A365944 Irregular triangle read by rows: T(N,k) (0 <= k <= 4*N^2) are coefficients of cluster density function for site percolation on a 2*N X 2*N 2D hexagonal lattice with periodic boundary conditions.

0, 4, 8, 4, 1, 0, 16, 216, 1344, 5100, 13152, 24304, 33056, 33464, 25264, 14248, 6112, 2084, 576, 120, 16, 1, 0, 36, 1152, 17820, 177426, 1277262, 7079808, 31421718, 114650784, 350474142, 910294965, 2031036588, 3927868917, 6635523816, 9862330248, 12985500660, 15247918890, 16066928502, 15272213638, 13143857346, 10260330822, 7263259932, 4653902178, 2690524944, 1397524095, 648817524, 267527043, 97208878, 30826998, 8428590, 1956162, 377550, 58923, 7140, 630, 36, 1

Offset

1,2

Comments

The cluster density function is (1/(4*N^2))*Sum_{k=0..4*N^2} T(N,k)*p^k*(1-p)^(4*N^2-k).

Links

Table of n, a(n) for n=1..59.

Stephan Mertens, Percolation

S. Mertens, I. Jensen, and R. M. Ziff, Universal features of cluster numbers in percolation, arXiv:1602.00644 [cond-mat.stat-mech], 2016-2017; Physical Review E 96 (2017) 052119.

Example

Triangle begins:
  0, 4, 8, 4, 1;
  0, 16, 216, 1344, 5100, 13152, 24304, 33056, 33464, 25264, 14248, 6112, 2084, 576, 120, 16, 1;
  0, 36, 1152, 17820, 177426, 1277262, 7079808, 31421718, 114650784, 350474142, 910294965, 2031036588, 3927868917, 6635523816, 9862330248, 12985500660, 15247918890, 16066928502, 15272213638, 13143857346, 10260330822, 7263259932, 4653902178, 2690524944, 1397524095, 648817524, 267527043, 97208878, 30826998, 8428590, 1956162, 377550, 58923, 7140, 630, 36, 1;
  ...

Crossrefs

Cf. A365940, A365941, A365942, A365943.

Sequence in context: A014457 A365941 A365946 * A092511 A045816 A085991

Adjacent sequences: A365941 A365942 A365943 * A365945 A365946 A365947

Keyword

nonn,tabf

Author

N. J. A. Sloane, Oct 09 2023

Extensions

The DATA shows three rows of the triangle.

Status

approved