Av(1234, 1342, 2413, 3124)
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Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x -1\right)^{6}}{x^{8}-9 x^{7}+34 x^{6}-74 x^{5}+93 x^{4}-72 x^{3}+34 x^{2}-9 x +1}\)
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Counting Sequence
1, 1, 2, 6, 20, 65, 204, 629, 1929, 5911, 18111, 55476, 169880, 520136, 1592538, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{8}-9 x^{7}+34 x^{6}-74 x^{5}+93 x^{4}-72 x^{3}+34 x^{2}-9 x +1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{6} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 204\)
\(\displaystyle a \! \left(7\right) = 629\)
\(\displaystyle a \! \left(n +8\right) = -a \! \left(n \right)+9 a \! \left(n +1\right)-34 a \! \left(n +2\right)+74 a \! \left(n +3\right)-93 a \! \left(n +4\right)+72 a \! \left(n +5\right)-34 a \! \left(n +6\right)+9 a \! \left(n +7\right), \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle -\frac{515069 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n +6}}{3938468}-\frac{515069 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n +6}}{3938468}-\frac{515069 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n +6}}{3938468}-\frac{515069 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n +6}}{3938468}-\frac{515069 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n +6}}{3938468}-\frac{515069 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =6\right)^{-n +6}}{3938468}-\frac{515069 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =7\right)^{-n +6}}{3938468}-\frac{515069 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =8\right)^{-n +6}}{3938468}+\frac{4704015 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n +5}}{3938468}+\frac{4704015 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n +5}}{3938468}+\frac{4704015 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n +5}}{3938468}+\frac{4704015 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n +5}}{3938468}+\frac{4704015 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n +5}}{3938468}+\frac{4704015 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =6\right)^{-n +5}}{3938468}+\frac{4704015 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =7\right)^{-n +5}}{3938468}+\frac{4704015 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =8\right)^{-n +5}}{3938468}-\frac{17903387 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n +4}}{3938468}-\frac{17903387 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n +4}}{3938468}-\frac{17903387 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n +4}}{3938468}-\frac{17903387 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n +4}}{3938468}-\frac{17903387 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n +4}}{3938468}-\frac{17903387 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =6\right)^{-n +4}}{3938468}-\frac{17903387 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =7\right)^{-n +4}}{3938468}-\frac{17903387 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =8\right)^{-n +4}}{3938468}+\frac{19347887 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n +3}}{1969234}+\frac{19347887 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n +3}}{1969234}+\frac{19347887 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n +3}}{1969234}+\frac{19347887 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n +3}}{1969234}+\frac{19347887 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n +3}}{1969234}+\frac{19347887 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =6\right)^{-n +3}}{1969234}+\frac{19347887 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =7\right)^{-n +3}}{1969234}+\frac{19347887 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =8\right)^{-n +3}}{1969234}-\frac{23667039 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n +2}}{1969234}-\frac{23667039 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n +2}}{1969234}-\frac{23667039 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n +2}}{1969234}-\frac{23667039 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n +2}}{1969234}-\frac{23667039 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n +2}}{1969234}-\frac{23667039 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =6\right)^{-n +2}}{1969234}-\frac{23667039 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =7\right)^{-n +2}}{1969234}-\frac{23667039 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =8\right)^{-n +2}}{1969234}+\frac{32902077 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n +1}}{3938468}+\frac{32902077 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n +1}}{3938468}+\frac{32902077 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n +1}}{3938468}+\frac{32902077 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n +1}}{3938468}+\frac{32902077 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n +1}}{3938468}+\frac{32902077 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =6\right)^{-n +1}}{3938468}+\frac{32902077 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =7\right)^{-n +1}}{3938468}+\frac{32902077 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =8\right)^{-n +1}}{3938468}+\frac{2123765 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n -1}}{3938468}+\frac{2123765 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n -1}}{3938468}+\frac{2123765 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n -1}}{3938468}+\frac{2123765 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n -1}}{3938468}+\frac{2123765 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n -1}}{3938468}+\frac{2123765 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =6\right)^{-n -1}}{3938468}+\frac{2123765 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =7\right)^{-n -1}}{3938468}+\frac{2123765 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =8\right)^{-n -1}}{3938468}-\frac{12378149 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =1\right)^{-n}}{3938468}-\frac{12378149 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =2\right)^{-n}}{3938468}-\frac{12378149 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =3\right)^{-n}}{3938468}-\frac{12378149 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =4\right)^{-n}}{3938468}-\frac{12378149 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =5\right)^{-n}}{3938468}-\frac{12378149 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =6\right)^{-n}}{3938468}-\frac{12378149 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =7\right)^{-n}}{3938468}-\frac{12378149 \mathit{RootOf} \left(Z^{8}-9 Z^{7}+34 Z^{6}-74 Z^{5}+93 Z^{4}-72 Z^{3}+34 Z^{2}-9 Z +1, \mathit{index} =8\right)^{-n}}{3938468}\)
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This specification was found using the strategy pack "Point Placements" and has 72 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ 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_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{23}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{46}\! \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_{12}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{43}\! \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_{12}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{12}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{54}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{12}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{12}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{66}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{67}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{12}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{12}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{69}\! \left(x \right)\\ \end{align*}\)