A157993 - OEIS

%I #24 Feb 16 2025 08:33:09

%S 0,0,0,786240,397543795824,3153491495915040,2897591335142535360,

%T 709217913680036905200,70921407068068519599840,

%U 3718329027062088400988544,119720148366778311215868480,2633253678249157711210367520,42653023518489941374251310800

%N Number of n-colorings of the Coxeter graph.

%C The Coxeter Graph is a nonhamiltonian cubic symmetric graph and has 28 vertices and 42 edges.

%C All terms are multiples of 336.

%H Alois P. Heinz, <a href="/A157993/b157993.txt">Table of n, a(n) for n = 0..1000</a>

%H Marc Timme, Frank van Bussel, Denny Fliegner, and Sebastian Stolzenberg, <a href="https://doi.org/10.1088/1367-2630/11/2/023001">Counting complex disordered states by efficient pattern matching: chromatic polynomials and Potts partition functions</a>, New Journal of Physics, Volume 11, February 2009.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticPolynomial.html">Chromatic Polynomial</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CoxeterGraph.html">Coxeter Graph</a>

%H <a href="/index/Rec#order_29">Index entries for linear recurrences with constant coefficients</a>, signature (29, -406, 3654, -23751, 118755, -475020, 1560780, -4292145, 10015005, -20030010, 34597290, -51895935, 67863915, -77558760, 77558760, -67863915, 51895935, -34597290, 20030010, -10015005, 4292145, -1560780, 475020, -118755, 23751, -3654, 406, -29, 1).

%F a(n) = n^28 -42*n^27 + ... (see Maple program).

%p a:= n-> n^28 -42*n^27 +861*n^26 -11480*n^25 +111930*n^24 -850668*n^23 +5245762*n^22 -26977443*n^21 +118014274*n^20 -445705967*n^19 +1469857872*n^18 -4270042980*n^17 +11001634164*n^16 -25266720456*n^15 +51908523754*n^14 -95589692821*n^13 +157862673577*n^12 -233517066853*n^11 +308423840605*n^10 -361701500512*n^9 +373419294214*n^8 -335133871598*n^7 +256750369239*n^6 -163506050813*n^5 +83144968151*n^4 -31635019987*n^3 +7989854148*n^2 -1000876932*n:

%p seq(a(n), n=0..30);

%t With[{poly = ChromaticPolynomial[GraphData["CoxeterGraph"]]}, Array[poly, 20]] (* _Eric W. Weisstein_, May 04 2022 *)

%K nonn,easy,changed

%O 0,4

%A _Alois P. Heinz_, Mar 10 2009