A365948 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 triangular lattice with periodic boundary conditions. This is for the probability that it wraps in both dimensions.

0, 1, 0, 0, 2, 4, 1, 0, 0, 0, 3, 27, 72, 78, 36, 9, 1, 0, 0, 0, 0, 4, 80, 568, 2144, 5034, 7456, 6872, 4208, 1812, 560, 120, 16, 1, 0, 0, 0, 0, 0, 5, 175, 2325, 17450, 86475, 307075, 817200, 1660050, 2569025, 3005250, 2681890, 1871800, 1046675, 476050, 176750, 53120, 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, 27, 72, 78, 36, 9, 1,
  0, 0, 0, 0, 4, 80, 568, 2144, 5034, 7456, 6872, 4208, 1812, 560, 120, 16, 1,
  0, 0, 0, 0, 0, 5, 175, 2325, 17450, 86475, 307075, 817200, 1660050, 2569025, 3005250, 2681890, 1871800, 1046675, 476050, 176750, 53120, 12650, 2300, 300, 25, 1,
...

Crossrefs

Cf. A365940-A365957, A366463-A366467.

Sequence in context: A188668 A366463 A366466 * A365950 A055599 A115407

Adjacent sequences: A365945 A365946 A365947 * A365949 A365950 A365951

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 12 2023

Status

approved