Av(13452, 13542, 14352, 31452, 31542, 34152, 34512, 35142, 35412, 41352, 43152, 43512)
Counting Sequence
1, 1, 2, 6, 24, 108, 517, 2575, 13200, 69180, 368993, 1996473, 10930943, 60449347, 337157867, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right) \left(x^{3}-4 x +4\right) F \left(x
\right)^{3}+x \left(x -2\right) \left(x^{2}+2 x -4\right) F \left(x
\right)^{2}+\left(2 x^{3}+2 x^{2}-9 x +1\right) F \! \left(x \right)+x^{2}+4 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 108\)
\(\displaystyle a(6) = 517\)
\(\displaystyle a(7) = 2575\)
\(\displaystyle a(8) = 13200\)
\(\displaystyle a(9) = 69180\)
\(\displaystyle a(10) = 368993\)
\(\displaystyle a(11) = 1996473\)
\(\displaystyle a(12) = 10930943\)
\(\displaystyle a(13) = 60449347\)
\(\displaystyle a{\left(n + 14 \right)} = \frac{115 \left(n + 1\right) \left(n + 2\right) a{\left(n \right)}}{16 \left(n + 14\right) \left(2 n + 29\right)} - \frac{\left(n + 2\right) \left(1423 n + 4282\right) a{\left(n + 1 \right)}}{32 \left(n + 14\right) \left(2 n + 29\right)} + \frac{\left(28 n^{2} + 750 n + 5021\right) a{\left(n + 13 \right)}}{\left(n + 14\right) \left(2 n + 29\right)} - \frac{\left(164 n^{2} + 4062 n + 25141\right) a{\left(n + 12 \right)}}{\left(n + 14\right) \left(2 n + 29\right)} + \frac{\left(2476 n^{2} + 55623 n + 312290\right) a{\left(n + 11 \right)}}{4 \left(n + 14\right) \left(2 n + 29\right)} - \frac{\left(3394 n^{2} - 25695 n - 193075\right) a{\left(n + 4 \right)}}{16 \left(n + 14\right) \left(2 n + 29\right)} + \frac{\left(3760 n^{2} + 31869 n + 61220\right) a{\left(n + 2 \right)}}{32 \left(n + 14\right) \left(2 n + 29\right)} - \frac{\left(4079 n^{2} + 74955 n + 225772\right) a{\left(n + 3 \right)}}{32 \left(n + 14\right) \left(2 n + 29\right)} + \frac{\left(6542 n^{2} + 116811 n + 520669\right) a{\left(n + 9 \right)}}{2 \left(n + 14\right) \left(2 n + 29\right)} - \frac{\left(6698 n^{2} + 134763 n + 677467\right) a{\left(n + 10 \right)}}{4 \left(n + 14\right) \left(2 n + 29\right)} + \frac{3 \left(6812 n^{2} + 43254 n + 30927\right) a{\left(n + 5 \right)}}{16 \left(n + 14\right) \left(2 n + 29\right)} + \frac{\left(35837 n^{2} + 475692 n + 1561051\right) a{\left(n + 7 \right)}}{8 \left(n + 14\right) \left(2 n + 29\right)} - \frac{\left(36497 n^{2} + 570276 n + 2219602\right) a{\left(n + 8 \right)}}{8 \left(n + 14\right) \left(2 n + 29\right)} - \frac{\left(48206 n^{2} + 505551 n + 1258654\right) a{\left(n + 6 \right)}}{16 \left(n + 14\right) \left(2 n + 29\right)}, \quad n \geq 14\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 108\)
\(\displaystyle a(6) = 517\)
\(\displaystyle a(7) = 2575\)
\(\displaystyle a(8) = 13200\)
\(\displaystyle a(9) = 69180\)
\(\displaystyle a(10) = 368993\)
\(\displaystyle a(11) = 1996473\)
\(\displaystyle a(12) = 10930943\)
\(\displaystyle a(13) = 60449347\)
\(\displaystyle a{\left(n + 14 \right)} = \frac{115 \left(n + 1\right) \left(n + 2\right) a{\left(n \right)}}{16 \left(n + 14\right) \left(2 n + 29\right)} - \frac{\left(n + 2\right) \left(1423 n + 4282\right) a{\left(n + 1 \right)}}{32 \left(n + 14\right) \left(2 n + 29\right)} + \frac{\left(28 n^{2} + 750 n + 5021\right) a{\left(n + 13 \right)}}{\left(n + 14\right) \left(2 n + 29\right)} - \frac{\left(164 n^{2} + 4062 n + 25141\right) a{\left(n + 12 \right)}}{\left(n + 14\right) \left(2 n + 29\right)} + \frac{\left(2476 n^{2} + 55623 n + 312290\right) a{\left(n + 11 \right)}}{4 \left(n + 14\right) \left(2 n + 29\right)} - \frac{\left(3394 n^{2} - 25695 n - 193075\right) a{\left(n + 4 \right)}}{16 \left(n + 14\right) \left(2 n + 29\right)} + \frac{\left(3760 n^{2} + 31869 n + 61220\right) a{\left(n + 2 \right)}}{32 \left(n + 14\right) \left(2 n + 29\right)} - \frac{\left(4079 n^{2} + 74955 n + 225772\right) a{\left(n + 3 \right)}}{32 \left(n + 14\right) \left(2 n + 29\right)} + \frac{\left(6542 n^{2} + 116811 n + 520669\right) a{\left(n + 9 \right)}}{2 \left(n + 14\right) \left(2 n + 29\right)} - \frac{\left(6698 n^{2} + 134763 n + 677467\right) a{\left(n + 10 \right)}}{4 \left(n + 14\right) \left(2 n + 29\right)} + \frac{3 \left(6812 n^{2} + 43254 n + 30927\right) a{\left(n + 5 \right)}}{16 \left(n + 14\right) \left(2 n + 29\right)} + \frac{\left(35837 n^{2} + 475692 n + 1561051\right) a{\left(n + 7 \right)}}{8 \left(n + 14\right) \left(2 n + 29\right)} - \frac{\left(36497 n^{2} + 570276 n + 2219602\right) a{\left(n + 8 \right)}}{8 \left(n + 14\right) \left(2 n + 29\right)} - \frac{\left(48206 n^{2} + 505551 n + 1258654\right) a{\left(n + 6 \right)}}{16 \left(n + 14\right) \left(2 n + 29\right)}, \quad n \geq 14\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 58 rules.
Finding the specification took 1117 seconds.
Copy 58 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_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{0}\! \left(x \right) F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{14}\! \left(x \right) F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13}\! \left(x \right) F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= \frac{F_{26}\! \left(x \right)}{F_{4}\! \left(x \right) F_{5}\! \left(x \right)}\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= -F_{48}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{4}\! \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_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{32}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= \frac{F_{34}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{35}\! \left(x \right) &= -F_{47}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= \frac{F_{37}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{16}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= \frac{F_{43}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= -F_{45}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{45}\! \left(x \right) &= x F_{45} \left(x \right)^{3}-x F_{45} \left(x \right)^{2}+x F_{45}\! \left(x \right)+1\\
F_{46}\! \left(x \right) &= F_{18}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= \frac{F_{50}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{50}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{14}\! \left(x \right) F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 74 rules.
Finding the specification took 7077 seconds.
Copy 74 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_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x F_{12} \left(x \right)^{3}+2 x F_{12} \left(x \right)^{2}+2 F_{12}\! \left(x \right) x +x\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= -F_{20}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= \frac{F_{19}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{20}\! \left(x \right) &= x F_{20} \left(x \right)^{3}-x F_{20} \left(x \right)^{2}+x F_{20}\! \left(x \right)+1\\
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_{24}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= \frac{F_{26}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= -F_{18}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{5}\! \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) &= \frac{F_{32}\! \left(x \right)}{F_{13}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{36}\! \left(x \right) &= -F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= -F_{49}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= \frac{F_{45}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= \frac{F_{48}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{48}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= \frac{F_{52}\! \left(x \right)}{F_{4}\! \left(x \right) F_{56}\! \left(x \right)}\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{60}\! \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_{13}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= \frac{F_{69}\! \left(x \right)}{F_{13}\! \left(x \right) F_{4}\! \left(x \right)}\\
F_{69}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{70}\! \left(x \right) &= -F_{71}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{47}\! \left(x \right) F_{60}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 58 rules.
Finding the specification took 2050 seconds.
Copy 58 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_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{0}\! \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_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right) F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{21}\! \left(x \right) F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= \frac{F_{28}\! \left(x \right)}{F_{4}\! \left(x \right) F_{5}\! \left(x \right)}\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= -F_{48}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= \frac{F_{31}\! \left(x \right)}{F_{4}\! \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_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{34}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{35}\! \left(x \right) &= -F_{47}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= \frac{F_{37}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{17}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= \frac{F_{43}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= -F_{45}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{45}\! \left(x \right) &= x F_{45} \left(x \right)^{3}-x F_{45} \left(x \right)^{2}+x F_{45}\! \left(x \right)+1\\
F_{46}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= \frac{F_{50}\! \left(x \right)}{F_{4}\! \left(x \right)}\\
F_{50}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{10}\! \left(x \right) F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{15}\! \left(x \right) F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
\end{align*}\)