I want to find a square root of a power series in the following way:

Let $$A(x)=a\_0+a\_1x+\cdots=\sum\_{j\ge 0}a\_jx^j\qquad \mathrm{and}\qquad B(x)=b\_0+b\_1x+\ldots=\sum\_{i\ge 0}b\_ix^i$$
such that $a\_0>0$ and $b\_0>0$ and $A=B^2$.

How can I express the coefficients $a\_j$ in terms of $b\_i$? I used the Cauchy product to get $a\_j=\sum\_{k=0}^jb\_kb\_{j-k}$, but have no idea how to continue.

Can someone help me out with finding a recursive relation between $a\_j$ and $b\_i$ with initial value for the sequence $(b\_i)$?