Av(12354, 12453, 12543, 13452, 13542, 14532, 21354, 21453, 21543, 23451, 23541, 24531, 31254, 31452, 31542, 32451, 32541, 34521, 41253, 41352, 41532, 42351, 42531, 43521, 51243, 51342, 51432, 52341, 52431, 53421)
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Generating Function
\(\displaystyle \frac{8 x^{5}-16 x^{4}+23 x^{3}-19 x^{2}+7 x -1}{\left(x -1\right)^{2} \left(2 x -1\right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 24, 90, 300, 910, 2576, 6930, 17940, 45078, 110616, 266266, 630812, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right)^{2} \left(2 x -1\right)^{3} F \! \left(x \right)-8 x^{5}+16 x^{4}-23 x^{3}+19 x^{2}-7 x +1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a{\left(n + 3 \right)} = - 2 \left(n - 3\right) + 8 a{\left(n \right)} - 12 a{\left(n + 1 \right)} + 6 a{\left(n + 2 \right)}, \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(8+\left(n -3\right) 2^{n}\right) n}{4} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 115 rules.

Finding the specification took 109 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_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{33}\! \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_{18}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{11}\! \left(x \right) &= 0\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \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_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{18}\! \left(x \right) &= x\\ F_{19}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{18}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{20}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{18}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{18}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{35}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{18}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{40}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{18}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{18}\! \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_{18}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{42}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{15}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{18}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{18}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{18}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{18}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{47}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{18}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{18}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{63}\! \left(x \right)+F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{18}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{18}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{71}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{18}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{18}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{68}\! \left(x \right)+F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{18}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{18}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{18}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{75}\! \left(x \right)+F_{76}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{18}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{18}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{90}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{18}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{18}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{92}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{18}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{11}\! \left(x \right)+F_{63}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{10}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{103}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{105}\! \left(x \right)+F_{11}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{18}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{112}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{11}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{11}\! \left(x \right)+F_{113}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{10}\! \left(x \right)\\ \end{align*}\)