Av(1243, 1342, 1423, 2134, 3241)
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Generating Function
\(\displaystyle -\frac{3 x^{8}-6 x^{7}+9 x^{6}-15 x^{5}+14 x^{4}-16 x^{3}+14 x^{2}-6 x +1}{\left(x^{3}+2 x -1\right)^{2} \left(x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 19, 54, 143, 362, 890, 2149, 5129, 12143, 28579, 66953, 156264, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+2 x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+3 x^{8}-6 x^{7}+9 x^{6}-15 x^{5}+14 x^{4}-16 x^{3}+14 x^{2}-6 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 143\)
\(\displaystyle a \! \left(7\right) = 362\)
\(\displaystyle a \! \left(8\right) = 890\)
\(\displaystyle a \! \left(n \right) = -4 a \! \left(n +2\right)+2 a \! \left(n +3\right)-4 a \! \left(n +4\right)+4 a \! \left(n +5\right)-a \! \left(n +6\right)-n \left(n +6\right), \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle \frac{\left(144 \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{165888}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{18432}-\frac{\mathrm{I} \sqrt{3}}{3456}-\frac{1}{3456}\right)^{-n} \left(n +\frac{59}{144}\right) \left(\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} \,2^{\frac{1}{3}} 3^{\frac{5}{6}}-8 \,6^{\frac{1}{3}}\right)^{n} \left(\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} \,2^{\frac{1}{3}} 3^{\frac{5}{6}}-8 \,6^{\frac{1}{3}}\right)^{n} \left(18+2 \sqrt{59}\, \sqrt{3}\right)^{\frac{n}{3}} \left(-4608 \left(\left(\frac{\mathrm{I} \sqrt{59}}{16}-\frac{3}{16}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{3 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{16}+\mathrm{I} \sqrt{3}-1\right) \left(-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{59}\, 3^{\frac{1}{6}}-8 \,6^{\frac{1}{3}}\right)\right)^{-n}-354 \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(\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}}\right)^{n} \left(\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(n^{2}+4 n -\frac{7}{2}\right) \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{331776}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{36864}-\frac{\mathrm{I} \sqrt{3}}{6912}-\frac{1}{6912}\right)^{-n} \left(-\frac{\left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{1152}+\frac{\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}}{1152}+\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}}}{9216}\right)^{n}+\left(-2 \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} \left(2^{\frac{1}{3}} \left(3^{\frac{1}{6}} \left(n -\frac{409}{1888}\right) \sqrt{59}-3 \left(n +\frac{1253}{288}\right) 3^{\frac{2}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}-8 \,2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\frac{9701 \sqrt{59}\, 3^{\frac{5}{6}}}{22656}+3^{\frac{1}{3}} \left(n -\frac{521}{128}\right)\right)\right) \left(3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-\left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{59}\, 3^{\frac{1}{6}}+8 \,6^{\frac{1}{3}}\right)^{-n} \left(18+2 \sqrt{59}\, \sqrt{3}\right)^{-\frac{n}{3}} \left(\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \sqrt{3}-48\right)^{-n} \left(-24 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+24 \,\mathrm{I} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\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}}\right)^{n}+\left(2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \left(n -\frac{409}{1888}\right) \sqrt{59}-9 \left(n +\frac{1253}{288}\right) \left(\mathrm{I} \,3^{\frac{1}{6}}+\frac{3^{\frac{2}{3}}}{3}\right)\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+8 \,2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\left(\frac{9701 \,\mathrm{I} \,3^{\frac{1}{3}}}{7552}-\frac{9701 \,3^{\frac{5}{6}}}{22656}\right) \sqrt{59}+\left(\mathrm{I} \,3^{\frac{5}{6}}-3^{\frac{1}{3}}\right) \left(n -\frac{521}{128}\right)\right)\right) \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{3 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{2}+8 \,\mathrm{I} \sqrt{3}-8\right)^{-n} \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}+144 \left(n +\frac{59}{144}\right) \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3456}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}}{72}-\frac{1}{72}\right)^{-n} \left(\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \sqrt{3}-48\right)^{-n}\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(\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}}\right)^{n}-\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(\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}}\right)^{n} \left(\left(2^{\frac{1}{3}} \left(\left(\mathrm{I} \,3^{\frac{2}{3}}-3^{\frac{1}{6}}\right) \left(n -\frac{409}{1888}\right) \sqrt{59}-9 \left(n +\frac{1253}{288}\right) \left(\mathrm{I} \,3^{\frac{1}{6}}-\frac{3^{\frac{2}{3}}}{3}\right)\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+8 \,2^{\frac{2}{3}} \left(\left(\frac{9701 \,\mathrm{I} \,3^{\frac{1}{3}}}{7552}+\frac{9701 \,3^{\frac{5}{6}}}{22656}\right) \sqrt{59}+\left(\mathrm{I} \,3^{\frac{5}{6}}+3^{\frac{1}{3}}\right) \left(n -\frac{521}{128}\right)\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{3 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{2}+8 \,\mathrm{I} \sqrt{3}-8\right)^{-n} \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}-144 \left(n +\frac{59}{144}\right) \left(\frac{\left(-3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3456}+\frac{\left(\mathrm{I}+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}}{72}-\frac{1}{72}\right)^{-n} \left(\left(3 \,\mathrm{I} \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \,3^{\frac{5}{6}} \left(\mathrm{I}-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+48 \,\mathrm{I} \sqrt{3}-48\right)^{-n}\right)\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}}}{1416}\)
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This specification was found using the strategy pack "Point Placements" and has 95 rules.

Found on January 18, 2022.

Finding the specification took 2 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_{21}\! \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_{17}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \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_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{23}\! \left(x \right) &= 0\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)+F_{34}\! \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_{33}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{61}\! \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_{14}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \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_{57}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{55}\! \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_{51}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \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_{48}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{61}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{45}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{74}\! \left(x \right)+F_{89}\! \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_{4}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= 2 F_{23}\! \left(x \right)+F_{45}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{85}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{59}\! \left(x \right)\\ \end{align*}\)