<s>[INST] Give a Sequence, Find Its Generating Function The Full Question
Find the generating function(closed form) of the following sequence:
$\binom{8}{1},2\binom{8}{2},3\binom{8}{3},\dots , 8\binom{8}{8}$
My Work
The open form of this generating function is:
$1\binom{8}{1}x^0+2\binom{8}{2}x^2 +3\binom{8}{3}x^3 + \cdots +8\binom{8}{8}x^8$
This can be re-written as:
$0\binom{8}{0}x^0+1\binom{8}{1}x^0+2\binom{8}{2}x^2 +3\binom{8}{3}x^3 + \cdots +8\binom{8}{8}x^8$
Which in Sigma Notation is:
$\sum_{k=0}^{8}k\binom{8}{k}x^k$
Part of this series is obviously binomial, which has a closed form of $(1+x)^8$ I don't know how to deal with the $k$ that is always being multiplied in. I thought I could maybe use the multiplication rule for sums, but it doesn't seem to apply here because we have $a_1b_1$ instead of $a_1b_n$
My Problem
How do we deal with the k that is being multiplied in there constantly? How can I get a nice closed form generating function with that $k$ getting in my way?
 [/INST] Hint: Say the generating function is $ f(x)$. What do you know about $ \int \frac{f(x) - 8}{x} \, dx $?
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