Av(1324, 2413, 2431)
Generating Function
\(\displaystyle \frac{\left(-x^{2}+3 x -1\right) \sqrt{1-4 x}-4 x^{3}+7 x^{2}-5 x +1}{2 \left(x^{2}-3 x +1\right) x^{2}}\)
Counting Sequence
1, 1, 2, 6, 21, 77, 285, 1053, 3875, 14212, 52021, 190301, 696532, 2553047, 9377034, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x^{2}-3 x +1\right)^{2} F \left(x
\right)^{2}+\left(x^{2}-3 x +1\right) \left(4 x^{3}-7 x^{2}+5 x -1\right) F \! \left(x \right)+4 x^{4}-13 x^{3}+16 x^{2}-7 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +5\right) = \frac{2 \left(3+2 n \right) a \! \left(n \right)}{7+n}+\frac{2 \left(89+25 n \right) a \! \left(2+n \right)}{7+n}-\frac{\left(63+25 n \right) a \! \left(n +1\right)}{7+n}-\frac{\left(163+35 n \right) a \! \left(n +3\right)}{7+n}+\frac{2 \left(29+5 n \right) a \! \left(n +4\right)}{7+n}, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +5\right) = \frac{2 \left(3+2 n \right) a \! \left(n \right)}{7+n}+\frac{2 \left(89+25 n \right) a \! \left(2+n \right)}{7+n}-\frac{\left(63+25 n \right) a \! \left(n +1\right)}{7+n}-\frac{\left(163+35 n \right) a \! \left(n +3\right)}{7+n}+\frac{2 \left(29+5 n \right) a \! \left(n +4\right)}{7+n}, \quad n \geq 5\)
This specification was found using the strategy pack "Point Placements" and has 23 rules.
Found on July 23, 2021.Finding the specification took 5 seconds.
Copy 23 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\
\end{align*}\)