Av(1234, 1243, 1432, 2314)
Generating Function
\(\displaystyle -\frac{\left(2 x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{2}}{2 x^{7}-x^{6}-4 x^{5}+12 x^{4}-16 x^{3}+13 x^{2}-6 x +1}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 201, 629, 1969, 6167, 19321, 60539, 189694, 594392, 1862481, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{7}-x^{6}-4 x^{5}+12 x^{4}-16 x^{3}+13 x^{2}-6 x +1\right) F \! \left(x \right)+\left(2 x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 201\)
\(\displaystyle a \! \left(n +7\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)+4 a \! \left(n +2\right)-12 a \! \left(n +3\right)+16 a \! \left(n +4\right)-13 a \! \left(n +5\right)+6 a \! \left(n +6\right), \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 201\)
\(\displaystyle a \! \left(n +7\right) = -2 a \! \left(n \right)+a \! \left(n +1\right)+4 a \! \left(n +2\right)-12 a \! \left(n +3\right)+16 a \! \left(n +4\right)-13 a \! \left(n +5\right)+6 a \! \left(n +6\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle -\frac{40319 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +2}}{376342}-\frac{40319 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +2}}{376342}-\frac{40319 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +2}}{376342}-\frac{40319 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +2}}{376342}-\frac{40319 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +2}}{376342}-\frac{40319 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +2}}{376342}-\frac{40319 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =7\right)^{-n +2}}{376342}-\frac{8839 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +3}}{1129026}-\frac{8839 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +3}}{1129026}-\frac{8839 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +3}}{1129026}-\frac{8839 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +3}}{1129026}-\frac{8839 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +3}}{1129026}-\frac{8839 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +3}}{1129026}-\frac{8839 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =7\right)^{-n +3}}{1129026}-\frac{53398 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +4}}{564513}-\frac{53398 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +4}}{564513}-\frac{53398 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +4}}{564513}-\frac{53398 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +4}}{564513}-\frac{53398 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +4}}{564513}-\frac{53398 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +4}}{564513}-\frac{53398 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =7\right)^{-n +4}}{564513}-\frac{29522 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +5}}{564513}-\frac{29522 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +5}}{564513}-\frac{29522 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +5}}{564513}-\frac{29522 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +5}}{564513}-\frac{29522 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +5}}{564513}-\frac{29522 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +5}}{564513}-\frac{29522 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =7\right)^{-n +5}}{564513}+\frac{41912 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +1}}{564513}+\frac{41912 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +1}}{564513}+\frac{41912 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +1}}{564513}+\frac{41912 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +1}}{564513}+\frac{41912 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +1}}{564513}+\frac{41912 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +1}}{564513}+\frac{41912 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =7\right)^{-n +1}}{564513}+\frac{20099 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n -1}}{376342}+\frac{20099 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n -1}}{376342}+\frac{20099 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n -1}}{376342}+\frac{20099 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n -1}}{376342}+\frac{20099 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n -1}}{376342}+\frac{20099 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n -1}}{376342}+\frac{20099 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =7\right)^{-n -1}}{376342}+\frac{18646 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n}}{564513}+\frac{18646 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n}}{564513}+\frac{18646 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n}}{564513}+\frac{18646 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n}}{564513}+\frac{18646 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n}}{564513}+\frac{18646 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n}}{564513}+\frac{18646 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-4 Z^{5}+12 Z^{4}-16 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =7\right)^{-n}}{564513}\)
This specification was found using the strategy pack "Point Placements" and has 151 rules.
Found on January 18, 2022.Finding the specification took 11 seconds.
This tree is too big to show here. Click to view tree on new page.
Copy 151 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \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_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{15}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{145}\! \left(x \right)+F_{15}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{42}\! \left(x \right) &= 0\\
F_{43}\! \left(x \right) &= F_{4}\! \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_{34}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{143}\! \left(x \right)+F_{15}\! \left(x \right)+F_{53}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{15}\! \left(x \right)+F_{62}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{4}\! \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_{56}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{66}\! \left(x \right) &= 0\\
F_{67}\! \left(x \right) &= 0\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{137}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{81}\! \left(x \right)+F_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{88}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{93}\! \left(x \right) &= 0\\
F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{15}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{104}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{15}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{130}\! \left(x \right)+F_{138}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{109}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{112}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{124}\! \left(x \right)+F_{128}\! \left(x \right)+F_{15}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{117}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{118}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{123}\! \left(x \right)+F_{15}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{116}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{123}\! \left(x \right) &= 0\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right) F_{4}\! \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_{133}\! \left(x \right) &= F_{134}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)\\
F_{136}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{137}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{132}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{142}\! \left(x \right) &= 0\\
F_{143}\! \left(x \right) &= F_{144}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{129}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{146}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{147}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{150}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{98}\! \left(x \right)\\
\end{align*}\)