A365949 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 square lattice with periodic boundary conditions. This is for the probability that it wraps in both dimensions.

0, 1, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 9, 42, 36, 9, 1, 0, 0, 0, 0, 0, 0, 0, 16, 328, 1504, 2960, 2992, 1668, 560, 120, 16, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 1510, 16300, 86925, 285200, 625550, 947740, 1004400, 754775, 412250, 168450, 52620, 12650, 2300, 300, 25, 1

Offset

1,6

Comments

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, 0, 4, 1,
  0, 0, 0, 0, 0, 9, 42, 36, 9, 1,
  0, 0, 0, 0, 0, 0, 0, 16, 328, 1504, 2960, 2992, 1668, 560, 120, 16, 1,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 1510, 16300, 86925, 285200, 625550, 947740, 1004400, 754775, 412250, 168450, 52620, 12650, 2300, 300, 25, 1,
...

Crossrefs

Cf. A365940-A365957, A366463-A366467.

Sequence in context: A320742 A093318 A255329 * A127560 A098172 A049759

Adjacent sequences: A365946 A365947 A365948 * A365950 A365951 A365952

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 12 2023

Status

approved