Av(1243, 1324, 2341, 4123)
Generating Function
\(\displaystyle \frac{-2 \left(x -\frac{1}{2}\right) \left(x^{2}+x -1\right) \left(x -1\right)^{5} \sqrt{1-4 x}+6 x^{9}-10 x^{8}-9 x^{7}+14 x^{6}+10 x^{5}-37 x^{4}+43 x^{3}-26 x^{2}+8 x -1}{4 \left(x -\frac{1}{2}\right) \left(x^{2}+x -1\right) \left(x -1\right)^{5} x}\)
Counting Sequence
1, 1, 2, 6, 20, 63, 187, 552, 1680, 5342, 17689, 60425, 211017, 748448, 2684799, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x^{2}+x -1\right)^{2} \left(x -1\right)^{10} F \left(x
\right)^{2}-\left(2 x -1\right) \left(x^{2}+x -1\right) \left(6 x^{9}-10 x^{8}-9 x^{7}+14 x^{6}+10 x^{5}-37 x^{4}+43 x^{3}-26 x^{2}+8 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+9 x^{17}-26 x^{16}-39 x^{15}+225 x^{14}-244 x^{13}-280 x^{12}+1290 x^{11}-1975 x^{10}+1259 x^{9}+881 x^{8}-3010 x^{7}+3654 x^{6}-2767 x^{5}+1424 x^{4}-501 x^{3}+116 x^{2}-16 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 63\)
\(\displaystyle a \! \left(6\right) = 187\)
\(\displaystyle a \! \left(7\right) = 552\)
\(\displaystyle a \! \left(8\right) = 1680\)
\(\displaystyle a \! \left(9\right) = 5342\)
\(\displaystyle a \! \left(10\right) = 17689\)
\(\displaystyle a \! \left(11\right) = 60425\)
\(\displaystyle a \! \left(12\right) = 211017\)
\(\displaystyle a \! \left(13\right) = 748448\)
\(\displaystyle a \! \left(n +7\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{n +8}+\frac{4 \left(3 n +4\right) a \! \left(n +1\right)}{n +8}-\frac{2 \left(24 n +61\right) a \! \left(n +2\right)}{n +8}+\frac{\left(16+3 n \right) a \! \left(n +3\right)}{n +8}+\frac{2 \left(23 n +104\right) a \! \left(n +4\right)}{n +8}-\frac{\left(35 n +198\right) a \! \left(n +5\right)}{n +8}+\frac{2 \left(5 n +34\right) a \! \left(n +6\right)}{n +8}-\frac{3 n^{5}-70 n^{4}+197 n^{3}+1846 n^{2}-3008 n -3600}{24 \left(n +8\right)}, \quad n \geq 14\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 63\)
\(\displaystyle a \! \left(6\right) = 187\)
\(\displaystyle a \! \left(7\right) = 552\)
\(\displaystyle a \! \left(8\right) = 1680\)
\(\displaystyle a \! \left(9\right) = 5342\)
\(\displaystyle a \! \left(10\right) = 17689\)
\(\displaystyle a \! \left(11\right) = 60425\)
\(\displaystyle a \! \left(12\right) = 211017\)
\(\displaystyle a \! \left(13\right) = 748448\)
\(\displaystyle a \! \left(n +7\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{n +8}+\frac{4 \left(3 n +4\right) a \! \left(n +1\right)}{n +8}-\frac{2 \left(24 n +61\right) a \! \left(n +2\right)}{n +8}+\frac{\left(16+3 n \right) a \! \left(n +3\right)}{n +8}+\frac{2 \left(23 n +104\right) a \! \left(n +4\right)}{n +8}-\frac{\left(35 n +198\right) a \! \left(n +5\right)}{n +8}+\frac{2 \left(5 n +34\right) a \! \left(n +6\right)}{n +8}-\frac{3 n^{5}-70 n^{4}+197 n^{3}+1846 n^{2}-3008 n -3600}{24 \left(n +8\right)}, \quad n \geq 14\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 212 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_{117}\! \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_{30}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{16}\! \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_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= x^{2}\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{12}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{37}\! \left(x \right) &= 0\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{12}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{44}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{12}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \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 \right)+F_{51}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{54}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{12}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{12}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{12}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{12}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{12}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{67}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{12}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{75}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{12}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{12}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{37}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{12}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{93}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{12}\! \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_{89}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{92}\! \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 \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{102}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{103}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{102}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{108}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{109}\! \left(x \right)+F_{113}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{112}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{113}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{119}\! \left(x \right)+F_{145}\! \left(x \right)+F_{210}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x , 1\right)\\
F_{120}\! \left(x , y\right) &= F_{121}\! \left(x , y\right) F_{122}\! \left(x , y\right)\\
F_{121}\! \left(x , y\right) &= y x\\
F_{122}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{120}\! \left(x , y\right)+F_{123}\! \left(x , y\right)+F_{125}\! \left(x , y\right)\\
F_{123}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{124}\! \left(x , y\right)\\
F_{124}\! \left(x , y\right) &= -\frac{-y F_{122}\! \left(x , y\right)+F_{122}\! \left(x , 1\right)}{-1+y}\\
F_{125}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{126}\! \left(x , y\right)\\
F_{126}\! \left(x , y\right) &= F_{127}\! \left(x \right)+F_{128}\! \left(x , y\right)\\
F_{127}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{128}\! \left(x , y\right) &= F_{129}\! \left(x , y\right)+F_{137}\! \left(x , y\right)\\
F_{129}\! \left(x , y\right) &= F_{130}\! \left(x , y\right)+F_{133}\! \left(x , y\right)\\
F_{130}\! \left(x , y\right) &= F_{131}\! \left(x , y\right)\\
F_{131}\! \left(x , y\right) &= F_{121}\! \left(x , y\right) F_{132}\! \left(x , y\right)\\
F_{132}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{130}\! \left(x , y\right)\\
F_{133}\! \left(x , y\right) &= F_{134}\! \left(x , y\right)+F_{136}\! \left(x , y\right)+F_{37}\! \left(x \right)\\
F_{134}\! \left(x , y\right) &= F_{121}\! \left(x , y\right) F_{135}\! \left(x , y\right)\\
F_{135}\! \left(x , y\right) &= F_{133}\! \left(x , y\right)+F_{63}\! \left(x \right)\\
F_{136}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{129}\! \left(x , y\right)\\
F_{137}\! \left(x , y\right) &= F_{138}\! \left(x , y\right)+F_{141}\! \left(x , y\right)\\
F_{138}\! \left(x , y\right) &= F_{139}\! \left(x , y\right)\\
F_{139}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{140}\! \left(x , y\right)\\
F_{140}\! \left(x , y\right) &= F_{130}\! \left(x , y\right)+F_{138}\! \left(x , y\right)\\
F_{141}\! \left(x , y\right) &= 2 F_{37}\! \left(x \right)+F_{142}\! \left(x , y\right)+F_{144}\! \left(x , y\right)\\
F_{142}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{143}\! \left(x , y\right)\\
F_{143}\! \left(x , y\right) &= F_{133}\! \left(x , y\right)+F_{141}\! \left(x , y\right)\\
F_{144}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{137}\! \left(x , y\right)\\
F_{145}\! \left(x \right) &= F_{12}\! \left(x \right) F_{146}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{119}\! \left(x \right)+F_{147}\! \left(x \right)+F_{150}\! \left(x \right)+F_{209}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x , 1\right)\\
F_{148}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{149}\! \left(x , y\right)\\
F_{149}\! \left(x , y\right) &= -\frac{-y F_{122}\! \left(x , y\right)+F_{122}\! \left(x , 1\right)}{-1+y}\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x , 1\right)\\
F_{151}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{152}\! \left(x , y\right)\\
F_{152}\! \left(x , y\right) &= -\frac{-y F_{153}\! \left(x , y\right)+F_{153}\! \left(x , 1\right)}{-1+y}\\
F_{153}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{120}\! \left(x , y\right)+F_{148}\! \left(x , y\right)+F_{151}\! \left(x , y\right)+F_{154}\! \left(x , y\right)\\
F_{154}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{155}\! \left(x , y\right)\\
F_{155}\! \left(x , y\right) &= F_{126}\! \left(x , y\right)+F_{156}\! \left(x , y\right)\\
F_{156}\! \left(x , y\right) &= F_{157}\! \left(x \right)+F_{185}\! \left(x , y\right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{165}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{163}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{12}\! \left(x \right) F_{161}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{162}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{37}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{12}\! \left(x \right) F_{164}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{175}\! \left(x \right)+F_{177}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{12}\! \left(x \right) F_{167}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{168}\! \left(x \right)+F_{171}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{12}\! \left(x \right) F_{170}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{168}\! \left(x \right)\\
F_{171}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{166}\! \left(x \right)+F_{172}\! \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_{61}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{172}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{12}\! \left(x \right) F_{176}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{165}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{12}\! \left(x \right) F_{178}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{179}\! \left(x \right)+F_{182}\! \left(x \right)\\
F_{179}\! \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_{179}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{182}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{177}\! \left(x \right)+F_{183}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{12}\! \left(x \right) F_{184}\! \left(x \right)\\
F_{184}\! \left(x \right) &= F_{182}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{185}\! \left(x , y\right) &= F_{186}\! \left(x , y\right)+F_{194}\! \left(x , y\right)\\
F_{186}\! \left(x , y\right) &= F_{187}\! \left(x , y\right)+F_{190}\! \left(x , y\right)\\
F_{187}\! \left(x , y\right) &= F_{188}\! \left(x , y\right)\\
F_{188}\! \left(x , y\right) &= F_{121}\! \left(x , y\right) F_{189}\! \left(x , y\right)\\
F_{189}\! \left(x , y\right) &= F_{187}\! \left(x , y\right)+F_{63}\! \left(x \right)\\
F_{190}\! \left(x , y\right) &= 2 F_{37}\! \left(x \right)+F_{191}\! \left(x , y\right)+F_{193}\! \left(x , y\right)\\
F_{191}\! \left(x , y\right) &= F_{121}\! \left(x , y\right) F_{192}\! \left(x , y\right)\\
F_{192}\! \left(x , y\right) &= F_{190}\! \left(x , y\right)+F_{77}\! \left(x \right)\\
F_{193}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{186}\! \left(x , y\right)\\
F_{194}\! \left(x , y\right) &= F_{195}\! \left(x , y\right)+F_{198}\! \left(x , y\right)\\
F_{195}\! \left(x , y\right) &= F_{196}\! \left(x , y\right)\\
F_{196}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{197}\! \left(x , y\right)\\
F_{197}\! \left(x , y\right) &= F_{187}\! \left(x , y\right)+F_{195}\! \left(x , y\right)\\
F_{198}\! \left(x , y\right) &= 3 F_{37}\! \left(x \right)+F_{199}\! \left(x , y\right)+F_{201}\! \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_{190}\! \left(x , y\right)+F_{198}\! \left(x , y\right)\\
F_{201}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{202}\! \left(x , y\right)\\
F_{202}\! \left(x , y\right) &= F_{203}\! \left(x , y\right)+F_{206}\! \left(x , y\right)\\
F_{203}\! \left(x , y\right) &= F_{204}\! \left(x , y\right)\\
F_{204}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{205}\! \left(x , y\right)\\
F_{205}\! \left(x , y\right) &= F_{187}\! \left(x , y\right)+F_{203}\! \left(x , y\right)\\
F_{206}\! \left(x , y\right) &= 3 F_{37}\! \left(x \right)+F_{201}\! \left(x , y\right)+F_{207}\! \left(x , y\right)\\
F_{207}\! \left(x , y\right) &= F_{12}\! \left(x \right) F_{208}\! \left(x , y\right)\\
F_{208}\! \left(x , y\right) &= F_{190}\! \left(x , y\right)+F_{206}\! \left(x , y\right)\\
F_{209}\! \left(x \right) &= F_{154}\! \left(x , 1\right)\\
F_{210}\! \left(x \right) &= F_{12}\! \left(x \right) F_{211}\! \left(x \right)\\
F_{211}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{157}\! \left(x \right)\\
\end{align*}\)