Av(1243, 2413, 2431, 4132)
Generating Function
\(\displaystyle \frac{\left(-3 x^{2}+3 x -1\right) \sqrt{1-4 x}-4 x^{4}+4 x^{3}+x^{2}-3 x +1}{2 x \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 214, 693, 2271, 7575, 25732, 88853, 311091, 1101821, 3940194, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{8} F \left(x
\right)^{2}+\left(2 x -1\right) \left(2 x^{3}-x^{2}-x +1\right) \left(x -1\right)^{4} F \! \left(x \right)+4 x^{7}-8 x^{6}+2 x^{5}+17 x^{4}-28 x^{3}+20 x^{2}-7 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 214\)
\(\displaystyle a \! \left(n +4\right) = -\frac{6 \left(2 n +5\right) a \! \left(n \right)}{n +5}+\frac{3 \left(9 n +20\right) a \! \left(n +1\right)}{n +5}-\frac{\left(22 n +65\right) a \! \left(n +2\right)}{n +5}+\frac{\left(31+8 n \right) a \! \left(n +3\right)}{n +5}+\frac{\left(n -2\right) \left(n -7\right)}{n +5}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 214\)
\(\displaystyle a \! \left(n +4\right) = -\frac{6 \left(2 n +5\right) a \! \left(n \right)}{n +5}+\frac{3 \left(9 n +20\right) a \! \left(n +1\right)}{n +5}-\frac{\left(22 n +65\right) a \! \left(n +2\right)}{n +5}+\frac{\left(31+8 n \right) a \! \left(n +3\right)}{n +5}+\frac{\left(n -2\right) \left(n -7\right)}{n +5}, \quad n \geq 7\)
This specification was found using the strategy pack "Point Placements" and has 27 rules.
Found on July 23, 2021.Finding the specification took 4 seconds.
Copy 27 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{5} \left(x \right)^{3}\\
F_{23}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{6}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{6}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)