Av(1234, 1243, 1342, 3412)
Generating Function
\(\displaystyle -\frac{9 x^{7}-35 x^{6}+77 x^{5}-97 x^{4}+75 x^{3}-35 x^{2}+9 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{6}}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 190, 524, 1356, 3330, 7842, 17870, 39700, 86508, 185782, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{2} \left(x -1\right)^{6} F \! \left(x \right)+9 x^{7}-35 x^{6}+77 x^{5}-97 x^{4}+75 x^{3}-35 x^{2}+9 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 190\)
\(\displaystyle a \! \left(7\right) = 524\)
\(\displaystyle a \! \left(n +2\right) = -\frac{n^{5}}{60}+\frac{n^{4}}{6}-\frac{7 n^{3}}{12}+\frac{11 n^{2}}{6}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{7 n}{5}+2, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 190\)
\(\displaystyle a \! \left(7\right) = 524\)
\(\displaystyle a \! \left(n +2\right) = -\frac{n^{5}}{60}+\frac{n^{4}}{6}-\frac{7 n^{3}}{12}+\frac{11 n^{2}}{6}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{7 n}{5}+2, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle -\frac{7 n^{3}}{12}-n^{2}-\frac{22 n}{5}-4-\frac{n^{5}}{60}+2^{-1+n} n +5 \,2^{n}\)
This specification was found using the strategy pack "Insertion Row And Col Placements" and has 81 rules.
Found on July 23, 2021.Finding the specification took 9 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_{15}\! \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_{31}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{15}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{12}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{22}\! \left(x \right) &= 0\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{15}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{15}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{32}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{15}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{15}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{15}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{13}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{13}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{15}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{13}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= 2 F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{13}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{13}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{13}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{64}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{15}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{15}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{15}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{15}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right) F_{15}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{20}\! \left(x \right)\\
\end{align*}\)