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A182368 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the square grid graph G_(n,n), highest powers first.

%I A182368 #31 Feb 18 2025 07:52:10
%S A182368 1,0,1,-4,6,-3,0,1,-12,66,-216,459,-648,594,-323,79,0,1,-24,276,-2015,
%T A182368 10437,-40614,122662,-292883,557782,-848056,1022204,-960627,682349,
%U A182368 -346274,112275,-17493,0,1,-40,780,-9864,90798,-647352,3714180,-17590911,69997383
%N A182368 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the square grid graph G_(n,n), highest powers first.
%C A182368 The square grid graph G_(n,n) has n^2 = A000290(n) vertices and 2*n*(n-1) = A046092(n-1) edges. The chromatic polynomial of G_(n,n) has n^2+1 = A002522(n) coefficients.
%H A182368 Alois P. Heinz, <a href="/A182368/b182368.txt">Rows n = 1..9, flattened</a>
%H A182368 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticPolynomial.html">Chromatic Polynomial</a>
%H A182368 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>
%H A182368 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%e A182368 3 example graphs:                          o---o---o
%e A182368 .                                          |   |   |
%e A182368 .                             o---o        o---o---o
%e A182368 .                             |   |        |   |   |
%e A182368 .                o            o---o        o---o---o
%e A182368 Graph:        G_(1,1)        G_(2,2)        G_(3,3)
%e A182368 Vertices:        1              4              9
%e A182368 Edges:           0              4             12
%e A182368 The square grid graph G_(2,2) is the cycle graph C_4 with chromatic polynomial q^4 -4*q^3 +6*q^2 -3*q => row 2 = [1, -4, 6, -3, 0].
%e A182368 Triangle T(n,k) begins:
%e A182368   1,    0;
%e A182368   1,   -4,     6,      -3,        0;
%e A182368   1,  -12,    66,    -216,      459,       -648,         594, ...
%e A182368   1,  -24,   276,   -2015,    10437,     -40614,      122662, ...
%e A182368   1,  -40,   780,   -9864,    90798,    -647352,     3714180, ...
%e A182368   1,  -60,  1770,  -34195,   486210,   -5421612,    49332660, ...
%e A182368   1,  -84,  3486,  -95248,  1926585,  -30755376,   403410654, ...
%e A182368   1, -112,  6216, -227871,  6205479, -133865298,  2382122274, ...
%e A182368   1, -144, 10296, -487280, 17169852, -480376848, 11114098408, ...
%e A182368   ...
%t A182368 Reverse /@ CoefficientList[Table[ChromaticPolynomial[GridGraph[{n, n}], x], {n, 5}], x] // Flatten (* _Eric W. Weisstein_, May 01 2017 *)
%Y A182368 Columns 0, 1 give: A000012, (-1)*A046092(n-1).
%Y A182368 Sums of absolute values of row elements give: A080690(n).
%Y A182368 Cf. A000290, A002522, A182406, A185442, A193233, A193277, A193283.
%K A182368 sign,look,tabf,hard
%O A182368 1,4
%A A182368 _Alois P. Heinz_, Apr 26 2012