Av(1234, 1243, 1432, 2413, 3142)
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Generating Function
\(\displaystyle \frac{\left(x -1\right) \left(x^{3}+2 x -1\right)}{x^{8}+2 x^{7}+3 x^{6}+3 x^{5}+x^{4}-3 x^{3}+4 x^{2}-4 x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 54, 150, 424, 1212, 3468, 9913, 28336, 81035, 231805, 663137, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{8}+2 x^{7}+3 x^{6}+3 x^{5}+x^{4}-3 x^{3}+4 x^{2}-4 x +1\right) F \! \left(x \right)-\left(x -1\right) \left(x^{3}+2 x -1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 150\)
\(\displaystyle a \! \left(7\right) = 424\)
\(\displaystyle a \! \left(n +8\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)-3 a \! \left(n +3\right)-a \! \left(n +4\right)+3 a \! \left(n +5\right)-4 a \! \left(n +6\right)+4 a \! \left(n +7\right), \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle -\frac{121435339 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +6}}{4773159364}-\frac{121435339 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +6}}{4773159364}-\frac{121435339 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +6}}{4773159364}-\frac{121435339 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +6}}{4773159364}-\frac{121435339 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +6}}{4773159364}-\frac{121435339 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +6}}{4773159364}-\frac{121435339 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +6}}{4773159364}-\frac{121435339 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +6}}{4773159364}-\frac{16926043 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +5}}{207528668}-\frac{16926043 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +5}}{207528668}-\frac{16926043 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +5}}{207528668}-\frac{16926043 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +5}}{207528668}-\frac{16926043 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +5}}{207528668}-\frac{16926043 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +5}}{207528668}-\frac{16926043 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +5}}{207528668}-\frac{16926043 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +5}}{207528668}-\frac{1370323 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +4}}{8584819}-\frac{1370323 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +4}}{8584819}-\frac{1370323 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +4}}{8584819}-\frac{1370323 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +4}}{8584819}-\frac{1370323 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +4}}{8584819}-\frac{1370323 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +4}}{8584819}-\frac{1370323 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +4}}{8584819}-\frac{1370323 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +4}}{8584819}-\frac{980003181 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +3}}{4773159364}-\frac{980003181 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +3}}{4773159364}-\frac{980003181 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +3}}{4773159364}-\frac{980003181 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +3}}{4773159364}-\frac{980003181 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +3}}{4773159364}-\frac{980003181 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +3}}{4773159364}-\frac{980003181 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +3}}{4773159364}-\frac{980003181 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +3}}{4773159364}-\frac{158406304 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +2}}{1193289841}-\frac{158406304 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +2}}{1193289841}-\frac{158406304 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +2}}{1193289841}-\frac{158406304 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +2}}{1193289841}-\frac{158406304 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +2}}{1193289841}-\frac{158406304 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +2}}{1193289841}-\frac{158406304 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +2}}{1193289841}-\frac{158406304 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +2}}{1193289841}+\frac{164492019 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +1}}{4773159364}+\frac{164492019 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +1}}{4773159364}+\frac{164492019 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +1}}{4773159364}+\frac{164492019 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +1}}{4773159364}+\frac{164492019 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +1}}{4773159364}+\frac{164492019 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +1}}{4773159364}+\frac{164492019 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n +1}}{4773159364}+\frac{164492019 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n +1}}{4773159364}+\frac{332574729 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n -1}}{4773159364}+\frac{332574729 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n -1}}{4773159364}+\frac{332574729 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n -1}}{4773159364}+\frac{332574729 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n -1}}{4773159364}+\frac{332574729 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n -1}}{4773159364}+\frac{332574729 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n -1}}{4773159364}+\frac{332574729 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n -1}}{4773159364}+\frac{332574729 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n -1}}{4773159364}+\frac{409636205 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n}}{4773159364}+\frac{409636205 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n}}{4773159364}+\frac{409636205 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n}}{4773159364}+\frac{409636205 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n}}{4773159364}+\frac{409636205 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n}}{4773159364}+\frac{409636205 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n}}{4773159364}+\frac{409636205 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =7\right)^{-n}}{4773159364}+\frac{409636205 \mathit{RootOf} \left(Z^{8}+2 Z^{7}+3 Z^{6}+3 Z^{5}+Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1, \mathit{index} =8\right)^{-n}}{4773159364}\)
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This specification was found using the strategy pack "Point Placements" and has 62 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_{19}\! \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_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \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_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{34}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \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_{37}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{60}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ \end{align*}\)