Av(1243, 1324, 2413)
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Generating Function
\(\displaystyle -\frac{\left(-3 x +1+\left(x -1\right) \sqrt{-4 x +1}\right) \left(x^{2}-3 x +1\right)}{2 x^{3} \left(x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 21, 76, 276, 1002, 3641, 13261, 48451, 177651, 653753, 2414426, 8947576, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{3} \left(x -1\right)^{2} F \left(x \right)^{2}-\left(x -1\right) \left(3 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+\left(x^{2}-3 x +1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +3\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{6+n}-\frac{\left(32+13 n \right) a \! \left(n +1\right)}{6+n}+\frac{\left(30+7 n \right) a \! \left(n +2\right)}{6+n}+\frac{3 n +6}{6+n}, \quad n \geq 5\)
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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 6 seconds.

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