Av(1243, 1324, 2143, 2431)
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Generating Function
\(\displaystyle \frac{\left(x^{3}-7 x^{2}+7 x -2\right) \sqrt{1-4 x}-7 x^{3}+13 x^{2}-9 x +2}{4 x^{3}-6 x^{2}+2 x}\)
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Counting Sequence
1, 1, 2, 6, 20, 67, 224, 752, 2547, 8722, 30211, 105793, 374189, 1335403, 4803709, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{2} F \left(x \right)^{2}+\left(2 x -1\right) \left(x -1\right) \left(7 x^{3}-13 x^{2}+9 x -2\right) F \! \left(x \right)+x^{6}-2 x^{5}+21 x^{4}-44 x^{3}+37 x^{2}-14 x +2 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 224\)
\(\displaystyle a \! \left(n +6\right) = -\frac{2 \left(-1+2 n \right) a \! \left(n \right)}{n +7}+\frac{5 \left(8+7 n \right) a \! \left(1+n \right)}{n +7}-\frac{\left(365+161 n \right) a \! \left(n +2\right)}{2 \left(n +7\right)}+\frac{\left(279+82 n \right) a \! \left(n +3\right)}{n +7}-\frac{6 \left(32+7 n \right) a \! \left(n +4\right)}{n +7}+\frac{\left(121+21 n \right) a \! \left(n +5\right)}{14+2 n}, \quad n \geq 7\)
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This specification was found using the strategy pack "Point Placements" and has 23 rules.

Found on July 23, 2021.

Finding the specification took 4 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{7} \left(x \right)^{2} F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\ \end{align*}\)