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
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
Crossrefs
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.
Comments