I'm trying to find the coefficient of $x^{20}$ in
$$(x^{2}+⋯+x^{6} )^{5}$$
My steps are
$$=x^{10}(1+⋯+x^{4} )^{5}$$
$$=\left(\dfrac {1-x^5} {1-x}\right)^{5} x^{10}$$
$$= (1-x^5)^5 * (1-x)^{-5} x^{10}$$
After that I am stuck, can someone show me how to use generating functions to finish this monster of a question? Please do not suggest using the multinomial theorem, I am learning generating functions right now so it would completely defeat the purpose of my learning if I used it :(. I'm thinking this is a convolution, but I don't know how it would look like multiplying two sequence together..Please, help is very much needed!
The answer is ${14\choose 10} – {5\choose 1} \times {9\choose 5} + {5\choose 2} $
Best Answer
what you are after if the coefficient of $x^{10}$ in the expansion $$\begin{align}[x^{10}](1-x)^{-5}(1-x^5)^5 &=[x^{10}] \left( 1 + {5 \choose 1}x + {6 \choose 2 }x^2 + \cdots \right)\left(1 - {5 \choose 1}x^5 + {5 \choose 2}x^{10} + \cdots\right)\\&={5 \choose 2} - {9 \choose 5} {5 \choose 1}+ {14 \choose 10}. \end{align} $$