%I #50 Sep 29 2024 15:05:07
%S 4,1,9,4,2,2,4,4,1,7,9,5,1,0,7,5,9,7,7,0,9,9,5,6,1,0,7,7,0,2,9,7,4,2,
%T 5,2,2,3,3,9,5,3,2,3,4,3,9,2,6,6,6,7,4,9,0,8,0,4,4,9,9,1,6,6,3,1,7,7,
%U 2,0,5,0,8,7,2,7,0,9,1,9,3,9,1,0,0,2,3,2,4,5,4,7,4,2,3,8,1,9,5,5,0,2,8,5,8
%N Decimal expansion of Product_{k>=0} (1 - 1/(2*4^k)).
%C This is the limiting probability that a large random symmetric binary matrix is nonsingular (cf. A086812, A048651). In other words, equals Lim_{n->oo} A086812(n)/A006125(n+1).- _H. Tracy Hall_, Sep 07 2024
%H John E. Cremona and R. W. K. Odoni, <a href="https://doi.org/10.1112/jlms/s2-39.1.16">Some density results for negative Pell equations; an application of graph theory</a>, Journal of the London Mathematical Society s2-39:1 (1989), pp. 16-28.
%F Equals lim inf_{n->oo} Product_{k=0..floor(log_4(n))} floor(n/4^k)*4^k/n.
%F Equals lim inf_{n->oo} A132028(n)/n^(1+floor(log_4(n)))*4^((1/2)*(1+floor(log_4(n)))*floor(log_4(n))).
%F Equals lim inf_{n->oo} A132028(n)/n^(1+floor(log_4(n)))*4^A000217(floor(log_4(n))).
%F Equals (1/2)*exp(-Sum_{n>0} (4^(-n)*(Sum_{k|n} 1/(k*2^k)))).
%F Equals lim inf_{n->oo} A132028(n)/A132028(n+1).
%F Equals Product_{k>0} (1-1/(2^k+1)). - _Robert G. Wilson v_, May 25 2011
%F From _Robert FERREOL_, Feb 23 2020: (Start)
%F Equals Product_{k>0} (1 + 1/2^k)^(-1) = 2/A081845.
%F Equals 1 + Sum_{n>=1} (-1)^n*2^(n*(n-1)/2)/((2-1)*(2^2-1)*...*(2^n-1)). (End)
%F From _Peter Bala_, Jan 15 2021: (Start)
%F Constant C = Sum_{n >= 0} 2^n/Product_{k = 1..n} (1 - 4^k).
%F Faster converging series:
%F 2*C = (1/2)*Sum_{n >= 0} 2^(-n)/Product_{k = 1..n} (1 - 4^k);
%F (2^4)*C = 7*Sum_{n >= 0} 2^(-3*n)/Product_{k = 1..n} (1 - 4^k);
%F (2^9)*C = 7*31*Sum_{n >= 0} 2^(-5*n)/Product_{k = 1..n} (1 - 4^k), and so on.
%F Slower converging series:
%F C = -Sum_{n >= 0} 2^(3*n)/Product_{k = 1..n} (1 - 4^k);
%F 7*C = Sum_{n >= 0} 2^(5*n)/Product_{k = 1..n} (1 - 4^k);
%F 7*31*C = -Sum_{n >= 0} 2^(7*n)/Product_{k = 1..n} (1 - 4^k), and so on. (End)
%F Equals Product_{n>=0} (1 - 1/A004171(n)). - _Amiram Eldar_, May 09 2023
%e 0.41942244179510759770995610770297425223395323439266674908044991663177...
%p evalf(1+sum((-1)^n*2^(n*(n-1)/2)/product(2^k-1, k=1..n), n=1..infinity), 120); # _Robert FERREOL_, Feb 23 2020
%t RealDigits[ Product[1 - 1/(2*4^i), {i, 0, 175}], 10, 111][[1]] (* _Robert G. Wilson v_, May 25 2011 *)
%t RealDigits[QPochhammer[1/2, 1/4], 10, 105][[1]] (* _Jean-François Alcover_, Nov 18 2015 *)
%o (PARI) prodinf(k=0,1-1.>>(2*k+1)) \\ _Charles R Greathouse IV_, Nov 16 2012
%Y Cf. A004171, A048651, A098844, A067080, A132019, A132026, A132028, A100221, A000217, A081845.
%K nonn,cons
%O 0,1
%A _Hieronymus Fischer_, Aug 14 2007
%E Name corrected by _Charles R Greathouse IV_, Nov 16 2012