A365953 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 triangular 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, 9, 81, 126, 84, 36, 9, 1, 0, 0, 0, 0, 12, 240, 1704, 5824, 10710, 11136, 8008, 4368, 1820, 560, 120, 16, 1, 0, 0, 0, 0, 0, 15, 525, 6975, 52350, 255875, 868195, 2098800, 3632800, 4541775, 4286850, 3243010, 2041200, 1081575, 480700, 177100, 53130, 12650, 2300, 300, 25, 1

Offset

1,5

Comments

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, 9, 81, 126, 84, 36, 9, 1,
  0, 0, 0, 0, 12, 240, 1704, 5824, 10710, 11136, 8008, 4368, 1820, 560, 120, 16, 1,
  0, 0, 0, 0, 0, 15, 525, 6975, 52350, 255875, 868195, 2098800, 3632800, 4541775, 4286850, 3243010, 2041200, 1081575, 480700, 177100, 53130, 12650, 2300, 300, 25, 1,
...

Crossrefs

Cf. A365940-A365957, A366463-A366467.

Sequence in context: A166978 A356547 A365956 * A365955 A368831 A158567

Adjacent sequences: A365950 A365951 A365952 * A365954 A365955 A365956

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 12 2023

Status

approved