A365947 Irregular triangle read by rows: T(n,k) (0 <= k <= n^2) are coefficients of cluster density function for site percolation on an n X n 2D nnsquare lattice with open boundary conditions.

0, 1, 0, 4, 6, 4, 1, 0, 9, 52, 128, 177, 154, 90, 36, 9, 1, 0, 16, 198, 1128, 3921, 9296, 15932, 20452, 20168, 15600, 9628, 4776, 1880, 564, 120, 16, 1, 0, 25, 528, 5308, 33785, 152766, 521962, 1399532, 3019536, 5334899, 7819026, 9603688, 9971750, 8823881, 6704598, 4401190, 2503378, 1230980, 518728, 184464, 54156, 12742, 2304, 300, 25, 1

Offset

1,4

Comments

An nnsquare lattice is a square lattice with additional next nearest neighbor links.

The cluster density function is (1/n^2)*Sum_{k=0..n^2} T(n,k)*p^k*(1-p)^(n^2-k).

Links

Table of n, a(n) for n=1..60.

Stephan Mertens, Percolation (Gives first 16 rows)

Example

Triangle begins:
  0, 1,
  0, 4, 6, 4, 1,
  0, 9, 52, 128, 177, 154, 90, 36, 9, 1,
  0, 16, 198, 1128, 3921, 9296, 15932, 20452, 20168, 15600, 9628, 4776, 1880, 564, 120, 16, 1,
  0, 25, 528, 5308, 33785, 152766, 521962, 1399532, 3019536, 5334899, 7819026, 9603688, 9971750, 8823881, 6704598, 4401190, 2503378, 1230980, 518728, 184464, 54156, 12742, 2304, 300, 25, 1,
...

Crossrefs

Cf. A365940-A365946.

Sequence in context: A350481 A155675 A365942 * A230207 A277949 A244081

Adjacent sequences: A365944 A365945 A365946 * A365948 A365949 A365950

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 10 2023

Status

approved