A387002 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional (d,2)-polyominoids of size n, 2 <= d <= n+1.

1, 2, 12, 6, 140, 320, 19, 1554, 10368, 13520, 63, 17622, 265344, 892864, 786432, 216, 206747, 6390484, 41998840, 89389920, 58383808, 760, 2503578, 152166240, 1749529040, 6773387520

Offset: 1

Author

Pontus von Brömssen, Aug 14 2025

Keywords

• nonn
• tabl
• more
• changed

Comments

A (d,2)-polyominoid consists of unit square cells with integer coordinates in the d-dimensional grid, where two cells are connected if they share an edge. The polyominoid is properly d-dimensional if it is not contained in a (d-1)-dimensional affine subspace.

Examples

			Triangle begins:
  n\d |     2          3           4           5          6        7  8  9 10 11
  ----+-------------------------------------------------------------------------
   1  |     1
   2  |     2         12
   3  |     6        140         320
   4  |    19       1554       10368       13520
   5  |    63      17622      265344      892864     786432
   6  |   216     206747     6390484    41998840   89389920 58383808
   7  |   760    2503578   152166240  1749529040 6773387520        ?  ?
   8  |  2725   31117536  3644734836 69246650605          ?        ?  ?  ?
   9  |  9910  394953243 88344741448           ?          ?        ?  ?  ?  ?
  10  | 36446 5098388985           ?           ?          ?        ?  ?  ?  ?  ?
		

Links

• Wikipedia, Polyominoid.
• Index entries for sequences related to polyominoes.

Crossrefs

Cf. A001168 (column d=2), A195739 (polyominoes), A385582 (polysticks), A385715, A387004 (free).

Formula

T(n,d) = Sum_{k=2..d} (-1)^(d-k)*binomial(d,k)*A385715(k,n), i.e., the n-th row is the inverse binomial transform of the n-th column of A385715 (with the convention that T(n,d) = A385715(d,n) = 0 when d <= 1).