Av(12345, 12354, 12435, 12453, 12534, 12543, 13425, 13524, 21345, 21354, 21435, 21453, 21534, 21543, 23415, 23514, 31425, 31452, 31524, 31542, 32415, 32514, 41523, 41532, 42513)
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Generating Function
\(\displaystyle -\frac{\left(2 x^{3}-2 x^{2}+3 x -1\right) \left(2 x -1\right)^{2}}{2 x^{7}+x^{6}-11 x^{5}+28 x^{4}-36 x^{3}+24 x^{2}-8 x +1}\)
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Counting Sequence
1, 1, 2, 6, 24, 95, 354, 1267, 4450, 15530, 54149, 188947, 659873, 2305755, 8058854, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{7}+x^{6}-11 x^{5}+28 x^{4}-36 x^{3}+24 x^{2}-8 x +1\right) F \! \left(x \right)+\left(2 x^{3}-2 x^{2}+3 x -1\right) \left(2 x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a(6) = 354\)
\(\displaystyle a{\left(n + 7 \right)} = - 2 a{\left(n \right)} - a{\left(n + 1 \right)} + 11 a{\left(n + 2 \right)} - 28 a{\left(n + 3 \right)} + 36 a{\left(n + 4 \right)} - 24 a{\left(n + 5 \right)} + 8 a{\left(n + 6 \right)}, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \frac{651663576 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +5}}{805952773}+\frac{651663576 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +5}}{805952773}+\frac{651663576 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +5}}{805952773}+\frac{651663576 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +5}}{805952773}+\frac{651663576 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +5}}{805952773}+\frac{651663576 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +5}}{805952773}+\frac{651663576 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +5}}{805952773}+\frac{573025472 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +4}}{805952773}+\frac{573025472 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +4}}{805952773}+\frac{573025472 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +4}}{805952773}+\frac{573025472 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +4}}{805952773}+\frac{573025472 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +4}}{805952773}+\frac{573025472 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +4}}{805952773}+\frac{573025472 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +4}}{805952773}-\frac{3379366172 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +3}}{805952773}-\frac{3379366172 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +3}}{805952773}-\frac{3379366172 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +3}}{805952773}-\frac{3379366172 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +3}}{805952773}-\frac{3379366172 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +3}}{805952773}-\frac{3379366172 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +3}}{805952773}-\frac{3379366172 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +3}}{805952773}+\frac{7859434347 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +2}}{805952773}+\frac{7859434347 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +2}}{805952773}+\frac{7859434347 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +2}}{805952773}+\frac{7859434347 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +2}}{805952773}+\frac{7859434347 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +2}}{805952773}+\frac{7859434347 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +2}}{805952773}+\frac{7859434347 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +2}}{805952773}-\frac{8677278424 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +1}}{805952773}-\frac{8677278424 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +1}}{805952773}-\frac{8677278424 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +1}}{805952773}-\frac{8677278424 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +1}}{805952773}-\frac{8677278424 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +1}}{805952773}-\frac{8677278424 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +1}}{805952773}-\frac{8677278424 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +1}}{805952773}-\frac{631841860 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n -1}}{805952773}-\frac{631841860 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n -1}}{805952773}-\frac{631841860 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n -1}}{805952773}-\frac{631841860 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n -1}}{805952773}-\frac{631841860 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n -1}}{805952773}-\frac{631841860 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n -1}}{805952773}-\frac{631841860 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n -1}}{805952773}+\frac{4281654209 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n}}{805952773}+\frac{4281654209 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n}}{805952773}+\frac{4281654209 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n}}{805952773}+\frac{4281654209 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n}}{805952773}+\frac{4281654209 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n}}{805952773}+\frac{4281654209 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n}}{805952773}+\frac{4281654209 \mathit{RootOf} \left(2 Z^{7}+Z^{6}-11 Z^{5}+28 Z^{4}-36 Z^{3}+24 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n}}{805952773}\)
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This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 124 rules.

Finding the specification took 106 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_{31}\! \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_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \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_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{16}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{18}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{16}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{36}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{16}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{16}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{53}\! \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_{16}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{49}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{16}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{16}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{16}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{58}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{16}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{16}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{16}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{64}\! \left(x \right)+F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{16}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{16}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{74}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{16}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{76}\! \left(x \right) &= 0\\ F_{77}\! \left(x \right) &= F_{16}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{82}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{16}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{16}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{18}\! \left(x \right)+F_{68}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{16}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{18}\! \left(x \right)+F_{68}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{16}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{18}\! \left(x \right)+F_{50}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{101}\! \left(x \right) &= 0\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{105}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{106}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{112}\! \left(x \right)+F_{18}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{112}\! \left(x \right) &= 0\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{117}\! \left(x \right)+F_{18}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{122}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{117}\! \left(x \right)+F_{18}\! \left(x \right)+F_{91}\! \left(x \right)\\ \end{align*}\)