Av(12345, 12354, 12435, 13245, 13254, 13425, 13452, 13524, 13542, 14235, 14325, 14352, 31245, 31254, 31425, 31452, 31524, 31542, 35124, 35142)
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Generating Function
\(\displaystyle \frac{\left(x -1\right) \left(2 x^{3}-2 x^{2}+3 x -1\right) \left(x^{2}-3 x +1\right)^{2}}{3 x^{8}-28 x^{7}+91 x^{6}-153 x^{5}+167 x^{4}-117 x^{3}+49 x^{2}-11 x +1}\)
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Counting Sequence
1, 1, 2, 6, 24, 100, 405, 1588, 6078, 22910, 85627, 318756, 1184874, 4403425, 16368973, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(3 x^{8}-28 x^{7}+91 x^{6}-153 x^{5}+167 x^{4}-117 x^{3}+49 x^{2}-11 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(2 x^{3}-2 x^{2}+3 x -1\right) \left(x^{2}-3 x +1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 405\)
\(\displaystyle a(7) = 1588\)
\(\displaystyle a(8) = 6078\)
\(\displaystyle a{\left(n + 8 \right)} = - 3 a{\left(n \right)} + 28 a{\left(n + 1 \right)} - 91 a{\left(n + 2 \right)} + 153 a{\left(n + 3 \right)} - 167 a{\left(n + 4 \right)} + 117 a{\left(n + 5 \right)} - 49 a{\left(n + 6 \right)} + 11 a{\left(n + 7 \right)}, \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle \left(\left\{\begin{array}{cc}\frac{2}{3} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)+\frac{1682813210364 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n}}{135913855817}+\frac{1682813210364 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n}}{135913855817}+\frac{1682813210364 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n}}{135913855817}+\frac{1682813210364 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n}}{135913855817}+\frac{1682813210364 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n}}{135913855817}+\frac{1682813210364 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n}}{135913855817}+\frac{1682813210364 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n}}{135913855817}+\frac{1682813210364 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n}}{135913855817}-\frac{186388728527 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n -1}}{135913855817}-\frac{186388728527 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n -1}}{135913855817}-\frac{186388728527 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n -1}}{135913855817}-\frac{186388728527 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n -1}}{135913855817}-\frac{186388728527 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n -1}}{135913855817}-\frac{186388728527 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n -1}}{135913855817}-\frac{186388728527 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n -1}}{135913855817}-\frac{186388728527 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n -1}}{135913855817}-\frac{5520003542429 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n +1}}{135913855817}-\frac{5520003542429 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n +1}}{135913855817}-\frac{5520003542429 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n +1}}{135913855817}-\frac{5520003542429 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n +1}}{135913855817}-\frac{5520003542429 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n +1}}{135913855817}-\frac{5520003542429 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n +1}}{135913855817}-\frac{5520003542429 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n +1}}{135913855817}-\frac{5520003542429 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n +1}}{135913855817}+\frac{9273391394974 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n +2}}{135913855817}+\frac{9273391394974 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n +2}}{135913855817}+\frac{9273391394974 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n +2}}{135913855817}+\frac{9273391394974 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n +2}}{135913855817}+\frac{9273391394974 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n +2}}{135913855817}+\frac{9273391394974 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n +2}}{135913855817}+\frac{9273391394974 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n +2}}{135913855817}+\frac{9273391394974 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n +2}}{135913855817}-\frac{9374180977152 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n +3}}{135913855817}-\frac{9374180977152 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n +3}}{135913855817}-\frac{9374180977152 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n +3}}{135913855817}-\frac{9374180977152 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n +3}}{135913855817}-\frac{9374180977152 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n +3}}{135913855817}-\frac{9374180977152 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n +3}}{135913855817}-\frac{9374180977152 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n +3}}{135913855817}-\frac{9374180977152 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n +3}}{135913855817}+\frac{6114854108486 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n +4}}{135913855817}+\frac{6114854108486 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n +4}}{135913855817}+\frac{6114854108486 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n +4}}{135913855817}+\frac{6114854108486 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n +4}}{135913855817}+\frac{6114854108486 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n +4}}{135913855817}+\frac{6114854108486 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n +4}}{135913855817}+\frac{6114854108486 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n +4}}{135913855817}+\frac{6114854108486 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n +4}}{135913855817}-\frac{2006936200224 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n +5}}{135913855817}-\frac{2006936200224 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n +5}}{135913855817}-\frac{2006936200224 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n +5}}{135913855817}-\frac{2006936200224 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n +5}}{135913855817}-\frac{2006936200224 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n +5}}{135913855817}-\frac{2006936200224 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n +5}}{135913855817}-\frac{2006936200224 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n +5}}{135913855817}-\frac{2006936200224 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n +5}}{135913855817}+\frac{222457861764 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n +6}}{135913855817}+\frac{222457861764 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n +6}}{135913855817}+\frac{222457861764 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n +6}}{135913855817}+\frac{222457861764 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n +6}}{135913855817}+\frac{222457861764 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n +6}}{135913855817}+\frac{222457861764 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n +6}}{135913855817}+\frac{222457861764 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n +6}}{135913855817}+\frac{222457861764 \mathit{RootOf} \left(3 Z^{8}-28 Z^{7}+91 Z^{6}-153 Z^{5}+167 Z^{4}-117 Z^{3}+49 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n +6}}{135913855817}\)
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This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 133 rules.

Finding the specification took 119 seconds.

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Copy 133 equations to clipboard:
\(\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_{19}\! \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_{32}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{19}\! \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_{20}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \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_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= x\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{21}\! \left(x \right) &= 0\\ F_{22}\! \left(x \right) &= F_{19}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{19}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{19}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{33}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{19}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{38}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{19}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{43}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{19}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{19}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{29}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{19}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{53}\! \left(x \right)+F_{86}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{19}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{19}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{19}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{60}\! \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_{19}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{68}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{69}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{19}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{19}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{19}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{79}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{53}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{19}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{19}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{19}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{90}\! \left(x \right) &= 0\\ F_{91}\! \left(x \right) &= F_{19}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{21}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{19}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{107}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{122}\! \left(x \right)+F_{126}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{121}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{112}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{109}\! \left(x \right)+F_{113}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \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_{116}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{117}\! \left(x \right)\\ F_{126}\! \left(x \right) &= 0\\ F_{127}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{128}\! \left(x \right)+F_{21}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{132}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{131}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{128}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{83}\! \left(x \right)\\ \end{align*}\)