Given two sequences $a_n$ and $b_n$ I want to compute a third sequence $c_n$. It is a mathematical operation. The first three terms look like this:
$ c_ 1 = {\frac {a_{{1}}b_{{0}}}{b_{{1}}a_{{0}}}} $
$ c_2 = {\frac {a_{{2}}b_{{0}}}{b_{{2}}a_{{0}}}}-\frac12 {\frac {{a_{{1}}}^{2}{b_{{0}}}^{2}}{{b_{{1}}}^{2}{a_{{0}}}^{2}}} $
$ c_3 = {\frac {a_{{3}}b_{{0}}}{b_{{3}}a_{{0}}}} - \frac13 {\frac {a_{{1}}{b_{{0}}}^{2}a_{{2}}}{b_{{1}}{a_{{0}}}^{2}b_{{2}}}} -\frac13 {\frac {{b_{{0}}}^{2} \left( 2a_{{2}}a_{{0}}{b_{{1}}}^{2} -{a_{{1}}}^{2}b_{{0}}b_{{2}} \right) a_{{1}}}{b_{{2}}{a_{{0}}}^{3}{b_{{1}}}^{3}}} $
These formulas are given by Maple in symbolic form after a longer computation. I assume that they can be generated by a simple algorithm. Yet I was unable to identify this algorithm from the formulas nor do I know what Maple really did. Perhaps these are well known formulas and someone can recognize them?