Av(12354, 12453, 13254, 13452, 14253, 14352, 15243, 15342, 21354, 21453, 23154, 23451, 24153, 24351, 25143, 25341, 31254, 31452, 32154, 32451, 34152, 34251, 35142, 35241, 41253, 41352, 42153, 42351, 43152, 43251, 45132, 45231, 51243, 51342, 52143, 52341, 53142, 53241, 54132, 54231)
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Generating Function
\(\displaystyle \frac{8 x^{5}-4 x^{4}+2 x^{3}-8 x^{2}+5 x -1}{\left(2 x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 24, 80, 240, 672, 1792, 4608, 11520, 28160, 67584, 159744, 372736, ...
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Implicit Equation for the Generating Function
\(\displaystyle -\left(2 x -1\right)^{3} F \! \left(x \right)+8 x^{5}-4 x^{4}+2 x^{3}-8 x^{2}+5 x -1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 80\)
\(\displaystyle a{\left(n + 3 \right)} = 8 a{\left(n \right)} - 12 a{\left(n + 1 \right)} + 6 a{\left(n + 2 \right)}, \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{2^{n} n \left(-1+n \right)}{8} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 94 rules.

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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_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{5}\! \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_{16}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{26}\! \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_{12}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{16}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{22}\! \left(x \right) &= 0\\ F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{16}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{16}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{22}\! \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_{24}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{16}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{16}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{16}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{16}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{16}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{49}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{56}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{16}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{61}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{16}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{61}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{16}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{69}\! \left(x \right)+F_{71}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{16}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{16}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{16}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{71}\! \left(x \right)+F_{80}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{16}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{16}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{16}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{69}\! \left(x \right)+F_{71}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{16}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{84}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{16}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{87}\! \left(x \right)\\ \end{align*}\)