Av(1243, 2314, 3214, 4123)
View Raw Data
Generating Function
\(\displaystyle -\frac{x^{8}-x^{6}+2 x^{5}+5 x^{4}+2 x^{3}+x^{2}-3 x +1}{\left(x^{2}-3 x +1\right) \left(x^{3}+x^{2}+x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 61, 174, 483, 1316, 3539, 9438, 25027, 66106, 174143, 457890, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+x^{8}-x^{6}+2 x^{5}+5 x^{4}+2 x^{3}+x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(6\right) = 174\)
\(\displaystyle a \! \left(7\right) = 483\)
\(\displaystyle a \! \left(8\right) = 1316\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)-3 a \! \left(n +3\right)+4 a \! \left(n +4\right), \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{\left(\left(\left(25 \sqrt{11}+5 \,\mathrm{I}\right) \sqrt{3}+75 \,\mathrm{I} \sqrt{11}+5\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-80+\left(\left(-110 \sqrt{11}-640 \,\mathrm{I}\right) \sqrt{3}+330 \,\mathrm{I} \sqrt{11}+640\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}}{1320}\\+\\\frac{\left(\left(\left(640 \,\mathrm{I}-110 \sqrt{11}\right) \sqrt{3}-330 \,\mathrm{I} \sqrt{11}+640\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-80+\left(\left(-5 \,\mathrm{I}+25 \sqrt{11}\right) \sqrt{3}-75 \,\mathrm{I} \sqrt{11}+5\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}}{1320}\\+\\\frac{\left(\left(220 \sqrt{11}\, \sqrt{3}-1280\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}-50 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-10 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}-80\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}}{1320}\\+\frac{\left(-4392 \sqrt{5}+10680\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{1320}+\frac{\left(4392 \sqrt{5}+10680\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{1320} & \text{otherwise} \end{array}\right.\)

This specification was found using the strategy pack "Point Placements" and has 191 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_{38}\! \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_{26}\! \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_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \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_{33}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{47}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{53}\! \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_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{54}\! \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_{39}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{170}\! \left(x \right)+F_{20}\! \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_{101}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{66}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{75}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \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_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{84}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{89}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{90}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{134}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{108}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{105}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{122}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{116}\! \left(x \right)\\ F_{116}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{117}\! \left(x \right)+F_{121}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{120}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{113}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{109}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{122}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{123}\! \left(x \right)+F_{130}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{128}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{126}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{132}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{129}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{129}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{134}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{135}\! \left(x \right)+F_{140}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{137}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{142}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{158}\! \left(x \right)\\ F_{143}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{144}\! \left(x \right)+F_{148}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{146}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{151}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{152}\! \left(x \right)\\ F_{152}\! \left(x \right) &= 3 F_{20}\! \left(x \right)+F_{153}\! \left(x \right)+F_{157}\! \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_{156}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{146}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{143}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{158}\! \left(x \right) &= 3 F_{20}\! \left(x \right)+F_{159}\! \left(x \right)+F_{166}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{160}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{161}\! \left(x \right)+F_{164}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{162}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{137}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)+F_{168}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{166}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{167}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{165}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{169}\! \left(x \right)\\ F_{169}\! \left(x \right) &= F_{165}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{171}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{172}\! \left(x \right)\\ F_{172}\! \left(x \right) &= F_{173}\! \left(x \right)+F_{182}\! \left(x \right)\\ F_{173}\! \left(x \right) &= F_{174}\! \left(x \right)+F_{20}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{175}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{176}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{173}\! \left(x \right)+F_{177}\! \left(x \right)\\ F_{177}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{178}\! \left(x \right)+F_{181}\! \left(x \right)\\ F_{178}\! \left(x \right) &= F_{179}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{179}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{180}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{181}\! \left(x \right)\\ F_{181}\! \left(x \right) &= F_{173}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{182}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{183}\! \left(x \right)+F_{187}\! \left(x \right)\\ F_{183}\! \left(x \right) &= F_{184}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{184}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{185}\! \left(x \right)\\ F_{185}\! \left(x \right) &= F_{186}\! \left(x \right)+F_{189}\! \left(x \right)\\ F_{186}\! \left(x \right) &= F_{187}\! \left(x \right)\\ F_{187}\! \left(x \right) &= F_{188}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{188}\! \left(x \right) &= F_{173}\! \left(x \right)+F_{186}\! \left(x \right)\\ F_{189}\! \left(x \right) &= F_{190}\! \left(x \right)\\ F_{190}\! \left(x \right) &= F_{186}\! \left(x \right) F_{4}\! \left(x \right)\\ \end{align*}\)