Av(1324, 2341, 4123)
Generating Function
\(\displaystyle \frac{-4 \left(x -1\right)^{6} \left(x -\frac{1}{2}\right)^{2} \sqrt{1-4 x}-8 x^{9}+22 x^{8}-38 x^{7}+91 x^{6}-152 x^{5}+157 x^{4}-104 x^{3}+43 x^{2}-10 x +1}{2 x \left(2 x -1\right)^{2} \left(x -1\right)^{6}}\)
Counting Sequence
1, 1, 2, 6, 21, 72, 230, 701, 2113, 6475, 20468, 66969, 226027, 782276, 2760094, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{4} \left(x -1\right)^{12} F \left(x
\right)^{2}+\left(8 x^{9}-22 x^{8}+38 x^{7}-91 x^{6}+152 x^{5}-157 x^{4}+104 x^{3}-43 x^{2}+10 x -1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)+16 x^{17}-72 x^{16}+45 x^{15}+738 x^{14}-4324 x^{13}+14094 x^{12}-31391 x^{11}+51202 x^{10}-63478 x^{9}+61033 x^{8}-45896 x^{7}+26968 x^{6}-12260 x^{5}+4226 x^{4}-1067 x^{3}+186 x^{2}-20 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 72\)
\(\displaystyle a \! \left(6\right) = 230\)
\(\displaystyle a \! \left(7\right) = 701\)
\(\displaystyle a \! \left(8\right) = 2113\)
\(\displaystyle a \! \left(9\right) = 6475\)
\(\displaystyle a \! \left(10\right) = 20468\)
\(\displaystyle a \! \left(11\right) = 66969\)
\(\displaystyle a \! \left(n +4\right) = -\frac{16 \left(2 n +1\right) a \! \left(n \right)}{n +5}+\frac{8 \left(7 n +11\right) a \! \left(n +1\right)}{n +5}-\frac{12 \left(3 n +8\right) a \! \left(n +2\right)}{n +5}+\frac{2 \left(5 n +19\right) a \! \left(n +3\right)}{n +5}+\frac{3 n^{6}-59 n^{5}+55 n^{4}+2895 n^{3}-9298 n^{2}+1964 n +1560}{120 n +600}, \quad n \geq 12\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 72\)
\(\displaystyle a \! \left(6\right) = 230\)
\(\displaystyle a \! \left(7\right) = 701\)
\(\displaystyle a \! \left(8\right) = 2113\)
\(\displaystyle a \! \left(9\right) = 6475\)
\(\displaystyle a \! \left(10\right) = 20468\)
\(\displaystyle a \! \left(11\right) = 66969\)
\(\displaystyle a \! \left(n +4\right) = -\frac{16 \left(2 n +1\right) a \! \left(n \right)}{n +5}+\frac{8 \left(7 n +11\right) a \! \left(n +1\right)}{n +5}-\frac{12 \left(3 n +8\right) a \! \left(n +2\right)}{n +5}+\frac{2 \left(5 n +19\right) a \! \left(n +3\right)}{n +5}+\frac{3 n^{6}-59 n^{5}+55 n^{4}+2895 n^{3}-9298 n^{2}+1964 n +1560}{120 n +600}, \quad n \geq 12\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 253 rules.
Found on July 23, 2021.Finding the specification took 5 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_{12}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{115}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{114}\! \left(x \right)+F_{48}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{12}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{12}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{32}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{12}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{12}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x , 1\right)\\
F_{50}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{102}\! \left(x , y\right)+F_{104}\! \left(x , y\right)+F_{112}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\
F_{51}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{52}\! \left(x , y\right)\\
F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\
F_{53}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{62}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)+F_{9}\! \left(x \right)\\
F_{55}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)+F_{60}\! \left(x , y\right)\\
F_{56}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)\\
F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right) F_{59}\! \left(x , y\right)\\
F_{58}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{56}\! \left(x , y\right)\\
F_{59}\! \left(x , y\right) &= y x\\
F_{60}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{55}\! \left(x , y\right)\\
F_{62}\! \left(x , y\right) &= F_{13}\! \left(x \right)+F_{63}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{59}\! \left(x , y\right) F_{66}\! \left(x , y\right)\\
F_{66}\! \left(x , y\right) &= F_{10}\! \left(x \right)+F_{64}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{63}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{70}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{23}\! \left(x \right)+F_{71}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= F_{72}\! \left(x , y\right)+F_{77}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{15}\! \left(x \right)+F_{73}\! \left(x , y\right)+F_{75}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{74}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{59}\! \left(x , y\right) F_{76}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{10}\! \left(x \right)+F_{72}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= 2 F_{15}\! \left(x \right)+F_{78}\! \left(x , y\right)+F_{80}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{79}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)+F_{77}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{71}\! \left(x , y\right)\\
F_{81}\! \left(x , y\right) &= F_{28}\! \left(x \right)+F_{82}\! \left(x , y\right)\\
F_{82}\! \left(x , y\right) &= F_{83}\! \left(x , y\right)+F_{88}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= 2 F_{15}\! \left(x \right)+F_{84}\! \left(x , y\right)+F_{86}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{85}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)+F_{83}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{59}\! \left(x , y\right) F_{87}\! \left(x , y\right)\\
F_{87}\! \left(x , y\right) &= F_{24}\! \left(x \right)+F_{83}\! \left(x , y\right)\\
F_{88}\! \left(x , y\right) &= 3 F_{15}\! \left(x \right)+F_{89}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\
F_{89}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{90}\! \left(x , y\right)\\
F_{90}\! \left(x , y\right) &= F_{67}\! \left(x , y\right)+F_{88}\! \left(x , y\right)\\
F_{91}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{92}\! \left(x , y\right)\\
F_{92}\! \left(x , y\right) &= F_{83}\! \left(x , y\right)+F_{93}\! \left(x , y\right)\\
F_{93}\! \left(x , y\right) &= 3 F_{15}\! \left(x \right)+F_{91}\! \left(x , y\right)+F_{94}\! \left(x , y\right)\\
F_{94}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{95}\! \left(x , y\right)\\
F_{95}\! \left(x , y\right) &= F_{96}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\
F_{96}\! \left(x , y\right) &= F_{97}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{98}\! \left(x , y\right)\\
F_{98}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)+F_{96}\! \left(x , y\right)\\
F_{99}\! \left(x , y\right) &= 3 F_{15}\! \left(x \right)+F_{100}\! \left(x , y\right)+F_{94}\! \left(x , y\right)\\
F_{100}\! \left(x , y\right) &= F_{101}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{101}\! \left(x , y\right) &= F_{83}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\
F_{102}\! \left(x , y\right) &= F_{103}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{103}\! \left(x , y\right) &= -\frac{-y F_{50}\! \left(x , y\right)+F_{50}\! \left(x , 1\right)}{-1+y}\\
F_{104}\! \left(x , y\right) &= F_{105}\! \left(x , y\right) F_{59}\! \left(x , y\right)\\
F_{105}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{104}\! \left(x , y\right)+F_{106}\! \left(x , y\right)+F_{110}\! \left(x , y\right)\\
F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right)+F_{109}\! \left(x , y\right)\\
F_{108}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\
F_{109}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\
F_{110}\! \left(x , y\right) &= F_{111}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{111}\! \left(x , y\right) &= -\frac{-y F_{105}\! \left(x , y\right)+F_{105}\! \left(x , 1\right)}{-1+y}\\
F_{112}\! \left(x , y\right) &= F_{113}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{113}\! \left(x , y\right) &= -\frac{-y F_{105}\! \left(x , y\right)+F_{105}\! \left(x , 1\right)}{-1+y}\\
F_{114}\! \left(x \right) &= F_{104}\! \left(x , 1\right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{142}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{125}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{12}\! \left(x \right) F_{121}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{12}\! \left(x \right) F_{122}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{129}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{12}\! \left(x \right) F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{126}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{132}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{12}\! \left(x \right) F_{131}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{129}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{12}\! \left(x \right) F_{133}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{137}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{12}\! \left(x \right) F_{134}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{140}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{139}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{12}\! \left(x \right) F_{137}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{197}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{155}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{12}\! \left(x \right) F_{146}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{148}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{149}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{150}\! \left(x \right)+F_{154}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{12}\! \left(x \right) F_{151}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{153}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{149}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{12}\! \left(x \right) F_{144}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{156}\! \left(x \right)+F_{191}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{12}\! \left(x \right) F_{157}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{166}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{159}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{160}\! \left(x \right)+F_{161}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{119}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{12}\! \left(x \right) F_{162}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{164}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{15}\! \left(x \right)+F_{161}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{167}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{168}\! \left(x \right)+F_{172}\! \left(x \right)+F_{184}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{12}\! \left(x \right) F_{169}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{170}\! \left(x \right)+F_{171}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{159}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{167}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{12}\! \left(x \right) F_{173}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{174}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{12}\! \left(x \right) F_{175}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{176}\! \left(x \right)+F_{178}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{177}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{160}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{173}\! \left(x \right)+F_{179}\! \left(x \right)\\
F_{179}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{172}\! \left(x \right)+F_{180}\! \left(x \right)\\
F_{180}\! \left(x \right) &= F_{12}\! \left(x \right) F_{181}\! \left(x \right)\\
F_{181}\! \left(x \right) &= F_{182}\! \left(x \right)+F_{183}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{177}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{179}\! \left(x \right)\\
F_{184}\! \left(x \right) &= F_{12}\! \left(x \right) F_{185}\! \left(x \right)\\
F_{185}\! \left(x \right) &= F_{186}\! \left(x \right)+F_{188}\! \left(x \right)\\
F_{186}\! \left(x \right) &= F_{187}\! \left(x \right)\\
F_{187}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{188}\! \left(x \right) &= F_{189}\! \left(x \right)\\
F_{189}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{184}\! \left(x \right)+F_{190}\! \left(x \right)\\
F_{190}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{191}\! \left(x \right) &= F_{12}\! \left(x \right) F_{192}\! \left(x \right)\\
F_{192}\! \left(x \right) &= F_{193}\! \left(x \right)+F_{194}\! \left(x \right)\\
F_{193}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{194}\! \left(x \right) &= F_{195}\! \left(x \right)+F_{196}\! \left(x \right)\\
F_{195}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{191}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{196}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{197}\! \left(x \right) &= F_{198}\! \left(x \right)+F_{221}\! \left(x \right)\\
F_{198}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{199}\! \left(x \right)+F_{219}\! \left(x \right)\\
F_{199}\! \left(x \right) &= F_{12}\! \left(x \right) F_{200}\! \left(x \right)\\
F_{200}\! \left(x \right) &= F_{201}\! \left(x \right)+F_{204}\! \left(x \right)\\
F_{201}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{202}\! \left(x \right)\\
F_{202}\! \left(x \right) &= F_{203}\! \left(x \right)\\
F_{203}\! \left(x \right) &= F_{12}\! \left(x \right) F_{126}\! \left(x \right)\\
F_{204}\! \left(x \right) &= F_{198}\! \left(x \right)+F_{205}\! \left(x \right)\\
F_{205}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{206}\! \left(x \right)+F_{210}\! \left(x \right)\\
F_{206}\! \left(x \right) &= F_{12}\! \left(x \right) F_{207}\! \left(x \right)\\
F_{207}\! \left(x \right) &= F_{208}\! \left(x \right)+F_{209}\! \left(x \right)\\
F_{208}\! \left(x \right) &= F_{202}\! \left(x \right)\\
F_{209}\! \left(x \right) &= F_{205}\! \left(x \right)\\
F_{210}\! \left(x \right) &= F_{12}\! \left(x \right) F_{211}\! \left(x \right)\\
F_{211}\! \left(x \right) &= F_{212}\! \left(x \right)\\
F_{212}\! \left(x \right) &= F_{12}\! \left(x \right) F_{213}\! \left(x \right)\\
F_{213}\! \left(x \right) &= F_{201}\! \left(x \right)+F_{214}\! \left(x \right)\\
F_{214}\! \left(x \right) &= F_{211}\! \left(x \right)+F_{215}\! \left(x \right)\\
F_{215}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{210}\! \left(x \right)+F_{216}\! \left(x \right)\\
F_{216}\! \left(x \right) &= F_{12}\! \left(x \right) F_{217}\! \left(x \right)\\
F_{217}\! \left(x \right) &= F_{208}\! \left(x \right)+F_{218}\! \left(x \right)\\
F_{218}\! \left(x \right) &= F_{215}\! \left(x \right)\\
F_{219}\! \left(x \right) &= F_{12}\! \left(x \right) F_{220}\! \left(x \right)\\
F_{220}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{211}\! \left(x \right)\\
F_{221}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{222}\! \left(x \right)+F_{246}\! \left(x \right)+F_{248}\! \left(x \right)\\
F_{222}\! \left(x \right) &= F_{12}\! \left(x \right) F_{223}\! \left(x \right)\\
F_{223}\! \left(x \right) &= F_{224}\! \left(x \right)+F_{227}\! \left(x \right)\\
F_{224}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{225}\! \left(x \right)\\
F_{225}\! \left(x \right) &= F_{226}\! \left(x \right)\\
F_{226}\! \left(x \right) &= F_{12}\! \left(x \right) F_{129}\! \left(x \right)\\
F_{227}\! \left(x \right) &= F_{221}\! \left(x \right)+F_{228}\! \left(x \right)\\
F_{228}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{229}\! \left(x \right)+F_{233}\! \left(x \right)+F_{245}\! \left(x \right)\\
F_{229}\! \left(x \right) &= F_{12}\! \left(x \right) F_{230}\! \left(x \right)\\
F_{230}\! \left(x \right) &= F_{231}\! \left(x \right)+F_{232}\! \left(x \right)\\
F_{231}\! \left(x \right) &= F_{225}\! \left(x \right)\\
F_{232}\! \left(x \right) &= F_{228}\! \left(x \right)\\
F_{233}\! \left(x \right) &= F_{12}\! \left(x \right) F_{234}\! \left(x \right)\\
F_{234}\! \left(x \right) &= F_{235}\! \left(x \right)\\
F_{235}\! \left(x \right) &= F_{12}\! \left(x \right) F_{236}\! \left(x \right)\\
F_{236}\! \left(x \right) &= F_{237}\! \left(x \right)+F_{239}\! \left(x \right)\\
F_{237}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{238}\! \left(x \right)\\
F_{238}\! \left(x \right) &= F_{226}\! \left(x \right)\\
F_{239}\! \left(x \right) &= F_{234}\! \left(x \right)+F_{240}\! \left(x \right)\\
F_{240}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{233}\! \left(x \right)+F_{241}\! \left(x \right)\\
F_{241}\! \left(x \right) &= F_{12}\! \left(x \right) F_{242}\! \left(x \right)\\
F_{242}\! \left(x \right) &= F_{243}\! \left(x \right)+F_{244}\! \left(x \right)\\
F_{243}\! \left(x \right) &= F_{238}\! \left(x \right)\\
F_{244}\! \left(x \right) &= F_{240}\! \left(x \right)\\
F_{245}\! \left(x \right) &= 0\\
F_{246}\! \left(x \right) &= F_{12}\! \left(x \right) F_{247}\! \left(x \right)\\
F_{247}\! \left(x \right) &= F_{173}\! \left(x \right)+F_{234}\! \left(x \right)\\
F_{248}\! \left(x \right) &= F_{12}\! \left(x \right) F_{249}\! \left(x \right)\\
F_{249}\! \left(x \right) &= F_{250}\! \left(x \right)+F_{251}\! \left(x \right)\\
F_{250}\! \left(x \right) &= F_{187}\! \left(x \right)\\
F_{251}\! \left(x \right) &= F_{252}\! \left(x \right)\\
F_{252}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{190}\! \left(x \right)+F_{248}\! \left(x \right)\\
\end{align*}\)