A366463 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 around the vertical dimension.

0, 1, 0, 0, 2, 4, 1, 0, 0, 0, 3, 18, 45, 63, 36, 9, 1, 0, 0, 0, 0, 4, 48, 280, 1008, 2558, 4480, 5088, 3664, 1744, 560, 120, 16, 1, 0, 0, 0, 0, 0, 5, 100, 1000, 6500, 30325, 107695, 302025, 678075, 1209075, 1679925, 1788325, 1452900, 910225, 446200, 172775, 52875, 12650, 2300, 300, 25, 1

Offset

1,5

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, 2, 4, 1,
  0, 0, 0, 3, 18, 45, 63, 36, 9, 1,
  0, 0, 0, 0, 4, 48, 280, 1008, 2558, 4480, 5088, 3664, 1744, 560, 120, 16, 1,
  0, 0, 0, 0, 0, 5, 100, 1000, 6500, 30325, 107695, 302025, 678075, 1209075, 1679925, 1788325, 1452900, 910225, 446200, 172775, 52875, 12650, 2300, 300, 25, 1,
...

Crossrefs

Cf. A365940-A365957, A366464-A366467.

Sequence in context: A163259 A309785 A188668 * A366466 A365948 A365950

Adjacent sequences: A366460 A366461 A366462 * A366464 A366465 A366466

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 12 2023

Status

approved