Av(1234, 1324, 1432, 3241)
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Generating Function
\(\displaystyle \frac{2 x^{10}+7 x^{9}+13 x^{8}+11 x^{7}-6 x^{6}-13 x^{5}-5 x^{4}-2 x^{3}+2 x -1}{\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 20, 61, 159, 380, 887, 2042, 4633, 10400, 23208, 51581, 114266, ...
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Implicit Equation for the Generating Function
\(\displaystyle -\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{5}+3 x^{4}+2 x^{3}+x^{2}+x -1\right) F \! \left(x \right)+2 x^{10}+7 x^{9}+13 x^{8}+11 x^{7}-6 x^{6}-13 x^{5}-5 x^{4}-2 x^{3}+2 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(6\right) = 159\)
\(\displaystyle a \! \left(7\right) = 380\)
\(\displaystyle a \! \left(8\right) = 887\)
\(\displaystyle a \! \left(9\right) = 2042\)
\(\displaystyle a \! \left(10\right) = 4633\)
\(\displaystyle a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{4}-a \! \left(n +1\right)+\frac{a \! \left(n +5\right)}{4}+\frac{a \! \left(n +6\right)}{2}-\frac{a \! \left(n +7\right)}{4}-2, \quad n \geq 11\)
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Explicit Closed Form
\(\displaystyle \frac{104169 \left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{3}+2 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2}+1\right) \left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)+1\right) \left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{3}+2 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{2}+1\right) \left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)+1\right) \left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{3}+2 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2}+1\right) \left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{3}+2 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{2}+1\right) \left(\frac{84518 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{1-n} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{104169}+\frac{47626 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2-n} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{104169}+\frac{45925 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +3} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{34723}+\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n +4} \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)+\frac{84518 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{1-n} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{104169}+\frac{47626 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{2-n} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{104169}+\frac{45925 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +3} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{34723}+\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n +4} \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)+\frac{84518 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{1-n} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{104169}+\frac{47626 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{2-n} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{104169}+\frac{45925 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n +3} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{34723}+\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n +4} \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)+\frac{84518 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{1-n} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{104169}+\frac{47626 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{2-n} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{104169}+\frac{45925 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n +3} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{34723}+\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n +4} \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)+\frac{84518 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{1-n} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)}{104169}+\frac{47626 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{2-n} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)}{104169}+\frac{45925 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n +3} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)}{34723}-\frac{20606 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{-n} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{104169}+\left(-\frac{20606 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right)^{-n} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{104169}+\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =2\right) \left(-\frac{20606 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)^{-n} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right) \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{104169}+\left(-\frac{20606 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)^{-n} \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)}{104169}+\left(-\frac{20606 \mathit{RootOf}\left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n}}{104169}+\left(\frac{51569 \left(\left\{\begin{array}{cc}10 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ 0 & \text{otherwise} \end{array}\right.\right)}{104169}+\left(-\frac{51569}{104169}+\frac{51569 \sqrt{5}}{104169}\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}-\frac{58936}{104169}+\left(-\frac{51569}{104169}-\frac{51569 \sqrt{5}}{104169}\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)+\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)^{-n +4}\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =4\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)\right)\right) \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)\right) \left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =5\right)+1\right) \left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{3}+2 \mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)^{2}+1\right) \left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =1\right)+1\right) \left(\mathit{RootOf} \left(Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+Z -1, \mathit{index} =3\right)+1\right)}{51569}\)
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This specification was found using the strategy pack "Point Placements" and has 409 rules.

Found on January 18, 2022.

Finding the specification took 23 seconds.

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Copy 409 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{20}\! \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_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= 0\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{33}\! \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_{11}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \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_{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_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{7}\! \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_{11}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{52}\! \left(x \right)+F_{60}\! \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_{47}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \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_{47}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{56}\! \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_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\ 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_{67}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{60}\! \left(x \right)+F_{69}\! \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_{72}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{77}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{130}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{102}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{89}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{4}\! \left(x \right) F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{101}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{89}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{98}\! \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_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{109}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{110}\! \left(x \right)\\ F_{110}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{102}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{108}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \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_{118}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{108}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{115}\! \left(x \right)\\ F_{119}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{120}\! \left(x \right)+F_{128}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{123}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{127}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{124}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{138}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{139}\! \left(x \right)\\ F_{139}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{128}\! \left(x \right)+F_{140}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{143}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{147}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{144}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{295}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{190}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{153}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{159}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{155}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{157}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{158}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{160}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{130}\! \left(x \right)+F_{161}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{162}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{165}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{164}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{181}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{168}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)+F_{171}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{170}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{172}\! \left(x \right)\\ F_{172}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{102}\! \left(x \right)+F_{173}\! \left(x \right)\\ F_{173}\! \left(x \right) &= F_{174}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{175}\! \left(x \right)+F_{176}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{170}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{177}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{178}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{178}\! \left(x \right) &= F_{179}\! \left(x \right)+F_{180}\! \left(x \right)\\ F_{179}\! \left(x \right) &= F_{170}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{177}\! \left(x \right)\\ F_{181}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{128}\! \left(x \right)+F_{182}\! \left(x \right)\\ F_{182}\! \left(x \right) &= F_{183}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{183}\! \left(x \right) &= F_{184}\! \left(x \right)+F_{185}\! \left(x \right)\\ F_{184}\! \left(x \right) &= F_{164}\! \left(x \right)\\ F_{185}\! \left(x \right) &= F_{186}\! \left(x \right)\\ F_{186}\! \left(x \right) &= F_{187}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{187}\! \left(x \right) &= F_{188}\! \left(x \right)+F_{189}\! \left(x \right)\\ F_{188}\! \left(x \right) &= F_{164}\! \left(x \right)\\ F_{189}\! \left(x \right) &= F_{186}\! \left(x \right)\\ F_{190}\! \left(x \right) &= F_{191}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{191}\! \left(x \right) &= F_{192}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{192}\! \left(x \right) &= F_{193}\! \left(x \right)+F_{22}\! \left(x \right)+F_{294}\! \left(x \right)\\ F_{193}\! \left(x \right) &= F_{194}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{194}\! \left(x \right) &= F_{195}\! \left(x \right)+F_{203}\! \left(x \right)\\ F_{195}\! \left(x \right) &= F_{196}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{196}\! \left(x \right) &= F_{197}\! \left(x \right)+F_{202}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{197}\! \left(x \right) &= F_{198}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{198}\! \left(x \right) &= F_{199}\! \left(x \right)\\ F_{199}\! \left(x \right) &= F_{200}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{200}\! \left(x \right) &= F_{201}\! \left(x \right)\\ F_{201}\! \left(x \right) &= x^{2}\\ F_{202}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{203}\! \left(x \right) &= F_{204}\! \left(x \right)+F_{223}\! \left(x \right)\\ F_{204}\! \left(x \right) &= F_{205}\! \left(x \right)\\ F_{205}\! \left(x \right) &= F_{206}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{206}\! \left(x \right) &= F_{207}\! \left(x \right)+F_{211}\! \left(x \right)\\ F_{207}\! \left(x \right) &= F_{208}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{208}\! \left(x \right) &= F_{209}\! \left(x \right)\\ F_{209}\! \left(x \right) &= F_{210}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{210}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{211}\! \left(x \right) &= F_{204}\! \left(x \right)+F_{212}\! \left(x \right)\\ F_{212}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{213}\! \left(x \right)+F_{221}\! \left(x \right)\\ F_{213}\! \left(x \right) &= F_{214}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{214}\! \left(x \right) &= F_{215}\! \left(x \right)+F_{216}\! \left(x \right)\\ F_{215}\! \left(x \right) &= F_{208}\! \left(x \right)\\ F_{216}\! \left(x \right) &= F_{217}\! \left(x \right)\\ F_{217}\! \left(x \right) &= F_{218}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{218}\! \left(x \right) &= F_{219}\! \left(x \right)+F_{220}\! \left(x \right)\\ F_{219}\! \left(x \right) &= F_{208}\! \left(x \right)\\ F_{220}\! \left(x \right) &= F_{217}\! \left(x \right)\\ F_{221}\! \left(x \right) &= F_{222}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{222}\! \left(x \right) &= F_{204}\! \left(x \right)\\ F_{223}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{224}\! \left(x \right)+F_{276}\! \left(x \right)\\ F_{224}\! \left(x \right) &= F_{225}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{225}\! \left(x \right) &= F_{226}\! \left(x \right)+F_{230}\! \left(x \right)\\ F_{226}\! \left(x \right) &= F_{196}\! \left(x \right)+F_{227}\! \left(x \right)\\ F_{227}\! \left(x \right) &= F_{228}\! \left(x \right)\\ F_{228}\! \left(x \right) &= F_{229}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{229}\! \left(x \right) &= F_{200}\! \left(x \right)\\ F_{230}\! \left(x \right) &= F_{231}\! \left(x \right)+F_{265}\! \left(x \right)\\ F_{231}\! \left(x \right) &= F_{232}\! \left(x \right)\\ F_{232}\! \left(x \right) &= F_{233}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{233}\! \left(x \right) &= F_{234}\! \left(x \right)+F_{238}\! \left(x \right)\\ F_{234}\! \left(x \right) &= F_{196}\! \left(x \right)+F_{235}\! \left(x \right)\\ F_{235}\! \left(x \right) &= F_{236}\! \left(x \right)\\ F_{236}\! \left(x \right) &= F_{237}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{237}\! \left(x \right) &= F_{200}\! \left(x \right)\\ F_{238}\! \left(x \right) &= F_{231}\! \left(x \right)+F_{239}\! \left(x \right)\\ F_{239}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{240}\! \left(x \right)+F_{248}\! \left(x \right)\\ F_{240}\! \left(x \right) &= F_{241}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{241}\! \left(x \right) &= F_{242}\! \left(x \right)+F_{243}\! \left(x \right)\\ F_{242}\! \left(x \right) &= F_{235}\! \left(x \right)\\ F_{243}\! \left(x \right) &= F_{244}\! \left(x \right)\\ F_{244}\! \left(x \right) &= F_{245}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{245}\! \left(x \right) &= F_{246}\! \left(x \right)+F_{247}\! \left(x \right)\\ F_{246}\! \left(x \right) &= F_{235}\! \left(x \right)\\ F_{247}\! \left(x \right) &= F_{244}\! \left(x \right)\\ F_{248}\! \left(x \right) &= F_{249}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{249}\! \left(x \right) &= F_{250}\! \left(x \right)\\ F_{250}\! \left(x \right) &= F_{251}\! \left(x \right)\\ F_{251}\! \left(x \right) &= F_{252}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{252}\! \left(x \right) &= F_{253}\! \left(x \right)+F_{255}\! \left(x \right)\\ F_{253}\! \left(x \right) &= F_{200}\! \left(x \right)+F_{254}\! \left(x \right)\\ F_{254}\! \left(x \right) &= F_{236}\! \left(x \right)\\ F_{255}\! \left(x \right) &= F_{250}\! \left(x \right)+F_{256}\! \left(x \right)\\ F_{256}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{248}\! \left(x \right)+F_{257}\! \left(x \right)\\ F_{257}\! \left(x \right) &= F_{258}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{258}\! \left(x \right) &= F_{259}\! \left(x \right)+F_{260}\! \left(x \right)\\ F_{259}\! \left(x \right) &= F_{254}\! \left(x \right)\\ F_{260}\! \left(x \right) &= F_{261}\! \left(x \right)\\ F_{261}\! \left(x \right) &= F_{262}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{262}\! \left(x \right) &= F_{263}\! \left(x \right)+F_{264}\! \left(x \right)\\ F_{263}\! \left(x \right) &= F_{254}\! \left(x \right)\\ F_{264}\! \left(x \right) &= F_{261}\! \left(x \right)\\ F_{265}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{266}\! \left(x \right)+F_{274}\! \left(x \right)\\ F_{266}\! \left(x \right) &= F_{267}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{267}\! \left(x \right) &= F_{268}\! \left(x \right)+F_{269}\! \left(x \right)\\ F_{268}\! \left(x \right) &= F_{227}\! \left(x \right)\\ F_{269}\! \left(x \right) &= F_{270}\! \left(x \right)\\ F_{270}\! \left(x \right) &= F_{271}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{271}\! \left(x \right) &= F_{272}\! \left(x \right)+F_{273}\! \left(x \right)\\ F_{272}\! \left(x \right) &= F_{227}\! \left(x \right)\\ F_{273}\! \left(x \right) &= F_{270}\! \left(x \right)\\ F_{274}\! \left(x \right) &= F_{275}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{275}\! \left(x \right) &= F_{250}\! \left(x \right)\\ F_{276}\! \left(x \right) &= F_{277}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{277}\! \left(x \right) &= F_{278}\! \left(x \right)\\ F_{278}\! \left(x \right) &= F_{204}\! \left(x \right)+F_{279}\! \left(x \right)\\ F_{279}\! \left(x \right) &= F_{280}\! \left(x \right)\\ F_{280}\! \left(x \right) &= F_{281}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{281}\! \left(x \right) &= F_{282}\! \left(x \right)+F_{284}\! \left(x \right)\\ F_{282}\! \left(x \right) &= F_{200}\! \left(x \right)+F_{283}\! \left(x \right)\\ F_{283}\! \left(x \right) &= F_{228}\! \left(x \right)\\ F_{284}\! \left(x \right) &= F_{250}\! \left(x \right)+F_{285}\! \left(x \right)\\ F_{285}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{274}\! \left(x \right)+F_{286}\! \left(x \right)\\ F_{286}\! \left(x \right) &= F_{287}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{287}\! \left(x \right) &= F_{288}\! \left(x \right)+F_{289}\! \left(x \right)\\ F_{288}\! \left(x \right) &= F_{283}\! \left(x \right)\\ F_{289}\! \left(x \right) &= F_{290}\! \left(x \right)\\ F_{290}\! \left(x \right) &= F_{291}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{291}\! \left(x \right) &= F_{292}\! \left(x \right)+F_{293}\! \left(x \right)\\ F_{292}\! \left(x \right) &= F_{283}\! \left(x \right)\\ F_{293}\! \left(x \right) &= F_{290}\! \left(x \right)\\ F_{294}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{295}\! \left(x \right) &= F_{296}\! \left(x \right)+F_{351}\! \left(x \right)\\ F_{296}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{297}\! \left(x \right)+F_{350}\! \left(x \right)\\ F_{297}\! \left(x \right) &= F_{298}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{298}\! \left(x \right) &= F_{299}\! \left(x \right)+F_{304}\! \left(x \right)\\ F_{299}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{300}\! \left(x \right)\\ F_{300}\! \left(x \right) &= F_{197}\! \left(x \right)+F_{22}\! \left(x \right)+F_{301}\! \left(x \right)\\ F_{301}\! \left(x \right) &= F_{302}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{302}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{303}\! \left(x \right)\\ F_{303}\! \left(x \right) &= F_{202}\! \left(x \right)\\ F_{304}\! \left(x \right) &= F_{305}\! \left(x \right)+F_{320}\! \left(x \right)\\ F_{305}\! \left(x \right) &= F_{306}\! \left(x \right)\\ F_{306}\! \left(x \right) &= F_{307}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{307}\! \left(x \right) &= F_{308}\! \left(x \right)+F_{310}\! \left(x \right)\\ F_{308}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{309}\! \left(x \right)\\ F_{309}\! \left(x \right) &= F_{209}\! \left(x \right)\\ F_{310}\! \left(x \right) &= F_{305}\! \left(x \right)+F_{311}\! \left(x \right)\\ F_{311}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{221}\! \left(x \right)+F_{312}\! \left(x \right)\\ F_{312}\! \left(x \right) &= F_{313}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{313}\! \left(x \right) &= F_{314}\! \left(x \right)+F_{315}\! \left(x \right)\\ F_{314}\! \left(x \right) &= F_{309}\! \left(x \right)\\ F_{315}\! \left(x \right) &= F_{316}\! \left(x \right)\\ F_{316}\! \left(x \right) &= F_{317}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{317}\! \left(x \right) &= F_{318}\! \left(x \right)+F_{319}\! \left(x \right)\\ F_{318}\! \left(x \right) &= F_{309}\! \left(x \right)\\ F_{319}\! \left(x \right) &= F_{316}\! \left(x \right)\\ F_{320}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{276}\! \left(x \right)+F_{321}\! \left(x \right)\\ F_{321}\! \left(x \right) &= F_{322}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{322}\! \left(x \right) &= F_{323}\! \left(x \right)+F_{325}\! \left(x \right)\\ F_{323}\! \left(x \right) &= F_{300}\! \left(x \right)+F_{324}\! \left(x \right)\\ F_{324}\! \left(x \right) &= F_{228}\! \left(x \right)\\ F_{325}\! \left(x \right) &= F_{326}\! \left(x \right)+F_{341}\! \left(x \right)\\ F_{326}\! \left(x \right) &= F_{327}\! \left(x \right)\\ F_{327}\! \left(x \right) &= F_{328}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{328}\! \left(x \right) &= F_{329}\! \left(x \right)+F_{331}\! \left(x \right)\\ F_{329}\! \left(x \right) &= F_{300}\! \left(x \right)+F_{330}\! \left(x \right)\\ F_{330}\! \left(x \right) &= F_{236}\! \left(x \right)\\ F_{331}\! \left(x \right) &= F_{326}\! \left(x \right)+F_{332}\! \left(x \right)\\ F_{332}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{248}\! \left(x \right)+F_{333}\! \left(x \right)\\ F_{333}\! \left(x \right) &= F_{334}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{334}\! \left(x \right) &= F_{335}\! \left(x \right)+F_{336}\! \left(x \right)\\ F_{335}\! \left(x \right) &= F_{330}\! \left(x \right)\\ F_{336}\! \left(x \right) &= F_{337}\! \left(x \right)\\ F_{337}\! \left(x \right) &= F_{338}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{338}\! \left(x \right) &= F_{339}\! \left(x \right)+F_{340}\! \left(x \right)\\ F_{339}\! \left(x \right) &= F_{330}\! \left(x \right)\\ F_{340}\! \left(x \right) &= F_{337}\! \left(x \right)\\ F_{341}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{274}\! \left(x \right)+F_{342}\! \left(x \right)\\ F_{342}\! \left(x \right) &= F_{343}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{343}\! \left(x \right) &= F_{344}\! \left(x \right)+F_{345}\! \left(x \right)\\ F_{344}\! \left(x \right) &= F_{324}\! \left(x \right)\\ F_{345}\! \left(x \right) &= F_{346}\! \left(x \right)\\ F_{346}\! \left(x \right) &= F_{347}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{347}\! \left(x \right) &= F_{348}\! \left(x \right)+F_{349}\! \left(x \right)\\ F_{348}\! \left(x \right) &= F_{324}\! \left(x \right)\\ F_{349}\! \left(x \right) &= F_{346}\! \left(x \right)\\ F_{350}\! \left(x \right) &= F_{191}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{351}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{352}\! \left(x \right)+F_{407}\! \left(x \right)\\ F_{352}\! \left(x \right) &= F_{353}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{353}\! \left(x \right) &= F_{354}\! \left(x \right)+F_{361}\! \left(x \right)\\ F_{354}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{355}\! \left(x \right)\\ F_{355}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{356}\! \left(x \right)+F_{359}\! \left(x \right)\\ F_{356}\! \left(x \right) &= F_{357}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{357}\! \left(x \right) &= F_{358}\! \left(x \right)\\ F_{358}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{359}\! \left(x \right) &= F_{360}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{360}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{361}\! \left(x \right) &= F_{362}\! \left(x \right)+F_{396}\! \left(x \right)\\ F_{362}\! \left(x \right) &= F_{363}\! \left(x \right)\\ F_{363}\! \left(x \right) &= F_{364}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{364}\! \left(x \right) &= F_{365}\! \left(x \right)+F_{369}\! \left(x \right)\\ F_{365}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{366}\! \left(x \right)\\ F_{366}\! \left(x \right) &= F_{367}\! \left(x \right)\\ F_{367}\! \left(x \right) &= F_{368}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{368}\! \left(x \right) &= F_{358}\! \left(x \right)\\ F_{369}\! \left(x \right) &= F_{362}\! \left(x \right)+F_{370}\! \left(x \right)\\ F_{370}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{371}\! \left(x \right)+F_{379}\! \left(x \right)\\ F_{371}\! \left(x \right) &= F_{372}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{372}\! \left(x \right) &= F_{373}\! \left(x \right)+F_{374}\! \left(x \right)\\ F_{373}\! \left(x \right) &= F_{366}\! \left(x \right)\\ F_{374}\! \left(x \right) &= F_{375}\! \left(x \right)\\ F_{375}\! \left(x \right) &= F_{376}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{376}\! \left(x \right) &= F_{377}\! \left(x \right)+F_{378}\! \left(x \right)\\ F_{377}\! \left(x \right) &= F_{366}\! \left(x \right)\\ F_{378}\! \left(x \right) &= F_{375}\! \left(x \right)\\ F_{379}\! \left(x \right) &= F_{380}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{380}\! \left(x \right) &= F_{381}\! \left(x \right)\\ F_{381}\! \left(x \right) &= F_{382}\! \left(x \right)\\ F_{382}\! \left(x \right) &= F_{383}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{383}\! \left(x \right) &= F_{384}\! \left(x \right)+F_{386}\! \left(x \right)\\ F_{384}\! \left(x \right) &= F_{358}\! \left(x \right)+F_{385}\! \left(x \right)\\ F_{385}\! \left(x \right) &= F_{367}\! \left(x \right)\\ F_{386}\! \left(x \right) &= F_{381}\! \left(x \right)+F_{387}\! \left(x \right)\\ F_{387}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{379}\! \left(x \right)+F_{388}\! \left(x \right)\\ F_{388}\! \left(x \right) &= F_{389}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{389}\! \left(x \right) &= F_{390}\! \left(x \right)+F_{391}\! \left(x \right)\\ F_{390}\! \left(x \right) &= F_{385}\! \left(x \right)\\ F_{391}\! \left(x \right) &= F_{392}\! \left(x \right)\\ F_{392}\! \left(x \right) &= F_{393}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{393}\! \left(x \right) &= F_{394}\! \left(x \right)+F_{395}\! \left(x \right)\\ F_{394}\! \left(x \right) &= F_{385}\! \left(x \right)\\ F_{395}\! \left(x \right) &= F_{392}\! \left(x \right)\\ F_{396}\! \left(x \right) &= 3 F_{22}\! \left(x \right)+F_{397}\! \left(x \right)+F_{405}\! \left(x \right)\\ F_{397}\! \left(x \right) &= F_{398}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{398}\! \left(x \right) &= F_{399}\! \left(x \right)+F_{400}\! \left(x \right)\\ F_{399}\! \left(x \right) &= F_{355}\! \left(x \right)\\ F_{400}\! \left(x \right) &= F_{401}\! \left(x \right)\\ F_{401}\! \left(x \right) &= F_{4}\! \left(x \right) F_{402}\! \left(x \right)\\ F_{402}\! \left(x \right) &= F_{403}\! \left(x \right)+F_{404}\! \left(x \right)\\ F_{403}\! \left(x \right) &= F_{355}\! \left(x \right)\\ F_{404}\! \left(x \right) &= F_{401}\! \left(x \right)\\ F_{405}\! \left(x \right) &= F_{4}\! \left(x \right) F_{406}\! \left(x \right)\\ F_{406}\! \left(x \right) &= F_{381}\! \left(x \right)\\ F_{407}\! \left(x \right) &= F_{4}\! \left(x \right) F_{408}\! \left(x \right)\\ F_{408}\! \left(x \right) &= F_{296}\! \left(x \right)\\ \end{align*}\)