Av(1234, 1243, 2314)
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Counting Sequence
1, 1, 2, 6, 21, 78, 301, 1197, 4875, 20235, 85294, 364131, 1571212, 6841633, 30025137, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{2} F \left(x \right)^{3}-x \left(2 x -1\right) F \left(x \right)^{2}+\left(x -1\right) F \! \left(x \right)-x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 78\)
\(\displaystyle a \! \left(6\right) = 301\)
\(\displaystyle a \! \left(7\right) = 1197\)
\(\displaystyle a \! \left(n +8\right) = \frac{128 n \left(2 n +1\right) a \! \left(n \right)}{25 \left(n +9\right) \left(n +8\right)}-\frac{8 \left(137 n^{2}+340 n +192\right) a \! \left(n +1\right)}{25 \left(n +9\right) \left(n +8\right)}+\frac{3 \left(629 n^{2}+2585 n +2496\right) a \! \left(n +2\right)}{25 \left(n +9\right) \left(n +8\right)}-\frac{2 \left(808 n^{2}+4145 n +4515\right) a \! \left(n +3\right)}{25 \left(n +9\right) \left(n +8\right)}+\frac{\left(437 n^{2}+877 n -5280\right) a \! \left(n +4\right)}{25 \left(n +9\right) \left(n +8\right)}+\frac{2 \left(266 n^{2}+3781 n +12885\right) a \! \left(n +5\right)}{25 \left(n +9\right) \left(n +8\right)}-\frac{\left(7 n +51\right) \left(17 n +112\right) a \! \left(n +6\right)}{5 \left(n +9\right) \left(n +8\right)}+\frac{4 \left(11 n +81\right) a \! \left(n +7\right)}{5 \left(n +9\right)}, \quad n \geq 8\)
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This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 20 rules.

Found on July 23, 2021.

Finding the specification took 7 seconds.

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