A365954 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 square lattice with periodic boundary conditions. This is for the probability that it wraps in either dimension.

0, 1, 0, 0, 4, 4, 1, 0, 0, 0, 6, 36, 81, 84, 36, 9, 1, 0, 0, 0, 0, 8, 96, 560, 2000, 4788, 7456, 7216, 4336, 1820, 560, 120, 16, 1, 0, 0, 0, 0, 0, 10, 200, 2000, 13000, 60625, 213880, 587750, 1269225, 2132950, 2734300, 2628910, 1901400, 1065675, 480150, 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, 4, 4, 1,
  0, 0, 0, 6, 36, 81, 84, 36, 9, 1,
  0, 0, 0, 0, 8, 96, 560, 2000, 4788, 7456, 7216, 4336, 1820, 560, 120, 16, 1,
  0, 0, 0, 0, 0, 10, 200, 2000, 13000, 60625, 213880, 587750, 1269225, 2132950, 2734300, 2628910, 1901400, 1065675, 480150, 177100, 53130, 12650, 2300, 300, 25, 1,
...

Crossrefs

Cf. A365940-A365957, A366463-A366467.

Sequence in context: A372005 A350491 A365957 * A260043 A366467 A366464

Adjacent sequences: A365951 A365952 A365953 * A365955 A365956 A365957

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 12 2023

Status

approved