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

1, 11, 1243, 490043, 463370491, 860454250571, 2769263554592683, 14178247400433059003, 108483732651999512059291, 1182804548772797481324575531, 17700419121823142496192223238923, 352709466470858225716888461028622363, 9127611521817307582541815420363992765691

Offset: 1

Author

Alois P. Heinz, May 04 2012

Keywords

• nonn

Comments

The chromatic invariant equals the absolute value of the first derivative of the chromatic polynomial evaluated at 1.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..100
• Eric Weisstein's World of Mathematics, Chromatic Invariant
• Eric Weisstein's World of Mathematics, Complete Tripartite Graph
• Wikipedia, Chromatic Polynomial

Crossrefs

Cf. A008277, A048144.

Programs

Maple

P:= n-> expand(add(add(Stirling2(n, k) *Stirling2(n, m)
        *mul(q-i, i=0..k+m-1) *(q-k-m)^n, m=1..n), k=1..n)):
a:= n-> abs(subs(q=1, diff(P(n), q))):
seq(a(n), n=1..15);

Mathematica

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 *)

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

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

Formula

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.

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

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