A391389 Site percolation series for square lattice: coefficients of the power series expansion in powers of q=1-p of the probability that a given open site belongs to the infinite cluster, where p is the probability that a site is open.

1, 0, 0, 0, -1, 0, -4, -8, -23, -28, -186, 48, -1301, 1412, -12292, 30384, -142441, 416612, -1613164, 5161772, -18738614, 62391348, -219708272, 744185860, -2587326809, 8835253824, -30515082648, 104424738432, -358958816131, 1227808860812, -4206527025296

Offset

0,7

Links

Table of n, a(n) for n=0..30.

A. R. Conway and A. J. Guttmann, On two-dimensional percolation, Journal of Physics A: Mathematical and General 28 (1995), 891-904. See Table 9. Apparently, there is a misprint in the value of a(30); it should be -4206527025296 rather than -4606527025296.

M. F. Sykes, D. S. Gaunt and M. Glen, Percolation processes in two dimensions. III. High density series expansions, Journal of Physics A: Mathematical and General 9 (1976), 715-724. See coefficients for expansion of P(p) for square site problem in the appendix.

Formula

a(n) = -Sum_{m>=1,k=max(0,n-m+1)..min(n,2*m+2)} A338210(m,k)*m*(-1)^(n-k)*binomial(m-1,n-k) for n >= 1.

a(n) = Sum_{k=0..n} A391388(k).

Crossrefs

Cf. A338210, A391388, A391391, A391393.

Sequence in context: A075688 A272563 A272152 * A306483 A274521 A026596

Adjacent sequences: A391386 A391387 A391388 * A391390 A391391 A391392

Keyword

sign

Author

Pontus von Brömssen, Dec 10 2025

Extensions

a(17)-a(30) from the Conway-Guttmann paper (with a correction for a(30)) added by Pontus von Brömssen, Dec 18 2025

Status

approved