Av(1234, 1342, 2314)
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Generating Function
\(\displaystyle -\frac{\left(-3 x^{2}+\left(x^{2}+x -1\right) \sqrt{1-4 x}+5 x -1\right) \left(x -1\right)}{2 x^{4}+8 x^{3}-18 x^{2}+12 x -2}\)
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Counting Sequence
1, 1, 2, 6, 21, 76, 277, 1015, 3743, 13893, 51874, 194693, 733983, 2777748, 10547615, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}+4 x^{3}-9 x^{2}+6 x -1\right) F \left(x \right)^{2}-\left(x -1\right) \left(3 x^{2}-5 x +1\right) F \! \left(x \right)+x \left(x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 277\)
\(\displaystyle a \! \left(7\right) = 1015\)
\(\displaystyle a \! \left(n +8\right) = -\frac{2 \left(1+2 n \right) a \! \left(n \right)}{n +8}-\frac{3 \left(2+5 n \right) a \! \left(n +1\right)}{n +8}+\frac{2 \left(31+24 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(-76+7 n \right) a \! \left(n +3\right)}{n +8}-\frac{\left(454+85 n \right) a \! \left(n +4\right)}{n +8}+\frac{21 \left(28+5 n \right) a \! \left(n +5\right)}{n +8}-\frac{\left(336+53 n \right) a \! \left(n +6\right)}{n +8}+\frac{2 \left(43+6 n \right) a \! \left(n +7\right)}{n +8}, \quad n \geq 8\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 27 rules.

Found on January 20, 2022.

Finding the specification took 49 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{13}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x \right)+F_{15}\! \left(x , y\right)\\ F_{10}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x , 1\right)\\ F_{14}\! \left(x , y\right) &= y x\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{14}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{13}\! \left(x \right) F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= \frac{y F_{17}\! \left(x , y\right)-F_{17}\! \left(x , 1\right)}{-1+y}\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{13}\! \left(x \right) F_{23}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{20}\! \left(x , 1\right)\\ F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{24}\! \left(x \right)\\ \end{align*}\)