VMO 1988 — Division A Question 2

Let $P(x)=a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$, where $n\ge3$. Suppose that the polynomial has $n$ real roots and that $a_0=1$, $a_1=-n$, $a_2=(n^2-n)/2$. Determine the coefficients $a_i$ for $i=3,4,\ldots,n$.