Av(1243, 2143, 2431)
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Generating Function
\(\displaystyle \frac{\left(-4 x^{4}+10 x^{3}-7 x^{2}+x \right) \sqrt{1-4 x}+24 x^{4}-58 x^{3}+57 x^{2}-19 x +2}{32 x^{4}-72 x^{3}+64 x^{2}-20 x +2}\)
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Counting Sequence
1, 1, 2, 6, 21, 77, 287, 1080, 4094, 15611, 59811, 230048, 887674, 3434510, 13319262, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x -1\right) \left(4 x^{3}-8 x^{2}+6 x -1\right) F \left(x \right)^{2}-\left(4 x -1\right) \left(6 x^{3}-13 x^{2}+11 x -2\right) F \! \left(x \right)+x^{5}+6 x^{4}-21 x^{3}+25 x^{2}-9 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 287\)
\(\displaystyle a \! \left(7\right) = 1080\)
\(\displaystyle a \! \left(n +7\right) = \frac{32 \left(2 n -1\right) a \! \left(n \right)}{n +6}-\frac{16 \left(19 n +11\right) a \! \left(n +1\right)}{n +6}+\frac{8 \left(75 n +113\right) a \! \left(n +2\right)}{n +6}-\frac{4 \left(157 n +374\right) a \! \left(n +3\right)}{n +6}+\frac{4 \left(91 n +298\right) a \! \left(n +4\right)}{n +6}-\frac{2 \left(56 n +233\right) a \! \left(n +5\right)}{n +6}+\frac{\left(86+17 n \right) a \! \left(n +6\right)}{n +6}, \quad n \geq 8\)
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This specification was found using the strategy pack "Point And Row Placements" and has 28 rules.

Found on July 23, 2021.

Finding the specification took 3 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_{14}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{14}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right) F_{3}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x \right) &= F_{14}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{14}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{3}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right) F_{23}\! \left(x \right)\\ \end{align*}\)