Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13452, 13542, 14253, 14352, 21345, 21354, 21435, 21453, 21534, 21543, 23451, 23541, 24351, 31452, 31542, 32451, 32541)
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Generating Function
\(\displaystyle \frac{x^{6}+2 x^{5}+5 x^{4}+9 x^{3}+3 x^{2}-5 x +1}{\left(x^{3}-x^{2}-3 x +1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 24, 95, 366, 1372, 5040, 18225, 65090, 230142, 807000, 2810119, 9727438, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-x^{2}-3 x +1\right)^{2} F \! \left(x \right)-x^{6}-2 x^{5}-5 x^{4}-9 x^{3}-3 x^{2}+5 x -1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a(6) = 366\)
\(\displaystyle a{\left(n + 6 \right)} = - a{\left(n \right)} + 2 a{\left(n + 1 \right)} + 5 a{\left(n + 2 \right)} - 8 a{\left(n + 3 \right)} - 7 a{\left(n + 4 \right)} + 6 a{\left(n + 5 \right)}, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ -\frac{n \left(\left(\left(\left(-\frac{i \sqrt{3}}{2}+\frac{3}{2}\right) \sqrt{37}+\frac{37 i \sqrt{3}}{4}+\frac{37}{4}\right) \left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{1}{3}}-74+\left(\left(-\frac{109 i \sqrt{3}}{400}-\frac{327}{400}\right) \sqrt{37}-\frac{703 i \sqrt{3}}{400}+\frac{703}{400}\right) \left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(3 i \sqrt{37}+i\right) \sqrt{3}+9 \sqrt{37}-1\right) \left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{i \left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{3}}{6}-\frac{\left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}\right)^{-n}+\left(\left(\left(-\frac{i \sqrt{3}}{2}-\frac{3}{2}\right) \sqrt{37}-\frac{37 i \sqrt{3}}{4}+\frac{37}{4}\right) \left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{1}{3}}-74+\left(\left(-\frac{109 i \sqrt{3}}{400}+\frac{327}{400}\right) \sqrt{37}+\frac{703 i \sqrt{3}}{400}+\frac{703}{400}\right) \left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(100 i \sqrt{3}-100\right) \left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{1}{3}}}{600}+\frac{1}{3}+\frac{\left(\left(3 i \sqrt{37}-i\right) \sqrt{3}-9 \sqrt{37}-1\right) \left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}\right)^{-n}+\left(\left(i \sqrt{37}\, \sqrt{3}-\frac{37}{2}\right) \left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{1}{3}}+\frac{109 i \left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{37}\, \sqrt{3}}{200}-\frac{703 \left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}}}{200}-74\right) \left(\frac{\left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}+\frac{1}{3}+\frac{\left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{i \left(1+3 i \sqrt{37}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{37}\, \sqrt{3}}{100}\right)^{-n}\right)}{111} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 77 rules.

Finding the specification took 94 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_{16}\! \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_{40}\! \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_{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_{19}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{16}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{20}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{16}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{16}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{38}\! \left(x \right) &= 0\\ F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{41}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{16}\! \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_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{16}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{16}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{48}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{55}\! \left(x \right)+F_{56}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{16}\! \left(x \right) F_{40}\! \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_{58}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{55}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{60}\! \left(x \right) &= 0\\ 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_{17}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{16}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{60}\! \left(x \right)+F_{61}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{16}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{16}\! \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_{11}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{56}\! \left(x \right)+F_{65}\! \left(x \right)+F_{70}\! \left(x \right)\\ \end{align*}\)