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A122215
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Denominators in infinite products for Pi/2, e and e^gamma (reduced).
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5
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1, 1, 3, 27, 3645, 61509375, 4204742431640625, 2396825584582984447479248046875, 3896237517467890187050354408614984136338676989907980896532535552978515625
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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) = denominator(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. A092799. Numerators are A122214. Unreduced denominators are A122217.
Sequence in context: A009039 A137092 A170921 * A122217 A068221 A068222
Adjacent sequences: A122212 A122213 A122214 * A122216 A122217 A122218
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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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