A365955 Irregular triangle read by rows: T(n,k) (0 <= k <= n^2) are coefficients of exact wrapping probability for site percolation on an n X n 2D nnsquare lattice with periodic boundary conditions. This is for the probability that it wraps in either dimension.

0, 1, 0, 0, 6, 4, 1, 0, 0, 0, 48, 117, 126, 84, 36, 9, 1, 0, 0, 0, 0, 160, 1504, 5496, 10352, 12662, 11424, 8008, 4368, 1820, 560, 120, 16, 1, 0, 0, 0, 0, 0, 520, 9000, 73250, 356450, 1135925, 2500910, 4024550, 5038100, 5160175, 4451000, 3268160, 2042950, 1081575, 480700, 177100, 53130, 12650, 2300, 300, 25, 1

Offset

1,5

Comments

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

The wrapping probability function is 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 7 rows)

Example

Triangle begins:
  0, 1,
  0, 0, 6, 4, 1,
  0, 0, 0, 48, 117, 126, 84, 36, 9, 1,
  0, 0, 0, 0, 160, 1504, 5496, 10352, 12662, 11424, 8008, 4368, 1820, 560, 120, 16, 1,
  0, 0, 0, 0, 0, 520, 9000, 73250, 356450, 1135925, 2500910, 4024550, 5038100, 5160175, 4451000, 3268160, 2042950, 1081575, 480700, 177100, 53130, 12650, 2300, 300, 25, 1,
...

Crossrefs

Cf. A365940-A365957, A366463-A366467.

Sequence in context: A356547 A365956 A365953 * A368831 A158567 A215832

Adjacent sequences: A365952 A365953 A365954 * A365956 A365957 A365958

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 12 2023

Status

approved