Av(1324, 2413, 2431, 4132)
Generating Function
\(\displaystyle \frac{\left(x^{3}+2 x^{2}-3 x +1\right) \sqrt{1-4 x}+2 x^{4}-5 x^{3}+3 x -1}{4 \left(x -\frac{1}{2}\right) \left(x -1\right)^{2} x}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 215, 701, 2310, 7727, 26259, 90573, 316559, 1119121, 3995385, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \left(x
\right)^{2}-\left(x^{3}-2 x^{2}-x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)+x^{7}-4 x^{6}+10 x^{5}-18 x^{3}+18 x^{2}-7 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 215\)
\(\displaystyle a \! \left(7\right) = 701\)
\(\displaystyle a \! \left(n +6\right) = \frac{4 \left(1+2 n \right) a \! \left(n \right)}{7+n}+\frac{2 \left(19+n \right) a \! \left(1+n \right)}{7+n}-\frac{\left(166+45 n \right) a \! \left(n +2\right)}{7+n}+\frac{\left(257+63 n \right) a \! \left(n +3\right)}{7+n}-\frac{\left(182+37 n \right) a \! \left(n +4\right)}{7+n}+\frac{\left(59+10 n \right) a \! \left(n +5\right)}{7+n}+\frac{1}{7+n}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 215\)
\(\displaystyle a \! \left(7\right) = 701\)
\(\displaystyle a \! \left(n +6\right) = \frac{4 \left(1+2 n \right) a \! \left(n \right)}{7+n}+\frac{2 \left(19+n \right) a \! \left(1+n \right)}{7+n}-\frac{\left(166+45 n \right) a \! \left(n +2\right)}{7+n}+\frac{\left(257+63 n \right) a \! \left(n +3\right)}{7+n}-\frac{\left(182+37 n \right) a \! \left(n +4\right)}{7+n}+\frac{\left(59+10 n \right) a \! \left(n +5\right)}{7+n}+\frac{1}{7+n}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements" and has 24 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 24 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{20}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{20}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{21}\! \left(x \right) F_{7}\! \left(x \right)\\
\end{align*}\)