OFFSET
0,1
COMMENTS
Limit of the probability that a random N X N matrix, with entries chosen independently and uniformly from the field F_3, is nonsingular [Morrison (2006)]. - L. Edson Jeffery, Jan 22 2012
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1200
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Eric Weisstein's World of Mathematics, Infinite Product.
FORMULA
exp(-Sum_{k > 0} sigma_1(k)/k/3^k) = exp(-Sum_{k > 0} A000203(k)/k/3^k). - Hieronymus Fischer, Aug 07 2007
Product_{k >= 1} (1 - 1/3^k) = (1/3; 1/3)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 27 2015
From Peter Bala, Jan 18 2021: (Start)
Constant C = (1 - 1/3)*Sum_{n >= 0} (-1/3)^n/Product_{k = 1..n} (3^k - 1);
C = (1 - 1/3)*(1 - 1/9)*Sum_{n >= 0} (-1/9)^n/Product_{k = 1..n} (3^k - 1);
C = (1 - 1/3)*(1 - 1/9)*(1 - 1/27)*Sum_{n >= 0} (-1/27)^n/Product_{k = 1..n} (3^k - 1), and so on. (End)
From Amiram Eldar, Feb 19 2022: (Start)
Equals sqrt(2*Pi/log(3)) * exp(log(3)/24 - Pi^2/(6*log(3))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(3))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027871(n). (End)
EXAMPLE
0.56012607792794894496979224331414001437973633379836...
MATHEMATICA
(3^(1/24)*EllipticThetaPrime[1, 0, 1/Sqrt[3]]^(1/3))/2^(1/3).
N[QPochhammer[1/3, 1/3]] (* G. C. Greubel, Nov 27 2015 *)
CROSSREFS
KEYWORD
AUTHOR
Eric W. Weisstein, Nov 09 2004
STATUS
approved