Av(1234, 1324, 1342, 1432, 3214)
Generating Function
\(\displaystyle -\frac{2 x^{3}+x^{2}-1}{\left(x -1\right) \left(4 x^{8}+16 x^{7}+24 x^{6}+21 x^{5}+11 x^{4}+6 x^{3}+2 x^{2}-1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 54, 139, 368, 1003, 2741, 7435, 20089, 54351, 147272, 399096, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(4 x^{8}+16 x^{7}+24 x^{6}+21 x^{5}+11 x^{4}+6 x^{3}+2 x^{2}-1\right) F \! \left(x \right)+2 x^{3}+x^{2}-1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 139\)
\(\displaystyle a \! \left(7\right) = 368\)
\(\displaystyle a \! \left(n +6\right) = -2 a \! \left(n \right)-8 a \! \left(n +1\right)-12 a \! \left(n +2\right)-\frac{21 a \! \left(n +3\right)}{2}-\frac{11 a \! \left(n +4\right)}{2}-3 a \! \left(n +5\right)+\frac{a \! \left(n +8\right)}{2}+1, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 139\)
\(\displaystyle a \! \left(7\right) = 368\)
\(\displaystyle a \! \left(n +6\right) = -2 a \! \left(n \right)-8 a \! \left(n +1\right)-12 a \! \left(n +2\right)-\frac{21 a \! \left(n +3\right)}{2}-\frac{11 a \! \left(n +4\right)}{2}-3 a \! \left(n +5\right)+\frac{a \! \left(n +8\right)}{2}+1, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \frac{2}{83}-\frac{7833679098640 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =1\right)^{-n +7}}{33365400498719}-\frac{7833679098640 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =2\right)^{-n +7}}{33365400498719}-\frac{7833679098640 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =3\right)^{-n +7}}{33365400498719}-\frac{7833679098640 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =4\right)^{-n +7}}{33365400498719}-\frac{7833679098640 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =5\right)^{-n +7}}{33365400498719}-\frac{7833679098640 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =6\right)^{-n +7}}{33365400498719}-\frac{7833679098640 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =7\right)^{-n +7}}{33365400498719}-\frac{7833679098640 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =8\right)^{-n +7}}{33365400498719}-\frac{15322822724416 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =1\right)^{-n +6}}{33365400498719}-\frac{15322822724416 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =2\right)^{-n +6}}{33365400498719}-\frac{15322822724416 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =3\right)^{-n +6}}{33365400498719}-\frac{15322822724416 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =4\right)^{-n +6}}{33365400498719}-\frac{15322822724416 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =5\right)^{-n +6}}{33365400498719}-\frac{15322822724416 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =6\right)^{-n +6}}{33365400498719}-\frac{15322822724416 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =7\right)^{-n +6}}{33365400498719}-\frac{15322822724416 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =8\right)^{-n +6}}{33365400498719}+\frac{2786490278376 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =1\right)^{-n +5}}{33365400498719}+\frac{2786490278376 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =2\right)^{-n +5}}{33365400498719}+\frac{2786490278376 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =3\right)^{-n +5}}{33365400498719}+\frac{2786490278376 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =4\right)^{-n +5}}{33365400498719}+\frac{2786490278376 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =5\right)^{-n +5}}{33365400498719}+\frac{2786490278376 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =6\right)^{-n +5}}{33365400498719}+\frac{2786490278376 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =7\right)^{-n +5}}{33365400498719}+\frac{2786490278376 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =8\right)^{-n +5}}{33365400498719}+\frac{3123308544528 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =1\right)^{-n +4}}{33365400498719}+\frac{3123308544528 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =2\right)^{-n +4}}{33365400498719}+\frac{3123308544528 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =3\right)^{-n +4}}{33365400498719}+\frac{3123308544528 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =4\right)^{-n +4}}{33365400498719}+\frac{3123308544528 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =5\right)^{-n +4}}{33365400498719}+\frac{3123308544528 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =6\right)^{-n +4}}{33365400498719}+\frac{3123308544528 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =7\right)^{-n +4}}{33365400498719}+\frac{3123308544528 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =8\right)^{-n +4}}{33365400498719}+\frac{3535182348040 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =1\right)^{-n +3}}{33365400498719}+\frac{3535182348040 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =2\right)^{-n +3}}{33365400498719}+\frac{3535182348040 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =3\right)^{-n +3}}{33365400498719}+\frac{3535182348040 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =4\right)^{-n +3}}{33365400498719}+\frac{3535182348040 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =5\right)^{-n +3}}{33365400498719}+\frac{3535182348040 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =6\right)^{-n +3}}{33365400498719}+\frac{3535182348040 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =7\right)^{-n +3}}{33365400498719}+\frac{3535182348040 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =8\right)^{-n +3}}{33365400498719}+\frac{91749436298 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =1\right)^{-n +2}}{33365400498719}+\frac{91749436298 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =2\right)^{-n +2}}{33365400498719}+\frac{91749436298 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =3\right)^{-n +2}}{33365400498719}+\frac{91749436298 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =4\right)^{-n +2}}{33365400498719}+\frac{91749436298 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =5\right)^{-n +2}}{33365400498719}+\frac{91749436298 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =6\right)^{-n +2}}{33365400498719}+\frac{91749436298 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =7\right)^{-n +2}}{33365400498719}+\frac{91749436298 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =8\right)^{-n +2}}{33365400498719}+\frac{8880852255467 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =1\right)^{-n +1}}{33365400498719}+\frac{8880852255467 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =2\right)^{-n +1}}{33365400498719}+\frac{8880852255467 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =3\right)^{-n +1}}{33365400498719}+\frac{8880852255467 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =4\right)^{-n +1}}{33365400498719}+\frac{8880852255467 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =5\right)^{-n +1}}{33365400498719}+\frac{8880852255467 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =6\right)^{-n +1}}{33365400498719}+\frac{8880852255467 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =7\right)^{-n +1}}{33365400498719}+\frac{8880852255467 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =8\right)^{-n +1}}{33365400498719}+\frac{1517035588165 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =1\right)^{-n -1}}{33365400498719}+\frac{1517035588165 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =2\right)^{-n -1}}{33365400498719}+\frac{1517035588165 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =3\right)^{-n -1}}{33365400498719}+\frac{1517035588165 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =4\right)^{-n -1}}{33365400498719}+\frac{1517035588165 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =5\right)^{-n -1}}{33365400498719}+\frac{1517035588165 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =6\right)^{-n -1}}{33365400498719}+\frac{1517035588165 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =7\right)^{-n -1}}{33365400498719}+\frac{1517035588165 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =8\right)^{-n -1}}{33365400498719}+\frac{48504444896 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =3\right)^{-n}}{401992777093}+\frac{48504444896 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =4\right)^{-n}}{401992777093}+\frac{48504444896 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =5\right)^{-n}}{401992777093}+\frac{48504444896 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =6\right)^{-n}}{401992777093}+\frac{48504444896 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =7\right)^{-n}}{401992777093}+\frac{48504444896 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =8\right)^{-n}}{401992777093}+\frac{48504444896 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =1\right)^{-n}}{401992777093}+\frac{48504444896 \mathit{RootOf} \left(4 Z^{8}+16 Z^{7}+24 Z^{6}+21 Z^{5}+11 Z^{4}+6 Z^{3}+2 Z^{2}-1, \mathit{index} =2\right)^{-n}}{401992777093}\)
This specification was found using the strategy pack "Point Placements" and has 111 rules.
Found on January 18, 2022.Finding the specification took 2 seconds.
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\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{48}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{49}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{44}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{60}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= x^{2}\\
F_{65}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{68}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{75}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{60}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{68}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{4}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{19}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{105}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{106}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{74}\! \left(x \right)\\
\end{align*}\)