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A185442 Triangle T(n,k), n>=1, 0<=k<=2n(n+1), read by rows: row n gives the coefficients of the chromatic polynomial of the Aztec diamond graph of order n, highest powers first.

%I A185442 #20 Feb 18 2025 08:03:24
%S A185442 1,-4,6,-3,0,1,-16,120,-555,1755,-3978,6588,-7965,6885,-4050,1458,
%T A185442 -243,0,1,-36,630,-7127,58476,-370128,1876942,-7818056,27208798,
%U A185442 -80059990,200769740,-431267475,795531116,-1260437072,1711682175,-1983112401,1945239399,-1597006926,1079055243,-585362106,245489859,-74816136,14762007,-1416933,0
%N A185442 Triangle T(n,k), n>=1, 0<=k<=2n(n+1), read by rows: row n gives the coefficients of the chromatic polynomial of the Aztec diamond graph of order n, highest powers first.
%C A185442 The Aztec diamond graph of order n has 2*n*(n+1) vertices with integer coordinates (x,y) obeying |x-1/2| + |y-1/2| <= n and (2*n)^2 edges connecting vertices having Euclidean distance 1. It can be derived from the Aztec diamond using vertices to represent tiles and edges to connect vertices of neighboring tiles. The chromatic polynomial has 2*n*(n+1)+1 coefficients.
%H A185442 Alois P. Heinz, <a href="/A185442/b185442.txt">Rows n = 1..7, flattened</a>
%H A185442 Propp, J., Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), <a href="http://www.msri.org/publications/books/Book38/contents.html">New Perspectives in Algebraic Combinatorics</a>
%H A185442 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AztecDiamond.html">Aztec Diamond</a>
%H A185442 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticPolynomial.html">Chromatic Polynomial</a>
%H A185442 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%e A185442 2 example graphs:      o-o
%e A185442 .                      | |
%e A185442 .                    o-o-o-o
%e A185442 .                    | | | |
%e A185442 .          o-o       o-o-o-o
%e A185442 .          | |         | |
%e A185442 .          o-o         o-o
%e A185442 Order:      1           2
%e A185442 Vertices:   4          12
%e A185442 Edges:      4          16
%e A185442 The Aztec diamond graph of order 1 is the cycle graph C_4 with chromatic polynomial q^4 -4*q^3 +6*q^2 -3*q => [1, -4, 6, -3, 0].
%e A185442 Triangle T(n,k) begins:
%e A185442   1,   -4,     6,      -3,        0;
%e A185442   1,  -16,   120,    -555,     1755,      -3978,        6588,  ...
%e A185442   1,  -36,   630,   -7127,    58476,    -370128,     1876942,  ...
%e A185442   1,  -64,  2016,  -41639,   633851,   -7578762,    74074918,  ...
%e A185442   1, -100,  4950, -161659,  3917248,  -75096624,  1186008180,  ...
%e A185442   1, -144, 10296, -487283, 17170275, -480406458, 11115470152,  ...
%e A185442   ...
%Y A185442 Cf. A046092, A016742, A006125, A007725.
%K A185442 sign,tabf,look,hard
%O A185442 1,2
%A A185442 _Alois P. Heinz_, Feb 03 2011