We just read off the coefficients of each \(x^n\) term. So \(a_0 = 3\) since the coefficient of \(x^0\) is 3 (\(x^0 = 1\) so this is the constant term). What is \(a_1\text{?}\) It is NOT 8, since 8 is the coefficient of \(x^2\text{,}\) so 8 is the term \(a_2\) of the sequence. To find \(a_1\) we need to look for the coefficient of \(x^1\) which in this case is 0. So \(a_1 = 0\text{.}\) Continuing, we have \(a_2 = 8\text{,}\) \(a_3 = 1\text{,}\) \(a_4 = 0\text{,}\) and \(a_5 = \frac{1}{7}\text{.}\) So we have the sequence
Note that when discussing generating functions, we always start our sequence with \(a_0\text{.}\)