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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. 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.)