Av(1234, 1342, 1423, 1432, 2413)
View Raw Data
Generating Function
\(\displaystyle -\frac{\left(x^{3}-2 x^{2}+3 x -1\right)^{2}}{x^{7}-5 x^{6}+12 x^{5}-21 x^{4}+24 x^{3}-18 x^{2}+7 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 60, 188, 586, 1822, 5658, 17557, 54452, 168820, 523276, 1621695, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}-5 x^{6}+12 x^{5}-21 x^{4}+24 x^{3}-18 x^{2}+7 x -1\right) F \! \left(x \right)+\left(x^{3}-2 x^{2}+3 x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 188\)
\(\displaystyle a \! \left(n +7\right) = a \! \left(n \right)-5 a \! \left(n +1\right)+12 a \! \left(n +2\right)-21 a \! \left(n +3\right)+24 a \! \left(n +4\right)-18 a \! \left(n +5\right)+7 a \! \left(n +6\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{424199 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n -1}}{819713}+\frac{424199 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n -1}}{819713}+\frac{424199 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n -1}}{819713}+\frac{424199 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n -1}}{819713}+\frac{424199 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n -1}}{819713}+\frac{424199 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n -1}}{819713}+\frac{424199 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n -1}}{819713}-\frac{1899606 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n}}{819713}-\frac{1899606 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n}}{819713}-\frac{1899606 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n}}{819713}-\frac{1899606 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n}}{819713}-\frac{1899606 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n}}{819713}-\frac{1899606 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n}}{819713}-\frac{1899606 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n}}{819713}-\frac{3112984 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +2}}{819713}-\frac{3112984 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +2}}{819713}-\frac{3112984 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +2}}{819713}-\frac{3112984 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +2}}{819713}-\frac{3112984 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +2}}{819713}-\frac{3112984 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +2}}{819713}-\frac{3112984 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +2}}{819713}+\frac{3196705 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +1}}{819713}+\frac{3196705 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +1}}{819713}+\frac{3196705 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +1}}{819713}+\frac{3196705 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +1}}{819713}+\frac{3196705 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +1}}{819713}+\frac{3196705 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +1}}{819713}+\frac{3196705 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +1}}{819713}+\frac{179840 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +5}}{819713}+\frac{179840 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +5}}{819713}+\frac{179840 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +5}}{819713}+\frac{179840 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +5}}{819713}+\frac{179840 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +5}}{819713}+\frac{179840 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +5}}{819713}+\frac{179840 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +5}}{819713}-\frac{844431 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +4}}{819713}-\frac{844431 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +4}}{819713}-\frac{844431 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +4}}{819713}-\frac{844431 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +4}}{819713}-\frac{844431 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +4}}{819713}-\frac{844431 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +4}}{819713}-\frac{844431 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +4}}{819713}+\frac{1891721 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +3}}{819713}+\frac{1891721 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +3}}{819713}+\frac{1891721 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +3}}{819713}+\frac{1891721 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +3}}{819713}+\frac{1891721 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +3}}{819713}+\frac{1891721 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +3}}{819713}+\frac{1891721 \mathit{RootOf} \left(Z^{7}-5 Z^{6}+12 Z^{5}-21 Z^{4}+24 Z^{3}-18 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +3}}{819713}\)

This specification was found using the strategy pack "Point Placements" and has 137 rules.

Found on January 18, 2022.

Finding the specification took 7 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_{17}\! \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_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ 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_{35}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{27}\! \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_{18}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{131}\! \left(x \right)+F_{19}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{47}\! \left(x \right) &= 0\\ F_{48}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{19}\! \left(x \right)+F_{50}\! \left(x \right)+F_{86}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{40}\! \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_{44}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)+F_{94}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{62}\! \left(x \right) &= 0\\ 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_{66}\! \left(x \right)\\ F_{66}\! \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) &= 2 F_{19}\! \left(x \right)+F_{69}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{75}\! \left(x \right)+F_{86}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)+F_{81}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{80}\! \left(x \right)\\ F_{85}\! \left(x \right) &= 0\\ F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{68}\! \left(x \right)\\ F_{89}\! \left(x \right) &= 0\\ F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{94}\! \left(x \right) &= 0\\ F_{95}\! \left(x \right) &= 0\\ F_{96}\! \left(x \right) &= F_{4}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{128}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{106}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{125}\! \left(x \right)+F_{19}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{113}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{19}\! \left(x \right)+F_{41}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{112}\! \left(x \right) &= 0\\ F_{113}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{19}\! \left(x \right)+F_{86}\! \left(x \right)+F_{89}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{119}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{118}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{120}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{19}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)+F_{85}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{124}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{118}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{120}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{130}\! \left(x \right) &= 0\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{136}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{2}\! \left(x \right)\\ \end{align*}\)