%I #21 Jan 20 2024 03:07:41
%S 0,0,0,0,0,0,0,5040,322560,10342080,216518400,3261535200,37026823680,
%T 325474269120,2264594492160,12789814237200,60389186457600,
%U 245221330273920,877374833287680,2821277454690240,8284633867238400,22503569636419200,57135310310453760
%N Number of n-colorings of the vertices of the 7-dimensional cross polytope such that no two adjacent vertices have the same color.
%H Peter Kagey, <a href="/A342075/b342075.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).
%F a(n) = -3597143040*n + 11590795728*n^2 - 15837356724*n^3 + 12355698460*n^4 - 6212542175*n^5 + 2144307578*n^6 - 526197678*n^7 + 93450369*n^8 - 12064836*n^9 + 1122618*n^10 - 73423*n^11 + 3206*n^12 - 84*n^13 + n^14.
%F a(n) = (n - 6)*(n - 5)*(n - 4)*(n - 3)*(n - 2)*(n - 1)*n*(n^7 - 63 n^6 + 1708 n^5 - 25795 n^4 + 234094 n^3 - 1275281 n^2 + 3858049 n - 4996032).
%F a(n) = Sum_{i=1..14} A334279(7,i)*n^i.
%F From _Chai Wah Wu_, Jan 19 2024: (Start)
%F a(n) = 15*a(n-1) - 105*a(n-2) + 455*a(n-3) - 1365*a(n-4) + 3003*a(n-5) - 5005*a(n-6) + 6435*a(n-7) - 6435*a(n-8) + 5005*a(n-9) - 3003*a(n-10) + 1365*a(n-11) - 455*a(n-12) + 105*a(n-13) - 15*a(n-14) + a(n-15) for n > 14.
%F G.f.: x^7*(-52370755920*x^7 - 27190754640*x^6 - 6557740560*x^5 - 959792400*x^4 - 92962800*x^3 - 6032880*x^2 - 246960*x - 5040)/(x - 1)^15. (End)
%t p = ChromaticPolynomial[CompleteGraph[Table[2, 7]], x];
%t Table[p /. x -> n, {n, 0, 50}]
%Y Analogous for k-dimensional cross polytope: A091940 (k=2), A115400 (k=3), A334281 (k=4), A342073 (k=5), A342074 (k=6).
%Y Cf. A334279, A342088.
%K nonn
%O 0,8
%A _Peter Kagey_, Feb 27 2021