Av(1342, 1423, 2431)
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Generating Function
\(\displaystyle \frac{\left(-x^{3}+3 x^{2}-2 x \right) \sqrt{1-4 x}+3 x^{3}-9 x^{2}+10 x -2}{2 x^{4}-8 x^{2}+10 x -2}\)
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Counting Sequence
1, 1, 2, 6, 21, 77, 288, 1091, 4172, 16069, 62240, 242152, 945536, 3703095, 14539109, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}-4 x^{2}+5 x -1\right) F \left(x \right)^{2}+\left(-3 x^{3}+9 x^{2}-10 x +2\right) F \! \left(x \right)+x^{3}-4 x^{2}+5 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 77\)
\(\displaystyle a \! \left(6\right) = 288\)
\(\displaystyle a \! \left(n +7\right) = -\frac{\left(1+2 n \right) a \! \left(n \right)}{n +6}+\frac{\left(32+13 n \right) a \! \left(n +1\right)}{12+2 n}+\frac{5 \left(-8+n \right) a \! \left(n +2\right)}{2 \left(n +6\right)}-\frac{7 \left(9+5 n \right) a \! \left(n +3\right)}{n +6}+\frac{\left(354+113 n \right) a \! \left(n +4\right)}{12+2 n}-\frac{2 \left(80+19 n \right) a \! \left(n +5\right)}{n +6}+\frac{3 \left(36+7 n \right) a \! \left(n +6\right)}{2 \left(n +6\right)}, \quad n \geq 7\)
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This specification was found using the strategy pack "Point Placements" and has 19 rules.

Found on January 17, 2022.

Finding the specification took 3 seconds.

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