Simplification of $a_0+\sum_{n=1}^{\infty} a_n\cdot \left[\sum_{k=0}^{\infty}b_k\cdot x^k\right]^n$ where I know the expressions for all the $a_n$

Question

I have this equation: $a_0+\sum_{n=1}^{\infty} a_n\cdot \left[\sum_{k=0}^{\infty}b_k\cdot x^k\right]^n$ where I know the expressions for all the $a_n$.

How can I simplify the multiplication of the two power series knowing that one series has an exponential $n$?

Thank you in advance for your answers.

Answer 1

Hint: Consider the multinomial coefficient as in

\begin{align*} \left(\sum_{k=0}^\infty b_kx^k\right)^n=\sum_{j=0}^\infty\left(\sum_{{k_1+\cdots+k_n=j}\atop{k_1,\ldots,k_n\geq 0}}\binom{j}{k_1,\ldots,k_n}\prod_{l=1}^n b_{k_l}\right)x^j \end{align*}