Av(1243, 1342, 2143, 2431)
Generating Function
\(\displaystyle \frac{\left(x^{4}-2 x^{3}+3 x^{2}-3 x +1\right) \sqrt{1-4 x}+2 x^{5}-7 x^{4}+14 x^{3}-11 x^{2}+5 x -1}{4 \left(x^{2}-2 x +2\right) \left(x -\frac{1}{2}\right) x^{2}}\)
Counting Sequence
1, 1, 2, 6, 20, 67, 225, 762, 2609, 9034, 31618, 111752, 398491, 1432211, 5183651, ...
Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x^{2}-2 x +2\right) \left(2 x -1\right)^{2} F \left(x
\right)^{2}-\left(2 x -1\right) \left(2 x^{5}-7 x^{4}+14 x^{3}-11 x^{2}+5 x -1\right) F \! \left(x \right)+x^{6}-4 x^{5}+12 x^{4}-17 x^{3}+14 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 225\)
\(\displaystyle a \! \left(7\right) = 762\)
\(\displaystyle a \! \left(8\right) = 2609\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{10+n}-\frac{\left(246+97 n \right) a \! \left(2+n \right)}{2 \left(10+n \right)}+\frac{\left(19 n +27\right) a \! \left(n +1\right)}{10+n}+\frac{\left(81 n +319\right) a \! \left(n +3\right)}{10+n}-\frac{\left(1012+191 n \right) a \! \left(n +4\right)}{2 \left(10+n \right)}+\frac{5 \left(31 n +202\right) a \! \left(n +5\right)}{2 \left(10+n \right)}-\frac{\left(590+77 n \right) a \! \left(n +6\right)}{2 \left(10+n \right)}+\frac{2 \left(5 n +44\right) a \! \left(n +7\right)}{10+n}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 225\)
\(\displaystyle a \! \left(7\right) = 762\)
\(\displaystyle a \! \left(8\right) = 2609\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{10+n}-\frac{\left(246+97 n \right) a \! \left(2+n \right)}{2 \left(10+n \right)}+\frac{\left(19 n +27\right) a \! \left(n +1\right)}{10+n}+\frac{\left(81 n +319\right) a \! \left(n +3\right)}{10+n}-\frac{\left(1012+191 n \right) a \! \left(n +4\right)}{2 \left(10+n \right)}+\frac{5 \left(31 n +202\right) a \! \left(n +5\right)}{2 \left(10+n \right)}-\frac{\left(590+77 n \right) a \! \left(n +6\right)}{2 \left(10+n \right)}+\frac{2 \left(5 n +44\right) a \! \left(n +7\right)}{10+n}, \quad n \geq 9\)
This specification was found using the strategy pack "Point Placements" and has 24 rules.
Found on July 23, 2021.Finding the specification took 4 seconds.
Copy 24 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{7} \left(x \right)^{2} F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= x\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{2}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right)\\
\end{align*}\)