Question

Prove that ratio of coefficient of x^10 in (1-x^2)^10

Answer 1

More generally, the coefficient of xn is given by [xn](1+x2(1−x))8=[xn]

8

∑

k=0 \mathchoice((((

8

k

\mathchoice))))x2k(1−x)k=(−1)n ∑n/3≤k≤min For n=10 we get \sum_{4\leq k\leq 5}\binom{8}{k}\binom{k}{10-2k}=\binom{8}{4}\binom{4}{2}+\binom{8}{5}\binom{5}{0}=476.