There is a unique sequence of integers $a_1,a_2,\dots ,a_{2027}$ such that

$\tan 2027x=\frac{a_1\tan x+a_3{\tan }^{3}x+a_5{\tan }^{5}x+\cdots +a_{2027}{\tan }^{2027}x}{1+a_2{\tan }^{2}x+a_4{\tan }^{4}x+\cdots +a_{2026}{\tan }^{2026}x}$

whenever $\tan 2027x$ is defined. What is $a_{2027}$?