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$$G(x) = \sum_{n=1}^\infty na_n x^n $$

Hello. I need to find a generating function for the summation above, where $a_n$ is the Fibonacci sequence.

I have found the generating function for the fib itself but am confused as to how I can progress further. I also tried plugging in the closed form of the fib sequence and tried to separate out variables to get a sum of sigmas, but it didn't help. Can someone guide me in the right direction?

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    $\begingroup$ hint: differentiate the generating function for the fibonacci numbers, then correct for the power of $x$. $\endgroup$ – lulu Feb 26 '16 at 16:46
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    $\begingroup$ Hint: $f(z) = \sum_{k=0}^\infty a_k z^k\quad\implies\quad z\frac{df(z)}{dz} = \sum_{k=1}^\infty ka_k z^k$. $\endgroup$ – achille hui Feb 26 '16 at 16:48
  • $\begingroup$ A very good solution is given here. $\endgroup$ – Hypergeometricx Feb 26 '16 at 17:04
  • $\begingroup$ Please transcribe the image into proper MathJax. Many here (myself included) won't click on random links. $\endgroup$ – vonbrand Feb 26 '16 at 18:03
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Hint:

Note that $$ G(x)=\sum_{n=1}^{\infty}na_nx^n=x\sum_{n=1}^{\infty}na_nx^{n-1}=x\frac{d}{dx}\left[\sum_{n=0}^{\infty}a_nx^n\right] $$

So, if you've already got the generating function for the Fibonacci sequence $(a_n)_{n=0}^{\infty}$, you can easily modify it to get the function that you're looking for.

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