Av(13425, 13524, 14523, 14532, 24531)
Generating Function
\(\displaystyle \frac{\left(4 x^{3}-22 x^{2}+x \right) \sqrt{1-8 x}+2 x^{4}-36 x^{3}+50 x^{2}+9 x +2}{2 \left(x +1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 24, 115, 618, 3584, 21920, 139365, 912450, 6112282, 41701080, 288791191, 2024964074, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x +1\right)^{4} F \left(x
\right)^{2}+\left(-2 x^{4}+36 x^{3}-50 x^{2}-9 x -2\right) F \! \left(x \right)+x^{4}-8 x^{3}+44 x^{2}+5 x +1 = 0\)
Recurrence
\(\displaystyle a{\left(0 \right)} = 1\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(3 \right)} = 6\)
\(\displaystyle a{\left(4 \right)} = 24\)
\(\displaystyle a{\left(n + 4 \right)} = \frac{16 \left(2 n + 1\right) a{\left(n \right)}}{n + 3} - \frac{6 \left(25 n - 36\right) a{\left(n + 2 \right)}}{n + 3} + \frac{\left(29 n + 28\right) a{\left(n + 3 \right)}}{n + 3} - \frac{4 \left(37 n + 132\right) a{\left(n + 1 \right)}}{n + 3}, \quad n \geq 5\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(3 \right)} = 6\)
\(\displaystyle a{\left(4 \right)} = 24\)
\(\displaystyle a{\left(n + 4 \right)} = \frac{16 \left(2 n + 1\right) a{\left(n \right)}}{n + 3} - \frac{6 \left(25 n - 36\right) a{\left(n + 2 \right)}}{n + 3} + \frac{\left(29 n + 28\right) a{\left(n + 3 \right)}}{n + 3} - \frac{4 \left(37 n + 132\right) a{\left(n + 1 \right)}}{n + 3}, \quad n \geq 5\)
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:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x , 1\right)\\
F_{5}\! \left(x , y\right) &= -\frac{-y F_{6}\! \left(x , y\right)+F_{6}\! \left(x , 1\right)}{-1+y}\\
F_{6}\! \left(x , y\right) &= x^{3} F_{6}\! \left(x , y\right)^{2} y^{3}+3 x^{2} F_{6}\! \left(x , y\right)^{2} y^{2}+8 x^{2} F_{6}\! \left(x , y\right) y^{2}+3 x F_{6}\! \left(x , y\right)^{2} y -20 x F_{6}\! \left(x , y\right) y +16 y x +F_{6}\! \left(x , y\right)^{2}\\
F_{7}\! \left(x \right) &= x\\
\end{align*}\)