Av(12345)
Counting Sequence
1, 1, 2, 6, 24, 119, 694, 4582, 33324, 261808, 2190688, 19318688, 178108704, 1705985883, 16891621166, ...

This specification was found using the strategy pack "Row Placements Tracked Fusion Expand Verified" and has 30 rules.

Found on January 22, 2022.

Finding the specification took 27 seconds.

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\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{23}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x , 1\right)\\ F_{4}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x , y_{0}\right)+F_{5}\! \left(x , y_{0}\right)\\ F_{5}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right) F_{6}\! \left(x , y_{0}\right)\\ F_{6}\! \left(x , y_{0}\right) &= F_{7}\! \left(x , 1, y_{0}\right)\\ F_{7}\! \left(x , y_{0}, y_{1}\right) &= F_{8}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{8}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x , y_{0}, y_{1}\right)+F_{26}\! \left(x , y_{0}, y_{1}\right)+F_{9}\! \left(x , y_{0}, y_{1}\right)\\ F_{9}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , y_{0}, y_{1}\right) F_{14}\! \left(x , y_{0}\right)\\ F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{0}, 1, y_{1}\right)\\ F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\ F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{14}\! \left(x , y_{0}\right)\\ F_{14}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{1}\right) F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{-y_{1} F_{17}\! \left(x , 1, y_{1}, y_{2}\right)+y_{0} F_{17}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\ F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0} y_{1}, y_{1}, y_{2}\right)\\ F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{2}\right) F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{-y_{2} F_{20}\! \left(x , y_{0}, 1, y_{2}\right)+y_{1} F_{20}\! \left(x , y_{0}, \frac{y_{1}}{y_{2}}, y_{2}\right)}{-y_{2}+y_{1}}\\ F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\ F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{23}\! \left(x \right)\\ F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{2} F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right)-F_{12}\! \left(x , y_{0}, y_{1}, 1\right)}{-1+y_{2}}\\ F_{23}\! \left(x \right) &= x\\ F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{1}\right) F_{25}\! \left(x , y_{0}, y_{1}\right)\\ F_{25}\! \left(x , y_{0}, y_{1}\right) &= \frac{-y_{1} F_{7}\! \left(x , 1, y_{1}\right)+y_{0} F_{7}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right)}{-y_{1}+y_{0}}\\ F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{23}\! \left(x \right) F_{27}\! \left(x , y_{0}, y_{1}\right)\\ F_{27}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{1} F_{8}\! \left(x , y_{0}, y_{1}\right)-F_{8}\! \left(x , y_{0}, 1\right)}{-1+y_{1}}\\ F_{28}\! \left(x , y_{0}\right) &= F_{23}\! \left(x \right) F_{29}\! \left(x , y_{0}\right)\\ F_{29}\! \left(x , y_{0}\right) &= \frac{y_{0} F_{4}\! \left(x , y_{0}\right)-F_{4}\! \left(x , 1\right)}{-1+y_{0}}\\ \end{align*}

This specification was found using the strategy pack "Row Placements Tracked Fusion Req Corrob Expand Verified" and has 38 rules.

Found on November 19, 2021.

Finding the specification took 174 seconds.

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\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{27}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{27}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y_{0}\right) &= F_{7}\! \left(x , 1, y_{0}\right)\\ F_{7}\! \left(x , y_{0}, y_{1}\right) &= F_{8}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{8}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x , y_{0}, y_{1}\right)+F_{30}\! \left(x , y_{0}, y_{1}\right)+F_{9}\! \left(x , y_{0}, y_{1}\right)\\ F_{9}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , y_{0}, y_{1}\right) F_{14}\! \left(x , y_{0}\right)\\ F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0} y_{1}, y_{1}, y_{2}\right)\\ F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{14}\! \left(x , y_{0}\right)\\ F_{14}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{1}\right) F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{y_{1} F_{11}\! \left(x , 1, y_{1}, y_{2}\right)-y_{0} F_{11}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\ F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{2}\right) F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{19}\! \left(x , y_{0}, y_{1}, 1, y_{2}\right)\\ F_{19}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{20}\! \left(x , y_{0}, 1, y_{1}, y_{2}, y_{3}\right)\\ F_{20}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}, y_{4}\right) &= -\frac{-y_{1} y_{2} F_{21}\! \left(x , y_{0}, y_{1}, \frac{y_{2}}{y_{4}}, y_{3} y_{4}\right)+y_{4} F_{21}\! \left(x , y_{0}, y_{1}, \frac{1}{y_{1}}, y_{3} y_{4}\right)}{y_{1} y_{2}-y_{4}}\\ F_{21}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{22}\! \left(x , y_{0}, y_{1}, 1, y_{2}, y_{3}\right)\\ F_{22}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}, y_{4}\right) &= F_{23}\! \left(x , y_{0} y_{1}, y_{2}, y_{1} y_{3}, y_{4}\right)\\ F_{23}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{24}\! \left(x , y_{0} y_{1}, y_{2}, y_{1} y_{3}\right)\\ F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{12}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\ F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{27}\! \left(x \right)\\ F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{2} F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{12}\! \left(x , y_{0}, y_{1}, 1\right)}{-1+y_{2}}\\ F_{27}\! \left(x \right) &= x\\ F_{28}\! \left(x , y_{0}, y_{1}\right) &= F_{14}\! \left(x , y_{1}\right) F_{29}\! \left(x , y_{0}, y_{1}\right)\\ F_{29}\! \left(x , y_{0}, y_{1}\right) &= -\frac{y_{1} F_{7}\! \left(x , 1, y_{1}\right)-y_{0} F_{7}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right)}{-y_{1}+y_{0}}\\ F_{30}\! \left(x , y_{0}, y_{1}\right) &= F_{27}\! \left(x \right) F_{31}\! \left(x , y_{0}, y_{1}\right)\\ F_{31}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{1} F_{8}\! \left(x , y_{0}, y_{1}\right)+F_{8}\! \left(x , y_{0}, 1\right)}{-1+y_{1}}\\ F_{32}\! \left(x \right) &= F_{27}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x , 1\right)\\ F_{34}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{35}\! \left(x , y_{0}\right)+F_{35}\! \left(x , 1\right)}{-1+y_{0}}\\ F_{35}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x , y_{0}\right)+F_{37}\! \left(x , y_{0}\right)\\ F_{36}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right) F_{6}\! \left(x , y_{0}\right)\\ F_{37}\! \left(x , y_{0}\right) &= F_{27}\! \left(x \right) F_{34}\! \left(x , y_{0}\right)\\ \end{align*}

This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Expand Verified" and has 34 rules.

Found on November 19, 2021.

Finding the specification took 92 seconds.

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\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{23}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\ F_{7}\! \left(x , y_{0}\right) &= F_{8}\! \left(x , 1, y_{0}\right)\\ F_{8}\! \left(x , y_{0}, y_{1}\right) &= F_{9}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{9}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y_{0}, y_{1}\right)+F_{24}\! \left(x , y_{0}, y_{1}\right)+F_{26}\! \left(x , y_{0}, y_{1}\right)\\ F_{10}\! \left(x , y_{0}, y_{1}\right) &= F_{11}\! \left(x , y_{0}, y_{1}\right) F_{15}\! \left(x , y_{0}\right)\\ F_{11}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{13}\! \left(x , y_{0} y_{1}, y_{1}, y_{2}\right)\\ F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{15}\! \left(x , y_{0}\right)\\ F_{15}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{15}\! \left(x , y_{1}\right) F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{y_{1} F_{12}\! \left(x , 1, y_{1}, y_{2}\right)-y_{0} F_{12}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\ F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{15}\! \left(x , y_{2}\right) F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{y_{2} F_{20}\! \left(x , y_{0}, 1, y_{2}\right)-y_{1} F_{20}\! \left(x , y_{0}, \frac{y_{1}}{y_{2}}, y_{2}\right)}{-y_{2}+y_{1}}\\ F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{13}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\ F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{23}\! \left(x \right)\\ F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{2} F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{13}\! \left(x , y_{0}, y_{1}, 1\right)}{-1+y_{2}}\\ F_{23}\! \left(x \right) &= x\\ F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , y_{1}\right) F_{25}\! \left(x , y_{0}, y_{1}\right)\\ F_{25}\! \left(x , y_{0}, y_{1}\right) &= -\frac{y_{1} F_{8}\! \left(x , 1, y_{1}\right)-y_{0} F_{8}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right)}{-y_{1}+y_{0}}\\ F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{23}\! \left(x \right) F_{27}\! \left(x , y_{0}, y_{1}\right)\\ F_{27}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{1} F_{9}\! \left(x , y_{0}, y_{1}\right)+F_{9}\! \left(x , y_{0}, 1\right)}{-1+y_{1}}\\ F_{28}\! \left(x \right) &= F_{23}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x , 1\right)\\ F_{30}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{31}\! \left(x , y_{0}\right)+F_{31}\! \left(x , 1\right)}{-1+y_{0}}\\ F_{31}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x , y_{0}\right)+F_{33}\! \left(x , y_{0}\right)\\ F_{32}\! \left(x , y_{0}\right) &= F_{15}\! \left(x , y_{0}\right) F_{7}\! \left(x , y_{0}\right)\\ F_{33}\! \left(x , y_{0}\right) &= F_{23}\! \left(x \right) F_{30}\! \left(x , y_{0}\right)\\ \end{align*}

This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 37 rules.

Found on June 09, 2021.

Finding the specification took 185 seconds.

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\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{21}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{21}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y_{0}\right) &= F_{7}\! \left(x , y_{0}, 1\right)\\ F_{7}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x , y_{0}, y_{1}\right)+F_{25}\! \left(x , y_{0}, y_{1}\right)+F_{8}\! \left(x , y_{0}, y_{1}\right)\\ F_{8}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{0}\right) F_{9}\! \left(x , y_{0}, y_{1}\right)\\ F_{9}\! \left(x , y_{0}, y_{1}\right) &= F_{10}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{10}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{11}\! \left(x , y_{0} y_{1}, y_{1}, y_{2}\right)\\ F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{13}\! \left(x , y_{0}\right)\\ F_{13}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{13}\! \left(x , y_{1}\right) F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{y_{1} F_{10}\! \left(x , 1, y_{1}, y_{2}\right)-y_{0} F_{10}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}, y_{2}\right)}{-y_{1}+y_{0}}\\ F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{13}\! \left(x , y_{2}\right) F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{y_{2} F_{18}\! \left(x , y_{0}, 1, y_{2}\right)-y_{1} F_{18}\! \left(x , y_{0}, \frac{y_{1}}{y_{2}}, y_{2}\right)}{-y_{2}+y_{1}}\\ F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{11}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\ F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{21}\! \left(x \right)\\ F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{2} F_{11}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{11}\! \left(x , y_{0}, y_{1}, 1\right)}{-1+y_{2}}\\ F_{21}\! \left(x \right) &= x\\ F_{22}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{1}\right) F_{23}\! \left(x , y_{0}, y_{1}\right)\\ F_{23}\! \left(x , y_{0}, y_{1}\right) &= -\frac{y_{1} F_{24}\! \left(x , 1, y_{1}\right)-y_{0} F_{24}\! \left(x , \frac{y_{0}}{y_{1}}, y_{1}\right)}{-y_{1}+y_{0}}\\ F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{7}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x \right) F_{26}\! \left(x , y_{0}, y_{1}\right)\\ F_{26}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{1} F_{7}\! \left(x , y_{0}, y_{1}\right)+F_{7}\! \left(x , y_{0}, 1\right)}{-1+y_{1}}\\ F_{27}\! \left(x \right) &= F_{21}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x , 1\right)\\ F_{29}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x , y_{0}\right)+F_{33}\! \left(x , y_{0}\right)+F_{35}\! \left(x , y_{0}\right)\\ F_{30}\! \left(x , y_{0}\right) &= F_{13}\! \left(x , y_{0}\right) F_{31}\! \left(x , y_{0}\right)\\ F_{31}\! \left(x , y_{0}\right) &= \frac{y_{0} F_{32}\! \left(x , y_{0}, 1\right)-F_{32}\! \left(x , y_{0}, \frac{1}{y_{0}}\right)}{-1+y_{0}}\\ F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{7}\! \left(x , y_{0}, y_{0} y_{1}\right)\\ F_{33}\! \left(x , y_{0}\right) &= F_{21}\! \left(x \right) F_{34}\! \left(x , y_{0}\right)\\ F_{34}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{6}\! \left(x , y_{0}\right)+F_{6}\! \left(x , 1\right)}{-1+y_{0}}\\ F_{35}\! \left(x , y_{0}\right) &= F_{21}\! \left(x \right) F_{36}\! \left(x , y_{0}\right)\\ F_{36}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{29}\! \left(x , y_{0}\right)+F_{29}\! \left(x , 1\right)}{-1+y_{0}}\\ \end{align*}