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%I #30 Mar 18 2024 03:25:45
%S 5,2,9,0,5,2,9,6,9,9,4,0,4,3,9,0,2,4,0,7,2,2,9,3,9,3,9,4,7,5,5,8,9,7,
%T 2,8,0,9,4,0,3,8,1,7,1,6,9,5,9,6,2,5,6,9,0,8,6,1,7,1,8,2,8,0,9,7,2,7,
%U 7,7,2,2,9,6,8,5,1,1,3,4,8,0,0,6,5,2,0,7,2,8,9,1,1,3,2,5,5,9,9,6,4,0,9,2
%N Decimal expansion of Sum_{n >= 1} |G_n|/n^2, where G_n are Gregory's coefficients.
%C Gregory's coefficients (A002206 and A002207) are also known as (reciprocal) logarithmic numbers, Bernoulli numbers of the second kind and Cauchy numbers of the first kind. First few coefficients are G_1=+1/2, G_2=-1/12, G_3=+1/24, G_4=-19/720, etc.
%D Bernard Candelpergher, Ramanujan summation of divergent series, Berlin: Springer, 2017. See p. 105, eq. (3.23).
%H Iaroslav V. Blagouchine and Marc-Antoine Coppo, <a href="https://doi.org/10.1007/s11139-018-9991-0">A note on some constants related to the zeta-function and their relationship with the Gregory coefficients</a>, The Ramanujan Journal, Vol. 47 (2018), pp. 457-473. See p. 470, eq. (37); <a href="https://arxiv.org/abs/1703.08601">arXiv preprint</a>, arXiv:1703.08601 [math.NT], 2017.
%H Mümün Can, Ayhan Dil, Levent Kargin, Mehmet Cenkci and Mutlu Güloglu, <a href="https://arxiv.org/abs/2109.01515">Generalizations of the Euler-Mascheroni constant associated with the hyperharmonic numbers</a>, arXiv:2109.01515 [math.NT], 2021.
%F Equals Integral_{x=0..1} (-li(1-x) + gamma + log(x))/x dx, where li(x) is the logarithmic integral.
%F Equals A131688 + gamma_1 + gamma^2/2 - zeta(2)/2, where gamma_1 = A082633 and gamma = A001620 (Candelpergher, 2017; Blagouchine and Coppo, 2018). - _Amiram Eldar_, Mar 18 2024
%e 0.5290529699404390240722939394755897280940381716959625...
%p evalf(int((-Li(1-x)+gamma+ln(x))/x, x = 0..1), 150)
%t N[Integrate[(-LogIntegral[1 - x] + EulerGamma + Log[x])/x, {x, 0, 1}], 150]
%Y Cf. A001620, A002206, A002207, A013661, A082633, A131688, A195189, A269330, A270857.
%K nonn,cons
%O 0,1
%A _Iaroslav V. Blagouchine_, Mar 24 2016
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