Av(1243, 2134, 2143, 2413)
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Generating Function
\(\displaystyle -\frac{\left(x^{2}-3 x +1\right)^{2}}{\left(x -1\right) \left(x^{4}-4 x^{3}+10 x^{2}-6 x +1\right)}\)
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Counting Sequence
1, 1, 2, 6, 20, 68, 231, 781, 2629, 8821, 29530, 98706, 329592, 1099792, 3668127, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{4}-4 x^{3}+10 x^{2}-6 x +1\right) F \! \left(x \right)+\left(x^{2}-3 x +1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +4\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+6 a \! \left(n +3\right)+1, \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \frac{\left(\left(\left(\left(-172 \sqrt{3}-92 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(1720 \sqrt{3}-680 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-2000 \sqrt{43}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+\left(5895 \sqrt{43}\, \sqrt{3}-17415\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+36000 \sqrt{3}\, \sqrt{43}\, \left(\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+4\right)\right) \sqrt{\left(-6 \sqrt{43}\, \sqrt{3}+262\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+320 \sqrt{3}\, \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+1600 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+25600}+371520000+\left(\left(-15480 \sqrt{3}-8280 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-154800 \left(\sqrt{3}-\frac{17 \sqrt{43}}{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}\right) \left(-\frac{\sqrt{960 \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}+\left(18 \sqrt{43}\, \sqrt{3}-786\right) 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}-4800 \,2^{\frac{1}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}-76800}}{240}+\frac{\left(-131 \,2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}-800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+6400\right) \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}}{864000}+\frac{\sqrt{-215 \,2^{\frac{1}{3}} \left(\sqrt{43}\, \sqrt{3}-97\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+34400+\left(-43 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+3311 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}{96000}+1\right)^{-n}}{2972160000}+\frac{\left(\left(\left(\left(172 \sqrt{3}+92 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(-1720 \sqrt{3}+680 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+2000 \sqrt{43}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+\left(-5895 \sqrt{43}\, \sqrt{3}+17415\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-36000 \sqrt{3}\, \sqrt{43}\, \left(\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+4\right)\right) \sqrt{\left(-6 \sqrt{43}\, \sqrt{3}+262\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+320 \sqrt{3}\, \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+1600 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+25600}+371520000+\left(\left(-15480 \sqrt{3}-8280 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-154800 \left(\sqrt{3}-\frac{17 \sqrt{43}}{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}\right) \left(\frac{\sqrt{960 \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}+\left(18 \sqrt{43}\, \sqrt{3}-786\right) 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}-4800 \,2^{\frac{1}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}-76800}}{240}+\frac{\left(-131 \,2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}-800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+6400\right) \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}}{864000}+\frac{\sqrt{-215 \,2^{\frac{1}{3}} \left(\sqrt{43}\, \sqrt{3}-97\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+34400+\left(-43 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+3311 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}{96000}+1\right)^{-n}}{2972160000}+\frac{\left(\left(\left(\left(-1834 \,\mathrm{I}+42 \,\mathrm{I} \sqrt{3}\, \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-11200 \,\mathrm{I} \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-94000 \,\mathrm{I}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+\left(100215 \,\mathrm{I} \sqrt{3}-6885 \,\mathrm{I} \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+612000 \,\mathrm{I} \sqrt{3}\, \left(\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{92}{17}\right)\right) \sqrt{\left(-6 \sqrt{43}\, \sqrt{3}+262\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+320 \sqrt{3}\, \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+1600 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+25600}+371520000+\left(\left(15480 \sqrt{3}+8280 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+154800 \left(\sqrt{3}-\frac{17 \sqrt{43}}{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}\right) \left(-\frac{\mathrm{I} \sqrt{960 \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}+\left(-18 \sqrt{43}\, \sqrt{3}+786\right) 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}+4800 \,2^{\frac{1}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+76800}}{240}+\frac{\left(131 \,2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}+800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-6400\right) \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}}{864000}-\frac{\sqrt{-215 \,2^{\frac{1}{3}} \left(\sqrt{43}\, \sqrt{3}-97\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+34400+\left(-43 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+3311 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}{96000}+1\right)^{-n}}{2972160000}+\frac{1}{2}+\frac{\left(\left(\left(\left(-42 \,\mathrm{I} \sqrt{3}\, \sqrt{43}+1834 \,\mathrm{I}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+11200 \,\mathrm{I} \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+94000 \,\mathrm{I}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+\left(-100215 \,\mathrm{I} \sqrt{3}+6885 \,\mathrm{I} \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}-612000 \,\mathrm{I} \sqrt{3}\, \left(\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{92}{17}\right)\right) \sqrt{\left(-6 \sqrt{43}\, \sqrt{3}+262\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+320 \sqrt{3}\, \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}+1600 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}+25600}+371520000+\left(\left(15480 \sqrt{3}+8280 \sqrt{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+154800 \left(\sqrt{3}-\frac{17 \sqrt{43}}{43}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \sqrt{\left(485-5 \sqrt{43}\, \sqrt{3}\right) \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{43}\, \sqrt{3}+77 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{2}{3}}+800}\right) \left(\frac{\mathrm{I} \sqrt{960 \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}+\left(-18 \sqrt{43}\, \sqrt{3}+786\right) 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}+4800 \,2^{\frac{1}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+76800}}{240}+\frac{\left(131 \,2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}+800 \left(262+6 \sqrt{43}\, \sqrt{3}\right)^{\frac{1}{3}}-6400\right) \sqrt{\left(-15 \sqrt{43}\, 2^{\frac{1}{3}} \sqrt{3}+1455 \,2^{\frac{1}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+2400+\left(-3 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+231 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}}{864000}-\frac{\sqrt{-215 \,2^{\frac{1}{3}} \left(\sqrt{43}\, \sqrt{3}-97\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{1}{3}}+34400+\left(-43 \,2^{\frac{2}{3}} \sqrt{3}\, \sqrt{43}+3311 \,2^{\frac{2}{3}}\right) \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}\, 2^{\frac{2}{3}} \left(3 \sqrt{43}\, \sqrt{3}+131\right)^{\frac{2}{3}}}{96000}+1\right)^{-n}}{2972160000}\)
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This specification was found using the strategy pack "Point Placements" and has 82 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{27}\! \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_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{26}\! \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_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{74}\! \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_{32}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{37}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{36}\! \left(x \right) &= 0\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{43}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{52}\! \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_{33}\! \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_{43}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{61}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{68}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{52}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \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_{79}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{29}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{44}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\ \end{align*}\)