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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:

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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

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

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    $\begingroup$ I thought the g.f. would be $\sum_{n=0}^\infty(3^n+7^n)x^n$.... $\endgroup$ – Lord Shark the Unknown Feb 7 at 22:51
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    $\begingroup$ It may be worth linking to your definition of generating functions, as there's more than one kind. $\endgroup$ – J.G. Feb 7 at 22:54
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    $\begingroup$ The series returned by Wolfram Alpha is just writing $3^n = e^{n\log 3}$ and using the Maclaurin series $e^x = \sum_{v \geq 0} \frac{x^v}{v!}$, so that's probably not what you want. $\endgroup$ – Riley Feb 7 at 22:58
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    $\begingroup$ I think the main confusion in this post is the difference between a generating function and a power series expansion of the $n$th term. $\endgroup$ – jmacmanus Feb 7 at 23:04
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    $\begingroup$ Because $\sum\limits_{n=0} x^n = \frac{1}{1-x}$ then $$f(x)=\sum\limits_{n=0}a_nx^n=\sum\limits_{n=0}\left(3^n+7^n\right)x^n= \sum\limits_{n=0}(3x)^n+\sum\limits_{n=0}(7x)^n=\frac{1}{1-3x}+\frac{1}{1-7x}$$ $\endgroup$ – rtybase Feb 7 at 23:07
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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 $\frac{1}{1-3x}+\frac{1}{1-7x}$.

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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.

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