Av(13452, 13524, 13542, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 51234, 51243, 51324, 51342, 51423, 51432, 52134, 52143)
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Generating Function
\(\displaystyle \frac{2 x^{7}-2 x^{6}+5 x^{5}+2 x^{4}-5 x^{3}+4 x -1}{2 x^{8}+2 x^{7}-8 x^{6}+7 x^{5}+8 x^{4}-6 x^{3}-3 x^{2}+5 x -1}\)
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Counting Sequence
1, 1, 2, 6, 24, 100, 409, 1655, 6670, 26857, 108141, 435477, 1753741, 7062752, 28443570, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{8}+2 x^{7}-8 x^{6}+7 x^{5}+8 x^{4}-6 x^{3}-3 x^{2}+5 x -1\right) F \! \left(x \right)-2 x^{7}+2 x^{6}-5 x^{5}-2 x^{4}+5 x^{3}-4 x +1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 409\)
\(\displaystyle a(7) = 1655\)
\(\displaystyle a{\left(n + 8 \right)} = 2 a{\left(n \right)} + 2 a{\left(n + 1 \right)} - 8 a{\left(n + 2 \right)} + 7 a{\left(n + 3 \right)} + 8 a{\left(n + 4 \right)} - 6 a{\left(n + 5 \right)} - 3 a{\left(n + 6 \right)} + 5 a{\left(n + 7 \right)}, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle -\frac{2698738997 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +6}}{56669200563}-\frac{2698738997 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +6}}{56669200563}-\frac{2698738997 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +6}}{56669200563}-\frac{2698738997 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +6}}{56669200563}-\frac{2698738997 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +6}}{56669200563}-\frac{2698738997 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +6}}{56669200563}-\frac{2698738997 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +6}}{56669200563}-\frac{2698738997 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +6}}{56669200563}-\frac{4064979733 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +5}}{56669200563}-\frac{4064979733 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +5}}{56669200563}-\frac{4064979733 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +5}}{56669200563}-\frac{4064979733 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +5}}{56669200563}-\frac{4064979733 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +5}}{56669200563}-\frac{4064979733 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +5}}{56669200563}-\frac{4064979733 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +5}}{56669200563}-\frac{4064979733 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +5}}{56669200563}+\frac{9994117493 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +4}}{56669200563}+\frac{9994117493 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +4}}{56669200563}+\frac{9994117493 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +4}}{56669200563}+\frac{9994117493 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +4}}{56669200563}+\frac{9994117493 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +4}}{56669200563}+\frac{9994117493 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +4}}{56669200563}+\frac{9994117493 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +4}}{56669200563}+\frac{9994117493 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +4}}{56669200563}-\frac{4618685113 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +3}}{113338401126}-\frac{4618685113 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +3}}{113338401126}-\frac{4618685113 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +3}}{113338401126}-\frac{4618685113 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +3}}{113338401126}-\frac{4618685113 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +3}}{113338401126}-\frac{4618685113 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +3}}{113338401126}-\frac{4618685113 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +3}}{113338401126}-\frac{4618685113 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +3}}{113338401126}-\frac{3697551091 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +2}}{18889733521}-\frac{3697551091 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +2}}{18889733521}-\frac{3697551091 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +2}}{18889733521}-\frac{3697551091 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +2}}{18889733521}-\frac{3697551091 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +2}}{18889733521}-\frac{3697551091 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +2}}{18889733521}-\frac{3697551091 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +2}}{18889733521}-\frac{3697551091 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +2}}{18889733521}+\frac{13474226195 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +1}}{113338401126}+\frac{13474226195 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +1}}{113338401126}+\frac{13474226195 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +1}}{113338401126}+\frac{13474226195 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +1}}{113338401126}+\frac{13474226195 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +1}}{113338401126}+\frac{13474226195 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n +1}}{113338401126}+\frac{13474226195 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n +1}}{113338401126}+\frac{13474226195 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n +1}}{113338401126}-\frac{1547411618 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n -1}}{56669200563}-\frac{1547411618 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n -1}}{56669200563}-\frac{1547411618 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n -1}}{56669200563}-\frac{1547411618 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n -1}}{56669200563}-\frac{1547411618 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n -1}}{56669200563}-\frac{1547411618 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n -1}}{56669200563}-\frac{1547411618 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n -1}}{56669200563}-\frac{1547411618 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n -1}}{56669200563}+\frac{7136522501 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n}}{37779467042}+\frac{7136522501 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n}}{37779467042}+\frac{7136522501 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n}}{37779467042}+\frac{7136522501 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n}}{37779467042}+\frac{7136522501 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n}}{37779467042}+\frac{7136522501 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =6\right)^{-n}}{37779467042}+\frac{7136522501 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =7\right)^{-n}}{37779467042}+\frac{7136522501 \mathit{RootOf} \left(2 Z^{8}+2 Z^{7}-8 Z^{6}+7 Z^{5}+8 Z^{4}-6 Z^{3}-3 Z^{2}+5 Z -1, \mathit{index} =8\right)^{-n}}{37779467042}\)
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This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 85 rules.

Finding the specification took 111 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_{20}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{75}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{7}\! \left(x \right) &= 0\\ F_{8}\! \left(x \right) &= F_{20}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{20}\! \left(x \right) &= x\\ F_{21}\! \left(x \right) &= F_{20}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{20}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{43}\! \left(x \right)+F_{44}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{20}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{20}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{20}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{20}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{43}\! \left(x \right) &= 0\\ F_{44}\! \left(x \right) &= F_{20}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{20}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{68}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{20}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{20}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{20}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{62}\! \left(x \right) &= 0\\ F_{63}\! \left(x \right) &= F_{20}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{20}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{20}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{20}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{20}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{7}\! \left(x \right)+F_{80}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{20}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{84}\! \left(x \right) &= 0\\ \end{align*}\)