A122216 Numerators in infinite products for Pi/2, e and e^gamma (unreduced).
1, 2, 4, 32, 4096, 201326592, 3283124128353091584, 26520146032764463901929624736590416838656, 840987221884558487834659180201583257033385988411167452990072842049923846092011283152896
Offset: 0
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
- 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 and 113 (2006), 670.
Programs
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Mathematica
a[n_] := Product[(2k)^Binomial[n, 2k-1], {k, 1, n/2 // Ceiling}]; Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Nov 18 2018 *)
Formula
a(n) = Product_{k=1..ceiling(n/2)} (2k)^binomial(n,2k-1).
Extensions
Offset and truncated term 840987221884... corrected by Jean-François Alcover, Nov 18 2018