Av(1342, 1423, 2431, 4132)
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Generating Function
\(\displaystyle \frac{\left(x^{3}+2 x^{2}-5 x +2\right) \sqrt{1-4 x}+2 x^{4}+x^{3}-6 x^{2}+7 x -2}{4 x^{2}-2 x}\)
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Counting Sequence
1, 1, 2, 6, 20, 66, 219, 734, 2489, 8542, 29655, 104056, 368660, 1317428, 4744046, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{2} x F \left(x \right)^{2}-\left(2 x -1\right) \left(2 x^{2}-3 x +2\right) \left(x^{2}+2 x -1\right) F \! \left(x \right)+x^{7}+2 x^{6}-2 x^{5}-3 x^{4}-4 x^{3}+15 x^{2}-10 x +2 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 66\)
\(\displaystyle a \! \left(6\right) = 219\)
\(\displaystyle a \! \left(7\right) = 734\)
\(\displaystyle a \! \left(n +5\right) = -\frac{2 \left(-3+2 n \right) a \! \left(n \right)}{n +6}-\frac{\left(7+5 n \right) a \! \left(1+n \right)}{n +6}+\frac{\left(112+51 n \right) a \! \left(n +2\right)}{12+2 n}-\frac{\left(83+24 n \right) a \! \left(n +3\right)}{n +6}+\frac{\left(80+17 n \right) a \! \left(n +4\right)}{12+2 n}, \quad n \geq 8\)
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This specification was found using the strategy pack "Point Placements" and has 31 rules.

Found on January 17, 2022.

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_{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_{27}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10} \left(x \right)^{3}\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{13}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{11} \left(x \right)^{2}\\ F_{24}\! \left(x \right) &= F_{11} \left(x \right)^{2} F_{10}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{11}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{22}\! \left(x \right)\\ \end{align*}\)