Av(12435, 12453, 12543, 14235, 14253, 14523, 15243, 15423, 21435, 21453, 21543, 24135, 24153, 24315, 41235, 41253, 41523, 42135, 42153, 42315)
View Raw Data
Generating Function
\(\displaystyle \frac{\left(4 x^{3}-5 x^{2}+4 x -1\right) \left(2 x -1\right)^{3}}{\left(x -1\right) \left(4 x^{8}-10 x^{7}-10 x^{6}+60 x^{5}-98 x^{4}+84 x^{3}-40 x^{2}+10 x -1\right)}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 24, 100, 400, 1530, 5684, 20806, 75730, 275362, 1001990, 3649894, 13306030, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(4 x^{8}-10 x^{7}-10 x^{6}+60 x^{5}-98 x^{4}+84 x^{3}-40 x^{2}+10 x -1\right) F \! \left(x \right)-\left(4 x^{3}-5 x^{2}+4 x -1\right) \left(2 x -1\right)^{3} = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 400\)
\(\displaystyle a(7) = 1530\)
\(\displaystyle a{\left(n + 8 \right)} = 4 a{\left(n \right)} - 10 a{\left(n + 1 \right)} - 10 a{\left(n + 2 \right)} + 60 a{\left(n + 3 \right)} - 98 a{\left(n + 4 \right)} + 84 a{\left(n + 5 \right)} - 40 a{\left(n + 6 \right)} + 10 a{\left(n + 7 \right)} + 2, \quad n \geq 8\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle 2+\frac{278019073311 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n +1}}{2154394729}+\frac{278019073311 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n +1}}{2154394729}+\frac{278019073311 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n +1}}{2154394729}+\frac{278019073311 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n +1}}{2154394729}+\frac{278019073311 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n +1}}{2154394729}+\frac{278019073311 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n +1}}{2154394729}+\frac{278019073311 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n +1}}{2154394729}+\frac{278019073311 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n +1}}{2154394729}-\frac{27083650464 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n +7}}{2154394729}-\frac{27083650464 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n +7}}{2154394729}-\frac{27083650464 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n +7}}{2154394729}-\frac{27083650464 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n +7}}{2154394729}-\frac{27083650464 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n +7}}{2154394729}-\frac{27083650464 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n +7}}{2154394729}-\frac{27083650464 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n +7}}{2154394729}-\frac{27083650464 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n +7}}{2154394729}+\frac{69332865212 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n +6}}{2154394729}+\frac{69332865212 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n +6}}{2154394729}+\frac{69332865212 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n +6}}{2154394729}+\frac{69332865212 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n +6}}{2154394729}+\frac{69332865212 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n +6}}{2154394729}+\frac{69332865212 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n +6}}{2154394729}+\frac{69332865212 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n +6}}{2154394729}+\frac{69332865212 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n +6}}{2154394729}+\frac{65446151206 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n +5}}{2154394729}+\frac{65446151206 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n +5}}{2154394729}+\frac{65446151206 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n +5}}{2154394729}+\frac{65446151206 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n +5}}{2154394729}+\frac{65446151206 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n +5}}{2154394729}+\frac{65446151206 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n +5}}{2154394729}+\frac{65446151206 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n +5}}{2154394729}+\frac{65446151206 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n +5}}{2154394729}-\frac{413422801360 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n +4}}{2154394729}-\frac{413422801360 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n +4}}{2154394729}-\frac{413422801360 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n +4}}{2154394729}-\frac{413422801360 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n +4}}{2154394729}-\frac{413422801360 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n +4}}{2154394729}-\frac{413422801360 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n +4}}{2154394729}-\frac{413422801360 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n +4}}{2154394729}-\frac{413422801360 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n +4}}{2154394729}-\frac{68836612318 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n}}{2154394729}-\frac{68836612318 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n}}{2154394729}-\frac{68836612318 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n}}{2154394729}-\frac{68836612318 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n}}{2154394729}-\frac{68836612318 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n}}{2154394729}-\frac{68836612318 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n}}{2154394729}-\frac{68836612318 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n}}{2154394729}-\frac{68836612318 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n}}{2154394729}+\frac{680817724538 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n +3}}{2154394729}+\frac{680817724538 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n +3}}{2154394729}+\frac{680817724538 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n +3}}{2154394729}+\frac{680817724538 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n +3}}{2154394729}+\frac{680817724538 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n +3}}{2154394729}+\frac{680817724538 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n +3}}{2154394729}+\frac{680817724538 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n +3}}{2154394729}+\frac{680817724538 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n +3}}{2154394729}+\frac{6934596935 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n -1}}{2154394729}+\frac{6934596935 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n -1}}{2154394729}+\frac{6934596935 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n -1}}{2154394729}+\frac{6934596935 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n -1}}{2154394729}+\frac{6934596935 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n -1}}{2154394729}+\frac{6934596935 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n -1}}{2154394729}+\frac{6934596935 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n -1}}{2154394729}+\frac{6934596935 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n -1}}{2154394729}-\frac{586898557602 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =1\right)^{-n +2}}{2154394729}-\frac{586898557602 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =2\right)^{-n +2}}{2154394729}-\frac{586898557602 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =3\right)^{-n +2}}{2154394729}-\frac{586898557602 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =4\right)^{-n +2}}{2154394729}-\frac{586898557602 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =5\right)^{-n +2}}{2154394729}-\frac{586898557602 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =6\right)^{-n +2}}{2154394729}-\frac{586898557602 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =7\right)^{-n +2}}{2154394729}-\frac{586898557602 \mathit{RootOf} \left(4 Z^{8}-10 Z^{7}-10 Z^{6}+60 Z^{5}-98 Z^{4}+84 Z^{3}-40 Z^{2}+10 Z -1, \mathit{index} =8\right)^{-n +2}}{2154394729}\)
Copy to clipboard:

This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 172 rules.

Finding the specification took 191 seconds.

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

Copy 172 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_{14}\! \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_{26}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{15}\! \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_{11}\! \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_{9}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{17}\! \left(x \right) &= 0\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{14}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{14}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{31}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{14}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{14}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{14}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{14}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{14}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{14}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{14}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{55}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{24}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{14}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{17}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{14}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{14}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{59}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{14}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{66}\! \left(x \right)+F_{76}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{14}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{80}\! \left(x \right)+F_{81}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{14}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{14}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{14}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= 2 F_{17}\! \left(x \right)+F_{89}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{14}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{14}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{14}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= 2 F_{17}\! \left(x \right)+F_{98}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{14}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{102}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{108}\! \left(x \right)+F_{17}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{111}\! \left(x \right) &= 2 F_{17}\! \left(x \right)+F_{112}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{111}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{122}\! \left(x \right)+F_{17}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{121}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{111}\! \left(x \right)\\ F_{122}\! \left(x \right) &= 0\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{127}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{17}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{128}\! \left(x \right)+F_{17}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{151}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{144}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{135}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{138}\! \left(x \right)+F_{17}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{141}\! \left(x \right) &= 2 F_{17}\! \left(x \right)+F_{142}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{14}\! \left(x \right) F_{143}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{14}\! \left(x \right) F_{145}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{14}\! \left(x \right) F_{149}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{147}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{154}\! \left(x \right)+F_{17}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{14}\! \left(x \right) F_{153}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{14}\! \left(x \right) F_{155}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{157}\! \left(x \right) &= 2 F_{17}\! \left(x \right)+F_{158}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{14}\! \left(x \right) F_{159}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{14}\! \left(x \right) F_{161}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)+F_{167}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{164}\! \left(x \right)+F_{17}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{14}\! \left(x \right) F_{165}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{157}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{168}\! \left(x \right)+F_{169}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{17}\! \left(x \right)+F_{170}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{14}\! \left(x \right) F_{171}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{30}\! \left(x \right)\\ \end{align*}\)