Av(1243, 1324, 1342, 2143, 4132)
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Generating Function
\(\displaystyle \frac{\left(-x^{3}+3 x^{2}-4 x +2\right) \sqrt{1-4 x}+5 x^{3}-7 x^{2}+6 x -2}{2 \left(x -1\right) x}\)
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Counting Sequence
1, 1, 2, 6, 19, 60, 193, 635, 2133, 7292, 25297, 88841, 315247, 1128561, 4071091, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right)^{2} x F \left(x \right)^{2}-\left(x -1\right) \left(5 x^{3}-7 x^{2}+6 x -2\right) F \! \left(x \right)+x^{6}+x^{4}-5 x^{3}+9 x^{2}-7 x +2 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 193\)
\(\displaystyle a \! \left(n +3\right) = \frac{\left(-3+2 n \right) a \! \left(n \right)}{n +4}-\frac{\left(4+9 n \right) a \! \left(n +1\right)}{2 \left(n +4\right)}+\frac{\left(12+5 n \right) a \! \left(n +2\right)}{n +4}+\frac{\frac{3 n}{2}+3}{n +4}, \quad n \geq 7\)
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This specification was found using the strategy pack "Point Placements" and has 18 rules.

Found on July 23, 2021.

Finding the specification took 3 seconds.

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