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%I #61 Oct 02 2022 23:08:56
%S 1,3,5,9,1,4,0,9,1,4,2,2,9,5,2,2,6,1,7,6,8,0,1,4,3,7,3,5,6,7,6,3,3,1,
%T 2,4,8,8,7,8,6,2,3,5,4,6,8,4,9,9,7,9,7,8,7,4,8,3,4,8,3,8,1,3,8,6,2,0,
%U 3,8,3,1,5,1,7,6,7,7,3,7,9,7,2,8,5,6,9,1,0,8,9,2,6,2,5,8,3,2,1
%N Decimal expansion of e/2.
%D Jolley, Summation of Series, Dover (1961) eq. (161) on page 30.
%H Harry J. Smith, <a href="/A019739/b019739.txt">Table of n, a(n) for n = 1..20000</a>
%H R. P. Millane., <a href="http://dx.doi.org/10.1016/0375-9601(92)90434-N">A product form of the Möbius transform</a>, Whistler Center for Carbohydrate Research, Purdue University, West Lafayette, USA.
%H Roger H. Moritz, <a href="http://dx.doi.org/10.1080/10511979108965613">Summing series</a>, PRIMUS, 1 (2) (2007) 212-219, Comment 2.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F e/2 = lim_{n->infinity} n*(e - (1+1/n)^n). - _Benoit Cloitre_, Sep 17 2002
%F e/2 = Product_{n>=1} ((1/n)!)^mu(n), where mu is the Mobius function is an unusual infinite product for this number: (see Millane ref.). - _John M. Campbell_, Jun 14 2011
%F 10*(this constant) = 5*exp(1) = Sum_{j>=0} j^3/j! [Jolley]. - _R. J. Mathar_, Oct 03 2011
%F Equals Sum_{j>=0} (1+j)/(1+2*j)!. - _Bruno Berselli_, May 25 2015
%F Equals the coefficient of x in Sum_{m>1} log((1 - x/m!)(1 - 2x/m!)...(1 - (m-1)x/m!)). - _M. F. Hasler_, Apr 01 2020
%F Equals Sum_{k>=1} k*(k-1)/(2 * k!). - _Amiram Eldar_, Aug 10 2020
%e 1.359140914229522617680143735676331248878623546849979787483483813862038... = A001113/2.
%t N[Product[((1/n)!)^MoebiusMu[n], {n, 1, 200000}]] (* _John M. Campbell_, Jun 14 2011 *)
%t RealDigits[E/2,10,120][[1]] (* _Harvey P. Dale_, Sep 18 2018 *)
%o (PARI) default(realprecision, 20080); x=exp(1)/2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019739.txt", n, " ", d)); \\ _Harry J. Smith_, May 10 2009
%o (PARI) digits(10^default(realprecision)*exp(1)\20) \\ _M. F. Hasler_, Apr 01 2020
%o (Magma) SetDefaultRealField(RealField(100)); Exp(1)/2; // _Vincenzo Librandi_, Apr 05 2020
%Y Cf. A006083 (continued fraction). - _Harry J. Smith_, May 10 2009
%K nonn,cons
%O 1,2
%A _N. J. A. Sloane_
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