Av(13452, 13542, 14352, 14532, 15342, 15432)
Generating Function
\(\displaystyle -2 x^{2}+2 x +1-x \sqrt{4 x^{2}-8 x +1}\)
Counting Sequence
1, 1, 2, 6, 24, 114, 600, 3372, 19824, 120426, 749976, 4762644, 30723792, 200778612, 1326360048, ...
Implicit Equation for the Generating Function
\(\displaystyle F \left(x
\right)^{2}+\left(4 x^{2}-4 x -2\right) F \! \left(x \right)-x^{2}+4 x +1 = 0\)
Recurrence
\(\displaystyle a{\left(0 \right)} = 1\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(n + 2 \right)} = - \frac{4 \left(n - 2\right) a{\left(n \right)}}{n + 1} + \frac{4 \left(2 n - 1\right) a{\left(n + 1 \right)}}{n + 1}, \quad n \geq 3\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(n + 2 \right)} = - \frac{4 \left(n - 2\right) a{\left(n \right)}}{n + 1} + \frac{4 \left(2 n - 1\right) a{\left(n + 1 \right)}}{n + 1}, \quad n \geq 3\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 136 rules.
Finding the specification took 11393 seconds.
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Copy 136 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_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{17}\! \left(x \right) &= \frac{F_{18}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{20}\! \left(x \right) &= x\\
F_{21}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{20}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= \frac{F_{28}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{20}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= \frac{F_{33}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{19}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{20}\! \left(x \right) F_{38}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{20}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= \frac{F_{48}\! \left(x \right)}{F_{20}\! \left(x \right) F_{70}\! \left(x \right)}\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{20}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= -F_{67}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{20}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= \frac{F_{55}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{57}\! \left(x \right) &= -F_{60}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= \frac{F_{59}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{59}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{2}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{20}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{2}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{20}\! \left(x \right) F_{70}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{71}\! \left(x \right) &= \frac{F_{72}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{72}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{2}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{20}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{20}\! \left(x \right) F_{71}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= -F_{86}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= \frac{F_{85}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{85}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{20}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= \frac{F_{89}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{89}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{0}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{20}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{0}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{20}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{20}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= -F_{108}\! \left(x \right)+F_{106}\! \left(x \right)\\
F_{106}\! \left(x \right) &= \frac{F_{107}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{107}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{17}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{109}\! \left(x \right) &= -F_{134}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= -F_{132}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= \frac{F_{112}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{0}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{20}\! \left(x \right) F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{2}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{129}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{123}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{20}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{122}\! \left(x \right) F_{20}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{0} \left(x \right)^{2}\\
\end{align*}\)
This specification was found using the strategy pack "Partial Row And Col Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 25 rules.
Finding the specification took 2802 seconds.
Copy 25 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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= y F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{15}\! \left(x , y\right)+F_{8}\! \left(x \right)+F_{9}\! \left(x , y\right)\\
F_{8}\! \left(x \right) &= 0\\
F_{10}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{8}\! \left(x \right)+F_{9}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{13}\! \left(x \right)\\
F_{12}\! \left(x , y\right) &= -\frac{-y F_{5}\! \left(x , y\right)+F_{5}\! \left(x , 1\right)}{-1+y}\\
F_{13}\! \left(x \right) &= x\\
F_{14}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{5}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= y F_{16}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{18}\! \left(x , y\right)+F_{8}\! \left(x \right)\\
F_{17}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{7}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= y F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{21}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{22}\! \left(x , y\right)+F_{23}\! \left(x , y\right)+F_{24}\! \left(x , y\right)+F_{8}\! \left(x \right)\\
F_{9}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{23}\! \left(x , y\right)+F_{8}\! \left(x \right)\\
F_{23}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{7}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= y F_{19}\! \left(x , y\right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 139 rules.
Finding the specification took 4773 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 139 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_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{17}\! \left(x \right) &= \frac{F_{18}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{18}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{19}\! \left(x \right) &= x\\
F_{20}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{19}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= \frac{F_{24}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= \frac{F_{27}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{27}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{19}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= \frac{F_{32}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{23}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{19}\! \left(x \right) F_{37}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{19}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{0}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{19}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{19}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= \frac{F_{52}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{19}\! \left(x \right) F_{76}\! \left(x \right)}\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{19}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= -F_{73}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{19}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= \frac{F_{61}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{63}\! \left(x \right) &= -F_{66}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= \frac{F_{65}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{65}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{2}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{19}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{2}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{19}\! \left(x \right) F_{51}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{2}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{19}\! \left(x \right) F_{51}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= -F_{92}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= -F_{88}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= \frac{F_{87}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{87}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{19}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= \frac{F_{91}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{91}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{0}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{19}\! \left(x \right) F_{98}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{19}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= -F_{107}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= \frac{F_{106}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{106}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{17}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{109}\! \left(x \right) &= -F_{126}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= -F_{125}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= \frac{F_{112}\! \left(x \right)}{F_{19}\! \left(x \right)}\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{0}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{19}\! \left(x \right) F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{19}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{128}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{0} \left(x \right)^{2}\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{136}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{19}\! \left(x \right) F_{53}\! \left(x \right)\\
\end{align*}\)