Av(1234, 1324, 2413, 3241, 4132)
Generating Function
\(\displaystyle \frac{x^{13}+3 x^{12}+x^{11}-7 x^{10}-14 x^{9}-12 x^{8}-x^{7}+8 x^{6}+7 x^{5}+2 x^{4}-4 x^{3}-4 x^{2}+4 x -1}{\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 52, 124, 272, 570, 1164, 2341, 4661, 9212, 18098, 35369, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} \left(x^{2}+x -1\right)^{2} F \! \left(x \right)+x^{13}+3 x^{12}+x^{11}-7 x^{10}-14 x^{9}-12 x^{8}-x^{7}+8 x^{6}+7 x^{5}+2 x^{4}-4 x^{3}-4 x^{2}+4 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(6\right) = 124\)
\(\displaystyle a \! \left(7\right) = 272\)
\(\displaystyle a \! \left(8\right) = 570\)
\(\displaystyle a \! \left(9\right) = 1164\)
\(\displaystyle a \! \left(10\right) = 2341\)
\(\displaystyle a \! \left(11\right) = 4661\)
\(\displaystyle a \! \left(12\right) = 9212\)
\(\displaystyle a \! \left(13\right) = 18098\)
\(\displaystyle a \! \left(n +4\right) = \frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{a \! \left(n +2\right)}{2}-\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(n +6\right)}{4}-\frac{a \! \left(n +7\right)}{4}+\frac{17 n}{4}-7, \quad n \geq 14\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 52\)
\(\displaystyle a \! \left(6\right) = 124\)
\(\displaystyle a \! \left(7\right) = 272\)
\(\displaystyle a \! \left(8\right) = 570\)
\(\displaystyle a \! \left(9\right) = 1164\)
\(\displaystyle a \! \left(10\right) = 2341\)
\(\displaystyle a \! \left(11\right) = 4661\)
\(\displaystyle a \! \left(12\right) = 9212\)
\(\displaystyle a \! \left(13\right) = 18098\)
\(\displaystyle a \! \left(n +4\right) = \frac{a \! \left(n \right)}{4}+\frac{3 a \! \left(n +1\right)}{4}+\frac{a \! \left(n +2\right)}{2}-\frac{a \! \left(n +3\right)}{2}+\frac{3 a \! \left(n +6\right)}{4}-\frac{a \! \left(n +7\right)}{4}+\frac{17 n}{4}-7, \quad n \geq 14\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 2 & n =2 \\ 6 & n =3 \\ 19 & n =4 \\ \frac{\left(\left(\left(-4875 \sqrt{11}+19525 \,\mathrm{I}\right) \sqrt{3}-14625 \,\mathrm{I} \sqrt{11}+19525\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+57200+\left(\left(37675 \,\mathrm{I}+6075 \sqrt{11}\right) \sqrt{3}-18225 \,\mathrm{I} \sqrt{11}-37675\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{13200}\\+\\\frac{\left(\left(\left(-37675 \,\mathrm{I}+6075 \sqrt{11}\right) \sqrt{3}+18225 \,\mathrm{I} \sqrt{11}-37675\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+57200+\left(\left(-19525 \,\mathrm{I}-4875 \sqrt{11}\right) \sqrt{3}+14625 \,\mathrm{I} \sqrt{11}+19525\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{13200}\\+\\\frac{\left(\left(-12150 \sqrt{11}\, \sqrt{3}+75350\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+9750 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-39050 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+57200\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{13200}\\+\frac{\left(\left(6600 n +12936\right) \sqrt{5}-14520 n -33000\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{13200}+\\\frac{\left(\left(-6600 n -12936\right) \sqrt{5}-14520 n -33000\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{13200}-\frac{17 n}{2}-3 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 184 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_{23}\! \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_{21}\! \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_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{32}\! \left(x \right) &= 0\\
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_{39}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{41}\! \left(x \right) &= 2 F_{32}\! \left(x \right)+F_{42}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \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_{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_{37}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= 2 F_{32}\! \left(x \right)+F_{46}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{49}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{32}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{68}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{11}\! \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_{4}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \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_{94}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{95}\! \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_{11}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= 2 F_{32}\! \left(x \right)+F_{100}\! \left(x \right)+F_{104}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{103}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \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_{111}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{96}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{112}\! \left(x \right)\\
F_{112}\! \left(x \right) &= 2 F_{32}\! \left(x \right)+F_{104}\! \left(x \right)+F_{113}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{110}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{113}\! \left(x \right)\\
F_{117}\! \left(x \right) &= 2 F_{32}\! \left(x \right)+F_{118}\! \left(x \right)+F_{144}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{124}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{126}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{126}\! \left(x \right) &= 3 F_{32}\! \left(x \right)+F_{127}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{130}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{121}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{127}\! \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_{133}\! \left(x \right) &= F_{134}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{138}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{139}\! \left(x \right)\\
F_{139}\! \left(x \right) &= 3 F_{32}\! \left(x \right)+F_{131}\! \left(x \right)+F_{140}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)+F_{143}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{137}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{140}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{134}\! \left(x \right)\\
F_{147}\! \left(x \right) &= F_{148}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)+F_{171}\! \left(x \right)\\
F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)+F_{168}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{151}\! \left(x \right) &= F_{152}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{155}\! \left(x \right)\\
F_{153}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{154}\! \left(x \right)\\
F_{154}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{68}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{155}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{156}\! \left(x \right)\\
F_{156}\! \left(x \right) &= 2 F_{32}\! \left(x \right)+F_{144}\! \left(x \right)+F_{157}\! \left(x \right)\\
F_{157}\! \left(x \right) &= F_{158}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{161}\! \left(x \right)\\
F_{159}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{160}\! \left(x \right)\\
F_{160}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{161}\! \left(x \right) &= F_{162}\! \left(x \right)+F_{163}\! \left(x \right)\\
F_{162}\! \left(x \right) &= F_{157}\! \left(x \right)\\
F_{163}\! \left(x \right) &= 3 F_{32}\! \left(x \right)+F_{131}\! \left(x \right)+F_{164}\! \left(x \right)\\
F_{164}\! \left(x \right) &= F_{165}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)+F_{167}\! \left(x \right)\\
F_{166}\! \left(x \right) &= F_{160}\! \left(x \right)\\
F_{167}\! \left(x \right) &= F_{164}\! \left(x \right)\\
F_{168}\! \left(x \right) &= F_{169}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{169}\! \left(x \right) &= F_{170}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{170}\! \left(x \right) &= F_{168}\! \left(x \right)\\
F_{171}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{180}\! \left(x \right)\\
F_{172}\! \left(x \right) &= F_{173}\! \left(x \right)\\
F_{173}\! \left(x \right) &= F_{174}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{174}\! \left(x \right) &= F_{175}\! \left(x \right)+F_{177}\! \left(x \right)\\
F_{175}\! \left(x \right) &= F_{176}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{176}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{177}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{178}\! \left(x \right)\\
F_{178}\! \left(x \right) &= F_{179}\! \left(x \right)\\
F_{179}\! \left(x \right) &= F_{172}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{180}\! \left(x \right) &= F_{181}\! \left(x \right)\\
F_{181}\! \left(x \right) &= F_{182}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{182}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{183}\! \left(x \right)\\
F_{183}\! \left(x \right) &= F_{181}\! \left(x \right)\\
\end{align*}\)