Av(12345, 12354, 12435, 12453, 12534, 12543, 13425, 13452, 13524, 13542, 14523, 14532, 21345, 21354, 21435, 21453, 21534, 21543, 23451, 23541, 24531, 31245, 31254, 31452, 31542, 32451, 32541, 41253, 41352, 42351)
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Generating Function
\(\displaystyle \frac{x^{8}-5 x^{7}+15 x^{6}-24 x^{5}+28 x^{4}-24 x^{3}+18 x^{2}-7 x +1}{\left(x^{4}-3 x^{3}+4 x^{2}-4 x +1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 24, 90, 318, 1092, 3680, 12213, 40030, 129888, 417972, 1335672, 4243022, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}-3 x^{3}+4 x^{2}-4 x +1\right)^{2} F \! \left(x \right)-x^{8}+5 x^{7}-15 x^{6}+24 x^{5}-28 x^{4}+24 x^{3}-18 x^{2}+7 x -1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 90\)
\(\displaystyle a(6) = 318\)
\(\displaystyle a(7) = 1092\)
\(\displaystyle a(8) = 3680\)
\(\displaystyle a{\left(n + 8 \right)} = - a{\left(n \right)} + 6 a{\left(n + 1 \right)} - 17 a{\left(n + 2 \right)} + 32 a{\left(n + 3 \right)} - 42 a{\left(n + 4 \right)} + 38 a{\left(n + 5 \right)} - 24 a{\left(n + 6 \right)} + 8 a{\left(n + 7 \right)}, \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle \frac{\left(-49 n -49\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -2}\right)}{331}+\frac{\left(171 n +196\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{331}+\frac{\left(149 n +147\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{331}-\frac{49 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{331}+\frac{\left(-102 n -196\right) \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-3 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{331}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
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This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 104 rules.

Finding the specification took 100 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_{17}\! \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_{47}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{17}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\ 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_{14}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= x\\ F_{18}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{17}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{17}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{17}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{17}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{17}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{17}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{39}\! \left(x \right)+F_{40}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{17}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{17}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{44}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{39}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{17}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{46}\! \left(x \right) &= 0\\ F_{47}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{48}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{17}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{17}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{17}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{58}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{17}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{17}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{60}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{17}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{67}\! \left(x \right)+F_{68}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{17}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{17}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{64}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{17}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{67}\! \left(x \right)+F_{77}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{17}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{17}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{17}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{68}\! \left(x \right)+F_{83}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{17}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{72}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{17}\! \left(x \right) F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{77}\! \left(x \right)+F_{79}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{17}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{17}\! \left(x \right) F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{102}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{60}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{68}\! \left(x \right)+F_{83}\! \left(x \right)+F_{96}\! \left(x \right)\\ \end{align*}\)