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?