A241420 Decimal expansion of D(1/2), where D(x) is the infinite product function defined in the formula section (or in the Finch reference).

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, 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, 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

Offset

1,2

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 20.

Eric Weisstein's MathWorld, Barnes G-Function

Eric Weisstein's MathWorld, Catalan's Constant

Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant

Formula

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

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.

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

Example

1.54912659257756216836957253384940992693702986341004836289999671...

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A090124 A338346 A097943 * A077142 A156057 A125057

Adjacent sequences: A241417 A241418 A241419 * A241421 A241422 A241423

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Aug 08 2014

Status

approved