Av(12345, 12354, 12435, 12453, 12534, 12543, 13425, 13524, 21345, 21354, 21435, 21453, 21534, 21543, 23415, 23514, 31245, 31254, 31425, 31524, 32415, 32514, 41235, 41325, 42315)
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Generating Function
\(\displaystyle \frac{x^{5}-6 x^{3}+4 x^{2}-4 x +1}{x^{5}+x^{4}-9 x^{3}+7 x^{2}-5 x +1}\)
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Counting Sequence
1, 1, 2, 6, 24, 95, 358, 1333, 4984, 18692, 70116, 262901, 985604, 3695081, 13853478, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}+x^{4}-9 x^{3}+7 x^{2}-5 x +1\right) F \! \left(x \right)-x^{5}+6 x^{3}-4 x^{2}+4 x -1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a{\left(n + 5 \right)} = - a{\left(n \right)} - a{\left(n + 1 \right)} + 9 a{\left(n + 2 \right)} - 7 a{\left(n + 3 \right)} + 5 a{\left(n + 4 \right)}, \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{581 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +1}}{6254}\\+\\\frac{1327 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +2}}{6254}\\-\\\frac{383 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +3}}{6254}\\-\\\frac{187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +4}}{6254}\\+\\\frac{\left(187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}-1683 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+1890\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +1}}{6254}\\+\\\frac{\left(187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-356\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +2}}{6254}\\+\\\frac{\left(187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-196\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +3}}{6254}\\+\\\frac{\left(\left(-187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+196\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+9 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+196\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+196 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+196 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+126\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +1}}{6254}\\+\\\frac{\left(\left(-187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+196\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+196 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-160\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +2}}{6254}\\+\\\frac{\left(\left(\left(187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-196\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-196 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+160\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-196 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+160\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+160 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+286\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +1}}{6254}\\+\\\frac{279 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n}}{3127}\\+\\\frac{\left(187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{4}+187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}-1683 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+1309 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-377\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n}}{6254}\\+\\\frac{\left(\left(-187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+196\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{3}+\left(-187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+9 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+196\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(-187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+9 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+1879 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-1764\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+196 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{3}+196 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}-1764 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+995\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n}}{6254}\\+\\\frac{\left(\left(\left(187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-196\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-196 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+160\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)^{2}+\left(\left(187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-196\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}-205 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-36\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)-196 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}-36 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+160\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(-196 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+160\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)^{2}+\left(-196 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}-36 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+160\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+160 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)^{2}+160 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-445\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n}}{6254}\\+\\\frac{\left(\left(\left(\left(-187 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)+196\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)+196 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-160\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(196 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-160\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-160 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-286\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =2\right)+\left(\left(196 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-160\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-160 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-286\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =1\right)+\left(-160 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-286\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =3\right)-286 \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =4\right)-731\right) \mathit{RootOf}\left(Z^{5}+Z^{4}-9 Z^{3}+7 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n}}{6254} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 121 rules.

Finding the specification took 132 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_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{18}\! \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_{16}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{24}\! \left(x \right) &= 0\\ F_{25}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{16}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{16}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{24}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{16}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{41}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{16}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{16}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{16}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{47}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{34}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{16}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{16}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{61}\! \left(x \right)+F_{62}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{16}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{16}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{61}\! \left(x \right)+F_{66}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{16}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{70}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{16}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{16}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{73}\! \left(x \right) &= 0\\ F_{74}\! \left(x \right) &= F_{16}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{16}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{70}\! \left(x \right)+F_{73}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{16}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{70}\! \left(x \right)+F_{73}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{16}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{16}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{16}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{16}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{47}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{16}\! \left(x \right) F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{105}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{24}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{24}\! \left(x \right)+F_{62}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{112}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{24}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{24}\! \left(x \right)+F_{66}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{118}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{120}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{28}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{24}\! \left(x \right)+F_{62}\! \left(x \right)+F_{87}\! \left(x \right)\\ \end{align*}\)