Av(1243, 1342, 2143, 4132)
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Generating Function
\(\displaystyle \frac{\left(x^{3}-2 x^{2}+2 x -1\right) \sqrt{1-4 x}+2 x^{4}-3 x^{3}+12 x^{2}-6 x +1}{2 x \left(x^{3}-2 x^{2}+5 x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 20, 68, 235, 824, 2924, 10479, 37868, 137818, 504663, 1857895, 6872089, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{3}-2 x^{2}+5 x -1\right) F \left(x \right)^{2}+\left(-2 x^{4}+3 x^{3}-12 x^{2}+6 x -1\right) F \! \left(x \right)+x^{4}+5 x^{2}-4 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 68\)
\(\displaystyle a \! \left(6\right) = 235\)
\(\displaystyle a \! \left(7\right) = 824\)
\(\displaystyle a \! \left(n +7\right) = \frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +8}-\frac{\left(26+17 n \right) a \! \left(1+n \right)}{n +8}+\frac{2 \left(49+24 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(238+75 n \right) a \! \left(n +3\right)}{n +8}+\frac{2 \left(163+36 n \right) a \! \left(n +4\right)}{n +8}-\frac{2 \left(125+21 n \right) a \! \left(n +5\right)}{n +8}+\frac{\left(78+11 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 8\)
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This specification was found using the strategy pack "Point And Row Placements" and has 25 rules.

Found on July 23, 2021.

Finding the specification took 8 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{12}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{13} \left(x \right)^{2} F_{12}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{13}\! \left(x \right) F_{21}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right) F_{21}\! \left(x \right)\\ \end{align*}\)