Av(12453, 12534, 12543, 13452, 13542, 14352, 21453, 21534, 21543, 23451, 23541, 24351, 31452, 31524, 31542, 32451, 32541, 41523, 41532, 42531)
Generating Function
\(\displaystyle -\frac{x^{7}+3 x^{6}+x^{5}-2 x^{4}-4 x^{3}+4 x -1}{2 x^{8}+5 x^{7}-6 x^{5}-3 x^{4}+5 x^{3}+3 x^{2}-5 x +1}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 407, 1639, 6575, 26355, 105642, 423497, 1697795, 6806554, 27287947, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{8}+5 x^{7}-6 x^{5}-3 x^{4}+5 x^{3}+3 x^{2}-5 x +1\right) F \! \left(x \right)+x^{7}+3 x^{6}+x^{5}-2 x^{4}-4 x^{3}+4 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 407\)
\(\displaystyle a(7) = 1639\)
\(\displaystyle a{\left(n + 1 \right)} = - \frac{2 a{\left(n \right)}}{5} + \frac{6 a{\left(n + 3 \right)}}{5} + \frac{3 a{\left(n + 4 \right)}}{5} - a{\left(n + 5 \right)} - \frac{3 a{\left(n + 6 \right)}}{5} + a{\left(n + 7 \right)} - \frac{a{\left(n + 8 \right)}}{5}, \quad n \geq 8\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 407\)
\(\displaystyle a(7) = 1639\)
\(\displaystyle a{\left(n + 1 \right)} = - \frac{2 a{\left(n \right)}}{5} + \frac{6 a{\left(n + 3 \right)}}{5} + \frac{3 a{\left(n + 4 \right)}}{5} - a{\left(n + 5 \right)} - \frac{3 a{\left(n + 6 \right)}}{5} + a{\left(n + 7 \right)} - \frac{a{\left(n + 8 \right)}}{5}, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \frac{40436512208 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +6}}{834427119781}+\frac{40436512208 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +6}}{834427119781}+\frac{40436512208 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +6}}{834427119781}+\frac{40436512208 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +6}}{834427119781}+\frac{40436512208 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +6}}{834427119781}+\frac{40436512208 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +6}}{834427119781}+\frac{40436512208 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n +6}}{834427119781}+\frac{40436512208 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =8\right)^{-n +6}}{834427119781}+\frac{69489474396 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +5}}{834427119781}+\frac{69489474396 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +5}}{834427119781}+\frac{69489474396 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +5}}{834427119781}+\frac{69489474396 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +5}}{834427119781}+\frac{69489474396 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +5}}{834427119781}+\frac{69489474396 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +5}}{834427119781}+\frac{69489474396 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n +5}}{834427119781}+\frac{69489474396 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =8\right)^{-n +5}}{834427119781}-\frac{90145181242 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +4}}{834427119781}-\frac{90145181242 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +4}}{834427119781}-\frac{90145181242 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +4}}{834427119781}-\frac{90145181242 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +4}}{834427119781}-\frac{90145181242 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +4}}{834427119781}-\frac{90145181242 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +4}}{834427119781}-\frac{90145181242 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n +4}}{834427119781}-\frac{90145181242 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =8\right)^{-n +4}}{834427119781}-\frac{192424727324 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +3}}{834427119781}-\frac{192424727324 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +3}}{834427119781}-\frac{192424727324 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +3}}{834427119781}-\frac{192424727324 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +3}}{834427119781}-\frac{192424727324 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +3}}{834427119781}-\frac{192424727324 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +3}}{834427119781}-\frac{192424727324 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n +3}}{834427119781}-\frac{192424727324 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =8\right)^{-n +3}}{834427119781}-\frac{57324554385 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +2}}{834427119781}-\frac{57324554385 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +2}}{834427119781}-\frac{57324554385 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +2}}{834427119781}-\frac{57324554385 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +2}}{834427119781}-\frac{57324554385 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +2}}{834427119781}-\frac{57324554385 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +2}}{834427119781}-\frac{57324554385 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n +2}}{834427119781}-\frac{57324554385 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =8\right)^{-n +2}}{834427119781}+\frac{177134088199 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +1}}{834427119781}+\frac{177134088199 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +1}}{834427119781}+\frac{177134088199 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +1}}{834427119781}+\frac{177134088199 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +1}}{834427119781}+\frac{177134088199 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +1}}{834427119781}+\frac{177134088199 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +1}}{834427119781}+\frac{177134088199 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n +1}}{834427119781}+\frac{177134088199 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =8\right)^{-n +1}}{834427119781}-\frac{28987672966 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n -1}}{834427119781}-\frac{28987672966 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n -1}}{834427119781}-\frac{28987672966 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n -1}}{834427119781}-\frac{28987672966 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n -1}}{834427119781}-\frac{28987672966 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n -1}}{834427119781}-\frac{28987672966 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n -1}}{834427119781}-\frac{28987672966 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n -1}}{834427119781}-\frac{28987672966 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =8\right)^{-n -1}}{834427119781}+\frac{161032491770 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n}}{834427119781}+\frac{161032491770 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n}}{834427119781}+\frac{161032491770 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n}}{834427119781}+\frac{161032491770 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n}}{834427119781}+\frac{161032491770 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n}}{834427119781}+\frac{161032491770 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n}}{834427119781}+\frac{161032491770 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n}}{834427119781}+\frac{161032491770 \mathit{RootOf} \left(2 Z^{8}+5 Z^{7}-6 Z^{5}-3 Z^{4}+5 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =8\right)^{-n}}{834427119781}\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 87 rules.
Finding the specification took 117 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_{18}\! \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_{7}\! \left(x \right)+F_{75}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{72}\! \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_{68}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{19}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= x\\
F_{19}\! \left(x \right) &= F_{18}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{18}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{18}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{27}\! \left(x \right) &= 0\\
F_{28}\! \left(x \right) &= F_{18}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{32}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{18}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{58}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{18}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{42}\! \left(x \right)+F_{43}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{42}\! \left(x \right) &= 0\\
F_{43}\! \left(x \right) &= F_{18}\! \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_{31}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{18}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{18}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{18}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{18}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{63}\! \left(x \right)+F_{66}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{18}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{66}\! \left(x \right) &= 0\\
F_{67}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{18}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{32}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{27}\! \left(x \right)+F_{43}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{18}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{7}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{18}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{58}\! \left(x \right)+F_{7}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{18}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{11}\! \left(x \right)\\
\end{align*}\)