Av(1324, 1432, 3412)
Generating Function
\(\displaystyle -\frac{5 x^{4}-11 x^{3}+13 x^{2}-6 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 21, 72, 232, 707, 2066, 5858, 16257, 44428, 120076, 321919, 857942, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+5 x^{4}-11 x^{3}+13 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+2 n +1, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +3\right) = 2 a \! \left(n \right)-7 a \! \left(n +1\right)+5 a \! \left(n +2\right)+2 n +1, \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \frac{\left(-\sqrt{5}+15\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(\sqrt{5}+15\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{10}+2 n -3 \,2^{n}+1\)
This specification was found using the strategy pack "Row Placements" and has 78 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_{23}\! \left(x \right) F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{23}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x \right)+F_{6}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{23}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{27}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{12}\! \left(x \right)+F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{23}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{18}\! \left(x \right) F_{23}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{23}\! \left(x \right) &= x\\
F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{23}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{18}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{24} \left(x \right)^{2} F_{23}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{21}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{11}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{23}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{23}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)+F_{53}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{23}\! \left(x \right) F_{24}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{41}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{23}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{23}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= 2 F_{27}\! \left(x \right)+F_{16}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{23}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{23}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{23}\! \left(x \right) F_{24}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{23}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{55}\! \left(x \right)+F_{60}\! \left(x \right)+F_{72}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{23}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)+F_{57}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{23}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{23}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{23}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{23}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{23}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{38}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{23}\! \left(x \right) F_{24}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{23}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{23}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{23}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{23}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{59}\! \left(x \right)+F_{62}\! \left(x \right)+F_{64}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{23}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{31}\! \left(x \right)+F_{59}\! \left(x \right)\\
\end{align*}\)