OFFSET
1,2
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 Jonathan 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.
Jonathan Sondow, A faster product for Pi and a new integral for ln Pi/2, arXiv:math/0401406 [math.NT], 2004.
Jonathan 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))).
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
For n >= 2, a(n) = numerator(exp((1/n)*Integral_{x=0..oo} (1-exp(-1/x))^n dx)). - Federico Provvedi, Jun 29 2023
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}]
Numerator@Exp@Join[{0}, Integrate[(1-Exp[-(#*x)^-1])^#, {x, 0, Infinity}]&/@Range[2, 10]] (* Federico Provvedi, Jun 29 2023 *)
PROG
(PARI) {a(n) = numerator(prod(k=1, n, k^((-1)^k*binomial(n-1, k-1))))} \\ Seiichi Manyama, Mar 10 2019
CROSSREFS
KEYWORD
frac,nonn
AUTHOR
Jonathan Sondow, Aug 26 2006
STATUS
approved