A122216 - OEIS

%I #28 Jun 03 2019 08:01:07

%S 1,2,4,32,4096,201326592,3283124128353091584,

%T 26520146032764463901929624736590416838656,

%U 840987221884558487834659180201583257033385988411167452990072842049923846092011283152896

%N Numerators in infinite products for Pi/2, e and e^gamma (unreduced).

%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 J. 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 J. 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 J. Sondow, <a href="http://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 and 113 (2006), 670.

%F a(n) = Product_{k=1..ceiling(n/2)} (2k)^binomial(n,2k-1).

%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 a[n_] := Product[(2k)^Binomial[n, 2k-1], {k, 1, n/2 // Ceiling}];

%t Table[a[n], {n, 0, 8}] (* _Jean-François Alcover_, Nov 18 2018 *)

%Y Cf. A092798. Denominators are A122217. Reduced numerators are A122214.

%K frac,nonn

%O 0,2

%A _Jonathan Sondow_, Aug 26 2006

%E Offset and truncated term 840987221884... corrected by _Jean-François Alcover_, Nov 18 2018