Av(12435, 12453, 12543, 14235, 14253, 14325, 21435, 21453, 21543, 24135, 24153, 24315, 41235, 41253, 41325, 42135, 42153, 42315, 43125, 43215)
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 Bottom" and has 172 rules.
Finding the specification took 261 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_{14}\! \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_{33}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{14}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{14}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{14}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{18}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{14}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{38}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{14}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{14}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{14}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{56}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{14}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{14}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{66}\! \left(x \right) &= 0\\
F_{67}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{68}\! \left(x \right)+F_{69}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{14}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{14}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{68}\! \left(x \right)+F_{73}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{14}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{76}\! \left(x \right) &= 0\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{68}\! \left(x \right)+F_{69}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{80}\! \left(x \right)+F_{84}\! \left(x \right)+F_{94}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{14}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{73}\! \left(x \right)+F_{76}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{14}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{14}\! \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_{72}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{14}\! \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_{87}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{94}\! \left(x \right) &= 0\\
F_{95}\! \left(x \right) &= 0\\
F_{96}\! \left(x \right) &= F_{14}\! \left(x \right) F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{106}\! \left(x \right)+F_{113}\! \left(x \right)+F_{18}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{105}\! \left(x \right) &= 0\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{109}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{109}\! \left(x \right)\\
F_{113}\! \left(x \right) &= 0\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{130}\! \left(x \right)+F_{18}\! \left(x \right)+F_{80}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{119}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{18}\! \left(x \right)+F_{80}\! \left(x \right)+F_{84}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{121}\! \left(x \right)+F_{128}\! \left(x \right)+F_{129}\! \left(x \right)+F_{18}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{120}\! \left(x \right) &= 0\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{124}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{124}\! \left(x \right)\\
F_{128}\! \left(x \right) &= 0\\
F_{129}\! \left(x \right) &= 0\\
F_{130}\! \left(x \right) &= 0\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{133}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{18}\! \left(x \right)+F_{73}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{137}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{138}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{14}\! \left(x \right) F_{140}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{142}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{138}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{14}\! \left(x \right) F_{144}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{159}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{14}\! \left(x \right) F_{149}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{151}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{18}\! \left(x \right)+F_{29}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{14}\! \left(x \right) F_{153}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{155}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)\\
F_{156}\! \left(x \right) &= F_{14}\! \left(x \right) F_{157}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{155}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{160}\! \left(x \right)+F_{18}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{14}\! \left(x \right) F_{161}\! \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_{164}\! \left(x \right)\\
F_{163}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{160}\! \left(x \right)+F_{18}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{165}\! \left(x \right)+F_{18}\! \left(x \right)+F_{94}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{14}\! \left(x \right) F_{166}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{167}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{164}\! \left(x \right)+F_{168}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{14}\! \left(x \right) F_{170}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{171}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{168}\! \left(x \right)\\
\end{align*}\)