Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13524, 14235, 14325, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 24135, 31245, 31254, 31452, 31542, 32145, 32154, 41253, 41352, 42153)
View Raw Data
Generating Function
\(\displaystyle -\frac{x^{10}+6 x^{9}+10 x^{8}+2 x^{7}-4 x^{6}-4 x^{5}+5 x^{3}+2 x^{2}+2 x -1}{\left(x +1\right) \left(x^{10}+3 x^{9}-5 x^{8}-14 x^{7}-7 x^{5}+7 x^{4}-7 x^{3}+3 x^{2}-4 x +1\right)}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 24, 90, 329, 1192, 4302, 15499, 55854, 201348, 725916, 2617194, 9436021, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x +1\right) \left(x^{10}+3 x^{9}-5 x^{8}-14 x^{7}-7 x^{5}+7 x^{4}-7 x^{3}+3 x^{2}-4 x +1\right) F \! \left(x \right)+x^{10}+6 x^{9}+10 x^{8}+2 x^{7}-4 x^{6}-4 x^{5}+5 x^{3}+2 x^{2}+2 x -1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a(6) = 329\)
\(\displaystyle a(7) = 1192\)
\(\displaystyle a(8) = 4302\)
\(\displaystyle a(9) = 15499\)
\(\displaystyle a(10) = 55854\)
\(\displaystyle a{\left(n + 5 \right)} = \frac{a{\left(n \right)}}{7} + \frac{4 a{\left(n + 1 \right)}}{7} - \frac{2 a{\left(n + 2 \right)}}{7} - \frac{19 a{\left(n + 3 \right)}}{7} - 2 a{\left(n + 4 \right)} - \frac{4 a{\left(n + 8 \right)}}{7} - \frac{a{\left(n + 9 \right)}}{7} - \frac{3 a{\left(n + 10 \right)}}{7} + \frac{a{\left(n + 11 \right)}}{7}, \quad n \geq 11\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle -\frac{28557947122622461 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +9}}{14553925365250029576}-\frac{28557947122622461 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +9}}{14553925365250029576}-\frac{28557947122622461 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +9}}{14553925365250029576}-\frac{28557947122622461 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +9}}{14553925365250029576}-\frac{28557947122622461 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +9}}{14553925365250029576}-\frac{28557947122622461 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +9}}{14553925365250029576}-\frac{28557947122622461 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +9}}{14553925365250029576}-\frac{28557947122622461 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +9}}{14553925365250029576}-\frac{28557947122622461 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +9}}{14553925365250029576}-\frac{28557947122622461 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =10\right)^{-n +9}}{14553925365250029576}-\frac{96485976510522139 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +8}}{43661776095750088728}-\frac{96485976510522139 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +8}}{43661776095750088728}-\frac{96485976510522139 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +8}}{43661776095750088728}-\frac{96485976510522139 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +8}}{43661776095750088728}-\frac{96485976510522139 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +8}}{43661776095750088728}-\frac{96485976510522139 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +8}}{43661776095750088728}-\frac{96485976510522139 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +8}}{43661776095750088728}-\frac{96485976510522139 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +8}}{43661776095750088728}-\frac{96485976510522139 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +8}}{43661776095750088728}-\frac{96485976510522139 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =10\right)^{-n +8}}{43661776095750088728}+\frac{1338311685267429415 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +7}}{43661776095750088728}+\frac{1338311685267429415 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +7}}{43661776095750088728}+\frac{1338311685267429415 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +7}}{43661776095750088728}+\frac{1338311685267429415 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +7}}{43661776095750088728}+\frac{1338311685267429415 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +7}}{43661776095750088728}+\frac{1338311685267429415 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +7}}{43661776095750088728}+\frac{1338311685267429415 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +7}}{43661776095750088728}+\frac{1338311685267429415 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +7}}{43661776095750088728}+\frac{1338311685267429415 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +7}}{43661776095750088728}+\frac{1338311685267429415 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =10\right)^{-n +7}}{43661776095750088728}+\frac{519157825470988267 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +6}}{10915444023937522182}+\frac{519157825470988267 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +6}}{10915444023937522182}+\frac{519157825470988267 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +6}}{10915444023937522182}+\frac{519157825470988267 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +6}}{10915444023937522182}+\frac{519157825470988267 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +6}}{10915444023937522182}+\frac{519157825470988267 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +6}}{10915444023937522182}+\frac{519157825470988267 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +6}}{10915444023937522182}+\frac{519157825470988267 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +6}}{10915444023937522182}+\frac{519157825470988267 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +6}}{10915444023937522182}+\frac{519157825470988267 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =10\right)^{-n +6}}{10915444023937522182}-\frac{1590741724595089135 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +5}}{21830888047875044364}-\frac{1590741724595089135 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +5}}{21830888047875044364}-\frac{1590741724595089135 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +5}}{21830888047875044364}-\frac{1590741724595089135 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +5}}{21830888047875044364}-\frac{1590741724595089135 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +5}}{21830888047875044364}-\frac{1590741724595089135 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +5}}{21830888047875044364}-\frac{1590741724595089135 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +5}}{21830888047875044364}-\frac{1590741724595089135 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +5}}{21830888047875044364}-\frac{1590741724595089135 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +5}}{21830888047875044364}-\frac{1590741724595089135 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =10\right)^{-n +5}}{21830888047875044364}-\frac{7392098671645951945 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +4}}{43661776095750088728}-\frac{7392098671645951945 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +4}}{43661776095750088728}-\frac{7392098671645951945 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +4}}{43661776095750088728}-\frac{7392098671645951945 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +4}}{43661776095750088728}-\frac{7392098671645951945 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +4}}{43661776095750088728}-\frac{7392098671645951945 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +4}}{43661776095750088728}-\frac{7392098671645951945 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +4}}{43661776095750088728}-\frac{7392098671645951945 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +4}}{43661776095750088728}-\frac{7392098671645951945 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +4}}{43661776095750088728}-\frac{7392098671645951945 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =10\right)^{-n +4}}{43661776095750088728}-\frac{7500161621749458589 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +3}}{43661776095750088728}-\frac{7500161621749458589 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +3}}{43661776095750088728}-\frac{7500161621749458589 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +3}}{43661776095750088728}-\frac{7500161621749458589 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +3}}{43661776095750088728}-\frac{7500161621749458589 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +3}}{43661776095750088728}-\frac{7500161621749458589 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +3}}{43661776095750088728}-\frac{7500161621749458589 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +3}}{43661776095750088728}-\frac{7500161621749458589 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +3}}{43661776095750088728}-\frac{7500161621749458589 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +3}}{43661776095750088728}-\frac{7500161621749458589 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =10\right)^{-n +3}}{43661776095750088728}-\frac{57303646139627879 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +2}}{14553925365250029576}-\frac{57303646139627879 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +2}}{14553925365250029576}-\frac{57303646139627879 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +2}}{14553925365250029576}-\frac{57303646139627879 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +2}}{14553925365250029576}-\frac{57303646139627879 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +2}}{14553925365250029576}-\frac{57303646139627879 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +2}}{14553925365250029576}-\frac{57303646139627879 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +2}}{14553925365250029576}-\frac{57303646139627879 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +2}}{14553925365250029576}-\frac{57303646139627879 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +2}}{14553925365250029576}-\frac{57303646139627879 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =10\right)^{-n +2}}{14553925365250029576}+\frac{2491563540893281819 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +1}}{43661776095750088728}+\frac{2491563540893281819 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +1}}{43661776095750088728}+\frac{2491563540893281819 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +1}}{43661776095750088728}+\frac{2491563540893281819 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +1}}{43661776095750088728}+\frac{2491563540893281819 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +1}}{43661776095750088728}+\frac{2491563540893281819 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +1}}{43661776095750088728}+\frac{2491563540893281819 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +1}}{43661776095750088728}+\frac{2491563540893281819 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +1}}{43661776095750088728}+\frac{2491563540893281819 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n +1}}{43661776095750088728}+\frac{2491563540893281819 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =10\right)^{-n +1}}{43661776095750088728}+\frac{828465295052207915 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n -1}}{43661776095750088728}+\frac{828465295052207915 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n -1}}{43661776095750088728}+\frac{828465295052207915 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n -1}}{43661776095750088728}+\frac{828465295052207915 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n -1}}{43661776095750088728}+\frac{828465295052207915 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n -1}}{43661776095750088728}+\frac{828465295052207915 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n -1}}{43661776095750088728}+\frac{828465295052207915 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n -1}}{43661776095750088728}+\frac{828465295052207915 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n -1}}{43661776095750088728}+\frac{828465295052207915 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n -1}}{43661776095750088728}+\frac{828465295052207915 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =10\right)^{-n -1}}{43661776095750088728}+\frac{1556683617454663477 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n}}{21830888047875044364}+\frac{1556683617454663477 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n}}{21830888047875044364}+\frac{1556683617454663477 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n}}{21830888047875044364}+\frac{1556683617454663477 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n}}{21830888047875044364}+\frac{1556683617454663477 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n}}{21830888047875044364}+\frac{\left(-1\right)^{-n}}{12}+\frac{1556683617454663477 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n}}{21830888047875044364}+\frac{1556683617454663477 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n}}{21830888047875044364}+\frac{1556683617454663477 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n}}{21830888047875044364}+\frac{1556683617454663477 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =9\right)^{-n}}{21830888047875044364}+\frac{1556683617454663477 \mathit{RootOf} \left(Z^{10}+3 Z^{9}-5 Z^{8}-14 Z^{7}-7 Z^{5}+7 Z^{4}-7 Z^{3}+3 Z^{2}-4 Z +1, \mathit{index} =10\right)^{-n}}{21830888047875044364}\)
Copy to clipboard:

This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 165 rules.

Finding the specification took 196 seconds.

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

Copy 165 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_{15}\! \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_{29}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{15}\! \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_{17}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= x\\ F_{16}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{15}\! \left(x \right) F_{6}\! \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_{15}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{18}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{15}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{15}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{15}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{44}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{45}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{15}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{15}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{15}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \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) &= F_{49}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{15}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{42}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{15}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{66}\! \left(x \right) &= x^{2}\\ F_{67}\! \left(x \right) &= F_{15}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{70}\! \left(x \right)+F_{71}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{15}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{15}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{73}\! \left(x \right) &= 0\\ F_{74}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{15}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{15}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{15}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{15}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{15}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{15}\! \left(x \right) F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{15}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{100}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{102}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{103}\! \left(x \right)+F_{104}\! \left(x \right)+F_{106}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{15}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{106}\! \left(x \right) &= 0\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{112}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{18}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{111}\! \left(x \right) &= x^{2}\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{114}\! \left(x \right)+F_{18}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{15}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{114}\! \left(x \right) &= 0\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{119}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{118}\! \left(x \right)+F_{18}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{15}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{118}\! \left(x \right) &= 0\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{121}\! \left(x \right)+F_{122}\! \left(x \right)+F_{18}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{15}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{121}\! \left(x \right) &= 0\\ F_{122}\! \left(x \right) &= 0\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{128}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{125}\! \left(x \right)+F_{18}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{129}\! \left(x \right)+F_{132}\! \left(x \right)+F_{133}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{132}\! \left(x \right) &= 0\\ F_{133}\! \left(x \right) &= 0\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{137}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{142}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{139}\! \left(x \right)+F_{18}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{122}\! \left(x \right)+F_{143}\! \left(x \right)+F_{18}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{144}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{150}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{18}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{151}\! \left(x \right)+F_{18}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{15}\! \left(x \right) F_{152}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{15}\! \left(x \right) F_{154}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{156}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{147}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{161}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)+F_{18}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{15}\! \left(x \right) F_{159}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{160}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{162}\! \left(x \right)+F_{18}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{15}\! \left(x \right) F_{163}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{164}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{94}\! \left(x \right)\\ \end{align*}\)