Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13452, 13524, 13542, 14235, 14253, 14325, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 23415, 23514, 24135, 24153, 24315, 24513, 25134, 25143, 25314, 25413, 31245, 31254, 31425, 31524, 32145, 32154, 32415, 32514, 34125, 34215, 35124, 35214, 41235, 41325, 42135, 42315, 43125, 43215)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(x +1\right)}{x^{8}+5 x^{7}+2 x^{6}-8 x^{5}-12 x^{4}-2 x^{3}-2 x^{2}-x +1}\)
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Counting Sequence
1, 1, 2, 6, 24, 60, 150, 399, 1145, 3132, 8420, 22716, 62128, 169536, 460885, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{8}+5 x^{7}+2 x^{6}-8 x^{5}-12 x^{4}-2 x^{3}-2 x^{2}-x +1\right) F \! \left(x \right)+\left(x -1\right) \left(x +1\right) = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 60\)
\(\displaystyle a(6) = 150\)
\(\displaystyle a(7) = 399\)
\(\displaystyle a{\left(n + 8 \right)} = - a{\left(n \right)} - 5 a{\left(n + 1 \right)} - 2 a{\left(n + 2 \right)} + 8 a{\left(n + 3 \right)} + 12 a{\left(n + 4 \right)} + 2 a{\left(n + 5 \right)} + 2 a{\left(n + 6 \right)} + a{\left(n + 7 \right)}, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle -\frac{67361983587 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +6}}{6077539662986}-\frac{67361983587 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +6}}{6077539662986}-\frac{67361983587 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +6}}{6077539662986}-\frac{67361983587 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +6}}{6077539662986}-\frac{67361983587 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +6}}{6077539662986}-\frac{67361983587 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +6}}{6077539662986}-\frac{67361983587 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +6}}{6077539662986}-\frac{67361983587 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +6}}{6077539662986}-\frac{165267818125 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +5}}{3038769831493}-\frac{165267818125 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +5}}{3038769831493}-\frac{165267818125 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +5}}{3038769831493}-\frac{165267818125 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +5}}{3038769831493}-\frac{165267818125 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +5}}{3038769831493}-\frac{165267818125 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +5}}{3038769831493}-\frac{165267818125 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +5}}{3038769831493}-\frac{165267818125 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +5}}{3038769831493}-\frac{98572167560 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +4}}{3038769831493}-\frac{98572167560 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +4}}{3038769831493}-\frac{98572167560 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +4}}{3038769831493}-\frac{98572167560 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +4}}{3038769831493}-\frac{98572167560 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +4}}{3038769831493}-\frac{98572167560 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +4}}{3038769831493}-\frac{98572167560 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +4}}{3038769831493}-\frac{98572167560 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +4}}{3038769831493}+\frac{42598046586 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +3}}{3038769831493}+\frac{42598046586 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +3}}{3038769831493}+\frac{42598046586 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +3}}{3038769831493}+\frac{42598046586 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +3}}{3038769831493}+\frac{42598046586 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +3}}{3038769831493}+\frac{42598046586 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +3}}{3038769831493}+\frac{42598046586 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +3}}{3038769831493}+\frac{42598046586 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +3}}{3038769831493}+\frac{282835668312 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +2}}{3038769831493}+\frac{282835668312 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +2}}{3038769831493}+\frac{282835668312 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +2}}{3038769831493}+\frac{282835668312 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +2}}{3038769831493}+\frac{282835668312 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +2}}{3038769831493}+\frac{282835668312 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +2}}{3038769831493}+\frac{282835668312 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +2}}{3038769831493}+\frac{282835668312 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +2}}{3038769831493}+\frac{389981665132 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +1}}{3038769831493}+\frac{389981665132 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +1}}{3038769831493}+\frac{389981665132 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +1}}{3038769831493}+\frac{389981665132 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +1}}{3038769831493}+\frac{389981665132 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +1}}{3038769831493}+\frac{389981665132 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +1}}{3038769831493}+\frac{389981665132 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +1}}{3038769831493}+\frac{389981665132 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +1}}{3038769831493}+\frac{194046080351 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n -1}}{6077539662986}+\frac{194046080351 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n -1}}{6077539662986}+\frac{194046080351 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n -1}}{6077539662986}+\frac{194046080351 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n -1}}{6077539662986}+\frac{194046080351 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n -1}}{6077539662986}+\frac{194046080351 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n -1}}{6077539662986}+\frac{194046080351 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n -1}}{6077539662986}+\frac{194046080351 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n -1}}{6077539662986}+\frac{704832431921 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n}}{3038769831493}+\frac{704832431921 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n}}{3038769831493}+\frac{704832431921 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n}}{3038769831493}+\frac{704832431921 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n}}{3038769831493}+\frac{704832431921 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n}}{3038769831493}+\frac{704832431921 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n}}{3038769831493}+\frac{704832431921 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n}}{3038769831493}+\frac{704832431921 \mathit{RootOf} \left(Z^{8}+5 Z^{7}+2 Z^{6}-8 Z^{5}-12 Z^{4}-2 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n}}{3038769831493}\)
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This specification was found using the strategy pack "Point Placements" and has 84 rules.

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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_{23}\! \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_{16}\! \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_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{29}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= x^{2}\\ F_{34}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \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_{40}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \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_{47}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{56}\! \left(x \right)+F_{66}\! \left(x \right)+F_{70}\! \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_{61}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{79}\! \left(x \right)+F_{80}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{79}\! \left(x \right) &= 0\\ F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{54}\! \left(x \right)\\ \end{align*}\)