Av(12345, 12354, 12453, 13245, 13254, 23145, 23154)
Counting Sequence
1, 1, 2, 6, 24, 113, 581, 3149, 17688, 102001, 600303, 3590921, 21768532, 133438243, 825696844, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) F \left(x
\right)^{4}+\left(-x +2\right) F \left(x
\right)^{3}+\left(-x -1\right) F \left(x
\right)^{2}+F \! \left(x \right)-1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 113\)
\(\displaystyle a \! \left(6\right) = 581\)
\(\displaystyle a \! \left(7\right) = 3149\)
\(\displaystyle a \! \left(8\right) = 17688\)
\(\displaystyle a \! \left(9\right) = 102001\)
\(\displaystyle a \! \left(10\right) = 600303\)
\(\displaystyle a \! \left(11\right) = 3590921\)
\(\displaystyle a \! \left(n +12\right) = \frac{1600 n \left(n +1\right) \left(2 n +1\right) a \! \left(n \right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{40 \left(n +1\right) \left(1814 n^{2}+4123 n +2403\right) a \! \left(n +1\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{30 \left(2178 n^{3}-12697 n^{2}-81881 n -96050\right) a \! \left(n +2\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{2 \left(376384 n^{3}+3722439 n^{2}+12425549 n +13912914\right) a \! \left(n +3\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{12 \left(532131 n^{3}+7201629 n^{2}+32240951 n +47807794\right) a \! \left(n +4\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{4 \left(4749038 n^{3}+75272808 n^{2}+397041913 n +697149048\right) a \! \left(n +5\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{6 \left(4978800 n^{3}+90328153 n^{2}+546634963 n +1103500758\right) a \! \left(n +6\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{\left(28463783 n^{3}+583249173 n^{2}+3987133012 n +9093976596\right) a \! \left(n +7\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{\left(17194963 n^{3}+393821559 n^{2}+3007073090 n +7655825904\right) a \! \left(n +8\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{2 \left(3278329 n^{3}+83200071 n^{2}+703342100 n +1980771132\right) a \! \left(n +9\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{2 \left(750095 n^{2}+13435303 n +60158832\right) a \! \left(n +10\right)}{8897 \left(n +12\right) \left(n +11\right)}+\frac{\left(182641 n +1732056\right) a \! \left(n +11\right)}{8897 n +106764}, \quad n \geq 12\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 24\)
\(\displaystyle a \! \left(5\right) = 113\)
\(\displaystyle a \! \left(6\right) = 581\)
\(\displaystyle a \! \left(7\right) = 3149\)
\(\displaystyle a \! \left(8\right) = 17688\)
\(\displaystyle a \! \left(9\right) = 102001\)
\(\displaystyle a \! \left(10\right) = 600303\)
\(\displaystyle a \! \left(11\right) = 3590921\)
\(\displaystyle a \! \left(n +12\right) = \frac{1600 n \left(n +1\right) \left(2 n +1\right) a \! \left(n \right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{40 \left(n +1\right) \left(1814 n^{2}+4123 n +2403\right) a \! \left(n +1\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{30 \left(2178 n^{3}-12697 n^{2}-81881 n -96050\right) a \! \left(n +2\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{2 \left(376384 n^{3}+3722439 n^{2}+12425549 n +13912914\right) a \! \left(n +3\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{12 \left(532131 n^{3}+7201629 n^{2}+32240951 n +47807794\right) a \! \left(n +4\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{4 \left(4749038 n^{3}+75272808 n^{2}+397041913 n +697149048\right) a \! \left(n +5\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{6 \left(4978800 n^{3}+90328153 n^{2}+546634963 n +1103500758\right) a \! \left(n +6\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{\left(28463783 n^{3}+583249173 n^{2}+3987133012 n +9093976596\right) a \! \left(n +7\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{\left(17194963 n^{3}+393821559 n^{2}+3007073090 n +7655825904\right) a \! \left(n +8\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}+\frac{2 \left(3278329 n^{3}+83200071 n^{2}+703342100 n +1980771132\right) a \! \left(n +9\right)}{8897 \left(n +10\right) \left(n +12\right) \left(n +11\right)}-\frac{2 \left(750095 n^{2}+13435303 n +60158832\right) a \! \left(n +10\right)}{8897 \left(n +12\right) \left(n +11\right)}+\frac{\left(182641 n +1732056\right) a \! \left(n +11\right)}{8897 n +106764}, \quad n \geq 12\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 134 rules.
Found on January 23, 2022.Finding the specification took 145 seconds.
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Copy 134 equations to clipboard:
\(\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_{23}\! \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_{20}\! \left(x , y\right)\\
F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{22}\! \left(x , y\right)\\
F_{21}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)\\
F_{23}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{24}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\
F_{25}\! \left(x \right) &= 0\\
F_{26}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\
F_{27}\! \left(x , y\right) &= F_{28}\! \left(x , y\right)+F_{31}\! \left(x , y\right)\\
F_{28}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{29}\! \left(x , y\right)\\
F_{29}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)+F_{25}\! \left(x \right)+F_{30}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{13}\! \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_{25}\! \left(x \right)+F_{26}\! \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_{25}\! \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_{36}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{24}\! \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_{24}\! \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_{130}\! \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_{54}\! \left(x , y\right)+F_{63}\! \left(x , y\right)\\
F_{54}\! \left(x , y\right) &= F_{55}\! \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_{60}\! \left(x , y\right)\\
F_{57}\! \left(x , y\right) &= F_{1}\! \left(x \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_{57}\! \left(x , y\right)\\
F_{60}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{61}\! \left(x , y\right)\\
F_{61}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)\\
F_{62}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{60}\! \left(x , y\right)\\
F_{63}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{64}\! \left(x , y\right)+F_{74}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{65}\! \left(x , y\right)\\
F_{65}\! \left(x , y\right) &= F_{66}\! \left(x , y\right)+F_{70}\! \left(x , y\right)\\
F_{66}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\
F_{67}\! \left(x , y\right) &= F_{68}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{69}\! \left(x , y\right)\\
F_{69}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\
F_{70}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)+F_{71}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= F_{72}\! \left(x , y\right)\\
F_{72}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{73}\! \left(x , y\right)\\
F_{73}\! \left(x , y\right) &= F_{61}\! \left(x , y\right)+F_{71}\! \left(x , y\right)\\
F_{74}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{75}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{76}\! \left(x , y\right)+F_{98}\! \left(x , y\right)\\
F_{76}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{77}\! \left(x , y\right)\\
F_{77}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{78}\! \left(x , y\right)+F_{88}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{79}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= F_{80}\! \left(x , y\right)+F_{83}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= F_{10}\! \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_{58}\! \left(x , y\right)\\
F_{83}\! \left(x , y\right) &= F_{84}\! \left(x , y\right)+F_{86}\! \left(x , y\right)\\
F_{84}\! \left(x , y\right) &= F_{85}\! \left(x , y\right)\\
F_{85}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{54}\! \left(x , y\right)\\
F_{86}\! \left(x , y\right) &= F_{87}\! \left(x , y\right)\\
F_{87}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{61}\! \left(x , y\right)\\
F_{88}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{89}\! \left(x , y\right)\\
F_{89}\! \left(x , y\right) &= F_{90}\! \left(x , y\right)+F_{92}\! \left(x , y\right)\\
F_{90}\! \left(x , y\right) &= F_{54}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\
F_{91}\! \left(x , y\right) &= F_{25}\! \left(x \right)+F_{78}\! \left(x , y\right)+F_{85}\! \left(x , y\right)\\
F_{92}\! \left(x , y\right) &= F_{77}\! \left(x , y\right)+F_{93}\! \left(x , y\right)\\
F_{93}\! \left(x , y\right) &= F_{94}\! \left(x , y\right)\\
F_{94}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{95}\! \left(x , y\right)\\
F_{95}\! \left(x , y\right) &= F_{96}\! \left(x , y\right)+F_{97}\! \left(x , y\right)\\
F_{96}\! \left(x , y\right) &= F_{91}\! \left(x , y\right)\\
F_{97}\! \left(x , y\right) &= F_{93}\! \left(x , y\right)\\
F_{98}\! \left(x , y\right) &= F_{63}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\
F_{99}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{108}\! \left(x , y\right)+F_{126}\! \left(x , y\right)+F_{25}\! \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_{105}\! \left(x , y\right)\\
F_{102}\! \left(x , y\right) &= F_{103}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\
F_{103}\! \left(x , y\right) &= F_{104}\! \left(x , y\right)\\
F_{104}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{67}\! \left(x , y\right)\\
F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)+F_{86}\! \left(x , y\right)\\
F_{106}\! \left(x , y\right) &= F_{107}\! \left(x , y\right)\\
F_{107}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{71}\! \left(x , y\right)\\
F_{108}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{109}\! \left(x , y\right)\\
F_{109}\! \left(x , y\right) &= F_{110}\! \left(x , y\right)+F_{114}\! \left(x , y\right)\\
F_{110}\! \left(x , y\right) &= F_{111}\! \left(x , y\right)+F_{91}\! \left(x , y\right)\\
F_{111}\! \left(x , y\right) &= F_{112}\! \left(x , y\right)+F_{113}\! \left(x , y\right)+F_{25}\! \left(x \right)+F_{94}\! \left(x , y\right)\\
F_{112}\! \left(x , y\right) &= 0\\
F_{113}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{77}\! \left(x , y\right)\\
F_{114}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{117}\! \left(x , y\right)\\
F_{115}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{108}\! \left(x , y\right)+F_{116}\! \left(x , y\right)+F_{25}\! \left(x \right)\\
F_{116}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{63}\! \left(x , y\right)\\
F_{117}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{119}\! \left(x , y\right)+F_{120}\! \left(x , y\right)+F_{125}\! \left(x , y\right)+F_{25}\! \left(x \right)\\
F_{118}\! \left(x , y\right) &= 0\\
F_{119}\! \left(x , y\right) &= 0\\
F_{120}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{121}\! \left(x , y\right)\\
F_{121}\! \left(x , y\right) &= F_{122}\! \left(x , y\right)+F_{123}\! \left(x , y\right)\\
F_{122}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)\\
F_{123}\! \left(x , y\right) &= F_{124}\! \left(x , y\right)\\
F_{124}\! \left(x , y\right) &= F_{120}\! \left(x , y\right)\\
F_{125}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{99}\! \left(x , y\right)\\
F_{126}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{127}\! \left(x , y\right)\\
F_{127}\! \left(x , y\right) &= F_{128}\! \left(x , y\right)+F_{129}\! \left(x , y\right)\\
F_{128}\! \left(x , y\right) &= F_{115}\! \left(x , y\right)+F_{63}\! \left(x , y\right)\\
F_{129}\! \left(x , y\right) &= F_{124}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\
F_{130}\! \left(x , y\right) &= -\frac{-y F_{131}\! \left(x , y\right)+F_{131}\! \left(x , 1\right)}{-1+y}\\
F_{132}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{131}\! \left(x , y\right)\\
F_{132}\! \left(x , y\right) &= F_{133}\! \left(x , y\right)\\
F_{6}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{133}\! \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 25 seconds.
Copy 34 equations to clipboard:
\(\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_{30}\! \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_{28}\! \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_{29}\! \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_{24}\! \left(x , y , z\right)+F_{28}\! \left(x , z , y\right)\\
F_{28}\! \left(x , y , z\right) &= F_{5}\! \left(x , z\right) F_{9}\! \left(x , y\right)\\
F_{29}\! \left(x , y , z\right) &= F_{10}\! \left(x , z\right) F_{9}\! \left(x , y\right)\\
F_{30}\! \left(x , y\right) &= F_{31}\! \left(x , 1, y\right)\\
F_{31}\! \left(x , y , z\right) &= F_{16}\! \left(x , y z , z\right)\\
F_{32}\! \left(x , y\right) &= F_{33}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{26}\! \left(x , y , 1\right)\\
\end{align*}\)