A345210 - OEIS

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A345210

Decimal expansion of Integral_{x=0..1} log(G(x+1)) dx, where G(x) is the Barnes G-function.

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

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OFFSET

-1,1

REFERENCES

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, 2012. See p. 53.

LINKS

Table of n, a(n) for n=-1..85.

Ernest William Barnes, The theory of the G-function, Quart. J. Math., Vol. 31 (1900), pp. 264-314. See p. 288.

Junesang Choi and Hari M. Srivastava, Sums associated with the zeta function, Journal of Mathematical Analysis and Applications, Vol. 206, No. 1 (1997), pp. 103-120. See eq. (2.55), p. 114.

Junesang Choi and H. M. Srivastava, Certain classes of series associated with the Zeta function and multiple Gamma functions, Journal of Computational and Applied Mathematics, Vol. 118, No. 1-2 (2000), pp. 87-109. See eq. (5.10), p. 97.

Junesang Choi, H. M. Srivastava and J. R. Quine, Some series involving the zeta function, Bulletin of the Australian Mathematical Society, Vol. 51, No. 3 (1995), pp. 383-393. See p. 386.

Eric Weisstein's World of Mathematics, Barnes G-Function.

Wikipedia, Barnes G-function.

FORMULA

Equals 1/12 + log(2*Pi)/4 - 2*log(A), where A is the Glaisher-Kinkelin constant (A074962) (Barnes, 1899).

Equals 1/12 + (1/4) * A061444 - 2 * A225746.

Equals 2*zeta'(-1) - zeta'(0)/2 - 1/12. - Vaclav Kotesovec, Jun 19 2021

EXAMPLE

0.04529364586810117912899221438391420106929264281515480574...

MATHEMATICA

RealDigits[1/12 + Log[2*Pi]/4 - 2*Log[Glaisher], 10, 100][[1]]

CROSSREFS

Cf. A061444, A074962, A225746.