Av(1234, 1324, 1342, 3214, 4132)
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Generating Function
\(\displaystyle \frac{\left(x +1\right) \left(2 x^{7}+4 x^{6}+7 x^{5}+6 x^{4}-x^{3}+2 x^{2}-2 x +1\right)}{\left(x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 52, 120, 266, 591, 1310, 2883, 6331, 13911, 30567, 67137, ...
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Implicit Equation for the Generating Function
\(\displaystyle -\left(x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+\left(x +1\right) \left(2 x^{7}+4 x^{6}+7 x^{5}+6 x^{4}-x^{3}+2 x^{2}-2 x +1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(6\right) = 120\)
\(\displaystyle a \! \left(7\right) = 266\)
\(\displaystyle a \! \left(8\right) = 591\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)+3 a \! \left(n +1\right)+2 a \! \left(n +2\right)+a \! \left(n +3\right)+a \! \left(4+n \right)+38, \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle \frac{\left(\left(-23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-677\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+\left(-23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}-746 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-2031\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)-677 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}-2031 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-624\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{1-n}}{973}+\frac{\left(\left(\left(23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+677\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+677 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+975\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+\left(677 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+975\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+975 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+2301\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{1-n}}{973}+\frac{\left(23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}+69 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+46 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+730\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{1-n}}{973}+\frac{\left(\left(-23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-677\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)-677 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-975\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2-n}}{973}+\frac{\left(23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+69 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+1056\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2-n}}{973}+\frac{\left(23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+677\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +3}}{973}+\frac{101 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{1-n}}{139}+\frac{1010 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2-n}}{973}+\frac{608 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +3}}{973}-\frac{23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +4}}{973}+\frac{\left(\left(\left(23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+677\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+677 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+975\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+\left(\left(23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+677\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+\left(23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}+1423 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+3006\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+677 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}+3006 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+2925\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+\left(677 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+975\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+\left(677 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}+3006 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+2925\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+975 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}+2925 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+1784\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n}}{973}+\frac{\left(\left(\left(\left(-23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-677\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-677 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-975\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+\left(-677 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-975\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-975 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-2301\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+\left(\left(-677 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-975\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-975 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-2301\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+\left(-975 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-2301\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)-2301 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)-5119\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n}}{973}+\frac{\left(\left(-23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-677\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}+\left(-23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}-746 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-2031\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+\left(-23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{3}-746 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}-2077 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-1354\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)-677 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{3}-2031 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}-1354 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)-166\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n}}{973}+\frac{\left(23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{4}+69 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}+46 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+23 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+511\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n}}{973}+\left(\left\{\begin{array}{cc}9 & n =0 \\ 2 & n =1\text{ or } n =2 \\ 0 & \text{otherwise} \end{array}\right.\right)+\frac{488 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n}}{973}-\frac{38}{7}\)
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This specification was found using the strategy pack "Point Placements" and has 121 rules.

Found on January 18, 2022.

Finding the specification took 2 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_{18}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{27}\! \left(x \right) &= 0\\ 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_{34}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{40}\! \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_{19}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{39}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{53}\! \left(x \right)+F_{97}\! \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_{71}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{57}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{59}\! \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_{23}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{32}\! \left(x \right)+F_{67}\! \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_{70}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \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_{19}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{82}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{83}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{76}\! \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_{45}\! \left(x \right)+F_{85}\! \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_{26}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{41}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{4}\! \left(x \right) F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{92}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{113}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{108}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{27}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{109}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{110}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{112}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{110}\! \left(x \right)+F_{41}\! \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_{19}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{27}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{117}\! \left(x \right)\\ \end{align*}\)