Av(12345, 12354, 12435, 12453, 12534, 12543, 13425, 13452, 13524, 13542, 14523, 14532, 21345, 21354, 21435, 21453, 21534, 21543, 31245, 31254)
Generating Function
\(\displaystyle \frac{x^{5}-4 x^{4}+7 x^{3}-6 x^{2}+5 x -1}{2 x^{5}-7 x^{4}+11 x^{3}-10 x^{2}+6 x -1}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 414, 1710, 7064, 29186, 120588, 498230, 2058518, 8505102, 35140218, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{5}-7 x^{4}+11 x^{3}-10 x^{2}+6 x -1\right) F \! \left(x \right)-x^{5}+4 x^{4}-7 x^{3}+6 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a{\left(n + 5 \right)} = 2 a{\left(n \right)} - 7 a{\left(n + 1 \right)} + 11 a{\left(n + 2 \right)} - 10 a{\left(n + 3 \right)} + 6 a{\left(n + 4 \right)}, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a{\left(n + 5 \right)} = 2 a{\left(n \right)} - 7 a{\left(n + 1 \right)} + 11 a{\left(n + 2 \right)} - 10 a{\left(n + 3 \right)} + 6 a{\left(n + 4 \right)}, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ -\frac{2648 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +1}}{39677}\\+\\\frac{10568 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +2}}{39677}\\-\\\frac{3488 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +3}}{39677}\\+\\\frac{552 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +4}}{39677}\\+\\\frac{\left(-1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{3}+3864 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}-6072 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)+224\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +1}}{79354}\\+\\\frac{\left(-1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}+3864 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)+15064\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +2}}{79354}\\+\\\frac{\left(-1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)-3112\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +3}}{79354}\\+\\\frac{\left(\left(1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)+3112\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)^{2}+\left(1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}-752 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)-10892\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)+3112 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}-10892 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)+17340\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +1}}{79354}\\+\\\frac{\left(\left(1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)+3112\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)+3112 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)+4172\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +2}}{79354}\\+\\\frac{\left(\left(\left(-1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)-3112\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)-3112 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)-4172\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)+\left(-3112 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)-4172\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)-4172 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)+31942\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +1}}{79354}\\+\\\frac{3370 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n}}{39677}\\+\\\frac{\left(-1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{4}+3864 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{3}-6072 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}+5520 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)+3428\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n}}{79354}\\+\\\frac{\left(\left(1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)+3112\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)^{3}+\left(1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}-752 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)-10892\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)^{2}+\left(1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{3}-752 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}-4820 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)+17116\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)+3112 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{3}-10892 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}+17116 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)-12132\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n}}{79354}\\+\\\frac{\left(\left(\left(-1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)-3112\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)-3112 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)-4172\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)^{2}+\left(\left(-1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)-3112\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}+\left(-1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)^{2}-2360 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)+6720\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)-3112 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)^{2}+6720 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)+14602\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)+\left(-3112 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)-4172\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)^{2}+\left(-3112 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)^{2}+6720 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)+14602\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)-4172 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)^{2}+14602 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)-35078\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n}}{79354}\\+\\\frac{\left(\left(\left(\left(1104 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)+3112\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)+3112 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)+4172\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)+\left(3112 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)+4172\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)+4172 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)-31942\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =2\right)+\left(\left(3112 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)+4172\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)+4172 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)-31942\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =1\right)+\left(4172 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)-31942\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =3\right)-31942 \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =4\right)+76719\right) \mathit{RootOf}\left(2 Z^{5}-7 Z^{4}+11 Z^{3}-10 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n}}{79354} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 144 rules.
Finding the specification took 337 seconds.
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Copy 144 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_{17}\! \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_{47}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{17}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= x\\
F_{18}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{20}\! \left(x \right) &= 0\\
F_{21}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{17}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{17}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{17}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{17}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{17}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{17}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{39}\! \left(x \right)+F_{40}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{17}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{17}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{44}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{39}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{17}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{46}\! \left(x \right) &= 0\\
F_{47}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{20}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{17}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{53}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{17}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{57}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{17}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{17}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{17}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{17}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{17}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{17}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{71}\! \left(x \right)+F_{76}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{17}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{75}\! \left(x \right)+F_{76}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{17}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{17}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{65}\! \left(x \right)+F_{76}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{17}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{85}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{17}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{87}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{17}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{17}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{62}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{65}\! \left(x \right)+F_{76}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{17}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{20}\! \left(x \right)+F_{85}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{102}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{103}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{101}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{104}\! \left(x \right) &= 0\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{111}\! \left(x \right)+F_{116}\! \left(x \right)+F_{117}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{20}\! \left(x \right)+F_{76}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{114}\! \left(x \right) &= 3 F_{20}\! \left(x \right)+F_{107}\! \left(x \right)+F_{115}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{113}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{116}\! \left(x \right) &= 0\\
F_{117}\! \left(x \right) &= 0\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{120}\! \left(x \right)\\
F_{119}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{120}\! \left(x \right) &= 3 F_{20}\! \left(x \right)+F_{107}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{126}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{125}\! \left(x \right)+F_{20}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{125}\! \left(x \right) &= 0\\
F_{126}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{127}\! \left(x \right)+F_{128}\! \left(x \right)+F_{133}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{17}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{131}\! \left(x \right) &= 3 F_{20}\! \left(x \right)+F_{127}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{130}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{133}\! \left(x \right) &= 0\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{140}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{137}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{20}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{20}\! \left(x \right)+F_{76}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{113}\! \left(x \right)\\
\end{align*}\)