Av(12435, 13425, 21435, 23415, 31425, 32415, 41325, 42315, 51324, 52314)
Generating Function
\(\displaystyle \frac{\left(-2 x^{3}+9 x^{2}-x \right) \sqrt{x^{2}-6 x +1}-2 x^{4}+15 x^{3}-18 x^{2}-9 x +2}{2 x^{2}-12 x +2}\)
Counting Sequence
1, 1, 2, 6, 24, 110, 540, 2758, 14448, 77022, 415860, 2267078, 12452616, 68814798, 382168332, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-6 x +1\right) F \left(x
\right)^{2}+\left(x -2\right) \left(2 x +1\right) \left(x^{2}-6 x +1\right) F \! \left(x \right)-2 x^{4}+3 x^{3}-17 x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a{\left(0 \right)} = 1\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(3 \right)} = 6\)
\(\displaystyle a{\left(4 \right)} = 24\)
\(\displaystyle a{\left(n + 4 \right)} = - \frac{2 \left(n - 2\right) a{\left(n \right)}}{n + 3} + \frac{3 \left(5 n + 8\right) a{\left(n + 3 \right)}}{n + 3} + \frac{3 \left(7 n - 6\right) a{\left(n + 1 \right)}}{n + 3} - \frac{3 \left(19 n + 9\right) a{\left(n + 2 \right)}}{n + 3}, \quad n \geq 5\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(3 \right)} = 6\)
\(\displaystyle a{\left(4 \right)} = 24\)
\(\displaystyle a{\left(n + 4 \right)} = - \frac{2 \left(n - 2\right) a{\left(n \right)}}{n + 3} + \frac{3 \left(5 n + 8\right) a{\left(n + 3 \right)}}{n + 3} + \frac{3 \left(7 n - 6\right) a{\left(n + 1 \right)}}{n + 3} - \frac{3 \left(19 n + 9\right) a{\left(n + 2 \right)}}{n + 3}, \quad n \geq 5\)
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:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= -\frac{-y F_{7}\! \left(x , y\right)+F_{7}\! \left(x , 1\right)}{-1+y}\\
F_{7}\! \left(x , y\right) &= x F_{7}\! \left(x , y\right) y +F_{7}\! \left(x , y\right)^{2}-2 F_{7}\! \left(x , y\right)+2\\
\end{align*}\)