A365952 Irregular triangle read by rows: T(N,k) (0 <= k <= 4*N^2) are coefficients of exact wrapping probability for site percolation on a 2*N X 2*N 2D hexagonal lattice with periodic boundary conditions. This is for the probability that it wraps in both dimensions.

0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 32, 160, 448, 588, 352, 104, 16, 1, 0, 0, 0, 0, 0, 0, 6, 180, 3186, 38832, 343125, 2284920, 11812665, 48453246, 160204284, 432061500, 959302998, 1766890134, 2717497054, 3511031418, 3832283250, 3552960132, 2812344534, 1909555974, 1116592323, 563847696, 246161187, 92809456, 30100680, 8335404, 1947336, 376992, 58905, 7140, 630, 36, 1

Offset

1,4

Comments

The wrapping probability function is Sum_{k=0..4*N^2} T(N,k)*p^k*(1-p)^(4*N^2-k).

Links

Table of n, a(n) for n=1..59.

Stephan Mertens, Percolation

Example

Triangle begins:
  0, 0, 0, 4, 1,
  0, 0, 0, 0, 0, 0, 0, 0, 4, 32, 160, 448, 588, 352, 104, 16, 1,
  0, 0, 0, 0, 0, 0, 6, 180, 3186, 38832, 343125, 2284920, 11812665, 48453246, 160204284, 432061500, 959302998, 1766890134, 2717497054, 3511031418, 3832283250, 3552960132, 2812344534, 1909555974, 1116592323, 563847696, 246161187, 92809456, 30100680, 8335404, 1947336, 376992, 58905, 7140, 630, 36, 1,
...

Crossrefs

Cf. A365940-A365957, A366463-A366467.

Sequence in context: A327113 A051390 A124120 * A324803 A320742 A093318

Adjacent sequences: A365949 A365950 A365951 * A365953 A365954 A365955

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 12 2023

Extensions

The DATA shows three rows of the triangle.

Status

approved