Coefficient of polynomials

Question

Could someone explain to me why $$ [x^{24}](1-2x^6)^{-31} = 2^4 \binom{4 + 31 - 1}{31 - 1} \, ? $$

Reads: The coefficient of $x^{24}$ in $(1-2x^6)^{-31} =$ ...

Answer 1

Apply the generalized binomial theorem (that allows for negative exponents) and notice that $\displaystyle (-1)^r\binom{-k}{r} = \binom{k+r-1}{k-1}$

Answer 2

The generating function for $\dbinom{n+k-1}k x^k$ is $\frac{1}{(1-x)^n}$. Then finding the $x^{24}$ coefficient of $\frac{1}{(1-2x^6)^{31}}$ is equivalent to finding the $x^4$ coefficient of $\frac{1}{(1-2x)^{31}}$. Then plug in the generating function and the answer follows.

The proof of the generating function can be found everywhere.