Av(132, 1234, 2341)
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Generating Function
\(\displaystyle \frac{x^{3}+2 x^{2}-3 x +1}{\left(2 x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 5, 12, 28, 64, 144, 320, 704, 1536, 3328, 7168, 15360, 32768, ...
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Implicit Equation for the Generating Function
\(\displaystyle -\left(2 x -1\right)^{2} F \! \left(x \right)+x^{3}+2 x^{2}-3 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 5\)
\(\displaystyle a \! \left(n +2\right) = -4 a \! \left(n \right)+4 a \! \left(n +1\right), \quad n \geq 4\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{2^{n} \left(n +2\right)}{8} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 40 rules.

Found on January 18, 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{25}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\ \end{align*}\)