Answer: 816

Small Hint:

Substitute $x=y-1$ before expanding

Big Hint:

Find only the coefficient of $y^2$ in each power and use the hockey-stick identity

Solution:

The polynomial is $\sum_{k=0}^{17}(-x)^k.$ Since $x=y-1,$ this becomes $$\sum_{k=0}^{17}(1-y)^k.$$ For $k\geq2,$ the coefficient of $y^2$ in $(1-y)^k$ is $\binom{k}{2}.$ Hence $$a_2=\sum_{k=2}^{17}\binom{k}{2} =\binom{18}{3}=816.$$