Av(1243, 3214)
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Generating Function
\(\displaystyle \frac{x^{5}-5 x^{4}+13 x^{3}-12 x^{2}+6 x -1}{4 x^{5}-13 x^{4}+22 x^{3}-17 x^{2}+7 x -1}\)
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Counting Sequence
1, 1, 2, 6, 22, 86, 338, 1318, 5110, 19770, 76466, 295810, 1144530, 4428622, 17136186, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{5}-13 x^{4}+22 x^{3}-17 x^{2}+7 x -1\right) F \! \left(x \right)-x^{5}+5 x^{4}-13 x^{3}+12 x^{2}-6 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(5\right) = 86\)
\(\displaystyle a \! \left(n +5\right) = 4 a \! \left(n \right)-13 a \! \left(n +1\right)+22 a \! \left(n +2\right)-17 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \left(\left\{\begin{array}{cc}\frac{1}{4} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)+\frac{28024 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +3}}{377305}+\frac{28024 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +3}}{377305}+\frac{28024 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +3}}{377305}+\frac{28024 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +3}}{377305}+\frac{28024 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +3}}{377305}-\frac{128066 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +2}}{377305}-\frac{128066 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +2}}{377305}-\frac{128066 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +2}}{377305}-\frac{128066 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +2}}{377305}-\frac{128066 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +2}}{377305}+\frac{240334 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +1}}{377305}+\frac{240334 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +1}}{377305}+\frac{240334 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +1}}{377305}+\frac{240334 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +1}}{377305}+\frac{240334 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +1}}{377305}+\frac{23662 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n -1}}{377305}+\frac{23662 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n -1}}{377305}+\frac{23662 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n -1}}{377305}+\frac{23662 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n -1}}{377305}+\frac{23662 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n -1}}{377305}-\frac{21452 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n}}{75461}-\frac{21452 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n}}{75461}-\frac{21452 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n}}{75461}-\frac{21452 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n}}{75461}-\frac{21452 \mathit{RootOf} \left(4 Z^{5}-13 Z^{4}+22 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n}}{75461}\)
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Heatmap

To create this heatmap, we sampled 1,000,000 permutations of length 300 uniformly at random. The color of the point \((i, j)\) represents how many permutations have value \(j\) at index \(i\) (darker = more).

This specification was found using the strategy pack "Point Placements" and has 137 rules.

Found on January 18, 2022.

Finding the specification took 15 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_{28}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{15}\! \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_{58}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{49}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{67}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{79}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{80}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{80}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{109}\! \left(x \right)+F_{120}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{4}\! \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_{70}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{100}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{107}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{103}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{114}\! \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_{113}\! \left(x \right) &= F_{111}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{118}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{114}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{130}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{129}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{128}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{123}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{136}\! \left(x \right)+F_{15}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{135}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{130}\! \left(x \right) F_{4}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 304 rules.

Found on April 28, 2021.

Finding the specification took 8 seconds.

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Copy 304 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_{12}\! \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_{45}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \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) &= F_{13}\! \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_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{12}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{34}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{12}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{231}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{12}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{176}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{12}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{12}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{12}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{12}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{12}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{66}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{12}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= x^{2}\\ F_{75}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{71}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{12}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{12}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{86}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{76}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{151}\! \left(x \right)+F_{88}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{12}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{12}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{88}\! \left(x \right)+F_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{12}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{12}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{94}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{12}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{112}\! \left(x \right)\\ F_{100}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{101}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{106}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{104}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{105}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{103}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{112}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{113}\! \left(x \right)+F_{117}\! \left(x \right)+F_{129}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{12}\! \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_{97}\! \left(x \right)\\ F_{120}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{113}\! \left(x \right)+F_{121}\! \left(x \right)+F_{122}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{119}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{122}\! \left(x \right) &= 0\\ F_{123}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{12}\! \left(x \right) F_{126}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{128}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{120}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{124}\! \left(x \right)\\ F_{129}\! \left(x \right) &= 0\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{134}\! \left(x \right)+F_{148}\! \left(x \right)+F_{15}\! \left(x \right)+F_{173}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{12}\! \left(x \right) F_{133}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{115}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{12}\! \left(x \right) F_{135}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{142}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{137}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{132}\! \left(x \right)+F_{138}\! \left(x \right)+F_{141}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{12}\! \left(x \right) F_{139}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{140}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{113}\! \left(x \right)+F_{138}\! \left(x \right)+F_{141}\! \left(x \right)\\ F_{141}\! \left(x \right) &= 0\\ F_{142}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{143}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{12}\! \left(x \right) F_{145}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{147}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{143}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{12}\! \left(x \right) F_{149}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{172}\! \left(x \right)\\ F_{150}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{151}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{12}\! \left(x \right) F_{152}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{154}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{164}\! \left(x \right)\\ F_{155}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{156}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{12}\! \left(x \right) F_{157}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{159}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{12}\! \left(x \right) F_{161}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)+F_{163}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{159}\! \left(x \right)\\ F_{164}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{113}\! \left(x \right)+F_{165}\! \left(x \right)+F_{175}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{12}\! \left(x \right) F_{166}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{167}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{164}\! \left(x \right)+F_{168}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{12}\! \left(x \right) F_{170}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)+F_{174}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{172}\! \left(x \right)\\ F_{172}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{113}\! \left(x \right)+F_{148}\! \left(x \right)+F_{173}\! \left(x \right)\\ F_{173}\! \left(x \right) &= 0\\ F_{174}\! \left(x \right) &= F_{168}\! \left(x \right)\\ F_{175}\! \left(x \right) &= 0\\ F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{178}\! \left(x \right)+F_{192}\! \left(x \right)+F_{226}\! \left(x \right)\\ F_{178}\! \left(x \right) &= F_{12}\! \left(x \right) F_{179}\! \left(x \right)\\ F_{179}\! \left(x \right) &= F_{180}\! \left(x \right)+F_{181}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{181}\! \left(x \right) &= F_{177}\! \left(x \right)+F_{182}\! \left(x \right)\\ F_{182}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{183}\! \left(x \right)+F_{187}\! \left(x \right)+F_{225}\! \left(x \right)\\ F_{183}\! \left(x \right) &= F_{12}\! \left(x \right) F_{184}\! \left(x \right)\\ F_{184}\! \left(x \right) &= F_{185}\! \left(x \right)+F_{186}\! \left(x \right)\\ F_{185}\! \left(x \right) &= F_{133}\! \left(x \right)\\ F_{186}\! \left(x \right) &= F_{182}\! \left(x \right)\\ F_{187}\! \left(x \right) &= F_{12}\! \left(x \right) F_{188}\! \left(x \right)\\ F_{188}\! \left(x \right) &= F_{189}\! \left(x \right)+F_{224}\! \left(x \right)\\ F_{189}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{190}\! \left(x \right)+F_{192}\! \left(x \right)\\ F_{190}\! \left(x \right) &= F_{12}\! \left(x \right) F_{191}\! \left(x \right)\\ F_{191}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{192}\! \left(x \right) &= F_{12}\! \left(x \right) F_{193}\! \left(x \right)\\ F_{193}\! \left(x \right) &= F_{194}\! \left(x \right)+F_{195}\! \left(x \right)\\ F_{194}\! \left(x \right) &= F_{189}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{195}\! \left(x \right) &= F_{196}\! \left(x \right)+F_{208}\! \left(x \right)\\ F_{196}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{197}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{197}\! \left(x \right) &= F_{12}\! \left(x \right) F_{198}\! \left(x \right)\\ F_{198}\! \left(x \right) &= F_{199}\! \left(x \right)+F_{202}\! \left(x \right)\\ F_{199}\! \left(x \right) &= F_{200}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{200}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{201}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{201}\! \left(x \right) &= F_{12}\! \left(x \right) F_{199}\! \left(x \right)\\ F_{202}\! \left(x \right) &= F_{196}\! \left(x \right)+F_{203}\! \left(x \right)\\ F_{203}\! \left(x \right) &= F_{204}\! \left(x \right)\\ F_{204}\! \left(x \right) &= F_{12}\! \left(x \right) F_{205}\! \left(x \right)\\ F_{205}\! \left(x \right) &= F_{206}\! \left(x \right)+F_{207}\! \left(x \right)\\ F_{206}\! \left(x \right) &= F_{200}\! \left(x \right)\\ F_{207}\! \left(x \right) &= F_{203}\! \left(x \right)\\ F_{208}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{209}\! \left(x \right)+F_{211}\! \left(x \right)+F_{223}\! \left(x \right)\\ F_{209}\! \left(x \right) &= F_{12}\! \left(x \right) F_{210}\! \left(x \right)\\ F_{210}\! \left(x \right) &= F_{114}\! \left(x \right)\\ F_{211}\! \left(x \right) &= F_{12}\! \left(x \right) F_{212}\! \left(x \right)\\ F_{212}\! \left(x \right) &= F_{213}\! \left(x \right)+F_{217}\! \left(x \right)\\ F_{213}\! \left(x \right) &= F_{189}\! \left(x \right)+F_{214}\! \left(x \right)\\ F_{214}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{209}\! \left(x \right)+F_{215}\! \left(x \right)+F_{216}\! \left(x \right)\\ F_{215}\! \left(x \right) &= F_{12}\! \left(x \right) F_{213}\! \left(x \right)\\ F_{216}\! \left(x \right) &= 0\\ F_{217}\! \left(x \right) &= F_{208}\! \left(x \right)+F_{218}\! \left(x \right)\\ F_{218}\! \left(x \right) &= F_{219}\! \left(x \right)\\ F_{219}\! \left(x \right) &= F_{12}\! \left(x \right) F_{220}\! \left(x \right)\\ F_{220}\! \left(x \right) &= F_{221}\! \left(x \right)+F_{222}\! \left(x \right)\\ F_{221}\! \left(x \right) &= F_{214}\! \left(x \right)\\ F_{222}\! \left(x \right) &= F_{218}\! \left(x \right)\\ F_{223}\! \left(x \right) &= 0\\ F_{224}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{187}\! \left(x \right)+F_{209}\! \left(x \right)+F_{225}\! \left(x \right)\\ F_{225}\! \left(x \right) &= 0\\ F_{226}\! \left(x \right) &= F_{12}\! \left(x \right) F_{227}\! \left(x \right)\\ F_{227}\! \left(x \right) &= F_{228}\! \left(x \right)+F_{268}\! \left(x \right)\\ F_{228}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{229}\! \left(x \right)+F_{231}\! \left(x \right)\\ F_{229}\! \left(x \right) &= F_{12}\! \left(x \right) F_{230}\! \left(x \right)\\ F_{230}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{231}\! \left(x \right) &= F_{12}\! \left(x \right) F_{232}\! \left(x \right)\\ F_{232}\! \left(x \right) &= F_{233}\! \left(x \right)+F_{234}\! \left(x \right)\\ F_{233}\! \left(x \right) &= F_{228}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{234}\! \left(x \right) &= F_{235}\! \left(x \right)+F_{256}\! \left(x \right)\\ F_{235}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{236}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{236}\! \left(x \right) &= F_{12}\! \left(x \right) F_{237}\! \left(x \right)\\ F_{237}\! \left(x \right) &= F_{238}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{238}\! \left(x \right) &= F_{235}\! \left(x \right)+F_{239}\! \left(x \right)\\ F_{239}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{240}\! \left(x \right)+F_{242}\! \left(x \right)+F_{252}\! \left(x \right)\\ F_{240}\! \left(x \right) &= F_{12}\! \left(x \right) F_{241}\! \left(x \right)\\ F_{241}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{242}\! \left(x \right) &= F_{12}\! \left(x \right) F_{243}\! \left(x \right)\\ F_{243}\! \left(x \right) &= F_{244}\! \left(x \right)+F_{246}\! \left(x \right)\\ F_{244}\! \left(x \right) &= F_{245}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{245}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{240}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{246}\! \left(x \right) &= F_{239}\! \left(x \right)+F_{247}\! \left(x \right)\\ F_{247}\! \left(x \right) &= F_{248}\! \left(x \right)\\ F_{248}\! \left(x \right) &= F_{12}\! \left(x \right) F_{249}\! \left(x \right)\\ F_{249}\! \left(x \right) &= F_{250}\! \left(x \right)+F_{251}\! \left(x \right)\\ F_{250}\! \left(x \right) &= F_{245}\! \left(x \right)\\ F_{251}\! \left(x \right) &= F_{247}\! \left(x \right)\\ F_{252}\! \left(x \right) &= F_{12}\! \left(x \right) F_{253}\! \left(x \right)\\ F_{253}\! \left(x \right) &= F_{254}\! \left(x \right)+F_{255}\! \left(x \right)\\ F_{254}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{255}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{252}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{256}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{257}\! \left(x \right)+F_{259}\! \left(x \right)+F_{279}\! \left(x \right)\\ F_{257}\! \left(x \right) &= F_{12}\! \left(x \right) F_{258}\! \left(x \right)\\ F_{258}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{259}\! \left(x \right) &= F_{12}\! \left(x \right) F_{260}\! \left(x \right)\\ F_{260}\! \left(x \right) &= F_{227}\! \left(x \right)+F_{261}\! \left(x \right)\\ F_{261}\! \left(x \right) &= F_{256}\! \left(x \right)+F_{262}\! \left(x \right)\\ F_{262}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{263}\! \left(x \right)+F_{265}\! \left(x \right)+F_{276}\! \left(x \right)+F_{301}\! \left(x \right)\\ F_{263}\! \left(x \right) &= F_{12}\! \left(x \right) F_{264}\! \left(x \right)\\ F_{264}\! \left(x \right) &= F_{133}\! \left(x \right)\\ F_{265}\! \left(x \right) &= F_{12}\! \left(x \right) F_{266}\! \left(x \right)\\ F_{266}\! \left(x \right) &= F_{267}\! \left(x \right)+F_{270}\! \left(x \right)\\ F_{267}\! \left(x \right) &= F_{268}\! \left(x \right)+F_{269}\! \left(x \right)\\ F_{268}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{192}\! \left(x \right)+F_{226}\! \left(x \right)+F_{257}\! \left(x \right)\\ F_{269}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{187}\! \left(x \right)+F_{225}\! \left(x \right)+F_{263}\! \left(x \right)\\ F_{270}\! \left(x \right) &= F_{262}\! \left(x \right)+F_{271}\! \left(x \right)\\ F_{271}\! \left(x \right) &= F_{272}\! \left(x \right)\\ F_{272}\! \left(x \right) &= F_{12}\! \left(x \right) F_{273}\! \left(x \right)\\ F_{273}\! \left(x \right) &= F_{274}\! \left(x \right)+F_{275}\! \left(x \right)\\ F_{274}\! \left(x \right) &= F_{269}\! \left(x \right)\\ F_{275}\! \left(x \right) &= F_{271}\! \left(x \right)\\ F_{276}\! \left(x \right) &= F_{12}\! \left(x \right) F_{277}\! \left(x \right)\\ F_{277}\! \left(x \right) &= F_{278}\! \left(x \right)+F_{300}\! \left(x \right)\\ F_{278}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{190}\! \left(x \right)+F_{279}\! \left(x \right)\\ F_{279}\! \left(x \right) &= F_{12}\! \left(x \right) F_{280}\! \left(x \right)\\ F_{280}\! \left(x \right) &= F_{281}\! \left(x \right)+F_{282}\! \left(x \right)\\ F_{281}\! \left(x \right) &= F_{254}\! \left(x \right)+F_{278}\! \left(x \right)\\ F_{282}\! \left(x \right) &= F_{283}\! \left(x \right)+F_{292}\! \left(x \right)\\ F_{283}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{284}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{284}\! \left(x \right) &= F_{12}\! \left(x \right) F_{285}\! \left(x \right)\\ F_{285}\! \left(x \right) &= F_{253}\! \left(x \right)+F_{286}\! \left(x \right)\\ F_{286}\! \left(x \right) &= F_{283}\! \left(x \right)+F_{287}\! \left(x \right)\\ F_{287}\! \left(x \right) &= F_{288}\! \left(x \right)\\ F_{288}\! \left(x \right) &= F_{12}\! \left(x \right) F_{289}\! \left(x \right)\\ F_{289}\! \left(x \right) &= F_{290}\! \left(x \right)+F_{291}\! \left(x \right)\\ F_{290}\! \left(x \right) &= F_{255}\! \left(x \right)\\ F_{291}\! \left(x \right) &= F_{287}\! \left(x \right)\\ F_{292}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{209}\! \left(x \right)+F_{293}\! \left(x \right)+F_{303}\! \left(x \right)\\ F_{293}\! \left(x \right) &= F_{12}\! \left(x \right) F_{294}\! \left(x \right)\\ F_{294}\! \left(x \right) &= F_{277}\! \left(x \right)+F_{295}\! \left(x \right)\\ F_{295}\! \left(x \right) &= F_{292}\! \left(x \right)+F_{296}\! \left(x \right)\\ F_{296}\! \left(x \right) &= F_{297}\! \left(x \right)\\ F_{297}\! \left(x \right) &= F_{12}\! \left(x \right) F_{298}\! \left(x \right)\\ F_{298}\! \left(x \right) &= F_{299}\! \left(x \right)+F_{302}\! \left(x \right)\\ F_{299}\! \left(x \right) &= F_{300}\! \left(x \right)\\ F_{300}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{209}\! \left(x \right)+F_{276}\! \left(x \right)+F_{301}\! \left(x \right)\\ F_{301}\! \left(x \right) &= 0\\ F_{302}\! \left(x \right) &= F_{296}\! \left(x \right)\\ F_{303}\! \left(x \right) &= 0\\ \end{align*}\)