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

1,2

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).

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.

Formula

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

Example

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

Crossrefs

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

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

Sequence in context: A360858 A181511 A115196 * A093346 A115597 A325007

Adjacent sequences: A385578 A385579 A385580 * A385582 A385583 A385584

Keyword

nonn,tabl

Author

Pontus von Brömssen, Jul 04 2025

Status

approved