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%I #20 May 09 2023 08:57:12
%S 9,0,9,7,2,6,2,6,8,9,0,5,9,9,4,8,8,8,6,3,6,3,6,2,0,4,6,9,7,7,0,8,0,2,
%T 4,9,1,2,0,7,9,1,6,9,1,9,4,1,0,1,4,2,7,4,3,2,6,1,5,4,4,4,1,2,8,6,9,0,
%U 2,4,5,7,6,6,1,9,5,4,1,6,2,0,2,6,0,0,0,5,5,3,8,8,8,8,1,0,8,5,1,4,8,3,9,7,1,9
%N Decimal expansion of Product_{k>0} (1-1/12^k).
%H G. C. Greubel, <a href="/A132268/b132268.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>.
%F Equals exp(-Sum_{n>0} sigma_1(n)/(n*12^n)) = exp(-Sum_{n>0} A000203(n)/(n*12^n)).
%F Equals (1/12; 1/12)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - _G. C. Greubel_, Dec 05 2015
%F From _Amiram Eldar_, May 09 2023: (Start)
%F Equals sqrt(2*Pi/log(12)) * exp(log(12)/24 - Pi^2/(6*log(12))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(12))) (McIntosh, 1995).
%F Equals Sum_{n>=0} (-1)^n/A027880(n). (End)
%e 0.9097262689059948886363620469770...
%t digits = 106; NProduct[1-1/12^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 18 2014 *)
%t N[QPochhammer[1/12,1/12]] (* _G. C. Greubel_, Dec 05 2015 *)
%o (PARI) prodinf(k=1, (1-1/12^k)) \\ _Michel Marcus_, Dec 05 2015
%Y Cf. A027880, A048651, A100220, A132019-A132026, A132034-A132038, A132265-A132267, A132323-A132326, A000203.
%K nonn,cons
%O 0,1
%A _Hieronymus Fischer_, Aug 20 2007
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