Av(12435, 12453, 12543, 15243, 21435, 21453, 21543, 24135, 24153, 24315, 24351, 24513, 24531, 25143, 25413, 25431, 51243, 52143, 52413, 52431)
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Generating Function
\(\displaystyle \frac{-6 x^{3} \sqrt{-4 x +1}+2 x^{2} \sqrt{-4 x +1}+16 x^{3}+8 x^{2}-7 x +1}{\left(-1+4 x \right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 24, 100, 420, 1764, 7392, 30888, 128700, 534820, 2217072, 9170616, 37858184, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(-1+4 x \right)^{3} F \left(x \right)^{2}-2 \left(x +1\right) \left(-1+4 x \right)^{3} F \! \left(x \right)+36 x^{6}+40 x^{5}+84 x^{4}-20 x^{3}-25 x^{2}+10 x -1 = 0\)
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Recurrence
\(\displaystyle a{\left(0 \right)} = 1\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(3 \right)} = 6\)
\(\displaystyle a{\left(n + 2 \right)} = - \frac{6 \left(2 n - 3\right) a{\left(n \right)}}{n} + \frac{\left(7 n - 4\right) a{\left(n + 1 \right)}}{n}, \quad n \geq 4\)
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This specification was found using the strategy pack "Point Placements Req Corrob" and has 29 rules.

Finding the specification took 156 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_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{5}\! \left(x \right) &= 4 F_{5} \left(x \right)^{2} x +x^{2}-8 F_{5}\! \left(x \right) x -F_{5} \left(x \right)^{2}+4 x +3 F_{5}\! \left(x \right)-1\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{7}\! \left(x \right) &= 4 x F_{7} \left(x \right)^{2}+x^{2}-F_{7} \left(x \right)^{2}+F_{7}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14} \left(x \right)^{2} F_{17}\! \left(x \right) F_{18}\! \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)^{2} F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= x\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{14}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{17}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= \frac{F_{28}\! \left(x \right)}{F_{17}\! \left(x \right)}\\ F_{28}\! \left(x \right) &= F_{23}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Req Corrob" and has 30 rules.

Finding the specification took 89 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_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{5}\! \left(x \right) &= 4 F_{5} \left(x \right)^{2} x +x^{2}-8 F_{5}\! \left(x \right) x -F_{5} \left(x \right)^{2}+4 x +3 F_{5}\! \left(x \right)-1\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{7}\! \left(x \right) &= 4 x F_{7} \left(x \right)^{2}+x^{2}-F_{7} \left(x \right)^{2}+F_{7}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14} \left(x \right)^{2} F_{17}\! \left(x \right) F_{18}\! \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)^{2} F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= x\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{14}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{17}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{17}\! \left(x \right)}\\ F_{29}\! \left(x \right) &= F_{24}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row And Col Placements Req Corrob" and has 21 rules.

Finding the specification took 30 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{12}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)+F_{18}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{12}\! \left(x \right) F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{9} \left(x \right)^{2} F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15} \left(x \right)^{2} F_{12}\! \left(x \right)\\ F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{12}\! \left(x \right)}\\ F_{16}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= 4 F_{17} \left(x \right)^{2} x +x^{2}-8 F_{17}\! \left(x \right) x -F_{17} \left(x \right)^{2}+4 x +3 F_{17}\! \left(x \right)-1\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{6}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{6}\! \left(x \right)\\ \end{align*}\)