OFFSET
1,2
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.3, p. 14.
Jolley, Summation of Series, Dover (1961) eq. (161) on page 30.
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..20000
R. P. Millane., A product form of the Möbius transform, Whistler Center for Carbohydrate Research, Purdue University, West Lafayette, USA.
Roger H. Moritz, Summing series, PRIMUS, 1 (2) (2007) 212-219, Comment 2.
FORMULA
e/2 = lim_{n->oo} n*(e - (1+1/n)^n). - Benoit Cloitre, Sep 17 2002
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
10*(this constant) = 5*exp(1) = Sum_{j>=0} j^3/j! [Jolley]. - R. J. Mathar, Oct 03 2011
Equals Sum_{j>=0} (1+j)/(1+2*j)!. - Bruno Berselli, May 25 2015
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
Equals Sum_{k>=1} k*(k-1)/(2 * k!). - Amiram Eldar, Aug 10 2020
EXAMPLE
1.359140914229522617680143735676331248878623546849979787483483813862038... = A001113/2.
MATHEMATICA
N[Product[((1/n)!)^MoebiusMu[n], {n, 1, 200000}]] (* John M. Campbell, Jun 14 2011 *)
RealDigits[E/2, 10, 120][[1]] (* Harvey P. Dale, Sep 18 2018 *)
PROG
(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
(PARI) digits(10^default(realprecision)*exp(1)\20) \\ M. F. Hasler, Apr 01 2020
(Magma) SetDefaultRealField(RealField(100)); Exp(1)/2; // Vincenzo Librandi, Apr 05 2020
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
STATUS
approved