Av(1243, 1324, 1342, 2413, 4132)
Generating Function
\(\displaystyle \frac{\left(-3 x^{4}+6 x^{3}-7 x^{2}+4 x -1\right) \sqrt{1-4 x}-2 x^{5}+7 x^{4}-8 x^{3}+7 x^{2}-4 x +1}{2 x \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 19, 59, 182, 570, 1833, 6060, 20520, 70825, 248145, 879814, 3149762, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{8} F \left(x
\right)^{2}+\left(2 x^{5}-7 x^{4}+8 x^{3}-7 x^{2}+4 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{9}+2 x^{8}-18 x^{7}+52 x^{6}-83 x^{5}+87 x^{4}-62 x^{3}+29 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 182\)
\(\displaystyle a \! \left(7\right) = 570\)
\(\displaystyle a \! \left(8\right) = 1833\)
\(\displaystyle a \! \left(9\right) = 6060\)
\(\displaystyle a \! \left(n +6\right) = -\frac{6 \left(2 n +1\right) a \! \left(n \right)}{n +7}+\frac{3 \left(13 n +12\right) a \! \left(n +1\right)}{n +7}-\frac{\left(61 n +135\right) a \! \left(n +2\right)}{n +7}+\frac{\left(194+57 n \right) a \! \left(n +3\right)}{n +7}-\frac{\left(31 n +143\right) a \! \left(n +4\right)}{n +7}+\frac{\left(52+9 n \right) a \! \left(n +5\right)}{n +7}+\frac{n^{2}-13 n +10}{2 n +14}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 182\)
\(\displaystyle a \! \left(7\right) = 570\)
\(\displaystyle a \! \left(8\right) = 1833\)
\(\displaystyle a \! \left(9\right) = 6060\)
\(\displaystyle a \! \left(n +6\right) = -\frac{6 \left(2 n +1\right) a \! \left(n \right)}{n +7}+\frac{3 \left(13 n +12\right) a \! \left(n +1\right)}{n +7}-\frac{\left(61 n +135\right) a \! \left(n +2\right)}{n +7}+\frac{\left(194+57 n \right) a \! \left(n +3\right)}{n +7}-\frac{\left(31 n +143\right) a \! \left(n +4\right)}{n +7}+\frac{\left(52+9 n \right) a \! \left(n +5\right)}{n +7}+\frac{n^{2}-13 n +10}{2 n +14}, \quad n \geq 10\)
This specification was found using the strategy pack "Point Placements" and has 25 rules.
Found on July 23, 2021.Finding the specification took 2 seconds.
Copy 25 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17} \left(x \right)^{2}\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{17}\! \left(x \right) F_{22}\! \left(x \right) F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{23}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)