Av(13542, 14532, 23541, 24531)
Generating Function
\(\displaystyle \frac{-x \sqrt{-8 x +1}-x +2}{4 x^{2}-4 x +2}\)
Counting Sequence
1, 1, 2, 6, 24, 116, 632, 3720, 23072, 148528, 983072, 6647776, 45727616, 318947136, 2250473344, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{2}-2 x +1\right) F \left(x
\right)^{2}+\left(-2+x \right) F \! \left(x \right)+x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a{\left(n + 3 \right)} = \frac{8 \left(2 n + 1\right) a{\left(n \right)}}{n + 2} - \frac{6 \left(3 n + 2\right) a{\left(n + 1 \right)}}{n + 2} + \frac{2 \left(5 n + 4\right) a{\left(n + 2 \right)}}{n + 2}, \quad n \geq 3\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a{\left(n + 3 \right)} = \frac{8 \left(2 n + 1\right) a{\left(n \right)}}{n + 2} - \frac{6 \left(3 n + 2\right) a{\left(n + 1 \right)}}{n + 2} + \frac{2 \left(5 n + 4\right) a{\left(n + 2 \right)}}{n + 2}, \quad n \geq 3\)
This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Tracked Component Fusion Req Corrob Symmetries" and has 51 rules.
Finding the specification took 67544 seconds.
Copy 51 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_{46}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{5}\! \left(x \right) &= x F_{5}\! \left(x \right)+F_{5} \left(x \right)^{2}-2 F_{5}\! \left(x \right)+2\\
F_{6}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{46}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{19}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{13}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= y x\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{46}\! \left(x \right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= -\frac{y \left(F_{29}\! \left(x , 1\right)-F_{29}\! \left(x , y\right)\right)}{y -1}\\
F_{10}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{4}\! \left(x \right)\\
F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{46}\! \left(x \right)\\
F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{46}\! \left(x \right)\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{19}\! \left(x , 1\right)\\
F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{16}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{42}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= -\frac{y \left(F_{45}\! \left(x , 1\right)-F_{45}\! \left(x , y\right)\right)}{y -1}\\
F_{19}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)+F_{6}\! \left(x \right)\\
F_{46}\! \left(x \right) &= x\\
F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= F_{46}\! \left(x \right) F_{49}\! \left(x , y\right)\\
F_{49}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\
F_{50}\! \left(x \right) &= F_{38}\! \left(x \right) F_{46}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Tracked Component Fusion Req Corrob Symmetries" and has 63 rules.
Finding the specification took 67544 seconds.
Copy 63 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_{58}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{5}\! \left(x \right) x +F_{5} \left(x \right)^{2}-2 F_{5}\! \left(x \right)+2\\
F_{6}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{58}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= y x\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)+F_{27}\! \left(x , y\right)+F_{7}\! \left(x \right)\\
F_{25}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{26}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{20}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{59}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{58}\! \left(x \right)\\
F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{9}\! \left(x \right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{37}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{37}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= -\frac{y \left(F_{40}\! \left(x , 1\right)-F_{40}\! \left(x , y\right)\right)}{-1+y}\\
F_{10}\! \left(x , y\right) &= F_{4}\! \left(x \right)+F_{40}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{13}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right) F_{58}\! \left(x \right)\\
F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{45}\! \left(x , y\right) F_{58}\! \left(x \right)\\
F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{30}\! \left(x , 1\right)\\
F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{41}\! \left(x , y\right) F_{53}\! \left(x , y\right)\\
F_{53}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)\\
F_{55}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{54}\! \left(x , y\right)\\
F_{55}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= -\frac{y \left(F_{57}\! \left(x , 1\right)-F_{57}\! \left(x , y\right)\right)}{-1+y}\\
F_{30}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)+F_{6}\! \left(x \right)\\
F_{58}\! \left(x \right) &= x\\
F_{59}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)\\
F_{60}\! \left(x , y\right) &= F_{58}\! \left(x \right) F_{61}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\
F_{62}\! \left(x \right) &= F_{50}\! \left(x \right) F_{58}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Req Corrob Symmetries" and has 64 rules.
Finding the specification took 31026 seconds.
Copy 64 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_{43}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{6}\! \left(x \right) x +F_{6} \left(x \right)^{2}-2 F_{6}\! \left(x \right)+2\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{8}\! \left(x \right) x +F_{8} \left(x \right)^{2}+x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x , 1\right)\\
F_{16}\! \left(x , y\right) &= F_{13}\! \left(x \right)+F_{15}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{44}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{6}\! \left(x \right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{33}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{29}\! \left(x , y\right) F_{33}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right) F_{6}\! \left(x \right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{33}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= y x\\
F_{34}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{33}\! \left(x , y\right) F_{37}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= 2 F_{27}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right) F_{43}\! \left(x \right)\\
F_{41}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{40}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= -\frac{y \left(F_{20}\! \left(x , 1\right)-F_{20}\! \left(x , y\right)\right)}{-1+y}\\
F_{43}\! \left(x \right) &= x\\
F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{43}\! \left(x \right) F_{46}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)+F_{49}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{34}\! \left(x , y\right)\\
F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{51}\! \left(x , y\right) F_{54}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{51}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)\\
F_{53}\! \left(x , y\right) &= -\frac{y \left(F_{15}\! \left(x , 1\right)-F_{15}\! \left(x , y\right)\right)}{-1+y}\\
F_{54}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\
F_{55}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= F_{43}\! \left(x \right) F_{57}\! \left(x , y\right)\\
F_{58}\! \left(x , y\right) &= F_{43}\! \left(x \right) F_{57}\! \left(x , y\right)\\
F_{58}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{43}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= \frac{F_{63}\! \left(x \right)}{F_{43}\! \left(x \right)}\\
F_{63}\! \left(x \right) &= F_{12}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Req Corrob Symmetries" and has 73 rules.
Finding the specification took 31026 seconds.
Copy 73 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_{52}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{6}\! \left(x \right) x +F_{6} \left(x \right)^{2}-2 F_{6}\! \left(x \right)+2\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{8}\! \left(x \right) x +F_{8} \left(x \right)^{2}+x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x , 1\right)\\
F_{16}\! \left(x , y\right) &= F_{13}\! \left(x \right)+F_{15}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{53}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{46}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{6}\! \left(x \right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{33}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{23}\! \left(x , y\right) F_{33}\! \left(x , y\right) F_{37}\! \left(x , y\right) F_{46}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{43}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right) F_{6}\! \left(x \right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{33}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= y x\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{37}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\
F_{38}\! \left(x , y\right) &= F_{39}\! \left(x \right)+F_{40}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\
F_{39}\! \left(x \right) &= 0\\
F_{40}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)+F_{38}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{33}\! \left(x , y\right) F_{45}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= 2 F_{27}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= F_{49}\! \left(x , y\right) F_{52}\! \left(x \right)\\
F_{50}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{49}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)\\
F_{51}\! \left(x , y\right) &= -\frac{y \left(F_{20}\! \left(x , 1\right)-F_{20}\! \left(x , y\right)\right)}{-1+y}\\
F_{52}\! \left(x \right) &= x\\
F_{53}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{52}\! \left(x \right) F_{55}\! \left(x , y\right)\\
F_{55}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)+F_{58}\! \left(x , y\right)\\
F_{57}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{46}\! \left(x , y\right)\\
F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\
F_{59}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{60}\! \left(x , y\right) F_{63}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{60}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)\\
F_{62}\! \left(x , y\right) &= -\frac{y \left(F_{15}\! \left(x , 1\right)-F_{15}\! \left(x , y\right)\right)}{-1+y}\\
F_{63}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{46}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{52}\! \left(x \right) F_{66}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{52}\! \left(x \right) F_{66}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{52}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= \frac{F_{72}\! \left(x \right)}{F_{52}\! \left(x \right)}\\
F_{72}\! \left(x \right) &= F_{12}\! \left(x \right)\\
\end{align*}\)