Av(1243, 1342, 2413, 2431, 3142)
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Generating Function
\(\displaystyle \frac{\left(-3 x^{2}+3 x -1\right) \sqrt{1-4 x}-2 x^{3}+3 x^{2}-3 x +1}{4 x^{3}-6 x^{2}+2 x}\)
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Counting Sequence
1, 1, 2, 6, 19, 60, 190, 609, 1986, 6599, 22325, 76761, 267675, 944725, 3368601, ...
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Implicit Equation for the Generating Function
\(\displaystyle x \left(2 x -1\right)^{2} \left(x -1\right)^{2} F \left(x \right)^{2}+\left(x -1\right) \left(x^{2}-x +1\right) \left(2 x -1\right)^{2} F \! \left(x \right)+x^{5}+6 x^{4}-15 x^{3}+14 x^{2}-6 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(n +5\right) = \frac{12 \left(1+2 n \right) a \! \left(n \right)}{n +6}-\frac{6 \left(15+11 n \right) a \! \left(1+n \right)}{n +6}+\frac{\left(174+71 n \right) a \! \left(n +2\right)}{n +6}-\frac{2 \left(69+19 n \right) a \! \left(n +3\right)}{n +6}+\frac{2 \left(24+5 n \right) a \! \left(n +4\right)}{n +6}, \quad n \geq 6\)
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This specification was found using the strategy pack "Point Placements" and has 20 rules.

Found on July 23, 2021.

Finding the specification took 2 seconds.

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