If $A = \sum_{n\geq 0} a_nx^n$ $B = \sum_{n\geq 0} b_nx^n$ and $xA = B+x$.
If I now want to express $a_n$ using $b_n$ how does the $x$-term in the RHS of $xA = B+x$ come into play? since Is $a_i = b_{i+1}$ for all $i \neq 1$ and $a_1 = b_2 +1$ $x$ has the generating function $0,1,0,0,0,0....$?