Av(12354, 12453, 12543, 13254, 13452, 13542, 14253, 14352, 14532, 15243, 15342, 15432)
Generating Function
\(\displaystyle \frac{2 x^{3}-5 x^{2}-\sqrt{12 x^{2}-8 x +1}+5 x +1}{x \left(2 x -3\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 108, 512, 2506, 12560, 64148, 332704, 1747748, 9280416, 49731768, 268613568, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -3\right)^{2} F \left(x
\right)^{2}+\left(-4 x^{3}+10 x^{2}-10 x -2\right) F \! \left(x \right)+x^{3}-2 x^{2}+3 x +2 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 3 \right)} = \frac{8 \left(n + 2\right) a{\left(n \right)}}{n + 4} - \frac{4 \left(13 n + 23\right) a{\left(n + 1 \right)}}{3 \left(n + 4\right)} + \frac{2 \left(13 n + 35\right) a{\left(n + 2 \right)}}{3 \left(n + 4\right)}, \quad n \geq 4\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 3 \right)} = \frac{8 \left(n + 2\right) a{\left(n \right)}}{n + 4} - \frac{4 \left(13 n + 23\right) a{\left(n + 1 \right)}}{3 \left(n + 4\right)} + \frac{2 \left(13 n + 35\right) a{\left(n + 2 \right)}}{3 \left(n + 4\right)}, \quad n \geq 4\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 120 rules.
Finding the specification took 52776 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_{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_{23}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{10}\! \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) &= F_{13}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{14}\! \left(x \right) &= \frac{F_{15}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= -F_{77}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= x\\
F_{24}\! \left(x \right) &= F_{23}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{23}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{23}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{25}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{13}\! \left(x \right) F_{23}\! \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) &= F_{38}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{27}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= \frac{F_{43}\! \left(x \right)}{F_{58}\! \left(x \right)}\\
F_{43}\! \left(x \right) &= -F_{53}\! \left(x \right)+F_{44}\! \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_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= -F_{8}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= \frac{F_{48}\! \left(x \right)}{F_{23}\! \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_{23}\! \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_{5}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{23}\! \left(x \right) F_{42}\! \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_{58}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{60}\! \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_{22}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{66}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{65}\! \left(x \right) &= 0\\
F_{66}\! \left(x \right) &= F_{23}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{23}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{23}\! \left(x \right) F_{27}\! \left(x \right) F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{23}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{23}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{13}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{84}\! \left(x \right) &= -F_{88}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= \frac{F_{86}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{23}\! \left(x \right) F_{90}\! \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_{42}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{23}\! \left(x \right) F_{42}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{13}\! \left(x \right) F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{23}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{101}\! \left(x \right) &= -F_{105}\! \left(x \right)+F_{102}\! \left(x \right)\\
F_{102}\! \left(x \right) &= \frac{F_{103}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{113}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{110}\! \left(x \right) &= 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_{107}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{23}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{110}\! \left(x \right) F_{23}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{107}\! \left(x \right) F_{13}\! \left(x \right) F_{23}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 123 rules.
Finding the specification took 15988 seconds.
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Copy 123 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_{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_{23}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{10}\! \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) &= F_{13}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{14}\! \left(x \right) &= \frac{F_{15}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= -F_{77}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= x\\
F_{24}\! \left(x \right) &= F_{23}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{23}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{23}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{25}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{13}\! \left(x \right) F_{23}\! \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) &= F_{38}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{27}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= \frac{F_{43}\! \left(x \right)}{F_{58}\! \left(x \right)}\\
F_{43}\! \left(x \right) &= -F_{53}\! \left(x \right)+F_{44}\! \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_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= -F_{8}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= \frac{F_{48}\! \left(x \right)}{F_{23}\! \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_{23}\! \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_{5}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{23}\! \left(x \right) F_{42}\! \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_{58}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{60}\! \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_{22}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{66}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{65}\! \left(x \right) &= 0\\
F_{66}\! \left(x \right) &= F_{23}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{23}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{23}\! \left(x \right) F_{27}\! \left(x \right) F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{23}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{23}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{13}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{84}\! \left(x \right) &= -F_{90}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= \frac{F_{86}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{49}\! \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_{94}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{0}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{23}\! \left(x \right) F_{42}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{13}\! \left(x \right) F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{103}\! \left(x \right) &= -F_{108}\! \left(x \right)+F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= \frac{F_{105}\! \left(x \right)}{F_{23}\! \left(x \right)}\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{106}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{107}\! \left(x \right)\\
F_{107}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{110}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{23}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{113}\! \left(x \right) F_{23}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{110}\! \left(x \right) F_{13}\! \left(x \right) F_{23}\! \left(x \right)\\
\end{align*}\)