Av(1243, 1324, 2143, 2431, 4132)
Generating Function
\(\displaystyle \frac{\left(-4 x^{3}+6 x^{2}-4 x +1\right) \sqrt{1-4 x}-2 x^{5}-4 x^{4}+6 x^{3}-6 x^{2}+4 x -1}{4 \left(x -\frac{1}{2}\right) \left(x -1\right)^{2} x}\)
Counting Sequence
1, 1, 2, 6, 19, 58, 177, 553, 1781, 5901, 20017, 69179, 242629, 861039, 3085137, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \left(x
\right)^{2}+\left(2 x -1\right) \left(2 x^{5}+4 x^{4}-6 x^{3}+6 x^{2}-4 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{9}+4 x^{8}-2 x^{7}+10 x^{6}-35 x^{5}+55 x^{4}-50 x^{3}+27 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 177\)
\(\displaystyle a \! \left(7\right) = 553\)
\(\displaystyle a \! \left(8\right) = 1781\)
\(\displaystyle a \! \left(n +6\right) = -\frac{16 \left(2 n +1\right) a \! \left(n \right)}{n +7}+\frac{8 \left(13 n +16\right) a \! \left(1+n \right)}{n +7}-\frac{4 \left(36 n +83\right) a \! \left(n +2\right)}{n +7}+\frac{2 \left(55 n +192\right) a \! \left(n +3\right)}{n +7}-\frac{16 \left(3 n +14\right) a \! \left(n +4\right)}{n +7}+\frac{\left(64+11 n \right) a \! \left(n +5\right)}{n +7}+\frac{3}{n +7}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 177\)
\(\displaystyle a \! \left(7\right) = 553\)
\(\displaystyle a \! \left(8\right) = 1781\)
\(\displaystyle a \! \left(n +6\right) = -\frac{16 \left(2 n +1\right) a \! \left(n \right)}{n +7}+\frac{8 \left(13 n +16\right) a \! \left(1+n \right)}{n +7}-\frac{4 \left(36 n +83\right) a \! \left(n +2\right)}{n +7}+\frac{2 \left(55 n +192\right) a \! \left(n +3\right)}{n +7}-\frac{16 \left(3 n +14\right) a \! \left(n +4\right)}{n +7}+\frac{\left(64+11 n \right) a \! \left(n +5\right)}{n +7}+\frac{3}{n +7}, \quad n \geq 9\)
This specification was found using the strategy pack "Point Placements" and has 25 rules.
Found on July 23, 2021.Finding the specification took 4 seconds.
Copy 25 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= x\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{18}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{5}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{5}\! \left(x \right)\\
\end{align*}\)