

A270859


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


3



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
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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



FORMULA

Equals Integral_{x=0..1} (li(1x) + 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(1x)+gamma+ln(x))/x, x = 0..1), 150)


MATHEMATICA

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


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



