A122216 Numerators in infinite products for Pi/2, e and e^gamma (unreduced).

1, 2, 4, 32, 4096, 201326592, 3283124128353091584, 26520146032764463901929624736590416838656, 840987221884558487834659180201583257033385988411167452990072842049923846092011283152896

Offset

0,2

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

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

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

a[n_] := Product[(2k)^Binomial[n, 2k-1], {k, 1, n/2 // Ceiling}];
Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Nov 18 2018 *)

Crossrefs

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

Sequence in context: A062740 A336832 A122214 * A100117 A073888 A114642

Adjacent sequences: A122213 A122214 A122215 * A122217 A122218 A122219

Keyword

frac,nonn

Author

Jonathan Sondow, Aug 26 2006

Extensions

Offset and truncated term 840987221884... corrected by Jean-François Alcover, Nov 18 2018

Status

approved