Av(1342, 1423, 2143, 2431)
Generating Function
\(\displaystyle \frac{\left(x^{2}+1\right) \left(x -1\right)^{2} \sqrt{1-4 x}+2 x^{5}-5 x^{4}+10 x^{3}-8 x^{2}+4 x -1}{2 x^{2} \left(x^{2}-2 x +2\right) \left(x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 218, 730, 2485, 8587, 30055, 106341, 379752, 1366964, 4954734, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x^{2}-2 x +2\right) \left(x -1\right)^{2} F \left(x
\right)^{2}-\left(x -1\right) \left(2 x -1\right) \left(x^{4}-2 x^{3}+4 x^{2}-2 x +1\right) F \! \left(x \right)+x^{6}-2 x^{5}+6 x^{4}-8 x^{3}+7 x^{2}-4 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 218\)
\(\displaystyle a \! \left(7\right) = 730\)
\(\displaystyle a \! \left(n +6\right) = -\frac{\left(1+2 n \right) a \! \left(n \right)}{8+n}-\frac{\left(50+23 n \right) a \! \left(2+n \right)}{2 \left(8+n \right)}+\frac{\left(16+13 n \right) a \! \left(n +1\right)}{16+2 n}+\frac{25 \left(n +4\right) a \! \left(n +3\right)}{2 \left(8+n \right)}-\frac{\left(116+21 n \right) a \! \left(n +4\right)}{2 \left(8+n \right)}+\frac{2 \left(20+3 n \right) a \! \left(n +5\right)}{8+n}+\frac{6}{8+n}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 218\)
\(\displaystyle a \! \left(7\right) = 730\)
\(\displaystyle a \! \left(n +6\right) = -\frac{\left(1+2 n \right) a \! \left(n \right)}{8+n}-\frac{\left(50+23 n \right) a \! \left(2+n \right)}{2 \left(8+n \right)}+\frac{\left(16+13 n \right) a \! \left(n +1\right)}{16+2 n}+\frac{25 \left(n +4\right) a \! \left(n +3\right)}{2 \left(8+n \right)}-\frac{\left(116+21 n \right) a \! \left(n +4\right)}{2 \left(8+n \right)}+\frac{2 \left(20+3 n \right) a \! \left(n +5\right)}{8+n}+\frac{6}{8+n}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements" and has 20 rules.
Found on July 23, 2021.Finding the specification took 2 seconds.
Copy 20 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{7} \left(x \right)^{2} F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= x\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{18}\! \left(x \right)\\
\end{align*}\)