Av(1324, 2413, 2431, 3142)
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Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \sqrt{-4 x +1}-2 x^{2}-2 x +1}{2 \left(x^{2}-3 x +1\right) x}\)
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Counting Sequence
1, 1, 2, 6, 20, 68, 232, 793, 2719, 9366, 32451, 113181, 397532, 1406291, 5009981, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right)^{2} x F \left(x \right)^{2}+\left(x^{2}-3 x +1\right) \left(2 x^{2}+2 x -1\right) F \! \left(x \right)+x^{3}+6 x^{2}-5 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -\frac{4 \left(3+2 n \right) a \! \left(n \right)}{5+n}+\frac{2 \left(28+15 n \right) a \! \left(1+n \right)}{5+n}-\frac{\left(77+27 n \right) a \! \left(n +2\right)}{5+n}+\frac{\left(35+9 n \right) a \! \left(n +3\right)}{5+n}, \quad n \geq 4\)
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This specification was found using the strategy pack "Point Placements" and has 38 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_{8}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{6}\! \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_{18}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{22}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= 0\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{32}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{35}\! \left(x \right) F_{8}\! \left(x \right)\\ \end{align*}\)