Let $a\_0,a\_1,\dots$ be the sequence satisfying
$$
\left(\sum\_{n=0}^\infty a\_n x^n\right)\left(\sum\_{n=0}^\infty \frac{x^n}{n+1}\right)=1.
$$
This means that $a\_0=1$ and $a\_{n+1}=-\sum\_{j=0}^n\frac{a\_j}{n+2-j}$.
One gets $a\_1=-\frac12$ and $a\_3=-\frac{13}{720}$.


The first and most important question is:
Is it true that all coefficients $a\_n$ for $n\ge 1$ are strictly negative?


Secondly, is there any connection to, say, Bernoulli numbers or any other sequence that carries a name?



Sign of coefficients
These numbers (with alternating signs) are called Bernoulli numbers of the second kind or Cauchy numbers. Two proofs that these numbers are negative can be found in <http://people.brandeis.edu/~gessel/homepage/slides/analysis-nec.pdf>. 


Another reference is 


[Merlini, Donatella; Sprugnoli, Renzo; Verri, M. Cecilia,
The Cauchy numbers. 
Discrete Math. 306 (2006), no. 16, 1906–1920.](http://www.dsi.unifi.it/~merlini/papers/Cauchy.pdf)


The determination of the sign of these numbers can be found much earlier in Charles Jordan's 
*Calculus of Finite Differences*, page 267.


