Av(13452, 13524, 13542, 14352, 14532, 15324, 15342, 15432, 31452, 31524, 31542, 35124, 35142, 35214, 51324, 51342, 51432, 53124, 53142, 53214)
Generating Function
\(\displaystyle \frac{x^{5}-x^{4}+10 x^{3}-8 x^{2}+5 x -1}{2 x^{5}-5 x^{4}+16 x^{3}-12 x^{2}+6 x -1}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 400, 1558, 6040, 23492, 91600, 357346, 1393664, 5434052, 21186864, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{5}-5 x^{4}+16 x^{3}-12 x^{2}+6 x -1\right) F \! \left(x \right)-x^{5}+x^{4}-10 x^{3}+8 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a{\left(n + 5 \right)} = 2 a{\left(n \right)} - 5 a{\left(n + 1 \right)} + 16 a{\left(n + 2 \right)} - 12 a{\left(n + 3 \right)} + 6 a{\left(n + 4 \right)}, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a{\left(n + 5 \right)} = 2 a{\left(n \right)} - 5 a{\left(n + 1 \right)} + 16 a{\left(n + 2 \right)} - 12 a{\left(n + 3 \right)} + 6 a{\left(n + 4 \right)}, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{731 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +1}}{69900}\\+\\\frac{20599 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +2}}{69900}\\-\\\frac{5891 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +3}}{69900}\\+\\\frac{1541 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +4}}{34950}\\+\\\frac{\left(-3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{3}+7705 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}-24656 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)+19223\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +1}}{69900}\\+\\\frac{\left(-3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}+7705 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)-4057\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +2}}{69900}\\+\\\frac{\left(-3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)+1814\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +3}}{69900}\\+\\\frac{\left(\left(3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)-1814\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{2}+\left(3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}-9519 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)+4535\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)-1814 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}+4535 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)+4711\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +1}}{69900}\\+\\\frac{\left(\left(3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)-1814\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)-1814 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)+478\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +2}}{69900}\\+\\\frac{\left(\left(\left(-3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)+1814\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)+1814 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)-478\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)+\left(1814 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)-478\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)-478 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)+5906\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +1}}{69900}\\+\\\frac{6433 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n}}{69900}\\+\\\frac{\left(-3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{4}+7705 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{3}-24656 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}+18492 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)-2813\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n}}{69900}\\+\\\frac{\left(\left(3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)-1814\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{3}+\left(3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}-9519 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)+4535\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{2}+\left(3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{3}-9519 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}+29191 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)-14512\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)-1814 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{3}+4535 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}-14512 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)+8071\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n}}{69900}\\+\\\frac{\left(\left(\left(-3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)+1814\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)+1814 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)-478\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{2}+\left(\left(-3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)+1814\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}+\left(-3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{2}+11333 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)-5013\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)+1814 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{2}-5013 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)+1195\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)+\left(1814 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)-478\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}+\left(1814 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{2}-5013 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)+1195\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)-478 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{2}+1195 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)+4247\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n}}{69900}\\+\\\frac{\left(\left(\left(\left(3082 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)-1814\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)-1814 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)+478\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)+\left(-1814 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)+478\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)+478 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)-5906\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)+\left(\left(-1814 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)+478\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)+478 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)-5906\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)+\left(478 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)-5906\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)-5906 \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)+19012\right) \mathit{RootOf}\left(2 Z^{5}-5 Z^{4}+16 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n}}{69900} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 66 rules.
Finding the specification took 120 seconds.
Copy 66 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_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{16}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{17}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x\\
F_{17}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{16}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{16}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{16}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{16}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{16}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{49}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{16}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{16}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{16}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{63}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{16}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{16}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{52}\! \left(x \right)\\
\end{align*}\)