Av(1324, 2143, 2413, 3241)
Generating Function
\(\displaystyle \frac{\left(-x^{5}+8 x^{4}-16 x^{3}+14 x^{2}-6 x +1\right) \sqrt{1-4 x}+3 x^{5}-14 x^{4}+18 x^{3}-14 x^{2}+6 x -1}{2 x \left(x^{2}-3 x +1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 214, 691, 2247, 7403, 24766, 84134, 289953, 1012325, 3575248, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{2}-3 x +1\right)^{2} \left(x -1\right)^{6} F \left(x
\right)^{2}-\left(3 x^{5}-14 x^{4}+18 x^{3}-14 x^{2}+6 x -1\right) \left(x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{10}-14 x^{9}+79 x^{8}-232 x^{7}+416 x^{6}-481 x^{5}+371 x^{4}-191 x^{3}+63 x^{2}-12 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 214\)
\(\displaystyle a \! \left(7\right) = 691\)
\(\displaystyle a \! \left(8\right) = 2247\)
\(\displaystyle a \! \left(9\right) = 7403\)
\(\displaystyle a \! \left(10\right) = 24766\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +9}+\frac{\left(76+45 n \right) a \! \left(n +1\right)}{n +9}-\frac{5 \left(35 n +89\right) a \! \left(n +2\right)}{n +9}+\frac{\left(1100+321 n \right) a \! \left(n +3\right)}{n +9}-\frac{\left(326 n +1455\right) a \! \left(n +4\right)}{n +9}+\frac{\left(1093+196 n \right) a \! \left(n +5\right)}{n +9}-\frac{\left(69 n +463\right) a \! \left(n +6\right)}{n +9}+\frac{\left(102+13 n \right) a \! \left(n +7\right)}{n +9}+\frac{3 n^{2}-3 n +2}{2 n +18}, \quad n \geq 11\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 214\)
\(\displaystyle a \! \left(7\right) = 691\)
\(\displaystyle a \! \left(8\right) = 2247\)
\(\displaystyle a \! \left(9\right) = 7403\)
\(\displaystyle a \! \left(10\right) = 24766\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(2 n +1\right) a \! \left(n \right)}{n +9}+\frac{\left(76+45 n \right) a \! \left(n +1\right)}{n +9}-\frac{5 \left(35 n +89\right) a \! \left(n +2\right)}{n +9}+\frac{\left(1100+321 n \right) a \! \left(n +3\right)}{n +9}-\frac{\left(326 n +1455\right) a \! \left(n +4\right)}{n +9}+\frac{\left(1093+196 n \right) a \! \left(n +5\right)}{n +9}-\frac{\left(69 n +463\right) a \! \left(n +6\right)}{n +9}+\frac{\left(102+13 n \right) a \! \left(n +7\right)}{n +9}+\frac{3 n^{2}-3 n +2}{2 n +18}, \quad n \geq 11\)
This specification was found using the strategy pack "Point Placements" and has 30 rules.
Found on July 23, 2021.Finding the specification took 5 seconds.
Copy 30 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_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{13}\! \left(x \right) &= 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_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \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)^{2} F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{23}\! \left(x \right) &= 0\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{16}\! \left(x \right) F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{0}\! \left(x \right) F_{17}\! \left(x \right) F_{18}\! \left(x \right)\\
\end{align*}\)