Av(1243, 1324, 1342, 1423, 2413, 4132)
Generating Function
\(\displaystyle \frac{\left(2 x^{4}-3 x^{3}+4 x^{2}-3 x +1\right) \sqrt{1-4 x}+2 x^{5}-6 x^{4}+5 x^{3}-4 x^{2}+3 x -1}{2 x \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 18, 54, 165, 520, 1692, 5657, 19322, 67101, 236106, 839681, 3012848, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{6} F \left(x
\right)^{2}-\left(2 x^{5}-6 x^{4}+5 x^{3}-4 x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{9}-2 x^{8}+x^{7}+9 x^{6}-21 x^{5}+27 x^{4}-25 x^{3}+16 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 165\)
\(\displaystyle a \! \left(7\right) = 520\)
\(\displaystyle a \! \left(8\right) = 1692\)
\(\displaystyle a \! \left(9\right) = 5657\)
\(\displaystyle a \! \left(n +6\right) = -\frac{4 \left(-1+2 n \right) a \! \left(n \right)}{7+n}+\frac{22 n a \! \left(1+n \right)}{7+n}-\frac{3 \left(19+11 n \right) a \! \left(n +2\right)}{7+n}+\frac{\left(113+35 n \right) a \! \left(n +3\right)}{7+n}-\frac{\left(105+23 n \right) a \! \left(n +4\right)}{7+n}+\frac{2 \left(23+4 n \right) a \! \left(n +5\right)}{7+n}+\frac{n -7}{7+n}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 54\)
\(\displaystyle a \! \left(6\right) = 165\)
\(\displaystyle a \! \left(7\right) = 520\)
\(\displaystyle a \! \left(8\right) = 1692\)
\(\displaystyle a \! \left(9\right) = 5657\)
\(\displaystyle a \! \left(n +6\right) = -\frac{4 \left(-1+2 n \right) a \! \left(n \right)}{7+n}+\frac{22 n a \! \left(1+n \right)}{7+n}-\frac{3 \left(19+11 n \right) a \! \left(n +2\right)}{7+n}+\frac{\left(113+35 n \right) a \! \left(n +3\right)}{7+n}-\frac{\left(105+23 n \right) a \! \left(n +4\right)}{7+n}+\frac{2 \left(23+4 n \right) a \! \left(n +5\right)}{7+n}+\frac{n -7}{7+n}, \quad n \geq 10\)
This specification was found using the strategy pack "Point Placements" and has 22 rules.
Found on July 23, 2021.Finding the specification took 2 seconds.
Copy 22 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{0}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right) F_{16}\! \left(x \right) F_{9}\! \left(x \right)\\
\end{align*}\)