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A212208 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 diagonal grid graph DG_(n,n), highest powers first.

%I A212208 #13 Feb 08 2017 18:31:31
%S A212208 1,0,1,-6,11,-6,0,1,-20,174,-859,2627,-5082,6048,-4023,1134,0,1,-42,
%T A212208 825,-10054,85011,-528254,2491825,-9084089,25795983,-57031153,
%U A212208 97292827,-125639547,118705077,-77301243,30931875,-5709042,0,1,-72,2492,-55183,877812
%N A212208 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 diagonal grid graph DG_(n,n), highest powers first.
%C A212208 The square diagonal grid graph DG_(n,n) has n^2 = A000290(n) vertices and 2*(n-1)*(2*n-1) = A002943(n-1) edges. The chromatic polynomial of DG_(n,n) has n^2+1 = A002522(n) coefficients.
%H A212208 Alois P. Heinz, <a href="/A212208/b212208.txt">Rows n = 1..7, flattened</a>
%H A212208 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%e A212208 3 example graphs:                          o---o---o
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%e A212208 .                             o---o        o---o---o
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%e A212208 .                             | X |        | X | X |
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%e A212208 .                o            o---o        o---o---o
%e A212208 Graph:        DG_(1,1)       DG_(2,2)       DG_(3,3)
%e A212208 Vertices:        1              4              9
%e A212208 Edges:           0              6             20
%e A212208 The square diagonal grid graph DG_(2,2) equals the complete graph K_4 and has chromatic polynomial q*(q-1)*(q-2)*(q-3) = q^4 -6*q^3 +11*q^2 -6*q => row 2 = [1, -6, 11, -6, 0].
%e A212208 Triangle T(n,k) begins:
%e A212208 1,    0;
%e A212208 1,   -6,    11,      -6,        0;
%e A212208 1,  -20,   174,    -859,     2627,      -5082, ...
%e A212208 1,  -42,   825,  -10054,    85011,    -528254, ...
%e A212208 1,  -72,  2492,  -55183,   877812,  -10676360, ...
%e A212208 1, -110,  5895, -205054,  5203946, -102687204, ...
%e A212208 1, -156, 11946, -598491, 22059705, -637802510, ...
%Y A212208 Columns 1-2 give: A000012, (-1)*A002943(n-1).
%Y A212208 Row sums (for n>1) and last elements of rows give: A000004, row lengths give: A002522.
%Y A212208 Cf. A000290, A212209.
%K A212208 sign,tabf
%O A212208 1,4
%A A212208 _Alois P. Heinz_, May 04 2012