Av(12345, 12354, 12435, 12453, 12534, 12543, 13425, 13524, 21345, 21354, 21435, 21453, 21534, 21543, 31245, 31254, 31452, 31542, 41253, 41352)
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Generating Function
\(\displaystyle \frac{2 x^{7}-6 x^{6}+12 x^{5}-21 x^{4}+22 x^{3}-17 x^{2}+7 x -1}{4 x^{7}-9 x^{6}+22 x^{5}-34 x^{4}+35 x^{3}-23 x^{2}+8 x -1}\)
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Counting Sequence
1, 1, 2, 6, 24, 100, 409, 1645, 6555, 26002, 102928, 407067, 1609301, 6361320, 25144059, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{7}-9 x^{6}+22 x^{5}-34 x^{4}+35 x^{3}-23 x^{2}+8 x -1\right) F \! \left(x \right)-2 x^{7}+6 x^{6}-12 x^{5}+21 x^{4}-22 x^{3}+17 x^{2}-7 x +1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 409\)
\(\displaystyle a(7) = 1645\)
\(\displaystyle a{\left(n + 7 \right)} = 4 a{\left(n \right)} - 9 a{\left(n + 1 \right)} + 22 a{\left(n + 2 \right)} - 34 a{\left(n + 3 \right)} + 35 a{\left(n + 4 \right)} - 23 a{\left(n + 5 \right)} + 8 a{\left(n + 6 \right)}, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle -\frac{1447376404 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +5}}{615674887}-\frac{1447376404 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +5}}{615674887}-\frac{1447376404 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +5}}{615674887}-\frac{1447376404 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +5}}{615674887}-\frac{1447376404 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +5}}{615674887}-\frac{1447376404 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +5}}{615674887}-\frac{1447376404 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +5}}{615674887}+\frac{7151566531 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +4}}{1847024661}+\frac{7151566531 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +4}}{1847024661}+\frac{7151566531 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +4}}{1847024661}+\frac{7151566531 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +4}}{1847024661}+\frac{7151566531 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +4}}{1847024661}+\frac{7151566531 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +4}}{1847024661}+\frac{7151566531 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +4}}{1847024661}-\frac{6465030123 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +3}}{615674887}-\frac{6465030123 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +3}}{615674887}-\frac{6465030123 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +3}}{615674887}-\frac{6465030123 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +3}}{615674887}-\frac{6465030123 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +3}}{615674887}-\frac{6465030123 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +3}}{615674887}-\frac{6465030123 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +3}}{615674887}+\frac{8354744672 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +2}}{615674887}+\frac{8354744672 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +2}}{615674887}+\frac{8354744672 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +2}}{615674887}+\frac{8354744672 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +2}}{615674887}+\frac{8354744672 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +2}}{615674887}+\frac{8354744672 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +2}}{615674887}+\frac{8354744672 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +2}}{615674887}-\frac{22251484330 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +1}}{1847024661}-\frac{22251484330 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +1}}{1847024661}-\frac{22251484330 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +1}}{1847024661}-\frac{22251484330 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +1}}{1847024661}-\frac{22251484330 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +1}}{1847024661}-\frac{22251484330 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n +1}}{1847024661}-\frac{22251484330 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n +1}}{1847024661}-\frac{1626584944 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n -1}}{1847024661}-\frac{1626584944 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n -1}}{1847024661}-\frac{1626584944 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n -1}}{1847024661}-\frac{1626584944 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n -1}}{1847024661}-\frac{1626584944 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n -1}}{1847024661}-\frac{1626584944 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n -1}}{1847024661}-\frac{1626584944 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n -1}}{1847024661}+\frac{10948222238 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n}}{1847024661}+\frac{10948222238 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n}}{1847024661}+\frac{10948222238 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n}}{1847024661}+\frac{10948222238 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n}}{1847024661}+\frac{10948222238 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n}}{1847024661}+\frac{10948222238 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =6\right)^{-n}}{1847024661}+\frac{10948222238 \mathit{RootOf} \left(4 Z^{7}-9 Z^{6}+22 Z^{5}-34 Z^{4}+35 Z^{3}-23 Z^{2}+8 Z -1, \mathit{index} =7\right)^{-n}}{1847024661}+\left(\left\{\begin{array}{cc}\frac{1}{2} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
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This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 168 rules.

Finding the specification took 512 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_{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_{163}\! \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_{156}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{52}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{17}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{17}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{17}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{17}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{64}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{65}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{17}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{17}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{67}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{68}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{17}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{72}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{17}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{17}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{73}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{17}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{79}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{65}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{17}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{82}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{83}\! \left(x \right)+F_{84}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{17}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{17}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{88}\! \left(x \right) &= 3 F_{20}\! \left(x \right)+F_{83}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{17}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{90}\! \left(x \right) &= 0\\ F_{91}\! \left(x \right) &= F_{17}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{60}\! \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_{97}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{63}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{149}\! \left(x \right)+F_{20}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{140}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{130}\! \left(x \right)+F_{20}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{108}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{109}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{112}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{113}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{111}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{114}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{115}\! \left(x \right)+F_{129}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{127}\! \left(x \right)\\ F_{118}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{119}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{124}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{123}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{126}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{127}\! \left(x \right) &= 3 F_{20}\! \left(x \right)+F_{128}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{117}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{129}\! \left(x \right) &= 0\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{135}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{134}\! \left(x \right)+F_{20}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{134}\! \left(x \right) &= 0\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{137}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{39}\! \left(x \right)+F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{138}\! \left(x \right)+F_{139}\! \left(x \right)+F_{20}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{138}\! \left(x \right) &= 0\\ F_{139}\! \left(x \right) &= 0\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{144}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{142}\! \left(x \right)+F_{20}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{143}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{111}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{147}\! \left(x \right)+F_{148}\! \left(x \right)+F_{20}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{117}\! \left(x \right)\\ F_{147}\! \left(x \right) &= 0\\ F_{148}\! \left(x \right) &= 0\\ F_{149}\! \left(x \right) &= F_{150}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{152}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{139}\! \left(x \right)+F_{145}\! \left(x \right)+F_{20}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{149}\! \left(x \right)+F_{20}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{160}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{20}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{159}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{130}\! \left(x \right)+F_{161}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{162}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{164}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)+F_{166}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{157}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{149}\! \left(x \right)+F_{161}\! \left(x \right)+F_{20}\! \left(x \right)\\ \end{align*}\)