Exponential generating function $A(x) = \sum_{n=0}^{\infty}\frac{a_n}{n!}x^n$

Question

Let $a_n = 3^n + 7^n$ for $n \in \mathbb{N}_0$

I know that the generating function of $a_n= 3^n+7^n = A(x) = \frac{1}{1 - 3x} + \frac{1}{1 - 7x}.$

But how can one calculate the exponential generating function

$$A(x) = \sum_{n=0}^{\infty}\frac{a_n}{n!}x^n$$

of the sequence $(a_n)_{n \in \mathbb{N}}$

Answer 1

$A(x)= \sum_{n=0}^{\infty}\frac{3^n}{n!}x^n+ \sum_{n=0}^{\infty}\frac{7^n}{n!}x^n= \sum_{n=0}^{\infty}\frac{(3x)^n}{n!}+ \sum_{n=0}^{\infty}\frac{(7x)^n}{n!}=e^{3x}+e^{7x}.$