Av(12345, 12354, 12435, 13245, 13254, 23145, 23154)
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Counting Sequence
1, 1, 2, 6, 24, 113, 582, 3164, 17838, 103276, 610304, 3665958, 22316734, 137375886, 853661444, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(3 x -2\right) F \left(x \right)^{4}+\left(-6 x +4\right) F \left(x \right)^{3}+2 x F \left(x \right)^{2}-3 F \! \left(x \right)+1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 113\)
\(\displaystyle a \! \left(6\right) = 582\)
\(\displaystyle a \! \left(7\right) = 3164\)
\(\displaystyle a \! \left(8\right) = 17838\)
\(\displaystyle a \! \left(n +9\right) = \frac{192 n \left(n +1\right) \left(2 n +1\right) a \! \left(n \right)}{407 \left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{16 \left(n +1\right) \left(230 n^{2}+1174 n +1257\right) a \! \left(n +1\right)}{407 \left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{16 \left(1231 n^{3}+6990 n^{2}+13499 n +8859\right) a \! \left(n +2\right)}{1221 \left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{\left(3617 n^{3}+98316 n^{2}+572425 n +929190\right) a \! \left(n +3\right)}{1221 \left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{\left(223387 n^{3}+2190678 n^{2}+6755264 n +6261150\right) a \! \left(n +4\right)}{2442 \left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{\left(418159 n^{3}+5275516 n^{2}+21635794 n +28540429\right) a \! \left(n +5\right)}{3256 \left(n +9\right) \left(n +8\right) \left(n +7\right)}+\frac{\left(3441661 n^{3}+55226586 n^{2}+294989687 n +525100746\right) a \! \left(n +6\right)}{39072 \left(n +9\right) \left(n +8\right) \left(n +7\right)}-\frac{\left(134687 n^{2}+1707842 n +5457366\right) a \! \left(n +7\right)}{3256 \left(n +8\right) \left(n +9\right)}+\frac{3 \left(12405 n +89393\right) a \! \left(n +8\right)}{3256 \left(n +9\right)}, \quad n \geq 9\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 147 rules.

Found on January 23, 2022.

Finding the specification took 128 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_{6}\! \left(x , 1\right)\\ F_{6}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{11}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= y x\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{20}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{17}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{13}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\ F_{22}\! \left(x \right) &= 0\\ F_{23}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{22}\! \left(x \right)+F_{27}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)\\ F_{30}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{22}\! \left(x \right)+F_{35}\! \left(x , y\right)+F_{36}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= 0\\ F_{36}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{37}\! \left(x , y\right)\\ F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{38}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\ F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{41}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{21}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{43}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right)+F_{45}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{40}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{48}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{49}\! \left(x , y\right)+F_{50}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{5}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{143}\! \left(x , y\right) F_{52}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{53}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{54}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{22}\! \left(x \right)+F_{55}\! \left(x , y\right)+F_{83}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{56}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{57}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{58}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{59}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{60}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)+F_{64}\! \left(x , y\right)\\ F_{61}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{62}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{63}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= y^{2} x^{2}\\ F_{64}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{65}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{58}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)+F_{74}\! \left(x , y\right)\\ F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{70}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)+F_{71}\! \left(x , y\right)\\ F_{71}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)+F_{72}\! \left(x , y\right)\\ F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right)\\ F_{73}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{68}\! \left(x , y\right)\\ F_{74}\! \left(x , y\right) &= F_{75}\! \left(x , y\right)\\ F_{75}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{76}\! \left(x , y\right)\\ F_{76}\! \left(x , y\right) &= F_{77}\! \left(x , y\right)+F_{80}\! \left(x , y\right)\\ F_{77}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{78}\! \left(x , y\right)\\ F_{78}\! \left(x , y\right) &= F_{79}\! \left(x , y\right)\\ F_{79}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{80}\! \left(x , y\right) &= F_{74}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\ F_{81}\! \left(x , y\right) &= F_{82}\! \left(x , y\right)\\ F_{82}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{74}\! \left(x , y\right)\\ F_{83}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{84}\! \left(x , y\right)\\ F_{84}\! \left(x , y\right) &= F_{85}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\ F_{85}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{86}\! \left(x , y\right)\\ F_{86}\! \left(x , y\right) &= F_{22}\! \left(x \right)+F_{87}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\ F_{87}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{88}\! \left(x , y\right)\\ F_{88}\! \left(x , y\right) &= F_{89}\! \left(x , y\right)+F_{90}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{62}\! \left(x , y\right)\\ F_{90}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{78}\! \left(x , y\right)\\ F_{91}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{92}\! \left(x , y\right)\\ F_{92}\! \left(x , y\right) &= F_{93}\! \left(x , y\right)+F_{95}\! \left(x , y\right)\\ F_{93}\! \left(x , y\right) &= F_{13}\! \left(x , y\right)+F_{94}\! \left(x , y\right)\\ F_{94}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{22}\! \left(x \right)+F_{87}\! \left(x , y\right)\\ F_{95}\! \left(x , y\right) &= F_{86}\! \left(x , y\right)+F_{96}\! \left(x , y\right)\\ F_{96}\! \left(x , y\right) &= F_{97}\! \left(x , y\right)\\ F_{97}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{86}\! \left(x , y\right)\\ F_{98}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\ F_{99}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{118}\! \left(x , y\right)+F_{139}\! \left(x , y\right)+F_{22}\! \left(x \right)\\ F_{100}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{101}\! \left(x , y\right)\\ F_{101}\! \left(x , y\right) &= F_{102}\! \left(x , y\right)+F_{108}\! \left(x , y\right)\\ F_{102}\! \left(x , y\right) &= F_{103}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\ F_{103}\! \left(x , y\right) &= 2 F_{22}\! \left(x \right)+F_{104}\! \left(x , y\right)+F_{66}\! \left(x , y\right)\\ F_{104}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{105}\! \left(x , y\right)\\ F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)+F_{107}\! \left(x , y\right)\\ F_{106}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)\\ F_{107}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)\\ F_{108}\! \left(x , y\right) &= F_{109}\! \left(x , y\right)+F_{113}\! \left(x , y\right)\\ F_{109}\! \left(x , y\right) &= 2 F_{22}\! \left(x \right)+F_{110}\! \left(x , y\right)+F_{73}\! \left(x , y\right)\\ F_{110}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{111}\! \left(x , y\right)\\ F_{111}\! \left(x , y\right) &= F_{112}\! \left(x , y\right)+F_{30}\! \left(x , y\right)\\ F_{112}\! \left(x , y\right) &= F_{72}\! \left(x , y\right)\\ F_{113}\! \left(x , y\right) &= 3 F_{22}\! \left(x \right)+F_{114}\! \left(x , y\right)+F_{82}\! \left(x , y\right)\\ F_{114}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{115}\! \left(x , y\right)\\ F_{115}\! \left(x , y\right) &= F_{116}\! \left(x , y\right)+F_{117}\! \left(x , y\right)\\ F_{116}\! \left(x , y\right) &= F_{78}\! \left(x , y\right)\\ F_{117}\! \left(x , y\right) &= F_{81}\! \left(x , y\right)\\ F_{118}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{119}\! \left(x , y\right)\\ F_{119}\! \left(x , y\right) &= F_{120}\! \left(x , y\right)+F_{127}\! \left(x , y\right)\\ F_{120}\! \left(x , y\right) &= F_{121}\! \left(x , y\right)+F_{94}\! \left(x , y\right)\\ F_{121}\! \left(x , y\right) &= F_{122}\! \left(x , y\right)+F_{123}\! \left(x , y\right)+F_{22}\! \left(x \right)+F_{97}\! \left(x , y\right)\\ F_{122}\! \left(x , y\right) &= 0\\ F_{123}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{124}\! \left(x , y\right)\\ F_{124}\! \left(x , y\right) &= F_{125}\! \left(x , y\right)+F_{126}\! \left(x , y\right)\\ F_{125}\! \left(x , y\right) &= F_{94}\! \left(x , y\right)\\ F_{126}\! \left(x , y\right) &= F_{96}\! \left(x , y\right)\\ F_{127}\! \left(x , y\right) &= F_{128}\! \left(x , y\right)+F_{130}\! \left(x , y\right)\\ F_{128}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{118}\! \left(x , y\right)+F_{129}\! \left(x , y\right)+F_{22}\! \left(x \right)\\ F_{129}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{54}\! \left(x , y\right)\\ F_{130}\! \left(x , y\right) &= F_{131}\! \left(x , y\right)+F_{132}\! \left(x , y\right)+F_{133}\! \left(x , y\right)+F_{138}\! \left(x , y\right)+F_{22}\! \left(x \right)\\ F_{131}\! \left(x , y\right) &= 0\\ F_{132}\! \left(x , y\right) &= 0\\ F_{133}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{134}\! \left(x , y\right)\\ F_{134}\! \left(x , y\right) &= F_{135}\! \left(x , y\right)+F_{136}\! \left(x , y\right)\\ F_{135}\! \left(x , y\right) &= F_{128}\! \left(x , y\right)\\ F_{136}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)\\ F_{137}\! \left(x , y\right) &= F_{138}\! \left(x , y\right)\\ F_{138}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{99}\! \left(x , y\right)\\ F_{139}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{140}\! \left(x , y\right)\\ F_{140}\! \left(x , y\right) &= F_{141}\! \left(x , y\right)+F_{142}\! \left(x , y\right)\\ F_{141}\! \left(x , y\right) &= F_{128}\! \left(x , y\right)+F_{54}\! \left(x , y\right)\\ F_{142}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\ F_{143}\! \left(x , y\right) &= -\frac{-y F_{144}\! \left(x , y\right)+F_{144}\! \left(x , 1\right)}{-1+y}\\ F_{145}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{144}\! \left(x , y\right)\\ F_{145}\! \left(x , y\right) &= F_{146}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{146}\! \left(x , y\right)\\ \end{align*}\)

This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 34 rules.

Found on January 22, 2022.

Finding the specification took 31 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_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\ F_{5}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{6}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{7}\! \left(x , y\right)\\ F_{7}\! \left(x , y\right) &= -\frac{-y F_{5}\! \left(x , y\right)+F_{5}\! \left(x , 1\right)}{-1+y}\\ F_{8}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= y x\\ F_{10}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x , y\right)+F_{28}\! \left(x , y\right)+F_{32}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , 1, y\right)\\ F_{12}\! \left(x , y , z\right) &= F_{13}\! \left(x , y z , z\right)\\ F_{13}\! \left(x , y , z\right) &= F_{14}\! \left(x , y , z\right) F_{3}\! \left(x \right)\\ F_{14}\! \left(x , y , z\right) &= -\frac{-y F_{15}\! \left(x , y , z\right)+F_{15}\! \left(x , 1, z\right)}{-1+y}\\ F_{15}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x , y , z\right)+F_{16}\! \left(x , y , z\right)+F_{31}\! \left(x , z , y\right)\\ F_{16}\! \left(x , y , z\right) &= F_{17}\! \left(x , y , z\right) F_{9}\! \left(x , y\right)\\ F_{17}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y , z\right)+F_{21}\! \left(x , y , z\right)+F_{24}\! \left(x , y , z\right)+F_{30}\! \left(x , z , y\right)\\ F_{18}\! \left(x , y , z\right) &= F_{19}\! \left(x , y , z\right) F_{3}\! \left(x \right)\\ F_{19}\! \left(x , y , z\right) &= \frac{y F_{20}\! \left(x , y , 1\right)-z F_{20}\! \left(x , y , \frac{z}{y}\right)}{-z +y}\\ F_{20}\! \left(x , y , z\right) &= F_{14}\! \left(x , y , y z \right)\\ F_{21}\! \left(x , y , z\right) &= F_{22}\! \left(x , y , z\right) F_{9}\! \left(x , y\right)\\ F_{22}\! \left(x , y , z\right) &= \frac{y F_{23}\! \left(x , y , 1\right)-z F_{23}\! \left(x , y , \frac{z}{y}\right)}{-z +y}\\ F_{23}\! \left(x , y , z\right) &= F_{17}\! \left(x , y , y z \right)\\ F_{24}\! \left(x , y , z\right) &= F_{25}\! \left(x , y , z\right) F_{9}\! \left(x , y\right)\\ F_{25}\! \left(x , y , z\right) &= \frac{y F_{26}\! \left(x , y , 1\right)-z F_{26}\! \left(x , y , \frac{z}{y}\right)}{-z +y}\\ F_{26}\! \left(x , y , z\right) &= F_{27}\! \left(x , y , y z \right)\\ F_{27}\! \left(x , y , z\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y , z\right)+F_{21}\! \left(x , y , z\right)+F_{28}\! \left(x , y\right)+F_{30}\! \left(x , z , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{26}\! \left(x , y , 1\right)\\ F_{30}\! \left(x , y , z\right) &= F_{10}\! \left(x , z\right) F_{9}\! \left(x , y\right)\\ F_{31}\! \left(x , y , z\right) &= F_{5}\! \left(x , z\right) F_{9}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , 1, y\right)\\ F_{33}\! \left(x , y , z\right) &= F_{16}\! \left(x , y z , z\right)\\ \end{align*}\)