Series solution — the printed key does not match the recurrence

The differential equation: $y"+e^xy=0$ has a series solution $y(x)=\sum_{k=0}^\infty a_kx^k$ which satisfies $y(0)=1$ and $y'(0)=0$ (printed as "$x(0)=1$" and "$x'(0)=0$" — a variable-name slip for the function $y$, since $x$ is the series variable itself and cannot meaningfully have a value "at $0$"). Then the values of the coefficients $a_0,a_1,a_2$ are: