|
%I #18 May 09 2023 08:57:09
%S 9,0,0,8,3,2,7,0,6,8,0,9,7,1,5,2,7,9,9,4,9,8,6,2,6,9,4,7,6,0,6,4,7,7,
%T 4,4,7,6,2,4,9,1,1,9,2,2,1,6,6,3,9,5,2,4,0,2,1,4,6,1,7,2,4,8,8,0,6,5,
%U 7,0,8,7,0,6,7,0,9,7,5,8,5,6,7,0,0,1,6,3,9,2,9,9,1,9,9,2,8,3,5,6,4,6,5,2,0
%N Decimal expansion of Product_{k>0} (1-1/11^k).
%H G. C. Greubel, <a href="/A132267/b132267.txt">Table of n, a(n) for n = 0..2000</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*11^n)) = exp(-Sum_{n>0} A000203(n)/(n*11^n)).
%F Equals (1/11; 1/11)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - _G. C. Greubel_, Dec 20 2015
%F From _Amiram Eldar_, May 09 2023: (Start)
%F Equals sqrt(2*Pi/log(11)) * exp(log(11)/24 - Pi^2/(6*log(11))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(11))) (McIntosh, 1995).
%F Equals Sum_{n>=0} (-1)^n/A027879(n). (End)
%e 0.900832706809715279949862694760...
%t digits = 105; NProduct[1-1/11^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[1/11, 1/11], 200] (* _G. C. Greubel_, Dec 20 2015 *)
%o (PARI) prodinf(x=1, 1-1/11^x) \\ _Altug Alkan_, Dec 20 2015
%Y Cf. A000203, A027879, A048651, A100220, A132019-A132026, A132034-A132038, A132265-A132268, A132323, A132324, A132325, A132326.
%K nonn,cons
%O 0,1
%A _Hieronymus Fischer_, Aug 20 2007
|
