Av(1234, 1243, 1324, 1342, 1423, 2134, 2314)
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Generating Function
\(\displaystyle \frac{-\sqrt{-4 x +1}\, x^{2}-x^{2}-\sqrt{-4 x +1}+1}{2 x \left(x^{4}+2 x^{2}-x +1\right)}\)
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Counting Sequence
1, 1, 2, 6, 17, 51, 161, 524, 1747, 5939, 20510, 71756, 253797, 906032, 3260380, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(x^{4}+2 x^{2}-x +1\right) F \left(x \right)^{2}+\left(x -1\right) \left(x +1\right) F \! \left(x \right)+1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 17\)
\(\displaystyle a \! \left(5\right) = 51\)
\(\displaystyle a \! \left(6\right) = 161\)
\(\displaystyle a \! \left(n +7\right) = \frac{2 \left(5+2 n \right) a \! \left(n \right)}{n +8}-\frac{\left(n +4\right) a \! \left(1+n \right)}{n +8}+\frac{2 \left(23+6 n \right) a \! \left(n +2\right)}{n +8}-\frac{\left(26+7 n \right) a \! \left(n +3\right)}{n +8}+\frac{\left(66+13 n \right) a \! \left(n +4\right)}{n +8}-\frac{\left(46+7 n \right) a \! \left(n +5\right)}{n +8}+\frac{\left(34+5 n \right) a \! \left(n +6\right)}{n +8}, \quad n \geq 7\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 26 rules.

Found on January 20, 2022.

Finding the specification took 11 seconds.

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