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A182553 Chromatic invariant of the complete tripartite graph K_(n,n,n).

%I A182553 #32 Feb 16 2025 08:33:13
%S A182553 1,11,1243,490043,463370491,860454250571,2769263554592683,
%T A182553 14178247400433059003,108483732651999512059291,
%U A182553 1182804548772797481324575531,17700419121823142496192223238923,352709466470858225716888461028622363,9127611521817307582541815420363992765691
%N A182553 Chromatic invariant of the complete tripartite graph K_(n,n,n).
%C A182553 The chromatic invariant equals the absolute value of the first derivative of the chromatic polynomial evaluated at 1.
%H A182553 Alois P. Heinz, <a href="/A182553/b182553.txt">Table of n, a(n) for n = 1..100</a>
%H A182553 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticInvariant.html">Chromatic Invariant</a>
%H A182553 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteTripartiteGraph.html">Complete Tripartite Graph</a>
%H A182553 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%F A182553 a(n) = |(d/dq P(n,q))_{q=1}| with P(n,q) = Sum_{k,m=1..n} S2(n,k) * S2(n,m) * (q-k-m)^n * Product_{i=0..k+m-1} (q-i) and S2 = A008277.
%F A182553 a(n) ~ (n-1)!^3 / (Pi * 3^(3/2) * (1 - log(3/2)) * (log(3/2))^(3*n-1)). - _Vaclav Kotesovec_, Sep 03 2014, updated Feb 18 2017
%F A182553 a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} (-1)^(n+i+j+k) * Stirling2(n,i) * Stirling2(n,j) * Stirling2(n,k) * (i+j+k-2)!. - _Andrew Howroyd_, Apr 22 2018
%p A182553 P:= n-> expand(add(add(Stirling2(n, k) *Stirling2(n, m)
%p A182553         *mul(q-i, i=0..k+m-1) *(q-k-m)^n, m=1..n), k=1..n)):
%p A182553 a:= n-> abs(subs(q=1, diff(P(n), q))):
%p A182553 seq(a(n), n=1..15);
%t A182553 Table[Sum[StirlingS2[n, k] StirlingS2[n, m] (-1)^(k + m + n) (1 - k - m)^n Gamma[k + m - 1], {k, n}, {m, n}], {n, 10}] (* _Eric W. Weisstein_, Apr 26 2017 *)
%o A182553 (PARI) a(n)={my(s=vector(n, k, stirling(n,k,2))); sum(i=1,n, sum(j=1,n, sum(k=1,n, (-1)^(n+i+j+k)*s[i]*s[j]*s[k]*(i+j+k-2)! )))} \\ _Andrew Howroyd_, Apr 22 2018
%o A182553 (PARI) a(n)={(-1)^n*subst(serlaplace(sum(k=1,n,stirling(n,k,2)*x^k)^3/x^2),x,-1)} \\ _Andrew Howroyd_, Apr 22 2018
%Y A182553 Cf. A008277, A048144.
%K A182553 nonn
%O A182553 1,2
%A A182553 _Alois P. Heinz_, May 04 2012