%I #26 Aug 01 2023 14:30:43

%S 1,0,1,-3,2,0,1,-9,32,-56,48,-16,0,1,-18,144,-672,2016,-4031,5368,

%T -4584,2272,-496,0,1,-30,419,-3612,21477,-93207,304555,-761340,

%U 1463473,-2152758,2385118,-1929184,1075936,-369824,58976,0

%N Triangle T(n,k), n>=1, 0<=k<=n*(n+1)/2, read by rows: row n gives the coefficients of the chromatic polynomial of the n X n X n triangular grid, highest powers first.

%C The n X n X n triangular grid has n 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(n) vertices and 3*A000217(n-1) edges altogether.

%H Alois P. Heinz, <a href="/A193283/b193283.txt">Rows n = 1..13, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a>

%e 4 example graphs: o

%e / \

%e o o---o

%e / \ / \ / \

%e o o---o o---o---o

%e / \ / \ / \ / \ / \ / \

%e o o---o o---o---o o---o---o---o

%e n: 1 2 3 4

%e Vertices: 1 3 6 10

%e Edges: 0 3 9 18

%e The 2 X 2 X 2 triangular grid is equal to the cycle graph C_3 with chromatic polynomial q^3 -3*q^2 +2*q => [1, -3, 2, 0].

%e Triangle T(n,k) begins:

%e 1, 0;

%e 1, -3, 2, 0;

%e 1, -9, 32, -56, 48, -16, 0;

%e 1, -18, 144, -672, 2016, -4031, 5368, ...

%e 1, -30, 419, -3612, 21477, -93207, 304555, ...

%e 1, -45, 965, -13115, 126720, -925528, 5303300, ...

%e ...

%Y Cf. A000217, A045943, A178435, A182797, A185442, A193233, A193277.

%K sign,hard,look,tabf

%O 1,4

%A _Alois P. Heinz_, Jul 20 2011