Av(1234, 1342, 1423, 3241)
Generating Function
\(\displaystyle \frac{x^{9}-4 x^{8}+12 x^{7}-28 x^{6}+43 x^{5}-55 x^{4}+49 x^{3}-27 x^{2}+8 x -1}{\left(2 x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 193, 554, 1528, 4075, 10559, 26696, 66107, 160873, 385841, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{5} F \! \left(x \right)-x^{9}+4 x^{8}-12 x^{7}+28 x^{6}-43 x^{5}+55 x^{4}-49 x^{3}+27 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 193\)
\(\displaystyle a \! \left(7\right) = 554\)
\(\displaystyle a \! \left(8\right) = 1528\)
\(\displaystyle a \! \left(9\right) = 4075\)
\(\displaystyle a \! \left(n +4\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)+\frac{\left(n +1\right) \left(n^{3}+3 n^{2}-4 n +48\right)}{12}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 193\)
\(\displaystyle a \! \left(7\right) = 554\)
\(\displaystyle a \! \left(8\right) = 1528\)
\(\displaystyle a \! \left(9\right) = 4075\)
\(\displaystyle a \! \left(n +4\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)+\frac{\left(n +1\right) \left(n^{3}+3 n^{2}-4 n +48\right)}{12}, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle 9+\frac{23 n^{2}}{24}+\frac{23 n}{4}+\frac{n^{3}}{4}+\frac{n^{4}}{24}-\frac{6994 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{59}+\frac{110 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{59}-\frac{13935 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{59}+\frac{7239 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{59}+\left(\left\{\begin{array}{cc}\frac{1}{2} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 377 rules.
Found on January 18, 2022.Finding the specification took 28 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_{17}\! \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_{11}\! \left(x \right)+F_{14}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{299}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{41}\! \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_{50}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{52}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{45}\! \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_{49}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{65}\! \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_{45}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \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_{73}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{74}\! \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_{88}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{102}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{99}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{105}\! \left(x \right)+F_{124}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{109}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{115}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{110}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{120}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{126}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{127}\! \left(x \right)+F_{254}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{141}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{130}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{131}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{134}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{37}\! \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_{138}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{227}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{151}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{150}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{152}\! \left(x \right)\\
F_{152}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{153}\! \left(x \right)+F_{175}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{154}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{160}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{156}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{159}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{138}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{161}\! \left(x \right)+F_{166}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{163}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{164}\! \left(x \right)+F_{165}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{146}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{161}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{168}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)+F_{170}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{156}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{172}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{173}\! \left(x \right)+F_{174}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{156}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{171}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{176}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)+F_{204}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{178}\! \left(x \right)\\
F_{178}\! \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_{182}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{147}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{178}\! \left(x \right)+F_{184}\! \left(x \right)\\
F_{184}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{175}\! \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_{189}\! \left(x \right)\\
F_{187}\! \left(x \right) &= F_{182}\! \left(x \right)+F_{188}\! \left(x \right)\\
F_{188}\! \left(x \right) &= F_{157}\! \left(x \right)\\
F_{189}\! \left(x \right) &= F_{190}\! \left(x \right)+F_{195}\! \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_{194}\! \left(x \right)\\
F_{193}\! \left(x \right) &= F_{182}\! \left(x \right)\\
F_{194}\! \left(x \right) &= F_{190}\! \left(x \right)\\
F_{195}\! \left(x \right) &= F_{196}\! \left(x \right)\\
F_{196}\! \left(x \right) &= F_{197}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{197}\! \left(x \right) &= F_{198}\! \left(x \right)+F_{199}\! \left(x \right)\\
F_{198}\! \left(x \right) &= F_{188}\! \left(x \right)\\
F_{199}\! \left(x \right) &= F_{200}\! \left(x \right)\\
F_{200}\! \left(x \right) &= F_{201}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{201}\! \left(x \right) &= F_{202}\! \left(x \right)+F_{203}\! \left(x \right)\\
F_{202}\! \left(x \right) &= F_{188}\! \left(x \right)\\
F_{203}\! \left(x \right) &= F_{200}\! \left(x \right)\\
F_{204}\! \left(x \right) &= F_{205}\! \left(x \right)\\
F_{205}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{206}\! \left(x \right)+F_{225}\! \left(x \right)\\
F_{206}\! \left(x \right) &= F_{207}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{207}\! \left(x \right) &= F_{208}\! \left(x \right)+F_{210}\! \left(x \right)\\
F_{208}\! \left(x \right) &= F_{209}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{209}\! \left(x \right) &= F_{139}\! \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_{31}\! \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_{209}\! \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_{209}\! \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_{178}\! \left(x \right)+F_{205}\! \left(x \right)\\
F_{227}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{228}\! \left(x \right)+F_{250}\! \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_{235}\! \left(x \right)\\
F_{230}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{231}\! \left(x \right)\\
F_{231}\! \left(x \right) &= F_{232}\! \left(x \right)\\
F_{232}\! \left(x \right) &= F_{233}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{233}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{234}\! \left(x \right)\\
F_{234}\! \left(x \right) &= F_{138}\! \left(x \right)\\
F_{235}\! \left(x \right) &= F_{236}\! \left(x \right)+F_{241}\! \left(x \right)\\
F_{236}\! \left(x \right) &= F_{237}\! \left(x \right)\\
F_{237}\! \left(x \right) &= F_{238}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{238}\! \left(x \right) &= F_{239}\! \left(x \right)+F_{240}\! \left(x \right)\\
F_{239}\! \left(x \right) &= F_{130}\! \left(x \right)\\
F_{240}\! \left(x \right) &= F_{236}\! \left(x \right)\\
F_{241}\! \left(x \right) &= F_{242}\! \left(x \right)\\
F_{242}\! \left(x \right) &= F_{243}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{243}\! \left(x \right) &= F_{244}\! \left(x \right)+F_{245}\! \left(x \right)\\
F_{244}\! \left(x \right) &= F_{231}\! \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_{231}\! \left(x \right)\\
F_{249}\! \left(x \right) &= F_{246}\! \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_{178}\! \left(x \right)\\
F_{253}\! \left(x \right) &= F_{205}\! \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_{283}\! \left(x \right)\\
F_{256}\! \left(x \right) &= F_{257}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{257}\! \left(x \right) &= F_{258}\! \left(x \right)\\
F_{258}\! \left(x \right) &= F_{259}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{259}\! \left(x \right) &= F_{260}\! \left(x \right)+F_{262}\! \left(x \right)\\
F_{260}\! \left(x \right) &= F_{261}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{261}\! \left(x \right) &= F_{131}\! \left(x \right)\\
F_{262}\! \left(x \right) &= F_{178}\! \left(x \right)+F_{263}\! \left(x \right)\\
F_{263}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{250}\! \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_{268}\! \left(x \right)\\
F_{266}\! \left(x \right) &= F_{261}\! \left(x \right)+F_{267}\! \left(x \right)\\
F_{267}\! \left(x \right) &= F_{232}\! \left(x \right)\\
F_{268}\! \left(x \right) &= F_{269}\! \left(x \right)+F_{274}\! \left(x \right)\\
F_{269}\! \left(x \right) &= F_{270}\! \left(x \right)\\
F_{270}\! \left(x \right) &= F_{271}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{271}\! \left(x \right) &= F_{272}\! \left(x \right)+F_{273}\! \left(x \right)\\
F_{272}\! \left(x \right) &= F_{261}\! \left(x \right)\\
F_{273}\! \left(x \right) &= F_{269}\! \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_{267}\! \left(x \right)\\
F_{278}\! \left(x \right) &= F_{279}\! \left(x \right)\\
F_{279}\! \left(x \right) &= F_{280}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{280}\! \left(x \right) &= F_{281}\! \left(x \right)+F_{282}\! \left(x \right)\\
F_{281}\! \left(x \right) &= F_{267}\! \left(x \right)\\
F_{282}\! \left(x \right) &= F_{279}\! \left(x \right)\\
F_{283}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{284}\! \left(x \right)\\
F_{284}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{225}\! \left(x \right)+F_{285}\! \left(x \right)\\
F_{285}\! \left(x \right) &= F_{286}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{286}\! \left(x \right) &= F_{287}\! \left(x \right)+F_{289}\! \left(x \right)\\
F_{287}\! \left(x \right) &= F_{288}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{288}\! \left(x \right) &= F_{139}\! \left(x \right)\\
F_{289}\! \left(x \right) &= F_{211}\! \left(x \right)+F_{290}\! \left(x \right)\\
F_{290}\! \left(x \right) &= F_{291}\! \left(x \right)\\
F_{291}\! \left(x \right) &= F_{292}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{292}\! \left(x \right) &= F_{293}\! \left(x \right)+F_{294}\! \left(x \right)\\
F_{293}\! \left(x \right) &= F_{288}\! \left(x \right)\\
F_{294}\! \left(x \right) &= F_{295}\! \left(x \right)\\
F_{295}\! \left(x \right) &= F_{296}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{296}\! \left(x \right) &= F_{297}\! \left(x \right)+F_{298}\! \left(x \right)\\
F_{297}\! \left(x \right) &= F_{288}\! \left(x \right)\\
F_{298}\! \left(x \right) &= F_{295}\! \left(x \right)\\
F_{299}\! \left(x \right) &= F_{300}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{300}\! \left(x \right) &= F_{301}\! \left(x \right)+F_{361}\! \left(x \right)\\
F_{301}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{302}\! \left(x \right)\\
F_{302}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{303}\! \left(x \right)+F_{360}\! \left(x \right)\\
F_{303}\! \left(x \right) &= F_{304}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{304}\! \left(x \right) &= F_{305}\! \left(x \right)+F_{307}\! \left(x \right)\\
F_{305}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{306}\! \left(x \right)\\
F_{306}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{307}\! \left(x \right) &= F_{308}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{308}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{254}\! \left(x \right)+F_{309}\! \left(x \right)\\
F_{309}\! \left(x \right) &= F_{310}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{310}\! \left(x \right) &= F_{311}\! \left(x \right)+F_{313}\! \left(x \right)\\
F_{311}\! \left(x \right) &= F_{306}\! \left(x \right)+F_{312}\! \left(x \right)\\
F_{312}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{131}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{313}\! \left(x \right) &= F_{314}\! \left(x \right)+F_{340}\! \left(x \right)\\
F_{314}\! \left(x \right) &= F_{315}\! \left(x \right)\\
F_{315}\! \left(x \right) &= F_{316}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{316}\! \left(x \right) &= F_{317}\! \left(x \right)+F_{319}\! \left(x \right)\\
F_{317}\! \left(x \right) &= F_{306}\! \left(x \right)+F_{318}\! \left(x \right)\\
F_{318}\! \left(x \right) &= F_{147}\! \left(x \right)\\
F_{319}\! \left(x \right) &= F_{314}\! \left(x \right)+F_{320}\! \left(x \right)\\
F_{320}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{175}\! \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_{325}\! \left(x \right)\\
F_{323}\! \left(x \right) &= F_{318}\! \left(x \right)+F_{324}\! \left(x \right)\\
F_{324}\! \left(x \right) &= F_{157}\! \left(x \right)\\
F_{325}\! \left(x \right) &= F_{326}\! \left(x \right)+F_{331}\! \left(x \right)\\
F_{326}\! \left(x \right) &= F_{327}\! \left(x \right)\\
F_{327}\! \left(x \right) &= F_{328}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{328}\! \left(x \right) &= F_{329}\! \left(x \right)+F_{330}\! \left(x \right)\\
F_{329}\! \left(x \right) &= F_{318}\! \left(x \right)\\
F_{330}\! \left(x \right) &= F_{326}\! \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_{335}\! \left(x \right)\\
F_{334}\! \left(x \right) &= F_{324}\! \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_{324}\! \left(x \right)\\
F_{339}\! \left(x \right) &= F_{336}\! \left(x \right)\\
F_{340}\! \left(x \right) &= 3 F_{19}\! \left(x \right)+F_{250}\! \left(x \right)+F_{341}\! \left(x \right)\\
F_{341}\! \left(x \right) &= F_{342}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{342}\! \left(x \right) &= F_{343}\! \left(x \right)+F_{345}\! \left(x \right)\\
F_{343}\! \left(x \right) &= F_{312}\! \left(x \right)+F_{344}\! \left(x \right)\\
F_{344}\! \left(x \right) &= F_{232}\! \left(x \right)\\
F_{345}\! \left(x \right) &= F_{346}\! \left(x \right)+F_{351}\! \left(x \right)\\
F_{346}\! \left(x \right) &= F_{347}\! \left(x \right)\\
F_{347}\! \left(x \right) &= F_{348}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{348}\! \left(x \right) &= F_{349}\! \left(x \right)+F_{350}\! \left(x \right)\\
F_{349}\! \left(x \right) &= F_{312}\! \left(x \right)\\
F_{350}\! \left(x \right) &= F_{346}\! \left(x \right)\\
F_{351}\! \left(x \right) &= F_{352}\! \left(x \right)\\
F_{352}\! \left(x \right) &= F_{353}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{353}\! \left(x \right) &= F_{354}\! \left(x \right)+F_{355}\! \left(x \right)\\
F_{354}\! \left(x \right) &= F_{344}\! \left(x \right)\\
F_{355}\! \left(x \right) &= F_{356}\! \left(x \right)\\
F_{356}\! \left(x \right) &= F_{357}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{357}\! \left(x \right) &= F_{358}\! \left(x \right)+F_{359}\! \left(x \right)\\
F_{358}\! \left(x \right) &= F_{344}\! \left(x \right)\\
F_{359}\! \left(x \right) &= F_{356}\! \left(x \right)\\
F_{360}\! \left(x \right) &= F_{301}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{361}\! \left(x \right) &= F_{302}\! \left(x \right)+F_{362}\! \left(x \right)\\
F_{362}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{124}\! \left(x \right)+F_{363}\! \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_{367}\! \left(x \right)\\
F_{365}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{366}\! \left(x \right)\\
F_{366}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{367}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{368}\! \left(x \right)\\
F_{368}\! \left(x \right) &= F_{369}\! \left(x \right)\\
F_{369}\! \left(x \right) &= F_{370}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{370}\! \left(x \right) &= F_{371}\! \left(x \right)+F_{372}\! \left(x \right)\\
F_{371}\! \left(x \right) &= F_{366}\! \left(x \right)\\
F_{372}\! \left(x \right) &= F_{373}\! \left(x \right)\\
F_{373}\! \left(x \right) &= F_{374}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{374}\! \left(x \right) &= F_{375}\! \left(x \right)+F_{376}\! \left(x \right)\\
F_{375}\! \left(x \right) &= F_{366}\! \left(x \right)\\
F_{376}\! \left(x \right) &= F_{373}\! \left(x \right)\\
\end{align*}\)