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%I #40 Feb 19 2022 07:05:18
%S 5,6,0,1,2,6,0,7,7,9,2,7,9,4,8,9,4,4,9,6,9,7,9,2,2,4,3,3,1,4,1,4,0,0,
%T 1,4,3,7,9,7,3,6,3,3,3,7,9,8,3,6,2,4,6,4,4,6,2,9,5,6,2,9,7,3,1,7,5,3,
%U 3,9,6,3,0,8,9,0,3,3,7,9,4,7,0,7,7,1,6,9,1,8,7,7,0,5,3,6,7,4,3,3,4,8
%N Decimal expansion of Product_{k>=1} (1 - 1/3^k).
%C 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
%H G. C. Greubel, <a href="/A100220/b100220.txt">Table of n, a(n) for n = 0..1200</a>
%H Richard J. McIntosh, <a href="https://doi.org/10.1112/jlms/51.1.120">Some Asymptotic Formulae for q-Hypergeometric Series</a>, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; <a href="https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.993.6895&rep=rep1&type=pdf">alternative link</a>.
%H Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>.
%F exp(-Sum_{k > 0} sigma_1(k)/k/3^k) = exp(-Sum_{k > 0} A000203(k)/k/3^k). - _Hieronymus Fischer_, Aug 07 2007
%F 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
%F From _Peter Bala_, Jan 18 2021: (Start)
%F Constant C = (1 - 1/3)*Sum_{n >= 0} (-1/3)^n/Product_{k = 1..n} (3^k - 1);
%F C = (1 - 1/3)*(1 - 1/9)*Sum_{n >= 0} (-1/9)^n/Product_{k = 1..n} (3^k - 1);
%F 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)
%F From _Amiram Eldar_, Feb 19 2022: (Start)
%F 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).
%F Equals Sum_{n>=0} (-1)^n/A027871(n). (End)
%e 0.56012607792794894496979224331414001437973633379836...
%t (3^(1/24)*EllipticThetaPrime[1, 0, 1/Sqrt[3]]^(1/3))/2^(1/3).
%t N[QPochhammer[1/3,1/3]] (* _G. C. Greubel_, Nov 27 2015 *)
%Y Cf. A048651, A027871.
%Y Cf. A000203, A100221, A100222, A132019, A132034, A132035, A132036, A132037, A132038, A258458.
%K nonn,cons,easy
%O 0,1
%A _Eric W. Weisstein_, Nov 09 2004
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