Av(1243, 2314, 3412)
Generating Function
\(\displaystyle -\frac{17 x^{8}-80 x^{7}+192 x^{6}-274 x^{5}+248 x^{4}-145 x^{3}+53 x^{2}-11 x +1}{\left(2 x -1\right)^{3} \left(-1+x \right)^{6}}\)
Counting Sequence
1, 1, 2, 6, 21, 73, 238, 722, 2054, 5541, 14323, 35788, 87043, 207201, 484772, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{3} \left(-1+x \right)^{6} F \! \left(x \right)+17 x^{8}-80 x^{7}+192 x^{6}-274 x^{5}+248 x^{4}-145 x^{3}+53 x^{2}-11 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 238\)
\(\displaystyle a \! \left(7\right) = 722\)
\(\displaystyle a \! \left(8\right) = 2054\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{5}}{120}-\frac{n^{4}}{6}+\frac{19 n^{3}}{24}-\frac{11 n^{2}}{6}+8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)+\frac{21 n}{5}-2, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 238\)
\(\displaystyle a \! \left(7\right) = 722\)
\(\displaystyle a \! \left(8\right) = 2054\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{5}}{120}-\frac{n^{4}}{6}+\frac{19 n^{3}}{24}-\frac{11 n^{2}}{6}+8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)+\frac{21 n}{5}-2, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \frac{\left(15 n^{2}+105 n -840\right) 2^{n}}{120}-\frac{n^{5}}{120}+\frac{n^{4}}{24}-\frac{n^{3}}{24}+\frac{35 n^{2}}{24}+\frac{71 n}{20}+8\)
This specification was found using the strategy pack "Point And Row Placements" and has 96 rules.
Found on July 23, 2021.Finding the specification took 11 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_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{7}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{6}\! \left(x \right) &= 0\\
F_{7}\! \left(x \right) &= F_{17}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{32}\! \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_{13}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= x\\
F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{17}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{17}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{17}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{30}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{17}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{29}\! \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_{17}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{17}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{46}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{17}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{17}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{17}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= 2 F_{6}\! \left(x \right)+F_{54}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{17}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{17}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{17}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{67}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{17}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{17}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{17}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{75}\! \left(x \right) &= 2 F_{6}\! \left(x \right)+F_{76}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{17}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{17}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{79}\! \left(x \right) &= 2 F_{6}\! \left(x \right)+F_{80}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{17}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{17}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{17}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{11}\! \left(x \right) F_{17}\! \left(x \right) F_{28}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{93}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{17}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{17}\! \left(x \right) F_{91}\! \left(x \right)\\
\end{align*}\)