Av(1234, 2341, 3412)
Generating Function
\(\displaystyle -\frac{18 x^{10}-134 x^{9}+446 x^{8}-918 x^{7}+1253 x^{6}-1161 x^{5}+736 x^{4}-315 x^{3}+87 x^{2}-14 x +1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{3} \left(-1+x \right)^{6}}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 259, 849, 2638, 7817, 22275, 61539, 166007, 439844, 1150070, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{3} \left(-1+x \right)^{6} F \! \left(x \right)+18 x^{10}-134 x^{9}+446 x^{8}-918 x^{7}+1253 x^{6}-1161 x^{5}+736 x^{4}-315 x^{3}+87 x^{2}-14 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 259\)
\(\displaystyle a \! \left(7\right) = 849\)
\(\displaystyle a \! \left(8\right) = 2638\)
\(\displaystyle a \! \left(9\right) = 7817\)
\(\displaystyle a \! \left(10\right) = 22275\)
\(\displaystyle a \! \left(n +5\right) = 8 a \! \left(n \right)-36 a \! \left(n +1\right)+50 a \! \left(n +2\right)-31 a \! \left(n +3\right)+9 a \! \left(n +4\right)-\frac{n \left(n^{4}-20 n^{3}+55 n^{2}+260 n -176\right)}{120}, \quad n \geq 11\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 259\)
\(\displaystyle a \! \left(7\right) = 849\)
\(\displaystyle a \! \left(8\right) = 2638\)
\(\displaystyle a \! \left(9\right) = 7817\)
\(\displaystyle a \! \left(10\right) = 22275\)
\(\displaystyle a \! \left(n +5\right) = 8 a \! \left(n \right)-36 a \! \left(n +1\right)+50 a \! \left(n +2\right)-31 a \! \left(n +3\right)+9 a \! \left(n +4\right)-\frac{n \left(n^{4}-20 n^{3}+55 n^{2}+260 n -176\right)}{120}, \quad n \geq 11\)
Explicit Closed Form
\(\displaystyle \frac{\left(36 \sqrt{5}+60\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{120}+\frac{\left(-36 \sqrt{5}+60\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{120}+2^{n +1}+\frac{\left(30 n^{2}-270 n \right) 2^{n}}{120}-\frac{n^{5}}{120}+\frac{n^{4}}{12}-\frac{n^{3}}{8}+\frac{11 n^{2}}{12}-\frac{13 n}{15}-2\)
This specification was found using the strategy pack "Insertion Point Placements" and has 198 rules.
Found on January 17, 2022.Finding the specification took 53 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 198 equations to clipboard:
\(\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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{11}\! \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_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{20}\! \left(x \right) &= 0\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12} \left(x \right)^{2}\\
F_{39}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{12} \left(x \right)^{2}\\
F_{48}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{78}\! \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_{62}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{11}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{66}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{75}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{70}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{52}\! \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_{109}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{20}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{88}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{90}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{94}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{4}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{20}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{101}\! \left(x \right) F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{107}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{20}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{106}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{12}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{181}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{164}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{133}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{126}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{125}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{130}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{132}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{134}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{158}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{155}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{142}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{12}\! \left(x \right) F_{141}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{144}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{12} \left(x \right)^{2} F_{18}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{12}\! \left(x \right) F_{147}\! \left(x \right)\\
F_{147}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{148}\! \left(x \right)+F_{150}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{152}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{154}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{152}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{12} \left(x \right)^{2} F_{157}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{160}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{161}\! \left(x \right)+F_{162}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{11}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{166}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)+F_{170}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{12}\! \left(x \right) F_{168}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{11}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)+F_{175}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{173}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{174}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{164}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{12}\! \left(x \right) F_{176}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{178}\! \left(x \right)+F_{180}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{179}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{179}\! \left(x \right) &= F_{177}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{180}\! \left(x \right) &= F_{176}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{181}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{182}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{183}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{184}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{184}\! \left(x \right) &= F_{185}\! \left(x \right)+F_{192}\! \left(x \right)\\
F_{185}\! \left(x \right) &= F_{186}\! \left(x \right)+F_{189}\! \left(x \right)\\
F_{186}\! \left(x \right) &= F_{187}\! \left(x \right)+F_{188}\! \left(x \right)\\
F_{187}\! \left(x \right) &= F_{12}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{188}\! \left(x \right) &= F_{12} \left(x \right)^{2} F_{11}\! \left(x \right)\\
F_{189}\! \left(x \right) &= F_{190}\! \left(x \right)+F_{191}\! \left(x \right)\\
F_{190}\! \left(x \right) &= F_{12}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{191}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{192}\! \left(x \right) &= F_{193}\! \left(x \right)+F_{195}\! \left(x \right)\\
F_{193}\! \left(x \right) &= F_{194}\! \left(x \right)\\
F_{194}\! \left(x \right) &= F_{12}\! \left(x \right) F_{186}\! \left(x \right)\\
F_{195}\! \left(x \right) &= F_{196}\! \left(x \right)+F_{197}\! \left(x \right)\\
F_{196}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{197}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{147}\! \left(x \right)\\
\end{align*}\)