Av(1234, 1342, 2143, 2413, 3214)
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Generating Function
\(\displaystyle \frac{\left(x -1\right)^{2}}{x^{8}-x^{7}+2 x^{5}-3 x^{4}-2 x^{3}+2 x^{2}-3 x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 50, 128, 337, 900, 2395, 6347, 16819, 44610, 118349, 313925, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{8}-x^{7}+2 x^{5}-3 x^{4}-2 x^{3}+2 x^{2}-3 x +1\right) F \! \left(x \right)-\left(x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 128\)
\(\displaystyle a \! \left(7\right) = 337\)
\(\displaystyle a \! \left(n +1\right) = a \! \left(n \right)+2 a \! \left(n +3\right)-3 a \! \left(n +4\right)-2 a \! \left(n +5\right)+2 a \! \left(n +6\right)-3 a \! \left(n +7\right)+a \! \left(n +8\right), \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle -\frac{3651851 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +6}}{123174239}-\frac{3651851 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +6}}{123174239}-\frac{3651851 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +6}}{123174239}-\frac{3651851 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +6}}{123174239}-\frac{3651851 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +6}}{123174239}-\frac{3651851 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +6}}{123174239}-\frac{3651851 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =7\right)^{-n +6}}{123174239}-\frac{3651851 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =8\right)^{-n +6}}{123174239}-\frac{9989451 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +5}}{246348478}-\frac{9989451 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =7\right)^{-n +5}}{246348478}-\frac{9989451 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =8\right)^{-n +5}}{246348478}-\frac{9989451 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +5}}{246348478}-\frac{9989451 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +5}}{246348478}-\frac{9989451 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +5}}{246348478}-\frac{9989451 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +5}}{246348478}-\frac{9989451 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +5}}{246348478}+\frac{3984197 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +4}}{123174239}+\frac{3984197 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +4}}{123174239}+\frac{3984197 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +4}}{123174239}+\frac{3984197 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +4}}{123174239}+\frac{3984197 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +4}}{123174239}+\frac{3984197 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +4}}{123174239}+\frac{3984197 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =7\right)^{-n +4}}{123174239}+\frac{3984197 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =8\right)^{-n +4}}{123174239}-\frac{906648 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +3}}{123174239}-\frac{906648 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +3}}{123174239}-\frac{906648 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +3}}{123174239}-\frac{906648 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +3}}{123174239}-\frac{906648 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +3}}{123174239}-\frac{906648 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +3}}{123174239}-\frac{906648 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =7\right)^{-n +3}}{123174239}-\frac{906648 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =8\right)^{-n +3}}{123174239}-\frac{2113990 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +2}}{123174239}-\frac{2113990 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +2}}{123174239}-\frac{2113990 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +2}}{123174239}-\frac{2113990 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +2}}{123174239}-\frac{2113990 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +2}}{123174239}-\frac{2113990 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +2}}{123174239}-\frac{2113990 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =7\right)^{-n +2}}{123174239}-\frac{2113990 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =8\right)^{-n +2}}{123174239}+\frac{15459818 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +1}}{123174239}+\frac{15459818 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +1}}{123174239}+\frac{15459818 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +1}}{123174239}+\frac{15459818 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +1}}{123174239}+\frac{15459818 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +1}}{123174239}+\frac{15459818 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +1}}{123174239}+\frac{15459818 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =7\right)^{-n +1}}{123174239}+\frac{15459818 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =8\right)^{-n +1}}{123174239}+\frac{4603213 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n -1}}{123174239}+\frac{4603213 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n -1}}{123174239}+\frac{4603213 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n -1}}{123174239}+\frac{4603213 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n -1}}{123174239}+\frac{4603213 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n -1}}{123174239}+\frac{4603213 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n -1}}{123174239}+\frac{4603213 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =7\right)^{-n -1}}{123174239}+\frac{4603213 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =8\right)^{-n -1}}{123174239}+\frac{55216331 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n}}{246348478}+\frac{55216331 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =7\right)^{-n}}{246348478}+\frac{55216331 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =8\right)^{-n}}{246348478}+\frac{55216331 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n}}{246348478}+\frac{55216331 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n}}{246348478}+\frac{55216331 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n}}{246348478}+\frac{55216331 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n}}{246348478}+\frac{55216331 \mathit{RootOf} \left(Z^{8}-Z^{7}+2 Z^{5}-3 Z^{4}-2 Z^{3}+2 Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n}}{246348478}\)
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This specification was found using the strategy pack "Point Placements" and has 74 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_{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_{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_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \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_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{60}\! \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_{26}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{36}\! \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_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \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_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \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_{59}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{48}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{27}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{57}\! \left(x \right)\\ \end{align*}\)