Av(1243, 1432, 2314, 3124, 3214)
View Raw Data
Generating Function
\(\displaystyle \frac{\left(x^{2}+x -1\right) \left(x -1\right)^{2}}{x^{8}+x^{7}-3 x^{6}-3 x^{5}+3 x^{4}+x^{3}-4 x^{2}+4 x -1}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 19, 54, 146, 397, 1093, 3020, 8332, 22957, 63243, 174266, 480249, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{8}+x^{7}-3 x^{6}-3 x^{5}+3 x^{4}+x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)-\left(x^{2}+x -1\right) \left(x -1\right)^{2} = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 146\)
\(\displaystyle a \! \left(7\right) = 397\)
\(\displaystyle a \! \left(n +8\right) = a \! \left(n \right)+a \! \left(n +1\right)-3 a \! \left(n +2\right)-3 a \! \left(n +3\right)+3 a \! \left(n +4\right)+a \! \left(n +5\right)-4 a \! \left(n +6\right)+4 a \! \left(n +7\right), \quad n \geq 8\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle \frac{238487874 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +6}}{3639954341}+\frac{238487874 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +6}}{3639954341}+\frac{238487874 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +6}}{3639954341}+\frac{238487874 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +6}}{3639954341}+\frac{238487874 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =8\right)^{-n +6}}{3639954341}+\frac{238487874 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +6}}{3639954341}+\frac{238487874 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +6}}{3639954341}+\frac{238487874 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +6}}{3639954341}+\frac{432773885 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +5}}{3639954341}+\frac{432773885 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +5}}{3639954341}+\frac{432773885 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +5}}{3639954341}+\frac{432773885 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +5}}{3639954341}+\frac{432773885 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +5}}{3639954341}+\frac{432773885 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +5}}{3639954341}+\frac{432773885 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +5}}{3639954341}+\frac{432773885 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =8\right)^{-n +5}}{3639954341}-\frac{471335374 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +4}}{3639954341}-\frac{471335374 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +4}}{3639954341}-\frac{471335374 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +4}}{3639954341}-\frac{471335374 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +4}}{3639954341}-\frac{471335374 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +4}}{3639954341}-\frac{471335374 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +4}}{3639954341}-\frac{471335374 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +4}}{3639954341}-\frac{471335374 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =8\right)^{-n +4}}{3639954341}-\frac{1186946263 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +3}}{3639954341}-\frac{1186946263 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +3}}{3639954341}-\frac{1186946263 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +3}}{3639954341}-\frac{1186946263 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +3}}{3639954341}-\frac{1186946263 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +3}}{3639954341}-\frac{1186946263 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +3}}{3639954341}-\frac{1186946263 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +3}}{3639954341}-\frac{1186946263 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =8\right)^{-n +3}}{3639954341}-\frac{121101255 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +2}}{3639954341}-\frac{121101255 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +2}}{3639954341}-\frac{121101255 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +2}}{3639954341}-\frac{121101255 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +2}}{3639954341}-\frac{121101255 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +2}}{3639954341}-\frac{121101255 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +2}}{3639954341}-\frac{121101255 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +2}}{3639954341}-\frac{121101255 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =8\right)^{-n +2}}{3639954341}+\frac{395394806 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +1}}{3639954341}+\frac{395394806 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +1}}{3639954341}+\frac{395394806 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +1}}{3639954341}+\frac{395394806 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +1}}{3639954341}+\frac{395394806 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n +1}}{3639954341}+\frac{395394806 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n +1}}{3639954341}+\frac{395394806 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n +1}}{3639954341}+\frac{395394806 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =8\right)^{-n +1}}{3639954341}+\frac{502849844 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n -1}}{3639954341}+\frac{502849844 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n -1}}{3639954341}+\frac{502849844 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n -1}}{3639954341}+\frac{502849844 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n -1}}{3639954341}+\frac{502849844 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n -1}}{3639954341}+\frac{502849844 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n -1}}{3639954341}+\frac{502849844 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n -1}}{3639954341}+\frac{502849844 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =8\right)^{-n -1}}{3639954341}-\frac{251405927 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n}}{3639954341}-\frac{251405927 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n}}{3639954341}-\frac{251405927 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n}}{3639954341}-\frac{251405927 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n}}{3639954341}-\frac{251405927 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n}}{3639954341}-\frac{251405927 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =6\right)^{-n}}{3639954341}-\frac{251405927 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =7\right)^{-n}}{3639954341}-\frac{251405927 \mathit{RootOf} \left(Z^{8}+Z^{7}-3 Z^{6}-3 Z^{5}+3 Z^{4}+Z^{3}-4 Z^{2}+4 Z -1, \mathit{index} =8\right)^{-n}}{3639954341}\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements" and has 60 rules.

Found on January 18, 2022.

Finding the specification took 1 seconds.

Copy to clipboard:

View tree on standalone page.

+ Expand

Copy 60 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_{21}\! \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_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{48}\! \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_{52}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{54}\! \left(x \right)\\ \end{align*}\)