A122214 Numerators in infinite products for Pi/2, e and e^gamma (reduced).

1, 2, 4, 32, 4096, 67108864, 4503599627370496, 2535301200456458802993406410752, 4084620902943761579745625423246687265522976897405582347410338578593480704

Offset

1,2

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, arXiv:math/0506319 [math.NT], 2005-2006; 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) = numerator(Product_{k=1..n} k^((-1)^k*binomial(n-1,k-1))).

For n >= 2, a(n) = numerator(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) = numerator(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}]
Numerator@Exp@Join[{0}, Integrate[(1-Exp[-(#*x)^-1])^#, {x, 0, Infinity}]&/@Range[2, 10]] (* Federico Provvedi, Jun 29 2023 *)

Prog

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

Crossrefs

Cf. A092798. Denominators are A122215. Unreduced numerators are A122216.

Sequence in context: A012509 A062740 A336832 * A122216 A100117 A073888

Adjacent sequences: A122211 A122212 A122213 * A122215 A122216 A122217

Keyword

frac,nonn

Author

Jonathan Sondow, Aug 26 2006

Status

approved