Av(1243, 1342, 1423, 1432, 4231)
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Generating Function
\(\displaystyle \frac{x^{9}-x^{8}+x^{7}+x^{6}-7 x^{5}+20 x^{4}-28 x^{3}+20 x^{2}-7 x +1}{\left(-1+2 x \right)^{2} \left(x -1\right)^{4}}\)
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Counting Sequence
1, 1, 2, 6, 19, 56, 153, 395, 978, 2347, 5499, 12643, 28628, 64015, 141653, ...
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Implicit Equation for the Generating Function
\(\displaystyle -\left(-1+2 x \right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+x^{9}-x^{8}+x^{7}+x^{6}-7 x^{5}+20 x^{4}-28 x^{3}+20 x^{2}-7 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 153\)
\(\displaystyle a \! \left(7\right) = 395\)
\(\displaystyle a \! \left(8\right) = 978\)
\(\displaystyle a \! \left(9\right) = 2347\)
\(\displaystyle a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right)+\frac{\left(-2+n \right) \left(n^{2}-10 n +39\right)}{6}, \quad n \geq 10\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{\left(81 n -306\right) 2^{n}}{96}+\frac{n^{3}}{6}-n^{2}+\frac{35 n}{6}-2 & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 32 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_{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_{29}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right)\\ 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) F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ 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_{13}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{20}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{13}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{13}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{10}\! \left(x \right) F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right) F_{20}\! \left(x \right)\\ \end{align*}\)