Generating function $3^n + 7^n$ for $n \in \mathbb{N}_0$

Question

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

How can one calculate the generating function of the sequence $(a_n)_{n\in \mathbb{N}}$?

Is that correct? If yes, how can one find that out?

In our Script $A(x) = \sum_{n=0}^{\infty} a_nx^n$ is a Power series and the generating function of the sequence $(a_n)_{n\in \mathbb{N}}$.

Equality, sum and product of two formal power series are defined as follows:

Can someone also tell me how one can calculate the exponential generating function:

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

For example, if $F(x) = \sum_{n=0}^{\infty} f_nx^n$ is the generating function of Fibonacci numbers, then $F(x) = \frac{x}{1-x-x^2}$ and

$$f_n = \frac{1}{\sqrt5} ((\frac{1+\sqrt5}{2})^n - (\frac{1-\sqrt5}{2})^n)$$

Answer 1

Wolfram Alpha gave an expression for $a_n$, but you want $\sum_{n\ge 0}a_n x^n$, which is a sum of two geometric series. For $|x|<\frac{1}{7}$, it converges to $\frac{1}{1-3x}+\frac{1}{1-7x}$.

Answer 2

Under your definitions, we would have $$A(x) = \sum_{k \geq 0} (3^k + 7^k) x^k = \sum_{k \geq 0} (3x)^k + \sum_{k \geq 0} (7x)^k = \frac{1}{1 - 3x} + \frac{1}{1 - 7x}.$$ What exactly this means depends on what you mean by "generating function." If you mean an actual power series, then this is an analytic function defined for $|x| < 1/7$. If you mean a formal power series, then it's just a nice way to write the sum.