Av(1234, 1243, 1324, 2314)
Counting Sequence
1, 1, 2, 6, 20, 71, 264, 1015, 4002, 16094, 65758, 272208, 1139182, 4811807, 20487096, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} F \left(x
\right)^{3}-x \left(x -1\right) F \left(x
\right)^{2}-F \! \left(x \right)+1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 71\)
\(\displaystyle a \! \left(n +6\right) = \frac{8 n \left(n +1\right) a \! \left(n \right)}{15 \left(n +7\right) \left(n +6\right)}-\frac{4 \left(7 n +12\right) \left(n +1\right) a \! \left(n +1\right)}{15 \left(n +7\right) \left(n +6\right)}+\frac{2 \left(4 n^{2}+7 n -6\right) a \! \left(n +2\right)}{15 \left(n +7\right) \left(n +6\right)}-\frac{2 \left(8 n^{2}-7 n -138\right) a \! \left(n +3\right)}{15 \left(n +7\right) \left(n +6\right)}+\frac{2 \left(11 n^{2}+77 n +114\right) a \! \left(n +4\right)}{15 \left(n +7\right) \left(n +6\right)}+\frac{\left(67 n +376\right) a \! \left(n +5\right)}{15 n +105}, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 71\)
\(\displaystyle a \! \left(n +6\right) = \frac{8 n \left(n +1\right) a \! \left(n \right)}{15 \left(n +7\right) \left(n +6\right)}-\frac{4 \left(7 n +12\right) \left(n +1\right) a \! \left(n +1\right)}{15 \left(n +7\right) \left(n +6\right)}+\frac{2 \left(4 n^{2}+7 n -6\right) a \! \left(n +2\right)}{15 \left(n +7\right) \left(n +6\right)}-\frac{2 \left(8 n^{2}-7 n -138\right) a \! \left(n +3\right)}{15 \left(n +7\right) \left(n +6\right)}+\frac{2 \left(11 n^{2}+77 n +114\right) a \! \left(n +4\right)}{15 \left(n +7\right) \left(n +6\right)}+\frac{\left(67 n +376\right) a \! \left(n +5\right)}{15 n +105}, \quad n \geq 6\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 19 rules.
Found on July 23, 2021.Finding the specification took 2 seconds.
Copy 19 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{7}\! \left(x , y\right)\\
F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x , y\right)\\
F_{8}\! \left(x , y\right) &= F_{9}\! \left(x , y\right)\\
F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{11}\! \left(x , y\right)\\
F_{10}\! \left(x , y\right) &= y x\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)+F_{13}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)\\
F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\
F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\
F_{15}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{8}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\
F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{4}\! \left(x \right) F_{7}\! \left(x , y\right)\\
F_{18}\! \left(x , y\right) &= \frac{y F_{6}\! \left(x , y\right)-F_{6}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)