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%I A132037 #16 Aug 06 2024 05:08:40
%S A132037 8,7,6,5,6,0,3,5,4,0,3,5,9,6,4,2,0,5,8,3,6,0,1,9,8,3,8,4,1,7,8,6,2,0,
%T A132037 1,0,1,0,6,6,3,5,1,0,1,1,7,4,6,7,1,8,3,3,6,1,4,9,3,5,2,8,0,1,5,8,7,0,
%U A132037 8,5,4,2,3,1,7,1,8,2,9,9,6,9,9,0,4,4,4,7,7,7,6,9,2,0,7,9,1,9,6,4,5,0,9
%N A132037 Decimal expansion of Product_{k>0} (1-1/9^k).
%H A132037 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/pdf/4f03a5e304ec19f8a725774525aecd2a78f4ad81">alternative link</a>.
%F A132037 Equals exp(-Sum_{n>0} sigma_1(n)/(n*9^n)) = exp(-Sum_{n>0} A000203(n)/(n*9^n)).
%F A132037 Equals Sum_{n>=0} A010815(n)/9^n. - _R. J. Mathar_, Mar 04 2009
%F A132037 From _Amiram Eldar_, May 09 2023: (Start)
%F A132037 Equals sqrt(Pi/log(3)) * exp(log(3)/12 - Pi^2/(12*log(3))) * Product_{k>=1} (1 - exp(-2*k*Pi^2/log(3))) (McIntosh, 1995).
%F A132037 Equals Sum_{n>=0} (-1)^n/A027877(n). (End)
%e A132037 0.8765603540359642058360198...
%t A132037 digits = 103; NProduct[1-1/9^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 18 2014 *)
%t A132037 RealDigits[QPochhammer[1/9], 10, 100][[1]] (* _Amiram Eldar_, May 09 2023 *)
%o A132037 (PARI) prodinf(k=1, 1 - 1/(9^k)) \\ _Amiram Eldar_, May 09 2023
%Y A132037 Cf. A010815, A027877, A048651, A100220, A132025, A132026, A132038, A098844, A067080, A000203.
%K A132037 nonn,cons
%O A132037 0,1
%A A132037 _Hieronymus Fischer_, Aug 14 2007

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