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%I #25 Aug 01 2023 14:18:32
%S 1,0,2,0,0,3,0,0,6,4,0,0,6,24,5,0,0,6,192,60,6,0,0,6,2112,1620,120,7,
%T 0,0,6,32640,98820,7680,210,8,0,0,6,718080,13638780,1574400,26250,336,
%U 9,0,0,6,22665216,4260983940,1034019840,13676250,72576,504,10
%N Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the k X k X k triangular grid.
%C The k X k X k triangular grid has k rows with i vertices in row i. Each vertex is connected to the neighbors in the same row and up to two vertices in each of the neighboring rows. The graph has A000217(k) vertices and 3*A000217(k-1) edges altogether.
%C The coefficients of the chromatic polynomials for the column sequences are given by the rows of A193283. - _Georg Fischer_, Jul 31 2023
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic polynomial</a>
%e Square array A(n,k) begins:
%e 1, 0, 0, 0, 0, 0, ...
%e 2, 0, 0, 0, 0, 0, ...
%e 3, 6, 6, 6, 6, 6, ...
%e 4, 24, 192, 2112, 32640, 718080, ...
%e 5, 60, 1620, 98820, 13638780, 4260983940, ...
%e 6, 120, 7680, 1574400, 1034019840, 2175789895680, ...
%Y Columns k=1-11 give: A000027, A007531, A182788, A182789, A182790, A182791, A182792, A182793, A182794, A182795, A182796.
%Y Rows n=1-10 give: A000007(k-1), A000038(k-1), A040006(k-1), A182798, A153467*4, A153468*5, A153469*6, A153470*7, A153471*8, A153472*9, A153473*10.
%Y Cf. A000217, A193283.
%K nonn,tabl
%O 1,3
%A _Alois P. Heinz_, Dec 02 2010
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