A241420 - OEIS

%I #18 Jan 17 2020 05:29:31

%S 1,5,4,9,1,2,6,5,9,2,5,7,7,5,6,2,1,6,8,3,6,9,5,7,2,5,3,3,8,4,9,4,0,9,

%T 9,2,6,9,3,7,0,2,9,8,6,3,4,1,0,0,4,8,3,6,2,8,9,9,9,9,6,7,1,0,3,9,9,8,

%U 3,8,0,0,8,3,6,5,4,3,2,9,8,7,4,0,6,5,1,1,4,0,9,2,0,7,0,0,8,0,6,1,5,4,6,4

%N Decimal expansion of D(1/2), where D(x) is the infinite product function defined in the formula section (or in the Finch reference).

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin Constant, p. 136.

%H G. C. Greubel, <a href="/A241420/b241420.txt">Table of n, a(n) for n = 1..10000</a>

%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 20.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/CatalansConstant.html">Catalan's Constant</a>

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html">Glaisher-Kinkelin Constant</a>

%F D(x) = lim_{n->infinity} ( prod_{k=1..2n+1} (1+x/k)^((-1)^(k+1)*k) ).

%F D(x) = (e^(x/2-1/4)*A^3*G((x+1)/2)^2*Gamma(x/2)^(x-2)*Gamma((x+1)/2)^(1-x)*(Gamma((x+1)/2)/Gamma(x/2))^x)/(2^(1/12)*G(x/2)^2), where A is the Glaisher-Kinkelin constant and G is the Barnes G-function.

%F D(1/2) = (e^(C/Pi)*A^3*sqrt(Gamma(3/4)/Gamma(1/4)))/2^(1/12), where C is Catalan's constant.

%e 1.54912659257756216836957253384940992693702986341004836289999671...

%t (E^(Catalan/Pi)*Glaisher^3*Sqrt[Gamma[3/4]/Gamma[1/4]])/2^(1/12) // RealDigits[#, 10, 104]& // First

%o (PARI) default(realprecision, 100); A=exp(1/12-zeta'(-1)); exp(Catalan/Pi)*A^3*sqrt(gamma(3/4)/gamma(1/4))/2^(1/12) \\ _G. C. Greubel_, Aug 24 2018

%Y Cf. A006752, A019610 (D(2)), A074962, A241421 (D(1)).

%K nonn,cons,easy

%O 1,2

%A _Jean-François Alcover_, Aug 08 2014