Av(1243, 1342, 2143, 2431, 3142)
Generating Function
\(\displaystyle \frac{\left(x -1\right)^{4} \sqrt{-4 x +1}+2 x^{6}-2 x^{5}+5 x^{4}-2 x^{3}-4 x^{2}+4 x -1}{4 \left(x -\frac{1}{2}\right) x \left(x^{4}-2 x^{3}+4 x^{2}-3 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 60, 192, 626, 2079, 7023, 24089, 83742, 294533, 1046424, 3750375, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{4}-2 x^{3}+4 x^{2}-3 x +1\right) \left(2 x -1\right)^{2} F \left(x
\right)^{2}-\left(2 x -1\right) \left(2 x^{6}-2 x^{5}+5 x^{4}-2 x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)+x^{7}+2 x^{5}+x^{4}-11 x^{3}+13 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 192\)
\(\displaystyle a \! \left(7\right) = 626\)
\(\displaystyle a \! \left(n +7\right) = \frac{4 \left(3+2 n \right) a \! \left(n \right)}{n +8}-\frac{2 \left(44+15 n \right) a \! \left(1+n \right)}{n +8}+\frac{\left(221+67 n \right) a \! \left(n +2\right)}{n +8}-\frac{5 \left(78+19 n \right) a \! \left(n +3\right)}{n +8}+\frac{80 \left(n +5\right) a \! \left(n +4\right)}{n +8}-\frac{3 \left(77+13 n \right) a \! \left(n +5\right)}{n +8}+\frac{\left(69+10 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 8\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 192\)
\(\displaystyle a \! \left(7\right) = 626\)
\(\displaystyle a \! \left(n +7\right) = \frac{4 \left(3+2 n \right) a \! \left(n \right)}{n +8}-\frac{2 \left(44+15 n \right) a \! \left(1+n \right)}{n +8}+\frac{\left(221+67 n \right) a \! \left(n +2\right)}{n +8}-\frac{5 \left(78+19 n \right) a \! \left(n +3\right)}{n +8}+\frac{80 \left(n +5\right) a \! \left(n +4\right)}{n +8}-\frac{3 \left(77+13 n \right) a \! \left(n +5\right)}{n +8}+\frac{\left(69+10 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 8\)
This specification was found using the strategy pack "Point Placements" and has 25 rules.
Found on July 23, 2021.Finding the specification took 5 seconds.
Copy 25 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{2}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right)\\
\end{align*}\)