Av(1234, 1243, 2314, 2341, 4213)
Generating Function
\(\displaystyle -\frac{2 x^{7}-3 x^{6}-2 x^{4}-4 x^{3}+9 x^{2}-5 x +1}{\left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 53, 131, 300, 654, 1382, 2863, 5857, 11887, 24004, 48318, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{2}+x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+2 x^{7}-3 x^{6}-2 x^{4}-4 x^{3}+9 x^{2}-5 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 131\)
\(\displaystyle a \! \left(7\right) = 300\)
\(\displaystyle a \! \left(n +3\right) = -n^{2}-2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+2 n +6, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 131\)
\(\displaystyle a \! \left(7\right) = 300\)
\(\displaystyle a \! \left(n +3\right) = -n^{2}-2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+2 n +6, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(\sqrt{5}-5\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}+\frac{\left(-\sqrt{5}-5\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{10}-n^{2}-2 n +3 \,2^{n} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 75 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 75 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_{24}\! \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_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{36}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{45}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \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) &= F_{16}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{64}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{63}\! \left(x \right)\\
\end{align*}\)