A270857 Decimal expansion of Sum_{n >= 1} G_n/n^2, where G_n are Gregory's coefficients.

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

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

			0.4826442216204626123794283911485757739701203962756656...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A270859, A269330, A001620, A002206, A002207, A195189.

Programs

Maple

evalf(int((Li(1+x)-gamma-ln(x))/x, x = 0..1), 120);

Mathematica

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

Formula

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

Extensions

Mathematica program corrected by Harvey P. Dale, Jul 05 2022