A365950 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 both dimensions.

0, 1, 0, 0, 2, 4, 1, 0, 0, 0, 6, 45, 90, 78, 36, 9, 1, 0, 0, 0, 0, 8, 160, 1240, 4400, 8202, 9440, 7448, 4272, 1812, 560, 120, 16, 1, 0, 0, 0, 0, 0, 10, 350, 5350, 45550, 238925, 819740, 1915800, 3190350, 3977875, 3879550, 3055790, 1982350, 1068575, 478700, 176900, 53120, 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, 2, 4, 1,
  0, 0, 0, 6, 45, 90, 78, 36, 9, 1,
  0, 0, 0, 0, 8, 160, 1240, 4400, 8202, 9440, 7448, 4272, 1812, 560, 120, 16, 1,
  0, 0, 0, 0, 0, 10, 350, 5350, 45550, 238925, 819740, 1915800, 3190350, 3977875, 3879550, 3055790, 1982350, 1068575, 478700, 176900, 53120, 12650, 2300, 300, 25, 1,
...

Crossrefs

Cf. A365940-A365957, A366463-A366467.

Sequence in context: A366463 A366466 A365948 * A055599 A115407 A010586

Adjacent sequences: A365947 A365948 A365949 * A365951 A365952 A365953

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 12 2023

Status

approved