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A132265 Decimal expansion of Product_{k>=0} (1 - 1/(2*11^k)).

%I A132265 #22 Feb 16 2025 08:33:06
%S A132265 4,7,5,1,0,4,1,2,7,5,0,7,6,0,3,1,0,5,3,9,7,5,6,4,4,4,7,2,9,4,6,9,7,6,
%T A132265 9,4,3,3,6,9,7,1,9,2,1,1,7,0,8,5,1,1,6,3,8,0,0,7,7,3,6,6,5,4,1,3,0,4,
%U A132265 7,5,4,4,5,7,2,4,8,7,7,3,7,2,3,0,8,4,3,7,6,9,3,7,4,4,1,6,8,2,4,9,8,2,2,7,3
%N A132265 Decimal expansion of Product_{k>=0} (1 - 1/(2*11^k)).
%H A132265 G. C. Greubel, <a href="/A132265/b132265.txt">Table of n, a(n) for n = 0..1500</a>
%H A132265 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>
%F A132265 lim inf Product_{k=0..floor(log_11(n))} floor(n/11^k)*11^k/n for n-->oo.
%F A132265 lim inf A132263(n)*11^((1+floor(log_11(n)))*floor(log_11(n))/2)/n^(1+floor(log_11(n))) for n-->oo.
%F A132265 lim inf A132263(n)*11^A000217(floor(log_11(n)))/n^(1+floor(log_11(n))) for n-->oo.
%F A132265 (1/2)*exp(-Sum_{n>0} 11^(-n)*Sum_{k|n} 1/(k*2^k)).
%F A132265 lim inf A132263(n)/A132263(n+1) = 0.47510412750760310539756444... for n-->oo.
%F A132265 Equals (1/2; 1/11)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - _G. C. Greubel_, Nov 30 2015
%e A132265 0.47510412750760310539756444...
%t A132265 digits = 105; NProduct[1-1/(2*11^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+10] // N[#, digits+10]& // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 18 2014 *)
%t A132265 N[QPochhammer[1/2,1/11]] (* _G. C. Greubel_, Nov 30 2015 *)
%o A132265 (PARI) prodinf(x=0, 1 - 1/(2*11^x)) \\ _Altug Alkan_, Dec 01 2015
%Y A132265 Cf. A000217, A132019-A132026, A132034-A132038, A132263, A132266-A132268, A132323-A132326.
%K A132265 nonn,cons
%O A132265 0,1
%A A132265 _Hieronymus Fischer_, Aug 20 2007