A365957 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 either dimension.

0, 0, 4, 4, 1, 0, 0, 0, 0, 4, 48, 272, 944, 2214, 3696, 4408, 3536, 1768, 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, 4, 4, 1,
  0, 0, 0, 0, 4, 48, 272, 944, 2214, 3696, 4408, 3536, 1768, 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-A365956, A366463-A366467.

Sequence in context: A102412 A372005 A350491 * A365954 A260043 A366467

Adjacent sequences: A365954 A365955 A365956 * A365958 A365959 A365960

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 12 2023

Extensions

The DATA shows three rows of the triangle.

Status

approved