A355997 Triangle read by rows: T(n,d) is the number of fixed orthoplex n-ominoes with cell centers determining d-space.

1, 0, 4, 0, 1, 32, 0, 0, 48, 400, 0, 0, 28, 1728, 6912, 0, 0, 8, 4240, 62720, 153664, 0, 0, 1, 7272, 344320, 2457600, 4194304, 0, 0, 0, 8720, 1465600, 23872320, 105815808, 136048896, 0, 0, 0, 7136, 5254576, 182691200, 1603840000, 5017600000, 512000000

Offset: 2

Author

Robert A. Russell, Jul 22 2022

Keywords

• nonn
• tabl

Comments

Orthoplex polyominoes are connected sets of cells of regular tilings with Schläfli symbols {}, {4}, {3,4}, {3,3,4}, {3,3,3,4}, etc. These are tilings of regular orthoplexes projected on their circumspheres. Orthoplex polyominoes are equivalent to multidimensional polyominoes that do not extend more than two units along any axis, i.e., fit within a 2^d cube. Two fixed polyominoes are identical only if one is a translation of the other.

Conjecture: T(n,n-4) = 2^(n-7) * n^(n-9) * (n-4) * (n-5) * (n-6) * (n^6-14*n^5+65*n^4-189*n^3+594*n^2-1305*n+6832) / 6 ~ A259015(n) / 8.

Examples

			Triangle begins with T(2,1):
n\d 1 2  3    4       5         6          7          8         9
2   1
3   0 4
4   0 1 32
5   0 0 48  400
6   0 0 28 1728    6912
7   0 0  8 4240   62720    153664
8   0 0  1 7272  344320   2457600    4194304
9   0 0  0 8720 1465600  23872320  105815808  136048896
10  0 0  0 7136 5254576 182691200 1603840000 5017600000 512000000
		

Links

• Robert A. Russell, Table of n, a(n) for n = 2..73

Crossrefs

Cf. A195739 (multidimensional).

Diagonals (with formulas) are A127670, A355998, A355999.