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A282247 a(n) = 1/(2*n) times the number of n-colorings of the complete tripartite graph K_(k,k,k).

%I A282247 #9 Feb 16 2025 08:33:40
%S A282247 0,0,1,66,9546,2995540,1569542955,1261871330286,1497794187367828,
%T A282247 2511721997105517288,5733323495739849790485,
%U A282247 17312353700125621441996450,67543299290149425529497170526,333695384900672678963632331684412,2052058288990669598319358806485894719
%N A282247 a(n) = 1/(2*n) times the number of n-colorings of the complete tripartite graph K_(k,k,k).
%H A282247 Alois P. Heinz, <a href="/A282247/b282247.txt">Table of n, a(n) for n = 1..174</a>
%H A282247 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteTripartiteGraph.html">Complete Tripartite Graph</a>
%H A282247 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%F A282247 a(n) = 1/(2*n) * Sum_{j,m=1..n} S2(n,j) * S2(n,m) * (n-j-m)^n * Product_{i=0..j+m-1} (n-i) with S2 = A008277.
%F A282247 a(n) = A212221(n,n).
%F A282247 a(n) ~ c * d^n * n!^3 / n^(5/2), where d = 2.1534859143209968... and c = 0.008659981748969... . - _Vaclav Kotesovec_, Feb 18 2017
%p A282247 a:= n-> add(add(Stirling2(n, k)*Stirling2(n, m)*
%p A282247     mul(n-i, i=0..k+m-1)*(n-k-m)^n, m=1..n), k=1..n)/(2*n):
%p A282247 seq(a(n), n=1..20);
%Y A282247 Main diagonal of A212221.
%Y A282247 Cf. A008277.
%K A282247 nonn
%O A282247 1,4
%A A282247 _Alois P. Heinz_, Feb 09 2017