Av(1234, 1243, 1432, 2143, 3214)
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Generating Function
\(\displaystyle \frac{x^{4}+3 x^{3}+x^{2}-1}{x^{9}-x^{8}+5 x^{6}+7 x^{5}+5 x^{4}+5 x^{3}+2 x^{2}+x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 53, 143, 393, 1090, 3019, 8344, 23059, 63745, 176239, 487237, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{9}-x^{8}+5 x^{6}+7 x^{5}+5 x^{4}+5 x^{3}+2 x^{2}+x -1\right) F \! \left(x \right)-x^{4}-3 x^{3}-x^{2}+1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 143\)
\(\displaystyle a \! \left(7\right) = 393\)
\(\displaystyle a \! \left(8\right) = 1090\)
\(\displaystyle a \! \left(n +1\right) = a \! \left(n \right)+5 a \! \left(n +3\right)+7 a \! \left(n +4\right)+5 a \! \left(n +5\right)+5 a \! \left(n +6\right)+2 a \! \left(n +7\right)+a \! \left(n +8\right)-a \! \left(n +9\right), \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle -\frac{14387596885 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =1\right)^{-n +7}}{23999589519916}-\frac{14387596885 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =2\right)^{-n +7}}{23999589519916}-\frac{14387596885 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =3\right)^{-n +7}}{23999589519916}-\frac{14387596885 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =4\right)^{-n +7}}{23999589519916}-\frac{14387596885 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =5\right)^{-n +7}}{23999589519916}-\frac{14387596885 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =6\right)^{-n +7}}{23999589519916}-\frac{14387596885 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =7\right)^{-n +7}}{23999589519916}-\frac{14387596885 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =8\right)^{-n +7}}{23999589519916}-\frac{14387596885 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =9\right)^{-n +7}}{23999589519916}-\frac{189835102969 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =1\right)^{-n +6}}{11999794759958}-\frac{189835102969 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =2\right)^{-n +6}}{11999794759958}-\frac{189835102969 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =3\right)^{-n +6}}{11999794759958}-\frac{189835102969 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =4\right)^{-n +6}}{11999794759958}-\frac{189835102969 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =5\right)^{-n +6}}{11999794759958}-\frac{189835102969 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =6\right)^{-n +6}}{11999794759958}-\frac{189835102969 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =7\right)^{-n +6}}{11999794759958}-\frac{189835102969 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =8\right)^{-n +6}}{11999794759958}-\frac{189835102969 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =9\right)^{-n +6}}{11999794759958}+\frac{215791351916 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =1\right)^{-n +5}}{5999897379979}+\frac{215791351916 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =2\right)^{-n +5}}{5999897379979}+\frac{215791351916 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =3\right)^{-n +5}}{5999897379979}+\frac{215791351916 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =4\right)^{-n +5}}{5999897379979}+\frac{215791351916 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =5\right)^{-n +5}}{5999897379979}+\frac{215791351916 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =6\right)^{-n +5}}{5999897379979}+\frac{215791351916 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =7\right)^{-n +5}}{5999897379979}+\frac{215791351916 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =8\right)^{-n +5}}{5999897379979}+\frac{215791351916 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =9\right)^{-n +5}}{5999897379979}-\frac{201457826791 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =1\right)^{-n +4}}{23999589519916}-\frac{201457826791 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =2\right)^{-n +4}}{23999589519916}-\frac{201457826791 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =3\right)^{-n +4}}{23999589519916}-\frac{201457826791 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =4\right)^{-n +4}}{23999589519916}-\frac{201457826791 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =5\right)^{-n +4}}{23999589519916}-\frac{201457826791 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =6\right)^{-n +4}}{23999589519916}-\frac{201457826791 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =7\right)^{-n +4}}{23999589519916}-\frac{201457826791 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =8\right)^{-n +4}}{23999589519916}-\frac{201457826791 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =9\right)^{-n +4}}{23999589519916}-\frac{1265734070237 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =1\right)^{-n +3}}{11999794759958}-\frac{1265734070237 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =2\right)^{-n +3}}{11999794759958}-\frac{1265734070237 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =3\right)^{-n +3}}{11999794759958}-\frac{1265734070237 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =4\right)^{-n +3}}{11999794759958}-\frac{1265734070237 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =5\right)^{-n +3}}{11999794759958}-\frac{1265734070237 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =6\right)^{-n +3}}{11999794759958}-\frac{1265734070237 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =7\right)^{-n +3}}{11999794759958}-\frac{1265734070237 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =8\right)^{-n +3}}{11999794759958}-\frac{1265734070237 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =9\right)^{-n +3}}{11999794759958}-\frac{446926415475 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =1\right)^{-n +2}}{23999589519916}-\frac{446926415475 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =2\right)^{-n +2}}{23999589519916}-\frac{446926415475 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =3\right)^{-n +2}}{23999589519916}-\frac{446926415475 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =4\right)^{-n +2}}{23999589519916}-\frac{446926415475 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =5\right)^{-n +2}}{23999589519916}-\frac{446926415475 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =6\right)^{-n +2}}{23999589519916}-\frac{446926415475 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =7\right)^{-n +2}}{23999589519916}-\frac{446926415475 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =8\right)^{-n +2}}{23999589519916}-\frac{446926415475 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =9\right)^{-n +2}}{23999589519916}+\frac{1232949747723 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =1\right)^{-n +1}}{11999794759958}+\frac{1232949747723 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =2\right)^{-n +1}}{11999794759958}+\frac{1232949747723 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =3\right)^{-n +1}}{11999794759958}+\frac{1232949747723 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =4\right)^{-n +1}}{11999794759958}+\frac{1232949747723 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =5\right)^{-n +1}}{11999794759958}+\frac{1232949747723 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =6\right)^{-n +1}}{11999794759958}+\frac{1232949747723 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =7\right)^{-n +1}}{11999794759958}+\frac{1232949747723 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =8\right)^{-n +1}}{11999794759958}+\frac{1232949747723 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =9\right)^{-n +1}}{11999794759958}+\frac{1285483749123 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =1\right)^{-n -1}}{23999589519916}+\frac{1285483749123 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =2\right)^{-n -1}}{23999589519916}+\frac{1285483749123 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =3\right)^{-n -1}}{23999589519916}+\frac{1285483749123 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =4\right)^{-n -1}}{23999589519916}+\frac{1285483749123 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =5\right)^{-n -1}}{23999589519916}+\frac{1285483749123 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =6\right)^{-n -1}}{23999589519916}+\frac{1285483749123 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =7\right)^{-n -1}}{23999589519916}+\frac{1285483749123 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =8\right)^{-n -1}}{23999589519916}+\frac{1285483749123 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =9\right)^{-n -1}}{23999589519916}+\frac{1703915866009 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =1\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =2\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =3\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =4\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =5\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =6\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =7\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =8\right)^{-n}}{11999794759958}+\frac{1703915866009 \mathit{RootOf} \left(Z^{9}-Z^{8}+5 Z^{6}+7 Z^{5}+5 Z^{4}+5 Z^{3}+2 Z^{2}+Z -1, \mathit{index} =9\right)^{-n}}{11999794759958}\)
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This specification was found using the strategy pack "Point Placements" and has 72 rules.

Found on January 18, 2022.

Finding the specification took 2 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_{55}\! \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_{29}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{28}\! \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_{18}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{35}\! \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_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{33}\! \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_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{54}\! \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_{65}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{66}\! \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_{33}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{40}\! \left(x \right)\\ \end{align*}\)