Av(2143, 3412, 234615)
Generating Function
\(\displaystyle -\frac{\left(3 \sqrt{1-4 x}\, x^{2}+2 x^{3}-4 \sqrt{1-4 x}\, x -9 x^{2}+\sqrt{1-4 x}+6 x -1\right) \left(3 x -1\right)^{2} \left(x -1\right)}{2 x^{6} \left(2 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 22, 86, 339, 1327, 5150, 19854, 76207, 291779, 1115824, 4265768, 16311631, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{6} \left(2 x -1\right)^{2} F \left(x
\right)^{2}+\left(x -1\right) \left(x^{2}-4 x +1\right) \left(3 x -1\right)^{2} \left(2 x -1\right)^{2} F \! \left(x \right)+\left(x -1\right)^{2} \left(3 x -1\right)^{4} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{12 \left(3+2 n \right) a \! \left(n \right)}{10+n}+\frac{2 \left(94+25 n \right) a \! \left(n +1\right)}{10+n}-\frac{7 \left(29+5 n \right) a \! \left(n +2\right)}{10+n}+\frac{\left(79+10 n \right) a \! \left(n +3\right)}{10+n}, \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{12 \left(3+2 n \right) a \! \left(n \right)}{10+n}+\frac{2 \left(94+25 n \right) a \! \left(n +1\right)}{10+n}-\frac{7 \left(29+5 n \right) a \! \left(n +2\right)}{10+n}+\frac{\left(79+10 n \right) a \! \left(n +3\right)}{10+n}, \quad n \geq 4\)
This specification was found using the strategy pack "Point And Row And Col Placements Tracked Fusion Expand Verified" and has 121 rules.
Found on January 23, 2022.Finding the specification took 610 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_{11}\! \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_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{62}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{38}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x , 1\right)\\
F_{20}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x , y\right)+F_{28}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{22}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= -F_{4}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= \frac{F_{26}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{26}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{27}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{28}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{29}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{35}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x , y\right)+F_{32}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{29}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{22}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{30}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= y x\\
F_{35}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= -\frac{-y F_{37}\! \left(x , y\right)+F_{37}\! \left(x , 1\right)}{-1+y}\\
F_{37}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{38}\! \left(x \right)+F_{41}\! \left(x , y\right)\\
F_{38}\! \left(x \right) &= F_{11}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= \frac{F_{40}\! \left(x \right)}{F_{11}\! \left(x \right)}\\
F_{40}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{41}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{36}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{34}\! \left(x , y\right) F_{43}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{34}\! \left(x , y\right) F_{44}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{48}\! \left(x \right)+F_{51}\! \left(x , y\right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{11}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{51}\! \left(x , y\right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\
F_{52}\! \left(x \right) &= 0\\
F_{53}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{54}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\
F_{55}\! \left(x , y\right) &= F_{34}\! \left(x , y\right) F_{47}\! \left(x , y\right)\\
F_{56}\! \left(x \right) &= F_{11}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x , 1\right)\\
F_{58}\! \left(x , y\right) &= -\frac{-y F_{37}\! \left(x , y\right)+F_{37}\! \left(x , 1\right)}{-1+y}\\
F_{59}\! \left(x \right) &= F_{11}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{11}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{11}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x , 1\right)\\
F_{64}\! \left(x , y\right) &= F_{105}\! \left(x , y\right)+F_{107}\! \left(x \right)+F_{65}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)+F_{8}\! \left(x \right)\\
F_{66}\! \left(x , y\right) &= F_{103}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{34}\! \left(x , y\right) F_{69}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{70}\! \left(x \right)+F_{95}\! \left(x , y\right)\\
F_{70}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{11}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{77}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{11}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{11}\! \left(x \right) F_{74}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{82}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{11}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{11}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{85}\! \left(x \right) &= -F_{86}\! \left(x \right)-F_{90}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{87}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{11}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{30}\! \left(x , 1\right)\\
F_{89}\! \left(x \right) &= F_{11}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{11}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{88}\! \left(x \right)+F_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{11}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{11}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{95}\! \left(x , y\right) &= F_{34}\! \left(x , y\right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{98}\! \left(x , y\right) &= F_{34}\! \left(x , y\right) F_{97}\! \left(x \right)\\
F_{99}\! \left(x , y\right) &= F_{71}\! \left(x \right)+F_{98}\! \left(x , y\right)\\
F_{99}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)\\
F_{100}\! \left(x , y\right) &= F_{101}\! \left(x , y\right) F_{11}\! \left(x \right)\\
F_{102}\! \left(x , y\right) &= F_{101}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\
F_{102}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)\\
F_{103}\! \left(x , y\right) &= F_{104}\! \left(x , y\right) F_{11}\! \left(x \right)\\
F_{64}\! \left(x , y\right) &= F_{104}\! \left(x , y\right)+F_{7}\! \left(x \right)\\
F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right) F_{11}\! \left(x \right)\\
F_{106}\! \left(x , y\right) &= -\frac{-y F_{64}\! \left(x , y\right)+F_{64}\! \left(x , 1\right)}{-1+y}\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{11}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{118}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x , 1\right)\\
F_{110}\! \left(x , y\right) &= F_{111}\! \left(x , y\right)+F_{114}\! \left(x , y\right)\\
F_{111}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{112}\! \left(x , y\right)+F_{113}\! \left(x , y\right)\\
F_{112}\! \left(x , y\right) &= F_{111}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\
F_{113}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{110}\! \left(x , y\right)\\
F_{114}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{115}\! \left(x , y\right)\\
F_{115}\! \left(x , y\right) &= F_{110}\! \left(x , y\right)+F_{116}\! \left(x , y\right)+F_{118}\! \left(x \right)\\
F_{116}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{117}\! \left(x , y\right)\\
F_{117}\! \left(x , y\right) &= -\frac{-y F_{115}\! \left(x , y\right)+F_{115}\! \left(x , 1\right)}{-1+y}\\
F_{118}\! \left(x \right) &= F_{11}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{11}\! \left(x \right) F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{117}\! \left(x , 1\right)\\
\end{align*}\)
This specification was found using the strategy pack "Insertion Row And Col Placements Tracked Fusion Req Corrob Expand Verified" and has 233 rules.
Found on January 22, 2022.Finding the specification took 535 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_{14}\! \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_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{17}\! \left(x \right) &= 0\\
F_{18}\! \left(x \right) &= F_{14}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{14}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{26}\! \left(x \right) &= -F_{17}\! \left(x \right)-F_{31}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= \frac{F_{30}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{30}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{14}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{24}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{14}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{12}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{14}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{14}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{14}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{20}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{14}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x , 1\right)\\
F_{66}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{68}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{70}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right) F_{22}\! \left(x \right) F_{71}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= F_{72}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right) F_{74}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= y x\\
F_{74}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{71}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= -\frac{y \left(F_{76}\! \left(x , 1\right)-F_{76}\! \left(x , y\right)\right)}{-1+y}\\
F_{76}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)+F_{77}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= F_{17}\! \left(x \right)+F_{78}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{79}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= F_{77}\! \left(x , y\right)+F_{80}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= F_{81}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= F_{73}\! \left(x , y\right) F_{82}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= F_{83}\! \left(x , y\right)+F_{84}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{80}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= F_{85}\! \left(x , y\right)+F_{86}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= F_{20}\! \left(x \right) F_{74}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{50}\! \left(x \right)+F_{77}\! \left(x , y\right)\\
F_{87}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{88}\! \left(x , y\right)\\
F_{88}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)+F_{89}\! \left(x , y\right)\\
F_{89}\! \left(x , y\right) &= F_{90}\! \left(x , y\right)\\
F_{90}\! \left(x , y\right) &= F_{20}\! \left(x \right) F_{22}\! \left(x \right) F_{71}\! \left(x , y\right)\\
F_{91}\! \left(x \right) &= F_{77}\! \left(x , 1\right)\\
F_{92}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{17}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{14}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{20}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{103}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{15}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x , 1\right)\\
F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right)\\
F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right) F_{14}\! \left(x \right)\\
F_{108}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{111}\! \left(x , y\right)\\
F_{109}\! \left(x , y\right) &= F_{110}\! \left(x , y\right)\\
F_{110}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{40}\! \left(x \right) F_{71}\! \left(x , y\right)\\
F_{111}\! \left(x , y\right) &= -\frac{y \left(F_{112}\! \left(x , 1\right)-F_{112}\! \left(x , y\right)\right)}{-1+y}\\
F_{112}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)+F_{113}\! \left(x , y\right)\\
F_{113}\! \left(x , y\right) &= F_{114}\! \left(x , y\right)\\
F_{114}\! \left(x , y\right) &= F_{115}\! \left(x , y\right) F_{73}\! \left(x , y\right)\\
F_{115}\! \left(x , y\right) &= F_{116}\! \left(x , y\right)+F_{117}\! \left(x , y\right)\\
F_{116}\! \left(x , y\right) &= F_{113}\! \left(x , y\right)+F_{27}\! \left(x \right)\\
F_{117}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{120}\! \left(x , y\right)\\
F_{118}\! \left(x , y\right) &= F_{119}\! \left(x , y\right)\\
F_{119}\! \left(x , y\right) &= F_{20} \left(x \right)^{2} F_{74}\! \left(x , y\right)\\
F_{120}\! \left(x , y\right) &= F_{58}\! \left(x \right)+F_{66}\! \left(x , y\right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{20} \left(x \right)^{2} F_{15}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{12}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{12} \left(x \right)^{2} F_{15}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{218}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{139}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{2}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{143}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{14}\! \left(x \right) F_{142}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{14}\! \left(x \right) F_{144}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{159}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{147}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{14}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{148}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{14}\! \left(x \right) F_{149}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{14}\! \left(x \right) F_{151}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{153}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{2}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{0}\! \left(x \right) F_{154}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{155}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{158}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{14}\! \left(x \right) F_{157}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{14}\! \left(x \right) F_{154}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{165}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{161}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)+F_{163}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{14}\! \left(x \right) F_{164}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{161}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{166}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)+F_{17}\! \left(x \right)+F_{170}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{168}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{14}\! \left(x \right) F_{169}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{166}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{14}\! \left(x \right) F_{171}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{172}\! \left(x \right)\\
F_{172}\! \left(x \right) &= -F_{161}\! \left(x \right)+F_{173}\! \left(x \right)\\
F_{173}\! \left(x \right) &= -F_{217}\! \left(x \right)+F_{174}\! \left(x \right)\\
F_{174}\! \left(x \right) &= -F_{10}\! \left(x \right)+F_{175}\! \left(x \right)\\
F_{175}\! \left(x \right) &= \frac{F_{176}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)\\
F_{177}\! \left(x \right) &= -F_{214}\! \left(x \right)+F_{178}\! \left(x \right)\\
F_{178}\! \left(x \right) &= -F_{197}\! \left(x \right)+F_{179}\! \left(x \right)\\
F_{179}\! \left(x \right) &= \frac{F_{180}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{180}\! \left(x \right) &= F_{181}\! \left(x \right)\\
F_{181}\! \left(x \right) &= F_{14}\! \left(x \right) F_{182}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{183}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{184}\! \left(x \right)\\
F_{184}\! \left(x \right) &= F_{185}\! \left(x \right)+F_{188}\! \left(x \right)\\
F_{185}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{17}\! \left(x \right)+F_{186}\! \left(x \right)\\
F_{186}\! \left(x \right) &= F_{14}\! \left(x \right) F_{187}\! \left(x \right)\\
F_{187}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{185}\! \left(x \right)\\
F_{188}\! \left(x \right) &= F_{189}\! \left(x \right)\\
F_{189}\! \left(x \right) &= F_{14}\! \left(x \right) F_{190}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{190}\! \left(x \right) &= -F_{193}\! \left(x \right)+F_{191}\! \left(x \right)\\
F_{191}\! \left(x \right) &= \frac{F_{192}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{192}\! \left(x \right) &= F_{140}\! \left(x \right)\\
F_{193}\! \left(x \right) &= F_{194}\! \left(x \right)\\
F_{194}\! \left(x \right) &= F_{195}\! \left(x \right)+F_{196}\! \left(x \right)\\
F_{195}\! \left(x \right) &= F_{12}\! \left(x \right) F_{2}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{196}\! \left(x \right) &= F_{140}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{197}\! \left(x \right) &= F_{198}\! \left(x \right)+F_{213}\! \left(x \right)\\
F_{198}\! \left(x \right) &= -F_{206}\! \left(x \right)+F_{199}\! \left(x \right)\\
F_{199}\! \left(x \right) &= -F_{203}\! \left(x \right)+F_{200}\! \left(x \right)\\
F_{200}\! \left(x \right) &= \frac{F_{201}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{201}\! \left(x \right) &= F_{202}\! \left(x \right)\\
F_{202}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{203}\! \left(x \right)\\
F_{203}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{204}\! \left(x \right)\\
F_{204}\! \left(x \right) &= \frac{F_{205}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{205}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{206}\! \left(x \right) &= F_{207}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{207}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{208}\! \left(x \right)+F_{210}\! \left(x \right)\\
F_{208}\! \left(x \right) &= F_{14}\! \left(x \right) F_{209}\! \left(x \right)\\
F_{209}\! \left(x \right) &= F_{207}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{210}\! \left(x \right) &= F_{14}\! \left(x \right) F_{211}\! \left(x \right)\\
F_{211}\! \left(x \right) &= F_{212}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{212}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{213}\! \left(x \right) &= F_{178}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{214}\! \left(x \right) &= F_{215}\! \left(x \right)\\
F_{215}\! \left(x \right) &= F_{14}\! \left(x \right) F_{216}\! \left(x \right)\\
F_{216}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{217}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{218}\! \left(x \right) &= F_{219}\! \left(x \right)\\
F_{219}\! \left(x \right) &= -F_{226}\! \left(x \right)+F_{220}\! \left(x \right)\\
F_{220}\! \left(x \right) &= -F_{223}\! \left(x \right)+F_{221}\! \left(x \right)\\
F_{221}\! \left(x \right) &= \frac{F_{222}\! \left(x \right)}{F_{14}\! \left(x \right)}\\
F_{222}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{223}\! \left(x \right) &= F_{224}\! \left(x \right)+F_{225}\! \left(x \right)\\
F_{224}\! \left(x \right) &= F_{0}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{225}\! \left(x \right) &= F_{15} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{226}\! \left(x \right) &= F_{227}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{227}\! \left(x \right) &= F_{12}\! \left(x \right) F_{228}\! \left(x \right)\\
F_{228}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{229}\! \left(x \right)\\
F_{229}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{230}\! \left(x \right)+F_{232}\! \left(x \right)\\
F_{230}\! \left(x \right) &= F_{14}\! \left(x \right) F_{231}\! \left(x \right)\\
F_{231}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{229}\! \left(x \right)\\
F_{232}\! \left(x \right) &= F_{14}\! \left(x \right) F_{228}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Row And Col Placements Tracked Fusion Expand Verified" and has 128 rules.
Found on January 22, 2022.Finding the specification took 362 seconds.
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Copy 128 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_{12}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= \frac{F_{9}\! \left(x \right)}{F_{12}\! \left(x \right)}\\
F_{9}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= \frac{F_{11}\! \left(x \right)}{F_{12}\! \left(x \right)}\\
F_{11}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{0}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{12}\! \left(x \right)}\\
F_{16}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x , 1\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x , y\right)+F_{53}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{24}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x , y\right)+F_{32}\! \left(x , y\right)+F_{52}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{26}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)+F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= -\frac{-y F_{29}\! \left(x , y\right)+F_{29}\! \left(x , 1\right)}{-1+y}\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{8}\! \left(x \right)\\
F_{30}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\
F_{31}\! \left(x , y\right) &= y x\\
F_{32}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{34}\! \left(x \right)+F_{51}\! \left(x , y\right)\\
F_{34}\! \left(x \right) &= \frac{F_{35}\! \left(x \right)}{F_{12}\! \left(x \right)}\\
F_{35}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{36}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{33}\! \left(x , 1\right)\\
F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{26}\! \left(x , 1\right)\\
F_{42}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{46}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{12}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{48}\! \left(x \right)+F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{12}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{12}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{51}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{31}\! \left(x , y\right)\\
F_{53}\! \left(x , y\right) &= F_{31}\! \left(x , y\right) F_{54}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)+F_{58}\! \left(x , y\right)\\
F_{55}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{56}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)\\
F_{57}\! \left(x , y\right) &= F_{31}\! \left(x , y\right) F_{55}\! \left(x , y\right)\\
F_{58}\! \left(x , y\right) &= F_{59}\! \left(x \right)+F_{62}\! \left(x , y\right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{12}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{62}\! \left(x , y\right) &= F_{63}\! \left(x \right)+F_{64}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\
F_{63}\! \left(x \right) &= 0\\
F_{64}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{65}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)+F_{62}\! \left(x , y\right)\\
F_{66}\! \left(x , y\right) &= F_{31}\! \left(x , y\right) F_{58}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{20}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{19}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{70}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= -\frac{-y F_{19}\! \left(x , y\right)+F_{19}\! \left(x , 1\right)}{-1+y}\\
F_{71}\! \left(x \right) &= F_{12}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{12}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{76}\! \left(x \right) &= \frac{F_{77}\! \left(x \right)}{F_{12}\! \left(x \right)}\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{3}\! \left(x \right)\\
F_{79}\! \left(x \right) &= -F_{82}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= \frac{F_{81}\! \left(x \right)}{F_{12}\! \left(x \right)}\\
F_{81}\! \left(x \right) &= -F_{0}\! \left(x \right)-F_{127}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{42}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{12}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{12}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{91}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{12}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{12}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{12}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{107}\! \left(x \right)+F_{84}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{12}\! \left(x \right) F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{103}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{103}\! \left(x \right) &= 2 F_{63}\! \left(x \right)+F_{104}\! \left(x \right)+F_{106}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{12}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{107}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{109}\! \left(x \right)-F_{123}\! \left(x \right)+F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= \frac{F_{42}\! \left(x \right)}{F_{12}\! \left(x \right)}\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{118}\! \left(x \right) &= 2 F_{63}\! \left(x \right)+F_{119}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{12}\! \left(x \right) F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{12}\! \left(x \right) F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{123}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{124}\! \left(x \right)-F_{96}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{12}\! \left(x \right) F_{125}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x , 1\right)\\
F_{126}\! \left(x , y\right) &= -\frac{-y F_{24}\! \left(x , y\right)+F_{24}\! \left(x , 1\right)}{-1+y}\\
F_{127}\! \left(x \right) &= F_{12}\! \left(x \right) F_{83}\! \left(x \right)\\
\end{align*}\)