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A212162 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the rhombic hexagonal square grid graph RH_(n,n), highest powers first.

%I A212162 #19 Feb 08 2017 19:06:41
%S A212162 1,0,1,-5,8,-4,0,1,-16,112,-448,1120,-1791,1786,-1012,248,0,1,-33,510,
%T A212162 -4898,32703,-160859,602408,-1749715,3975561,-7068408,9755858,
%U A212162 -10265148,7968348,-4304712,1445104,-226720,0,1,-56,1508,-25992,321994,-3051871,23000726,-141421592,722137763,-3101089711
%N A212162 Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the rhombic hexagonal square grid graph RH_(n,n), highest powers first.
%C A212162 T differs from A212194 first at (n,k) = (5,10): T(5,10) = -3101089711, A212194(5,10) = -3101089710.
%C A212162 The rhombic hexagonal square grid graph RH_(n,n) has n^2 = A000290(n) vertices and (n-1)*(3*n-1) = A045944(n-1) edges. The chromatic polynomial of RH_(n,n) has n^2+1 = A002522(n) coefficients.
%H A212162 Alois P. Heinz, <a href="/A212162/b212162.txt">Rows n = 1..8, flattened</a>
%H A212162 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%e A212162 3 example graphs:                        o--o--o
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%e A212162 .                            o--o        o--o--o
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%e A212162 .               o            o--o        o--o--o
%e A212162 Graph:       RH_(1,1)      RH_(2,2)      RH_(3,3)
%e A212162 Vertices:       1             4             9
%e A212162 Edges:          0             5            16
%e A212162 The rhombic hexagonal square grid graph RH_(2,2) has chromatic polynomial q*(q-1)*(q-2)^2 = q^4 -5*q^3 +8*q^2 -4*q => row 2 = [1, -5, 8, -4, 0].
%e A212162 Triangle T(n,k) begins:
%e A212162 1,    0;
%e A212162 1,   -5,     8,      -4,        0;
%e A212162 1,  -16,   112,    -448,     1120,      -1791, ...
%e A212162 1,  -33,   510,   -4898,    32703,    -160859, ...
%e A212162 1,  -56,  1508,  -25992,   321994,   -3051871, ... , -3101089711, ...
%e A212162 1,  -85,  3520,  -94620,  1855860,  -28306676, ...
%e A212162 1, -120,  7068, -272344,  7720110, -171656543, ...
%e A212162 1, -161, 12782, -667058, 25738055, -783003395, ...
%Y A212162 Columns 1-2 give: A000012, (-1)*A045944(n-1).
%Y A212162 Row sums (for n>1) and last elements of rows give: A000004, row lengths give: A002522.
%Y A212162 Cf. A000290, A212163, A212194.
%K A212162 sign,tabf
%O A212162 1,4
%A A212162 _Alois P. Heinz_, May 02 2012