Av(1234, 1432, 2413, 4132, 4213)
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Generating Function
\(\displaystyle \frac{2 x^{7}+2 x^{6}-4 x^{5}-6 x^{4}-3 x^{3}+2 x -1}{\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 51, 120, 266, 565, 1163, 2342, 4638, 9067, 17547, 33684, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+2 x^{7}+2 x^{6}-4 x^{5}-6 x^{4}-3 x^{3}+2 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 51\)
\(\displaystyle a \! \left(6\right) = 120\)
\(\displaystyle a \! \left(7\right) = 266\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right)+8, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \frac{3363 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-2 Z^{3}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +4}\right)}{55}+\frac{10803 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-2 Z^{3}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{110}-\frac{14 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-2 Z^{3}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{11}-\frac{6736 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-2 Z^{3}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{55}-\frac{2953 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-2 Z^{3}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{22}+\frac{5644 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{6}+Z^{5}-Z^{4}-2 Z^{3}-Z^{2}+3 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{55}+\left(\left\{\begin{array}{cc}2 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)

This specification was found using the strategy pack "Point Placements" and has 66 rules.

Found on January 18, 2022.

Finding the specification took 1 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_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{38}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{49}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \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_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{48}\! \left(x \right)\\ \end{align*}\)