Av(1234, 1243, 2134, 2341)
Generating Function
\(\displaystyle \frac{\left(-2 x^{5}+2 x^{4}+3 x^{3}-6 x^{2}+4 x -1\right) \sqrt{1-4 x}+6 x^{5}-6 x^{4}+x^{3}+4 x^{2}-4 x +1}{2 x \left(2 x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 20, 68, 229, 766, 2567, 8672, 29622, 102389, 357973, 1264690, 4509858, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \left(x
\right)^{2}-\left(2 x -1\right) \left(6 x^{5}-6 x^{4}+x^{3}+4 x^{2}-4 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+4 x^{10}-24 x^{8}+50 x^{7}-31 x^{6}-32 x^{5}+77 x^{4}-69 x^{3}+34 x^{2}-9 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 68\)
\(\displaystyle a \! \left(6\right) = 229\)
\(\displaystyle a \! \left(7\right) = 766\)
\(\displaystyle a \! \left(8\right) = 2567\)
\(\displaystyle a \! \left(9\right) = 8672\)
\(\displaystyle a \! \left(10\right) = 29622\)
\(\displaystyle a \! \left(n +7\right) = -\frac{8 \left(2 n -1\right) a \! \left(n \right)}{n +8}+\frac{4 \left(7 n -6\right) a \! \left(n +1\right)}{n +8}+\frac{2 \left(5 n +47\right) a \! \left(n +2\right)}{n +8}-\frac{8 \left(8 n +33\right) a \! \left(n +3\right)}{n +8}+\frac{\left(345+71 n \right) a \! \left(n +4\right)}{n +8}-\frac{\left(38 n +223\right) a \! \left(n +5\right)}{n +8}+\frac{\left(69+10 n \right) a \! \left(n +6\right)}{n +8}+\frac{3 \left(n -2\right) \left(n -5\right)}{2 \left(n +8\right)}, \quad n \geq 11\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 68\)
\(\displaystyle a \! \left(6\right) = 229\)
\(\displaystyle a \! \left(7\right) = 766\)
\(\displaystyle a \! \left(8\right) = 2567\)
\(\displaystyle a \! \left(9\right) = 8672\)
\(\displaystyle a \! \left(10\right) = 29622\)
\(\displaystyle a \! \left(n +7\right) = -\frac{8 \left(2 n -1\right) a \! \left(n \right)}{n +8}+\frac{4 \left(7 n -6\right) a \! \left(n +1\right)}{n +8}+\frac{2 \left(5 n +47\right) a \! \left(n +2\right)}{n +8}-\frac{8 \left(8 n +33\right) a \! \left(n +3\right)}{n +8}+\frac{\left(345+71 n \right) a \! \left(n +4\right)}{n +8}-\frac{\left(38 n +223\right) a \! \left(n +5\right)}{n +8}+\frac{\left(69+10 n \right) a \! \left(n +6\right)}{n +8}+\frac{3 \left(n -2\right) \left(n -5\right)}{2 \left(n +8\right)}, \quad n \geq 11\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 219 rules.
Found on July 23, 2021.Finding the specification took 4 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_{4}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \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_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{12}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{24}\! \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_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{12}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{12}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{12}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{12}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{12}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{12}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{73}\! \left(x \right)+F_{77}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{12}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{73}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{12}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{12}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{12}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{12}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{12}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{91}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{92}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{12}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{12}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{12}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{99}\! \left(x , 1\right)\\
F_{99}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{100}\! \left(x , y\right)+F_{102}\! \left(x , y\right)+F_{104}\! \left(x , y\right)\\
F_{100}\! \left(x , y\right) &= F_{101}\! \left(x , y\right) F_{99}\! \left(x , y\right)\\
F_{101}\! \left(x , y\right) &= y x\\
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_{99}\! \left(x , y\right)+F_{99}\! \left(x , 1\right)}{-1+y}\\
F_{104}\! \left(x , y\right) &= F_{105}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)+F_{5}\! \left(x \right)\\
F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right)+F_{153}\! \left(x , y\right)\\
F_{107}\! \left(x , y\right) &= F_{108}\! \left(x , y\right)+F_{136}\! \left(x , y\right)\\
F_{108}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{112}\! \left(x , y\right)\\
F_{109}\! \left(x , y\right) &= F_{110}\! \left(x , y\right)\\
F_{110}\! \left(x , y\right) &= F_{101}\! \left(x , y\right) F_{111}\! \left(x , y\right)\\
F_{111}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{109}\! \left(x , y\right)\\
F_{112}\! \left(x , y\right) &= F_{113}\! \left(x , y\right)+F_{118}\! \left(x , y\right)+F_{15}\! \left(x \right)\\
F_{113}\! \left(x , y\right) &= F_{101}\! \left(x , y\right) F_{114}\! \left(x , y\right)\\
F_{114}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{22}\! \left(x \right)\\
F_{115}\! \left(x , y\right) &= F_{113}\! \left(x , y\right)+F_{116}\! \left(x , y\right)+F_{15}\! \left(x \right)\\
F_{116}\! \left(x , y\right) &= F_{117}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{117}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{115}\! \left(x , y\right)\\
F_{118}\! \left(x , y\right) &= F_{119}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{119}\! \left(x , y\right) &= F_{120}\! \left(x , y\right)+F_{125}\! \left(x , y\right)\\
F_{120}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{121}\! \left(x , y\right)\\
F_{121}\! \left(x , y\right) &= F_{122}\! \left(x , y\right)+F_{124}\! \left(x , y\right)+F_{15}\! \left(x \right)\\
F_{122}\! \left(x , y\right) &= F_{101}\! \left(x , y\right) F_{123}\! \left(x , y\right)\\
F_{123}\! \left(x , y\right) &= F_{12}\! \left(x \right)+F_{121}\! \left(x , y\right)\\
F_{124}\! \left(x , y\right) &= F_{109}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{125}\! \left(x , y\right) &= F_{112}\! \left(x , y\right)+F_{126}\! \left(x , y\right)\\
F_{126}\! \left(x , y\right) &= F_{127}\! \left(x , y\right)+F_{131}\! \left(x , y\right)+F_{135}\! \left(x , y\right)+F_{15}\! \left(x \right)\\
F_{127}\! \left(x , y\right) &= F_{101}\! \left(x , y\right) F_{128}\! \left(x , y\right)\\
F_{128}\! \left(x , y\right) &= F_{129}\! \left(x , y\right)+F_{29}\! \left(x \right)\\
F_{129}\! \left(x , y\right) &= 2 F_{15}\! \left(x \right)+F_{127}\! \left(x , y\right)+F_{130}\! \left(x , y\right)\\
F_{130}\! \left(x , y\right) &= F_{115}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{131}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{132}\! \left(x , y\right)\\
F_{132}\! \left(x , y\right) &= F_{133}\! \left(x , y\right)+F_{134}\! \left(x , y\right)\\
F_{133}\! \left(x , y\right) &= F_{121}\! \left(x , y\right)\\
F_{134}\! \left(x , y\right) &= F_{126}\! \left(x , y\right)\\
F_{135}\! \left(x , y\right) &= F_{112}\! \left(x , y\right) F_{12}\! \left(x \right)\\
F_{136}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)+F_{140}\! \left(x , y\right)\\
F_{137}\! \left(x , y\right) &= F_{138}\! \left(x , y\right)\\
F_{138}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{139}\! \left(x , y\right)\\
F_{139}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{137}\! \left(x , y\right)\\
F_{140}\! \left(x , y\right) &= F_{141}\! \left(x , y\right)\\
F_{141}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{142}\! \left(x , y\right)\\
F_{142}\! \left(x , y\right) &= F_{143}\! \left(x , y\right)+F_{146}\! \left(x , y\right)\\
F_{143}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)+F_{144}\! \left(x , y\right)\\
F_{144}\! \left(x , y\right) &= F_{145}\! \left(x , y\right)\\
F_{145}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{137}\! \left(x , y\right)\\
F_{146}\! \left(x , y\right) &= F_{140}\! \left(x , y\right)+F_{147}\! \left(x , y\right)\\
F_{147}\! \left(x , y\right) &= 3 F_{15}\! \left(x \right)+F_{148}\! \left(x , y\right)+F_{152}\! \left(x , y\right)\\
F_{148}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{149}\! \left(x , y\right)\\
F_{149}\! \left(x , y\right) &= F_{150}\! \left(x , y\right)+F_{151}\! \left(x , y\right)\\
F_{150}\! \left(x , y\right) &= F_{144}\! \left(x , y\right)\\
F_{151}\! \left(x , y\right) &= F_{147}\! \left(x , y\right)\\
F_{152}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{140}\! \left(x , y\right)\\
F_{153}\! \left(x , y\right) &= F_{154}\! \left(x , y\right)+F_{204}\! \left(x , y\right)\\
F_{154}\! \left(x , y\right) &= F_{155}\! \left(x , y\right)+F_{163}\! \left(x , y\right)\\
F_{155}\! \left(x , y\right) &= F_{15}\! \left(x \right)+F_{156}\! \left(x , y\right)+F_{158}\! \left(x , y\right)\\
F_{156}\! \left(x , y\right) &= F_{101}\! \left(x , y\right) F_{157}\! \left(x , y\right)\\
F_{157}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)+F_{22}\! \left(x \right)\\
F_{158}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{159}\! \left(x , y\right)\\
F_{159}\! \left(x , y\right) &= F_{139}\! \left(x , y\right)+F_{160}\! \left(x , y\right)\\
F_{160}\! \left(x , y\right) &= F_{155}\! \left(x , y\right)+F_{161}\! \left(x , y\right)\\
F_{161}\! \left(x , y\right) &= F_{162}\! \left(x , y\right)\\
F_{162}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{160}\! \left(x , y\right)\\
F_{163}\! \left(x , y\right) &= F_{15}\! \left(x \right)+F_{164}\! \left(x , y\right)+F_{169}\! \left(x , y\right)+F_{186}\! \left(x , y\right)\\
F_{164}\! \left(x , y\right) &= F_{101}\! \left(x , y\right) F_{165}\! \left(x , y\right)\\
F_{165}\! \left(x , y\right) &= F_{166}\! \left(x , y\right)+F_{50}\! \left(x \right)\\
F_{166}\! \left(x , y\right) &= F_{167}\! \left(x , y\right)\\
F_{167}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{168}\! \left(x , y\right)\\
F_{168}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)+F_{166}\! \left(x , y\right)\\
F_{169}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{170}\! \left(x , y\right)\\
F_{170}\! \left(x , y\right) &= F_{171}\! \left(x , y\right)+F_{172}\! \left(x , y\right)\\
F_{171}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{166}\! \left(x , y\right)\\
F_{172}\! \left(x , y\right) &= F_{173}\! \left(x , y\right)+F_{179}\! \left(x , y\right)\\
F_{173}\! \left(x , y\right) &= F_{15}\! \left(x \right)+F_{164}\! \left(x , y\right)+F_{169}\! \left(x , y\right)+F_{174}\! \left(x , y\right)\\
F_{174}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{175}\! \left(x , y\right)\\
F_{175}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)+F_{176}\! \left(x , y\right)\\
F_{176}\! \left(x , y\right) &= 2 F_{15}\! \left(x \right)+F_{174}\! \left(x , y\right)+F_{177}\! \left(x , y\right)\\
F_{177}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{178}\! \left(x , y\right)\\
F_{178}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{176}\! \left(x , y\right)\\
F_{179}\! \left(x , y\right) &= F_{180}\! \left(x , y\right)\\
F_{180}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{181}\! \left(x , y\right)\\
F_{181}\! \left(x , y\right) &= F_{182}\! \left(x , y\right)+F_{185}\! \left(x , y\right)\\
F_{182}\! \left(x , y\right) &= F_{183}\! \left(x , y\right)\\
F_{183}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{184}\! \left(x , y\right)\\
F_{184}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)+F_{182}\! \left(x , y\right)\\
F_{185}\! \left(x , y\right) &= F_{180}\! \left(x , y\right)\\
F_{186}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{187}\! \left(x , y\right)\\
F_{187}\! \left(x , y\right) &= F_{188}\! \left(x , y\right)+F_{192}\! \left(x , y\right)\\
F_{188}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)+F_{189}\! \left(x , y\right)\\
F_{189}\! \left(x , y\right) &= 2 F_{15}\! \left(x \right)+F_{145}\! \left(x , y\right)+F_{190}\! \left(x , y\right)\\
F_{190}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{191}\! \left(x , y\right)\\
F_{191}\! \left(x , y\right) &= F_{121}\! \left(x , y\right)+F_{189}\! \left(x , y\right)\\
F_{192}\! \left(x , y\right) &= F_{193}\! \left(x , y\right)+F_{194}\! \left(x , y\right)\\
F_{193}\! \left(x , y\right) &= 2 F_{15}\! \left(x \right)+F_{177}\! \left(x , y\right)+F_{186}\! \left(x , y\right)\\
F_{194}\! \left(x , y\right) &= 2 F_{15}\! \left(x \right)+F_{195}\! \left(x , y\right)+F_{199}\! \left(x , y\right)+F_{203}\! \left(x , y\right)\\
F_{195}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{196}\! \left(x , y\right)\\
F_{196}\! \left(x , y\right) &= F_{129}\! \left(x , y\right)+F_{197}\! \left(x , y\right)\\
F_{197}\! \left(x , y\right) &= 3 F_{15}\! \left(x \right)+F_{195}\! \left(x , y\right)+F_{198}\! \left(x , y\right)\\
F_{198}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{176}\! \left(x , y\right)\\
F_{199}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{200}\! \left(x , y\right)\\
F_{200}\! \left(x , y\right) &= F_{201}\! \left(x , y\right)+F_{202}\! \left(x , y\right)\\
F_{201}\! \left(x , y\right) &= F_{189}\! \left(x , y\right)\\
F_{202}\! \left(x , y\right) &= F_{194}\! \left(x , y\right)\\
F_{203}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{193}\! \left(x , y\right)\\
F_{204}\! \left(x , y\right) &= F_{161}\! \left(x , y\right)+F_{205}\! \left(x , y\right)\\
F_{205}\! \left(x , y\right) &= F_{206}\! \left(x , y\right)\\
F_{206}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{207}\! \left(x , y\right)\\
F_{207}\! \left(x , y\right) &= F_{208}\! \left(x , y\right)+F_{211}\! \left(x , y\right)\\
F_{208}\! \left(x , y\right) &= F_{182}\! \left(x , y\right)+F_{209}\! \left(x , y\right)\\
F_{209}\! \left(x , y\right) &= F_{210}\! \left(x , y\right)\\
F_{210}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{182}\! \left(x , y\right)\\
F_{211}\! \left(x , y\right) &= F_{212}\! \left(x , y\right)+F_{213}\! \left(x , y\right)\\
F_{212}\! \left(x , y\right) &= F_{206}\! \left(x , y\right)\\
F_{213}\! \left(x , y\right) &= 4 F_{15}\! \left(x \right)+F_{214}\! \left(x , y\right)+F_{218}\! \left(x , y\right)\\
F_{214}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{215}\! \left(x , y\right)\\
F_{215}\! \left(x , y\right) &= F_{216}\! \left(x , y\right)+F_{217}\! \left(x , y\right)\\
F_{216}\! \left(x , y\right) &= F_{209}\! \left(x , y\right)\\
F_{217}\! \left(x , y\right) &= F_{213}\! \left(x , y\right)\\
F_{218}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{212}\! \left(x , y\right)\\
\end{align*}\)