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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 $0,1,0,0,0,0....$?

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up vote 2 down vote accepted

Almost: your indexing is off by $1$.

From $xA=B+x$ you have $$x\sum_{n\ge 0}a_nx^n=x+\sum_{n\ge 0}b_nx^n\;,$$ so $$\sum_{n\ge 0}a_nx^{n+1}=b_0+(b_1+1)x+\sum_{n\ge 2}b_nx^n\;.$$ Equivalently, $$a_0x+\sum_{n\ge 2}a_{n-1}x^n=b_0+(b_1+1)x+\sum_{n\ge 2}b_nx^n\;,$$ and now it’s easy to equate coefficients: $b_0=0,a_0=b_1+1$, and $a_n=b_{n+1}$ for $n\ge 1$.

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