A365942 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 nnsquare lattice with periodic boundary conditions.

0, 1, 0, 4, 6, 4, 1, 0, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 16, 176, 848, 2448, 5056, 8440, 11584, 12890, 11440, 8008, 4368, 1820, 560, 120, 16, 1, 0, 25, 500, 4700, 27500, 112145, 339650, 801500, 1543750, 2538800, 3675220, 4717300, 5330625, 5250975, 4472200, 3271820, 2043375, 1081600, 480700, 177100, 53130, 12650, 2300, 300, 25, 1

Offset

1,4

Comments

An nnsquare lattice is a square lattice with additional next nearest neighbor links.

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, 6, 4, 1,
  0, 9, 36, 84, 126, 126, 84, 36, 9, 1,
  0, 16, 176, 848, 2448, 5056, 8440, 11584, 12890, 11440, 8008, 4368, 1820, 560, 120, 16, 1,
...

Crossrefs

Cf. A365940-A365944.

Sequence in context: A219234 A350481 A155675 * A365947 A230207 A277949

Adjacent sequences: A365939 A365940 A365941 * A365943 A365944 A365945

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 09 2023

Status

approved