Generating functions to find a coefficient

Question

Use generating functions to find the coefficient of $x^{15}$ in $\frac{x^3-5x }{(1-x)^3}$.

Answer 1

Notice that $$\frac{1}{(1-x)^3} = \frac{1}{2}\frac{d^2}{dx^2}\left(\frac{1}{1-x}\right) = \frac{1}{2}\frac{d^2}{dx^2}\left(\sum_{n\geqslant 0}x^n\right)=\sum_{n\geqslant 0}\frac{(n+1)(n+2)}{2}x^n.$$ Write $[x^k]f(x)$ for the coefficient of $x^k$ in the generating function $f$. Then, $$[x^{15}]\left\{\frac{x^3-5x}{(1-x)^3}\right\}=[x^{12}]\left\{\frac{1}{(1-x)^3}\right\}-5[x^{14}]\left\{\frac{1}{(1-x)^3}\right\}=13\cdot7 - 5\cdot 15\cdot8=-509.$$