Av(1243, 1342, 1423, 2431, 4132)
Generating Function
\(\displaystyle \frac{\left(-2 x^{3}+4 x^{2}-3 x +1\right) \sqrt{1-4 x}-2 x^{5}-2 x^{4}+4 x^{3}-4 x^{2}+3 x -1}{2 x \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 57, 172, 537, 1739, 5802, 19798, 68713, 241661, 859058, 3081120, ...
Implicit Equation for the Generating Function
\(\displaystyle x \left(x -1\right)^{6} F \left(x
\right)^{2}+\left(2 x^{5}+2 x^{4}-4 x^{3}+4 x^{2}-3 x +1\right) \left(x -1\right)^{3} F \! \left(x \right)+x^{9}+2 x^{8}-3 x^{7}+4 x^{6}-12 x^{5}+22 x^{4}-24 x^{3}+16 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(6\right) = 172\)
\(\displaystyle a \! \left(7\right) = 537\)
\(\displaystyle a \! \left(8\right) = 1739\)
\(\displaystyle a \! \left(n +4\right) = -\frac{4 \left(2 n +1\right) a \! \left(n \right)}{n +5}+\frac{2 \left(9 n +10\right) a \! \left(n +1\right)}{n +5}-\frac{8 \left(2 n +5\right) a \! \left(n +2\right)}{n +5}+\frac{\left(26+7 n \right) a \! \left(n +3\right)}{n +5}-\frac{3 n^{2}-3 n +2}{2 \left(n +5\right)}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(6\right) = 172\)
\(\displaystyle a \! \left(7\right) = 537\)
\(\displaystyle a \! \left(8\right) = 1739\)
\(\displaystyle a \! \left(n +4\right) = -\frac{4 \left(2 n +1\right) a \! \left(n \right)}{n +5}+\frac{2 \left(9 n +10\right) a \! \left(n +1\right)}{n +5}-\frac{8 \left(2 n +5\right) a \! \left(n +2\right)}{n +5}+\frac{\left(26+7 n \right) a \! \left(n +3\right)}{n +5}-\frac{3 n^{2}-3 n +2}{2 \left(n +5\right)}, \quad n \geq 9\)
This specification was found using the strategy pack "Point Placements" and has 19 rules.
Found on July 23, 2021.Finding the specification took 1 seconds.
Copy 19 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{13}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{10}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{14} \left(x \right)^{2}\\
\end{align*}\)