A241421 Decimal expansion of D(1), where D(x) is the infinite product function defined in the formula section (or in the Finch reference).

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Offset

1,1

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..5000

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

Eric Weisstein's MathWorld, Barnes G-Function

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) = A^6/(2^(1/6)*sqrt(Pi)).

Example

2.23588559550896986428396479931189064484515912285952474779344797826...

Mathematica

RealDigits[Glaisher^6/(2^(1/6)*Sqrt[Pi]), 10, 104] // First

Prog

(PARI) default(realprecision, 100); A=exp(1/12-zeta'(-1)); A^6/(2^(1/6)* sqrt(Pi)) \\ G. C. Greubel, Aug 24 2018

Crossrefs

Cf. A006752, A019610 (D(2)), A074962, A241420 (D(1/2)).

Sequence in context: A127678 A199962 A114990 * A157176 A276429 A111181

Adjacent sequences: A241418 A241419 A241420 * A241422 A241423 A241424

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Aug 08 2014

Status

approved