A365951 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 union jack lattice with periodic boundary conditions. This is for the probability that it wraps in both dimensions.

0, 0, 2, 4, 1, 0, 0, 0, 0, 4, 48, 332, 1360, 3385, 5344, 5476, 3760, 1756, 560, 120, 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

1,3

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, 2, 4, 1,
  0, 0, 0, 0, 4, 48, 332, 1360, 3385, 5344, 5476, 3760, 1756, 560, 120, 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: A309598 A305570 A287651 * A163259 A309785 A188668

Adjacent sequences: A365948 A365949 A365950 * A365952 A365953 A365954

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 12 2023

Extensions

The DATA shows three rows of the triangle.

Status

approved