Av(1234, 1342, 1432, 2431, 3214)
View Raw Data
Generating Function
\(\displaystyle -\frac{x^{8}+4 x^{7}+12 x^{6}+16 x^{5}+6 x^{4}+x^{3}+1}{x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 19, 49, 99, 210, 471, 1045, 2282, 4998, 10993, 24161, 53041, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+x^{8}+4 x^{7}+12 x^{6}+16 x^{5}+6 x^{4}+x^{3}+1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 49\)
\(\displaystyle a \! \left(6\right) = 99\)
\(\displaystyle a \! \left(7\right) = 210\)
\(\displaystyle a \! \left(8\right) = 471\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)+3 a \! \left(n +1\right)+2 a \! \left(n +2\right)+a \! \left(n +3\right)+a \! \left(4+n \right), \quad n \geq 9\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle \frac{\left(\left(-4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-10131\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+\left(-4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}-24324 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-30393\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)-10131 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}-30393 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-9775\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{1-n}}{7367}+\frac{\left(\left(\left(4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+10131\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+10131 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+17593\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+\left(10131 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+17593\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+17593 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+43004\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{1-n}}{7367}+\frac{\left(4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}+14193 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+9462 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+10487\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{1-n}}{7367}+\frac{\left(\left(-4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-10131\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)-10131 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-17593\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2-n}}{7367}+\frac{\left(4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+14193 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+12800\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2-n}}{7367}+\frac{\left(4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+10131\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{3-n}}{7367}+\frac{5756 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{1-n}}{7367}+\frac{3338 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2-n}}{7367}-\frac{4062 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3-n}}{7367}-\frac{4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +4}}{7367}+\frac{\left(\left(\left(4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+10131\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+10131 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+17593\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+\left(\left(4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+10131\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+\left(4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}+34455 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+47986\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+10131 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}+47986 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+52779\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+\left(10131 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+17593\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+\left(10131 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}+47986 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+52779\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+17593 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}+52779 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+33494\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n}}{7367}+\frac{\left(\left(\left(\left(-4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-10131\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-10131 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-17593\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+\left(-10131 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-17593\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-17593 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-43004\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+\left(\left(-10131 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-17593\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-17593 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-43004\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+\left(-17593 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-43004\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-43004 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-95518\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n}}{7367}+\frac{\left(\left(-4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-10131\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}+\left(-4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}-24324 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-30393\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+\left(-4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{3}-24324 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}-39855 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-20262\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)-10131 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{3}-30393 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}-20262 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-1692\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n}}{7367}+\frac{\left(4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{4}+14193 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}+9462 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+4731 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+8439\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n}}{7367}-\left(\left\{\begin{array}{cc}-8 & n =0 \\ 7 & n =1 \\ 1 & n =2\text{ or } n =3 \\ 0 & \text{otherwise} \end{array}\right.\right)+\frac{3708 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n}}{7367}\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements" and has 88 rules.

Found on January 18, 2022.

Finding the specification took 1 seconds.

This tree is too big to show here. Click to view tree on new page.

Copy 88 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_{20}\! \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_{14}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \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_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{22}\! \left(x \right) &= 0\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{46}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{47}\! \left(x \right)+F_{60}\! \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_{43}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{71}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{47}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{17}\! \left(x \right)\\ \end{align*}\)