Av(1234, 1243, 3412)
Generating Function
\(\displaystyle -\frac{12 x^{7}-36 x^{6}+78 x^{5}-98 x^{4}+75 x^{3}-35 x^{2}+9 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{6}}\)
Counting Sequence
1, 1, 2, 6, 21, 73, 238, 718, 2013, 5301, 13266, 31886, 74269, 168841, 376750, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)+12 x^{7}-36 x^{6}+78 x^{5}-98 x^{4}+75 x^{3}-35 x^{2}+9 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 238\)
\(\displaystyle a \! \left(7\right) = 718\)
\(\displaystyle a \! \left(n +2\right) = -\frac{n^{5}}{30}+\frac{5 n^{4}}{12}-\frac{4 n^{3}}{3}+\frac{37 n^{2}}{12}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{32 n}{15}+2, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 238\)
\(\displaystyle a \! \left(7\right) = 718\)
\(\displaystyle a \! \left(n +2\right) = -\frac{n^{5}}{30}+\frac{5 n^{4}}{12}-\frac{4 n^{3}}{3}+\frac{37 n^{2}}{12}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{32 n}{15}+2, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle 5+2^{n +1} n -4 \,2^{n}-\frac{2 n^{3}}{3}+\frac{5 n^{2}}{12}+\frac{n}{5}+\frac{n^{4}}{12}-\frac{n^{5}}{30}\)
This specification was found using the strategy pack "Insertion Point Placements" and has 142 rules.
Found on January 17, 2022.Finding the specification took 33 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_{22}\! \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_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{13}\! \left(x \right) &= 0\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \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_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{31}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{31}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{36}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{37}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{46}\! \left(x \right)+F_{47}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{46}\! \left(x \right) &= 0\\
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_{50}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{19}\! \left(x \right) F_{7}\! \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_{60}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{19}\! \left(x \right) F_{21}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{19}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{19}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{76}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{18}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{86}\! \left(x \right)+F_{88}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{85}\! \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_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{19} \left(x \right)^{2} F_{21}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{21} \left(x \right)^{2} F_{64}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{106}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \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_{117}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{111}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{116}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{112}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{123}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{124}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{128}\! \left(x \right)\\
F_{128}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{129}\! \left(x \right)+F_{131}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{127}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{135}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{134}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{137}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{113}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{140}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{19}\! \left(x \right) F_{21}\! \left(x \right)\\
\end{align*}\)