Av(1234, 1432, 2431, 3241)
Generating Function
\(\displaystyle \frac{5 x^{8}+12 x^{7}-7 x^{6}-11 x^{5}-7 x^{4}-3 x^{3}+2 x -1}{\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 61, 158, 358, 777, 1622, 3296, 6573, 12918, 25102, 48345, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+5 x^{8}+12 x^{7}-7 x^{6}-11 x^{5}-7 x^{4}-3 x^{3}+2 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(6\right) = 158\)
\(\displaystyle a \! \left(7\right) = 358\)
\(\displaystyle a \! \left(8\right) = 777\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)+10, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(6\right) = 158\)
\(\displaystyle a \! \left(7\right) = 358\)
\(\displaystyle a \! \left(8\right) = 777\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)+10, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(\left(\left(-135 \sqrt{11}+385 \,\mathrm{I}\right) \sqrt{3}-405 \,\mathrm{I} \sqrt{11}+385\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+4400+\left(\left(285 \sqrt{11}+1705 \,\mathrm{I}\right) \sqrt{3}-855 \,\mathrm{I} \sqrt{11}-1705\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{660}\\+\\\frac{\left(\left(\left(285 \sqrt{11}-1705 \,\mathrm{I}\right) \sqrt{3}+855 \,\mathrm{I} \sqrt{11}-1705\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+4400+\left(\left(-135 \sqrt{11}-385 \,\mathrm{I}\right) \sqrt{3}+405 \,\mathrm{I} \sqrt{11}+385\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{660}\\+\\\frac{\left(\left(-570 \sqrt{11}\, \sqrt{3}+3410\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+270 \sqrt{11}\, \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-770 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+4400\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{660}\\+\frac{\left(3366 \sqrt{5}-4950\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{660}+5+\\\frac{\left(-3366 \sqrt{5}-4950\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{660} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 505 rules.
Found on January 18, 2022.Finding the specification took 33 seconds.
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Copy 505 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_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \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_{18}\! \left(x \right)+F_{268}\! \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_{55}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{35}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \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_{11}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{50}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{67}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \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_{60}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{88}\! \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_{102}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{108}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{107}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{103}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{112}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{113}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{123}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{124}\! \left(x \right)+F_{229}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{136}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{127}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{128}\! \left(x \right)+F_{134}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{203}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{145}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{147}\! \left(x \right)\\
F_{147}\! \left(x \right) &= 3 F_{18}\! \left(x \right)+F_{148}\! \left(x \right)+F_{169}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{154}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{151}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{153}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{145}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{160}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{157}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{159}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{141}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{155}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{161}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{162}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{164}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{151}\! \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_{168}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{151}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{165}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{170}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)+F_{198}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{172}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{173}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{174}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{175}\! \left(x \right)+F_{177}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{176}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{142}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{178}\! \left(x \right)\\
F_{178}\! \left(x \right) &= 3 F_{18}\! \left(x \right)+F_{169}\! \left(x \right)+F_{179}\! \left(x \right)\\
F_{179}\! \left(x \right) &= F_{180}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{180}\! \left(x \right) &= F_{181}\! \left(x \right)+F_{183}\! \left(x \right)\\
F_{181}\! \left(x \right) &= F_{176}\! \left(x \right)+F_{182}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{152}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{184}\! \left(x \right)+F_{189}\! \left(x \right)\\
F_{184}\! \left(x \right) &= F_{185}\! \left(x \right)\\
F_{185}\! \left(x \right) &= F_{186}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{186}\! \left(x \right) &= F_{187}\! \left(x \right)+F_{188}\! \left(x \right)\\
F_{187}\! \left(x \right) &= F_{176}\! \left(x \right)\\
F_{188}\! \left(x \right) &= F_{184}\! \left(x \right)\\
F_{189}\! \left(x \right) &= F_{190}\! \left(x \right)\\
F_{190}\! \left(x \right) &= F_{191}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{191}\! \left(x \right) &= F_{192}\! \left(x \right)+F_{193}\! \left(x \right)\\
F_{192}\! \left(x \right) &= F_{182}\! \left(x \right)\\
F_{193}\! \left(x \right) &= F_{194}\! \left(x \right)\\
F_{194}\! \left(x \right) &= F_{195}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{195}\! \left(x \right) &= F_{196}\! \left(x \right)+F_{197}\! \left(x \right)\\
F_{196}\! \left(x \right) &= F_{182}\! \left(x \right)\\
F_{197}\! \left(x \right) &= F_{194}\! \left(x \right)\\
F_{198}\! \left(x \right) &= F_{199}\! \left(x \right)\\
F_{199}\! \left(x \right) &= F_{200}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{200}\! \left(x \right) &= F_{201}\! \left(x \right)+F_{202}\! \left(x \right)\\
F_{201}\! \left(x \right) &= F_{132}\! \left(x \right)\\
F_{202}\! \left(x \right) &= F_{199}\! \left(x \right)\\
F_{203}\! \left(x \right) &= 3 F_{18}\! \left(x \right)+F_{204}\! \left(x \right)+F_{225}\! \left(x \right)\\
F_{204}\! \left(x \right) &= F_{205}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{205}\! \left(x \right) &= F_{206}\! \left(x \right)+F_{210}\! \left(x \right)\\
F_{206}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{207}\! \left(x \right)\\
F_{207}\! \left(x \right) &= F_{208}\! \left(x \right)\\
F_{208}\! \left(x \right) &= F_{209}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{209}\! \left(x \right) &= F_{131}\! \left(x \right)\\
F_{210}\! \left(x \right) &= F_{211}\! \left(x \right)+F_{216}\! \left(x \right)\\
F_{211}\! \left(x \right) &= F_{212}\! \left(x \right)\\
F_{212}\! \left(x \right) &= F_{213}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{213}\! \left(x \right) &= F_{214}\! \left(x \right)+F_{215}\! \left(x \right)\\
F_{214}\! \left(x \right) &= F_{127}\! \left(x \right)\\
F_{215}\! \left(x \right) &= F_{211}\! \left(x \right)\\
F_{216}\! \left(x \right) &= F_{217}\! \left(x \right)\\
F_{217}\! \left(x \right) &= F_{218}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{218}\! \left(x \right) &= F_{219}\! \left(x \right)+F_{220}\! \left(x \right)\\
F_{219}\! \left(x \right) &= F_{207}\! \left(x \right)\\
F_{220}\! \left(x \right) &= F_{221}\! \left(x \right)\\
F_{221}\! \left(x \right) &= F_{222}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{222}\! \left(x \right) &= F_{223}\! \left(x \right)+F_{224}\! \left(x \right)\\
F_{223}\! \left(x \right) &= F_{207}\! \left(x \right)\\
F_{224}\! \left(x \right) &= F_{221}\! \left(x \right)\\
F_{225}\! \left(x \right) &= F_{226}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{226}\! \left(x \right) &= F_{227}\! \left(x \right)+F_{228}\! \left(x \right)\\
F_{227}\! \left(x \right) &= F_{172}\! \left(x \right)\\
F_{228}\! \left(x \right) &= F_{199}\! \left(x \right)\\
F_{229}\! \left(x \right) &= F_{230}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{230}\! \left(x \right) &= F_{231}\! \left(x \right)+F_{258}\! \left(x \right)\\
F_{231}\! \left(x \right) &= F_{232}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{232}\! \left(x \right) &= F_{233}\! \left(x \right)\\
F_{233}\! \left(x \right) &= F_{234}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{234}\! \left(x \right) &= F_{235}\! \left(x \right)+F_{237}\! \left(x \right)\\
F_{235}\! \left(x \right) &= F_{236}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{236}\! \left(x \right) &= F_{128}\! \left(x \right)\\
F_{237}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{238}\! \left(x \right)\\
F_{238}\! \left(x \right) &= 3 F_{18}\! \left(x \right)+F_{225}\! \left(x \right)+F_{239}\! \left(x \right)\\
F_{239}\! \left(x \right) &= F_{240}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{240}\! \left(x \right) &= F_{241}\! \left(x \right)+F_{243}\! \left(x \right)\\
F_{241}\! \left(x \right) &= F_{236}\! \left(x \right)+F_{242}\! \left(x \right)\\
F_{242}\! \left(x \right) &= F_{208}\! \left(x \right)\\
F_{243}\! \left(x \right) &= F_{244}\! \left(x \right)+F_{249}\! \left(x \right)\\
F_{244}\! \left(x \right) &= F_{245}\! \left(x \right)\\
F_{245}\! \left(x \right) &= F_{246}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{246}\! \left(x \right) &= F_{247}\! \left(x \right)+F_{248}\! \left(x \right)\\
F_{247}\! \left(x \right) &= F_{236}\! \left(x \right)\\
F_{248}\! \left(x \right) &= F_{244}\! \left(x \right)\\
F_{249}\! \left(x \right) &= F_{250}\! \left(x \right)\\
F_{250}\! \left(x \right) &= F_{251}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{251}\! \left(x \right) &= F_{252}\! \left(x \right)+F_{253}\! \left(x \right)\\
F_{252}\! \left(x \right) &= F_{242}\! \left(x \right)\\
F_{253}\! \left(x \right) &= F_{254}\! \left(x \right)\\
F_{254}\! \left(x \right) &= F_{255}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{255}\! \left(x \right) &= F_{256}\! \left(x \right)+F_{257}\! \left(x \right)\\
F_{256}\! \left(x \right) &= F_{242}\! \left(x \right)\\
F_{257}\! \left(x \right) &= F_{254}\! \left(x \right)\\
F_{258}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{259}\! \left(x \right)\\
F_{259}\! \left(x \right) &= F_{260}\! \left(x \right)\\
F_{260}\! \left(x \right) &= F_{261}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{261}\! \left(x \right) &= F_{262}\! \left(x \right)+F_{263}\! \left(x \right)\\
F_{262}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{263}\! \left(x \right) &= F_{264}\! \left(x \right)\\
F_{264}\! \left(x \right) &= F_{265}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{265}\! \left(x \right) &= F_{266}\! \left(x \right)+F_{267}\! \left(x \right)\\
F_{266}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{267}\! \left(x \right) &= F_{264}\! \left(x \right)\\
F_{268}\! \left(x \right) &= F_{269}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{269}\! \left(x \right) &= F_{270}\! \left(x \right)+F_{372}\! \left(x \right)\\
F_{270}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{271}\! \left(x \right)\\
F_{271}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{272}\! \left(x \right)+F_{329}\! \left(x \right)\\
F_{272}\! \left(x \right) &= F_{273}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{273}\! \left(x \right) &= F_{274}\! \left(x \right)+F_{276}\! \left(x \right)\\
F_{274}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{275}\! \left(x \right)\\
F_{275}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{24}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{276}\! \left(x \right) &= F_{277}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{277}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{229}\! \left(x \right)+F_{278}\! \left(x \right)\\
F_{278}\! \left(x \right) &= F_{279}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{279}\! \left(x \right) &= F_{280}\! \left(x \right)+F_{282}\! \left(x \right)\\
F_{280}\! \left(x \right) &= F_{275}\! \left(x \right)+F_{281}\! \left(x \right)\\
F_{281}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{128}\! \left(x \right)+F_{134}\! \left(x \right)\\
F_{282}\! \left(x \right) &= F_{283}\! \left(x \right)+F_{309}\! \left(x \right)\\
F_{283}\! \left(x \right) &= F_{284}\! \left(x \right)\\
F_{284}\! \left(x \right) &= F_{285}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{285}\! \left(x \right) &= F_{286}\! \left(x \right)+F_{288}\! \left(x \right)\\
F_{286}\! \left(x \right) &= F_{275}\! \left(x \right)+F_{287}\! \left(x \right)\\
F_{287}\! \left(x \right) &= F_{142}\! \left(x \right)\\
F_{288}\! \left(x \right) &= F_{283}\! \left(x \right)+F_{289}\! \left(x \right)\\
F_{289}\! \left(x \right) &= 3 F_{18}\! \left(x \right)+F_{169}\! \left(x \right)+F_{290}\! \left(x \right)\\
F_{290}\! \left(x \right) &= F_{291}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{291}\! \left(x \right) &= F_{292}\! \left(x \right)+F_{294}\! \left(x \right)\\
F_{292}\! \left(x \right) &= F_{287}\! \left(x \right)+F_{293}\! \left(x \right)\\
F_{293}\! \left(x \right) &= F_{152}\! \left(x \right)\\
F_{294}\! \left(x \right) &= F_{295}\! \left(x \right)+F_{300}\! \left(x \right)\\
F_{295}\! \left(x \right) &= F_{296}\! \left(x \right)\\
F_{296}\! \left(x \right) &= F_{297}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{297}\! \left(x \right) &= F_{298}\! \left(x \right)+F_{299}\! \left(x \right)\\
F_{298}\! \left(x \right) &= F_{287}\! \left(x \right)\\
F_{299}\! \left(x \right) &= F_{295}\! \left(x \right)\\
F_{300}\! \left(x \right) &= F_{301}\! \left(x \right)\\
F_{301}\! \left(x \right) &= F_{302}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{302}\! \left(x \right) &= F_{303}\! \left(x \right)+F_{304}\! \left(x \right)\\
F_{303}\! \left(x \right) &= F_{293}\! \left(x \right)\\
F_{304}\! \left(x \right) &= F_{305}\! \left(x \right)\\
F_{305}\! \left(x \right) &= F_{306}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{306}\! \left(x \right) &= F_{307}\! \left(x \right)+F_{308}\! \left(x \right)\\
F_{307}\! \left(x \right) &= F_{293}\! \left(x \right)\\
F_{308}\! \left(x \right) &= F_{305}\! \left(x \right)\\
F_{309}\! \left(x \right) &= 3 F_{18}\! \left(x \right)+F_{225}\! \left(x \right)+F_{310}\! \left(x \right)\\
F_{310}\! \left(x \right) &= F_{311}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{311}\! \left(x \right) &= F_{312}\! \left(x \right)+F_{314}\! \left(x \right)\\
F_{312}\! \left(x \right) &= F_{281}\! \left(x \right)+F_{313}\! \left(x \right)\\
F_{313}\! \left(x \right) &= F_{208}\! \left(x \right)\\
F_{314}\! \left(x \right) &= F_{315}\! \left(x \right)+F_{320}\! \left(x \right)\\
F_{315}\! \left(x \right) &= F_{316}\! \left(x \right)\\
F_{316}\! \left(x \right) &= F_{317}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{317}\! \left(x \right) &= F_{318}\! \left(x \right)+F_{319}\! \left(x \right)\\
F_{318}\! \left(x \right) &= F_{281}\! \left(x \right)\\
F_{319}\! \left(x \right) &= F_{315}\! \left(x \right)\\
F_{320}\! \left(x \right) &= F_{321}\! \left(x \right)\\
F_{321}\! \left(x \right) &= F_{322}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{322}\! \left(x \right) &= F_{323}\! \left(x \right)+F_{324}\! \left(x \right)\\
F_{323}\! \left(x \right) &= F_{313}\! \left(x \right)\\
F_{324}\! \left(x \right) &= F_{325}\! \left(x \right)\\
F_{325}\! \left(x \right) &= F_{326}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{326}\! \left(x \right) &= F_{327}\! \left(x \right)+F_{328}\! \left(x \right)\\
F_{327}\! \left(x \right) &= F_{313}\! \left(x \right)\\
F_{328}\! \left(x \right) &= F_{325}\! \left(x \right)\\
F_{329}\! \left(x \right) &= F_{330}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{330}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{331}\! \left(x \right)\\
F_{331}\! \left(x \right) &= F_{332}\! \left(x \right)\\
F_{332}\! \left(x \right) &= F_{333}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{333}\! \left(x \right) &= F_{334}\! \left(x \right)+F_{340}\! \left(x \right)\\
F_{334}\! \left(x \right) &= F_{335}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{335}\! \left(x \right) &= F_{336}\! \left(x \right)\\
F_{336}\! \left(x \right) &= F_{337}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{337}\! \left(x \right) &= F_{338}\! \left(x \right)+F_{339}\! \left(x \right)\\
F_{338}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{339}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{340}\! \left(x \right) &= F_{331}\! \left(x \right)+F_{341}\! \left(x \right)\\
F_{341}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{342}\! \left(x \right)+F_{363}\! \left(x \right)\\
F_{342}\! \left(x \right) &= F_{343}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{343}\! \left(x \right) &= F_{344}\! \left(x \right)+F_{348}\! \left(x \right)\\
F_{344}\! \left(x \right) &= F_{335}\! \left(x \right)+F_{345}\! \left(x \right)\\
F_{345}\! \left(x \right) &= F_{346}\! \left(x \right)\\
F_{346}\! \left(x \right) &= F_{347}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{347}\! \left(x \right) &= F_{339}\! \left(x \right)\\
F_{348}\! \left(x \right) &= F_{349}\! \left(x \right)+F_{354}\! \left(x \right)\\
F_{349}\! \left(x \right) &= F_{350}\! \left(x \right)\\
F_{350}\! \left(x \right) &= F_{351}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{351}\! \left(x \right) &= F_{352}\! \left(x \right)+F_{353}\! \left(x \right)\\
F_{352}\! \left(x \right) &= F_{335}\! \left(x \right)\\
F_{353}\! \left(x \right) &= F_{349}\! \left(x \right)\\
F_{354}\! \left(x \right) &= F_{355}\! \left(x \right)\\
F_{355}\! \left(x \right) &= F_{356}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{356}\! \left(x \right) &= F_{357}\! \left(x \right)+F_{358}\! \left(x \right)\\
F_{357}\! \left(x \right) &= F_{345}\! \left(x \right)\\
F_{358}\! \left(x \right) &= F_{359}\! \left(x \right)\\
F_{359}\! \left(x \right) &= F_{360}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{360}\! \left(x \right) &= F_{361}\! \left(x \right)+F_{362}\! \left(x \right)\\
F_{361}\! \left(x \right) &= F_{345}\! \left(x \right)\\
F_{362}\! \left(x \right) &= F_{359}\! \left(x \right)\\
F_{363}\! \left(x \right) &= F_{364}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{364}\! \left(x \right) &= F_{365}\! \left(x \right)+F_{366}\! \left(x \right)\\
F_{365}\! \left(x \right) &= F_{331}\! \left(x \right)\\
F_{366}\! \left(x \right) &= F_{367}\! \left(x \right)\\
F_{367}\! \left(x \right) &= F_{368}\! \left(x \right)\\
F_{368}\! \left(x \right) &= F_{369}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{369}\! \left(x \right) &= F_{370}\! \left(x \right)+F_{371}\! \left(x \right)\\
F_{370}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{371}\! \left(x \right) &= F_{367}\! \left(x \right)\\
F_{372}\! \left(x \right) &= F_{373}\! \left(x \right)+F_{399}\! \left(x \right)\\
F_{373}\! \left(x \right) &= F_{374}\! \left(x \right)\\
F_{374}\! \left(x \right) &= F_{375}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{375}\! \left(x \right) &= F_{376}\! \left(x \right)+F_{378}\! \left(x \right)\\
F_{376}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{377}\! \left(x \right)\\
F_{377}\! \left(x \right) &= F_{336}\! \left(x \right)\\
F_{378}\! \left(x \right) &= F_{373}\! \left(x \right)+F_{379}\! \left(x \right)\\
F_{379}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{363}\! \left(x \right)+F_{380}\! \left(x \right)\\
F_{380}\! \left(x \right) &= F_{381}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{381}\! \left(x \right) &= F_{382}\! \left(x \right)+F_{384}\! \left(x \right)\\
F_{382}\! \left(x \right) &= F_{377}\! \left(x \right)+F_{383}\! \left(x \right)\\
F_{383}\! \left(x \right) &= F_{346}\! \left(x \right)\\
F_{384}\! \left(x \right) &= F_{385}\! \left(x \right)+F_{390}\! \left(x \right)\\
F_{385}\! \left(x \right) &= F_{386}\! \left(x \right)\\
F_{386}\! \left(x \right) &= F_{387}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{387}\! \left(x \right) &= F_{388}\! \left(x \right)+F_{389}\! \left(x \right)\\
F_{388}\! \left(x \right) &= F_{377}\! \left(x \right)\\
F_{389}\! \left(x \right) &= F_{385}\! \left(x \right)\\
F_{390}\! \left(x \right) &= F_{391}\! \left(x \right)\\
F_{391}\! \left(x \right) &= F_{392}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{392}\! \left(x \right) &= F_{393}\! \left(x \right)+F_{394}\! \left(x \right)\\
F_{393}\! \left(x \right) &= F_{383}\! \left(x \right)\\
F_{394}\! \left(x \right) &= F_{395}\! \left(x \right)\\
F_{395}\! \left(x \right) &= F_{396}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{396}\! \left(x \right) &= F_{397}\! \left(x \right)+F_{398}\! \left(x \right)\\
F_{397}\! \left(x \right) &= F_{383}\! \left(x \right)\\
F_{398}\! \left(x \right) &= F_{395}\! \left(x \right)\\
F_{399}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{400}\! \left(x \right)+F_{503}\! \left(x \right)\\
F_{400}\! \left(x \right) &= F_{4}\! \left(x \right) F_{401}\! \left(x \right)\\
F_{401}\! \left(x \right) &= F_{402}\! \left(x \right)+F_{410}\! \left(x \right)\\
F_{402}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{403}\! \left(x \right)\\
F_{403}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{404}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{404}\! \left(x \right) &= F_{4}\! \left(x \right) F_{405}\! \left(x \right)\\
F_{405}\! \left(x \right) &= F_{406}\! \left(x \right)+F_{407}\! \left(x \right)\\
F_{406}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{407}\! \left(x \right) &= F_{408}\! \left(x \right)\\
F_{408}\! \left(x \right) &= F_{4}\! \left(x \right) F_{409}\! \left(x \right)\\
F_{409}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{410}\! \left(x \right) &= F_{411}\! \left(x \right)+F_{477}\! \left(x \right)\\
F_{411}\! \left(x \right) &= F_{412}\! \left(x \right)\\
F_{412}\! \left(x \right) &= F_{4}\! \left(x \right) F_{413}\! \left(x \right)\\
F_{413}\! \left(x \right) &= F_{414}\! \left(x \right)+F_{420}\! \left(x \right)\\
F_{414}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{415}\! \left(x \right)\\
F_{415}\! \left(x \right) &= F_{416}\! \left(x \right)\\
F_{416}\! \left(x \right) &= F_{4}\! \left(x \right) F_{417}\! \left(x \right)\\
F_{417}\! \left(x \right) &= F_{418}\! \left(x \right)+F_{419}\! \left(x \right)\\
F_{418}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{419}\! \left(x \right) &= F_{408}\! \left(x \right)\\
F_{420}\! \left(x \right) &= F_{411}\! \left(x \right)+F_{421}\! \left(x \right)\\
F_{421}\! \left(x \right) &= 3 F_{18}\! \left(x \right)+F_{422}\! \left(x \right)+F_{443}\! \left(x \right)\\
F_{422}\! \left(x \right) &= F_{4}\! \left(x \right) F_{423}\! \left(x \right)\\
F_{423}\! \left(x \right) &= F_{424}\! \left(x \right)+F_{428}\! \left(x \right)\\
F_{424}\! \left(x \right) &= F_{415}\! \left(x \right)+F_{425}\! \left(x \right)\\
F_{425}\! \left(x \right) &= F_{426}\! \left(x \right)\\
F_{426}\! \left(x \right) &= F_{4}\! \left(x \right) F_{427}\! \left(x \right)\\
F_{427}\! \left(x \right) &= F_{419}\! \left(x \right)\\
F_{428}\! \left(x \right) &= F_{429}\! \left(x \right)+F_{434}\! \left(x \right)\\
F_{429}\! \left(x \right) &= F_{430}\! \left(x \right)\\
F_{430}\! \left(x \right) &= F_{4}\! \left(x \right) F_{431}\! \left(x \right)\\
F_{431}\! \left(x \right) &= F_{432}\! \left(x \right)+F_{433}\! \left(x \right)\\
F_{432}\! \left(x \right) &= F_{415}\! \left(x \right)\\
F_{433}\! \left(x \right) &= F_{429}\! \left(x \right)\\
F_{434}\! \left(x \right) &= F_{435}\! \left(x \right)\\
F_{435}\! \left(x \right) &= F_{4}\! \left(x \right) F_{436}\! \left(x \right)\\
F_{436}\! \left(x \right) &= F_{437}\! \left(x \right)+F_{438}\! \left(x \right)\\
F_{437}\! \left(x \right) &= F_{425}\! \left(x \right)\\
F_{438}\! \left(x \right) &= F_{439}\! \left(x \right)\\
F_{439}\! \left(x \right) &= F_{4}\! \left(x \right) F_{440}\! \left(x \right)\\
F_{440}\! \left(x \right) &= F_{441}\! \left(x \right)+F_{442}\! \left(x \right)\\
F_{441}\! \left(x \right) &= F_{425}\! \left(x \right)\\
F_{442}\! \left(x \right) &= F_{439}\! \left(x \right)\\
F_{443}\! \left(x \right) &= F_{4}\! \left(x \right) F_{444}\! \left(x \right)\\
F_{444}\! \left(x \right) &= F_{445}\! \left(x \right)+F_{472}\! \left(x \right)\\
F_{445}\! \left(x \right) &= F_{446}\! \left(x \right)\\
F_{446}\! \left(x \right) &= F_{447}\! \left(x \right)\\
F_{447}\! \left(x \right) &= F_{4}\! \left(x \right) F_{448}\! \left(x \right)\\
F_{448}\! \left(x \right) &= F_{449}\! \left(x \right)+F_{451}\! \left(x \right)\\
F_{449}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{450}\! \left(x \right)\\
F_{450}\! \left(x \right) &= F_{416}\! \left(x \right)\\
F_{451}\! \left(x \right) &= F_{446}\! \left(x \right)+F_{452}\! \left(x \right)\\
F_{452}\! \left(x \right) &= 3 F_{18}\! \left(x \right)+F_{443}\! \left(x \right)+F_{453}\! \left(x \right)\\
F_{453}\! \left(x \right) &= F_{4}\! \left(x \right) F_{454}\! \left(x \right)\\
F_{454}\! \left(x \right) &= F_{455}\! \left(x \right)+F_{457}\! \left(x \right)\\
F_{455}\! \left(x \right) &= F_{450}\! \left(x \right)+F_{456}\! \left(x \right)\\
F_{456}\! \left(x \right) &= F_{426}\! \left(x \right)\\
F_{457}\! \left(x \right) &= F_{458}\! \left(x \right)+F_{463}\! \left(x \right)\\
F_{458}\! \left(x \right) &= F_{459}\! \left(x \right)\\
F_{459}\! \left(x \right) &= F_{4}\! \left(x \right) F_{460}\! \left(x \right)\\
F_{460}\! \left(x \right) &= F_{461}\! \left(x \right)+F_{462}\! \left(x \right)\\
F_{461}\! \left(x \right) &= F_{450}\! \left(x \right)\\
F_{462}\! \left(x \right) &= F_{458}\! \left(x \right)\\
F_{463}\! \left(x \right) &= F_{464}\! \left(x \right)\\
F_{464}\! \left(x \right) &= F_{4}\! \left(x \right) F_{465}\! \left(x \right)\\
F_{465}\! \left(x \right) &= F_{466}\! \left(x \right)+F_{467}\! \left(x \right)\\
F_{466}\! \left(x \right) &= F_{456}\! \left(x \right)\\
F_{467}\! \left(x \right) &= F_{468}\! \left(x \right)\\
F_{468}\! \left(x \right) &= F_{4}\! \left(x \right) F_{469}\! \left(x \right)\\
F_{469}\! \left(x \right) &= F_{470}\! \left(x \right)+F_{471}\! \left(x \right)\\
F_{470}\! \left(x \right) &= F_{456}\! \left(x \right)\\
F_{471}\! \left(x \right) &= F_{468}\! \left(x \right)\\
F_{472}\! \left(x \right) &= F_{473}\! \left(x \right)\\
F_{473}\! \left(x \right) &= F_{4}\! \left(x \right) F_{474}\! \left(x \right)\\
F_{474}\! \left(x \right) &= F_{475}\! \left(x \right)+F_{476}\! \left(x \right)\\
F_{475}\! \left(x \right) &= F_{408}\! \left(x \right)\\
F_{476}\! \left(x \right) &= F_{473}\! \left(x \right)\\
F_{477}\! \left(x \right) &= 3 F_{18}\! \left(x \right)+F_{478}\! \left(x \right)+F_{499}\! \left(x \right)\\
F_{478}\! \left(x \right) &= F_{4}\! \left(x \right) F_{479}\! \left(x \right)\\
F_{479}\! \left(x \right) &= F_{480}\! \left(x \right)+F_{484}\! \left(x \right)\\
F_{480}\! \left(x \right) &= F_{403}\! \left(x \right)+F_{481}\! \left(x \right)\\
F_{481}\! \left(x \right) &= F_{482}\! \left(x \right)\\
F_{482}\! \left(x \right) &= F_{4}\! \left(x \right) F_{483}\! \left(x \right)\\
F_{483}\! \left(x \right) &= F_{407}\! \left(x \right)\\
F_{484}\! \left(x \right) &= F_{485}\! \left(x \right)+F_{490}\! \left(x \right)\\
F_{485}\! \left(x \right) &= F_{486}\! \left(x \right)\\
F_{486}\! \left(x \right) &= F_{4}\! \left(x \right) F_{487}\! \left(x \right)\\
F_{487}\! \left(x \right) &= F_{488}\! \left(x \right)+F_{489}\! \left(x \right)\\
F_{488}\! \left(x \right) &= F_{403}\! \left(x \right)\\
F_{489}\! \left(x \right) &= F_{485}\! \left(x \right)\\
F_{490}\! \left(x \right) &= F_{491}\! \left(x \right)\\
F_{491}\! \left(x \right) &= F_{4}\! \left(x \right) F_{492}\! \left(x \right)\\
F_{492}\! \left(x \right) &= F_{493}\! \left(x \right)+F_{494}\! \left(x \right)\\
F_{493}\! \left(x \right) &= F_{481}\! \left(x \right)\\
F_{494}\! \left(x \right) &= F_{495}\! \left(x \right)\\
F_{495}\! \left(x \right) &= F_{4}\! \left(x \right) F_{496}\! \left(x \right)\\
F_{496}\! \left(x \right) &= F_{497}\! \left(x \right)+F_{498}\! \left(x \right)\\
F_{497}\! \left(x \right) &= F_{481}\! \left(x \right)\\
F_{498}\! \left(x \right) &= F_{495}\! \left(x \right)\\
F_{499}\! \left(x \right) &= F_{4}\! \left(x \right) F_{500}\! \left(x \right)\\
F_{500}\! \left(x \right) &= F_{501}\! \left(x \right)+F_{502}\! \left(x \right)\\
F_{501}\! \left(x \right) &= F_{446}\! \left(x \right)\\
F_{502}\! \left(x \right) &= F_{473}\! \left(x \right)\\
F_{503}\! \left(x \right) &= F_{4}\! \left(x \right) F_{504}\! \left(x \right)\\
F_{504}\! \left(x \right) &= F_{367}\! \left(x \right)+F_{373}\! \left(x \right)\\
\end{align*}\)