A122217 Denominators in infinite products for Pi/2, e and e^gamma (unreduced).

1, 1, 3, 27, 3645, 184528125, 3065257232666015625, 25071642180724968784488737583160400390625, 802200753381108669054307548505058630413812174354826201039259103708900511264801025390625

Offset

0,3

Links

Table of n, a(n) for n=0..8.

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, 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..floor(n/2)+1} (2k-1)^binomial(n,2k-2).

Example

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) *
...

Mathematica

Table[Product[(2k-1)^Binomial[n, 2k-2], {k, 1+Floor[n/2]}], {n, 0, 8}] (* T. D. Noe, Nov 16 2006 *)

Crossrefs

Cf. A092799. Numerators are A122216. Reduced denominators are A122215.

Sequence in context: A009039 A137092 A122215 * A316368 A068221 A068222

Adjacent sequences: A122214 A122215 A122216 * A122218 A122219 A122220

Keyword

frac,nonn

Author

Jonathan Sondow, Aug 26 2006

Extensions

Corrected by T. D. Noe, Nov 16 2006

Status

approved