A383735 Array read by antidiagonals, where each row is the cluster series for percolation on the cells of a certain type of polyominoids.

1, 0, 1, 0, 2, 1, 0, 2, 0, 1, 0, 2, 0, 2, 1, 0, 2, 0, 2, 4, 1, 0, 2, 0, 2, 12, 6, 1, 0, 2, 0, 2, 24, 18, 0, 1, 0, 2, 0, 2, 52, 48, 0, 4, 1, 0, 2, 0, 2, 108, 126, 0, 12, 4, 1, 0, 2, 0, 2, 224, 300, 0, 24, 12, 8, 1, 0, 2, 0, 2, 412, 762, 0, 52, 24, 32, 0, 1

Offset: 1

Author

Pontus von Brömssen, May 10 2025

Keywords

• nonn
• tabl

Comments

T(n,k) is the coefficient of p^(k+1), k >= 0, in the power series expansion of the expected finite size of the cluster containing a given cell for percolation with probability p on the polyominoid cells corresponding to row n of A366766. If the given cell is not open, its cluster is empty. Equivalently, T(n,k) can be taken to be the coefficient of p^k if we condition on the event that the given cell is open.

See A366766 for details on how the polyominoids are specified and on the ordering of the rows.

Examples

			Array begins:
  n\k| 0  1  2   3   4    5    6    7     8     9     10     11      12
  ---+-----------------------------------------------------------------
   1 | 1  0  0   0   0    0    0    0     0     0      0      0       0
   2 | 1  2  2   2   2    2    2    2     2     2      2      2       2
   3 | 1  0  0   0   0    0    0    0     0     0      0      0       0
   4 | 1  2  2   2   2    2    2    2     2     2      2      2       2
   5 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
   6 | 1  6 18  48 126  300  762 1668  4216  8668  21988  43058  110832
   7 | 1  0  0   0   0    0    0    0     0     0      0      0       0
   8 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
   9 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
  10 | 1  8 32 108 348 1068 3180 9216 26452 73708 206872 563200 1555460
  11 | 1  0  0   0   0    0    0    0     0     0      0      0       0
  12 | 1  2  2   2   2    2    2    2     2     2      2      2       2
		

Links

• Index entries for sequences related to cluster series.

Crossrefs

Cf. A366766, A366767, A366768.

Rows include:

n | sequence for row n

---+-------------------

1 | A000007

2 | A040000

3 | A000007

4 | A040000

5 | A003203

6 | A003198

7 | A000007

8 | A003203

9 | A003203

10 | A003201

11 | A000007

12 | A040000

13 | A383737

14 | A003207

15 | A000007

16 | A003203

17 | A383737

18 | A383736

19 | A003203

20 | A003201

...

31 | A000007

32 | A003211

33 | A003209

34 | A036396

35 | A003210

...

38 | A036402

39 | A000007

40 | A040000

...

43 | A000007

44 | A003203

...

47 | A003203

48 | A003201

Formula

T(n,k) = [p^k] Sum_P m^2*p^(m-1)*(1-p)^j / binomial(D,d) = Sum_P m^2*(-1)^(k-m+1)*binomial(j,k-m+1) / binomial(D,d), where the sum is over all fixed polyominoids P (corresponding to row n of A366766), m is the number of cells of P, and j is the number of cells that are not in P but are adjacent to a cell in P; d is the dimension of the cells and D is the dimension of the ambient space. It is sufficient to take the sums over those P that have at most k+1 cells.