Av(13254, 13524, 13542)
Counting Sequence
1, 1, 2, 6, 24, 117, 652, 3988, 26112, 180126, 1295090, 9631656, 73676572, 577180996, 4615090192, ...
Heatmap
To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point \((i, j)\) represents how many permutations have value \(j\) at index \(i\) (darker = more).
This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Expand Verified" and has 30 rules.
Found on November 06, 2021.Finding the specification took 816 seconds.
Copy 30 equations to clipboard:
\(\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_{4}\! \left(x \right)\\
F_{3}\! \left(x \right) &= x\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= -\frac{-y F_{5}\! \left(x , y\right)+F_{5}\! \left(x , 1\right)}{-1+y}\\
F_{8}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= y x\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , 1, y\right)\\
F_{11}\! \left(x , y , z\right) &= F_{12}\! \left(x , y , z\right)+F_{5}\! \left(x , z\right)\\
F_{12}\! \left(x , y , z\right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x , y , z\right)+F_{17}\! \left(x , y , z\right)+F_{28}\! \left(x , z , y\right)\\
F_{13}\! \left(x \right) &= 0\\
F_{14}\! \left(x , y , z\right) &= F_{15}\! \left(x , y , z\right) F_{3}\! \left(x \right)\\
F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , z\right)\\
F_{16}\! \left(x , y , z\right) &= \frac{y z \left(F_{12}\! \left(x , y , z\right)-F_{12}\! \left(x , \frac{1}{z}, z\right)\right)}{y z -1}\\
F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right) F_{9}\! \left(x , z\right)\\
F_{18}\! \left(x , y , z\right) &= F_{19}\! \left(x , y , z\right)+F_{23}\! \left(x , y , z\right)\\
F_{19}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y , z\right)+F_{20}\! \left(x , y , z\right)+F_{22}\! \left(x , z , y\right)\\
F_{20}\! \left(x , y , z\right) &= F_{21}\! \left(x , y , z\right) F_{3}\! \left(x \right)\\
F_{21}\! \left(x , y , z\right) &= \frac{y z F_{19}\! \left(x , y , z\right)-F_{19}\! \left(x , \frac{1}{z}, z\right)}{y z -1}\\
F_{22}\! \left(x , y , z\right) &= F_{19}\! \left(x , z , y\right) F_{9}\! \left(x , y\right)\\
F_{23}\! \left(x , y , z\right) &= F_{24}\! \left(x , y z , z\right)\\
F_{24}\! \left(x , y , z\right) &= F_{25}\! \left(x , y , z\right)\\
F_{25}\! \left(x , y , z\right) &= \frac{y F_{26}\! \left(x , y , 1\right)-z F_{26}\! \left(x , y , \frac{z}{y}\right)}{-z +y}\\
F_{26}\! \left(x , y , z\right) &= F_{27}\! \left(x , y , y z \right)\\
F_{12}\! \left(x , y , z\right) &= F_{27}\! \left(x , y z , z\right)\\
F_{28}\! \left(x , y , z\right) &= F_{29}\! \left(x , y , z\right)\\
F_{29}\! \left(x , y , z\right) &= F_{12}\! \left(x , z , y\right) F_{9}\! \left(x , y\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Req Corrob" and has 38 rules.
Found on July 27, 2021.Finding the specification took 5394 seconds.
Copy 38 equations to clipboard:
\(\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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{9}\! \left(x , y_{0}\right)+F_{9}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{9}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y_{0}\right)+F_{11}\! \left(x , y_{0}\right)\\
F_{10}\! \left(x , y_{0}\right) &= F_{4}\! \left(x \right) F_{8}\! \left(x , y_{0}\right)\\
F_{11}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{13}\! \left(x , y_{0}\right)\\
F_{12}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{13}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right)+F_{9}\! \left(x , y_{0}\right)\\
F_{14}\! \left(x , y_{0}\right) &= F_{15}\! \left(x , y_{0}, 1\right)\\
F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x , y_{0}, y_{1}\right)+F_{25}\! \left(x , y_{0}, y_{1}\right)+F_{34}\! \left(x , y_{0}, y_{1}\right)\\
F_{16}\! \left(x \right) &= 0\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , y_{0}, y_{1}\right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}, y_{1}\right)\\
F_{19}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{20}\! \left(x , y_{0}, y_{1}\right)\\
F_{20}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{21}\! \left(x , y_{0}, 1, y_{1}\right)-F_{21}\! \left(x , y_{0}, \frac{1}{y_{0}}, y_{1}\right)}{-1+y_{0}}\\
F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0} y_{1}, y_{0} y_{2}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{22}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{23}\! \left(x , y_{0}, y_{1}\right)\\
F_{24}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , y_{0}, y_{1}\right)\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{26}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{27}\! \left(x , y_{0}, y_{1}\right)\\
F_{27}\! \left(x , y_{0}, y_{1}\right) &= F_{28}\! \left(x , y_{0}, y_{1}\right)+F_{32}\! \left(x , y_{0}, y_{1}\right)\\
F_{28}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x , y_{0}, y_{1}\right)+F_{29}\! \left(x , y_{0}, y_{1}\right)+F_{31}\! \left(x , y_{1}, y_{0}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{30}\! \left(x , y_{0}, y_{1}\right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{0} F_{28}\! \left(x , y_{0}, y_{1}\right)+F_{28}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{28}\! \left(x , y_{1}, y_{0}\right)\\
F_{32}\! \left(x , y_{0}, y_{1}\right) &= F_{33}\! \left(x , y_{0}, y_{1}\right)\\
F_{33}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{15}\! \left(x , y_{0}, 1\right)-y_{1} F_{15}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\
F_{34}\! \left(x , y_{0}, y_{1}\right) &= F_{35}\! \left(x , y_{0}, y_{1}\right)\\
F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{1}\right) F_{15}\! \left(x , y_{0}, y_{1}\right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Req Corrob Expand Verified" and has 32 rules.
Found on November 06, 2021.Finding the specification took 3416 seconds.
Copy 32 equations to clipboard:
\(\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_{4}\! \left(x \right)\\
F_{3}\! \left(x \right) &= x\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y_{0}\right)+F_{8}\! \left(x , y_{0}\right)\\
F_{6}\! \left(x , y_{0}\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y_{0}\right)\\
F_{7}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{5}\! \left(x , y_{0}\right)+F_{5}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{8}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\
F_{9}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{10}\! \left(x , y_{0}\right) &= F_{11}\! \left(x , y_{0}\right)+F_{5}\! \left(x , y_{0}\right)\\
F_{11}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}, y_{1}, y_{0}\right)\\
F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{13}\! \left(x , y_{0}, y_{1}, 1, y_{2}\right)\\
F_{13}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{14}\! \left(x , y_{2}, y_{3}\right)\\
F_{14}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , y_{0}, y_{1}, 1\right)\\
F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{30}\! \left(x , y_{1}, y_{2}, y_{0}\right)\\
F_{16}\! \left(x \right) &= 0\\
F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{3}\! \left(x \right)\\
F_{18}\! \left(x , y_{0}, y_{1}, 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_{0} y_{1} \left(F_{15}\! \left(x , y_{0}, y_{1}, y_{2}\right)-F_{15}\! \left(x , \frac{1}{y_{1}}, y_{1}, y_{2}\right)\right)}{y_{0} y_{1}-1}\\
F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{9}\! \left(x , y_{1}\right)\\
F_{21}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{25}\! \left(x , y_{1}, y_{2}, y_{0}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{3}\! \left(x \right)\\
F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} y_{1} F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right)-F_{22}\! \left(x , \frac{1}{y_{1}}, y_{1}, y_{2}\right)}{y_{0} y_{1}-1}\\
F_{25}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{22}\! \left(x , y_{2}, y_{0}, y_{1}\right) F_{9}\! \left(x , y_{1}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{27}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\
F_{27}\! \left(x , y_{0}, y_{1}\right) &= F_{28}\! \left(x , y_{0}, y_{1}\right)\\
F_{28}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{29}\! \left(x , y_{0}, 1\right)-y_{1} F_{29}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{15}\! \left(x , 1, y_{0}, y_{1}\right)\\
F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\
F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{15}\! \left(x , y_{2}, y_{0}, y_{1}\right) F_{9}\! \left(x , y_{1}\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row And Col Placements Tracked Fusion" and has 40 rules.
Found on July 27, 2021.Finding the specification took 1489 seconds.
Copy 40 equations to clipboard:
\(\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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\
F_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{11}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{8}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{13}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= y x\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y , 1\right)\\
F_{14}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y , z\right)+F_{25}\! \left(x , y , z\right)+F_{35}\! \left(x , y , z\right)\\
F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , z\right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x , y , z\right) &= F_{17}\! \left(x , y , z\right)+F_{8}\! \left(x , y z \right)\\
F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y , z\right)\\
F_{18}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{19}\! \left(x , y , z\right)\\
F_{19}\! \left(x , y , z\right) &= F_{20}\! \left(x , y , y z \right)\\
F_{20}\! \left(x , y , z\right) &= -\frac{-y F_{21}\! \left(x , y , z\right)+F_{21}\! \left(x , 1, z\right)}{-1+y}\\
F_{22}\! \left(x , y , z\right) &= F_{21}\! \left(x , y , y z \right)\\
F_{23}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{22}\! \left(x , y , z\right)\\
F_{23}\! \left(x , y , z\right) &= F_{24}\! \left(x , y , z\right)\\
F_{14}\! \left(x , y , z\right) &= F_{24}\! \left(x , y , z\right)+F_{9}\! \left(x , y z \right)\\
F_{25}\! \left(x , y , z\right) &= F_{26}\! \left(x , y , y z \right)\\
F_{26}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{27}\! \left(x , y , z\right)\\
F_{27}\! \left(x , y , z\right) &= F_{28}\! \left(x , y , z\right)+F_{32}\! \left(x , y , z\right)\\
F_{28}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x , y , z\right)+F_{29}\! \left(x , y , z\right)+F_{31}\! \left(x , z , y\right)\\
F_{29}\! \left(x , y , z\right) &= F_{30}\! \left(x , y , z\right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x , y , z\right) &= -\frac{-y F_{28}\! \left(x , y , z\right)+F_{28}\! \left(x , 1, z\right)}{-1+y}\\
F_{31}\! \left(x , y , z\right) &= F_{12}\! \left(x , y\right) F_{28}\! \left(x , z , y\right)\\
F_{33}\! \left(x , y , z\right) &= F_{32}\! \left(x , y , z\right)+F_{34}\! \left(x , y , z\right)\\
F_{33}\! \left(x , y , z\right) &= \frac{y F_{14}\! \left(x , y , 1\right)-z F_{14}\! \left(x , y , \frac{z}{y}\right)}{-z +y}\\
F_{34}\! \left(x , y , z\right) &= \frac{y F_{9}\! \left(x , y\right)-z F_{9}\! \left(x , z\right)}{-z +y}\\
F_{35}\! \left(x , y , z\right) &= F_{12}\! \left(x , z\right) F_{36}\! \left(x , y , z\right)\\
F_{36}\! \left(x , y , z\right) &= F_{13}\! \left(x , y z \right)+F_{37}\! \left(x , y , z\right)\\
F_{37}\! \left(x , y , z\right) &= F_{24}\! \left(x , y , z\right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row And Col Placements Tracked Fusion Req Corrob" and has 40 rules.
Found on July 27, 2021.Finding the specification took 7957 seconds.
Copy 40 equations to clipboard:
\(\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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{9}\! \left(x , y_{0}\right)+F_{9}\! \left(x , 1\right)}{-1+y_{0}}\\
F_{9}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y_{0}\right)+F_{11}\! \left(x , y_{0}\right)\\
F_{10}\! \left(x , y_{0}\right) &= F_{4}\! \left(x \right) F_{8}\! \left(x , y_{0}\right)\\
F_{11}\! \left(x , y_{0}\right) &= F_{12}\! \left(x , y_{0}\right) F_{13}\! \left(x , y_{0}\right)\\
F_{12}\! \left(x , y_{0}\right) &= y_{0} x\\
F_{13}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}, 1\right)\\
F_{14}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y_{0}, y_{1}\right)+F_{25}\! \left(x , y_{0}, y_{1}\right)+F_{35}\! \left(x , y_{0}, y_{1}\right)\\
F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , y_{0}, y_{1}\right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x , y_{0}, y_{1}\right) &= F_{17}\! \left(x , y_{0}, y_{1}\right)+F_{8}\! \left(x , y_{0} y_{1}\right)\\
F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , y_{0}, y_{1}\right)\\
F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{19}\! \left(x , y_{0}, y_{1}\right)\\
F_{19}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{20}\! \left(x , y_{0}, 1, y_{1}\right)-F_{20}\! \left(x , y_{0}, \frac{1}{y_{0}}, y_{1}\right)}{-1+y_{0}}\\
F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{0} y_{1}, y_{0} y_{2}\right)\\
F_{22}\! \left(x , y_{0}, y_{1}\right) &= F_{21}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{22}\! \left(x , y_{0}, y_{1}\right)\\
F_{23}\! \left(x , y_{0}, y_{1}\right) &= F_{24}\! \left(x , y_{0}, y_{1}\right)\\
F_{14}\! \left(x , y_{0}, y_{1}\right) &= F_{24}\! \left(x , y_{0}, y_{1}\right)+F_{9}\! \left(x , y_{0} y_{1}\right)\\
F_{25}\! \left(x , y_{0}, y_{1}\right) &= F_{26}\! \left(x , y_{0}, y_{0} y_{1}\right)\\
F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{27}\! \left(x , y_{0}, y_{1}\right)\\
F_{27}\! \left(x , y_{0}, y_{1}\right) &= F_{28}\! \left(x , y_{0}, y_{1}\right)+F_{32}\! \left(x , y_{0}, y_{1}\right)\\
F_{28}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x , y_{0}, y_{1}\right)+F_{29}\! \left(x , y_{0}, y_{1}\right)+F_{31}\! \left(x , y_{1}, y_{0}\right)\\
F_{29}\! \left(x , y_{0}, y_{1}\right) &= F_{30}\! \left(x , y_{0}, y_{1}\right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{0} F_{28}\! \left(x , y_{0}, y_{1}\right)+F_{28}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\
F_{31}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}\right) F_{28}\! \left(x , y_{1}, y_{0}\right)\\
F_{33}\! \left(x , y_{0}, y_{1}\right) &= F_{32}\! \left(x , y_{0}, y_{1}\right)+F_{34}\! \left(x , y_{0}, y_{1}\right)\\
F_{33}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{14}\! \left(x , y_{0}, 1\right)-y_{1} F_{14}\! \left(x , y_{0}, \frac{y_{1}}{y_{0}}\right)}{-y_{1}+y_{0}}\\
F_{34}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{9}\! \left(x , y_{0}\right)-y_{1} F_{9}\! \left(x , y_{1}\right)}{-y_{1}+y_{0}}\\
F_{35}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{1}\right) F_{36}\! \left(x , y_{0}, y_{1}\right)\\
F_{36}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{0} y_{1}\right)+F_{37}\! \left(x , y_{0}, y_{1}\right)\\
F_{37}\! \left(x , y_{0}, y_{1}\right) &= F_{24}\! \left(x , y_{0}, y_{1}\right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\
\end{align*}\)