A122214 Numerators in infinite products for Pi/2, e and e^gamma (reduced).
1, 2, 4, 32, 4096, 67108864, 4503599627370496, 2535301200456458802993406410752, 4084620902943761579745625423246687265522976897405582347410338578593480704
Offset: 1
Examples
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) * ....
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 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.
Programs
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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 *) -
PARI
{a(n) = numerator(prod(k=1, n, k^((-1)^k*binomial(n-1,k-1))))} \\ Seiichi Manyama, Mar 10 2019
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