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A122214
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Numerators in infinite products for Pi/2, e and e^gamma (reduced).
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6
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1, 2, 4, 32, 4096, 67108864, 4503599627370496, 2535301200456458802993406410752, 4084620902943761579745625423246687265522976897405582347410338578593480704
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| For n>=2 the n-th term of this sequence of rational numbers equals exp(-2*integral(x=0..1, x^(2*n-1)/log(1-x^2) ) ) (see Mathematica code below). [From John M. Campbell, Jul 18 2011]
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REFERENCES
| J. Sondow, A faster product for Pi and a new integral for ln Pi/2, Amer. Math. Monthly 112 (2005) 729-734.
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LINKS
| J. Baez, This Week's Finds in Mathematical Physics
J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008) 247-270.
J. Sondow, A faster product for Pi and a new integral for ln Pi/2
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FORMULA
| a(n) = numerator(product(k=1..n, k^((-1)^k*binomial(n-1,k-1)))).
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EXAMPLE
| Pi/2 = (2/1)^(1/2) * (4/3)^(1/4) * (32/27)^(1/8) * (4096/3645)^(1/16) * ...,
e = (2/1)^(1/1) * (4/3)^(1/2) * (32/27)^(1/3) * (4096/3645)^(1/4) * ... and
e^gamma = (2/1)^(1/2) * (4/3)^(1/3) * (32/27)^(1/4) * (4096/3645)^(1/5) * ....
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MATHEMATICA
| Table[Exp[-2*Integrate[x^(2n-1)/Log[1-x^2], {x, 0, 1}]], {n, 2, 8}]
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CROSSREFS
| Cf. A092798. Denominators are A122215. Unreduced numerators are A122216.
Sequence in context: A118992 A012509 A062740 * A122216 A100117 A073888
Adjacent sequences: A122211 A122212 A122213 * A122215 A122216 A122217
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KEYWORD
| frac,nonn
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AUTHOR
| Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 26 2006
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