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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). 3
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 (list; constant; graph; refs; listen; history; text; internal format)
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: A215337 A090124 A097943 * A077142 A156057 A125057

Adjacent sequences:  A241417 A241418 A241419 * A241421 A241422 A241423

KEYWORD

nonn,cons,easy

AUTHOR

Jean-Fran├žois Alcover, Aug 08 2014

STATUS

approved

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Last modified February 20 22:47 EST 2020. Contains 332086 sequences. (Running on oeis4.)