Av(1432, 2143, 3214)
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Generating Function
\(\displaystyle \frac{\left(x^{4}-9 x^{3}+12 x^{2}-6 x +1\right) \left(-1+x \right)^{4}}{5 x^{8}-44 x^{7}+128 x^{6}-208 x^{5}+209 x^{4}-132 x^{3}+51 x^{2}-11 x +1}\)
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Counting Sequence
1, 1, 2, 6, 21, 76, 273, 964, 3356, 11587, 39866, 137055, 471326, 1621698, 5581897, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(5 x^{8}-44 x^{7}+128 x^{6}-208 x^{5}+209 x^{4}-132 x^{3}+51 x^{2}-11 x +1\right) F \! \left(x \right)-\left(x^{4}-9 x^{3}+12 x^{2}-6 x +1\right) \left(-1+x \right)^{4} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 273\)
\(\displaystyle a \! \left(7\right) = 964\)
\(\displaystyle a \! \left(8\right) = 3356\)
\(\displaystyle a \! \left(n +8\right) = -5 a \! \left(n \right)+44 a \! \left(n +1\right)-128 a \! \left(n +2\right)+208 a \! \left(n +3\right)-209 a \! \left(n +4\right)+132 a \! \left(n +5\right)-51 a \! \left(n +6\right)+11 a \! \left(n +7\right), \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle -\frac{482750725 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n +6}}{185536383}-\frac{482750725 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n +6}}{185536383}-\frac{482750725 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n +6}}{185536383}-\frac{482750725 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n +6}}{185536383}-\frac{482750725 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n +6}}{185536383}-\frac{482750725 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n +6}}{185536383}-\frac{482750725 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n +6}}{185536383}-\frac{482750725 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n +6}}{185536383}+\frac{4026054590 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n +5}}{185536383}+\frac{4026054590 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n +5}}{185536383}+\frac{4026054590 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n +5}}{185536383}+\frac{4026054590 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n +5}}{185536383}+\frac{4026054590 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n +5}}{185536383}+\frac{4026054590 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n +5}}{185536383}+\frac{4026054590 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n +5}}{185536383}+\frac{4026054590 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n +5}}{185536383}-\frac{10481128598 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n +4}}{185536383}-\frac{10481128598 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n +4}}{185536383}-\frac{10481128598 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n +4}}{185536383}-\frac{10481128598 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n +4}}{185536383}-\frac{10481128598 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n +4}}{185536383}-\frac{10481128598 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n +4}}{185536383}-\frac{10481128598 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n +4}}{185536383}-\frac{10481128598 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n +4}}{185536383}+\frac{15075243493 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n +3}}{185536383}+\frac{15075243493 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n +3}}{185536383}+\frac{15075243493 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n +3}}{185536383}+\frac{15075243493 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n +3}}{185536383}+\frac{15075243493 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n +3}}{185536383}+\frac{15075243493 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n +3}}{185536383}+\frac{15075243493 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n +3}}{185536383}+\frac{15075243493 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n +3}}{185536383}-\frac{12863997899 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n +2}}{185536383}-\frac{12863997899 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n +2}}{185536383}-\frac{12863997899 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n +2}}{185536383}-\frac{12863997899 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n +2}}{185536383}-\frac{12863997899 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n +2}}{185536383}-\frac{12863997899 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n +2}}{185536383}-\frac{12863997899 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n +2}}{185536383}-\frac{12863997899 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n +2}}{185536383}+\frac{2122346689 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n +1}}{61845461}+\frac{2122346689 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n +1}}{61845461}+\frac{2122346689 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n +1}}{61845461}+\frac{2122346689 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n +1}}{61845461}+\frac{2122346689 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n +1}}{61845461}+\frac{2122346689 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n +1}}{61845461}+\frac{2122346689 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n +1}}{61845461}+\frac{2122346689 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n +1}}{61845461}+\frac{62557958 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n -1}}{61845461}+\frac{62557958 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n -1}}{61845461}+\frac{62557958 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n -1}}{61845461}+\frac{62557958 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n -1}}{61845461}+\frac{62557958 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n -1}}{61845461}+\frac{62557958 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n -1}}{61845461}+\frac{62557958 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n -1}}{61845461}+\frac{62557958 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n -1}}{61845461}-\frac{1681531585 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =1\right)^{-n}}{185536383}-\frac{1681531585 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =2\right)^{-n}}{185536383}-\frac{1681531585 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =3\right)^{-n}}{185536383}-\frac{1681531585 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =4\right)^{-n}}{185536383}-\frac{1681531585 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =5\right)^{-n}}{185536383}-\frac{1681531585 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =6\right)^{-n}}{185536383}-\frac{1681531585 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =7\right)^{-n}}{185536383}-\frac{1681531585 \mathit{RootOf} \left(5 Z^{8}-44 Z^{7}+128 Z^{6}-208 Z^{5}+209 Z^{4}-132 Z^{3}+51 Z^{2}-11 Z +1, \mathit{index} =8\right)^{-n}}{185536383}+\left(\left\{\begin{array}{cc}\frac{1}{5} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
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This specification was found using the strategy pack "Insertion Point Placements" and has 114 rules.

Found on July 23, 2021.

Finding the specification took 54 seconds.

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Copy 114 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_{16}\! \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_{16}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{23}\! \left(x \right) &= 0\\ F_{24}\! \left(x \right) &= F_{16}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{16}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{16}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{16}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{16}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{16}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{14} \left(x \right)^{2}\\ F_{49}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{16}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{14}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{14}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right) F_{28}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{16}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{36}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{14}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{13}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{14}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{13}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{16}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{14}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{13}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{16}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{13}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{13}\! \left(x \right) F_{21}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{16}\! \left(x \right) F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{103}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{102}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{13}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{106}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{14}\! \left(x \right) F_{21}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{112}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{2}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{31}\! \left(x \right)\\ \end{align*}\)