Av(1324, 1342, 3124, 3412)
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Generating Function
\(\displaystyle -\frac{11 x^{5}-29 x^{4}+38 x^{3}-25 x^{2}+8 x -1}{\left(2 x -1\right) \left(3 x -1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 202, 608, 1800, 5299, 15606, 46098, 136652, 406389, 1211650, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(3 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+11 x^{5}-29 x^{4}+38 x^{3}-25 x^{2}+8 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(n +2\right) = -\frac{n^{3}}{3}+\frac{3 n^{2}}{2}+5 a \! \left(n +1\right)-6 a \! \left(n \right)-\frac{13 n}{6}+3, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle -\frac{n^{3}}{6}-\frac{4 n}{3}-\frac{1}{4}+2^{n}+\frac{3^{n}}{4}\)

This specification was found using the strategy pack "Point Placements" and has 57 rules.

Found on July 23, 2021.

Finding the specification took 6 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_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \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_{8}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{33}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{39}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{44}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{16}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{54}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{52}\! \left(x \right) F_{8}\! \left(x \right)\\ \end{align*}\)