Av(1234, 1243, 1342, 4123)
Generating Function
\(\displaystyle \frac{\left(-2 x^{5}+10 x^{4}-16 x^{3}+14 x^{2}-6 x +1\right) \sqrt{1-4 x}+12 x^{5}-34 x^{4}+42 x^{3}-26 x^{2}+8 x -1}{4 x^{2} \left(x -1\right)^{2} \left(x -\frac{1}{2}\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 68, 231, 786, 2693, 9320, 32611, 115320, 411721, 1482374, 5376412, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \left(x
\right)^{2}-\left(6 x^{4}-14 x^{3}+14 x^{2}-6 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)+4 x^{9}-5 x^{8}-30 x^{7}+124 x^{6}-216 x^{5}+219 x^{4}-137 x^{3}+52 x^{2}-11 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 68\)
\(\displaystyle a \! \left(6\right) = 231\)
\(\displaystyle a \! \left(7\right) = 786\)
\(\displaystyle a \! \left(8\right) = 2693\)
\(\displaystyle a \! \left(9\right) = 9320\)
\(\displaystyle a \! \left(n +8\right) = -\frac{8 \left(2 n -1\right) a \! \left(n \right)}{10+n}-\frac{2 \left(141 n +299\right) a \! \left(2+n \right)}{10+n}+\frac{12 \left(9 n +8\right) a \! \left(n +1\right)}{10+n}+\frac{6 \left(68 n +235\right) a \! \left(n +3\right)}{10+n}-\frac{2 \left(183 n +878\right) a \! \left(n +4\right)}{10+n}+\frac{2 \left(103 n +629\right) a \! \left(n +5\right)}{10+n}-\frac{14 \left(5 n +37\right) a \! \left(n +6\right)}{10+n}+\frac{\left(113+13 n \right) a \! \left(n +7\right)}{10+n}+\frac{2}{10+n}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 68\)
\(\displaystyle a \! \left(6\right) = 231\)
\(\displaystyle a \! \left(7\right) = 786\)
\(\displaystyle a \! \left(8\right) = 2693\)
\(\displaystyle a \! \left(9\right) = 9320\)
\(\displaystyle a \! \left(n +8\right) = -\frac{8 \left(2 n -1\right) a \! \left(n \right)}{10+n}-\frac{2 \left(141 n +299\right) a \! \left(2+n \right)}{10+n}+\frac{12 \left(9 n +8\right) a \! \left(n +1\right)}{10+n}+\frac{6 \left(68 n +235\right) a \! \left(n +3\right)}{10+n}-\frac{2 \left(183 n +878\right) a \! \left(n +4\right)}{10+n}+\frac{2 \left(103 n +629\right) a \! \left(n +5\right)}{10+n}-\frac{14 \left(5 n +37\right) a \! \left(n +6\right)}{10+n}+\frac{\left(113+13 n \right) a \! \left(n +7\right)}{10+n}+\frac{2}{10+n}, \quad n \geq 10\)
This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 202 rules.
Found on July 23, 2021.Finding the specification took 8 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_{7}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{101}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)+F_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{5}\! \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) &= y x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{12}\! \left(x \right)+F_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{10}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{14}\! \left(x , y\right) &= -\frac{-y F_{15}\! \left(x , y\right)+F_{15}\! \left(x , 1\right)}{-1+y}\\
F_{15}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)+F_{16}\! \left(x , y\right)+F_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{16}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x , 1\right)\\
F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{19}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x , y\right)+F_{34}\! \left(x , y\right)+F_{51}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{22}\! \left(x \right)+F_{25}\! \left(x , y\right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{22}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{25}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\
F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\
F_{30}\! \left(x \right) &= 0\\
F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{32}\! \left(x , y\right) &= F_{23}\! \left(x \right)+F_{29}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x \right)+F_{42}\! \left(x , y\right)\\
F_{36}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{39}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{37}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{42}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)+F_{43}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\
F_{44}\! \left(x , y\right) &= F_{45}\! \left(x , y\right)\\
F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{46}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{44}\! \left(x , y\right)\\
F_{47}\! \left(x , y\right) &= 2 F_{30}\! \left(x \right)+F_{48}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\
F_{48}\! \left(x , y\right) &= F_{49}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{49}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\
F_{50}\! \left(x , y\right) &= F_{43}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{52}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{53}\! \left(x , y\right)+F_{54}\! \left(x \right)+F_{77}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\
F_{53}\! \left(x , y\right) &= F_{52}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x , 1\right)\\
F_{55}\! \left(x , y\right) &= F_{56}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{56}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{57}\! \left(x , y\right)+F_{74}\! \left(x , y\right)+F_{76}\! \left(x , y\right)\\
F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)+F_{63}\! \left(x , y\right)\\
F_{59}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{60}\! \left(x , y\right)\\
F_{60}\! \left(x , y\right) &= F_{23}\! \left(x \right)+F_{61}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)\\
F_{62}\! \left(x , y\right) &= F_{60}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{64}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{23}\! \left(x \right)+F_{65}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{30}\! \left(x \right)+F_{66}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\
F_{66}\! \left(x , y\right) &= F_{67}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{67}\! \left(x , y\right) &= F_{26}\! \left(x , y\right)+F_{65}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{64}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{38}\! \left(x \right)+F_{70}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= 2 F_{30}\! \left(x \right)+F_{71}\! \left(x , y\right)+F_{73}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= F_{7}\! \left(x \right) F_{72}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{69}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{7}\! \left(x \right) F_{75}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= -\frac{-y F_{56}\! \left(x , y\right)+F_{56}\! \left(x , 1\right)}{-1+y}\\
F_{76}\! \left(x , y\right) &= F_{56}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= F_{7}\! \left(x \right) F_{78}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= -\frac{-y F_{52}\! \left(x , y\right)+F_{52}\! \left(x , 1\right)}{-1+y}\\
F_{79}\! \left(x , y\right) &= F_{7}\! \left(x \right) F_{80}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= F_{81}\! \left(x \right)+F_{91}\! \left(x , y\right)\\
F_{81}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{7}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{87}\! \left(x \right) &= 2 F_{30}\! \left(x \right)+F_{88}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{7}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{7}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{91}\! \left(x , y\right) &= F_{42}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\
F_{92}\! \left(x , y\right) &= F_{43}\! \left(x , y\right)+F_{93}\! \left(x , y\right)\\
F_{93}\! \left(x , y\right) &= F_{94}\! \left(x , y\right)+F_{97}\! \left(x , y\right)\\
F_{94}\! \left(x , y\right) &= F_{95}\! \left(x , y\right)\\
F_{95}\! \left(x , y\right) &= F_{7}\! \left(x \right) F_{96}\! \left(x , y\right)\\
F_{96}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)+F_{94}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= 3 F_{30}\! \left(x \right)+F_{100}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\
F_{98}\! \left(x , y\right) &= F_{7}\! \left(x \right) F_{99}\! \left(x , y\right)\\
F_{99}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)+F_{97}\! \left(x , y\right)\\
F_{100}\! \left(x , y\right) &= F_{7}\! \left(x \right) F_{93}\! \left(x , y\right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{103}\! \left(x \right)+F_{135}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{117}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{107}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{115}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{23}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{134}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{122}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{128}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{129}\! \left(x \right) &= 2 F_{30}\! \left(x \right)+F_{130}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{129}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{129}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x , 1\right)\\
F_{137}\! \left(x , y\right) &= F_{138}\! \left(x , y\right)+F_{139}\! \left(x , y\right)\\
F_{138}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\
F_{139}\! \left(x , y\right) &= F_{140}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\
F_{140}\! \left(x , y\right) &= F_{141}\! \left(x , y\right)+F_{38}\! \left(x \right)\\
F_{141}\! \left(x , y\right) &= F_{142}\! \left(x , y\right)+F_{143}\! \left(x , y\right)+F_{145}\! \left(x , y\right)+F_{30}\! \left(x \right)\\
F_{142}\! \left(x , y\right) &= 0\\
F_{143}\! \left(x , y\right) &= F_{144}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{144}\! \left(x , y\right) &= F_{141}\! \left(x , y\right)+F_{65}\! \left(x , y\right)\\
F_{145}\! \left(x , y\right) &= F_{140}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{146}\! \left(x \right) &= F_{147}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x , 1\right)\\
F_{148}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{149}\! \left(x , y\right)+F_{183}\! \left(x , y\right)+F_{199}\! \left(x , y\right)+F_{201}\! \left(x , y\right)\\
F_{149}\! \left(x , y\right) &= F_{150}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{150}\! \left(x , y\right) &= F_{151}\! \left(x , y\right)+F_{164}\! \left(x , y\right)\\
F_{151}\! \left(x , y\right) &= F_{152}\! \left(x , y\right)+F_{163}\! \left(x , y\right)\\
F_{152}\! \left(x , y\right) &= F_{153}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\
F_{153}\! \left(x , y\right) &= F_{107}\! \left(x \right)+F_{154}\! \left(x , y\right)\\
F_{154}\! \left(x , y\right) &= F_{155}\! \left(x , y\right)+F_{30}\! \left(x \right)+F_{62}\! \left(x , y\right)\\
F_{155}\! \left(x , y\right) &= F_{156}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{156}\! \left(x , y\right) &= F_{157}\! \left(x , y\right)+F_{158}\! \left(x , y\right)\\
F_{157}\! \left(x , y\right) &= F_{8}\! \left(x , y\right)\\
F_{158}\! \left(x , y\right) &= F_{159}\! \left(x , y\right)\\
F_{159}\! \left(x , y\right) &= F_{160}\! \left(x , y\right)+F_{162}\! \left(x , y\right)+F_{30}\! \left(x \right)\\
F_{160}\! \left(x , y\right) &= F_{161}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{161}\! \left(x , y\right) &= F_{159}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{162}\! \left(x , y\right) &= F_{23}\! \left(x \right) F_{8}\! \left(x , y\right)\\
F_{163}\! \left(x , y\right) &= F_{23}\! \left(x \right)\\
F_{164}\! \left(x , y\right) &= F_{165}\! \left(x , y\right)+F_{182}\! \left(x , y\right)\\
F_{165}\! \left(x , y\right) &= F_{166}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\
F_{166}\! \left(x , y\right) &= F_{119}\! \left(x \right)+F_{167}\! \left(x , y\right)\\
F_{167}\! \left(x , y\right) &= F_{168}\! \left(x , y\right)+F_{170}\! \left(x , y\right)+F_{30}\! \left(x \right)+F_{73}\! \left(x , y\right)\\
F_{168}\! \left(x , y\right) &= F_{169}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{169}\! \left(x , y\right) &= F_{154}\! \left(x , y\right)+F_{167}\! \left(x , y\right)\\
F_{170}\! \left(x , y\right) &= F_{171}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{171}\! \left(x , y\right) &= F_{172}\! \left(x , y\right)+F_{176}\! \left(x , y\right)\\
F_{172}\! \left(x , y\right) &= F_{173}\! \left(x , y\right)\\
F_{173}\! \left(x , y\right) &= F_{174}\! \left(x , y\right)\\
F_{174}\! \left(x , y\right) &= F_{175}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{175}\! \left(x , y\right) &= F_{173}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{176}\! \left(x , y\right) &= F_{177}\! \left(x , y\right)\\
F_{177}\! \left(x , y\right) &= 2 F_{30}\! \left(x \right)+F_{178}\! \left(x , y\right)+F_{180}\! \left(x , y\right)\\
F_{178}\! \left(x , y\right) &= F_{179}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{179}\! \left(x , y\right) &= F_{159}\! \left(x , y\right)+F_{177}\! \left(x , y\right)\\
F_{180}\! \left(x , y\right) &= F_{181}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{181}\! \left(x , y\right) &= F_{173}\! \left(x , y\right)+F_{177}\! \left(x , y\right)\\
F_{182}\! \left(x , y\right) &= F_{38}\! \left(x \right)\\
F_{183}\! \left(x , y\right) &= F_{184}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{184}\! \left(x , y\right) &= F_{185}\! \left(x , y\right)+F_{58}\! \left(x , y\right)\\
F_{185}\! \left(x , y\right) &= F_{186}\! \left(x , y\right)+F_{63}\! \left(x , y\right)\\
F_{186}\! \left(x , y\right) &= F_{140}\! \left(x , y\right)+F_{187}\! \left(x , y\right)\\
F_{187}\! \left(x , y\right) &= F_{188}\! \left(x \right)+F_{194}\! \left(x , y\right)\\
F_{188}\! \left(x \right) &= F_{189}\! \left(x \right)+F_{190}\! \left(x \right)+F_{192}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{189}\! \left(x \right) &= 0\\
F_{190}\! \left(x \right) &= F_{191}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{191}\! \left(x \right) &= F_{188}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{192}\! \left(x \right) &= F_{193}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{193}\! \left(x \right) &= F_{188}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{194}\! \left(x , y\right) &= 2 F_{30}\! \left(x \right)+F_{195}\! \left(x , y\right)+F_{196}\! \left(x , y\right)+F_{198}\! \left(x , y\right)\\
F_{195}\! \left(x , y\right) &= 0\\
F_{196}\! \left(x , y\right) &= F_{197}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{197}\! \left(x , y\right) &= F_{194}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\
F_{198}\! \left(x , y\right) &= F_{187}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{199}\! \left(x , y\right) &= F_{200}\! \left(x , y\right) F_{7}\! \left(x \right)\\
F_{200}\! \left(x , y\right) &= -\frac{-y F_{148}\! \left(x , y\right)+F_{148}\! \left(x , 1\right)}{-1+y}\\
F_{201}\! \left(x , y\right) &= F_{148}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
\end{align*}\)