Av(12345, 12354, 12435, 12453, 12534, 12543, 13452, 13542, 21345, 21354, 21435, 21453, 21534, 21543, 31245, 31254, 31425, 31524, 41235, 41325)
Generating Function
\(\displaystyle -\frac{2 x^{5}+x^{4}+5 x^{3}-10 x^{2}+6 x -1}{4 x^{6}+x^{5}-x^{4}-12 x^{3}+15 x^{2}-7 x +1}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 409, 1651, 6632, 26619, 106878, 429287, 1724612, 6928898, 27838281, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{6}+x^{5}-x^{4}-12 x^{3}+15 x^{2}-7 x +1\right) F \! \left(x \right)+2 x^{5}+x^{4}+5 x^{3}-10 x^{2}+6 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a{\left(n + 6 \right)} = - 4 a{\left(n \right)} - a{\left(n + 1 \right)} + a{\left(n + 2 \right)} + 12 a{\left(n + 3 \right)} - 15 a{\left(n + 4 \right)} + 7 a{\left(n + 5 \right)}, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a{\left(n + 6 \right)} = - 4 a{\left(n \right)} - a{\left(n + 1 \right)} + a{\left(n + 2 \right)} + 12 a{\left(n + 3 \right)} - 15 a{\left(n + 4 \right)} + 7 a{\left(n + 5 \right)}, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle -\frac{92646271 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +1}}{80135769}-\frac{92646271 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +1}}{80135769}-\frac{92646271 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +1}}{80135769}-\frac{92646271 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +1}}{80135769}-\frac{92646271 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +1}}{80135769}-\frac{92646271 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +1}}{80135769}-\frac{4042203 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n -1}}{26711923}-\frac{4042203 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n -1}}{26711923}-\frac{4042203 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n -1}}{26711923}-\frac{4042203 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n -1}}{26711923}-\frac{4042203 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n -1}}{26711923}-\frac{4042203 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n -1}}{26711923}+\frac{78022039 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n}}{80135769}+\frac{78022039 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n}}{80135769}+\frac{78022039 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n}}{80135769}+\frac{78022039 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n}}{80135769}+\frac{78022039 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n}}{80135769}+\frac{78022039 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n}}{80135769}+\frac{10776856 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +3}}{26711923}+\frac{10776856 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +3}}{26711923}+\frac{10776856 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +3}}{26711923}+\frac{10776856 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +3}}{26711923}+\frac{10776856 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +3}}{26711923}+\frac{10776856 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +3}}{26711923}+\frac{34770368 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +4}}{80135769}+\frac{34770368 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +4}}{80135769}+\frac{34770368 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +4}}{80135769}+\frac{34770368 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +4}}{80135769}+\frac{34770368 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +4}}{80135769}+\frac{34770368 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +4}}{80135769}+\frac{15167330 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +2}}{80135769}+\frac{15167330 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +2}}{80135769}+\frac{15167330 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +2}}{80135769}+\frac{15167330 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +2}}{80135769}+\frac{15167330 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +2}}{80135769}+\frac{15167330 \mathit{RootOf} \left(4 Z^{6}+Z^{5}-Z^{4}-12 Z^{3}+15 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +2}}{80135769}\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 180 rules.
Finding the specification took 197 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_{16}\! \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_{43}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \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_{16}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{14}\! \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) &= x\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{16}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{27}\! \left(x \right) &= 0\\
F_{28}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{16}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{31}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{16}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{16}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{16}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{174}\! \left(x \right)+F_{27}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{16}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{167}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{27}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{16}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{16}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{16}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{54}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{35}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{16}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{16}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{41}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{16}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{70}\! \left(x \right) &= 0\\
F_{71}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{27}\! \left(x \right)+F_{72}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{16}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{16}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{27}\! \left(x \right)+F_{72}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{16}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{81}\! \left(x \right)+F_{82}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{16}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{16}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{85}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{81}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{86}\! \left(x \right) &= 0\\
F_{87}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{100}\! \left(x \right)+F_{108}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{16}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{90}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{16}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{16}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{107}\! \left(x \right)\\
F_{102}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{103}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{106}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{107}\! \left(x \right) &= 3 F_{27}\! \left(x \right)+F_{100}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{108}\! \left(x \right) &= 0\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{115}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{27}\! \left(x \right)+F_{81}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{122}\! \left(x \right)+F_{27}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{122}\! \left(x \right) &= 0\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{126}\! \left(x \right)+F_{127}\! \left(x \right)+F_{27}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{126}\! \left(x \right) &= 0\\
F_{127}\! \left(x \right) &= 0\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{27}\! \left(x \right)+F_{81}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{135}\! \left(x \right)+F_{136}\! \left(x \right)+F_{27}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{135}\! \left(x \right) &= 0\\
F_{136}\! \left(x \right) &= 0\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{140}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{142}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{27}\! \left(x \right)+F_{41}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{127}\! \left(x \right)+F_{133}\! \left(x \right)+F_{27}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{156}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{151}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{27}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{153}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{162}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{27}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{159}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{161}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{163}\! \left(x \right)+F_{27}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{16}\! \left(x \right) F_{164}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{166}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{168}\! \left(x \right)+F_{171}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)+F_{27}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{16}\! \left(x \right) F_{170}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{172}\! \left(x \right)+F_{27}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{16}\! \left(x \right) F_{173}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{16}\! \left(x \right) F_{175}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{176}\! \left(x \right)+F_{177}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{168}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{178}\! \left(x \right)+F_{179}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{31}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{179}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{172}\! \left(x \right)+F_{27}\! \left(x \right)+F_{73}\! \left(x \right)\\
\end{align*}\)