Av(1243, 1432, 2413, 4132)
Generating Function
\(\displaystyle -\frac{\left(-x^{3}+2 x^{2}+\left(x^{3}-4 x^{2}+3 x -1\right) \sqrt{1-4 x}-3 x +1\right) \left(x -1\right)^{3}}{2 x \left(x^{6}-8 x^{5}+23 x^{4}-29 x^{3}+20 x^{2}-7 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 216, 713, 2392, 8157, 28209, 98706, 348862, 1243818, 4468895, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{6}-8 x^{5}+23 x^{4}-29 x^{3}+20 x^{2}-7 x +1\right) F \left(x
\right)^{2}-\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(x -1\right)^{6} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 216\)
\(\displaystyle a \! \left(7\right) = 713\)
\(\displaystyle a \! \left(8\right) = 2392\)
\(\displaystyle a \! \left(9\right) = 8157\)
\(\displaystyle a \! \left(10\right) = 28209\)
\(\displaystyle a \! \left(n +11\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +12}-\frac{\left(76+53 n \right) a \! \left(1+n \right)}{n +12}+\frac{\left(706+293 n \right) a \! \left(n +2\right)}{n +12}-\frac{2 \left(1528+443 n \right) a \! \left(n +3\right)}{n +12}+\frac{8 \left(928+205 n \right) a \! \left(n +4\right)}{n +12}-\frac{\left(11180+1999 n \right) a \! \left(n +5\right)}{n +12}+\frac{10 \left(1112+167 n \right) a \! \left(n +6\right)}{n +12}-\frac{\left(7476+967 n \right) a \! \left(n +7\right)}{n +12}+\frac{\left(3374+383 n \right) a \! \left(n +8\right)}{n +12}-\frac{3 \left(326+33 n \right) a \! \left(n +9\right)}{n +12}+\frac{\left(164+15 n \right) a \! \left(n +10\right)}{n +12}, \quad n \geq 11\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 216\)
\(\displaystyle a \! \left(7\right) = 713\)
\(\displaystyle a \! \left(8\right) = 2392\)
\(\displaystyle a \! \left(9\right) = 8157\)
\(\displaystyle a \! \left(10\right) = 28209\)
\(\displaystyle a \! \left(n +11\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +12}-\frac{\left(76+53 n \right) a \! \left(1+n \right)}{n +12}+\frac{\left(706+293 n \right) a \! \left(n +2\right)}{n +12}-\frac{2 \left(1528+443 n \right) a \! \left(n +3\right)}{n +12}+\frac{8 \left(928+205 n \right) a \! \left(n +4\right)}{n +12}-\frac{\left(11180+1999 n \right) a \! \left(n +5\right)}{n +12}+\frac{10 \left(1112+167 n \right) a \! \left(n +6\right)}{n +12}-\frac{\left(7476+967 n \right) a \! \left(n +7\right)}{n +12}+\frac{\left(3374+383 n \right) a \! \left(n +8\right)}{n +12}-\frac{3 \left(326+33 n \right) a \! \left(n +9\right)}{n +12}+\frac{\left(164+15 n \right) a \! \left(n +10\right)}{n +12}, \quad n \geq 11\)
This specification was found using the strategy pack "Point Placements" and has 21 rules.
Found on July 23, 2021.Finding the specification took 5 seconds.
Copy 21 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{0}\! \left(x \right) F_{12}\! \left(x \right) F_{16}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right)\\
\end{align*}\)