Av(1243, 1324, 1342, 1432, 2143)
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Generating Function
\(\displaystyle \frac{\left(-x -1\right) \sqrt{-4 x +1}+2 x^{2}+5 x -1}{2 x^{2}+8 x -2}\)
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Counting Sequence
1, 1, 2, 6, 19, 63, 215, 749, 2650, 9490, 34318, 125104, 459152, 1694914, 6287896, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+4 x -1\right) F \left(x \right)^{2}+\left(-2 x^{2}-5 x +1\right) F \! \left(x \right)+x \left(x +2\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +4\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +4}-\frac{\left(16+19 n \right) a \! \left(n +1\right)}{n +4}-\frac{\left(26+7 n \right) a \! \left(n +2\right)}{n +4}+\frac{\left(24+7 n \right) a \! \left(n +3\right)}{n +4}, \quad n \geq 5\)
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This specification was found using the strategy pack "Point Placements" and has 13 rules.

Found on July 23, 2021.

Finding the specification took 4 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{10}\! \left(x \right) F_{7}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{9}\! \left(x \right) &= x\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{9}\! \left(x \right)\\ \end{align*}\)