# Study recursion with undefined coefficients

I have this recursion

$$b_n=-a_n-\sum_{k=1}^{n-1}a_kb_{n-k}$$

for undefined coefficients $a_k\in\Bbb C$. I want to see the structure of the coefficients $b_n$ in terms of the $a_k$ only. How I can do that in mathematica?

This code

RSolve[b[n] == -a[n] - Sum[b[n - k]*a[k], {k, 1, n - 1}], b[n], n]


wont help.

P.S.: the recursion, probably, is not solvable as a closed expression defined by the $a_k$, I dont know... but I want to see some examples to guess it structure, if possible.

b[n_] := b[n] = -a[n] - Sum[a[k] b[n - k], {k, n - 1}]