Av(1342, 1432, 2413, 4123)
Generating Function
\(\displaystyle \frac{x^{5}-2 x^{4}+x^{3}-x^{2}+3 x -1}{x^{5}-2 x^{4}+2 x^{3}-3 x^{2}+4 x -1}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 205, 658, 2111, 6772, 21725, 69695, 223585, 717272, 2301045, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-2 x^{4}+2 x^{3}-3 x^{2}+4 x -1\right) F \! \left(x \right)-x^{5}+2 x^{4}-x^{3}+x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-2 a \! \left(n +1\right)+2 a \! \left(n +2\right)-3 a \! \left(n +3\right)+4 a \! \left(4+n \right), \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-2 a \! \left(n +1\right)+2 a \! \left(n +2\right)-3 a \! \left(n +3\right)+4 a \! \left(4+n \right), \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ -\frac{2005 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +1}}{19961}\\+\\\frac{2506 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +2}}{19961}\\-\\\frac{3424 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +3}}{19961}\\+\\\frac{1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n +4}}{19961}\\+\\\frac{\left(-1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}+3228 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}-3228 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)+2837\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +1}}{19961}\\+\\\frac{\left(-1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+3228 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)-722\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +2}}{19961}\\+\\\frac{\left(-1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)-196\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n +3}}{19961}\\+\\\frac{\left(\left(1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)+196\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}-3032 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)-392\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =2\right)+196 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}-392 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)+3229\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +1}}{19961}\\+\\\frac{\left(\left(1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)+196\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =2\right)+196 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)-1114\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n +2}}{19961}\\+\\\frac{\left(\left(\left(-1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)-196\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)-196 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)+1114\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-196 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)+1114\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)+1114 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)+1001\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n +1}}{19961}\\+\\\frac{4226 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{-n}}{19961}\\+\\\frac{\left(-1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{4}+3228 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}-3228 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+4842 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)-2230\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =2\right)^{-n}}{19961}\\+\\\frac{\left(\left(1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)+196\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =2\right)^{3}+\left(1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}-3032 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)-392\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}-3032 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+2836 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)+392\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =2\right)+196 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{3}-392 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+392 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)-2818\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)^{-n}}{19961}\\+\\\frac{\left(\left(\left(-1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)-196\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)-196 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)+1114\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =2\right)^{2}+\left(\left(-1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)-196\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(-1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}+2836 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)+1506\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)-196 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}+1506 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)-2228\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(-196 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)+1114\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)^{2}+\left(-196 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}+1506 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)-2228\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)+1114 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)^{2}-2228 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)-590\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =4\right)^{-n}}{19961}\\+\\\frac{\left(\left(\left(\left(1614 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =4\right)+196\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)+196 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =4\right)-1114\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(196 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =4\right)-1114\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)-1114 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =4\right)-1001\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =2\right)+\left(\left(196 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =4\right)-1114\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)-1114 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =4\right)-1001\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =1\right)+\left(-1114 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =4\right)-1001\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =3\right)-1001 \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =4\right)+1412\right) \mathit{RootOf}\left(Z^{5}-2 Z^{4}+2 Z^{3}-3 Z^{2}+4 Z -1, \mathit{index} =5\right)^{-n}}{19961} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 87 rules.
Found on January 18, 2022.Finding the specification took 1 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_{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_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{20}\! \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_{26}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{28}\! \left(x \right) &= 0\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= 2 F_{28}\! \left(x \right)+F_{37}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{48}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \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_{55}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{58}\! \left(x \right)+F_{75}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= 2 F_{28}\! \left(x \right)+F_{66}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{74}\! \left(x \right) &= 0\\
F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\
\end{align*}\)