Av(1342, 4123)
The requested set is not in the database, but a symmetry of it is.
Generating Function
\(\displaystyle \frac{\left(-6 x^{5}+15 x^{4}-14 x^{3}+6 x^{2}-x \right) \sqrt{1-4 x}-20 x^{6}+110 x^{5}-201 x^{4}+180 x^{3}-86 x^{2}+21 x -2}{8 x^{7}-64 x^{6}+196 x^{5}-282 x^{4}+220 x^{3}-96 x^{2}+22 x -2}\)
Counting Sequence
1, 1, 2, 6, 22, 87, 352, 1434, 5861, 24019, 98677, 406291, 1676009, 6924618, 28646875, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{7}-32 x^{6}+98 x^{5}-141 x^{4}+110 x^{3}-48 x^{2}+11 x -1\right) F \left(x
\right)^{2}+\left(20 x^{6}-110 x^{5}+201 x^{4}-180 x^{3}+86 x^{2}-21 x +2\right) F \! \left(x \right)+25 x^{5}-66 x^{4}+71 x^{3}-38 x^{2}+10 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(5\right) = 87\)
\(\displaystyle a \! \left(6\right) = 352\)
\(\displaystyle a \! \left(7\right) = 1434\)
\(\displaystyle a \! \left(8\right) = 5861\)
\(\displaystyle a \! \left(9\right) = 24019\)
\(\displaystyle a \! \left(10\right) = 98677\)
\(\displaystyle a \! \left(11\right) = 406291\)
\(\displaystyle a \! \left(n +12\right) = -\frac{48 \left(3+2 n \right) a \! \left(n \right)}{11+n}+\frac{3 \left(71+7 n \right) a \! \left(11+n \right)}{11+n}+\frac{1032 \left(n +2\right) a \! \left(n +1\right)}{11+n}-\frac{4 \left(3089+1187 n \right) a \! \left(n +2\right)}{11+n}+\frac{396 \left(103+31 n \right) a \! \left(n +3\right)}{11+n}-\frac{2 \left(41453+10080 n \right) a \! \left(n +4\right)}{11+n}+\frac{\left(111043+22471 n \right) a \! \left(n +5\right)}{11+n}-\frac{4 \left(25509+4403 n \right) a \! \left(n +6\right)}{11+n}+\frac{2 \left(32779+4923 n \right) a \! \left(n +7\right)}{11+n}-\frac{8 \left(3682+489 n \right) a \! \left(n +8\right)}{11+n}+\frac{\left(9067+1079 n \right) a \! \left(n +9\right)}{11+n}-\frac{2 \left(909+98 n \right) a \! \left(n +10\right)}{11+n}, \quad n \geq 12\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 22\)
\(\displaystyle a \! \left(5\right) = 87\)
\(\displaystyle a \! \left(6\right) = 352\)
\(\displaystyle a \! \left(7\right) = 1434\)
\(\displaystyle a \! \left(8\right) = 5861\)
\(\displaystyle a \! \left(9\right) = 24019\)
\(\displaystyle a \! \left(10\right) = 98677\)
\(\displaystyle a \! \left(11\right) = 406291\)
\(\displaystyle a \! \left(n +12\right) = -\frac{48 \left(3+2 n \right) a \! \left(n \right)}{11+n}+\frac{3 \left(71+7 n \right) a \! \left(11+n \right)}{11+n}+\frac{1032 \left(n +2\right) a \! \left(n +1\right)}{11+n}-\frac{4 \left(3089+1187 n \right) a \! \left(n +2\right)}{11+n}+\frac{396 \left(103+31 n \right) a \! \left(n +3\right)}{11+n}-\frac{2 \left(41453+10080 n \right) a \! \left(n +4\right)}{11+n}+\frac{\left(111043+22471 n \right) a \! \left(n +5\right)}{11+n}-\frac{4 \left(25509+4403 n \right) a \! \left(n +6\right)}{11+n}+\frac{2 \left(32779+4923 n \right) a \! \left(n +7\right)}{11+n}-\frac{8 \left(3682+489 n \right) a \! \left(n +8\right)}{11+n}+\frac{\left(9067+1079 n \right) a \! \left(n +9\right)}{11+n}-\frac{2 \left(909+98 n \right) a \! \left(n +10\right)}{11+n}, \quad n \geq 12\)
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 "Col Placements Tracked Fusion" and has 40 rules.
Found on April 26, 2021.Finding the specification took 18 seconds.
Copy 40 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_{4}\! \left(x \right)\\
F_{3}\! \left(x \right) &= x\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{3}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\
F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{7}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= y x\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{11}\! \left(x , y\right) &= -\frac{-y F_{7}\! \left(x , y\right)+F_{7}\! \left(x , 1\right)}{-1+y}\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{13}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{14}\! \left(x , y\right)+F_{15}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{14}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x , 1\right)\\
F_{17}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y\right)+F_{20}\! \left(x , y\right)+F_{34}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{19}\! \left(x , y\right) &= -\frac{-y F_{17}\! \left(x , y\right)+F_{17}\! \left(x , 1\right)}{-1+y}\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{15}\! \left(x \right)+F_{20}\! \left(x , y\right)+F_{22}\! \left(x \right)+F_{29}\! \left(x , y\right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{23}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{1}\! \left(x \right)+F_{15}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{3}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{26}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{26}\! \left(x \right) F_{3}\! \left(x \right) F_{31}\! \left(x , y\right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{32}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{31}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{34}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{35}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{34}\! \left(x , y\right)+F_{36}\! \left(x , y\right)\\
F_{36}\! \left(x , y\right) &= F_{37}\! \left(x , y\right)\\
F_{37}\! \left(x , y\right) &= F_{31}\! \left(x , y\right) F_{4}\! \left(x \right) F_{9}\! \left(x , y\right)\\
F_{38}\! \left(x \right) &= F_{3}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{38}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Row And Col Placements Tracked Fusion" and has 47 rules.
Found on April 26, 2021.Finding the specification took 211 seconds.
Copy 47 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_{4}\! \left(x \right)\\
F_{3}\! \left(x \right) &= x\\
F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{46}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{3}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\
F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{7}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= y x\\
F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{11}\! \left(x , y\right) &= -\frac{-y F_{7}\! \left(x , y\right)+F_{7}\! \left(x , 1\right)}{-1+y}\\
F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x \right)+F_{44}\! \left(x , y\right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{3}\! \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_{25}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{22}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{27}\! \left(x \right) &= 0\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{25}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{21}\! \left(x \right) F_{3}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{3}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{3}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{22}\! \left(x \right) F_{3}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{4}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{3}\! \left(x \right) F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{22}\! \left(x \right) F_{9}\! \left(x , y\right)\\
F_{46}\! \left(x \right) &= F_{14}\! \left(x \right) F_{3}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Insertion Row And Col Placements Tracked Fusion" and has 118 rules.
Found on April 26, 2021.Finding the specification took 258 seconds.
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Copy 118 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_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
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_{23}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{26}\! \left(x \right) F_{8}\! \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_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x , 1\right)\\
F_{40}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right) F_{49}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)+F_{44}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{2}\! \left(x \right) F_{46}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{47}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= F_{46}\! \left(x , y\right) F_{49}\! \left(x , y\right)\\
F_{49}\! \left(x , y\right) &= y x\\
F_{50}\! \left(x , y\right) &= F_{105}\! \left(x , y\right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{58}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{57}\! \left(x \right) &= 0\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{64}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{76}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{8}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{8}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{66}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{57}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{15}\! \left(x \right) F_{8}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{14}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{14}\! \left(x \right) F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{50}\! \left(x , 1\right)\\
F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)\\
F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right)+F_{112}\! \left(x , y\right)\\
F_{108}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{111}\! \left(x , y\right)\\
F_{109}\! \left(x , y\right) &= F_{110}\! \left(x , y\right)\\
F_{110}\! \left(x , y\right) &= F_{47}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{111}\! \left(x , y\right) &= -\frac{y \left(F_{41}\! \left(x , 1\right)-F_{41}\! \left(x , y\right)\right)}{-1+y}\\
F_{112}\! \left(x , y\right) &= F_{113}\! \left(x , y\right)+F_{115}\! \left(x , y\right)\\
F_{113}\! \left(x , y\right) &= F_{114}\! \left(x , y\right)\\
F_{114}\! \left(x , y\right) &= F_{47}\! \left(x , y\right) F_{89}\! \left(x \right)\\
F_{115}\! \left(x , y\right) &= -\frac{y \left(F_{105}\! \left(x , 1\right)-F_{105}\! \left(x , y\right)\right)}{-1+y}\\
F_{116}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{117}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{103}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Col Placements Tracked Fusion" and has 57 rules.
Found on April 26, 2021.Finding the specification took 96 seconds.
Copy 57 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_{4}\! \left(x \right)\\
F_{3}\! \left(x \right) &= x\\
F_{4}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{3}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\
F_{8}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= y x\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{12}\! \left(x , y\right) &= -\frac{-y F_{8}\! \left(x , y\right)+F_{8}\! \left(x , 1\right)}{-1+y}\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{14}\! \left(x \right) &= 0\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x , 1\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{14}\! \left(x \right)+F_{20}\! \left(x , y\right)+F_{21}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{22}\! \left(x , y\right) &= -\frac{-y F_{19}\! \left(x , y\right)+F_{19}\! \left(x , 1\right)}{-1+y}\\
F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{3}\! \left(x \right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{54}\! \left(x , y\right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{29}\! \left(x \right)+F_{4}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{3}\! \left(x \right) F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{3}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{3}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{3}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{3}\! \left(x \right) F_{32}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{3}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{3}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{3}\! \left(x \right) F_{33}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{4}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{3}\! \left(x \right) F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{54}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)\\
F_{55}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{24}\! \left(x , y\right) F_{33}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{25}\! \left(x \right) F_{3}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Insertion Point Placements Tracked Fusion" and has 143 rules.
Found on April 26, 2021.Finding the specification took 145 seconds.
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Copy 143 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_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
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_{23}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{26}\! \left(x \right) F_{8}\! \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_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x , 1\right)\\
F_{40}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)\\
F_{42}\! \left(x , y\right) &= F_{43}\! \left(x , y\right) F_{49}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)+F_{44}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{2}\! \left(x \right) F_{46}\! \left(x , y\right)\\
F_{46}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{47}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= F_{46}\! \left(x , y\right) F_{49}\! \left(x , y\right)\\
F_{49}\! \left(x , y\right) &= y x\\
F_{50}\! \left(x , y\right) &= F_{128}\! \left(x , y\right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{58}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{57}\! \left(x \right) &= 0\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{64}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{76}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{8}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{8}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{66}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{8}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{15}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{2}\! \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_{8}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{108}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{104}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{115}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{113}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{15}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{15} \left(x \right)^{2} F_{14}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{126}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{50}\! \left(x , 1\right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{15} \left(x \right)^{2} F_{14}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{128}\! \left(x , y\right) &= F_{129}\! \left(x , y\right)\\
F_{129}\! \left(x , y\right) &= F_{130}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{130}\! \left(x , y\right) &= F_{131}\! \left(x , y\right)+F_{135}\! \left(x , y\right)\\
F_{131}\! \left(x , y\right) &= F_{132}\! \left(x , y\right)+F_{134}\! \left(x , y\right)\\
F_{132}\! \left(x , y\right) &= F_{133}\! \left(x , y\right)\\
F_{133}\! \left(x , y\right) &= F_{47}\! \left(x , y\right) F_{55}\! \left(x \right)\\
F_{134}\! \left(x , y\right) &= -\frac{y \left(F_{41}\! \left(x , 1\right)-F_{41}\! \left(x , y\right)\right)}{-1+y}\\
F_{135}\! \left(x , y\right) &= F_{136}\! \left(x , y\right)+F_{138}\! \left(x , y\right)\\
F_{136}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)\\
F_{137}\! \left(x , y\right) &= F_{47}\! \left(x , y\right) F_{89}\! \left(x \right)\\
F_{138}\! \left(x , y\right) &= -\frac{y \left(F_{128}\! \left(x , 1\right)-F_{128}\! \left(x , y\right)\right)}{-1+y}\\
F_{139}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{140}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{142}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{91}\! \left(x \right)\\
\end{align*}\)