Av(1324, 1342, 3124, 3412)
Generating Function
\(\displaystyle -\frac{11 x^{5}-29 x^{4}+38 x^{3}-25 x^{2}+8 x -1}{\left(2 x -1\right) \left(3 x -1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 202, 608, 1800, 5299, 15606, 46098, 136652, 406389, 1211650, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(3 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+11 x^{5}-29 x^{4}+38 x^{3}-25 x^{2}+8 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(n +2\right) = -\frac{n^{3}}{3}+\frac{3 n^{2}}{2}+5 a \! \left(n +1\right)-6 a \! \left(n \right)-\frac{13 n}{6}+3, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(n +2\right) = -\frac{n^{3}}{3}+\frac{3 n^{2}}{2}+5 a \! \left(n +1\right)-6 a \! \left(n \right)-\frac{13 n}{6}+3, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle -\frac{n^{3}}{6}-\frac{4 n}{3}-\frac{1}{4}+2^{n}+\frac{3^{n}}{4}\)
This specification was found using the strategy pack "Point Placements" and has 57 rules.
Found on July 23, 2021.Finding the specification took 6 seconds.
Copy 57 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_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{33}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{39}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{44}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{16}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{54}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{52}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)