A092798 Numerator of partial products in an approximation of Pi/2.
2, 16, 8192, 274877906944, 5070602400912917605986812821504, 115792089237316195423570985008687907853269984665640564039457584007913129639936
Offset: 1
Examples
The first approximations are 2^(1/2), (16/3)^(1/4), (8192/243)^(1/8), (274877906944/215233605)^(1/16).
Links
- J. Guillera and J. Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008) 247-270; arXiv:math/0506319 [math.NT], 2005-2006.
- 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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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), ", "))
Formula
a(n) = Product_{k=1..n+1} A122214(k)^2^(n-k+1). - Jonathan Sondow, Sep 13 2006
a(n) = Numerator(Product_{k=1..n+1} (A122216(k)/A122217(k))^2^(n-k+1)). - Jonathan Sondow, Sep 13 2006