Av(1243, 1342, 4231)
Generating Function
\(\displaystyle -\frac{28 x^{7}-111 x^{6}+204 x^{5}-214 x^{4}+136 x^{3}-52 x^{2}+11 x -1}{\left(2 x -1\right)^{4} \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 259, 852, 2669, 7997, 23043, 64190, 173677, 458255, 1183139, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{4} \left(x -1\right)^{4} F \! \left(x \right)+28 x^{7}-111 x^{6}+204 x^{5}-214 x^{4}+136 x^{3}-52 x^{2}+11 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 259\)
\(\displaystyle a \! \left(7\right) = 852\)
\(\displaystyle a \! \left(n +4\right) = -\frac{n^{3}}{6}+\frac{3 n^{2}}{2}-16 a \! \left(n \right)+32 a \! \left(n +1\right)-24 a \! \left(n +2\right)+8 a \! \left(n +3\right)-\frac{10 n}{3}+5, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 259\)
\(\displaystyle a \! \left(7\right) = 852\)
\(\displaystyle a \! \left(n +4\right) = -\frac{n^{3}}{6}+\frac{3 n^{2}}{2}-16 a \! \left(n \right)+32 a \! \left(n +1\right)-24 a \! \left(n +2\right)+8 a \! \left(n +3\right)-\frac{10 n}{3}+5, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \frac{\left(n^{3}-6 n^{2}+5 n +96\right) 2^{n}}{24}-\frac{n^{3}}{6}-\frac{n^{2}}{2}-\frac{10 n}{3}-3\)
This specification was found using the strategy pack "Row Placements" and has 355 rules.
Found on July 23, 2021.Finding the specification took 15 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_{25}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{348}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{25}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{283}\! \left(x \right)+F_{331}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{25}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{275}\! \left(x \right)+F_{276}\! \left(x \right)+F_{277}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{25}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{272}\! \left(x \right)+F_{273}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{193}\! \left(x \right)+F_{195}\! \left(x \right)+F_{253}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{108}\! \left(x \right)+F_{14}\! \left(x \right)+F_{181}\! \left(x \right)+F_{182}\! \left(x \right)+F_{193}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{25}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{16}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= x\\
F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{25}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{25}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{30}\! \left(x \right) F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{25}\! \left(x \right) F_{26}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{16}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{25}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{46}\! \left(x \right) &= 0\\
F_{47}\! \left(x \right) &= F_{25}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{25}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{54}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{25}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{25}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{25}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{62}\! \left(x \right) &= 2 F_{46}\! \left(x \right)+F_{63}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{25}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{25}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{25}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{25}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= 2 F_{46}\! \left(x \right)+F_{78}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{25}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{25}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= 2 F_{46}\! \left(x \right)+F_{85}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{25}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{25}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{25}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{93}\! \left(x \right) &= 3 F_{46}\! \left(x \right)+F_{94}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{25}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{25}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= 2 F_{46}\! \left(x \right)+F_{101}\! \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) &= F_{53}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{104}\! \left(x \right) &= 3 F_{46}\! \left(x \right)+F_{105}\! \left(x \right)+F_{107}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{103}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{155}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{139}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{125}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{123}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{120}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{129}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{128}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{135}\! \left(x \right) &= 3 F_{46}\! \left(x \right)+F_{136}\! \left(x \right)+F_{138}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{131}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)\\
F_{140}\! \left(x \right) &= 2 F_{46}\! \left(x \right)+F_{141}\! \left(x \right)+F_{143}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{150}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{146}\! \left(x \right) &= 2 F_{46}\! \left(x \right)+F_{147}\! \left(x \right)+F_{149}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{145}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{151}\! \left(x \right)\\
F_{151}\! \left(x \right) &= 3 F_{46}\! \left(x \right)+F_{152}\! \left(x \right)+F_{154}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{153}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{151}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{150}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{170}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{157}\! \left(x \right)\\
F_{157}\! \left(x \right) &= 3 F_{46}\! \left(x \right)+F_{158}\! \left(x \right)+F_{160}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{159}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{157}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{161}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{162}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{166}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{164}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{165}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{163}\! \left(x \right)\\
F_{166}\! \left(x \right) &= 4 F_{46}\! \left(x \right)+F_{167}\! \left(x \right)+F_{169}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{168}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{166}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{162}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)\\
F_{171}\! \left(x \right) &= 3 F_{46}\! \left(x \right)+F_{172}\! \left(x \right)+F_{174}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{173}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{171}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{175}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{176}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{177}\! \left(x \right)\\
F_{177}\! \left(x \right) &= 4 F_{46}\! \left(x \right)+F_{178}\! \left(x \right)+F_{180}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{179}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{179}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{177}\! \left(x \right)\\
F_{180}\! \left(x \right) &= F_{176}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{181}\! \left(x \right) &= F_{25}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{183}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{184}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{184}\! \left(x \right) &= F_{185}\! \left(x \right)+F_{189}\! \left(x \right)+F_{190}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{185}\! \left(x \right) &= F_{186}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{186}\! \left(x \right) &= F_{185}\! \left(x \right)+F_{187}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{187}\! \left(x \right) &= F_{188}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{188}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{189}\! \left(x \right) &= F_{184}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{190}\! \left(x \right) &= F_{191}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{191}\! \left(x \right) &= F_{188}\! \left(x \right)+F_{192}\! \left(x \right)\\
F_{192}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{193}\! \left(x \right) &= F_{194}\! \left(x \right)\\
F_{194}\! \left(x \right) &= F_{186}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{195}\! \left(x \right) &= F_{196}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{196}\! \left(x \right) &= F_{197}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{197}\! \left(x \right) &= F_{198}\! \left(x \right)+F_{215}\! \left(x \right)\\
F_{198}\! \left(x \right) &= F_{199}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{199}\! \left(x \right) &= F_{200}\! \left(x \right)\\
F_{200}\! \left(x \right) &= F_{201}\! \left(x \right)+F_{203}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{201}\! \left(x \right) &= F_{202}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{202}\! \left(x \right) &= F_{200}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{203}\! \left(x \right) &= F_{204}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{204}\! \left(x \right) &= F_{205}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{205}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{206}\! \left(x \right)\\
F_{206}\! \left(x \right) &= F_{207}\! \left(x \right)+F_{212}\! \left(x \right)+F_{214}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{207}\! \left(x \right) &= F_{208}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{208}\! \left(x \right) &= F_{209}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{209}\! \left(x \right) &= 2 F_{46}\! \left(x \right)+F_{207}\! \left(x \right)+F_{210}\! \left(x \right)\\
F_{210}\! \left(x \right) &= F_{211}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{211}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{209}\! \left(x \right)\\
F_{212}\! \left(x \right) &= F_{213}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{213}\! \left(x \right) &= F_{206}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{214}\! \left(x \right) &= F_{205}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{215}\! \left(x \right) &= F_{216}\! \left(x \right)+F_{217}\! \left(x \right)\\
F_{216}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{140}\! \left(x \right)\\
F_{217}\! \left(x \right) &= F_{218}\! \left(x \right)\\
F_{218}\! \left(x \right) &= F_{219}\! \left(x \right)+F_{241}\! \left(x \right)+F_{243}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{219}\! \left(x \right) &= F_{220}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{220}\! \left(x \right) &= F_{221}\! \left(x \right)+F_{222}\! \left(x \right)\\
F_{221}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{47}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{222}\! \left(x \right) &= 2 F_{46}\! \left(x \right)+F_{219}\! \left(x \right)+F_{223}\! \left(x \right)\\
F_{223}\! \left(x \right) &= F_{224}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{224}\! \left(x \right) &= F_{211}\! \left(x \right)+F_{225}\! \left(x \right)\\
F_{225}\! \left(x \right) &= F_{226}\! \left(x \right)+F_{232}\! \left(x \right)\\
F_{226}\! \left(x \right) &= 2 F_{46}\! \left(x \right)+F_{227}\! \left(x \right)+F_{230}\! \left(x \right)\\
F_{227}\! \left(x \right) &= F_{228}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{228}\! \left(x \right) &= F_{229}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{229}\! \left(x \right) &= F_{227}\! \left(x \right)\\
F_{230}\! \left(x \right) &= F_{231}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{231}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{226}\! \left(x \right)\\
F_{232}\! \left(x \right) &= 2 F_{46}\! \left(x \right)+F_{233}\! \left(x \right)+F_{238}\! \left(x \right)+F_{240}\! \left(x \right)\\
F_{233}\! \left(x \right) &= F_{234}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{234}\! \left(x \right) &= F_{235}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{235}\! \left(x \right) &= 3 F_{46}\! \left(x \right)+F_{233}\! \left(x \right)+F_{236}\! \left(x \right)\\
F_{236}\! \left(x \right) &= F_{237}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{237}\! \left(x \right) &= F_{229}\! \left(x \right)+F_{235}\! \left(x \right)\\
F_{238}\! \left(x \right) &= F_{239}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{239}\! \left(x \right) &= F_{209}\! \left(x \right)+F_{232}\! \left(x \right)\\
F_{240}\! \left(x \right) &= F_{225}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{241}\! \left(x \right) &= F_{242}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{242}\! \left(x \right) &= F_{200}\! \left(x \right)+F_{218}\! \left(x \right)\\
F_{243}\! \left(x \right) &= F_{244}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{244}\! \left(x \right) &= F_{205}\! \left(x \right)+F_{245}\! \left(x \right)\\
F_{245}\! \left(x \right) &= F_{246}\! \left(x \right)+F_{249}\! \left(x \right)\\
F_{246}\! \left(x \right) &= 2 F_{46}\! \left(x \right)+F_{227}\! \left(x \right)+F_{247}\! \left(x \right)\\
F_{247}\! \left(x \right) &= F_{248}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{248}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{246}\! \left(x \right)\\
F_{249}\! \left(x \right) &= 2 F_{46}\! \left(x \right)+F_{233}\! \left(x \right)+F_{250}\! \left(x \right)+F_{252}\! \left(x \right)\\
F_{250}\! \left(x \right) &= F_{25}\! \left(x \right) F_{251}\! \left(x \right)\\
F_{251}\! \left(x \right) &= F_{206}\! \left(x \right)+F_{249}\! \left(x \right)\\
F_{252}\! \left(x \right) &= F_{245}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{253}\! \left(x \right) &= F_{25}\! \left(x \right) F_{254}\! \left(x \right)\\
F_{254}\! \left(x \right) &= F_{255}\! \left(x \right)+F_{262}\! \left(x \right)\\
F_{255}\! \left(x \right) &= F_{256}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{256}\! \left(x \right) &= F_{257}\! \left(x \right)+F_{261}\! \left(x \right)\\
F_{257}\! \left(x \right) &= F_{258}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{258}\! \left(x \right) &= F_{259}\! \left(x \right)\\
F_{259}\! \left(x \right) &= F_{25}\! \left(x \right) F_{260}\! \left(x \right)\\
F_{260}\! \left(x \right) &= F_{258}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{261}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{262}\! \left(x \right) &= F_{263}\! \left(x \right)+F_{266}\! \left(x \right)\\
F_{263}\! \left(x \right) &= F_{264}\! \left(x \right)+F_{265}\! \left(x \right)\\
F_{264}\! \left(x \right) &= F_{200}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{265}\! \left(x \right) &= F_{200}\! \left(x \right)\\
F_{266}\! \left(x \right) &= F_{267}\! \left(x \right)+F_{271}\! \left(x \right)\\
F_{267}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{268}\! \left(x \right)\\
F_{268}\! \left(x \right) &= F_{269}\! \left(x \right)\\
F_{269}\! \left(x \right) &= F_{25}\! \left(x \right) F_{270}\! \left(x \right)\\
F_{270}\! \left(x \right) &= F_{200}\! \left(x \right)+F_{268}\! \left(x \right)\\
F_{271}\! \left(x \right) &= F_{218}\! \left(x \right)\\
F_{272}\! \left(x \right) &= F_{25}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{273}\! \left(x \right) &= F_{25}\! \left(x \right) F_{274}\! \left(x \right)\\
F_{274}\! \left(x \right) &= F_{263}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{275}\! \left(x \right) &= F_{11}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{276}\! \left(x \right) &= F_{13}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{277}\! \left(x \right) &= F_{278}\! \left(x \right)\\
F_{278}\! \left(x \right) &= F_{16}\! \left(x \right) F_{25}\! \left(x \right) F_{279}\! \left(x \right)\\
F_{279}\! \left(x \right) &= F_{280}\! \left(x \right)+F_{281}\! \left(x \right)\\
F_{280}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{281}\! \left(x \right) &= F_{282}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{282}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{283}\! \left(x \right) &= F_{25}\! \left(x \right) F_{284}\! \left(x \right)\\
F_{284}\! \left(x \right) &= F_{285}\! \left(x \right)+F_{304}\! \left(x \right)\\
F_{285}\! \left(x \right) &= F_{286}\! \left(x \right)+F_{297}\! \left(x \right)\\
F_{286}\! \left(x \right) &= F_{287}\! \left(x \right)+F_{290}\! \left(x \right)\\
F_{287}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{288}\! \left(x \right)\\
F_{288}\! \left(x \right) &= F_{289}\! \left(x \right)\\
F_{289}\! \left(x \right) &= F_{25}\! \left(x \right) F_{280}\! \left(x \right)\\
F_{290}\! \left(x \right) &= F_{288}\! \left(x \right)+F_{291}\! \left(x \right)\\
F_{291}\! \left(x \right) &= F_{292}\! \left(x \right)+F_{296}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{292}\! \left(x \right) &= F_{25}\! \left(x \right) F_{293}\! \left(x \right)\\
F_{293}\! \left(x \right) &= F_{294}\! \left(x \right)+F_{295}\! \left(x \right)\\
F_{294}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{295}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{296}\! \left(x \right) &= F_{25}\! \left(x \right) F_{281}\! \left(x \right)\\
F_{297}\! \left(x \right) &= F_{298}\! \left(x \right)+F_{299}\! \left(x \right)\\
F_{298}\! \left(x \right) &= F_{288}\! \left(x \right)\\
F_{299}\! \left(x \right) &= F_{300}\! \left(x \right)\\
F_{300}\! \left(x \right) &= F_{301}\! \left(x \right)\\
F_{301}\! \left(x \right) &= F_{25}\! \left(x \right) F_{302}\! \left(x \right)\\
F_{302}\! \left(x \right) &= F_{294}\! \left(x \right)+F_{303}\! \left(x \right)\\
F_{303}\! \left(x \right) &= F_{258}\! \left(x \right)\\
F_{304}\! \left(x \right) &= F_{305}\! \left(x \right)+F_{323}\! \left(x \right)\\
F_{305}\! \left(x \right) &= F_{306}\! \left(x \right)+F_{310}\! \left(x \right)\\
F_{306}\! \left(x \right) &= F_{307}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{307}\! \left(x \right) &= F_{308}\! \left(x \right)\\
F_{308}\! \left(x \right) &= F_{25}\! \left(x \right) F_{309}\! \left(x \right)\\
F_{309}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{310}\! \left(x \right) &= F_{311}\! \left(x \right)+F_{315}\! \left(x \right)\\
F_{311}\! \left(x \right) &= F_{201}\! \left(x \right)+F_{312}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{312}\! \left(x \right) &= F_{25}\! \left(x \right) F_{313}\! \left(x \right)\\
F_{313}\! \left(x \right) &= F_{264}\! \left(x \right)+F_{314}\! \left(x \right)\\
F_{314}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{218}\! \left(x \right)\\
F_{315}\! \left(x \right) &= 2 F_{46}\! \left(x \right)+F_{316}\! \left(x \right)+F_{320}\! \left(x \right)\\
F_{316}\! \left(x \right) &= F_{25}\! \left(x \right) F_{317}\! \left(x \right)\\
F_{317}\! \left(x \right) &= F_{318}\! \left(x \right)+F_{319}\! \left(x \right)\\
F_{318}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{319}\! \left(x \right) &= F_{140}\! \left(x \right)\\
F_{320}\! \left(x \right) &= F_{25}\! \left(x \right) F_{321}\! \left(x \right)\\
F_{321}\! \left(x \right) &= F_{216}\! \left(x \right)+F_{322}\! \left(x \right)\\
F_{322}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{171}\! \left(x \right)\\
F_{323}\! \left(x \right) &= F_{324}\! \left(x \right)+F_{325}\! \left(x \right)\\
F_{324}\! \left(x \right) &= F_{311}\! \left(x \right)\\
F_{325}\! \left(x \right) &= F_{326}\! \left(x \right)\\
F_{326}\! \left(x \right) &= F_{327}\! \left(x \right)\\
F_{327}\! \left(x \right) &= F_{25}\! \left(x \right) F_{328}\! \left(x \right)\\
F_{328}\! \left(x \right) &= F_{329}\! \left(x \right)+F_{330}\! \left(x \right)\\
F_{329}\! \left(x \right) &= F_{200}\! \left(x \right)\\
F_{330}\! \left(x \right) &= F_{268}\! \left(x \right)\\
F_{331}\! \left(x \right) &= F_{332}\! \left(x \right)\\
F_{332}\! \left(x \right) &= F_{25}\! \left(x \right) F_{333}\! \left(x \right)\\
F_{333}\! \left(x \right) &= F_{334}\! \left(x \right)+F_{337}\! \left(x \right)\\
F_{334}\! \left(x \right) &= F_{287}\! \left(x \right)+F_{335}\! \left(x \right)\\
F_{335}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{336}\! \left(x \right)\\
F_{336}\! \left(x \right) &= F_{296}\! \left(x \right)+F_{46}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{337}\! \left(x \right) &= F_{338}\! \left(x \right)\\
F_{338}\! \left(x \right) &= F_{25}\! \left(x \right) F_{339}\! \left(x \right)\\
F_{339}\! \left(x \right) &= F_{340}\! \left(x \right)+F_{344}\! \left(x \right)\\
F_{340}\! \left(x \right) &= F_{341}\! \left(x \right)+F_{342}\! \left(x \right)\\
F_{341}\! \left(x \right) &= F_{30} \left(x \right)^{2} F_{40}\! \left(x \right)\\
F_{342}\! \left(x \right) &= F_{343}\! \left(x \right)\\
F_{343}\! \left(x \right) &= F_{25}\! \left(x \right) F_{31}\! \left(x \right) F_{41}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{344}\! \left(x \right) &= F_{345}\! \left(x \right)+F_{347}\! \left(x \right)\\
F_{345}\! \left(x \right) &= F_{342}\! \left(x \right)+F_{346}\! \left(x \right)\\
F_{346}\! \left(x \right) &= F_{30} \left(x \right)^{2} F_{41}\! \left(x \right)\\
F_{347}\! \left(x \right) &= F_{31}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{348}\! \left(x \right) &= F_{25}\! \left(x \right) F_{349}\! \left(x \right)\\
F_{349}\! \left(x \right) &= F_{350}\! \left(x \right)+F_{352}\! \left(x \right)\\
F_{350}\! \left(x \right) &= F_{287}\! \left(x \right)+F_{351}\! \left(x \right)\\
F_{351}\! \left(x \right) &= F_{288}\! \left(x \right)+F_{300}\! \left(x \right)\\
F_{352}\! \left(x \right) &= F_{353}\! \left(x \right)+F_{354}\! \left(x \right)\\
F_{353}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{311}\! \left(x \right)\\
F_{354}\! \left(x \right) &= F_{311}\! \left(x \right)+F_{326}\! \left(x \right)\\
\end{align*}\)