So I am using exponential generating functions and have a question about taking the product of more than 2 exponential generating functions.
I know that the product of 2 exponential generating functions is
What I'm having a hard time with is that taking the product of more than 2 gets messy with the sum notation. So my question; is there a more efficient way to determine the coefficient of \$$a_n$ given that \$$a_n$ is a product of more than 2 exponential generating functions?
$$\left(\sum_{k=0}^\infty a_{k}^{(1)}\frac{x^k}{k!}\right)\cdots\left(\sum_{k=0}^\infty a_k^{(r)}\frac{x^k}{k!}\right)=\sum_{n=0}^\infty\left[\sum_{k_1+\cdots+k_r=n}\binom{n}{k_1,\cdots,k_r}a_{k_1}^{(1)}\cdots a_{k_r}^{(r)}\right]\frac{x^n}{n!}$$