Av(1243, 1432, 3214)
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Generating Function
\(\displaystyle -\frac{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{4}}{x^{8}-4 x^{7}+18 x^{6}-35 x^{5}+51 x^{4}-47 x^{3}+26 x^{2}-8 x +1}\)
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Counting Sequence
1, 1, 2, 6, 21, 73, 241, 766, 2399, 7514, 23648, 74706, 236352, 747770, 2364773, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{8}-4 x^{7}+18 x^{6}-35 x^{5}+51 x^{4}-47 x^{3}+26 x^{2}-8 x +1\right) F \! \left(x \right)+\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{4} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 241\)
\(\displaystyle a \! \left(7\right) = 766\)
\(\displaystyle a \! \left(n +8\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-18 a \! \left(n +2\right)+35 a \! \left(n +3\right)-51 a \! \left(n +4\right)+47 a \! \left(n +5\right)-26 a \! \left(n +6\right)+8 a \! \left(n +7\right), \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle -\frac{9949964691 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +6}}{75243945211}-\frac{9949964691 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +6}}{75243945211}-\frac{9949964691 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +6}}{75243945211}-\frac{9949964691 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +6}}{75243945211}-\frac{9949964691 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +6}}{75243945211}-\frac{9949964691 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +6}}{75243945211}-\frac{9949964691 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +6}}{75243945211}-\frac{9949964691 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +6}}{75243945211}+\frac{34366338489 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +5}}{75243945211}+\frac{34366338489 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +5}}{75243945211}+\frac{34366338489 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +5}}{75243945211}+\frac{34366338489 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +5}}{75243945211}+\frac{34366338489 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +5}}{75243945211}+\frac{34366338489 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +5}}{75243945211}+\frac{34366338489 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +5}}{75243945211}+\frac{34366338489 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +5}}{75243945211}-\frac{159892967682 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +4}}{75243945211}-\frac{159892967682 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +4}}{75243945211}-\frac{159892967682 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +4}}{75243945211}-\frac{159892967682 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +4}}{75243945211}-\frac{159892967682 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +4}}{75243945211}-\frac{159892967682 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +4}}{75243945211}-\frac{159892967682 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +4}}{75243945211}-\frac{159892967682 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +4}}{75243945211}+\frac{259435485695 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +3}}{75243945211}+\frac{259435485695 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +3}}{75243945211}+\frac{259435485695 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +3}}{75243945211}+\frac{259435485695 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +3}}{75243945211}+\frac{259435485695 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +3}}{75243945211}+\frac{259435485695 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +3}}{75243945211}+\frac{259435485695 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +3}}{75243945211}+\frac{259435485695 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +3}}{75243945211}-\frac{359740963329 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +2}}{75243945211}-\frac{359740963329 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +2}}{75243945211}-\frac{359740963329 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +2}}{75243945211}-\frac{359740963329 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +2}}{75243945211}-\frac{359740963329 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +2}}{75243945211}-\frac{359740963329 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +2}}{75243945211}-\frac{359740963329 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +2}}{75243945211}-\frac{359740963329 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +2}}{75243945211}+\frac{261480651042 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n +1}}{75243945211}+\frac{261480651042 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n +1}}{75243945211}+\frac{261480651042 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n +1}}{75243945211}+\frac{261480651042 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n +1}}{75243945211}+\frac{261480651042 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n +1}}{75243945211}+\frac{261480651042 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n +1}}{75243945211}+\frac{261480651042 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n +1}}{75243945211}+\frac{261480651042 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n +1}}{75243945211}+\frac{20425731213 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n -1}}{75243945211}+\frac{20425731213 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n -1}}{75243945211}+\frac{20425731213 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n -1}}{75243945211}+\frac{20425731213 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n -1}}{75243945211}+\frac{20425731213 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n -1}}{75243945211}+\frac{20425731213 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n -1}}{75243945211}+\frac{20425731213 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n -1}}{75243945211}+\frac{20425731213 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n -1}}{75243945211}-\frac{100130251439 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =1\right)^{-n}}{75243945211}-\frac{100130251439 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =2\right)^{-n}}{75243945211}-\frac{100130251439 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =3\right)^{-n}}{75243945211}-\frac{100130251439 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =4\right)^{-n}}{75243945211}-\frac{100130251439 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =5\right)^{-n}}{75243945211}-\frac{100130251439 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =6\right)^{-n}}{75243945211}-\frac{100130251439 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =7\right)^{-n}}{75243945211}-\frac{100130251439 \mathit{RootOf} \left(Z^{8}-4 Z^{7}+18 Z^{6}-35 Z^{5}+51 Z^{4}-47 Z^{3}+26 Z^{2}-8 Z +1, \mathit{index} =8\right)^{-n}}{75243945211}\)
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This specification was found using the strategy pack "Point Placements" and has 117 rules.

Found on January 18, 2022.

Finding the specification took 4 seconds.

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Copy 117 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{20}\! \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_{13}\! \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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \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_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{43}\! \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_{49}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{55}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{56}\! \left(x \right)+F_{79}\! \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_{47}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{66}\! \left(x \right)+F_{75}\! \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_{70}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{56}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{60}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= 3 F_{15}\! \left(x \right)+F_{83}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{15}\! \left(x \right)+F_{60}\! \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_{103}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{41}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{4}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{104}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{105}\! \left(x \right)+F_{113}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{107}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{113}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\ \end{align*}\)