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A365943
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 union jack lattice with periodic boundary conditions.
1
0, 4, 7, 4, 1, 0, 16, 192, 1040, 3404, 7608, 12460, 15688, 15737, 12808, 8452, 4456, 1828, 560, 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).
REFERENCES
S. Mertens, I. Jensen, and R. M. Ziff, Universal features of cluster numbers in percolation, Physical Review E 96 (2017) 052119.
EXAMPLE
Triangle begins:
0, 4, 7, 4, 1,
0, 16, 192, 1040, 3404, 7608, 12460, 15688, 15737, 12808, 8452, 4456, 1828, 560, 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
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Oct 09 2023
EXTENSIONS
The DATA shows three rows of the triangle.
STATUS
approved