Av(12345, 12354, 12435, 13425, 21345, 21354, 21435, 23415, 31245, 31254, 31425, 32415, 41235, 41253, 41325, 42315, 51234, 51243, 51324, 52314)
Counting Sequence
1, 1, 2, 6, 24, 100, 426, 1848, 8120, 36018, 160940, 723338, 3266496, 14809366, 67365298, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(-4 x^{4}+12 x^{3}-4 x^{2}+24 x -5\right) F \left(x
\right)^{3}+\left(x +3\right) \left(4 x^{4}-12 x^{3}+4 x^{2}-24 x +5\right) F \left(x
\right)^{2}+\left(-8 x^{5}+12 x^{4}+36 x^{3}+37 x^{2}+62 x -15\right) F \! \left(x \right)+\left(4 x^{2}+4 x -1\right) \left(x^{3}-3 x^{2}-x -5\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 426\)
\(\displaystyle a(7) = 1848\)
\(\displaystyle a{\left(n + 7 \right)} = - \frac{3 n \left(7 n + 11\right) a{\left(n + 1 \right)}}{5 \left(n + 5\right) \left(n + 6\right)} + \frac{4 \left(n - 1\right) \left(n + 1\right) a{\left(n \right)}}{5 \left(n + 5\right) \left(n + 6\right)} + \frac{7 \left(n + 1\right) \left(14 n + 45\right) a{\left(n + 2 \right)}}{10 \left(n + 5\right) \left(n + 6\right)} + \frac{3 \left(31 n + 133\right) a{\left(n + 6 \right)}}{10 \left(n + 6\right)} + \frac{\left(89 n^{2} + 493 n + 660\right) a{\left(n + 4 \right)}}{5 \left(n + 5\right) \left(n + 6\right)} - \frac{\left(91 n^{2} + 459 n + 590\right) a{\left(n + 3 \right)}}{5 \left(n + 5\right) \left(n + 6\right)} - \frac{\left(493 n^{2} + 3625 n + 6414\right) a{\left(n + 5 \right)}}{20 \left(n + 5\right) \left(n + 6\right)}, \quad n \geq 8\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 100\)
\(\displaystyle a(6) = 426\)
\(\displaystyle a(7) = 1848\)
\(\displaystyle a{\left(n + 7 \right)} = - \frac{3 n \left(7 n + 11\right) a{\left(n + 1 \right)}}{5 \left(n + 5\right) \left(n + 6\right)} + \frac{4 \left(n - 1\right) \left(n + 1\right) a{\left(n \right)}}{5 \left(n + 5\right) \left(n + 6\right)} + \frac{7 \left(n + 1\right) \left(14 n + 45\right) a{\left(n + 2 \right)}}{10 \left(n + 5\right) \left(n + 6\right)} + \frac{3 \left(31 n + 133\right) a{\left(n + 6 \right)}}{10 \left(n + 6\right)} + \frac{\left(89 n^{2} + 493 n + 660\right) a{\left(n + 4 \right)}}{5 \left(n + 5\right) \left(n + 6\right)} - \frac{\left(91 n^{2} + 459 n + 590\right) a{\left(n + 3 \right)}}{5 \left(n + 5\right) \left(n + 6\right)} - \frac{\left(493 n^{2} + 3625 n + 6414\right) a{\left(n + 5 \right)}}{20 \left(n + 5\right) \left(n + 6\right)}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 8 rules.
Finding the specification took 1 seconds.
Copy 8 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= -\frac{-y F_{7}\! \left(x , y\right)+F_{7}\! \left(x , 1\right)}{-1+y}\\
F_{7}\! \left(x , y\right) &= y^{2} x^{2} F_{7}\! \left(x , y\right)^{3}-x^{2} F_{7}\! \left(x , y\right)^{2} y^{2}+y x F_{7}\! \left(x , y\right)^{2}+1\\
\end{align*}\)
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:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x , 1\right)\\
F_{6}\! \left(x , y\right) &= -\frac{-y F_{7}\! \left(x , y\right)+F_{7}\! \left(x , 1\right)}{-1+y}\\
F_{7}\! \left(x , y\right) &= y^{2} x^{2} F_{7}\! \left(x , y\right)^{3}-x^{2} F_{7}\! \left(x , y\right)^{2} y^{2}+y x F_{7}\! \left(x , y\right)^{2}+1\\
\end{align*}\)