Av(1342, 2314, 2413)
Generating Function
\(\displaystyle -\frac{4 \left(x -\frac{1}{2}\right) \left(\left(x -\frac{1}{2}\right) \sqrt{-4 x +1}+x^{3}-4 x^{2}+3 x -\frac{1}{2}\right)}{2 x^{5}-16 x^{4}+44 x^{3}-42 x^{2}+16 x -2}\)
Counting Sequence
1, 1, 2, 6, 21, 77, 286, 1067, 3993, 14992, 56488, 213600, 810449, 3084733, 11774727, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-8 x^{4}+22 x^{3}-21 x^{2}+8 x -1\right) F \left(x
\right)^{2}+\left(2 x -1\right) \left(2 x^{3}-8 x^{2}+6 x -1\right) F \! \left(x \right)+x \left(2 x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 286\)
\(\displaystyle a \! \left(n +7\right) = \frac{4 \left(2 n +5\right) a \! \left(n \right)}{n +7}-\frac{10 \left(7 n +19\right) a \! \left(n +1\right)}{n +7}+\frac{3 \left(75 n +229\right) a \! \left(n +2\right)}{n +7}-\frac{4 \left(77 n +284\right) a \! \left(n +3\right)}{n +7}+\frac{4 \left(53 n +236\right) a \! \left(n +4\right)}{n +7}-\frac{11 \left(7 n +37\right) a \! \left(n +5\right)}{n +7}+\frac{2 \left(7 n +43\right) a \! \left(n +6\right)}{n +7}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 286\)
\(\displaystyle a \! \left(n +7\right) = \frac{4 \left(2 n +5\right) a \! \left(n \right)}{n +7}-\frac{10 \left(7 n +19\right) a \! \left(n +1\right)}{n +7}+\frac{3 \left(75 n +229\right) a \! \left(n +2\right)}{n +7}-\frac{4 \left(77 n +284\right) a \! \left(n +3\right)}{n +7}+\frac{4 \left(53 n +236\right) a \! \left(n +4\right)}{n +7}-\frac{11 \left(7 n +37\right) a \! \left(n +5\right)}{n +7}+\frac{2 \left(7 n +43\right) a \! \left(n +6\right)}{n +7}, \quad n \geq 7\)
This specification was found using the strategy pack "Point Placements" and has 21 rules.
Found on July 23, 2021.Finding the specification took 11 seconds.
Copy 21 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= x\\
F_{8}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{0}\! \left(x \right) F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12} \left(x \right)^{2} F_{7}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{7}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{0}\! \left(x \right) F_{12}\! \left(x \right)\\
\end{align*}\)