A385581 Square array read by antidiagonals: T(n,d) is the number of fixed d-dimensional polysticks of size n.

1, 2, 1, 3, 6, 1, 4, 15, 22, 1, 5, 28, 95, 88, 1, 6, 45, 252, 681, 372, 1, 7, 66, 525, 2600, 5277, 1628, 1, 8, 91, 946, 7065, 29248, 43086, 7312, 1, 9, 120, 1547, 15696, 104097, 349132, 365313, 33466, 1, 10, 153, 2360, 30513, 285828, 1632915, 4351944, 3186444, 155446, 1

Offset: 1

Author

Pontus von Brömssen, Jul 04 2025

Keywords

• nonn
• tabl

Comments

The first 17 antidiagonals are from Mertens and Moore (2018), either directly from Table 1 or computed using the perimeter polynomials in Appendix A. T(14,5) is the only unknown value in the 18th antidiagonal.

T(13,6) = 14054816418877200 (Mertens and Moore).

Examples

			Table begins:
  n\d| 1     2       3        4         5          6          7           8
  ---+---------------------------------------------------------------------
  1  | 1     2       3        4         5          6          7           8
  2  | 1     6      15       28        45         66         91         120
  3  | 1    22      95      252       525        946       1547        2360
  4  | 1    88     681     2600      7065      15696      30513       53936
  5  | 1   372    5277    29248    104097     285828     661549     1356384
  6  | 1  1628   43086   349132   1632915    5551480   15314936    36449288
  7  | 1  7312  365313  4351944  26817465  113045832  372033993  1028383408
  8  | 1 33466 3186444 56062681 456137580 2386821009 9377038237 30118187174
		

Links

• Pontus von Brömssen, Table of n, a(n) for n = 1..153 (first 17 antidiagonals)
• Stephan Mertens and Cristopher Moore, Series expansion of the percolation threshold on hypercubic lattices, J. Phys. A: Math. Theor., 51 (2018), 475001; arXiv:1805.02701 [cond-mat.stat-mech], 2018.
• Index entries for sequences related to polyominoes.

Crossrefs

Cf. A000384 (row n=2), A385291 (polyominoes), A385582, A385583 (free).

Columns: A096267 (d=2), A365560 (d=3), A365562 (d=4), A365564 (d=5).

Formula

T(n,d) = Sum_{k=1..d} binomial(n,k)*A385582(n,k) (with A385582(n,k) = 0 if d > n).