Av(13254, 13524, 31254, 31524, 32154, 32514, 32541, 35124, 35214, 35241)
Counting Sequence
1, 1, 2, 6, 24, 110, 540, 2765, 14584, 78669, 431930, 2405744, 13559328, 77190464, 443190608, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x -1\right)^{2} F \left(x
\right)^{3}-x \left(3 x -11\right) \left(x -1\right)^{2} F \left(x
\right)^{2}+\left(x -1\right) \left(3 x^{3}-17 x^{2}+21 x +1\right) F \! \left(x \right)-x^{4}+5 x^{3}-14 x^{2}+10 x +1 = 0\)
Recurrence
\(\displaystyle a{\left(0 \right)} = 1\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(3 \right)} = 6\)
\(\displaystyle a{\left(4 \right)} = 24\)
\(\displaystyle a{\left(n + 4 \right)} = - \frac{22 \left(n + 1\right) \left(2 n + 1\right) a{\left(n \right)}}{5 \left(n + 3\right) \left(n + 5\right)} - \frac{3 \left(169 n + 281\right) a{\left(n + 2 \right)}}{20 \left(n + 5\right)} + \frac{17 \left(n + 4\right) \left(23 n + 55\right) a{\left(n + 3 \right)}}{40 \left(n + 3\right) \left(n + 5\right)} + \frac{\left(1015 n^{2} + 3077 n + 2076\right) a{\left(n + 1 \right)}}{40 \left(n + 3\right) \left(n + 5\right)}, \quad n \geq 5\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(3 \right)} = 6\)
\(\displaystyle a{\left(4 \right)} = 24\)
\(\displaystyle a{\left(n + 4 \right)} = - \frac{22 \left(n + 1\right) \left(2 n + 1\right) a{\left(n \right)}}{5 \left(n + 3\right) \left(n + 5\right)} - \frac{3 \left(169 n + 281\right) a{\left(n + 2 \right)}}{20 \left(n + 5\right)} + \frac{17 \left(n + 4\right) \left(23 n + 55\right) a{\left(n + 3 \right)}}{40 \left(n + 3\right) \left(n + 5\right)} + \frac{\left(1015 n^{2} + 3077 n + 2076\right) a{\left(n + 1 \right)}}{40 \left(n + 3\right) \left(n + 5\right)}, \quad n \geq 5\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 8 rules.
Finding the specification took 0 seconds.
Copy 8 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{6}\! \left(x , y\right) &= y^{4} x^{4} F_{6}\! \left(x , y\right)^{3}+5 x^{3} F_{6}\! \left(x , y\right)^{2} y^{3}-11 y^{2} x^{2} F_{6}\! \left(x , y\right)^{2}+3 x^{2} F_{6}\! \left(x , y\right) y^{2}+10 x F_{6}\! \left(x , y\right) y -9 y x +1\\
F_{7}\! \left(x \right) &= x\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 8 rules.
Finding the specification took 0 seconds.
Copy 8 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{6}\! \left(x , y\right) &= y^{4} x^{4} F_{6}\! \left(x , y\right)^{3}+5 x^{3} F_{6}\! \left(x , y\right)^{2} y^{3}-11 y^{2} x^{2} F_{6}\! \left(x , y\right)^{2}+3 x^{2} F_{6}\! \left(x , y\right) y^{2}+10 x F_{6}\! \left(x , y\right) y -9 y x +1\\
F_{7}\! \left(x \right) &= x\\
\end{align*}\)