Av(1324, 2143, 2431)
Generating Function
\(\displaystyle \frac{\left(-x^{4}+8 x^{3}-16 x^{2}+10 x -2\right) \sqrt{1-4 x}+3 x^{4}-18 x^{3}+24 x^{2}-12 x +2}{2 x \left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 265, 927, 3229, 11253, 39355, 138362, 489440, 1742576, 6244395, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{4} F \left(x
\right)^{2}-\left(x^{2}-3 x +1\right) \left(3 x^{4}-18 x^{3}+24 x^{2}-12 x +2\right) \left(x -1\right)^{2} F \! \left(x \right)+x^{8}-14 x^{7}+73 x^{6}-183 x^{5}+255 x^{4}-202 x^{3}+90 x^{2}-21 x +2 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 265\)
\(\displaystyle a \! \left(7\right) = 927\)
\(\displaystyle a \! \left(8\right) = 3229\)
\(\displaystyle a \! \left(n +8\right) = -\frac{\left(1+2 n \right) a \! \left(n \right)}{n +9}+\frac{\left(86+49 n \right) a \! \left(1+n \right)}{2 n +18}-\frac{\left(611+220 n \right) a \! \left(n +2\right)}{2 \left(n +9\right)}+\frac{3 \left(597+160 n \right) a \! \left(n +3\right)}{2 \left(n +9\right)}-\frac{\left(2629+563 n \right) a \! \left(n +4\right)}{2 \left(n +9\right)}+\frac{\left(1049+185 n \right) a \! \left(n +5\right)}{n +9}-\frac{2 \left(229+34 n \right) a \! \left(n +6\right)}{n +9}+\frac{\left(102+13 n \right) a \! \left(n +7\right)}{n +9}+\frac{5}{2 \left(n +9\right)}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 75\)
\(\displaystyle a \! \left(6\right) = 265\)
\(\displaystyle a \! \left(7\right) = 927\)
\(\displaystyle a \! \left(8\right) = 3229\)
\(\displaystyle a \! \left(n +8\right) = -\frac{\left(1+2 n \right) a \! \left(n \right)}{n +9}+\frac{\left(86+49 n \right) a \! \left(1+n \right)}{2 n +18}-\frac{\left(611+220 n \right) a \! \left(n +2\right)}{2 \left(n +9\right)}+\frac{3 \left(597+160 n \right) a \! \left(n +3\right)}{2 \left(n +9\right)}-\frac{\left(2629+563 n \right) a \! \left(n +4\right)}{2 \left(n +9\right)}+\frac{\left(1049+185 n \right) a \! \left(n +5\right)}{n +9}-\frac{2 \left(229+34 n \right) a \! \left(n +6\right)}{n +9}+\frac{\left(102+13 n \right) a \! \left(n +7\right)}{n +9}+\frac{5}{2 \left(n +9\right)}, \quad n \geq 9\)
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:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{7} \left(x \right)^{2} F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= x\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{11}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15} \left(x \right)^{2} F_{11}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{16}\! \left(x \right)\\
\end{align*}\)