We consider two power series, $A(x) = \sum_{n\geq 0} a_nx^n$ and $B(x) = \sum_{n \geq 0}b_nx^n$ where $B(x)^2 = A(x)$.
Given these conditions, how might one determine a recurrence for $b_n$ in terms of $a_k$? For instance I have proved that $2A(X)B(x)' = A(x)'B(x)$ for $'$ denoting the derivative of the power series, would this be useful to determine the recurrence?