  We prove upper and lower bounds for all the coefficients in the Hilbert
Polynomial of a graded Gorenstein algebra $S=R/I$ with a quasi-pure resolution
over $R$. The bounds are in terms of the minimal and the maximal shifts in the
resolution of $R$ . These bounds are analogous to the bounds for the
multiplicity found in \cite{S} and are stronger than the bounds for the Cohen
Macaulay algebras found in \cite{HZ}.
