Av(12435, 12453, 12543, 14235, 14253, 14325, 14352, 14523, 14532, 15243, 15423, 15432, 41235, 41253, 41325, 41352, 41523, 41532, 43125, 43152)
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Generating Function
\(\displaystyle \frac{\left(x +1\right) \left(2 x -1\right)^{3} \left(x -1\right)^{6}}{16 x^{14}-48 x^{13}+48 x^{12}-58 x^{11}+168 x^{10}-338 x^{9}+414 x^{8}-318 x^{7}+61 x^{6}+200 x^{5}-269 x^{4}+170 x^{3}-61 x^{2}+12 x -1}\)
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Counting Sequence
1, 1, 2, 6, 24, 100, 400, 1488, 5280, 18350, 63686, 222596, 783920, 2772902, 9819934, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(16 x^{14}-48 x^{13}+48 x^{12}-58 x^{11}+168 x^{10}-338 x^{9}+414 x^{8}-318 x^{7}+61 x^{6}+200 x^{5}-269 x^{4}+170 x^{3}-61 x^{2}+12 x -1\right) F \! \left(x \right)-\left(x +1\right) \left(2 x -1\right)^{3} \left(x -1\right)^{6} = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 400\)
\(\displaystyle a(7) = 1488\)
\(\displaystyle a(8) = 5280\)
\(\displaystyle a(9) = 18350\)
\(\displaystyle a(10) = 63686\)
\(\displaystyle a(11) = 222596\)
\(\displaystyle a(12) = 783920\)
\(\displaystyle a(13) = 2772902\)
\(\displaystyle a{\left(n + 14 \right)} = 16 a{\left(n \right)} - 48 a{\left(n + 1 \right)} + 48 a{\left(n + 2 \right)} - 58 a{\left(n + 3 \right)} + 168 a{\left(n + 4 \right)} - 338 a{\left(n + 5 \right)} + 414 a{\left(n + 6 \right)} - 318 a{\left(n + 7 \right)} + 61 a{\left(n + 8 \right)} + 200 a{\left(n + 9 \right)} - 269 a{\left(n + 10 \right)} + 170 a{\left(n + 11 \right)} - 61 a{\left(n + 12 \right)} + 12 a{\left(n + 13 \right)}, \quad n \geq 14\)
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Explicit Closed Form
\(\displaystyle \frac{1025063522533885204426837222072 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =1\right)^{-n +10}}{67567740159465211696527336101}+\frac{1025063522533885204426837222072 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =2\right)^{-n +10}}{67567740159465211696527336101}+\frac{1025063522533885204426837222072 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =3\right)^{-n +10}}{67567740159465211696527336101}+\frac{1025063522533885204426837222072 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =4\right)^{-n +10}}{67567740159465211696527336101}+\frac{1025063522533885204426837222072 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =5\right)^{-n +10}}{67567740159465211696527336101}+\frac{1025063522533885204426837222072 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =6\right)^{-n +10}}{67567740159465211696527336101}+\frac{1025063522533885204426837222072 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =7\right)^{-n +10}}{67567740159465211696527336101}+\frac{1025063522533885204426837222072 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =8\right)^{-n +10}}{67567740159465211696527336101}+\frac{1025063522533885204426837222072 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =9\right)^{-n +10}}{67567740159465211696527336101}+\frac{1025063522533885204426837222072 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =10\right)^{-n +10}}{67567740159465211696527336101}+\frac{1025063522533885204426837222072 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =11\right)^{-n +10}}{67567740159465211696527336101}+\frac{1025063522533885204426837222072 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =12\right)^{-n +10}}{67567740159465211696527336101}+\frac{1025063522533885204426837222072 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =13\right)^{-n +10}}{67567740159465211696527336101}+\frac{1025063522533885204426837222072 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =14\right)^{-n +10}}{67567740159465211696527336101}-\frac{26211567561192013365050273888703 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =8\right)^{-n +9}}{1283787063029839022234019385919}-\frac{26211567561192013365050273888703 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =9\right)^{-n +9}}{1283787063029839022234019385919}-\frac{26211567561192013365050273888703 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =10\right)^{-n +9}}{1283787063029839022234019385919}-\frac{26211567561192013365050273888703 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =11\right)^{-n +9}}{1283787063029839022234019385919}-\frac{26211567561192013365050273888703 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =12\right)^{-n +9}}{1283787063029839022234019385919}-\frac{26211567561192013365050273888703 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =13\right)^{-n +9}}{1283787063029839022234019385919}-\frac{26211567561192013365050273888703 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =14\right)^{-n +9}}{1283787063029839022234019385919}-\frac{26211567561192013365050273888703 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =1\right)^{-n +9}}{1283787063029839022234019385919}-\frac{26211567561192013365050273888703 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =2\right)^{-n +9}}{1283787063029839022234019385919}-\frac{26211567561192013365050273888703 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =3\right)^{-n +9}}{1283787063029839022234019385919}-\frac{26211567561192013365050273888703 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =4\right)^{-n +9}}{1283787063029839022234019385919}-\frac{26211567561192013365050273888703 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =5\right)^{-n +9}}{1283787063029839022234019385919}-\frac{26211567561192013365050273888703 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =6\right)^{-n +9}}{1283787063029839022234019385919}-\frac{26211567561192013365050273888703 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =7\right)^{-n +9}}{1283787063029839022234019385919}+\frac{87423235355019730069065107293611 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =1\right)^{-n +8}}{1283787063029839022234019385919}+\frac{87423235355019730069065107293611 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =2\right)^{-n +8}}{1283787063029839022234019385919}+\frac{87423235355019730069065107293611 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =3\right)^{-n +8}}{1283787063029839022234019385919}+\frac{87423235355019730069065107293611 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =4\right)^{-n +8}}{1283787063029839022234019385919}+\frac{87423235355019730069065107293611 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =5\right)^{-n +8}}{1283787063029839022234019385919}+\frac{87423235355019730069065107293611 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =6\right)^{-n +8}}{1283787063029839022234019385919}+\frac{87423235355019730069065107293611 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =7\right)^{-n +8}}{1283787063029839022234019385919}+\frac{87423235355019730069065107293611 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =8\right)^{-n +8}}{1283787063029839022234019385919}+\frac{87423235355019730069065107293611 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =9\right)^{-n +8}}{1283787063029839022234019385919}+\frac{87423235355019730069065107293611 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =10\right)^{-n +8}}{1283787063029839022234019385919}+\frac{87423235355019730069065107293611 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =11\right)^{-n +8}}{1283787063029839022234019385919}+\frac{87423235355019730069065107293611 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =12\right)^{-n +8}}{1283787063029839022234019385919}+\frac{87423235355019730069065107293611 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =13\right)^{-n +8}}{1283787063029839022234019385919}+\frac{87423235355019730069065107293611 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =14\right)^{-n +8}}{1283787063029839022234019385919}+\frac{3046207003167622705126152734489 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =1\right)^{-n -1}}{2567574126059678044468038771838}+\frac{3046207003167622705126152734489 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =2\right)^{-n -1}}{2567574126059678044468038771838}+\frac{3046207003167622705126152734489 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =3\right)^{-n -1}}{2567574126059678044468038771838}+\frac{3046207003167622705126152734489 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =4\right)^{-n -1}}{2567574126059678044468038771838}+\frac{3046207003167622705126152734489 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =5\right)^{-n -1}}{2567574126059678044468038771838}+\frac{3046207003167622705126152734489 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =6\right)^{-n -1}}{2567574126059678044468038771838}+\frac{3046207003167622705126152734489 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =7\right)^{-n -1}}{2567574126059678044468038771838}+\frac{3046207003167622705126152734489 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =8\right)^{-n -1}}{2567574126059678044468038771838}+\frac{3046207003167622705126152734489 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =9\right)^{-n -1}}{2567574126059678044468038771838}+\frac{3046207003167622705126152734489 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 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Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =14\right)^{-n -1}}{2567574126059678044468038771838}-\frac{14208502965383690072076547984881 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =1\right)^{-n +4}}{2567574126059678044468038771838}-\frac{14208502965383690072076547984881 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =2\right)^{-n +4}}{2567574126059678044468038771838}-\frac{14208502965383690072076547984881 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =3\right)^{-n 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Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =9\right)^{-n +5}}{67567740159465211696527336101}-\frac{6318836505394328905017102089332 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =10\right)^{-n +5}}{67567740159465211696527336101}-\frac{6318836505394328905017102089332 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =11\right)^{-n +5}}{67567740159465211696527336101}-\frac{6318836505394328905017102089332 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =12\right)^{-n +5}}{67567740159465211696527336101}-\frac{6318836505394328905017102089332 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =13\right)^{-n +5}}{67567740159465211696527336101}-\frac{6318836505394328905017102089332 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =14\right)^{-n +5}}{67567740159465211696527336101}+\frac{182306399378304246847645766263983 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =1\right)^{-n +6}}{1283787063029839022234019385919}+\frac{182306399378304246847645766263983 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =2\right)^{-n +6}}{1283787063029839022234019385919}+\frac{182306399378304246847645766263983 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =3\right)^{-n +6}}{1283787063029839022234019385919}+\frac{182306399378304246847645766263983 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =4\right)^{-n +6}}{1283787063029839022234019385919}+\frac{182306399378304246847645766263983 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =5\right)^{-n +6}}{1283787063029839022234019385919}+\frac{182306399378304246847645766263983 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =6\right)^{-n +6}}{1283787063029839022234019385919}+\frac{182306399378304246847645766263983 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =7\right)^{-n +6}}{1283787063029839022234019385919}+\frac{182306399378304246847645766263983 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =8\right)^{-n +6}}{1283787063029839022234019385919}+\frac{182306399378304246847645766263983 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =9\right)^{-n +6}}{1283787063029839022234019385919}+\frac{182306399378304246847645766263983 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =10\right)^{-n +6}}{1283787063029839022234019385919}+\frac{182306399378304246847645766263983 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =11\right)^{-n +6}}{1283787063029839022234019385919}+\frac{182306399378304246847645766263983 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =12\right)^{-n +6}}{1283787063029839022234019385919}+\frac{182306399378304246847645766263983 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =13\right)^{-n +6}}{1283787063029839022234019385919}+\frac{182306399378304246847645766263983 \mathit{RootOf} \left(16 Z^{14}-48 Z^{13}+48 Z^{12}-58 Z^{11}+168 Z^{10}-338 Z^{9}+414 Z^{8}-318 Z^{7}+61 Z^{6}+200 Z^{5}-269 Z^{4}+170 Z^{3}-61 Z^{2}+12 Z -1, \mathit{index} =14\right)^{-n +6}}{1283787063029839022234019385919}\)
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This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 212 rules.

Finding the specification took 174 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_{19}\! \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_{34}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{19}\! \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_{29}\! \left(x \right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{16}\! \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_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= x\\ F_{20}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{19}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{19}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{21}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{19}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{19}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{188}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{19}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{12}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{19}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{12}\! \left(x \right)+F_{45}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{19}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{19}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{19}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{19}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{45}\! \left(x \right)+F_{58}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{19}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{19}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{19}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{19}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{58}\! \left(x \right)+F_{70}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{19}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{19}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{76}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{19}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{19}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{19}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{78}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{19}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{19}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{70}\! \left(x \right)+F_{87}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{19}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{76}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{19}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{95}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{103}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{84}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{107}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{12}\! \left(x \right)+F_{176}\! \left(x \right)+F_{180}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{168}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{12}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{126}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{121}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{19}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{136}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{128}\! \left(x \right)+F_{130}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{133}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{134}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{130}\! \left(x \right)+F_{135}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{137}\! \left(x \right)+F_{139}\! \left(x \right)+F_{159}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{116}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{143}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{121}\! \left(x \right)+F_{125}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{145}\! \left(x \right)+F_{146}\! \left(x \right)+F_{150}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{19}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{149}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{142}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{144}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{151}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{153}\! \left(x \right)\\ F_{153}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{154}\! \left(x \right)+F_{155}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{19}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{156}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{158}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{124}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{153}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{160}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{161}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{162}\! \left(x \right)\\ F_{162}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{163}\! \left(x \right)+F_{164}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{120}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{165}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{167}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{124}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{153}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{12}\! \left(x \right)+F_{169}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{170}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{172}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{173}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{173}\! \left(x \right) &= F_{174}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{175}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{172}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{177}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{178}\! \left(x \right)+F_{179}\! \left(x \right)\\ F_{178}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{179}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{181}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{181}\! \left(x \right) &= F_{182}\! \left(x \right)\\ F_{182}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{183}\! \left(x \right)\\ F_{183}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{173}\! \left(x \right)+F_{184}\! \left(x \right)\\ F_{184}\! \left(x \right) &= F_{185}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{185}\! \left(x \right) &= F_{186}\! \left(x \right)+F_{187}\! \left(x \right)\\ F_{186}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{187}\! \left(x \right) &= F_{95}\! \left(x \right)\\ F_{188}\! \left(x \right) &= F_{189}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{189}\! \left(x \right) &= F_{190}\! \left(x \right)+F_{196}\! \left(x \right)\\ F_{190}\! \left(x \right) &= F_{191}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{191}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{192}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{192}\! \left(x \right) &= F_{19}\! \left(x \right) F_{193}\! \left(x \right)\\ F_{193}\! \left(x \right) &= F_{194}\! \left(x \right)+F_{195}\! \left(x \right)\\ F_{194}\! \left(x \right) &= F_{191}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{195}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{183}\! \left(x \right)\\ F_{196}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{197}\! \left(x \right)\\ F_{197}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{176}\! \left(x \right)+F_{198}\! \left(x \right)+F_{208}\! \left(x \right)\\ F_{198}\! \left(x \right) &= F_{19}\! \left(x \right) F_{199}\! \left(x \right)\\ F_{199}\! \left(x \right) &= F_{200}\! \left(x \right)\\ F_{200}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{201}\! \left(x \right)\\ F_{201}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{198}\! \left(x \right)+F_{202}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{202}\! \left(x \right) &= F_{19}\! \left(x \right) F_{203}\! \left(x \right)\\ F_{203}\! \left(x \right) &= F_{204}\! \left(x \right)\\ F_{204}\! \left(x \right) &= F_{205}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{205}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{206}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{206}\! \left(x \right) &= F_{19}\! \left(x \right) F_{207}\! \left(x \right)\\ F_{207}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{205}\! \left(x \right)\\ F_{208}\! \left(x \right) &= F_{19}\! \left(x \right) F_{209}\! \left(x \right)\\ F_{209}\! \left(x \right) &= F_{210}\! \left(x \right)\\ F_{210}\! \left(x \right) &= F_{211}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{211}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{184}\! \left(x \right)+F_{206}\! \left(x \right)\\ \end{align*}\)