Power of Series

Question

In taking the power of a series

$$\left(\sum_{k=0}^{\infty} a_k x^k \right)^n = \sum_{k=0}^{\infty} c_k x^k$$

do you know an expression for $c_k$ solely in terms of the coefficients $a_k$?

Answer 1

$c_k$ is the sum of $a_{k_1} a_{k_2} \ldots a_{k_n}$ over all ordered $n$-tuples of nonnegative integers $(k_1, \ldots, k_n)$ whose sum is $k$.

Answer 2

This is repeated application of Cauchy Product : https://en.wikipedia.org/wiki/Cauchy_product

Answer 3

The form you seem to be seeking is $$c_k=\sum_{\substack i_0+i_1+i_2+\cdots+i_k=n\\i_1+2i_2+\cdots+ki_k=k}\binom{n}{i_0,\cdots,i_k}\prod_{j=0}^k a_j^{i_j}$$