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A122214 Numerators in infinite products for Pi/2, e and e^gamma (reduced). 6
1, 2, 4, 32, 4096, 67108864, 4503599627370496, 2535301200456458802993406410752, 4084620902943761579745625423246687265522976897405582347410338578593480704 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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). - John M. Campbell, Jul 18 2011

REFERENCES

Mohammad K. Azarian, Euler's Number Via Difference Equations, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095 - 1102.

J. Sondow, A faster product for Pi and a new integral for ln Pi/2, Amer. Math. Monthly 112 (2005) 729-734.

LINKS

Table of n, a(n) for n=1..9.

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

FORMULA

a(n) = numerator(product(k=1..n, k^((-1)^k*binomial(n-1,k-1)))).

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) * ....

MATHEMATICA

Table[Exp[-2*Integrate[x^(2n-1)/Log[1-x^2], {x, 0, 1}]], {n, 2, 8}]

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

KEYWORD

frac,nonn

AUTHOR

Jonathan Sondow, Aug 26 2006

STATUS

approved

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Last modified February 23 17:23 EST 2018. Contains 299584 sequences. (Running on oeis4.)