A365946 Irregular triangle read by rows: T(n,k) (0 <= k <= n^2) are coefficients of cluster density function for site percolation on an n X n 2D square lattice with open boundary conditions.

0, 1, 0, 4, 8, 4, 1, 0, 9, 60, 168, 256, 230, 124, 40, 9, 1, 0, 16, 216, 1344, 5105, 13212, 24618, 34012, 35347, 27748, 16412, 7284, 2442, 632, 124, 16, 1, 0, 25, 544, 5644, 37136, 173866, 616009, 1714798, 3843649, 7053191, 10716779, 13586810, 14446850, 12928519, 9764490, 6242558, 3390724, 1570643, 621183, 208557, 58479, 13306, 2352, 302, 25, 1

Offset

1,4

Comments

The cluster density function is (1/n^2)*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 16 rows)

Example

Triangle begins:
  0, 1,
  0, 4, 8, 4, 1,
  0, 9, 60, 168, 256, 230, 124, 40, 9, 1,
  0, 16, 216, 1344, 5105, 13212, 24618, 34012, 35347, 27748, 16412, 7284, 2442, 632, 124, 16, 1,
  0, 25, 544, 5644, 37136, 173866, 616009, 1714798, 3843649, 7053191, 10716779, 13586810, 14446850, 12928519, 9764490, 6242558, 3390724, 1570643, 621183, 208557, 58479, 13306, 2352, 302, 25, 1,
...

Crossrefs

Cf. A365940-A365947.

Sequence in context: A014457 A376546 A365941 * A365944 A092511 A045816

Adjacent sequences: A365943 A365944 A365945 * A365947 A365948 A365949

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 10 2023

Status

approved