A365941 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 periodic boundary conditions.

0, 1, 0, 4, 8, 4, 1, 0, 9, 54, 132, 171, 135, 84, 36, 9, 1, 0, 16, 192, 1040, 3340, 7104, 10872, 13136, 13500, 11584, 8024, 4368, 1820, 560, 120, 16, 1, 0, 25, 550, 5750, 37975, 177635, 625300, 1717700, 3767900, 6698700, 9746040, 11687850, 11647575, 9773725, 7044650, 4449220, 2480625, 1211300, 510150, 181900, 53630, 12675, 2300, 300, 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).

References

S. Mertens, I. Jensen, and R. M. Ziff, Universal features of cluster numbers in percolation, Physical Review E 96 (2017) 052119.

Links

Table of n, a(n) for n=1..60.

Stephan Mertens, Percolation (Gives first 7 rows)

Example

Triangle begins:
  0,1,
  0,4,8,4,1,
  0,9,54,132,171,135,84,36,9,1,
  0,16,192,1040,3340,7104,10872,13136,13500,11584,8024,4368,1820,560,120,16,1,
  0,25,550,5750,37975,177635,625300,1717700,3767900,6698700,9746040,11687850,11647575,9773725,7044650,4449220,2480625,1211300,510150,181900,53630,12675,2300,300,25,1,
...

Crossrefs

Cf. A365940-A365944.

Sequence in context: A124002 A014457 A376546 * A365946 A365944 A092511

Adjacent sequences: A365938 A365939 A365940 * A365942 A365943 A365944

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 09 2023

Status

approved