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

0,1

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.

References

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

Links

Table of n, a(n) for n=0..103.

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.

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

Example

0.5290529699404390240722939394755897280940381716959625...

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]

Crossrefs

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

Sequence in context: A357114 A078335 A021658 * A248749 A248751 A021193

Adjacent sequences: A270856 A270857 A270858 * A270860 A270861 A270862

Keyword

nonn,cons

Author

Iaroslav V. Blagouchine, Mar 24 2016

Status

approved