Av(12435, 12453, 14235, 14253, 14325, 14352, 14523, 14532)
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Generating Function
\(\displaystyle \frac{4 x -5+\sqrt{8 x^{2}-8 x +1}}{4 x -4}\)
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Counting Sequence
1, 1, 2, 6, 24, 112, 568, 3032, 16768, 95200, 551616, 3248704, 19389824, 117021824, 712934784, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -2\right) F \left(x \right)^{2}+\left(-4 x +5\right) F \! \left(x \right)+x -3 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 112\)
\(\displaystyle a(6) = 568\)
\(\displaystyle a(7) = 3032\)
\(\displaystyle a(8) = 16768\)
\(\displaystyle a(9) = 95200\)
\(\displaystyle a(10) = 551616\)
\(\displaystyle a{\left(n + 3 \right)} = \frac{8 n a{\left(n \right)}}{n + 3} + \frac{3 \left(3 n + 5\right) a{\left(n + 2 \right)}}{n + 3} - \frac{4 \left(4 n + 3\right) a{\left(n + 1 \right)}}{n + 3}, \quad n \geq 11\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 107 rules.

Finding the specification took 8396 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_{38}\! \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 , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= x^{2} F_{7}\! \left(x , y\right)^{2} y^{2}+2 x^{2} F_{7}\! \left(x , y\right) y^{2}+x^{2} y^{2}-2 x F_{7}\! \left(x , y\right)^{2} y -3 x F_{7}\! \left(x , y\right) y -y x +2 F_{7}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{38}\! \left(x \right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{12}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= -\frac{y \left(F_{6}\! \left(x , 1\right)-F_{6}\! \left(x , y\right)\right)}{-1+y}\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{105}\! \left(x , y\right) F_{14}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{103}\! \left(x , y\right) F_{14}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(y x \right)\\ F_{18}\! \left(x \right) &= -F_{20}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{20}\! \left(x \right) &= x^{2} F_{20} \left(x \right)^{2}-2 x F_{20} \left(x \right)^{2}+F_{20}\! \left(x \right) x +2 F_{20}\! \left(x \right)-1\\ F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{41}\! \left(x \right)}\\ F_{22}\! \left(x \right) &= -F_{94}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{25}\! \left(x \right) &= \frac{F_{26}\! \left(x \right)}{F_{38}\! \left(x \right)}\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= -F_{39}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x , 1\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{58}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{35}\! \left(x \right)\\ F_{32}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= y x\\ F_{35}\! \left(x \right) &= F_{1}\! \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_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= x\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{38}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{47}\! \left(x \right) &= 0\\ F_{48}\! \left(x \right) &= F_{38}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{38}\! \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_{38}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{38}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{56}\! \left(x , y\right) &= F_{40}\! \left(x \right) F_{57}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= -x^{2} y^{2}+y x +x^{5} F_{57}\! \left(x , y\right)^{2} y^{2}-3 x^{4} F_{57}\! \left(x , y\right)^{2} y^{2}-2 x^{4} F_{57}\! \left(x , y\right)^{2} y -2 x^{4} F_{57}\! \left(x , y\right) y^{2}+3 x^{3} F_{57}\! \left(x , y\right)^{2} y^{2}+x^{4} F_{57}\! \left(x , y\right) y +6 x^{3} F_{57}\! \left(x , y\right)^{2} y +4 x^{3} F_{57}\! \left(x , y\right) y^{2}-x^{2} F_{57}\! \left(x , y\right)^{2} y^{2}+x^{3} F_{57}\! \left(x , y\right) y -6 x^{2} F_{57}\! \left(x , y\right)^{2} y -2 x^{2} F_{57}\! \left(x , y\right) y^{2}-5 x^{2} F_{57}\! \left(x , y\right) y +2 x F_{57}\! \left(x , y\right)^{2} y +3 x F_{57}\! \left(x , y\right) y +x^{3} F_{57}\! \left(x , y\right)-3 x^{2} F_{57}\! \left(x , y\right)+3 F_{57}\! \left(x , y\right) x -x^{3} y +x^{3} y^{2}-x +2 x^{2}-x^{3}\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{38}\! \left(x \right) F_{60}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)+F_{93}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{38}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{38}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{72}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{38}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{38}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{38}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{80}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{38}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{38}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{38}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{38}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{88}\! \left(x \right)+F_{89}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{38}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{38}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{38}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{92}\! \left(x , y\right) &= F_{57}\! \left(x , y\right) F_{64}\! \left(x \right)\\ F_{93}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{38}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{36}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{41}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= x^{5} F_{97} \left(x \right)^{2}-5 x^{4} F_{97} \left(x \right)^{2}-x^{4} F_{97}\! \left(x \right)+9 x^{3} F_{97} \left(x \right)^{2}+6 x^{3} F_{97}\! \left(x \right)-7 x^{2} F_{97} \left(x \right)^{2}-x^{3}-10 x^{2} F_{97}\! \left(x \right)+2 x F_{97} \left(x \right)^{2}+x^{2}+6 F_{97}\! \left(x \right) x\\ F_{98}\! \left(x , y\right) &= F_{99}\! \left(x , y\right)\\ F_{99}\! \left(x , y\right) &= F_{100}\! \left(x , y\right) F_{14}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\ F_{101}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\ F_{101}\! \left(x , y\right) &= F_{102}\! \left(x , y\right)\\ F_{102}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{103}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{104}\! \left(x , y\right)\\ F_{104}\! \left(x , y\right) &= x^{2} F_{104}\! \left(x , y\right)^{2} y^{2}-2 y x F_{104}\! \left(x , y\right)^{2}+x F_{104}\! \left(x , y\right) y +2 F_{104}\! \left(x , y\right)-1\\ F_{105}\! \left(x , y\right) &= -\frac{-y F_{106}\! \left(x , y\right)+F_{106}\! \left(x , 1\right)}{-1+y}\\ F_{106}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x , y\right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 21 rules.

Finding the specification took 1663 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= F_{6}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= x^{2} F_{6}\! \left(x , y\right)^{2} y^{2}-2 y x F_{6}\! \left(x , y\right)^{2}+x F_{6}\! \left(x , y\right) y +2 F_{6}\! \left(x , y\right)-1\\ F_{7}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{20}\! \left(x \right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{11}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= -\frac{-y F_{5}\! \left(x , y\right)+F_{5}\! \left(x , 1\right)}{-1+y}\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{13}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{19}\! \left(x , y\right) F_{6}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{6}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= x^{2} F_{18}\! \left(x , y\right)^{2} y^{2}+2 x^{2} F_{18}\! \left(x , y\right) y^{2}+x^{2} y^{2}-2 x F_{18}\! \left(x , y\right)^{2} y -3 x F_{18}\! \left(x , y\right) y -y x +2 F_{18}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= y x\\ F_{20}\! \left(x \right) &= x\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 106 rules.

Finding the specification took 6458 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_{38}\! \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 , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= x^{2} F_{7}\! \left(x , y\right)^{2} y^{2}+2 x^{2} F_{7}\! \left(x , y\right) y^{2}+x^{2} y^{2}-2 x F_{7}\! \left(x , y\right)^{2} y -3 x F_{7}\! \left(x , y\right) y -y x +2 F_{7}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{38}\! \left(x \right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{12}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= -\frac{y \left(F_{6}\! \left(x , 1\right)-F_{6}\! \left(x , y\right)\right)}{-1+y}\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{104}\! \left(x , y\right) F_{14}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{102}\! \left(x , y\right) F_{14}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{97}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(y x \right)\\ F_{18}\! \left(x \right) &= -F_{20}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{20}\! \left(x \right) &= x^{2} F_{20} \left(x \right)^{2}-2 x F_{20} \left(x \right)^{2}+F_{20}\! \left(x \right) x +2 F_{20}\! \left(x \right)-1\\ F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{41}\! \left(x \right)}\\ F_{22}\! \left(x \right) &= -F_{93}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{25}\! \left(x \right) &= \frac{F_{26}\! \left(x \right)}{F_{38}\! \left(x \right)}\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= -F_{39}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x , 1\right)\\ F_{29}\! \left(x , y\right) &= F_{30}\! \left(x , y\right)+F_{58}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{35}\! \left(x \right)\\ F_{32}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{34}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= y x\\ F_{35}\! \left(x \right) &= F_{1}\! \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_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= x\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{38}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{47}\! \left(x \right) &= 0\\ F_{48}\! \left(x \right) &= F_{38}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{38}\! \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_{38}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{38}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{56}\! \left(x , y\right) &= F_{40}\! \left(x \right) F_{57}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= -x^{3} y +x^{3} y^{2}+x^{3} F_{57}\! \left(x , y\right)-3 x^{2} F_{57}\! \left(x , y\right)+3 F_{57}\! \left(x , y\right) x -x^{2} y^{2}+y x -2 x^{4} F_{57}\! \left(x , y\right) y^{2}+x^{5} F_{57}\! \left(x , y\right)^{2} y^{2}-3 x^{4} F_{57}\! \left(x , y\right)^{2} y^{2}+6 x^{3} F_{57}\! \left(x , y\right)^{2} y +4 x^{3} F_{57}\! \left(x , y\right) y^{2}+3 x^{3} F_{57}\! \left(x , y\right)^{2} y^{2}+x^{4} F_{57}\! \left(x , y\right) y -2 x^{4} F_{57}\! \left(x , y\right)^{2} y +x^{3} F_{57}\! \left(x , y\right) y +2 x F_{57}\! \left(x , y\right)^{2} y -x^{2} F_{57}\! \left(x , y\right)^{2} y^{2}-2 x^{2} F_{57}\! \left(x , y\right) y^{2}-5 x^{2} F_{57}\! \left(x , y\right) y +3 x F_{57}\! \left(x , y\right) y -6 x^{2} F_{57}\! \left(x , y\right)^{2} y -x +2 x^{2}-x^{3}\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{38}\! \left(x \right) F_{60}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{38}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{38}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{73}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{38}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{38}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{38}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{38}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{38}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{38}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{87}\! \left(x \right)+F_{88}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{38}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{38}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{38}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{91}\! \left(x , y\right) &= F_{57}\! \left(x , y\right) F_{64}\! \left(x \right)\\ F_{92}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{38}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{36}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{41}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= x^{5} F_{96} \left(x \right)^{2}-5 x^{4} F_{96} \left(x \right)^{2}-x^{4} F_{96}\! \left(x \right)+9 x^{3} F_{96} \left(x \right)^{2}+6 x^{3} F_{96}\! \left(x \right)-7 x^{2} F_{96} \left(x \right)^{2}-x^{3}-10 x^{2} F_{96}\! \left(x \right)+2 x F_{96} \left(x \right)^{2}+x^{2}+6 F_{96}\! \left(x \right) x\\ F_{97}\! \left(x , y\right) &= F_{98}\! \left(x , y\right)\\ F_{98}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{34}\! \left(x , y\right) F_{99}\! \left(x , y\right)\\ F_{100}\! \left(x , y\right) &= F_{97}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\ F_{100}\! \left(x , y\right) &= F_{101}\! \left(x , y\right)\\ F_{101}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{102}\! \left(x , y\right) &= F_{103}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\ F_{103}\! \left(x , y\right) &= x^{2} F_{103}\! \left(x , y\right)^{2} y^{2}-2 y x F_{103}\! \left(x , y\right)^{2}+x F_{103}\! \left(x , y\right) y +2 F_{103}\! \left(x , y\right)-1\\ F_{104}\! \left(x , y\right) &= -\frac{-y F_{105}\! \left(x , y\right)+F_{105}\! \left(x , 1\right)}{-1+y}\\ F_{105}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x , y\right)\\ \end{align*}\)