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%I #48 Aug 25 2023 10:00:14
%S 1,2,4,32,4096,67108864,4503599627370496,
%T 2535301200456458802993406410752,
%U 4084620902943761579745625423246687265522976897405582347410338578593480704
%N Numerators in infinite products for Pi/2, e and e^gamma (reduced).
%H Seiichi Manyama, <a href="/A122214/b122214.txt">Table of n, a(n) for n = 1..12</a>
%H Mohammad K. Azarian, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/21-24-2012/azarianIJCMS21-24-2012.pdf">Euler's Number Via Difference Equations</a>, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095-1102.
%H J. Baez, <a href="http://math.ucr.edu/home/baez/week230.html">This Week's Finds in Mathematical Physics</a>
%H J. Guillera and Jonathan Sondow, <a href="https://arxiv.org/abs/math/0506319">Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent</a>, arXiv:math/0506319 [math.NT], 2005-2006; Ramanujan J. 16 (2008) 247-270.
%H Jonathan Sondow, <a href="https://arxiv.org/abs/math/0401406">A faster product for Pi and a new integral for ln Pi/2</a>, arXiv:math/0401406 [math.NT], 2004.
%H Jonathan Sondow, <a href="https://www.jstor.org/stable/30037575">A faster product for Pi and a new integral for ln Pi/2</a>, Amer. Math. Monthly 112 (2005) 729-734.
%F a(n) = numerator(Product_{k=1..n} k^((-1)^k*binomial(n-1,k-1))).
%F For n >= 2, a(n) = numerator(exp(-2*Integral_{x=0..1} x^(2*n-1)/log(1-x^2) dx)) (see Mathematica code below). - _John M. Campbell_, Jul 18 2011
%F For n >= 2, a(n) = numerator(exp((1/n)*Integral_{x=0..oo} (1-exp(-1/x))^n dx)). - _Federico Provvedi_, Jun 29 2023
%e Pi/2 = (2/1)^(1/2) * (4/3)^(1/4) * (32/27)^(1/8) * (4096/3645)^(1/16) * ...,
%e e = (2/1)^(1/1) * (4/3)^(1/2) * (32/27)^(1/3) * (4096/3645)^(1/4) * ... and
%e e^gamma = (2/1)^(1/2) * (4/3)^(1/3) * (32/27)^(1/4) * (4096/3645)^(1/5) * ....
%t Table[Exp[-2*Integrate[x^(2n-1)/Log[1-x^2],{x,0,1}]],{n,2,8}]
%t Numerator@Exp@Join[{0},Integrate[(1-Exp[-(#*x)^-1])^#,{x,0,Infinity}]&/@Range[2,10]] (* _Federico Provvedi_, Jun 29 2023 *)
%o (PARI) {a(n) = numerator(prod(k=1, n, k^((-1)^k*binomial(n-1,k-1))))} \\ _Seiichi Manyama_, Mar 10 2019
%Y Cf. A092798. Denominators are A122215. Unreduced numerators are A122216.
%K frac,nonn
%O 1,2
%A _Jonathan Sondow_, Aug 26 2006
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