Where should $n$ in generating function start?

Question

For a sequence of numbers, the generating function has the form $$ g(x)=\sum_{n\space\geq\space0}a_nx^n $$ If $n$ starts from $2$ instead, should $\displaystyle g(x)=\sum_{n\space\geq\space2}a_{n+2}x^{n+2} $, (or $\displaystyle g(x)=\sum_{n\space\geq\space0}a_{n+1}x^{n+1}-1-a_1x$)? I'm not super clear if where n starts is conventionally chosen as $0$.