Av(1243, 2143, 2413, 4132)
Generating Function
\(\displaystyle \frac{-2 \left(x -1\right)^{3} \left(x -\frac{1}{2}\right) \left(x^{2}-x +1\right) \sqrt{1-4 x}+12 x^{6}-37 x^{5}+58 x^{4}-51 x^{3}+27 x^{2}-8 x +1}{8 x^{6}-18 x^{5}+20 x^{4}-10 x^{3}+2 x^{2}}\)
Counting Sequence
1, 1, 2, 6, 20, 68, 233, 806, 2817, 9941, 35385, 126921, 458362, 1665445, 6084515, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(4 x^{4}-9 x^{3}+10 x^{2}-5 x +1\right) F \left(x
\right)^{2}+\left(-12 x^{6}+37 x^{5}-58 x^{4}+51 x^{3}-27 x^{2}+8 x -1\right) F \! \left(x \right)+x^{7}+2 x^{6}-13 x^{5}+28 x^{4}-31 x^{3}+20 x^{2}-7 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 68\)
\(\displaystyle a \! \left(6\right) = 233\)
\(\displaystyle a \! \left(7\right) = 806\)
\(\displaystyle a \! \left(8\right) = 2817\)
\(\displaystyle a \! \left(9\right) = 9941\)
\(\displaystyle a \! \left(n +9\right) = \frac{16 \left(-1+2 n \right) a \! \left(n \right)}{11+n}+\frac{2 \left(450+197 n \right) a \! \left(2+n \right)}{11+n}-\frac{4 \left(39+40 n \right) a \! \left(n +1\right)}{11+n}-\frac{\left(2195+609 n \right) a \! \left(n +3\right)}{11+n}+\frac{2 \left(1563+319 n \right) a \! \left(n +4\right)}{11+n}-\frac{\left(2855+463 n \right) a \! \left(n +5\right)}{11+n}+\frac{6 \left(281+38 n \right) a \! \left(n +6\right)}{11+n}-\frac{6 \left(103+12 n \right) a \! \left(n +7\right)}{11+n}+\frac{\left(127+13 n \right) a \! \left(n +8\right)}{11+n}, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 68\)
\(\displaystyle a \! \left(6\right) = 233\)
\(\displaystyle a \! \left(7\right) = 806\)
\(\displaystyle a \! \left(8\right) = 2817\)
\(\displaystyle a \! \left(9\right) = 9941\)
\(\displaystyle a \! \left(n +9\right) = \frac{16 \left(-1+2 n \right) a \! \left(n \right)}{11+n}+\frac{2 \left(450+197 n \right) a \! \left(2+n \right)}{11+n}-\frac{4 \left(39+40 n \right) a \! \left(n +1\right)}{11+n}-\frac{\left(2195+609 n \right) a \! \left(n +3\right)}{11+n}+\frac{2 \left(1563+319 n \right) a \! \left(n +4\right)}{11+n}-\frac{\left(2855+463 n \right) a \! \left(n +5\right)}{11+n}+\frac{6 \left(281+38 n \right) a \! \left(n +6\right)}{11+n}-\frac{6 \left(103+12 n \right) a \! \left(n +7\right)}{11+n}+\frac{\left(127+13 n \right) a \! \left(n +8\right)}{11+n}, \quad n \geq 10\)
This specification was found using the strategy pack "Point Placements" and has 20 rules.
Found on July 23, 2021.Finding the specification took 9 seconds.
Copy 20 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{9} \left(x \right)^{3}\\
F_{17}\! \left(x \right) &= F_{18} \left(x \right)^{2} F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right)\\
\end{align*}\)