Av(1234, 1432, 2143, 3214)
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Generating Function
\(\displaystyle \frac{x^{5}+x^{3}+x^{2}+x -1}{x^{9}+x^{8}+x^{6}+6 x^{5}+4 x^{4}+2 x^{3}+x^{2}+2 x -1}\)
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Counting Sequence
1, 1, 2, 6, 20, 59, 165, 466, 1334, 3828, 10959, 31335, 89608, 256330, 733300, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{9}+x^{8}+x^{6}+6 x^{5}+4 x^{4}+2 x^{3}+x^{2}+2 x -1\right) F \! \left(x \right)-x^{5}-x^{3}-x^{2}-x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 165\)
\(\displaystyle a \! \left(7\right) = 466\)
\(\displaystyle a \! \left(8\right) = 1334\)
\(\displaystyle a \! \left(n +1\right) = -a \! \left(n \right)-a \! \left(n +3\right)-6 a \! \left(n +4\right)-4 a \! \left(n +5\right)-2 a \! \left(n +6\right)-a \! \left(n +7\right)-2 a \! \left(n +8\right)+a \! \left(n +9\right), \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle -\frac{15096011956 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +7}}{1942431559805}-\frac{15096011956 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +7}}{1942431559805}-\frac{15096011956 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +7}}{1942431559805}-\frac{15096011956 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +7}}{1942431559805}-\frac{15096011956 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +7}}{1942431559805}-\frac{15096011956 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n +7}}{1942431559805}-\frac{15096011956 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =7\right)^{-n +7}}{1942431559805}-\frac{15096011956 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =8\right)^{-n +7}}{1942431559805}-\frac{15096011956 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =9\right)^{-n +7}}{1942431559805}-\frac{3650272553 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +6}}{388486311961}-\frac{3650272553 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +6}}{388486311961}-\frac{3650272553 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +6}}{388486311961}-\frac{3650272553 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +6}}{388486311961}-\frac{3650272553 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +6}}{388486311961}-\frac{3650272553 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n +6}}{388486311961}-\frac{3650272553 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =7\right)^{-n +6}}{388486311961}-\frac{3650272553 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =8\right)^{-n +6}}{388486311961}-\frac{3650272553 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =9\right)^{-n +6}}{388486311961}+\frac{12864084588 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +5}}{388486311961}+\frac{12864084588 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +5}}{388486311961}+\frac{12864084588 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +5}}{388486311961}+\frac{12864084588 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +5}}{388486311961}+\frac{12864084588 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +5}}{388486311961}+\frac{12864084588 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n +5}}{388486311961}+\frac{12864084588 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =7\right)^{-n +5}}{388486311961}+\frac{12864084588 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =8\right)^{-n +5}}{388486311961}+\frac{12864084588 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =9\right)^{-n +5}}{388486311961}+\frac{27005976409 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +4}}{1942431559805}+\frac{27005976409 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +4}}{1942431559805}+\frac{27005976409 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +4}}{1942431559805}+\frac{27005976409 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +4}}{1942431559805}+\frac{27005976409 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +4}}{1942431559805}+\frac{27005976409 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n +4}}{1942431559805}+\frac{27005976409 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =7\right)^{-n +4}}{1942431559805}+\frac{27005976409 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =8\right)^{-n +4}}{1942431559805}+\frac{27005976409 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =9\right)^{-n +4}}{1942431559805}-\frac{19719062589 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +3}}{388486311961}-\frac{19719062589 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +3}}{388486311961}-\frac{19719062589 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +3}}{388486311961}-\frac{19719062589 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +3}}{388486311961}-\frac{19719062589 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +3}}{388486311961}-\frac{19719062589 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n +3}}{388486311961}-\frac{19719062589 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =7\right)^{-n +3}}{388486311961}-\frac{19719062589 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =8\right)^{-n +3}}{388486311961}-\frac{19719062589 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =9\right)^{-n +3}}{388486311961}+\frac{2493295711 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +2}}{1942431559805}+\frac{2493295711 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +2}}{1942431559805}+\frac{2493295711 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +2}}{1942431559805}+\frac{2493295711 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +2}}{1942431559805}+\frac{2493295711 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +2}}{1942431559805}+\frac{2493295711 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n +2}}{1942431559805}+\frac{2493295711 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =7\right)^{-n +2}}{1942431559805}+\frac{2493295711 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =8\right)^{-n +2}}{1942431559805}+\frac{2493295711 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =9\right)^{-n +2}}{1942431559805}+\frac{246512553577 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +1}}{1942431559805}+\frac{246512553577 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +1}}{1942431559805}+\frac{246512553577 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +1}}{1942431559805}+\frac{246512553577 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +1}}{1942431559805}+\frac{246512553577 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +1}}{1942431559805}+\frac{246512553577 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n +1}}{1942431559805}+\frac{246512553577 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =7\right)^{-n +1}}{1942431559805}+\frac{246512553577 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =8\right)^{-n +1}}{1942431559805}+\frac{246512553577 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =9\right)^{-n +1}}{1942431559805}+\frac{82322837216 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n -1}}{1942431559805}+\frac{82322837216 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n -1}}{1942431559805}+\frac{82322837216 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n -1}}{1942431559805}+\frac{82322837216 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n -1}}{1942431559805}+\frac{82322837216 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n -1}}{1942431559805}+\frac{82322837216 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n -1}}{1942431559805}+\frac{82322837216 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =7\right)^{-n -1}}{1942431559805}+\frac{82322837216 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =8\right)^{-n -1}}{1942431559805}+\frac{82322837216 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =9\right)^{-n -1}}{1942431559805}+\frac{261371694072 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n}}{1942431559805}+\frac{261371694072 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n}}{1942431559805}+\frac{261371694072 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n}}{1942431559805}+\frac{261371694072 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n}}{1942431559805}+\frac{261371694072 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n}}{1942431559805}+\frac{261371694072 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =6\right)^{-n}}{1942431559805}+\frac{261371694072 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =7\right)^{-n}}{1942431559805}+\frac{261371694072 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =8\right)^{-n}}{1942431559805}+\frac{261371694072 \mathit{RootOf} \left(Z^{9}+Z^{8}+Z^{6}+6 Z^{5}+4 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =9\right)^{-n}}{1942431559805}\)
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This specification was found using the strategy pack "Point Placements" and has 104 rules.

Found on January 18, 2022.

Finding the specification took 2 seconds.

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Copy 104 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_{28}\! \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_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ 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_{22}\! \left(x \right) &= x^{2}\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{95}\! \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_{46}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{43}\! \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_{38}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{52}\! \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_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{52}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{50}\! \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_{18}\! \left(x \right)+F_{51}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{30}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{74}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \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_{39}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{84}\! \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_{50}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{65}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{59}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{18}\! \left(x \right)+F_{74}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{73}\! \left(x \right)\\ \end{align*}\)