Av(1243, 1324, 1342, 1432, 2413)
Generating Function
\(\displaystyle -\frac{2 \left(x -1\right)^{2} \left(\left(x^{2}-x +\frac{1}{2}\right) \sqrt{1-4 x}+x -\frac{1}{2}\right)}{8 x^{5}-18 x^{4}+20 x^{3}-10 x^{2}+2 x}\)
Counting Sequence
1, 1, 2, 6, 19, 60, 192, 628, 2098, 7135, 24627, 86069, 304026, 1083845, 3894878, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(4 x^{4}-9 x^{3}+10 x^{2}-5 x +1\right) F \left(x
\right)^{2}+\left(2 x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(x -1\right)^{4} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 192\)
\(\displaystyle a \! \left(7\right) = 628\)
\(\displaystyle a \! \left(n +8\right) = -\frac{16 \left(1+2 n \right) a \! \left(n \right)}{9+n}+\frac{4 \left(61+36 n \right) a \! \left(1+n \right)}{9+n}-\frac{18 \left(46+17 n \right) a \! \left(n +2\right)}{9+n}+\frac{4 \left(365+98 n \right) a \! \left(n +3\right)}{9+n}-\frac{25 \left(62+13 n \right) a \! \left(n +4\right)}{9+n}+\frac{\left(1033+177 n \right) a \! \left(n +5\right)}{9+n}-\frac{\left(420+61 n \right) a \! \left(n +6\right)}{9+n}+\frac{\left(95+12 n \right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 192\)
\(\displaystyle a \! \left(7\right) = 628\)
\(\displaystyle a \! \left(n +8\right) = -\frac{16 \left(1+2 n \right) a \! \left(n \right)}{9+n}+\frac{4 \left(61+36 n \right) a \! \left(1+n \right)}{9+n}-\frac{18 \left(46+17 n \right) a \! \left(n +2\right)}{9+n}+\frac{4 \left(365+98 n \right) a \! \left(n +3\right)}{9+n}-\frac{25 \left(62+13 n \right) a \! \left(n +4\right)}{9+n}+\frac{\left(1033+177 n \right) a \! \left(n +5\right)}{9+n}-\frac{\left(420+61 n \right) a \! \left(n +6\right)}{9+n}+\frac{\left(95+12 n \right) a \! \left(n +7\right)}{9+n}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements" and has 20 rules.
Found on July 23, 2021.Finding the specification took 3 seconds.
Copy 20 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{5} \left(x \right)^{2} F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{0}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\
\end{align*}\)