Av(1234, 1243, 1342, 2413, 3241)
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Generating Function
\(\displaystyle -\frac{4 x^{6}-9 x^{5}+10 x^{4}-15 x^{3}+14 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 19, 54, 141, 350, 842, 1985, 4617, 10639, 24349, 55437, 125696, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{3}+2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+4 x^{6}-9 x^{5}+10 x^{4}-15 x^{3}+14 x^{2}-6 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 141\)
\(\displaystyle a \! \left(n +4\right) = -\frac{n^{2}}{2}-2 a \! \left(n \right)+a \! \left(n +1\right)-4 a \! \left(n +2\right)+4 a \! \left(n +3\right)-\frac{3 n}{2}+4, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \frac{197 \left(\frac{32096 \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{27}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{16 \,\mathrm{I} \sqrt{3}}{9}-\frac{16}{9}\right)^{-n} \left(\frac{\left(-\mathrm{I} \sqrt{59}+3\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{1}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,\mathrm{I} \,3^{\frac{5}{6}} 2^{\frac{1}{3}}+8 \,6^{\frac{1}{3}}}{\sqrt{59}\, 3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,6^{\frac{1}{3}}-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}\right)^{n} \left(18+2 \sqrt{59}\, \sqrt{3}\right)^{\frac{n}{3}} \left(\left(-64 \,\mathrm{I} \sqrt{59}-192\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+576 \,3^{\frac{1}{6}} \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-512 \,\mathrm{I} \,3^{\frac{5}{6}} 2^{\frac{1}{3}}-512 \,6^{\frac{1}{3}}\right)^{n}}{197}+\frac{1916 \left(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(6 \,\mathrm{I} \sqrt{59}-18\right) 18^{\frac{1}{3}}-54 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-6 \,\mathrm{I} \sqrt{59}-18\right) 18^{\frac{1}{3}}+54 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(2^{\frac{2}{3}} \left(\sqrt{59}\, 3^{\frac{5}{6}}-\frac{1947 \,3^{\frac{1}{3}}}{479}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{591 \,2^{\frac{1}{3}} \left(\sqrt{59}\, 3^{\frac{1}{6}}-\frac{5605 \,3^{\frac{2}{3}}}{591}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{958}\right) \left(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(2 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}-6 \,18^{\frac{1}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+\left(12 \,\mathrm{I}-4 \sqrt{3}\right) \sqrt{59}-36 \,\mathrm{I} \sqrt{3}+36\right)^{-n}}{591}-\frac{11328 \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{9}+\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{16 \,\mathrm{I} \sqrt{3}}{3}-\frac{16}{3}\right)^{-n} \left(\left(n^{2}+4 n -\frac{7}{2}\right) \left(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-6 \,\mathrm{I} \sqrt{59}-18\right) 18^{\frac{1}{3}}+54 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}-\frac{17 \,576^{n}}{6}\right) \left(-\frac{16 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}+\frac{16 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{3}+\frac{\left(\left(2 \,\mathrm{I} \sqrt{59}-6\right) 18^{\frac{1}{3}}-18 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{3}\right)^{n}}{197}+\frac{32096 \,576^{n} \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{9}+\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{16 \,\mathrm{I} \sqrt{3}}{3}-\frac{16}{3}\right)^{-n} \left(-\frac{16 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{16 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{3}+\frac{\left(\left(-2 \,\mathrm{I} \sqrt{59}-6\right) 18^{\frac{1}{3}}+18 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{3}\right)^{n}}{197}-\frac{90624 \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{128}+\frac{9 \left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{128}-\frac{3 \,\mathrm{I} \sqrt{3}}{8}-\frac{3}{8}\right)^{-n} \left(-\frac{3 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{4}+\frac{3 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{4}+\frac{\left(\left(3 \,\mathrm{I} \sqrt{59}-9\right) 18^{\frac{1}{3}}-27 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+3 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{32}\right)^{n} \left(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-6 \,\mathrm{I} \sqrt{59}-18\right) 18^{\frac{1}{3}}+54 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{197}+\left(\left(3^{2 n +\frac{2}{3}} \left(\mathrm{I} \sqrt{59}-\frac{5605}{591}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} 2^{6 n +\frac{1}{3}}+\frac{958 \,2^{6 n +\frac{2}{3}} \left(\mathrm{I} \sqrt{59}+\frac{649}{479}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{2 n +\frac{1}{3}}}{197}\right) \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(-\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(-3^{2 n +\frac{2}{3}} \left(\mathrm{I} \sqrt{59}+\frac{5605}{591}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} 2^{6 n +\frac{1}{3}}-\frac{958 \,2^{6 n +\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\mathrm{I} \sqrt{59}-\frac{649}{479}\right) 3^{2 n +\frac{1}{3}}}{197}\right) \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{12}+\frac{\left(\left(\mathrm{I} \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{96}\right)^{n}+\left(-\frac{1298 \,2^{\frac{2}{3}} \left(\frac{479 \sqrt{59}}{1947}+\mathrm{I}\right) 3^{\frac{5}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{197}-\frac{5605 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \left(-\frac{197 \sqrt{59}}{5605}+\mathrm{I}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{197}\right) \left(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-6 \,\mathrm{I} \sqrt{59}-18\right) 18^{\frac{1}{3}}+54 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}+\frac{1298 \left(\left(\mathrm{I}-\frac{479 \sqrt{59}}{1947}\right) 2^{\frac{2}{3}} 3^{\frac{5}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{95 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\mathrm{I}+\frac{197 \sqrt{59}}{5605}\right)}{22}\right) \left(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(6 \,\mathrm{I} \sqrt{59}-18\right) 18^{\frac{1}{3}}-54 \,\mathrm{I} \,2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{197}\right) \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}-\frac{\mathrm{I} \sqrt{3}}{12}-\frac{1}{12}\right)^{-n}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \sqrt{3}-48\right)^{-n}}{45312}\)
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This specification was found using the strategy pack "Point Placements" and has 159 rules.

Found on January 18, 2022.

Finding the specification took 5 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_{17}\! \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_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{45}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{58}\! \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_{57}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{48}\! \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_{65}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{67}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{4}\! \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_{79}\! \left(x \right)\\ F_{90}\! \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_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{4}\! \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_{70}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{96}\! \left(x \right)\\ F_{100}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{101}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{105}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{110}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{106}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{119}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{116}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{151}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{150}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{129}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{13}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{130}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{131}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{135}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{134}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{141}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{140}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{127}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{136}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{143}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{145}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{134}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{149}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{134}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{146}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{158}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{154}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)+F_{157}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{150}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{152}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{96}\! \left(x \right)\\ \end{align*}\)