Av(1234, 1342, 2341, 3142, 3241)
Generating Function
\(\displaystyle -\frac{2 x^{6}+5 x^{5}-20 x^{4}+28 x^{3}-20 x^{2}+7 x -1}{\left(2 x -1\right)^{2} \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 19, 58, 166, 446, 1137, 2782, 6596, 15266, 34679, 77634, 171778, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+2 x^{6}+5 x^{5}-20 x^{4}+28 x^{3}-20 x^{2}+7 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 166\)
\(\displaystyle a \! \left(n +2\right) = -\frac{n^{3}}{6}+2 n^{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{23 n}{6}+4, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 166\)
\(\displaystyle a \! \left(n +2\right) = -\frac{n^{3}}{6}+2 n^{2}-4 a \! \left(n \right)+4 a \! \left(n +1\right)-\frac{23 n}{6}+4, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 5+2^{n} n -7 \,2^{-1+n}+\frac{n}{6}+n^{2}-\frac{n^{3}}{6} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 51 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 51 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_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{19}\! \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_{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_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{21}\! \left(x \right) &= 0\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{28}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \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_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{33}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{41}\! \left(x \right)\\
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_{2}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{41}\! \left(x \right)\\
\end{align*}\)