A122215 Denominators in infinite products for Pi/2, e and e^gamma (reduced).

1, 1, 3, 27, 3645, 61509375, 4204742431640625, 2396825584582984447479248046875, 3896237517467890187050354408614984136338676989907980896532535552978515625

Offset

1,3

Links

Seiichi Manyama, Table of n, a(n) for n = 1..12

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 Jonathan Sondow, Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent, Ramanujan J. 16 (2008) 247-270.

Jonathan Sondow, A faster product for Pi and a new integral for ln Pi/2, arXiv:math/0401406 [math.NT], 2004.

Jonathan Sondow, A faster product for Pi and a new integral for ln Pi/2, Amer. Math. Monthly 112 (2005) 729-734.

Formula

a(n) = denominator(Product_{k=1..n} k^((-1)^k*binomial(n-1,k-1))).

For n>=2, a(n) = denominator(exp(-2 * Integral_{x=0..1} x^(2*n-1)/log(1-x^2) dx)) (see Mathematica code below). - John M. Campbell, Jul 18 2011

For n>=2, a(n) = denominator(exp((1/n)*Integral_{x=0..oo} (1-exp(-1/x))^n dx)). - Federico Provvedi, Jun 29 2023

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[Exp[-2*Integrate[x^(2n-1)/Log[1-x^2], {x, 0, 1}]], {n, 2, 8}]
Denominator@Table[Product[k^((-1)^k Binomial[n-1, k-1]), {k, 1, n}], {n, 1, 10}] (* Vladimir Reshetnikov, May 29 2016 *)

Prog

(PARI) {a(n) = denominator(prod(k=1, n, k^((-1)^k*binomial(n-1, k-1))))} \\ Seiichi Manyama, Mar 10 2019

Crossrefs

Cf. A092799. Numerators are A122214. Unreduced denominators are A122217.

Sequence in context: A078233 A009039 A137092 * A122217 A316368 A068221

Adjacent sequences: A122212 A122213 A122214 * A122216 A122217 A122218

Keyword

frac,nonn

Author

Jonathan Sondow, Aug 26 2006

Status

approved