Graded algebras: what is the name of the coeffients

Question

Let $R=\oplus_i R_i,$ $R_0=\mathbb{C}$ be graded finitelly generaded algebra with the Krull dimension $r$ and the Hilbert series $H(A,t).$ Then the Laurent series expansion of $H(A,t)$ at $t=1$ has the form $$ H(A,t)=\frac{a_0}{(t-1)^r}+\frac{a_1}{(t-1)^{r-1}}+\cdots $$

Questions:

1. What are the correct name of the numbers $a_0,a_1,... $ ?
2. Why the numbers are so important and useful?
3. The same questions in the case if $A$ is Cohen-Macaulay and Gorestein algebra.