Av(13425, 13452, 13542, 31425, 31452, 31542)
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Generating Function
\(\displaystyle \frac{\left(-7 x^{2}+x \right) \sqrt{x^{2}-6 x +1}-3 x^{3}-4 x^{2}+13 x -2}{20 x^{3}-26 x^{2}+16 x -2}\)
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Counting Sequence
1, 1, 2, 6, 24, 114, 596, 3290, 18760, 109186, 644188, 3836682, 23005904, 138641666, 838667012, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(10 x^{3}-13 x^{2}+8 x -1\right) F \left(x \right)^{2}+\left(3 x^{3}+4 x^{2}-13 x +2\right) F \! \left(x \right)-x^{3}+7 x^{2}+5 x -1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 114\)
\(\displaystyle a{\left(n + 6 \right)} = - \frac{70 n a{\left(n \right)}}{n + 5} + \frac{\left(21 n + 82\right) a{\left(n + 5 \right)}}{n + 5} - \frac{2 \left(80 n + 233\right) a{\left(n + 4 \right)}}{n + 5} - \frac{5 \left(149 n + 253\right) a{\left(n + 2 \right)}}{n + 5} + \frac{2 \left(265 n + 558\right) a{\left(n + 3 \right)}}{n + 5} + \frac{\left(521 n + 650\right) a{\left(n + 1 \right)}}{n + 5}, \quad n \geq 6\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 62 rules.

Finding the specification took 3091 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{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_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{12}\! \left(x \right) &= \frac{F_{13}\! \left(x \right)}{F_{23}\! \left(x \right)}\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{2}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{23}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{18}\! \left(x \right) &= \frac{F_{19}\! \left(x \right)}{F_{23}\! \left(x \right)}\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= -F_{22}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{22}\! \left(x \right) x +F_{22} \left(x \right)^{2}+x\\ F_{23}\! \left(x \right) &= x\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x , 1\right)\\ F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{5}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= y F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{23}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{23}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= \frac{F_{35}\! \left(x \right)}{F_{23}\! \left(x \right)}\\ F_{35}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{23}\! \left(x \right) F_{34}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x , 1\right)\\ F_{39}\! \left(x , y\right) &= -\frac{-F_{40}\! \left(x , y\right) y +F_{40}\! \left(x , 1\right)}{-1+y}\\ F_{40}\! \left(x , y\right) &= x F_{40}\! \left(x , y\right) y +y x +F_{40}\! \left(x , y\right)^{2}\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{23}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{23}\! \left(x \right) F_{45}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{45}\! \left(x \right) &= \frac{F_{46}\! \left(x \right)}{F_{23}\! \left(x \right)}\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{23}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{22}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{23}\! \left(x \right) F_{24}\! \left(x \right) F_{54}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{54}\! \left(x \right) &= \frac{F_{55}\! \left(x \right)}{F_{23}\! \left(x \right) F_{56}\! \left(x \right)}\\ F_{55}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{23}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x , 1\right)\\ F_{60}\! \left(x , y\right) &= -\frac{-y F_{61}\! \left(x , y\right)+F_{61}\! \left(x , 1\right)}{-1+y}\\ F_{61}\! \left(x , y\right) &= F_{24}\! \left(y x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Req Corrob" and has 62 rules.

Finding the specification took 1114 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{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_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{12}\! \left(x \right) &= \frac{F_{13}\! \left(x \right)}{F_{23}\! \left(x \right)}\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{2}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{23}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{18}\! \left(x \right) &= \frac{F_{19}\! \left(x \right)}{F_{23}\! \left(x \right)}\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= -F_{22}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{22}\! \left(x \right) x +F_{22} \left(x \right)^{2}+x\\ F_{23}\! \left(x \right) &= x\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{24} \left(x \right)^{2} F_{23}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{21}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{23}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{33}\! \left(x \right) &= \frac{F_{34}\! \left(x \right)}{F_{23}\! \left(x \right)}\\ F_{34}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{23}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= \frac{F_{38}\! \left(x \right)}{F_{23}\! \left(x \right) F_{42}\! \left(x \right)}\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{23}\! \left(x \right) F_{42}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{44}\! \left(x \right) &= \frac{F_{45}\! \left(x \right)}{F_{23}\! \left(x \right)}\\ F_{45}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{21}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{23}\! \left(x \right) F_{24}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{23}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{24}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= \frac{F_{56}\! \left(x \right)}{F_{23}\! \left(x \right)}\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{23}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{24}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{23}\! \left(x \right) F_{37}\! \left(x \right) F_{42}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Req Corrob" and has 138 rules.

Finding the specification took 2992 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{4}\! \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_{5}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= \frac{F_{17}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= -F_{25}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= \frac{F_{24}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{24}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{25}\! \left(x \right) x +F_{25} \left(x \right)^{2}-2 F_{25}\! \left(x \right)+2\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{23}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{28}\! \left(x \right) x +F_{28} \left(x \right)^{2}+x\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{34} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{28}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{34}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= \frac{F_{44}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{44}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{133}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= \frac{F_{47}\! \left(x \right)}{F_{120}\! \left(x \right)}\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= -F_{129}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= \frac{F_{50}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{65}\! \left(x \right) &= \frac{F_{66}\! \left(x \right)}{F_{25}\! \left(x \right)}\\ F_{66}\! \left(x \right) &= -F_{67}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{67}\! \left(x \right) &= -F_{72}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= \frac{F_{69}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= -F_{71}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{71}\! \left(x \right) &= x^{2} F_{71} \left(x \right)^{2}+4 x^{2} F_{71}\! \left(x \right)+4 x F_{71} \left(x \right)^{2}+4 x^{2}-5 x F_{71}\! \left(x \right)-F_{71} \left(x \right)^{2}-x +2 F_{71}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{28}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{73}\! \left(x \right) &= -F_{128}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{28}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= \frac{F_{77}\! \left(x \right)}{F_{25}\! \left(x \right)}\\ F_{77}\! \left(x \right) &= -F_{108}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= \frac{F_{81}\! \left(x \right)}{F_{25}\! \left(x \right)}\\ F_{81}\! \left(x \right) &= -F_{95}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= \frac{F_{83}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= -F_{87}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= \frac{F_{86}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{86}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= \frac{F_{90}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= -F_{23}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= -F_{93}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{58}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= \frac{F_{97}\! \left(x \right)}{F_{25}\! \left(x \right) F_{4}\! \left(x \right)}\\ F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{106}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{103}\! \left(x \right) F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{108}\! \left(x \right) &= -F_{116}\! \left(x \right)+F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= -F_{74}\! \left(x \right)+F_{110}\! \left(x \right)\\ F_{110}\! \left(x \right) &= \frac{F_{111}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)\\ F_{112}\! \left(x \right) &= -F_{113}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{113}\! \left(x \right) &= -F_{71}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{114}\! \left(x \right) &= -F_{115}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{115}\! \left(x \right) &= x^{2} F_{115} \left(x \right)^{2}+2 x^{2} F_{115}\! \left(x \right)+4 x F_{115} \left(x \right)^{2}+x^{2}-13 x F_{115}\! \left(x \right)-F_{115} \left(x \right)^{2}+8 x +4 F_{115}\! \left(x \right)-2\\ F_{116}\! \left(x \right) &= -F_{118}\! \left(x \right)+F_{117}\! \left(x \right)\\ F_{117}\! \left(x \right) &= -F_{78}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123} \left(x \right)^{2} F_{71}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{123} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{120}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{54}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{130}\! \left(x \right) &= \frac{F_{131}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{133}\! \left(x \right) F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ \end{align*}\)