###### Av(1243, 1342)
Generating Function
$$\displaystyle -\frac{x}{2}+\frac{3}{2}-\frac{\sqrt{x^{2}-6 x +1}}{2}$$
Counting Sequence
1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, ...
Implicit Equation for the Generating Function
$$\displaystyle F \left(x \right)^{2}+\left(x -3\right) F \! \left(x \right)+2 = 0$$
Recurrence
$$\displaystyle a \! \left(0\right) = 1$$
$$\displaystyle a \! \left(1\right) = 1$$
$$\displaystyle a \! \left(n +2\right) = -\frac{\left(n -1\right) a \! \left(n \right)}{n +2}+\frac{3 \left(2 n +1\right) a \! \left(n +1\right)}{n +2}, \quad n \geq 2$$
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 And Row Placements Tracked Fusion" and has 18 rules.

Found on April 25, 2021.

Finding the specification took 223 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_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{0}\! \left(x \right) F_{15}\! \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_{16}\! \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_{15}\! \left(x \right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x \right) F_{7}\! \left(x \right) F_{8}\! \left(x , y\right)\\ F_{15}\! \left(x \right) &= x\\ F_{16}\! \left(x , y\right) &= F_{15}\! \left(x \right) F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= \frac{y F_{8}\! \left(x , y\right)-F_{8}\! \left(x , 1\right)}{-1+y}\\ \end{align*}

### This specification was found using the strategy pack "Point And Col Placements Tracked Fusion" and has 53 rules.

Found on April 25, 2021.

Finding the specification took 1263 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_{7}\! \left(x , 1\right)\\ F_{7}\! \left(x , y_{0}\right) &= F_{8}\! \left(x , y_{0}\right)\\ F_{8}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}\right)+F_{12}\! \left(x , y_{0}\right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= 0\\ F_{10}\! \left(x , y_{0}\right) &= F_{11}\! \left(x , y_{0}\right) F_{3}\! \left(x \right)\\ F_{11}\! \left(x , y_{0}\right) &= -\frac{y_{0} \left(F_{8}\! \left(x , 1\right)-F_{8}\! \left(x , y_{0}\right)\right)}{-1+y_{0}}\\ F_{12}\! \left(x , y_{0}\right) &= F_{13}\! \left(x , y_{0}\right) F_{14}\! \left(x , y_{0}\right)\\ F_{13}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{14}\! \left(x , y_{0}\right) &= F_{15}\! \left(x , y_{0}, 1\right)\\ F_{15}\! \left(x , y_{0}, y_{1}\right) &= F_{16}\! \left(x , y_{0}, y_{0} y_{1}\right)\\ F_{16}\! \left(x , y_{0}, y_{1}\right) &= F_{17}\! \left(x , y_{0}, y_{1}\right)+F_{25}\! \left(x , y_{0}, y_{1}\right)\\ F_{17}\! \left(x , y_{0}, y_{1}\right) &= F_{18}\! \left(x , y_{0}, y_{1}\right)\\ F_{18}\! \left(x , y_{0}, y_{1}\right) &= F_{19}\! \left(x , y_{0}\right) F_{22}\! \left(x , y_{1}\right)\\ F_{19}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x , y_{0}\right)\\ F_{20}\! \left(x , y_{0}\right) &= F_{21}\! \left(x , y_{0}\right)\\ F_{21}\! \left(x , y_{0}\right) &= F_{13}\! \left(x , y_{0}\right) F_{19}\! \left(x , y_{0}\right) F_{22}\! \left(x , y_{0}\right)\\ F_{22}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x , y_{0}\right)\\ F_{23}\! \left(x , y_{0}\right) &= F_{24}\! \left(x , y_{0}\right)\\ F_{24}\! \left(x , y_{0}\right) &= F_{13}\! \left(x , y_{0}\right) F_{22}\! \left(x , y_{0}\right)\\ F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{25}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{26}\! \left(x , y_{0}, y_{1}\right) &= F_{27}\! \left(x , y_{1}\right)+F_{37}\! \left(x , y_{0}, y_{1}\right)\\ F_{27}\! \left(x , y_{0}\right) &= F_{28}\! \left(x , y_{0}\right)\\ F_{28}\! \left(x , y_{0}\right) &= F_{22}\! \left(x , y_{0}\right) F_{29}\! \left(x , y_{0}\right) F_{3}\! \left(x \right)\\ F_{29}\! \left(x , y_{0}\right) &= F_{30}\! \left(x , 1, y_{0}\right)\\ F_{30}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x , y_{0}, y_{1}\right)+F_{33}\! \left(x , y_{0}, y_{1}\right)+F_{36}\! \left(x , y_{1}, y_{0}\right)\\ F_{31}\! \left(x , y_{0}, y_{1}\right) &= F_{3}\! \left(x \right) F_{32}\! \left(x , y_{0}, y_{1}\right)\\ F_{32}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{0} F_{30}\! \left(x , y_{0}, y_{1}\right)+F_{30}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\ F_{33}\! \left(x , y_{0}, y_{1}\right) &= F_{34}\! \left(x , y_{0}, 1, y_{1}\right)\\ F_{34}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{13}\! \left(x , y_{1}\right) F_{35}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{35}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{0} F_{15}\! \left(x , y_{0} y_{1}, 1\right)-y_{2} F_{15}\! \left(x , y_{0} y_{1}, \frac{y_{2}}{y_{0}}\right)}{-y_{2}+y_{0}}\\ F_{36}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{0}\right) F_{16}\! \left(x , y_{1}, y_{0}\right)\\ F_{37}\! \left(x , y_{0}, y_{1}\right) &= F_{38}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{38}\! \left(x , y_{0}, y_{1}\right) &= 2 F_{9}\! \left(x \right)+F_{39}\! \left(x , y_{0}, y_{1}\right)+F_{49}\! \left(x , y_{0}, y_{1}\right)\\ F_{39}\! \left(x , y_{0}, y_{1}\right) &= F_{40}\! \left(x , y_{0}, 1, y_{1}\right)\\ F_{40}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{3}\! \left(x \right) F_{41}\! \left(x , y_{0}, y_{1}, y_{2}\right)\\ F_{41}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{42}\! \left(x , y_{0} y_{1}, y_{1} y_{2}\right)\\ F_{42}\! \left(x , y_{0}, y_{1}\right) &= F_{43}\! \left(x , y_{0}, y_{1}\right)\\ F_{43}\! \left(x , y_{0}, y_{1}\right) &= F_{44}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{44}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{1} \left(F_{45}\! \left(x , y_{0}, y_{2}\right)-F_{45}\! \left(x , y_{1}, y_{2}\right)\right)}{-y_{1}+y_{0}}\\ F_{46}\! \left(x , y_{0}, y_{1}\right) &= F_{45}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{46}\! \left(x , y_{0}, y_{1}\right) &= F_{37}\! \left(x , y_{0}, y_{1}\right)+F_{47}\! \left(x , y_{0}, y_{1}\right)\\ F_{47}\! \left(x , y_{0}, y_{1}\right) &= F_{48}\! \left(x , y_{0}, y_{1}\right)\\ F_{48}\! \left(x , y_{0}, y_{1}\right) &= F_{20}\! \left(x , y_{1}\right) F_{22}\! \left(x , y_{1}\right)\\ F_{50}\! \left(x , y_{0}, y_{1}\right) &= F_{49}\! \left(x , y_{0} y_{1}, y_{1}\right)\\ F_{50}\! \left(x , y_{0}, y_{1}\right) &= F_{13}\! \left(x , y_{1}\right) F_{51}\! \left(x , y_{0}, y_{1}\right)\\ F_{51}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{52}\! \left(x , y_{0} y_{1}, 1\right)-F_{52}\! \left(x , y_{0} y_{1}, \frac{1}{y_{0}}\right)}{-1+y_{0}}\\ F_{52}\! \left(x , y_{0}, y_{1}\right) &= F_{25}\! \left(x , y_{0}, y_{0} y_{1}\right)\\ \end{align*}

### This specification was found using the strategy pack "Col Placements Tracked Fusion Req Corrob" and has 47 rules.

Found on April 25, 2021.

Finding the specification took 208 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_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y_{0}\right)+F_{8}\! \left(x , y_{0}\right)\\ F_{6}\! \left(x , y_{0}\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y_{0}\right)\\ F_{7}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{5}\! \left(x , y_{0}\right)+F_{5}\! \left(x , 1\right)}{-1+y_{0}}\\ F_{8}\! \left(x , y_{0}\right) &= F_{10}\! \left(x , y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\ F_{9}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{10}\! \left(x , y_{0}\right) &= F_{11}\! \left(x , y_{0}, 1\right)\\ F_{11}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , 1, y_{0}, y_{1}\right)\\ F_{12}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{35}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{39}\! \left(x , y_{1} y_{2}\right)\\ F_{13}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{3}\! \left(x \right)\\ F_{14}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{15}\! \left(x , 1, y_{0}, y_{1}, y_{2}\right)\\ F_{15}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{16}\! \left(x , y_{0}, y_{1} y_{2}, y_{2} y_{3}\right)\\ F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{16}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\ F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{29}\! \left(x , y_{1}, y_{2}, y_{0}\right)+F_{33}\! \left(x , y_{2}, y_{0}\right)\\ F_{18}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{3}\! \left(x \right)\\ F_{19}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{-y_{0} F_{17}\! \left(x , y_{0}, y_{1}, y_{2}\right)+F_{17}\! \left(x , 1, y_{1}, y_{2}\right)}{-1+y_{0}}\\ F_{20}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\ F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{21}\! \left(x , y_{0}, y_{1}, y_{0} y_{2}\right)\\ F_{22}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{23}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{9}\! \left(x , y_{0}\right)\\ F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{23}\! \left(x , y_{0}, y_{0} y_{1}, y_{2}\right)\\ F_{24}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{25}\! \left(x , y_{0}, y_{1}, y_{0} y_{2}\right)\\ F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{25}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\ F_{26}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= \frac{y_{1} y_{2} F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right)-F_{27}\! \left(x , y_{0}, \frac{1}{y_{2}}, y_{2}\right)}{y_{1} y_{2}-1}\\ F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{28}\! \left(x , 1, y_{0}, y_{1}, y_{2}\right)\\ F_{28}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= \frac{F_{12}\! \left(x , y_{0}, y_{1}, y_{2} y_{3}\right) y_{1} y_{2}-F_{12}\! \left(x , y_{0}, y_{1}, \frac{y_{3}}{y_{1}}\right)}{y_{1} y_{2}-1}\\ F_{29}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{30}\! \left(x , y_{2}, y_{0}, y_{1}\right) F_{9}\! \left(x , y_{1}\right)\\ F_{30}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= -\frac{y_{1} y_{2} F_{31}\! \left(x , 1, y_{1}, y_{2}\right)-y_{0} F_{31}\! \left(x , \frac{y_{0}}{y_{1} y_{2}}, y_{1}, y_{2}\right)}{-y_{1} y_{2}+y_{0}}\\ F_{31}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{32}\! \left(x , y_{0} y_{2}, y_{1}, y_{2}\right)\\ F_{32}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{28}\! \left(x , y_{0}, y_{1}, 1, y_{2}\right)\\ F_{33}\! \left(x , y_{0}, y_{1}\right) &= F_{34}\! \left(x , y_{1}, y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\ F_{34}\! \left(x , y_{0}, y_{1}\right) &= F_{12}\! \left(x , y_{0}, 1, y_{1}\right)\\ F_{35}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{36}\! \left(x , y_{0}, y_{1}, y_{2}\right) F_{9}\! \left(x , y_{1}\right)\\ F_{36}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{37}\! \left(x , y_{0}, y_{1}, 1, y_{2}\right)\\ F_{37}\! \left(x , y_{0}, y_{1}, y_{2}, y_{3}\right) &= F_{38}\! \left(x , y_{0} y_{1}, y_{2}, y_{1} y_{3}\right)\\ F_{27}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{38}\! \left(x , y_{0}, y_{1} y_{2}, y_{2}\right)\\ F_{39}\! \left(x , y_{0}\right) &= F_{40}\! \left(x , y_{0}\right) F_{9}\! \left(x , y_{0}\right)\\ F_{40}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{39}\! \left(x , y_{0}\right)+F_{41}\! \left(x , y_{0}\right)\\ F_{41}\! \left(x , y_{0}\right) &= F_{3}\! \left(x \right) F_{42}\! \left(x , y_{0}\right)\\ F_{42}\! \left(x , y_{0}\right) &= F_{43}\! \left(x , 1, y_{0}\right)\\ F_{43}\! \left(x , y_{0}, y_{1}\right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x , y_{1}, y_{0}\right)+F_{44}\! \left(x , y_{0}, y_{1}\right)+F_{46}\! \left(x , y_{0}, y_{1}\right)\\ F_{44}\! \left(x , y_{0}, y_{1}\right) &= F_{3}\! \left(x \right) F_{45}\! \left(x , y_{0}, y_{1}\right)\\ F_{45}\! \left(x , y_{0}, y_{1}\right) &= -\frac{-y_{0} F_{43}\! \left(x , y_{0}, y_{1}\right)+F_{43}\! \left(x , 1, y_{1}\right)}{-1+y_{0}}\\ F_{46}\! \left(x , y_{0}, y_{1}\right) &= F_{35}\! \left(x , y_{0}, 1, y_{1}\right)\\ \end{align*}

### This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Isolated Req Corrob" and has 26 rules.

Found on April 25, 2021.

Finding the specification took 123 seconds.

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\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{0}\! \left(x \right) F_{16}\! \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_{4}\! \left(x \right)+F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{15}\! \left(x , y\right)\\ F_{11}\! \left(x \right) &= 0\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{14}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{9}\! \left(x , y\right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{16}\! \left(x \right) F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)+F_{21}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{7}\! \left(x \right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x , y\right) F_{25}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= -\frac{y \left(F_{9}\! \left(x , 1\right)-F_{9}\! \left(x , y\right)\right)}{-1+y}\\ \end{align*}

### This specification was found using the strategy pack "Point And Row And Col Placements Tracked Fusion Isolated Req Corrob" and has 29 rules.

Found on April 25, 2021.

Finding the specification took 4267 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_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{0}\! \left(x \right) F_{19}\! \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_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{13}\! \left(x \right) &= 0\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{16}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= y x\\ F_{16}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= x\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{19}\! \left(x \right) F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{7}\! \left(x \right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{25}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= -\frac{y \left(F_{11}\! \left(x , 1\right)-F_{11}\! \left(x , y\right)\right)}{-1+y}\\ \end{align*}