Av(1243, 1342, 2134, 2314)
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Generating Function
\(\displaystyle -\frac{\left(-3 x^{2}+\left(x^{2}-x +1\right) \sqrt{1-4 x}+3 x -1\right) \left(x -1\right)^{2}}{2 x^{2} \left(x^{3}-x +1\right)}\)
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Counting Sequence
1, 1, 2, 6, 20, 67, 225, 764, 2631, 9181, 32402, 115453, 414754, 1500578, 5463000, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{2} \left(x^{3}-x +1\right) F \left(x \right)^{2}-\left(3 x^{2}-3 x +1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(x -1\right)^{4} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 67\)
\(\displaystyle a \! \left(6\right) = 225\)
\(\displaystyle a \! \left(n +7\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{9+n}-\frac{\left(47+6 n \right) a \! \left(2+n \right)}{9+n}+\frac{\left(9 n +26\right) a \! \left(n +1\right)}{9+n}-\frac{\left(-15+7 n \right) a \! \left(n +3\right)}{9+n}+\frac{6 \left(3 n +11\right) a \! \left(n +4\right)}{9+n}-\frac{4 \left(4 n +23\right) a \! \left(n +5\right)}{9+n}+\frac{\left(7 n +52\right) a \! \left(n +6\right)}{9+n}, \quad n \geq 7\)
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This specification was found using the strategy pack "Row Placements Tracked Fusion" and has 27 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_{10}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{10}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)+F_{21}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{0}\! \left(x \right) F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x , 1\right)\\ F_{12}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{13}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x , y\right)+F_{16}\! \left(x , y\right)+F_{19}\! \left(x , y\right)+F_{21}\! \left(x \right)\\ F_{14}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= y x\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{0}\! \left(x \right) F_{10}\! \left(x \right) F_{18}\! \left(x , y\right) F_{8}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= \frac{y F_{13}\! \left(x , y\right)-F_{13}\! \left(x , 1\right)}{-1+y}\\ F_{21}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x , 1\right)\\ F_{23}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{24}\! \left(x , y\right)+F_{25}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{23}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{10}\! \left(x \right) F_{26}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= \frac{y F_{23}\! \left(x , y\right)-F_{23}\! \left(x , 1\right)}{-1+y}\\ \end{align*}\)