A182797 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.

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, 0, 0, 6, 32640, 98820, 7680, 210, 8, 0, 0, 6, 718080, 13638780, 1574400, 26250, 336, 9, 0, 0, 6, 22665216, 4260983940, 1034019840, 13676250, 72576, 504, 10

Offset: 1

Author

Alois P. Heinz, Dec 02 2010

Keywords

• nonn
• tabl

Comments

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.

The coefficients of the chromatic polynomials for the column sequences are given by the rows of A193283. - Georg Fischer, Jul 31 2023

Examples

			Square array A(n,k) begins:
  1,   0,    0,       0,          0,             0,  ...
  2,   0,    0,       0,          0,             0,  ...
  3,   6,    6,       6,          6,             6,  ...
  4,  24,  192,    2112,      32640,        718080,  ...
  5,  60, 1620,   98820,   13638780,    4260983940,  ...
  6, 120, 7680, 1574400, 1034019840, 2175789895680,  ...
		

Links

• Wikipedia, Triangular grid graph
• Wikipedia, Chromatic polynomial

Crossrefs

Columns k=1-11 give: A000027, A007531, A182788, A182789, A182790, A182791, A182792, A182793, A182794, A182795, A182796.

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.

Cf. A000217, A193283.