Suppose that $$P[x]=\sum_{n=0}^{\infty}a_nx^n$$ is a formal power series. Therefore, the issues of convergence are supposed to be ignored for now.
I'm looking for a direct formula that allows me to establish an identity of the form$$\sum_{n=0}^{\infty}b_nx^n=\left(\sum_{n=0}^{\infty}a_nx^n\right)^2$$
More precisely, I want $b_n$ to be the coefficient of $x^n$ in $P[x]^2$. No recurrence relations for $b_n$ are allowed. Express $b_n$ as a function $a_i$'s.
By definition, the product of two formal power series $\sum_{n=0}^\infty c_nx^n$ and $\sum_{n=0}^\infty d_n x^n$ is the formal power series $\sum_{n=0}^\infty b_nx^n$, where $b_n=\sum_{i=0}^nc_id_{n-i}$. To square your $P[x]$, you just want to multiply two copies of $P[x]$, so you want to take $c_n=d_n=a_n$ for all $n$. This gives $$b_n=\sum_{i=0}^na_ia_{n-i}.$$