Av(1243, 1324, 2413, 2431, 3142, 4132)
Generating Function
\(\displaystyle \frac{\left(-2 x^{3}+5 x^{2}-4 x +1\right) \sqrt{1-4 x}-2 x^{4}+2 x^{3}-5 x^{2}+4 x -1}{4 x \left(x -\frac{1}{2}\right) \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 18, 53, 158, 486, 1550, 5109, 17298, 59799, 210048, 746983, 2682618, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \left(x
\right)^{2}+\left(2 x -1\right) \left(2 x^{4}-2 x^{3}+5 x^{2}-4 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{7}+2 x^{6}-15 x^{5}+37 x^{4}-43 x^{3}+26 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 158\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{4+n}-\frac{4 \left(5 n +8\right) a \! \left(1+n \right)}{4+n}+\frac{2 \left(4 n +11\right) a \! \left(n +2\right)}{4+n}+\frac{n \left(3 n -13\right)}{4+n}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 158\)
\(\displaystyle a \! \left(n +3\right) = \frac{8 \left(2 n +1\right) a \! \left(n \right)}{4+n}-\frac{4 \left(5 n +8\right) a \! \left(1+n \right)}{4+n}+\frac{2 \left(4 n +11\right) a \! \left(n +2\right)}{4+n}+\frac{n \left(3 n -13\right)}{4+n}, \quad n \geq 7\)
This specification was found using the strategy pack "Point Placements" and has 22 rules.
Found on July 23, 2021.Finding the specification took 1 seconds.
Copy 22 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{5} \left(x \right)^{3}\\
F_{13}\! \left(x \right) &= F_{20} \left(x \right)^{2} F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)