 150137280*x^5 - 35996400*x^4 - 5126400*x^3 - 464400*x^2 - 25920*x - 720)/(x - 1)^13. (End)
%t A342074 p = ChromaticPolynomial[CompleteGraph[Table[2, 6]], x];
%t A342074 Table[p /. x -> n, {n, 0, 50}]
%Y A342074 Analogous for k-dimensional cross polytope: A091940 (k=2), A115400 (k=3), A334281 (k=4), A342073 (k=5), A342075 (k=7).
%Y A342074 Cf. A334279, A342088.
%K A342074 nonn
%O A342074 0,7
%A A342074 _Peter Kagey_, Feb 27 2021