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A122214 Numerators in infinite products for Pi/2, e and e^gamma (reduced). 7
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

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..12

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

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, arXiv:math/0506319 [math.NT], 2005-2006; Ramanujan J. 16 (2008) 247-270.

J. Sondow, A faster product for Pi and a new integral for ln Pi/2, arXiv:math/0401406 [math.NT], 2004.

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

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}]

PROG

(PARI) {a(n) = numerator(prod(k=1, n, k^((-1)^k*binomial(n-1, k-1))))} \\ Seiichi Manyama, Mar 10 2019

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 October 16 13:51 EDT 2019. Contains 328093 sequences. (Running on oeis4.)