Let $S$ be a graded $k$-algebra generated by homogeneous elements $f_1,\dots,f_r$ with degrees $d_1,\dots,d_r$. The following is Proposition 1.9 in "Mukai, An Introduction to Invariants and Moduli" (with minor changes!):
The Hilbert series of S is equal to $$ P(t) = \frac{F(t)}{(1-t^{d_1})\dots(1-t^{d_r})},$$ for some $F(t)\in \mathbb{Z}[t].$
I think there is a flaw in Mukai's proof ($f_i$'s are not necessary nonzero-divisor in $S/k[f_1,\dots,f_{i-1}]$ and then the map $h\mapsto f_i h$ is not injective in this ring and induction does not work...).
There is also a proof in the wikipedia page for the case $d_1=d_2=\dots =d_r= 1$, but I want a proof for the general case.
Thanks!