A270859 Decimal expansion of Sum_{n >= 1} |G_n|/n^2, where G_n are Gregory's coefficients.

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, 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, 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

Offset: 0

Author

Iaroslav V. Blagouchine, Mar 24 2016

Keywords

• nonn
• cons

Comments

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.

Examples

			0.5290529699404390240722939394755897280940381716959625...
		

References

• Bernard Candelpergher, Ramanujan summation of divergent series, Berlin: Springer, 2017. See p. 105, eq. (3.23).

Links

• Iaroslav V. Blagouchine and Marc-Antoine Coppo, A note on some constants related to the zeta-function and their relationship with the Gregory coefficients, The Ramanujan Journal, Vol. 47 (2018), pp. 457-473. See p. 470, eq. (37); arXiv preprint, arXiv:1703.08601 [math.NT], 2017.
• Mümün Can, Ayhan Dil, Levent Kargin, Mehmet Cenkci and Mutlu Güloglu, Generalizations of the Euler-Mascheroni constant associated with the hyperharmonic numbers, arXiv:2109.01515 [math.NT], 2021.

Crossrefs

Cf. A001620, A002206, A002207, A013661, A082633, A131688, A195189, A269330, A270857.

Programs

Maple

evalf(int((-Li(1-x)+gamma+ln(x))/x, x = 0..1), 150)

Mathematica

N[Integrate[(-LogIntegral[1 - x] + EulerGamma + Log[x])/x, {x, 0, 1}], 150]

Formula

Equals Integral_{x=0..1} (-li(1-x) + gamma + log(x))/x dx, where li(x) is the logarithmic integral.

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