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

1,4

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.

Links

Alois P. Heinz, Rows n = 1..13, flattened

Wikipedia, Chromatic Polynomial

Wikipedia, Triangular grid graph

Example

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, ...
  ...

Crossrefs

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

Sequence in context: A142071 A350448 A291680 * A193277 A118972 A171224

Adjacent sequences: A193280 A193281 A193282 * A193284 A193285 A193286

Keyword

sign,hard,look,tabf

Author

Alois P. Heinz, Jul 20 2011

Status

approved