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