A092798 Numerator of partial products in an approximation of Pi/2.

2, 16, 8192, 274877906944, 5070602400912917605986812821504, 115792089237316195423570985008687907853269984665640564039457584007913129639936

Offset

1,1

Links

Table of n, a(n) for n=1..6.

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.

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

Example

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

Prog

(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), ", "))

Crossrefs

Denominators are in A092799.

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

Sequence in context: A306729 A325049 A334912 * A333540 A258169 A325048

Adjacent sequences: A092795 A092796 A092797 * A092799 A092800 A092801

Keyword

nonn,easy,frac

Author

Ralf Stephan, Mar 05 2004

Status

approved