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A019739
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Decimal expansion of e/2.
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9
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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, 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, 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
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OFFSET
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1,2
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REFERENCES
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Jolley, Summation of Series, Dover (1961) eq. (161) on page 30.
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LINKS
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Roger H. Moritz, Summing series, PRIMUS, 1 (2) (2007) 212-219, Comment 2.
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FORMULA
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e/2 = lim_{n->infinity} 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 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
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EXAMPLE
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1.359140914229522617680143735676331248878623546849979787483483813862038... = A001113/2.
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MATHEMATICA
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N[Product[((1/n)!)^MoebiusMu[n], {n, 1, 200000}]] (* John M. Campbell, Jun 14 2011 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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