A365956 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 either dimension.

0, 0, 6, 4, 1, 0, 0, 0, 0, 68, 640, 2644, 6320, 9773, 10304, 7788, 4352, 1820, 560, 120, 16, 1, 0, 0, 0, 0, 0, 0, 462, 12456, 162810, 1370172, 8316297, 38643624, 142444695, 426043278, 1049742000, 2153912760, 3711426624, 5409742482, 6716638430, 7153883766, 6584406876, 5274811392, 3701882250, 2287146006, 1247224353, 600206400, 254134359, 94140796, 30260304, 8347680, 1947792, 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, 6, 4, 1,
  0, 0, 0, 0, 68, 640, 2644, 6320, 9773, 10304, 7788, 4352, 1820, 560, 120, 16, 1,
  0, 0, 0, 0, 0, 0, 462, 12456, 162810, 1370172, 8316297, 38643624, 142444695, 426043278, 1049742000, 2153912760, 3711426624, 5409742482, 6716638430, 7153883766, 6584406876, 5274811392, 3701882250, 2287146006, 1247224353, 600206400, 254134359, 94140796, 30260304, 8347680, 1947792, 376992, 58905, 7140, 630, 36, 1,
...

Crossrefs

Cf. A365940-A365957, A366463-A366467.

Sequence in context: A323525 A166978 A356547 * A365953 A365955 A368831

Adjacent sequences: A365953 A365954 A365955 * A365957 A365958 A365959

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 12 2023

Extensions

The DATA shows three rows of the triangle.

Status

approved