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A193283 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.

%I A193283 #26 Aug 01 2023 14:30:43
%S A193283 1,0,1,-3,2,0,1,-9,32,-56,48,-16,0,1,-18,144,-672,2016,-4031,5368,
%T A193283 -4584,2272,-496,0,1,-30,419,-3612,21477,-93207,304555,-761340,
%U A193283 1463473,-2152758,2385118,-1929184,1075936,-369824,58976,0
%N A193283 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 A193283 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 A193283 Alois P. Heinz, <a href="/A193283/b193283.txt">Rows n = 1..13, flattened</a>
%H A193283 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%H A193283 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_graph#Other_kinds">Triangular grid graph</a>
%e A193283 4 example graphs:                           o
%e A193283                                            / \
%e A193283                               o           o---o
%e A193283                              / \         / \ / \
%e A193283                     o       o---o       o---o---o
%e A193283                    / \     / \ / \     / \ / \ / \
%e A193283               o   o---o   o---o---o   o---o---o---o
%e A193283   n:          1     2         3             4
%e A193283   Vertices:   1     3         6            10
%e A193283   Edges:      0     3         9            18
%e A193283 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 A193283 Triangle T(n,k) begins:
%e A193283   1,   0;
%e A193283   1,  -3,   2,      0;
%e A193283   1,  -9,  32,    -56,     48,     -16,       0;
%e A193283   1, -18, 144,   -672,   2016,   -4031,    5368, ...
%e A193283   1, -30, 419,  -3612,  21477,  -93207,  304555, ...
%e A193283   1, -45, 965, -13115, 126720, -925528, 5303300, ...
%e A193283   ...
%Y A193283 Cf. A000217, A045943, A178435, A182797, A185442, A193233, A193277.
%K A193283 sign,hard,look,tabf
%O A193283 1,4
%A A193283 _Alois P. Heinz_, Jul 20 2011