Av(1234, 1342, 2413, 2431, 3241)
Generating Function
\(\displaystyle -\frac{4 x^{8}+2 x^{7}-4 x^{6}+x^{5}+x^{4}+4 x^{3}-9 x^{2}+5 x -1}{\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 19, 55, 147, 364, 846, 1872, 3988, 8251, 16690, 33173, 65034, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+4 x^{8}+2 x^{7}-4 x^{6}+x^{5}+x^{4}+4 x^{3}-9 x^{2}+5 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 147\)
\(\displaystyle a \! \left(7\right) = 364\)
\(\displaystyle a \! \left(8\right) = 846\)
\(\displaystyle a \! \left(n +5\right) = -\frac{n^{3}}{2}+\frac{9 n^{2}}{2}-a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)+9 n +16, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 147\)
\(\displaystyle a \! \left(7\right) = 364\)
\(\displaystyle a \! \left(8\right) = 846\)
\(\displaystyle a \! \left(n +5\right) = -\frac{n^{3}}{2}+\frac{9 n^{2}}{2}-a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)+9 n +16, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \frac{\left(\left(\left(-2760 \sqrt{3}-8280 \,\mathrm{I}\right) \sqrt{11}+10780 \,\mathrm{I} \sqrt{3}+10780\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+32120+\left(\left(-10935 \,\mathrm{I}+3645 \sqrt{3}\right) \sqrt{11}+22495 \,\mathrm{I} \sqrt{3}-22495\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}}{5280}+\frac{\left(\left(\left(8280 \,\mathrm{I}-2760 \sqrt{3}\right) \sqrt{11}-10780 \,\mathrm{I} \sqrt{3}+10780\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+32120+\left(\left(10935 \,\mathrm{I}+3645 \sqrt{3}\right) \sqrt{11}-22495 \,\mathrm{I} \sqrt{3}-22495\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}}{5280}+\frac{\left(\left(5520 \sqrt{11}\, \sqrt{3}-21560\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-7290 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}+44990 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+32120\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}}{5280}+\frac{\left(40128 \sqrt{5}-89760\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5280}+\frac{\left(-40128 \sqrt{5}-89760\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{5280}-\frac{n^{3}}{4}+\frac{3 n^{2}}{2}+\frac{15 n}{4}+\frac{67}{4}\)
This specification was found using the strategy pack "Point Placements" and has 281 rules.
Found on January 18, 2022.Finding the specification took 14 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_{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_{19}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{216}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{21}\! \left(x \right) &= 0\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{26}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{48}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{60}\! \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_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{55}\! \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_{51}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{69}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{80}\! \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_{95}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{92}\! \left(x \right)\\
F_{96}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{187}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{100}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{101}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{104}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{163}\! \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_{113}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= 3 F_{21}\! \left(x \right)+F_{115}\! \left(x \right)+F_{136}\! \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_{121}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{112}\! \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_{37}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{126}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{112}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{122}\! \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_{130}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{118}\! \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_{134}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{132}\! \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_{138}\! \left(x \right) &= F_{139}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{142}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{143}\! \left(x \right)\\
F_{143}\! \left(x \right) &= 3 F_{21}\! \left(x \right)+F_{136}\! \left(x \right)+F_{144}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{148}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{154}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{153}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{149}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{156}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{158}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{147}\! \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_{147}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{159}\! \left(x \right)\\
F_{163}\! \left(x \right) &= 3 F_{21}\! \left(x \right)+F_{164}\! \left(x \right)+F_{185}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{165}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{170}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{167}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{168}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{169}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)+F_{176}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{172}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{173}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{174}\! \left(x \right)+F_{175}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{171}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{178}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{179}\! \left(x \right)+F_{180}\! \left(x \right)\\
F_{179}\! \left(x \right) &= F_{167}\! \left(x \right)\\
F_{180}\! \left(x \right) &= F_{181}\! \left(x \right)\\
F_{181}\! \left(x \right) &= F_{182}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{183}\! \left(x \right)+F_{184}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{167}\! \left(x \right)\\
F_{184}\! \left(x \right) &= F_{181}\! \left(x \right)\\
F_{185}\! \left(x \right) &= F_{186}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{186}\! \left(x \right) &= F_{138}\! \left(x \right)\\
F_{187}\! \left(x \right) &= F_{188}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{188}\! \left(x \right) &= F_{189}\! \left(x \right)\\
F_{189}\! \left(x \right) &= F_{190}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{190}\! \left(x \right) &= F_{191}\! \left(x \right)\\
F_{191}\! \left(x \right) &= F_{192}\! \left(x \right) F_{4}\! \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_{29}\! \left(x \right)\\
F_{194}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{195}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{196}\! \left(x \right)\\
F_{196}\! \left(x \right) &= 3 F_{21}\! \left(x \right)+F_{185}\! \left(x \right)+F_{197}\! \left(x \right)\\
F_{197}\! \left(x \right) &= F_{198}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{198}\! \left(x \right) &= F_{199}\! \left(x \right)+F_{201}\! \left(x \right)\\
F_{199}\! \left(x \right) &= F_{194}\! \left(x \right)+F_{200}\! \left(x \right)\\
F_{200}\! \left(x \right) &= F_{168}\! \left(x \right)\\
F_{201}\! \left(x \right) &= F_{202}\! \left(x \right)+F_{207}\! \left(x \right)\\
F_{202}\! \left(x \right) &= F_{203}\! \left(x \right)\\
F_{203}\! \left(x \right) &= F_{204}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{204}\! \left(x \right) &= F_{205}\! \left(x \right)+F_{206}\! \left(x \right)\\
F_{205}\! \left(x \right) &= F_{194}\! \left(x \right)\\
F_{206}\! \left(x \right) &= F_{202}\! \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_{210}\! \left(x \right)+F_{211}\! \left(x \right)\\
F_{210}\! \left(x \right) &= F_{200}\! \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_{200}\! \left(x \right)\\
F_{215}\! \left(x \right) &= F_{212}\! \left(x \right)\\
F_{216}\! \left(x \right) &= F_{217}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{217}\! \left(x \right) &= F_{218}\! \left(x \right)\\
F_{218}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{219}\! \left(x \right)\\
F_{219}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{220}\! \left(x \right)+F_{279}\! \left(x \right)\\
F_{220}\! \left(x \right) &= F_{221}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{221}\! \left(x \right) &= F_{222}\! \left(x \right)+F_{226}\! \left(x \right)\\
F_{222}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{223}\! \left(x \right)\\
F_{223}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{224}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{224}\! \left(x \right) &= F_{225}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{225}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{226}\! \left(x \right) &= F_{227}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{227}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{187}\! \left(x \right)+F_{228}\! \left(x \right)\\
F_{228}\! \left(x \right) &= F_{229}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{229}\! \left(x \right) &= F_{230}\! \left(x \right)+F_{232}\! \left(x \right)\\
F_{230}\! \left(x \right) &= F_{223}\! \left(x \right)+F_{231}\! \left(x \right)\\
F_{231}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{101}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{232}\! \left(x \right) &= F_{233}\! \left(x \right)+F_{259}\! \left(x \right)\\
F_{233}\! \left(x \right) &= F_{234}\! \left(x \right)\\
F_{234}\! \left(x \right) &= F_{235}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{235}\! \left(x \right) &= F_{236}\! \left(x \right)+F_{238}\! \left(x \right)\\
F_{236}\! \left(x \right) &= F_{223}\! \left(x \right)+F_{237}\! \left(x \right)\\
F_{237}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{238}\! \left(x \right) &= F_{233}\! \left(x \right)+F_{239}\! \left(x \right)\\
F_{239}\! \left(x \right) &= 3 F_{21}\! \left(x \right)+F_{136}\! \left(x \right)+F_{240}\! \left(x \right)\\
F_{240}\! \left(x \right) &= F_{241}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{241}\! \left(x \right) &= F_{242}\! \left(x \right)+F_{244}\! \left(x \right)\\
F_{242}\! \left(x \right) &= F_{237}\! \left(x \right)+F_{243}\! \left(x \right)\\
F_{243}\! \left(x \right) &= F_{119}\! \left(x \right)\\
F_{244}\! \left(x \right) &= F_{245}\! \left(x \right)+F_{250}\! \left(x \right)\\
F_{245}\! \left(x \right) &= F_{246}\! \left(x \right)\\
F_{246}\! \left(x \right) &= F_{247}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{247}\! \left(x \right) &= F_{248}\! \left(x \right)+F_{249}\! \left(x \right)\\
F_{248}\! \left(x \right) &= F_{237}\! \left(x \right)\\
F_{249}\! \left(x \right) &= F_{245}\! \left(x \right)\\
F_{250}\! \left(x \right) &= F_{251}\! \left(x \right)\\
F_{251}\! \left(x \right) &= F_{252}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{252}\! \left(x \right) &= F_{253}\! \left(x \right)+F_{254}\! \left(x \right)\\
F_{253}\! \left(x \right) &= F_{243}\! \left(x \right)\\
F_{254}\! \left(x \right) &= F_{255}\! \left(x \right)\\
F_{255}\! \left(x \right) &= F_{256}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{256}\! \left(x \right) &= F_{257}\! \left(x \right)+F_{258}\! \left(x \right)\\
F_{257}\! \left(x \right) &= F_{243}\! \left(x \right)\\
F_{258}\! \left(x \right) &= F_{255}\! \left(x \right)\\
F_{259}\! \left(x \right) &= 3 F_{21}\! \left(x \right)+F_{185}\! \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_{264}\! \left(x \right)\\
F_{262}\! \left(x \right) &= F_{231}\! \left(x \right)+F_{263}\! \left(x \right)\\
F_{263}\! \left(x \right) &= F_{168}\! \left(x \right)\\
F_{264}\! \left(x \right) &= F_{265}\! \left(x \right)+F_{270}\! \left(x \right)\\
F_{265}\! \left(x \right) &= F_{266}\! \left(x \right)\\
F_{266}\! \left(x \right) &= F_{267}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{267}\! \left(x \right) &= F_{268}\! \left(x \right)+F_{269}\! \left(x \right)\\
F_{268}\! \left(x \right) &= F_{231}\! \left(x \right)\\
F_{269}\! \left(x \right) &= F_{265}\! \left(x \right)\\
F_{270}\! \left(x \right) &= F_{271}\! \left(x \right)\\
F_{271}\! \left(x \right) &= F_{272}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{272}\! \left(x \right) &= F_{273}\! \left(x \right)+F_{274}\! \left(x \right)\\
F_{273}\! \left(x \right) &= F_{263}\! \left(x \right)\\
F_{274}\! \left(x \right) &= F_{275}\! \left(x \right)\\
F_{275}\! \left(x \right) &= F_{276}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{276}\! \left(x \right) &= F_{277}\! \left(x \right)+F_{278}\! \left(x \right)\\
F_{277}\! \left(x \right) &= F_{263}\! \left(x \right)\\
F_{278}\! \left(x \right) &= F_{275}\! \left(x \right)\\
F_{279}\! \left(x \right) &= F_{280}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{280}\! \left(x \right) &= F_{2}\! \left(x \right)\\
\end{align*}\)