Av(1324, 2143)
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Generating Function
\(\displaystyle \frac{-3 x^{2}-x^{2} \sqrt{-4 x +1}+5 x -1}{4 x^{3}-8 x^{2}+6 x -1}\)
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Counting Sequence
1, 1, 2, 6, 22, 88, 366, 1552, 6652, 28696, 124310, 540040, 2350820, 10248248, 44725516, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{3}-8 x^{2}+6 x -1\right) F \left(x \right)^{2}+\left(6 x^{2}-10 x +2\right) F \! \left(x \right)+x^{2}+4 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{8 \left(2 n +1\right) a \! \left(n \right)}{2+n}-\frac{4 \left(8 n +7\right) a \! \left(2+n \right)}{2+n}+\frac{12 \left(3 n +2\right) a \! \left(n +1\right)}{2+n}+\frac{2 \left(5 n +7\right) a \! \left(n +3\right)}{2+n}, \quad n \geq 4\)
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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 "Row And Col Placements" and has 33 rules.

Found on April 26, 2021.

Finding the specification took 60 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_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ 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_{10}\! \left(x \right) F_{3}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{7} \left(x \right)^{2} F_{3}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{3}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{3}\! \left(x \right) F_{30}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{23}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{22}\! \left(x \right) F_{3}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{3}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{3}\! \left(x \right) F_{30}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Row And Col Placements Req Corrob" and has 33 rules.

Found on April 26, 2021.

Finding the specification took 78 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_{9}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right) F_{9}\! \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_{6} \left(x \right)^{2} F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{30}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \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_{6}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{19}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{22}\! \left(x \right) F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right) F_{30}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{30} \left(x \right)^{2} F_{9}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Row And Col Placements" and has 70 rules.

Found on April 26, 2021.

Finding the specification took 92 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_{6}\! \left(x \right)+F_{9}\! \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_{6} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right) F_{6}\! \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_{21}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{1}\! \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_{54}\! \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_{25}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{19}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{18}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{19}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{37}\! \left(x \right) &= 0\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{18}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{15}\! \left(x \right) F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{15} \left(x \right)^{2}\\ F_{61}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{15}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{15}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{17}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{64}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Req Corrob" and has 45 rules.

Found on April 26, 2021.

Finding the specification took 116 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \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_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{23} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{23}\! \left(x \right) F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{33}\! \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_{0}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{17}\! \left(x \right) F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{17}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{42} \left(x \right)^{2} F_{4}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row And Col Placements" and has 52 rules.

Found on April 26, 2021.

Finding the specification took 52 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_{16}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right) F_{20}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{26}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{16}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{16}\! \left(x \right) F_{26}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{16}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{16}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{39}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{26}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{16}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{16}\! \left(x \right) F_{31}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{16}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{16}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{16}\! \left(x \right) F_{47}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{19} \left(x \right)^{2} F_{16}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{20} \left(x \right)^{2} F_{16}\! \left(x \right) F_{7}\! \left(x \right)\\ \end{align*}\)