Av(12435, 12453, 12543, 14235, 14253, 14325, 21435, 21453, 21543, 24135, 24153, 24513, 25143, 25413, 41235, 41253, 41325, 42135, 42153, 42513)
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Generating Function
\(\displaystyle \frac{\left(4 x^{3}-5 x^{2}+4 x -1\right) \left(x^{2}-3 x +1\right)^{2}}{\left(x -1\right) \left(2 x^{7}+10 x^{6}-52 x^{5}+87 x^{4}-78 x^{3}+39 x^{2}-10 x +1\right)}\)
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Counting Sequence
1, 1, 2, 6, 24, 100, 402, 1560, 5922, 22218, 82874, 308288, 1145482, 4254332, 15798942, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(2 x^{7}+10 x^{6}-52 x^{5}+87 x^{4}-78 x^{3}+39 x^{2}-10 x +1\right) F \! \left(x \right)-\left(4 x^{3}-5 x^{2}+4 x -1\right) \left(x^{2}-3 x +1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 402\)
\(\displaystyle a(7) = 1560\)
\(\displaystyle a{\left(n + 7 \right)} = - 2 a{\left(n \right)} - 10 a{\left(n + 1 \right)} + 52 a{\left(n + 2 \right)} - 87 a{\left(n + 3 \right)} + 78 a{\left(n + 4 \right)} - 39 a{\left(n + 5 \right)} + 10 a{\left(n + 6 \right)} - 2, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle 2-\frac{125490028 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =1\right)^{-n +6}}{22458833}-\frac{125490028 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =2\right)^{-n +6}}{22458833}-\frac{125490028 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =3\right)^{-n +6}}{22458833}-\frac{125490028 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =4\right)^{-n +6}}{22458833}-\frac{125490028 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =5\right)^{-n +6}}{22458833}-\frac{125490028 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =6\right)^{-n +6}}{22458833}-\frac{125490028 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =7\right)^{-n +6}}{22458833}-\frac{619261352 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =1\right)^{-n +5}}{22458833}-\frac{619261352 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =2\right)^{-n +5}}{22458833}-\frac{619261352 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =3\right)^{-n +5}}{22458833}-\frac{619261352 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =4\right)^{-n +5}}{22458833}-\frac{619261352 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =5\right)^{-n +5}}{22458833}-\frac{619261352 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =6\right)^{-n +5}}{22458833}-\frac{619261352 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =7\right)^{-n +5}}{22458833}+\frac{3308762214 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =1\right)^{-n +4}}{22458833}+\frac{3308762214 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =2\right)^{-n +4}}{22458833}+\frac{3308762214 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =3\right)^{-n +4}}{22458833}+\frac{3308762214 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =4\right)^{-n +4}}{22458833}+\frac{3308762214 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =5\right)^{-n +4}}{22458833}+\frac{3308762214 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =6\right)^{-n +4}}{22458833}+\frac{3308762214 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =7\right)^{-n +4}}{22458833}-\frac{5645792806 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =1\right)^{-n +3}}{22458833}-\frac{5645792806 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =2\right)^{-n +3}}{22458833}-\frac{5645792806 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =3\right)^{-n +3}}{22458833}-\frac{5645792806 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =4\right)^{-n +3}}{22458833}-\frac{5645792806 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =5\right)^{-n +3}}{22458833}-\frac{5645792806 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =6\right)^{-n +3}}{22458833}-\frac{5645792806 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =7\right)^{-n +3}}{22458833}+\frac{5118171406 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =1\right)^{-n +2}}{22458833}+\frac{5118171406 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =2\right)^{-n +2}}{22458833}+\frac{5118171406 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =3\right)^{-n +2}}{22458833}+\frac{5118171406 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =4\right)^{-n +2}}{22458833}+\frac{5118171406 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =5\right)^{-n +2}}{22458833}+\frac{5118171406 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =6\right)^{-n +2}}{22458833}+\frac{5118171406 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =7\right)^{-n +2}}{22458833}-\frac{2580302448 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =1\right)^{-n +1}}{22458833}-\frac{2580302448 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =2\right)^{-n +1}}{22458833}-\frac{2580302448 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =3\right)^{-n +1}}{22458833}-\frac{2580302448 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =4\right)^{-n +1}}{22458833}-\frac{2580302448 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =5\right)^{-n +1}}{22458833}-\frac{2580302448 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =6\right)^{-n +1}}{22458833}-\frac{2580302448 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =7\right)^{-n +1}}{22458833}-\frac{61778111 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =1\right)^{-n -1}}{22458833}-\frac{61778111 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =2\right)^{-n -1}}{22458833}-\frac{61778111 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =3\right)^{-n -1}}{22458833}-\frac{61778111 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =4\right)^{-n -1}}{22458833}-\frac{61778111 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =5\right)^{-n -1}}{22458833}-\frac{61778111 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =6\right)^{-n -1}}{22458833}-\frac{61778111 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =7\right)^{-n -1}}{22458833}+\frac{650608791 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =1\right)^{-n}}{22458833}+\frac{650608791 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =2\right)^{-n}}{22458833}+\frac{650608791 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =3\right)^{-n}}{22458833}+\frac{650608791 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =4\right)^{-n}}{22458833}+\frac{650608791 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =5\right)^{-n}}{22458833}+\frac{650608791 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =6\right)^{-n}}{22458833}+\frac{650608791 \mathit{RootOf} \left(2 Z^{7}+10 Z^{6}-52 Z^{5}+87 Z^{4}-78 Z^{3}+39 Z^{2}-10 Z +1, \mathit{index} =7\right)^{-n}}{22458833}\)
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This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 127 rules.

Finding the specification took 156 seconds.

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Copy 127 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_{22}\! \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_{6}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{6}\! \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_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{16}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{15}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{12}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{15}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{27}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{15}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{31}\! \left(x \right)+F_{33}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{15}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{15}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \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_{40}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{33}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{15}\! \left(x \right) F_{26}\! \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_{47}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{49}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{15}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{15}\! \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_{50}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{56}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{15}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{15}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{15}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{15}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{12}\! \left(x \right)+F_{70}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{15}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{15}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{37}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{15}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{21}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{15}\! \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_{15}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{80}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{44}\! \left(x \right)+F_{86}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{15}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{44}\! \left(x \right)+F_{72}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{15}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{49}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{15}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{85}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{107}\! \left(x \right)+F_{12}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{106}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{102}\! \left(x \right)\\ F_{107}\! \left(x \right) &= 0\\ F_{108}\! \left(x \right) &= F_{89}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{112}\! \left(x \right)\\ F_{112}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{113}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{122}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{12}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{118}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{124}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{125}\! \left(x \right)+F_{86}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{26}\! \left(x \right)\\ \end{align*}\)