Av(1243, 1324, 1432, 2143)
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Generating Function
\(\displaystyle \frac{-2 x^{4}+8 x^{3}-x \sqrt{-4 x +1}-14 x^{2}+11 x -2}{2 x^{5}-8 x^{4}+18 x^{3}-20 x^{2}+12 x -2}\)
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Counting Sequence
1, 1, 2, 6, 20, 70, 253, 934, 3498, 13236, 50470, 193597, 746152, 2886940, 11205837, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-4 x^{4}+9 x^{3}-10 x^{2}+6 x -1\right) F \left(x \right)^{2}+\left(2 x^{4}-8 x^{3}+14 x^{2}-11 x +2\right) F \! \left(x \right)+x^{3}-4 x^{2}+5 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 70\)
\(\displaystyle a \! \left(n +6\right) = -\frac{2 \left(7+2 n \right) a \! \left(n \right)}{5+n}+\frac{2 \left(22+5 n \right) a \! \left(5+n \right)}{5+n}+\frac{\left(61+17 n \right) a \! \left(n +1\right)}{5+n}-\frac{2 \left(73+20 n \right) a \! \left(n +2\right)}{5+n}+\frac{\left(185+49 n \right) a \! \left(n +3\right)}{5+n}-\frac{2 \left(67+17 n \right) a \! \left(n +4\right)}{5+n}, \quad n \geq 6\)
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This specification was found using the strategy pack "Point Placements" and has 24 rules.

Found on July 23, 2021.

Finding the specification took 7 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{0}\! \left(x \right) F_{12}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ \end{align*}\)