A365940 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 triangular lattice with periodic boundary conditions.

0, 1, 0, 4, 7, 4, 1, 0, 9, 45, 90, 126, 126, 84, 36, 9, 1, 0, 16, 192, 1040, 3340, 7104, 10872, 13136, 13500, 11584, 8024, 4368, 1820, 560, 120, 16, 1, 0, 25, 525, 5225, 32725, 144390, 476025, 1214400, 2454525, 4009875, 5420535, 6256650, 6371225, 5810550, 4714050, 3354100, 2064550, 1085475, 481150, 177125, 53130, 12650, 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, 7, 4, 1,
0, 9, 45, 90, 126, 126, 84, 36, 9, 1,
0, 16, 192, 1040, 3340, 7104, 10872, 13136, 13500, 11584, 8024, 4368, 1820, 560, 120, 16, 1,
...

Crossrefs

Cf. A365941-A365944.

Sequence in context: A063378 A280547 A301930 * A365945 A365943 A201412

Adjacent sequences: A365937 A365938 A365939 * A365941 A365942 A365943

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 09 2023

Status

approved