%I #19 Jun 03 2019 08:03:29

%S 2,16,8192,274877906944,5070602400912917605986812821504,

%T 115792089237316195423570985008687907853269984665640564039457584007913129639936

%N Numerator of partial products in an approximation of Pi/2.

%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>, Ramanujan J. 16 (2008) 247-270; arXiv:math/0506319 [math.NT], 2005-2006.

%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..n+1} A122214(k)^2^(n-k+1). - _Jonathan Sondow_, Sep 13 2006

%F a(n) = Numerator(Product_{k=1..n+1} (A122216(k)/A122217(k))^2^(n-k+1)). - _Jonathan Sondow_, Sep 13 2006

%e The first approximations are 2^(1/2), (16/3)^(1/4), (8192/243)^(1/8), (274877906944/215233605)^(1/16).

%o (PARI) for(m=1, 7, p=1; for(n=1, m, p=p*p*(prod(k=1, ceil(n/2), (2*k)^binomial(n, 2*k-1))/(prod(k=1, floor(n/2)+1, (2*k-1)^binomial(n, 2*k-2))))); print1(numerator(p), ", "))

%Y Denominators are in A092799.

%Y Cf. A000246, A001900, A001901, A001902, A122214, A122216.

%K nonn,easy,frac

%O 1,1

%A _Ralf Stephan_, Mar 05 2004