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.

1, 0, 1, -3, 2, 0, 1, -9, 32, -56, 48, -16, 0, 1, -18, 144, -672, 2016, -4031, 5368, -4584, 2272, -496, 0, 1, -30, 419, -3612, 21477, -93207, 304555, -761340, 1463473, -2152758, 2385118, -1929184, 1075936, -369824, 58976, 0

Offset: 1

Author

Alois P. Heinz, Jul 20 2011

Keywords

• sign
• hard
• look
• tabf

Comments

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.

Examples

			4 example graphs:                           o
                                           / \
                              o           o---o
                             / \         / \ / \
                    o       o---o       o---o---o
                   / \     / \ / \     / \ / \ / \
              o   o---o   o---o---o   o---o---o---o
  n:          1     2         3             4
  Vertices:   1     3         6            10
  Edges:      0     3         9            18
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].
Triangle T(n,k) begins:
  1,   0;
  1,  -3,   2,      0;
  1,  -9,  32,    -56,     48,     -16,       0;
  1, -18, 144,   -672,   2016,   -4031,    5368, ...
  1, -30, 419,  -3612,  21477,  -93207,  304555, ...
  1, -45, 965, -13115, 126720, -925528, 5303300, ...
  ...
		

Links

• Alois P. Heinz, Rows n = 1..13, flattened
• Wikipedia, Chromatic Polynomial
• Wikipedia, Triangular grid graph

Crossrefs

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