Av(13425, 13452, 13542, 15342, 31425, 31452, 31542)
Generating Function
\(\displaystyle \frac{\left(-3 x^{3}-7 x^{2}+x \right) \sqrt{x^{2}-6 x +1}+3 x^{4}-28 x^{3}+22 x^{2}+9 x -2}{26 x^{4}-30 x^{3}+4 x^{2}+12 x -2}\)
Counting Sequence
1, 1, 2, 6, 24, 113, 580, 3122, 17287, 97469, 556360, 3203701, 18567896, 108147601, 632329025, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(13 x^{4}-15 x^{3}+2 x^{2}+6 x -1\right) F \left(x
\right)^{2}+\left(-3 x^{4}+28 x^{3}-22 x^{2}-9 x +2\right) F \! \left(x \right)-3 x^{3}+18 x^{2}+3 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 113\)
\(\displaystyle a(6) = 580\)
\(\displaystyle a(7) = 3122\)
\(\displaystyle a{\left(n + 8 \right)} = - \frac{39 n a{\left(n \right)}}{n + 7} + \frac{\left(19 n + 110\right) a{\left(n + 7 \right)}}{n + 7} - \frac{4 \left(29 n + 134\right) a{\left(n + 6 \right)}}{n + 7} - \frac{9 \left(85 n + 332\right) a{\left(n + 3 \right)}}{n + 7} + \frac{2 \left(94 n + 299\right) a{\left(n + 5 \right)}}{n + 7} + \frac{3 \left(119 n + 491\right) a{\left(n + 4 \right)}}{n + 7} + \frac{\left(188 n + 169\right) a{\left(n + 1 \right)}}{n + 7} + \frac{\left(349 n + 1651\right) a{\left(n + 2 \right)}}{n + 7}, \quad n \geq 8\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 113\)
\(\displaystyle a(6) = 580\)
\(\displaystyle a(7) = 3122\)
\(\displaystyle a{\left(n + 8 \right)} = - \frac{39 n a{\left(n \right)}}{n + 7} + \frac{\left(19 n + 110\right) a{\left(n + 7 \right)}}{n + 7} - \frac{4 \left(29 n + 134\right) a{\left(n + 6 \right)}}{n + 7} - \frac{9 \left(85 n + 332\right) a{\left(n + 3 \right)}}{n + 7} + \frac{2 \left(94 n + 299\right) a{\left(n + 5 \right)}}{n + 7} + \frac{3 \left(119 n + 491\right) a{\left(n + 4 \right)}}{n + 7} + \frac{\left(188 n + 169\right) a{\left(n + 1 \right)}}{n + 7} + \frac{\left(349 n + 1651\right) a{\left(n + 2 \right)}}{n + 7}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 54 rules.
Finding the specification took 1611 seconds.
Copy 54 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_{22}\! \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_{22}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{35}\! \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_{22}\! \left(x \right)\\
F_{12}\! \left(x \right) &= \frac{F_{13}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= -F_{29}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{0}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{22}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{20}\! \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) &= x\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x , 1\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\
F_{25}\! \left(x , y\right) &= x F_{25}\! \left(x , y\right) y +F_{25}\! \left(x , y\right)^{2}-2 F_{25}\! \left(x , y\right)+2\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)^{2} F_{28}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= y x\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{22}\! \left(x \right) F_{23}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= \frac{F_{33}\! \left(x \right)}{F_{22}\! \left(x \right)}\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{34}\! \left(x \right) x +F_{34} \left(x \right)^{2}+x\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{22}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{19}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x , 1\right)\\
F_{40}\! \left(x , y\right) &= -\frac{-F_{25}\! \left(x , y\right) y +F_{25}\! \left(x , 1\right)}{-1+y}\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{22}\! \left(x \right) F_{43}\! \left(x \right) F_{49}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{43}\! \left(x \right) &= \frac{F_{44}\! \left(x \right)}{F_{22}\! \left(x \right) F_{23}\! \left(x \right) F_{49}\! \left(x \right)}\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{22}\! \left(x \right) F_{23}\! \left(x \right) F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= \frac{F_{47}\! \left(x \right)}{F_{22}\! \left(x \right) F_{5}\! \left(x \right)}\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{22}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x , 1\right)\\
F_{53}\! \left(x , y\right) &= -\frac{-y F_{24}\! \left(x , y\right)+F_{24}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 91 rules.
Finding the specification took 3286 seconds.
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Copy 91 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_{25}\! \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_{7}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{6}\! \left(x \right) x +F_{6} \left(x \right)^{2}+x\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{25}\! \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_{4}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{18}\! \left(x \right) x +F_{18} \left(x \right)^{2}-2 F_{18}\! \left(x \right)+2\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{25}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{43}\! \left(x \right)}\\
F_{22}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= \frac{F_{24}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{24}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{25}\! \left(x \right) &= x\\
F_{26}\! \left(x \right) &= -F_{29}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= \frac{F_{28}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{28}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{21}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{25}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{25}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{39}\! \left(x \right) &= 0\\
F_{40}\! \left(x \right) &= F_{25}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{25}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{44}\! \left(x \right) &= \frac{F_{45}\! \left(x \right)}{F_{0}\! \left(x \right)}\\
F_{45}\! \left(x \right) &= -F_{48}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= \frac{F_{47}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{47}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{48}\! \left(x \right) &= -F_{51}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= \frac{F_{50}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{50}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{2}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{25}\! \left(x \right) F_{44}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= \frac{F_{55}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{55}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{25}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{5}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= \frac{F_{61}\! \left(x \right)}{F_{25}\! \left(x \right) F_{4}\! \left(x \right) F_{65}\! \left(x \right)}\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right) F_{65}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{65}\! \left(x \right) &= \frac{F_{66}\! \left(x \right)}{F_{18}\! \left(x \right) F_{25}\! \left(x \right) F_{37}\! \left(x \right)}\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{37} \left(x \right)^{2} F_{25}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= \frac{F_{69}\! \left(x \right)}{F_{25}\! \left(x \right) F_{37}\! \left(x \right)}\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= -F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{25}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{25}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{44}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= \frac{F_{80}\! \left(x \right)}{F_{25}\! \left(x \right) F_{44}\! \left(x \right)}\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{25}\! \left(x \right) F_{44}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= \frac{F_{83}\! \left(x \right)}{F_{25}\! \left(x \right)}\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{25}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{44}\! \left(x \right) F_{62}\! \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) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{25}\! \left(x \right) F_{60}\! \left(x \right) F_{65}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 89 rules.
Finding the specification took 1781 seconds.
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Copy 89 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_{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_{32}\! \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) &= \frac{F_{25}\! \left(x \right)}{F_{0}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= -F_{28}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= \frac{F_{27}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{27}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{28}\! \left(x \right) &= -F_{31}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= \frac{F_{30}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{30}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{2}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{21}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{23}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{21}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{23}\! \left(x \right) F_{4}\! \left(x \right) F_{43}\! \left(x \right)}\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right) F_{43}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{43}\! \left(x \right) &= \frac{F_{44}\! \left(x \right)}{F_{23}\! \left(x \right) F_{5}\! \left(x \right) F_{62}\! \left(x \right)}\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{62} \left(x \right)^{2} F_{23}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= \frac{F_{47}\! \left(x \right)}{F_{23}\! \left(x \right) F_{62}\! \left(x \right)}\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= \frac{F_{50}\! \left(x \right)}{F_{68}\! \left(x \right)}\\
F_{50}\! \left(x \right) &= -F_{53}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= \frac{F_{52}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{52}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{53}\! \left(x \right) &= -F_{54}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{49}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{23}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{23}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{64}\! \left(x \right) &= 0\\
F_{65}\! \left(x \right) &= F_{23}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{23}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{23}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{23}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{24}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= \frac{F_{78}\! \left(x \right)}{F_{23}\! \left(x \right) F_{24}\! \left(x \right)}\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{23}\! \left(x \right) F_{24}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= \frac{F_{81}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{23}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{24}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{23}\! \left(x \right) F_{38}\! \left(x \right) F_{43}\! \left(x \right)\\
\end{align*}\)