Finding coefficient of $x_n$ term in exponential generating functions

Question

I'm working on some problems that involve generating functions, and I've been able to understand when exponential vs. ordinary generating functions are used and how to write a generating function to solve a given problem. Where I get stuck is finding the coefficient of $x_n$ for exponential generating functions. Here is an example of what I have tried, but I don't know where to go from there. Any advice would be appreciated.

$$G(x) = (1 + \frac{x^2}{2!} + \frac{x^4}{4!} + ...) (1 + x + \frac{x^2}{2!} +\frac{x^3}{3!} + ...) (1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...) (1+x) $$ $$= \frac{e^x + e^{-x}}{2} e^x e^x (1+x)$$ $$= \frac{1}{2} (e^{3x} +e^x)(1+x) $$ $$= \frac{1}{2} (\sum_{n=0}^{\infty} 3^n \frac{x^n}{n!} + \sum_{n=0}^{\infty} \frac{x^n}{n!}) (1+x) = \frac{1}{2} (\sum_{n=0}^{\infty} (3^n + 1) \frac{x^n}{n!}) (1+x)$$

Answer 1

In this example, the coefficient of $x^n$ is $$\frac12\left(\frac{3^n+1}{n!}+\frac{3^{n-1}+1}{(n-1)!}\right).$$

Answer 2

You're very close. Let $a_{-1}=0$. Then:

$$(1+x)\sum_{n=0}^\infty a_nx^n=\sum_{n=0}a_nx^n+\sum_{n=0}^\infty a_{n-1} x^{n}=\sum_{n=0}^\infty (a_n+a_{n-1})x^n$$