Av(1243, 1324, 2143, 2413, 4132)
Generating Function
\(\displaystyle \frac{\left(x -1\right)^{4} \sqrt{-4 x +1}-5 x^{4}+6 x^{3}-6 x^{2}+4 x -1}{4 x \left(x -1\right)^{2} \left(x -\frac{1}{2}\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 59, 183, 577, 1863, 6163, 20835, 71734, 250713, 887098, 3170859, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \left(x
\right)^{2}+\left(2 x -1\right) \left(5 x^{4}-6 x^{3}+6 x^{2}-4 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{8}-2 x^{7}+15 x^{6}-39 x^{5}+56 x^{4}-50 x^{3}+27 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(n +3\right) = \frac{4 \left(-1+2 n \right) a \! \left(n \right)}{n +4}-\frac{2 \left(10+7 n \right) a \! \left(n +1\right)}{n +4}+\frac{\left(19+7 n \right) a \! \left(n +2\right)}{n +4}-\frac{3 \left(4 n -5\right)}{n +4}, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(n +3\right) = \frac{4 \left(-1+2 n \right) a \! \left(n \right)}{n +4}-\frac{2 \left(10+7 n \right) a \! \left(n +1\right)}{n +4}+\frac{\left(19+7 n \right) a \! \left(n +2\right)}{n +4}-\frac{3 \left(4 n -5\right)}{n +4}, \quad n \geq 6\)
This specification was found using the strategy pack "Point Placements" and has 24 rules.
Found on July 23, 2021.Finding the specification took 4 seconds.
Copy 24 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{14}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21} \left(x \right)^{2} F_{14}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{0}\! \left(x \right) F_{19}\! \left(x \right) F_{21}\! \left(x \right)\\
\end{align*}\)