Av(1234, 1243, 1432, 2314, 3142)
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Generating Function
\(\displaystyle \frac{\left(x -1\right)^{3}}{\left(x^{2}-x +1\right) \left(x^{5}+2 x^{4}-x^{2}+3 x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 54, 147, 400, 1097, 3019, 8309, 22855, 62849, 172826, 475266, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-x +1\right) \left(x^{5}+2 x^{4}-x^{2}+3 x -1\right) F \! \left(x \right)-\left(x -1\right)^{3} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 147\)
\(\displaystyle a \! \left(n +7\right) = a \! \left(n \right)+a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+4 a \! \left(n +4\right)-5 a \! \left(n +5\right)+4 a \! \left(n +6\right), \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \frac{36 \,\mathrm{I}}{377} \left(\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +4} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{59 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +3} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{18}-\frac{29 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +1} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{18}+\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +4} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{119 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +2} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{36}-\frac{29 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +1} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{18}-\frac{47 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{36}+\frac{119 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +2} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{36}+\frac{119 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +2} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{36}-\frac{29 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +1} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{18}-\frac{29 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +1} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)}{18}+\frac{59 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +3} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{18}+\frac{59 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +3} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{18}+\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +4} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+\frac{119 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +2} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{36}+\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +4} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)+\frac{59 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +3} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{18}+\frac{59 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +3} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{18}+\frac{119 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +2} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{36}-\frac{29 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +1} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{18}+\left(-\frac{47 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{36}+\left(-\frac{47 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)}{36}+\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right) \left(-\frac{47 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)}{36}+\left(-\frac{47 \,\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n}}{36}-\frac{377 \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right) \sqrt{3}\, \left(-\left(\frac{1}{2}+\frac{\mathrm{I} \sqrt{3}}{2}\right)^{-n}+\left(\frac{1}{2}-\frac{\mathrm{I} \sqrt{3}}{2}\right)^{-n}\right)}{108}+\mathrm{I} \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +4}\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)\right)\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)\right) \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)\right) \left(\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{4}+2 \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{3}-\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)+3\right) \left(\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)^{4}+2 \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)^{3}-\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)+3\right) \left(\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)^{4}+2 \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)^{3}-\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)+3\right) \left(\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{4}+2 \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{3}-\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)+3\right) \left(\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{4}+2 \mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{3}-\mathit{RootOf} \left(Z^{5}+2 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)+3\right)\)
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This specification was found using the strategy pack "Point Placements" and has 59 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_{19}\! \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_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \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_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{56}\! \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_{12}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{57}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ \end{align*}\)