Av(13425, 13452, 13542, 14235, 14253, 14325, 14352, 14523, 14532, 15342, 15423, 15432, 31425, 31452, 31542, 34125, 34152, 41235, 41253, 41325, 41352, 41523, 41532, 43125, 43152)
View Raw Data
Generating Function
\(\displaystyle \frac{\left(x +1\right) \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{3}}{2 x^{11}-7 x^{10}-11 x^{9}+51 x^{8}-48 x^{7}+56 x^{6}-41 x^{5}-14 x^{4}+44 x^{3}-30 x^{2}+9 x -1}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 24, 95, 363, 1323, 4729, 16809, 59786, 212922, 758779, 2703927, 9633000, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{11}-7 x^{10}-11 x^{9}+51 x^{8}-48 x^{7}+56 x^{6}-41 x^{5}-14 x^{4}+44 x^{3}-30 x^{2}+9 x -1\right) F \! \left(x \right)-\left(x +1\right) \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{3} = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a(6) = 363\)
\(\displaystyle a(7) = 1323\)
\(\displaystyle a(8) = 4729\)
\(\displaystyle a(9) = 16809\)
\(\displaystyle a(10) = 59786\)
\(\displaystyle a{\left(n + 11 \right)} = 2 a{\left(n \right)} - 7 a{\left(n + 1 \right)} - 11 a{\left(n + 2 \right)} + 51 a{\left(n + 3 \right)} - 48 a{\left(n + 4 \right)} + 56 a{\left(n + 5 \right)} - 41 a{\left(n + 6 \right)} - 14 a{\left(n + 7 \right)} + 44 a{\left(n + 8 \right)} - 30 a{\left(n + 9 \right)} + 9 a{\left(n + 10 \right)}, \quad n \geq 11\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle -\frac{27132848034443717374347 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +6}}{29959562526356440880305}-\frac{27132848034443717374347 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +6}}{29959562526356440880305}-\frac{27132848034443717374347 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +6}}{29959562526356440880305}-\frac{27132848034443717374347 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +6}}{29959562526356440880305}-\frac{27132848034443717374347 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +6}}{29959562526356440880305}-\frac{27132848034443717374347 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +6}}{29959562526356440880305}-\frac{27132848034443717374347 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +6}}{29959562526356440880305}-\frac{27132848034443717374347 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +6}}{29959562526356440880305}-\frac{27132848034443717374347 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +6}}{29959562526356440880305}-\frac{27132848034443717374347 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =10\right)^{-n +6}}{29959562526356440880305}-\frac{27132848034443717374347 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =11\right)^{-n +6}}{29959562526356440880305}-\frac{11961310889827891668716 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +5}}{29959562526356440880305}-\frac{11961310889827891668716 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +5}}{29959562526356440880305}-\frac{11961310889827891668716 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +5}}{29959562526356440880305}-\frac{11961310889827891668716 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +5}}{29959562526356440880305}-\frac{11961310889827891668716 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +5}}{29959562526356440880305}-\frac{11961310889827891668716 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +5}}{29959562526356440880305}-\frac{11961310889827891668716 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +5}}{29959562526356440880305}-\frac{11961310889827891668716 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +5}}{29959562526356440880305}-\frac{11961310889827891668716 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +5}}{29959562526356440880305}-\frac{11961310889827891668716 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =10\right)^{-n +5}}{29959562526356440880305}-\frac{11961310889827891668716 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =11\right)^{-n +5}}{29959562526356440880305}-\frac{37539837981417472045583 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +4}}{29959562526356440880305}-\frac{37539837981417472045583 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +4}}{29959562526356440880305}-\frac{37539837981417472045583 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +4}}{29959562526356440880305}-\frac{37539837981417472045583 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +4}}{29959562526356440880305}-\frac{37539837981417472045583 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +4}}{29959562526356440880305}-\frac{37539837981417472045583 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +4}}{29959562526356440880305}-\frac{37539837981417472045583 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +4}}{29959562526356440880305}-\frac{37539837981417472045583 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +4}}{29959562526356440880305}-\frac{37539837981417472045583 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +4}}{29959562526356440880305}-\frac{37539837981417472045583 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =10\right)^{-n +4}}{29959562526356440880305}-\frac{37539837981417472045583 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =11\right)^{-n +4}}{29959562526356440880305}-\frac{4563473562272079331556 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +3}}{5991912505271288176061}-\frac{4563473562272079331556 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +3}}{5991912505271288176061}-\frac{4563473562272079331556 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +3}}{5991912505271288176061}-\frac{4563473562272079331556 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +3}}{5991912505271288176061}-\frac{4563473562272079331556 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +3}}{5991912505271288176061}-\frac{4563473562272079331556 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +3}}{5991912505271288176061}-\frac{4563473562272079331556 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +3}}{5991912505271288176061}-\frac{4563473562272079331556 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +3}}{5991912505271288176061}-\frac{4563473562272079331556 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +3}}{5991912505271288176061}-\frac{4563473562272079331556 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =10\right)^{-n +3}}{5991912505271288176061}-\frac{4563473562272079331556 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =11\right)^{-n +3}}{5991912505271288176061}+\frac{22670655121207522257019 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +2}}{29959562526356440880305}+\frac{22670655121207522257019 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +2}}{29959562526356440880305}+\frac{22670655121207522257019 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +2}}{29959562526356440880305}+\frac{22670655121207522257019 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +2}}{29959562526356440880305}+\frac{22670655121207522257019 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +2}}{29959562526356440880305}+\frac{22670655121207522257019 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +2}}{29959562526356440880305}+\frac{22670655121207522257019 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +2}}{29959562526356440880305}+\frac{22670655121207522257019 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +2}}{29959562526356440880305}+\frac{22670655121207522257019 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +2}}{29959562526356440880305}+\frac{22670655121207522257019 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =10\right)^{-n +2}}{29959562526356440880305}+\frac{22670655121207522257019 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =11\right)^{-n +2}}{29959562526356440880305}-\frac{5641593284545022393231 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +1}}{29959562526356440880305}-\frac{5641593284545022393231 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +1}}{29959562526356440880305}-\frac{5641593284545022393231 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +1}}{29959562526356440880305}-\frac{5641593284545022393231 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +1}}{29959562526356440880305}-\frac{5641593284545022393231 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +1}}{29959562526356440880305}-\frac{5641593284545022393231 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +1}}{29959562526356440880305}-\frac{5641593284545022393231 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +1}}{29959562526356440880305}-\frac{5641593284545022393231 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +1}}{29959562526356440880305}-\frac{5641593284545022393231 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +1}}{29959562526356440880305}-\frac{5641593284545022393231 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =10\right)^{-n +1}}{29959562526356440880305}-\frac{5641593284545022393231 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =11\right)^{-n +1}}{29959562526356440880305}+\frac{2252853776856898763997 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =10\right)^{-n -1}}{29959562526356440880305}+\frac{2252853776856898763997 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =11\right)^{-n -1}}{29959562526356440880305}+\frac{2252853776856898763997 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n -1}}{29959562526356440880305}+\frac{2252853776856898763997 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n -1}}{29959562526356440880305}+\frac{2252853776856898763997 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n -1}}{29959562526356440880305}+\frac{2252853776856898763997 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n -1}}{29959562526356440880305}+\frac{2252853776856898763997 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n -1}}{29959562526356440880305}+\frac{2252853776856898763997 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n -1}}{29959562526356440880305}+\frac{2252853776856898763997 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n -1}}{29959562526356440880305}+\frac{2252853776856898763997 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n -1}}{29959562526356440880305}+\frac{2252853776856898763997 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n -1}}{29959562526356440880305}-\frac{1982208568300405093902 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n}}{29959562526356440880305}-\frac{1982208568300405093902 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n}}{29959562526356440880305}-\frac{1982208568300405093902 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n}}{29959562526356440880305}-\frac{1982208568300405093902 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n}}{29959562526356440880305}-\frac{1982208568300405093902 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n}}{29959562526356440880305}-\frac{1982208568300405093902 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n}}{29959562526356440880305}-\frac{1982208568300405093902 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n}}{29959562526356440880305}-\frac{1982208568300405093902 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n}}{29959562526356440880305}-\frac{1982208568300405093902 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n}}{29959562526356440880305}-\frac{1982208568300405093902 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =10\right)^{-n}}{29959562526356440880305}-\frac{1982208568300405093902 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =11\right)^{-n}}{29959562526356440880305}+\frac{20002286568320269388252 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +7}}{29959562526356440880305}+\frac{20002286568320269388252 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +7}}{29959562526356440880305}+\frac{20002286568320269388252 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +7}}{29959562526356440880305}+\frac{20002286568320269388252 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +7}}{29959562526356440880305}+\frac{20002286568320269388252 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +7}}{29959562526356440880305}+\frac{20002286568320269388252 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +7}}{29959562526356440880305}+\frac{20002286568320269388252 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +7}}{29959562526356440880305}+\frac{20002286568320269388252 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +7}}{29959562526356440880305}+\frac{20002286568320269388252 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +7}}{29959562526356440880305}+\frac{20002286568320269388252 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =10\right)^{-n +7}}{29959562526356440880305}+\frac{20002286568320269388252 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =11\right)^{-n +7}}{29959562526356440880305}+\frac{4011801358634711954093 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +8}}{29959562526356440880305}+\frac{4011801358634711954093 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +8}}{29959562526356440880305}+\frac{4011801358634711954093 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +8}}{29959562526356440880305}+\frac{4011801358634711954093 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +8}}{29959562526356440880305}+\frac{4011801358634711954093 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +8}}{29959562526356440880305}+\frac{4011801358634711954093 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +8}}{29959562526356440880305}+\frac{4011801358634711954093 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +8}}{29959562526356440880305}+\frac{4011801358634711954093 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +8}}{29959562526356440880305}+\frac{4011801358634711954093 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +8}}{29959562526356440880305}+\frac{4011801358634711954093 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =10\right)^{-n +8}}{29959562526356440880305}+\frac{4011801358634711954093 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =11\right)^{-n +8}}{29959562526356440880305}-\frac{2077181679396326549042 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =1\right)^{-n +9}}{29959562526356440880305}-\frac{2077181679396326549042 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =2\right)^{-n +9}}{29959562526356440880305}-\frac{2077181679396326549042 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =3\right)^{-n +9}}{29959562526356440880305}-\frac{2077181679396326549042 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =4\right)^{-n +9}}{29959562526356440880305}-\frac{2077181679396326549042 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =5\right)^{-n +9}}{29959562526356440880305}-\frac{2077181679396326549042 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =6\right)^{-n +9}}{29959562526356440880305}-\frac{2077181679396326549042 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =7\right)^{-n +9}}{29959562526356440880305}-\frac{2077181679396326549042 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =8\right)^{-n +9}}{29959562526356440880305}-\frac{2077181679396326549042 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =9\right)^{-n +9}}{29959562526356440880305}-\frac{2077181679396326549042 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =10\right)^{-n +9}}{29959562526356440880305}-\frac{2077181679396326549042 \mathit{RootOf} \left(2 Z^{11}-7 Z^{10}-11 Z^{9}+51 Z^{8}-48 Z^{7}+56 Z^{6}-41 Z^{5}-14 Z^{4}+44 Z^{3}-30 Z^{2}+9 Z -1, \mathit{index} =11\right)^{-n +9}}{29959562526356440880305}\)
Copy to clipboard:

This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 114 rules.

Finding the specification took 112 seconds.

This tree is too big to show here. Click to view tree on new page.

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